Robust and Optimal Switching LinearParameter-Varying ControlbyPan ZhaoB.Eng., Beihang University, 2009M.Eng., Beihang University, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2018© Pan Zhao 2018The following individuals certify that they have read, and recommend to the Faculty of Graduate and Post-doctoral Studies for acceptance, the dissertation entitled:Robust and Optimal Switching Linear Parameter-Varying Controlsubmitted by Pan Zhao in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Dr. Ryozo Nagamune, Mechanical EngineeringSupervisorDr. Mu Chiao, Mechanical EngineeringCo-SupervisorDr. Boris Stoeber, Mechanical EngineeringSupervisory Committee MemberDr. Hongshen Ma, Mechanical EngineeringUniversity ExaminerDr. Shahriar Mirabbasi, Electrical and Computer EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Dr. A. Srikantha Phani, Mechanical EngineeringSupervisory Committee MemberDr. Bhushan Gopaluni, Chemical and Biological EngineeringSupervisory Committee MemberiiAbstractLinear parameter-varying (LPV) control is a systematic way for gain-scheduling control of a nonlinear ortime-varying system that has parameter-dependent dynamics variations in its operating region. However,when the dynamics variations are large, LPV control may give the conservative performance. One way toreduce the conservatism is switching LPV (SLPV) control, in which we partition the operating region intosub-regions, design one local LPV controller for each sub-region, and switch among those local controllersaccording to some switching rules.On the one hand, this thesis makes three theoretical contributions to the SLPV control theory. Firstly,this thesis proposes a new approach to designing SLPV controllers with guaranteed stability and perfor-mance even when the scheduling parameters cannot be exactly measured. Secondly, this thesis presentstwo algorithms to optimize the switching surfaces (SSs) that can further improve the performance of anSLPV controller. One algorithm is based on sequentially optimizing the SSs and the SLPV controller for thestate-feedback case. The other one is based on particle swarm optimization and can be used for both state-feedback and output-feedback cases. Finally, this thesis introduces a novel approach to designing SLPVcontrollers that could yield significantly improved local performance in some sub-regions without much sac-rifice of the worst-case performance. This is different from the traditional approach that often leads to similarperformance in all the sub-regions.On the other hand, this thesis addresses two practical problems using the theoretic approaches developedin this thesis. One is control of miniaturized optical image stabilizers with product variations. Specifically,multiple parameter-dependent robust (MPDR) controllers are designed to adapt to the product variations,while being robust against the uncertainties in measurement of the scheduling parameters that characterizethe dynamics variation. Experimental results validate the advantages of the proposed MPDR controllersover a conventional robust µ-synthesis controller. The other application is control of a floating offshore windturbine on a semi-submersible platform. SLPV controllers are designed for regulating the power and thegenerator speed and reducing the platform motion. The superior performance of the SLPV controllers isdemonstrated in high-fidelity simulations.iiiLay SummaryMany engineering systems have varying dynamics in their operating range. In case the dynamics variationcan be characterized by some measurable parameters, a gain-scheduling (GS) controller can adapt to thevarying dynamics utilizing those parameters, called scheduling parameters. When the operating range islarge, GS control may yield conservative results. One remedy is switching GS (SGS) control, in which wepartition the operating range into sub-regions, design one local GS controller for each sub-region, and switchamong those local controllers.In practice, the measurements of scheduling parameters may not be accurate due to sensor drift andnoise. Therefore, we develop a novel approach to designing SGS controllers that are guaranteed to workeven using inaccurate measurements. Moreover, different partitions of the operating range for a system canyield SGS controllers with different performance. Thus, we devise algorithms to optimize the partition tofurther improve the performance of an SGS controller.ivPrefaceThe results presented in this thesis are based on my original ideas and discussions with my supervisorDr. Ryozo Nagamune. I solely completed all the research work and wrote the thesis under the supervi-sion of Dr. Nagamune. The results in Chapter 5 also resulted from the discussion with my co-supervisorDr. Mu Chiao and the collaboration with Dr. Nagamune’s former student Alireza Alizadegen, and Dr. Chiao’sformer student Kaiwen Yuan.• The SLPV control under uncertain scheduling parameters proposed in Chapter 2 has been publishedas a journal paper:– P. Zhao and R. Nagamune, “Switching LPV controller design under uncertain scheduling param-eters,” Automatica, 2017, 76: 243–250.• The optimal switching surface design for state-feedback SLPV control proposed in Chapter 3 has beenpublished as a conference paper:– P. Zhao and R. Nagamune, “Optimal switching surface design for state-feedback SLPV control,”Proceedings of American Control Conference (ACC), pp. 817–822, 2015.• The results in Chapter 4 have been published in one conference paper and one journal paper:– P. Zhao and R. Nagamune, “Optimal switching surface design for SLPV control and its appli-cation to air-fuel ratio control of an automotive engine,” Proceedings of IEEE Conference onControl Technology and Applications (CCTA), pp. 898–903, 2017.– P. Zhao and R. Nagamune, “Switching linear parameter-varying control with improved localperformance and optimized switching surfaces,” International Journal of Robust and NonlinearControl, 2018, 28:3403–3421.• Control of the miniaturized optical image stabilizers explained in Chapter 5 have been published as ajournal paper:– P. Zhao, R. Nagamune and M. Chiao, “Multiple parameter-dependent robust control of miniatur-ized optical image stabilizers,” Control Engineering Practice, 2018, 76: 1–11.• Control of a floating offshore wind turbine presented in Chapter 6 will be summarized into a paper andsubmitted to a peer-reviewed journal:– P. Zhao and R. Nagamune, “Switching LPV control of a floating offshore wind turbine aboverated wind speed.”vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Notations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 LPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 SLPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Uncertain Scheduling Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Switching Surface Design in SLPV Control . . . . . . . . . . . . . . . . . . . . . . 51.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Significance of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 SLPV Control with Uncertain Scheduling Parameters . . . . . . . . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Constraints on Switching Surfaces for ADT Switching . . . . . . . . . . . . . . . . . . . . 122.4 Admissible Region Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14viTable of Contents2.5 SLPV Controller Design under Uncertain Scheduling Parameters . . . . . . . . . . . . . . 152.5.1 Hysteresis Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.2 ADT Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.3 Solving PDLMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Optimal Switching Surface Design for State-Feedback SLPV Control . . . . . . . . . . . . . 263.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Basic Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 A Simultaneous Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 State-Feedback SLPV Synthesis using Slack Variables . . . . . . . . . . . . . . . . 303.3.2 A Sequential Design Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 SLPV Control with Improved Local Performance and Optimized Switching Surfaces . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Review of SLPV control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.1 Plant Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.2 SLPV Control and Switching Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 384.2.3 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 SLPV Control with Improved Local Performance . . . . . . . . . . . . . . . . . . . . . . . 414.3.1 A New Criterion to Evaluate the Performance of an SLPV Controller . . . . . . . . 414.3.2 Conditions for SLPV Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.3 Design of SLPV Controllers with Improved Local Performance . . . . . . . . . . . 464.3.4 Revisit of the Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 SLPV Control with Optimized Switching Surfaces . . . . . . . . . . . . . . . . . . . . . . 484.4.1 A Switching Surface Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.2 Switching Surface Optimization Based on Particle Swarm Optimization . . . . . . . 494.4.3 Revisit of the Motivating Example for SS optimization . . . . . . . . . . . . . . . . 514.5 Application to Air-Fuel Ratio Control of an Automotive Engine . . . . . . . . . . . . . . . 524.5.1 Modeling of the Air-Fuel Ratio Dynamics . . . . . . . . . . . . . . . . . . . . . . 534.5.2 Controller Design and Computation Results . . . . . . . . . . . . . . . . . . . . . 544.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59viiTable of Contents5 Application 1: Miniaturized Optical Image Stabilizers . . . . . . . . . . . . . . . . . . . . . 605.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Conceptual Miniaturized OIS and Large-Scale Prototypes . . . . . . . . . . . . . . . . . . 625.2.1 Miniaturized Lens-Tilting OIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.2 Control Objectives for OIS Control System . . . . . . . . . . . . . . . . . . . . . . 635.2.3 OIS Control Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.4 Large-Scale OIS Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Feedback Control Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 Modeling of Large-Scale OIS Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4.1 Transfer Function Model for a Single OIS . . . . . . . . . . . . . . . . . . . . . . 675.4.2 Model Set for Multiple OIS’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.3 Identified Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.4 State-Space Model Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4.5 Parameter Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.5 Multiple Parameter-Dependent Robust Control of OIS’s . . . . . . . . . . . . . . . . . . . 705.5.1 Implementation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5.2 MPDR Controllers Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.6 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.6.2 Assumptions for Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . 735.6.3 Designed controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.6.4 Frequency-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.6.5 Robust Stability and Time-Domain Tracking Performance . . . . . . . . . . . . . . 755.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 Application 2: Floating Offshore Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . 796.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2.1 5 MW Baseline Wind Turbine System . . . . . . . . . . . . . . . . . . . . . . . . 826.2.2 Control Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Dynamics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3.1 FAST Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3.2 Simplified Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3.3 Linearized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.4.1 Nacelle Yaw Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.4.2 Generator Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.4.3 Model Re-Derivation and Simplification . . . . . . . . . . . . . . . . . . . . . . . 866.4.4 Blade Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92viiiTable of Contents6.5.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.5.2 Results Without Measurement Uncertainty of Wind Speed . . . . . . . . . . . . . . 936.5.3 Results Under Measurement Uncertainty of Wind Speed . . . . . . . . . . . . . . . 1016.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Limitations and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108AppendicesA Multiplicative Uncertainty in Measurement of Scheduling Parameters . . . . . . . . . . . . 116B Construction of an SLPV Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C Parameters and Details of OIS Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 119C.1 Parameters and Functions Used in Experiments . . . . . . . . . . . . . . . . . . . . . . . . 119C.1.1 Actuator Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119C.1.2 Band-Stop Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120C.2 Experimental Investigation of the Translational Motion under Pitch Actuation . . . . . . . . 121C.3 Details of MPDR Controllers Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.3.1 Admissible Regions for Local Controllers . . . . . . . . . . . . . . . . . . . . . . 122C.3.2 MPDR Controllers Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122D Influence of Considering the Wave Disturbance in Controller Design . . . . . . . . . . . . . 125E State-Feedback SLPV Control with Uncertain Scheduling Parameters . . . . . . . . . . . . 128ixList of Tables2.1 Selection of Lyapunov variables and SS conditions for hysteresis switching under practicalvalidity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 γ value given by non-switching and hysteresis switching LPV controllers . . . . . . . . . . 222.3 γ value given by ADT switching controllers with X0, Y (j)(θˆ) and ε = 3.162 · 10−5 . . . . 223.1 Optimization results for 2-subset case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Optimization results for 4-subset case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1 Performance of different SLPV controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Performance of proposed SLPV controllers with optimized SSs (γ∗ and γ0b are defined inTable 4.1.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Different partitions of the parameter set and associated SSs . . . . . . . . . . . . . . . . . . 554.4 Performance of different SLPV controllers with heuristically selected SSs . . . . . . . . . . 554.5 Performance of SLPV controllers with optimized SSs (N1 = 3, N2 = 1) . . . . . . . . . . . 565.1 Identified parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Weights parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Designed controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4 Closed-loop stability and controller order . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.1 Properties of the NREL 5 MW baseline WTS . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Saturation and rate-limit on control inputs of the WTS . . . . . . . . . . . . . . . . . . . . 93C.1 Parameters of magnetic actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119xList of Figures1.1 A simple LPV control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A SLPV control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Switching instants under inexact measurement of a scheduling parameter . . . . . . . . . . 52.1 Partition of a two-dimensional parameter set Θ (N = 4) . . . . . . . . . . . . . . . . . . . 112.2 Modification of nominal SSs for ADT switching to accommodate measurement uncertainties 132.3 Θ(j)i and admissible value of θi at switching . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Admissible region of (θi, θˆi) for σ = j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Scheduling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Disturbance input and Channel 1 of z(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Rectangular subsets and SSs for a two-dimensional parameter set (N1 = 3, N2 = 2) . . . . 283.2 Block diagram of a state-feedback SLPV controller K(θ) for an LPV plant G(θ) . . . . . . 283.3 Switching surfaces and γ value obtained using brute-force search . . . . . . . . . . . . . . . 343.4 γ value VS iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Subsets and SSs (s = 2, N1 = N2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Performance of different SLPV controllers. SLPV 0(x): traditional SLPV controller, SLPV1(x): proposed SLPV controller with improved local performance . . . . . . . . . . . . . . 414.3 Comparison of the performance of two SLPV controllers (s = 1, N = 2) . . . . . . . . . . 424.4 Performance of SLPV controllers with optimized SSs. SLPV 0(c): traditional SLPV con-troller with optimized SSs, SLPV 1(c): proposed SLPV controller with improved local per-formance and optimized SSs under the setting of γ¯ = γ∗, SLPV 1(d): proposed SLPV con-troller with improved local performance and optimized SSs under the setting of γ¯ = 1.02γ0c 524.5 Global best in each iteration of Algorithm 1 for the proposed SLPV controller with γ¯ = γ∗ . 524.6 Brute-force search result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Controller synthesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.8 L2-gain bounds comparison for the air-fuel ratio control example . . . . . . . . . . . . . . . 574.9 Profiles of air flow and engine speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.10 Trajectory of the engine operating point. The red dash-dot lines in the right figure denote theoptimized SSs while the blue dashed lines denote the bounds of θ1 . . . . . . . . . . . . . . 58xiList of Figures4.11 Switching signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.12 Disturbance rejection performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Mechanical layout of a miniaturized OIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Block diagram of the OIS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3 Demonstration of beam width variation and unbalanced forces (δb and δF,i represent thedeviations from the nominal beam width b and nominal force output F , respectively.) . . . . 645.4 Large-scale 3-DOF prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Block diagram of the OIS controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.6 A comparison between measured frequency response and frequency response of the modelset (5.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.7 Implementation of the MPDR controllers for a specific product for pitch control . . . . . . . 715.8 A synthesis structure with the generalized plant and MPDR controllers . . . . . . . . . . . . 725.9 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.10 Subsets from partitioning the set of the natural frequencies . . . . . . . . . . . . . . . . . . 745.11 Frequency response of different controllers including the band-stop filter for LP 3 . . . . . . 755.12 Quantitative comparison of different controllers. Top: tracking performance, bottom: controlinput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.13 Tracking performance of different controllers. Reference (solid blue, only in the top sub-figure), Kcla (dash-dot black), Kµ (solid green), KM (dash-dot red) . . . . . . . . . . . . . 776.1 Different types of foundations for offshore wind turbines (Source: Principle Power). TLP:tension-leg platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 The baseline WTS with a semi-submersible floating platform . . . . . . . . . . . . . . . . . 826.3 Open-loop frequency response of the 15th order model (6.9). Note that in this figure, δu :=δβ, δv := δvx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.4 Open-loop frequency response of the 9th order model (6.10). Note that in this figure, δu :=δβ, δv := δvx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.5 Synthesis structure of the blade pitch controller. The input δw does not exist when the wavedisturbance is ignored in controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.6 Cost function and SSVs in PSO iterations. A “local best” refers to the best position of anindividual particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.7 Profiles of wind with a constant mean speed and wave. Top: wind speed in x axis . . . . . . 946.8 Performance of different controllers in regulating the platform pitch, generator speed andpower under the wind and wave profiles shown in Fig. 6.7 . . . . . . . . . . . . . . . . . . . 956.9 Control inputs and switching signal (of the SLPV controller) under the wind and wave pro-files shown in Fig. 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.10 Normalized root-mean-squared error yielded by different controllers under the wind andwave profiles shown in Fig. 6.7. Performance of the baseline controller was utilized as thebasis for normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97xiiList of Figures6.11 Profiles of wind with a varying mean speed and wave. Top: wind speed in x axis . . . . . . 986.12 Performance of different controllers in generator-speed and power regulation under the windand wave profiles shown in Fig. 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.13 Performance of different controllers in platform motion regulation under the wind and waveprofiles shown in Fig. 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.14 Control inputs and switching signal (of the SLPV controller) under the wind and wave pro-files shown in Fig. 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.15 Normalized root-mean-squared error yielded by different controllers under the wind andwave profiles shown in Fig. 6.11. Performance of theH∞ controller was utilized as the basisfor normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.16 Actual and measured (filtered) wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.17 Performance of different controllers in generator-speed and power regulation under measure-ment uncertainty of wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.18 Performance of different controllers in platform motion regulation under measurement un-certainty of wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.19 Control inputs and switching signals of different controllers. Note that the switching signalsof the two SLPV controllers were the same due to use of the same SSs . . . . . . . . . . . . 1036.20 Normalized root-mean-squared error yielded by different controllers under measurement un-certainty of wind speed. Performance of the H∞ controller was utilized as the basis fornormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104C.1 Frequency response of the band-stop filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.2 Displacement measurements at two symmetric points under the same currents . . . . . . . . 122D.1 Singular value plots of the open-loop system and the closed-loop systems consisting of dif-ferent controllers. OL: open loop, CL: closed loop, K1: H∞ controller ignoring wave dis-turbance, K2: H∞ controller considering wave disturbance . . . . . . . . . . . . . . . . . . 126D.2 Performance of controllers considering and ignoring the wave disturbance . . . . . . . . . . 127xiiiList of Notations and SymbolsR, Rn, Rm×n The real numbers, real n-dimensional vectors, real m× n matricesSn n× n real symmetric matricesZN Integer set defined as {1, 2, · · · , N}I An identity matrix of appropriate dimensionIn An n× n identity matrix0 A zero matrix of appropriate dimension〈X〉 Shorthand for X +XTdiag(X1, . . . , Xk) A block-diagonal matrix comprising X1, . . . , XkMT Transpose of a matrix MM ≥ 0 M is symmetric and positive semidefiniteM > 0 M is symmetric and positive definiteM > N M and N are symmetric and M −N > 0M ≥ N M and N are symmetric and M −N ≥ 0? An off-diagonal block induced by symmetry in a symmetric matrix, i.e.[M11 ?M21 M22]:=[M11 MT21M21 M22]× Cartesian product⊗ Kronecker product∅ An empty setA \B The difference of sets A and B, defined as {x : x ∈ A and x 6∈ B}A ∩B The intersection of sets A and B, defined as {x : x ∈ A and x ∈ B}int(Θ) The interior of a set Θver(Θ) Set of vertices of a polytopic set ΘΘi Projected set of Θ onto the i-th coordinatexivList of AcronymsADT average dwell timeFAST Fatigue, Aerodynamics, Structures, and TurbulenceGS gain schedulingLMI linear matrix inequalityLP lens platformLPV linear parameter-varyingLQR linear quadratic regulatorLTI linear time-invariantMPDR multiple parameter-dependent robustNREL National Renewable Energy LaboratoryOIS optical image stabilizerPDLF parameter-dependent Lyapunov functionPDLMI parameter-dependent linear matrix inequalityPI proportional-integralPID proportional-integral-derivativePSO particle swarm optimizationRMS root mean squareRMSE root-mean-squared errorSLPV switching linear parameter-varyingSS switching surfaceSSV switching surface variableTLP tension-leg platformWTS wind turbine systemxvAcknowledgementsFirst of all, I would like to thank my supervisor, Prof. Ryozo Nagamune. He introduced me to the field ofrobust and gain-scheduling control and gave me invaluable guidance in addressing problems arising fromcontrol engineering. This work would be impossible without his consistent support, patience and inspiringdiscussions throughout my PhD study. His diligence, meticulous attitude and constant kindness to everyonearound him motivated me to be a hard-working, rigorous and tolerant person. Working with Prof. Nagamuneis one of the most enjoyable and rewarding experiences in my life.I am thankful to my co-supervisor, Prof. Mu Chiao. He not only helped me in completing the opti-cal image stabilizer project, but also gave me valuable advice on improving my communication and inter-personal skills. I would also like to thank my supervising committee members, Prof. Bhushan Gopaluni,Prof. A. Srikantha Phani and Prof. Boris Stoeber, for their consistent support and instructive feedback for myresearch work. I am also grateful to Prof. Bei Lu at Shanghai Jiao Tong University for her detailed commentson my thesis.My sincere appreciation also goes to the current and previous members of Control Engineering Lab atUBC for the pleasant working atmosphere. Special thanks go to Chenlu Han and Jeffery Homer for theirhelp in wind turbine modeling control, and to Jihoon Lim for his help in automotive engine control. Specialthanks also go to Masashi Karasawa and Alireza Alizadegan for their help in the optical image stabilizerproject. I am also grateful to Mahammadreza Rostam, Sara Hosseinirad, Tiancheng Zhong, Shuo Kang,Ran Fan, Anderson Soares, Cristopher Cortes, Eduardo Escobar, Marcin Mirski, Ali Cherom Kheirabadi,Dr. Masih Hanifzadegan, Arman Zandinia and Dillon Melamed for being friendly and supportive labmates.I have been so fortunate to work with them.I also want to thank all my friends who have always been there to provide their encouragement and help.Some of them deserve special attention, including Teng Li, Dr. Qiugang Lu, Hui Tian, Dr. Peng Shi, Mr.Qinghua Zhu, Dr. Ting Yue, Dr. Kenichi Katoh, Dr. Yunfei Zhang and Dr. Min Xia.The financial support of the Vanier Canada Graduate Scholarship from Natural Sciences and EngineeringResearch Council of Canada is gratefully acknowledged as well.My family deserves the greatest gratitude for their irreplaceable support, patience and unconditional love.My mother is always an example of being industrious, thrifty and perseverant. My sister has helped a lot inlooking after my parents while I was not around them. My special thanks go to my wife, Jie, whose love andcompany has tremendously helped shape my life as it is. She helped me grow by constantly challenging meand keeping me out of my comfort zones. She has done much more than she should do in taking care of ournewborn daughter, Emma (Zejia). Last but not least, I thank Emma for her smiles to warm my heart.xviDedicationThis thesis is dedicated to my family.xviiChapter 1Introduction1.1 BackgroundLinear parameter-varying (LPV) systems are linear dynamic systems whose state-space matrices depend onexogenous varying parameters, in the form of{x˙ = A(θ(t))x(t) +B(θ(t))u(t),y = C(θ(t))x(t) +D(θ(t))u(t),(1.1)where x(t) is the state vector, u(t) is an input vector, y(t) is an output vector and θ(t) is a vector of exogenousparameters that can be time-varying and is available in real-time by measurement or estimation. The vectorθ(t) is usually assumed to belong to a set, Θ, that represents the operating range of the systems. LPV systemwas introduced for the analysis and design of gain-scheduling (GS) control systems [1–3]. In short, gainscheduling is a control design approach for constructing a nonlinear controller for a nonlinear plant with afamily of linear controllers. Prior to the advent of LPV techniques, GS was usually realized with a “heuristic”approach, in which the operating range of a system was divided into small regions where the plant was treatedas a linear time-invariant (LTI) system. An LTI controller was then designed for each region and the overallcontrol law was obtained via interpolation or switching of those LTI controllers. This approach, however,disregarded the time-varying nature of the plant and thus could not guarantee satisfactory performance oreven stability along all possible trajectories of the operating point [2, 4].The LPV technique provides a systematic way for gain scheduling with theoretical performance guar-antee. To apply this technique to a nonlinear system, the first step is to obtain an LPV model to describethe plant dynamics, i.e. a linear description or approximation of the nonlinear plant parameterized by thescheduling parameters. There are two approaches to this [5]. One is the classical linearization approach,which is based on Jacobian linearization of the nonlinear plant about a family of operating points. This ap-proach yields a family of linearized plants parameterized by scheduling parameters. The other is the quasi-LPV description, in which the plant dynamics are rewritten to disguise the nonlinearities as time-varyingparameters that are treated as scheduling parameters.After obtaining the LPV model, an LPV controller together with a Lyapunov function can be obtainedby solving an optimization problem involving (parameter-dependent) linear matrix inequalities (LMIs) thatguarantee the stability and performance of the closed-loop system under all admissible trajectories of thescheduling parameters. In fact, a prespecified parameterization (e.g. affine or polynomial) is normallyassumed for the controller and the Lyapunov function, and the decision matrices that define the parameteri-zation are what actually appear in the LMIs and to be solved for. With the specified parameterization and the11.2. Literature Reviewdecision matrices obtained from solving the LMI optimization problem, the state-space matrices of an LPVcontroller can be computed online using the measured values of scheduling parameters. Figure 1.1 depicts asimple LPV control system for reference tracking, where r is the reference signal, u is the control input, y isthe output, P (θ) is an LPV plant in the form of (1.1) and K(θ) is an LPV controller parameterized by θ.—LPV controller LPV plantFigure 1.1: A simple LPV control systemIn LPV paradigm, an LPV controller is generally used to control an LPV plant. When designing the LPVcontroller, we need to guarantee the stability and performance of the closed-loop system for all θ belonging tothe parameter variation set Θ. However, for plants having significant dynamics variation in Θ, a single LPVcontroller with a (constant or parameter-independent) Lyapunov function may lead to conservative results,or even worse, the problem of finding a single stabilizing LPV controller may be infeasible. To resolvethese problems, Lu and Wu [6] proposed the switching LPV (SLPV) control. As shown in Fig. 1.2, theidea of SLPV control is to divide the parameter variation set Θ into several subsets separated by switchingsurfaces (SSs), to design a local LPV controller K(j)(θ) for each subset Θ(j) and to switch among theselocal controllers according to some switching rules. SLPV control has been successfully applied to variousengineering systems, such as aircraft [7, 8], automotive engine [9], ball-screw drive [10] and nuclear steamgenerator [11].3LPV plant— Switching LPV controllerSwitchingSurfacesFigure 1.2: A SLPV control system1.2 Literature ReviewIn this section, we review the literature related to LPV and SLPV.21.2. Literature Review1.2.1 LPVLPV system was introduced in [1] for analysis and design of GS control systems and has received consider-able attention in the last three decades [2–4, 12–16]. Prior to the advent of LPV techniques, GS were usuallyrealized with a “heuristic” approach. In this approach, a number of operating points were selected within theoperating range of a nonlinear or time-varying plant, an LTI controller was then designed for each operatingpoint and the overall control law was obtained via interpolation or switching of those LTI controllers. Thisapproach, however, disregarded the time-varying nature of the original plant and thus could not guaranteesatisfactory performance or even stability along all possible trajectories of the operating point [2, 4].One approach to rigorously address the GS problem is linear-fractional-transformation (LFT) GS tech-niques [17–20] that employ constant Lyapunov functions, as opposed to one which depends on the schedul-ing parameters. Another approach is LPV GS techniques. For LPV techniques, the so-called quadratic GStechniques were first proposed in [21] and [12]. These techniques provide H∞ performance guarantee withconstant LPV functions. However, according to [13], both the LFT GS and the quadratic GS techniques maylead to conservative performance because the constant Lyapunov functions in those approaches allow therate of variation of scheduling parameters to be arbitrarily fast. To reduce the conservativeness, LPV designapproaches adopting parameter-dependent Lyapunov functions (PDLFs) were proposed in [22] and [14],which allow the rate of variation of scheduling parameters to be explicitly incorporated in controller design.According to [14], such approaches can be further divided into two categories. One is based on the so-calledbasic characterization which allows the incorporation of multiple specifications into the design problemsuch as pole placement, mixed H2/H∞, etc. The basic technique is also desired for controller implemen-tation. Another one is based on the projected solvability conditions, which involve less variables in offlinecontroller design but more computation in online controller implementation. As mentioned in [14], to ensurethe practical validity of the designed LPV controllers, i.e. controller implementation does not require timederivatives of scheduling parameters, some of the Lyapunov variables have to be constant. To release thisstructural constraint on the Lyapunov function, another approach was developed in [16] for design of output-feedback LPV controllers. It was proven in [16] that this method is no more conservative than the methodssummarized in [14] although the line search involved significantly increases the computational complexity.1.2.2 SLPVSLPV control was first proposed by Lu and Wu [6, 7] to reduce the conservatism of LPV control. In thisapproach, the parameter space is divided into several subsets and an LPV controller is designed for eachsubset using a switched parameter-dependent Lyapunov function. Local controllers are switched accordingto some switching rules such as the hysteresis switching.Another approach for designing an SLPV controller was proposed in [23]. In this approach, Youlaparameterization is utilized to decompose the controller design into two steps. One is for ensuring stabilityunder switching while the other one is for enhancing the local performance, i.e. the performance in the localsubsets. In this way, each local controller can be designed independently of each other without sacrificing thestability. However, the use of a constant Lyapunov function makes the approach overly conservative, and thehigh order of the resulting controller limits the use of this approach. To decouple the objectives of stability31.2. Literature Reviewguarantee and local performance improvement, a different approach based on separate Lyapunov functionsfor the two objectives were proposed for discrete-time state-feedback SLPV control [24]. It is shown thatthis approach has the potential to significantly improve the local performance compared to using a commonLyapunov function for the two objectives.A potential drawback of switching controllers including SLPV controllers is that there may be discon-tinuities in control signal when switches occur. The sharp change of control signal may cause mechanicaldamage and actuator saturation. To mitigate this problem, smooth SLPV control were investigated in [25, 26]for the state-feedback case. For output-feedback case, smooth switching was achieved in [27] by constrain-ing the derivative of controller matrices at switching instants and interpolating adjacent local controllers inoverlapped regions. Another approach was proposed in [8], in which the projected solvability conditions andinterpolation are used.The advantages of SLPV control have been demonstrated through application to various engineeringsystems, such as aircraft [7, 8], automotive engine [9], ball-screw drive [10], nuclear steam generator [11]and optical image stabilizers [28].1.2.3 Uncertain Scheduling ParametersImplementation of LPV controllers needs online measurement or estimation of scheduling parameters. How-ever, for some applications, exact values of the scheduling parameters may not be available due to sensordrift and noise, or parameter estimation errors. Uncertainties on scheduling parameters have been consid-ered in LPV controller design for both state-feedback cases [15, 29] and output-feedback cases [30, 31]. Thepaper [30] dealt with continuous-time polytopic LPV systems with multiplicative uncertainties, i.e. uncer-tainties with bounds proportional to actual values of scheduling parameters. However, the method cannothandle bias errors that are common in many measurement systems. In [29], robust stabilization problem wasconsidered for polytopic LPV systems considering additive uncertainties, i.e. uncertainties whose boundsare independent of actual values of scheduling parameters. Robust state-feedback LPV controller design wasproposed in [15] to handle additive uncertainties. A more general approach was proposed in [31] for design ofcontinuous-time output-feedback LPV controllers against both additive and multiplicative uncertainties butnot coexistence of them. Coexistence of additive and multiplicative uncertainties were addressed [32, 33]for output-feedback LPV controller design.Despite the aforementioned research results, to the best of the author’s knowledge, there is no workreported on design of SLPV controllers incorporating measurement uncertainties in scheduling parametersto date except [34]. This work is important because large dynamics and measurement uncertainties cancoexist in some applications. In [34], the authors proposed a method of designing robust SLPV controllersagainst additive uncertainties. But it just dealt with the state-feedback case, not the output-feedback case.Furthermore, the admissible region of the actual and measured scheduling parameters for each subset is notfully investigated, which may lead to conservative results.41.3. Research Objectives1.2.4 Switching Surface Design in SLPV ControlIn most previous work on SLPV controller design, subsets and SSs are usually prespecified. However, dif-ferent SSs may lead to significantly different performance. Even though trial and error methods can be usedto search for satisfactory SSs, optimization-based methods that are able to find the optimal SSs in an efficientway are more desired. Up to now, the only work that addresses the optimal SS design problem is [35], whichproposed a gradient-based optimization algorithm to optimize the SSs. However, the computational cost ofthis algorithm is high due to numerical evaluation of the gradient of the objective function. In addition, themethod is a local search method that can easily get stuck in a local optimum.1.3 Research ObjectivesDespite the fact that SLPV control has been investigated a lot in the last decade, there are still some openproblems from the author’s perspective. First, uncertainties in the measurement of the scheduling parametershave not been considered in SLPV control, although they have been dealt with in LPV control, as explained inSection 1.2.3. Measurement uncertainties, if not accounted in SLPV controller design, may lead to degradedperformance and loss of stability of the closed-loop system. To understand this issue, see Fig. 1.3 for anillustration where the trajectories of both the actual and measured scheduling parameters for an SLPV controlsystem are given. For the main results of this thesis presented in Chapters 2 to 4 , we assume that there aretwo switching surfaces between any two adjacement subsets. Here, to simply express the consequences ofignoring the measurement uncertainty, we just show one switching surface S(1,2), which is for determiningthe switch from K(1) to K(2). Assume that a switching event happens at t1 when the measured schedulingparameter crosses the switching surface S(1,2) and the local LPV controller K2(θ) is activated subsequently.Nevertheless, the actual parameter will not move into Θ(2) until t2. Considering that K2(θ) is exclusivelydesigned for Θ(2), the stability and the performance of the closed-loop system cannot be guaranteed betweent1 and t2.MeasuredActualFigure 1.3: Switching instants under inexact measurement of a scheduling parameterSecond, in most of previous work on SLPV, the SSs that determine the partition of the parameter vari-ation set are generally prespecified in SLPV controller design. However, the selection of SSs may play an51.3. Research Objectivesimportant role in determining the performance of an SLPV controller. In other words, different SSs mayyield a significant difference in the performance of an SLPV controller. Although a gradient-based opti-mization algorithm was proposed in [35] for designing the SSs, the high compuational cost due to numericalevaluation of the gradient and the local-search property limit its use. Therefore, the SS design problem hasnot been fully solved from the author’s perspective.Finally, the performance of an LPV control system is usually characterized by the upper bound of theL2-gain of the closed-loop system. To the best of the author’s knowledge, in all the previous work on SLPVcontrol, the SLPV controllers were designed by solving an optimization problem minimizing the worst-case L2-gain bound, i.e. the maximum of the local L2-gain bounds in all the subsets. We refer to thisdesign approach as the traditional approach. It has been observed in our experiences that this approachtends to yield almost identical local L2-gain bounds in all the subsets, which are sometimes only marginallysmaller than the L2-gain bound yielded by a non-switching LPV controller. This means that the performanceimprovement brought by the traditional SLPV control over the LPV control in the whole operating range ofthese plants can be marginal, despite the much higher computational cost in designing the SLPV controllers.Based on the literature review and the open problems stated above, we define the research objectives ofthis thesis as follows.The first research objective is to propose an approach to SLPV controller design under uncertain schedul-ing parameters with guaranteed stability and performance. The second one is to develop algorithms to opti-mize the SSs to further improve the performance of an SLPV controller. Our third objective is to propose anew approach to designing SLPV controllers with improved local performance without much sacrificing theworst-case performance over all subsets.Moreover, to validate the approaches to be proposed, we will apply them to two practical problems,namely control of miniaturized optical image stabilizers and control of a floating offshore wind turbine.Each of the two applications will be briefly explained as follows.Image blur as a result of involuntary hand shaking when taking photos has always been an issue for con-sumers. As a powerful technique to mitigate the image blur, optical image stabilizers (OIS’s) have been verycommon in high-end cameras. On the other hand, there is a tendency to popularize miniaturized OISs in mo-bile phones. However, product variation in fabrication of these micro-scale devices is considerable [36, 37]and can cause significant variation of dynamics from one product to another. To deal with the dynamicsvariations, we will apply the approach to SLPV control using uncertain scheduling parameters, and experi-mentally demonstrate its superior performance.Wind power has been the world’s fastest-growing renewable energy source in recent years [38]. Oneof the recent trends in the wind energy industry is to place large turbines offshore on floating platforms indeep water [39]. However, the dynamics of the wind turbine is nonlinear with respect to the wind speed.Moreover, the wind and wave disturbances cause motions of the platform, and consequently impose loadingon the turbine structures. This will increase the maintenance cost and reduce the turbine’s lifespan. To dealwith the nonlinear turbine dynamics and to mitigate the platform motion and improve the power quality,control techniques based on the theoretical methods developed in this thesis will be proposed and validatedusing medium-fidelity simulations.61.4. Outline of This ThesisTo summarize, the research objectives of this thesis are to (1) develop a novel approach to SLPV controlusing uncertain scheduling parameters, (2) develop an approach to optimizing the SSs in SLPV controllerdesign, (3) develop a new approach to SLPV control with improved local performance, and apply the pro-posed theoretical methods to (4) control of miniaturized optical image stabilizers with product variations,and (5) control of a floating offshore wind turbine.1.4 Outline of This ThesisThis thesis is organized as follows.Chapter 2 is devoted to the development of a novel approach to designing output-feedback SLPV con-trollers under inexact measurement of scheduling parameters. The SLPV controllers are robustly designed sothat the stability and L2-gain performance of the switched closed-loop system can be guaranteed even undercontroller switching determined by the measured (not actual) scheduling parameters. As for the switch-ing rules, both hysteresis switching and average-dwell-time (ADT) switching are considered. Solvabilityconditions for the controller design problem are expressed in terms of parameter-dependent linear matrixinequalities and nonconvex SS conditions, with a line search parameter. The nonconvex conditions canbe convexified for hysteresis switching controller design by imposing practical validity constraints, and forADT switching controller design by adding equality constraints on some of Lyapunov variables on the SSs.The effectiveness of the proposed approach is demonstrated with a numerical example.Chapter 3 presents the optimal SS design for state-feedback SLPV control. For LPV plants with poly-nomial parameter-dependence, the problem of simultaneous design of a state-feedback SLPV controller andSSs is formulated as an optimization problem involving bilinear matrix inequalities (BMIs). An algorithmis then proposed to solve the BMI problem by sequentially updating SS variables and controller variableswhile fixing the other. Although the algorithm cannot guarantee to find the global optimum or even a localoptimum, a numerical example demonstrates its effectiveness.Chapter 4 introduces a new method to design SLPV controllers with improved local performance, andanother algorithm to optimize the SSs. The design approach utilizes the weighted average of the local L2-gain bounds (representing the local performance) as the cost function to be minimized, while the maximumof the local L2-gain bounds (representing the worst-case performance over all subsets) is bounded with atuning parameter. The tuning parameter is useful for taking the trade-off between the local performance andthe worst-case performance. An algorithm based on particle swarm optimization is proposed to optimize theSSs, which has a great potential to find the global optimum. The proposed design methods are validated onboth a numerical example and a practical problem of air-fuel ratio control of an automotive engine.Chapter 5 demonstrates an application of the method developed in Chapter 2 to control of miniaturizedoptical image stabilizers with product variations. The dynamics of batch-fabricated OIS’s with inevitableproduct variations is represented by a set of linear models, parameterized by two product-dependent naturalfrequencies and one uncertain gain. It turns out that the natural frequencies for each OIS product are difficultto determine accurately, and thus assumed to be estimated with errors. The method developed in Chapter 2is employed to design multiple robust controllers parameterized by estimated natural frequencies, which are71.5. Significance of This Thesisrobust against both the estimation errors of the natural frequencies and the gain uncertainty. Experimen-tal results on large-scale prototypes demonstrate that the proposed controllers outperform a conventionalparameter-independent robust controller as well as a single parameter-dependent robust controller.Chapter 6 deals with control of a floating offshore wind turbine on a semi-submersible platform aboverated wind speed using the methods developed in Chapters 2 and 4. By linearizing a control-oriented nonlin-ear model in [40] at a series of operating points, a family of LTI models are obtained and used to design theSLPV controllers as well as an LPV controller for comparison. SSs are optimized in SLPV controller designutilizing the algorithm developed in Chapter 4. Uncertainties in measurement of the wind speed are alsoinvestigated in both simulation and controller design. The SLPV controllers are validated in medium-fidelitysimulations based on FAST and compared with other controllers including a baseline controller, an LPVcontroller, and an H∞ controllers in power regulation and platform motion mitigation.1.5 Significance of This ThesisIn the theoretical side, this thesis contributes to the SLPV control theory in three aspects. Firstly, the thesisproposes an approach to designing SLPV controllers using uncertain scheduling parameters. This contribu-tion makes the SLPV control technique applicable to more broad engineering systems where only inexactmeasurement of scheduling parameters are available. Secondly, this thesis proposes a new approach to de-signing SLPV controllers with improved local performance. As opposed to the traditional approach thatoften leads to similar performance in all the subsets, the new approach has potential to significantly improvethe performance in some subsets without much sacrifice of the worst-case performance. Finally, this thesispresents algorithms to optimize the SSs in SLPV controller design. The proposed algorithms are easy toimplement and have great potential to further improve the performance of an SLPV controller.In the application side, this thesis addresses two practical problems using the theoretical methods de-veloped in this thesis. For control of miniaturized optical image stabilizers, utilizing the method developedin Chapter 2, multiple parameter-dependent robust controllers are designed and validated to outperform aconventional µ synthesis controller by experimental results. The controllers have relatively low order andcan be implemented in mass-produced miniaturized OISs for mobile phones for enhanced performance. Forcontrol of a floating offshore wind turbine, based on the methods developed in Chapter 2 and Chapter 4,SLPV controllers are designed and their superior performance in both power and generator speed regulationand platform motion suppression is demonstrated in medium-fidelity simulations. Furthermore, the SLPVcontroller considering the measurement uncertainty of the wind speed is guaranteed to work even when thewind speed cannot be exactly measured for gain scheduling. The developed solutions have good practicalityand have great potential to help reduce the cost of offshore wind energy by reducing the maintenance costand improving the power capture of the wind turbines.8Chapter 2SLPV Control with Uncertain SchedulingParameters2.1 IntroductionIn LPV control applications, exact values of scheduling parameters may not be available for adaptationof controller parameters, due to sensor drift and noise, or parameter estimation errors. Uncertainties onscheduling parameters have been considered in LPV controller design for both state-feedback cases [15, 29]and output-feedback cases [30, 31]. [30] dealt with multiplicative uncertainties. However, the method cannothandle bias errors that are common in many measurement systems. A more general approach was proposedin [31], which considered additive uncertainties. Coexistence of additive and multiplicative uncertaintieswere addressed in [32, 33].Despite the aforementioned research results, there is no work reported on design of SLPV controllersincorporating measurement uncertainties in scheduling parameters. This work is important because largeparameter variation can exist in applications where significant uncertainties in scheduling parameter mea-surement are unavoidable. Switching among local controllers can only be based on measured schedulingparameters. It is not trivial how to design local controllers and switching rule properly so that switchingbased on measured scheduling parameters will not cause performance degradation or instability. The mainpurpose of this chapter is to accomplish this practically-important but still-missing work.The main contribution of this chapter is to propose a systematic approach to the design of output-feedback SLPV controllers with guaranteedL2-gain performance, under measurement uncertainties in schedul-ing parameters. In our scheme, both local controller adaptation and switching among local controllers dependon measured (not actual) scheduling parameters (denoted as θˆ hereafter). For robustness guarantee, the ad-missible regions of the actual scheduling parameters (denoted as θ hereafter) and θˆ for local controllers aredetermined. Using the admissible regions, the design problem is formulated as an optimization problem in-volving parameter-dependent linear matrix inequalities (PDLMIs) with a line search parameter. Other minorcontributions of this chapter include:• Convexification of originally nonconvex SSs conditions for hysteresis switching by considering practi-cal validity constraints (i.e., the controller does not depend on the derivative of scheduling parameters).• Modification of ADT switching rule for the inexact measurement case, and acquisition of convex SSconditions for ADT switching that is missing in [6].This chapter is organized as follows. Section 2.2 gives preliminaries and formulate the SLPV controllerdesign problem with uncertain scheduling parameters. Section 2.3 presents the constraints on SSs for ADT92.2. Problem Settingswitching. To restate the problem in terms of PDLMIs with respect to θ and θˆ, we explicitly show theadmissible regions of (θ, θˆ) in Section 2.4. The main theorems for design of SLPV controllers againstmeasurement uncertainties are given in Section 2.5. Section 2.6 presents a numerical example to illustratethe efficacy of the proposed method.2.2 Problem SettingWe consider the following generalized LPV plant:x˙ = A(θ)x+B1(θ)w +B2u,z = C1(θ)x+D11(θ)w +D12(θ)u,y = C2x+D21(θ)w,(2.1)where θ := [θ1, · · · , θs]T is the vector of scheduling parameters which are measurable in real time, x(t) ∈Rn is the state vector, u(t) ∈ Rnu is the control input vector, w(t) ∈ Rnw is the vector of exogenous inputs,y(t) ∈ Rny is the measured output vector, and z(t) ∈ Rnz is the vector of output signals related to theperformance of the control system. The matrices in (2.1) are assumed to have compatible dimensions and tobe dependent on θ except B2 and C2.Remark 2.1. The assumption of constant B2 and C2 is needed for on-line calculation of controller matriceswithout the actual scheduling parameter vector θ (see (2.20)). This assumption can always be satisfied bypre-filtering of the control inputs u and/or post-filtering of the measured outputs y. See [12, p. 1255] fordetails. As long as the bandwidth of filters is much higher than the plant, the effects of the filters on theperformance of the closed-loop system can be neglected.Additionally, the parameter vector θ and its rate of variation θ˙ are supposed to be bounded by hyper-rectangular sets as:θ ∈ Θ, Θ := {θ ∈ Rs : θi ≤ θi ≤ θi, i ∈ Zs},θ˙ ∈ Ωθ, Ωθ := {θ˙ ∈ Rs : ωi ≤ θ˙i ≤ ωi, i ∈ Zs}.(2.2)For the main results of this chapter, we consider additive uncertainties in scheduling parameter measurementwhile necessary modifications of the results for multiplicative uncertainties are given in the Appendix. Withadditive uncertainties, the actual scheduling parameter θi is measured as θˆi = θi + δi, where δi representsthe uncertainty in the measurement of θi. The vectors θˆ := [θˆ1, · · · , θˆs]T and δ := [δ1, · · · , δs]T denote themeasured scheduling parameters and the uncertainties in the scheduling parameters, respectively. Anotherassumption is that the uncertainty δi is independent of each other, and its bound is independent of the actualscheduling parameter θi. The bounds of uncertainties and their rate of variation can generally be estimated apriori, and are assumed to satisfyδ ∈ ∆, ∆ := {δ ∈ Rs : δi ≤ δi ≤ δi, i ∈ Zs},δ˙ ∈ Ωδ, Ωδ := {δ˙ ∈ Rs : νi ≤ δ˙i ≤ νi, i ∈ Zs},(2.3)where δi < 0 < δi is presumed without loss of generality. From (2.2) and (2.3), we have˙ˆθ = θ˙ + δ˙ ∈ Ωθˆ,102.2. Problem SettingFigure 2.1: Partition of a two-dimensional parameter set Θ (N = 4)whereΩθˆ := {v ∈ Rs : ωi + νi ≤ vi ≤ ωi + νi, i ∈ Zs}. (2.4)Suppose that the parameter set Θ is partitioned into N closed hyper-rectangular subsets {Θ(j)}j∈ZN bydividing Θi (i.e. the projection of Θ onto the ith axis) into Ni intervals with overlapped segments betweenany adjacent intervals, where N = Πsi=1Ni and Θ =⋃j∈ZN Θ(j). Here, hyper-rectangular subsets arechosen for their simplicity, while subsets of other shapes, e.g. triangular ones, are also potential choices. SeeFig. 2.1 for an illustration. The subsets are separated by a family of SSs. Let S(j,k) denote the SSs specifyingthe one-directional move of θˆ from subset Θ(j) to subset Θ(k). Here we define that two sets are adjacent ifthe interior of their intersection is non-empty. Moreover, two sets Θ(j) and Θ(k) are defined to be adjacentin θκ direction (for some κ ∈ Zs) if they are adjacent and their projected sets onto θκ are different, i.e.int(Θ(j) ∩Θ(k)) 6= ∅, Θ(j)κ 6= Θ(k)κ . (2.5)By this definition, in Fig. 2.1, Θ(1) is adjacent to Θ(2) in θ1 direction, to Θ(6) in θ2 direction and to Θ(5) inboth θ1 and θ2 direction.For the LPV system (2.1), we consider an SLPV controller consisting of a family of full-order output-feedback LPV controllers in the form of[x˙Ku]=[A(j)K (θˆ) B(j)K (θˆ)C(j)K (θˆ) D(j)K (θˆ)]︸ ︷︷ ︸=:K(j)(θˆ)[xKy], j ∈ ZN , (2.6)where xK(t) ∈ Rn is the controller state vector, and a local LPV controller K(j)(θˆ) ∈ R(n+nu)×(n+ny)takes charge of a specific parameter subset Θ(j). Note that controller matrices are functions of measuredscheduling parameters θˆ, not of actual scheduling parameters θ. Local controllers are switched accordingto switching rules such as hysteresis switching and ADT switching, when the trajectory of θˆ hits one of theSSs. A switching signal σ designates the active local controller, and is generated based on θˆ. Hysteresisswitching occurs as follows: Let σ(0) = j if θˆ(0) ∈ Θ(j) (pick an arbitrary j if there exist multiple j). Forany t > 0, if σ(t−) = j (where σ(t−) is the left limit of σ at time t) and θˆ(t) ∈ Θ(j), keep σ(t) = j. On theother hand, if σ(t−) = j, θˆ(t−) ∈ S(j,k), and θˆ(t) ∈ Θ(k) \Θ(j), then let σ(t) = k.112.3. Constraints on Switching Surfaces for ADT SwitchingUnder SLPV control, the closed-loop system becomes a switched LPV system, which can be describedas {x˙cl =A(σ)cl (θ, θˆ)xcl +B(σ)cl (θ, θˆ)w,z =C(σ)cl (θ, θˆ)xcl +D(σ)cl (θ, θˆ)w,(2.7)where xcl := [xT xTK ]T ∈ R2n, and the expressions of A(σ)cl , B(σ)cl , C(σ)cl and D(σ)cl in terms of plant andcontroller matrices can be readily obtained.The controller design problem tackled in this chapter is stated below.Problem 2.1. Given an LPV plant (2.1) with scheduling parameters θ satisfying (2.2) and the parametersubsets {Θ(j)}j∈ZN , as well as the measurement uncertainties δ satisfying (2.3), find an SLPV controller(2.6) with a switching rule that enforces stability and minimizes the L2-gain bound of the closed-loop system(2.7) for all admissible trajectories θ(·) and δ(·).2.3 Constraints on Switching Surfaces for ADT SwitchingFor ADT switching without considering measurement uncertainties [6], there is only one SS, which werefer to as nominal SS, introduced for any two adjacent subsets. In this case, overlapped area of the twoadjacent subsets is the SS, and has no interior in contrast to our assumption (2.5). A switching signal σ isdefined to have average dwell time (ADT) τ which is not less than τ0 > 0 if there exists a finite positivenumber N0 such thatNσ(T, t) ≤ N0 + T − tτ0, ∀T ≥ t ≥ 0, (2.8)where Nσ(T, t) is the number of switches on time interval [t, T ], and N0 is called the chatter bound.Without measurement uncertainties, the ADT (denoted by τ nom) of switching signals is estimated basedon the trajectories of actual scheduling parameters θ and nominal SSs, and used in controller design. How-ever, when there are uncertainties in scheduling parameter measurement, the uncertainties may cause extraswitches, thereby making the ADT τ of the switching signals based on θˆ smaller than the nominal valueτ nom. As a result, the controller designed based on τ nom may not work. To resolve this issue, for subsetsΘ(j) and Θ(k) that are adjacent in θκ direction, we will introduce two SSs S(j,k) and S(k,j) satisfyingS(k,j)κ − δκ ≤ Snomκ ≤ S(j,k)κ − δκ, (2.9)where Snomκ is the projection onto θκ of the nominal SS Snom between Θ(j) and Θ(k), and S(k,j)κ < S(j,k)κ isassumed. See Fig. 2.2 for an illustration. Hereafter, σ denotes a switching signal based on θˆ and modifiedSSs, while σnom denotes a switching signal based on θ and nominal SSs. With the modified SSs, we have thefollowing lemma and theorem.Lemma 2.1. Considering switches between two adjacent subsets, for any trajectories θ(·) and θˆ(·) satisfying(2.3), there must be at least one switch that happens based on σnom between any two successive switchesbased on σ.122.3. Constraints on Switching Surfaces for ADT SwitchingNominal ModifiedNominal ModifiedFigure 2.2: Modification of nominal SSs for ADT switching to accommodate measurement uncertaintiesProof. We prove this lemma by considering the switches between two subsets that are adjacent in onlyone direction, while the proof for the case of two subsets adjacent in multiple directions can be derivedanalogously. We use the notations in Fig. 2.2. Assume a switching from σ = j to σ = k happens at t1,which means θˆκ(t1) > S(j,k)κ . Then we haveθκ(t1) ≥ θˆκ(t1)− δκ > S(j,k)κ − δκ ≥ Snomκ ,where the first and third inequalities are from (2.3) and (2.9), respectively. This indicates that σnom(t1) = k.Suppose next switching from σ = k to σ = j happens at t2. Then we have θˆκ(t2) < S(k,j)κ , and thusθκ(t2) ≤ θˆκ(t2)− δκ < S(k,j)κ − δκ ≤ Snomκ ,where the assumptions in (2.3) and (2.9) are used again. This implies that σnom(t2) = j. Comparing σnomvalue at t1 and t2, we conclude that at least one switching from σnom = k to σnom = j happens in the interval(t1, t2].Theorem 2.2. For any trajectories θ(·) and θˆ(·) satisfying (2.3), the ADT τ of a switching signal σ is notless than the ADT τ nom of a switching signal σnom, i.e. τ ≥ τ nom.Proof. Let Nσnom(T, t) and Nσ(T, t) denote the number of switchings at time interval [t, T ] based on σnomand σ, respectively. According to (2.8), there exists Nnom0 > 0 satisfyingNσnom(T, t) ≤ Nnom0 +T − tτ nom, ∀T ≥ t ≥ 0. (2.10)If switching happens between only one pair of (i.e. two) adjacent subsets, then we have Nσ(T, t) ≤Nσnom(T, t) + 1 from Lemma 2.1. If switching happens between m(≥ 1) pairs of adjacent subsets, thenwe have Nσ(T, t) ≤ Nσnom(T, t) +m, which with (2.10) will further yieldsNσ(T, t) ≤ Nnom0 +m+T − tτ nom≤ N0 + T − tτ nom, ∀T ≥ t ≥ 0,where N0 := Nnom0 + m. Based on the definition of ADT, the switching signal σ has ADT that is not lessthan τ nom.132.4. Admissible Region DeterminationRemark 2.2. Theorem 2.2 implies that the ADT estimated based on nominal SSs and actual schedulingparameters can be utilized as it is in SLPV controller design when scheduling parameters are not exactlymeasurable.With two SSs introduced for any two adjacent subsets, switching signal σ for ADT switching is generatedin the same way as for hysteresis switching. However, as will be explained in Section 5, the constrainton Lyapunov functions at SSs for ADT switching is different from that for hysteresis switching. Underhysteresis switching, Lyapunov functions need to be monotonically decreasing at switching, while suchrequirement can be relaxed for ADT switching.2.4 Admissible Region DeterminationTo express the design problem in terms of LMIs with respect to θ and θˆ, we need to determine the admissibleregion of (θ, θˆ) for σ = j, which is denoted as Φ(j) hereafter. Since there is no essential difference inpartition of Θ and σ generation between hysteresis switching and ADT switching, we will not distinguishthem for determination of Φ(j). The region Φ(j) consists of all possible pairs of (θ, θˆ) when the localcontroller K(j)(θˆ) is active. Note that when K(j)(θˆ) is active, θˆ should be in subset Θ(j), or in other words,θˆi should stay in Θ(j)i for all i ∈ Zs. Therefore Φ(j) can be obtained as the Cartesian product:Φ(j) = Φ(j)1 × · · · × Φ(j)s , (2.11)where Φ(j)i is the projected set of Φ(j) onto the i-th coordinate, and includes all admissible pairs of (θi, θˆi)to make θˆ lie in subset Θ(j). The expression of Φ(j)i isΦ(j)i := {(θi, θˆi) : θi + δi = θˆi ∈ Θ(j)i , δi ∈ ∆i, θi ∈ Θi}. (2.12)To explicitly show the shape of Φ(j)i , let us consider four specific cases of Θ(j)i , illustrated in Fig. 2.3, andlisted as follows:Case 1: Θ(j)i = [θi, S(j,p)i ] for some p 6= j,Case 2: Θ(j)i = [S(j,p)i , S(j,q)i ] for some p, q 6= j,Case 3: Θ(j)i = [S(j,q)i , θi] for some q 6= j,Case 4: Θ(j)i = [θi, θi], i.e. Θi is not partitioned.For Cases 1–3 shown in Fig. 2.3, the value of θi right before switching from σ = j should be inside theshaded region. An example of θi and θˆi at switching is also given in Fig. 2.3. The admissible region Φ(j)i isobtained, by utilizing (2.12), as polytopes depicted in Fig. 2.4142.5. SLPV Controller Design under Uncertain Scheduling ParametersFigure 2.3: Θ(j)i and admissible value of θi at switching2.5 SLPV Controller Design under Uncertain Scheduling Parameters2.5.1 Hysteresis SwitchingWe first give a lemma that can be obtained from [6, Section 3.1]. The proof is omitted for brevity.Lemma 2.3. Suppose that there exists a family of parameter-dependent positive-definite matrices Z(j)(θˆ) ∈S2n such that (2.13) holds for any j ∈ ZN and (2.14) holds for any adjacent Θ(j) and Θ(k). Then, with theSLPV controller (2.6) with hysteresis switching rule, the closed-loop system (2.7) is exponentially stable andits performance ||z||2 < γ||w||2 is achieved with γ = maxj∈ZN {γ(j)} for all admissible trajectories θ(·)and δ(·). 〈A(j)cl (θ, θˆ)Z(j)(θˆ)〉− Z˙(j)(θˆ) ? B(j)cl (θ, θˆ)C(j)cl (θ, θˆ)Z(j)(θˆ) −γ(j)Inz D(j)cl (θ, θˆ)? ? −γ(j)Inw < 0,∀(θ, θˆ, ˙ˆθ) ∈ Φ(j) × ver(Ωθˆ), (2.13)Z(j)(θˆ) ≤ Z(k)(θˆ), ∀θˆ ∈ S(j,k). (2.14)The following theorem is obtained based on Lemma 2.3 and [31, Theorem 1]. For brevity, hereafter, weomit the dependence of plant matrices on θ, and the dependence of other matrices on θˆ.Theorem 2.4. Suppose that there exist a family of parameter-dependent matricesX(j) ∈ Sn and Y (j) ∈ Sn,parameter-dependent matrices Aˆ(j)K , Bˆ(j)K , Cˆ(j)K and Dˆ(j)K , and a positive scalar ε such that conditions (2.15)and (2.16) hold for any j ∈ ZN , and one of (2.18) and (2.19) holds for any adjacent Θ(j) and Θ(k). Then,with the SLPV controller (2.6) obtained by (2.20), the closed-loop system (2.7) is exponentially stable, andits performance ||z||2 < γ||w||2 is achieved with γ = maxj∈ZN {γ(j)} for all admissible trajectories θ(·)and δ(·).152.5. SLPV Controller Design under Uncertain Scheduling Parameters(a) Case 1 (b) Case 2(c) Case 3 (d) Case 4Figure 2.4: Admissible region of (θi, θˆi) for σ = j[Subset conditions] [X(j) InIn Y(j)]> 0, ∀θˆ ∈ Θ(j), (2.15)[ H(θ, θˆ) ?[εX(j) (A(θ)−A(θˆ))TY (j) 0 0]−εIn]< 0,∀(θ, θˆ, ˙ˆθ) ∈ Φ(j) × ver(Ωθˆ), (2.16)whereH(θ, θˆ) :=[M11 ?M21 M22]+[diag(−X˙(j), Y˙ (j)) 00 0], (2.17)162.5. SLPV Controller Design under Uncertain Scheduling ParametersM11 :=[〈AX(j) +B2Cˆ(j)K 〉 ?Aˆ(j)K + [A+B2Dˆ(j)K C2]T 〈Y (j)A+ Bˆ(j)K C2〉],M21 :=[[B1 +B2Dˆ(j)K D21]T [Y (j)B1 + Bˆ(j)K D21]TC1X(j) +D12Cˆ(j)K C1 +D12Dˆ(j)K C2],M22 :=[−γ(j)Inw ?D11 +D12Dˆ(j)K D21 −γ(j)Inz].[Switching surface conditions]{Y (j) ≥ Y (k),X(j) − Y (j)−1 ≤ X(k) − Y (k)−1, ∀θˆ ∈ S(j,k), (2.18){X(j) ≤ X(k),Y (j) −X(j)−1 ≥ Y (k) −X(k)−1, ∀θˆ ∈ S(j,k). (2.19)[Controller reconstruction]A(j)K =N(j)−1(Aˆ(j)K − Y˙ (j)X(j) − N˙ (j)M (j)T−Y (j)(A(θˆ)−B2Dˆ(j)K C2)X(j)−Bˆ(j)K C2X(j) − Y (j)B2Cˆ(j)K)M (j)−T,B(j)K =N(j)−1(Bˆ(j)K − Y (j)B2Dˆ(j)K),C(j)K =(Cˆ(j)K − Dˆ(j)K C2X(j))M (j)−T,D(j)K = Dˆ(j)K ,(2.20)where N (j) and M (j) are from the factorizationN (j)M (j)T= I − Y (j)X(j). (2.21)Proof. The parameter-dependent Lyapunov function (PDLF) for the closed-loop system (2.7) is selected tobe a piecewise-continuous function depending on θˆ, defined asV (σ)(xcl, θˆ) := xTclP(σ)(θˆ)xcl, (2.22)where {P (j)}j∈ZN is a family of positive definite parameter-dependent matrices. The matrix P (j) and itsinverse can be parameterized asP (j) =[Y (j) N (j)N (j)T?], P (j)−1=[X(j) M (j)M (j)T?], (2.23)where ‘?’ denotes irrelevant matrix blocks. The matrix inequality (2.15) guarantees that Lyapunov matrices{P (j)}j∈ZN are positive definite in their subsets (see [41] for details). Setting Z(j) := P (j)−1, the inequality(2.16) leads to (2.13) according to [31], and (2.18) is equivalent to (2.14) through congruence transformation,when M (j) and N (j) satisfying (2.21) are chosen to be X(j)−Y (j)−1 and−Y (j), respectively. If we choose172.5. SLPV Controller Design under Uncertain Scheduling ParametersM (j) = −X(j) andN (j) = Y (j)−X(j)−1, (2.19) can be obtained as another equivalent condition for (2.14).Thus, from Lemma 2.3, the statements are proved. It is worth mentioning that the variable ε in (2.16) was originally introduced in [31] for LPV synthe-sis considering measurement uncertainty of scheduling parameters, which helped converting binear matrixinequality (BMI) conditions into LMI conditions. Note that the constraints (2.18) and (2.19) are noncon-vex. However, if we impose practical validity constraint, these nonconvex conditions can be simplified andbecome convex, as explained next. To be practically valid, the SLPV controller should not depend on thederivative of θˆ. Therefore, as suggested in [14], at least one of X(j) and Y (j) must be constant.When X(j) (Y (j)) are selected to be constant for each j, all X(j) (Y (j)), j ∈ ZN , must be the same inorder to satisfy condition (2.18) or (2.19). Take X(j) for example. Given constant matrices X(j) and X(k)for some j, k, either (2.18) or (2.19) indicates that X(j) = X(k) when we consider the constraints at bothS(j,k) and S(k,j). After examining the switching between any two adjacent subsets, we conclude that allconstant X(j), j ∈ ZN have to be the same.Using the constant matrix, denoted by X0, (2.18) and (2.19) reduce to convex equality constraintsY (j) = Y (k), ∀θˆ ∈ S(j,k), (2.24)and convex inequality constraintsY (j) ≥ Y (k), ∀θˆ ∈ S(j,k), (2.25)respectively. Of course, (2.25) is less conservative than (2.24), and thus should be chosen. Analogously,using a constant matrix Y0 for all Y (j), we obtain convex SS conditions X(j) ≤ X(k) for any θˆ ∈ S(j,k)from (2.18). All allowable choices of X(j) and Y (j) for practical validity, with associated SS conditions,are summarized in Table 2.1.Table 2.1: Selection of Lyapunov variables and SS conditions for hysteresis switching under practical validityconstraintsVariables X(j), Y (j) Variables M (j), N (j) SS ConditionX(j), Y0 X(j) − Y0−1, −Y0 X(j) ≤ X(k)X0, Y (j) −X0, Y (j) −X0−1 Y (j) ≥ Y (k)X0, Y0 any M (j), N (j) from (2.21) —2.5.2 ADT SwitchingWhen hysteresis switching is used for switching controller design, the stability and performance are guaran-teed no matter how frequently the switching events occur. However, if we know that only a small number ofswitches can happen in average over a finite time interval, then ADT switching may yield better performancethan hysteresis switching. This is because the Lyapunov function is no longer needed to be monotonically182.5. SLPV Controller Design under Uncertain Scheduling Parametersdecreasing during switching under ADT switching logic. The following theorem is obtained based on theresults in [6, Theorem 3] and [31, Theorem 8].Theorem 2.5. Given scalars λ0 > 0 and µ > 1, suppose there exist a family of parameter-dependentmatricesX(j) ∈ Sn and Y (j) ∈ Sn, parameter-dependent matrices Aˆ(j)K , Bˆ(j)K , Cˆ(j)K and Dˆ(j)K , and a positivescalar ε such that convex conditions (2.15) and (2.27) hold for any j ∈ ZN , and one of (2.28) and (2.29)holds for any adjacent Θ(j) and Θ(k). Then, with the switching controller (2.6) obtained by (2.20), and byreplacingA(j)K withA(j)K +(λ0/2)In, the closed-loop system (2.7) is exponentially stable, and its performance||z||2 < γ||w||2 is achieved with γ = maxj∈ZN {γ(j)} for all admissible trajectories θ(·) and δ(·) and forevery switching signal σ with ADTτ > (ln µ)/λ0. (2.26)[Subset conditions] (2.15) andH(θ, θˆ) + diag([λ0X(j) ?λ02 In λ0Y(j)],0,0)?[εX(j) (A(θ)−A(θˆ))TY (j) 0 0]−εIn < 0, ∀(θ, θˆ, ˙ˆθ) ∈ Φ(j) × ver(Ωθˆ), (2.27)whereH(θ, θˆ) is defined in (2.17).[SS conditions] Y (j) = Y (k),[µX(k) −X(j) ?√µ− 1In Y (j)]≥ 0, ∀θˆ ∈ S(j,k), (2.28)X(j) = X(k),[µY (j) − Y (k) ?√µ− 1In X(j)]≥ 0, ∀θˆ ∈ S(j,k). (2.29)Proof. Here we just prove that both (2.28) and (2.29) are sufficient conditions for µP (j) ≥ P (k), or equiv-alently P (j)−1 ≤ µP (k)−1, with the parameterization of P (j) and P (j)−1 given in (2.23). The proofof (2.27) is straightforward from [6, 31]. Schur complement to the inequality condition in (2.28) givesµX(k) −X(j) − (µ− 1)Y (j)−1 ≥ 0, which, together with the equality condition in (2.28) and µ > 1, leadstodiag(Y (j)−1, X(j) − Y (j)−1)≤ µ diag(Y (k)−1, X(k) − Y (k)−1). (2.30)This inequality is equivalent to P (j)−1 ≤ µP (k)−1 through congruence transformation, whenM (j) andN (j)satisfying (2.21) are chosen to be X(j) − Y (j)−1 and −Y (j), respectively. If we choose M (j) = −X(j)and N (j) = Y (j) − X(j)−1, another equivalent nonconvex condition for µP (j) ≥ P (k) can be obtainedanalogously, which is guaranteed by the convex condition (2.29).One can see that the nonconvex SS condition (2.30) cannot be convexified by imposing practical validityconstraints because constant matrices X(j) or Y (j), j ∈ ZN do not need to be the same due to the existenceof µ. Therefore, the equality constraints X(j) = X(k) or Y (j) = Y (k) are made in order to attain convex192.5. SLPV Controller Design under Uncertain Scheduling Parametersconditions (2.28) or (2.29). Note that convex SS conditions for ADT switching controller design are notavailable in [6]. Depending on selection of constant X(j) or Y (j), and (2.28) or (2.29), there are totallyfour conditions, each of which is sufficient to guarantee the stability of the switched closed-loop system.However, it is unclear which condition among the four conditions provides the least conservative controllerbefore actual controller design and performance comparisons.Unlike hysteresis switching, the Lyapunov function does not need to be monotonically decreasing inADT switching case. Instead, it can increase by µ times compared to its value before switching. However,the subset condition (2.27) for ADT switching becomes more restrictive compared to its counterpart (2.16)for hysteresis switching due to addition of the extra block containing λ0. The extra block is introduced tocompensate possible increase of the Lyapunov function. From the perspective of (2.27), small λ0 is desired,corresponding to small µ, while large µ is desired when considering SS conditions (2.28) and (2.29). Hence,there is a trade-off between small λ0 and large µ. For fixed λ0, as τ becomes larger, so does µ. Therefore,ADT switching is most suitable for systems with large ADT.2.5.3 Solving PDLMIsPDLMIs appear in Theorem 2.4 and Theorem 2.5, which will lead to an infinite-dimensional and infinitelyconstrained problem for general parameter dependence of the plant. A standard approach to the selectionof finite-dimensional optimization functions is to imitate the parameter dependence of the plant in the op-timization variables X(j), Y (j), Aˆ(j)K , · · · , Dˆ(j)K [14]. In order to obtain a finite number of LMIs, griddingtechnique generally has to be employed for general parameter dependence of the plant [14]. Under spe-cial circumstances like affine or polynomial parameter dependence of the plant, techniques such as multi-convexity concepts [42], coefficient check [43] and SOS relaxations [44], can be employed to obtain a finitenumber of sufficient LMI conditions.Since a scalar optimization variable ε is introduced in Theorem 2.4 and Theorem 2.5, and coupled withmatrix optimization variables X(j), a line search for ε has to be implemented. Actually, to reduce theconservatism, one can use different ε(j) for different subset Θ(j). In this case, the line search will be time-consuming due to increased computational complexity. Alternatively, noting that ε(j) are only coupled withX(j), one can alternately optimize, via an LMI solver, ε(j) orX(j) while fixing the other. For ADT switchingcontroller design, due to coupling between µ and X(j) in (2.28) or Y (j) in (2.29), a line search or alternativeoptimization is also needed for µ.202.6. A Numerical Example2.6 A Numerical ExampleIn this section, we apply the proposed SLPV control synthesis technique to a numerical example1 with thefollowing state-space matrices (see (2.1)):A(θ) =0 1 + θ21 0 0−104(θ21 + 0.5θ1 + 1.25) θ2 − 10 0 0−1 0 −1 0−8 0 0 −28.723 ,B1 =[0 0 1 8]T, B2 =[0 106 0 0]T,C1 =[−0.1 0 0 6.8220 0 0 0], D11 =[0.10], D12 =[00.1],C2 =[−1 0 0 0], D21 = 1,where θi ∈ Θi = [−2, 2] and |θ˙i| ≤ 2 for i = 1, 2. This example roughly represents a mass-spring-dampersystem which is to be controlled to track a reference position, where θ1 is the natural frequency. A first-orderweighting function and an integrator are added, increasing the order of the generalized plant to four. Weassume that θi is measured with uncertainty δi ∈ ∆i = [−ξ, ξ], and its rate |δ˙i| ≤ 20.For designing SLPV controllers, we partitioned Θi into Θ(1)i = [−2, 0.2] and Θ(2)i = [−0.2, 2] fori = 1, 2. For simplicity, all the parameter-dependent matrices X(j), Y (j), Aˆ(j)K , · · · , Dˆ(j)K are parameterizedto be affine with respect to θˆ. For instance, when X(j) is chosen to be parameter-dependent, then it isparameterized asX(j)(θˆ) = X(j)0 + θˆ1X(j)1 + θˆ2X(j)2 .The software package LMI Lab [45] and gridding technique were used to solve PDLMIs. For gridding ofthe involved polytopes, we used a rather coarse gridding that includes their vertices and centers. The resultsof γ minimization for different uncertain bounds ξ and parameterization of Lyapunov matrices are shown inTable 2.2, where X0 (Y0) means that X(j) (Y (j)) are constant and the same for all subsets (see Table 2.1).The line search for ε was conducted with 13 logarithmically equi-spaced points between 10−5 and 102 fornon-switching LPV controller design, and the obtained ε was used for SLPV controller design as well. Theε obtained by line searches for different cases is given in parentheses in Table 2.2. One can see that theSLPV controller always gives better γ-value than the non-switching LPV controller, and the improvementbrought by switching became more significant when the uncertainties in scheduling parameter measurementincreased.All the above computations were conducted in Matlab 2015b on a PC with Intel i5-3470 CPU and 16GB RAM running Win7 64-bit OS. After ε is fixed, the computation time for designing an SLPV controllerwas about 60 minutes when using a PDLF, and 6 minutes when using constant Lyapunov functions. If wefix the scheduling parameter θ2 to be zero, i.e., we drop the term θ2 in A matrix, then the computational time1Matlab code is available at http://cel.mech.ubc.ca/software-files/212.6. A Numerical ExampleTable 2.2: γ value given by non-switching and hysteresis switching LPV controllersX0, Y0 X(j)(θˆ), Y0 X0, Y(j)(θˆ)ξ NS S NS S NS S0 0.1643 0.1642 0.1641 0.1622 0.1512 0.14640.01 5.376e4 4.235e3 2.207 1.819 2.095 1.716(3.162) (10−5) (10−5)0.1 2.007e6 5.177e3 50.717 13.748 46.406 10.751(3.162) (3.162 · 10−5) (3.162 · 10−5)0.2 2.567e6 7.161e3 202.530 76.449 182.605 26.191(1) (3.162 · 10−5) (3.162 · 10−5)* NS: non-switching, S: switching, aeb:= a× 10bfor designing an SLPV controller decreased significantly to around 20 seconds when using a PDLF and 6seconds when using a constant Lyapunov function. These sharp decreases of computation time are due toreduction of both optimization variables and grid points for solving PDLMIs.We also tried design of ADT switching controllers using X0 and Y (j)(θˆ) as well as SS condition (2.29).The line search for µ was implemented with 11 logarithmically equi-spaced points between 100.01 and 102.The γ-value obtained for different ADT τ and uncertainty bounds ξ are displayed in Table 2.3, where thecorresponding µ value is given in the parentheses. Note that when ξ = 0, the subsets Θ(1)i = [−2, 0] andΘ(2)i = [0, 2] for i = 1, 2 were used. In comparison to the last column of Table 2.2, one can see that, forlarge τ , ADT switching control yielded a better γ-value than hysteresis switching control, while we couldnot find a feasible ADT switching controller for small τ .Table 2.3: γ value given by ADT switching controllers with X0, Y (j)(θˆ) and ε = 3.162 · 10−5ξ τ = 100 τ = 1 τ = 0.010 0.1451 (1.618) 0.1451 (1.023) Infeasible0.2 25.228 (63.241) 25.908 (2.559) InfeasibleTime-domain simulations were performed for |δi| < 0.2. For comparisons, we tested three controllersdesigned with X0 and Y (j)(θˆ), which areK1: Hysteresis switching controller considering δ designed using the proposed approach,K2: ADT switching controller considering δ designed with τ = 1 using the proposed approach, andK3: Hysteresis switching controller ignoring δ designed using the approach in [6].222.6. A Numerical Example0 1 2 3 4 5Time (s)-2-101231323^13^20 1 2 3 4 5Time (s)012345<Figure 2.5: Scheduling parameters0 1 2 3 4 5Time (s)-202 31323^13^20 1 2 3 4 5Time (s)05<Figure 2.6: Switching signal232.6. A Numerical ExampleFigure 2.7: Disturbance input and Channel 1 of z(t)Switching instantStep instantFigure 2.8: Control inputTrajectories of the scheduling parameters are depicted in Fig. 2.5, where the actual scheduling parametersare measured with bias error and random noise. Fig. 2.6 illustrates the switching signal. Fig. 2.7 shows the242.7. Conclusiontrajectories ofw and z1 while control input is given in Fig. 2.8. It can be seen thatK3 could not even stabilizethe plant, due to ignorance of uncertainties in the scheduling parameter measurement. On the other hand,both K1 and K2 achieved the closed-loop stability, and furthermore, yielded smaller z1-value than K3 evenbefore the signal z1 started diverging at around 4 seconds. In Fig. 2.8, we can also see that there were jumpsin control input during switching instants.2.7 ConclusionThis chapter presented an approach to designing SLPV controllers with guaranteed stability and performancewhen the scheduling parameters are measured with uncertainties. As a key step in our approach, admissibleregions of actual and measured scheduling parameters for all local controllers were obtained according tothe switching rules based on measured scheduling parameters. Convexified conditions on switching surfaceswere gained for hysteresis switching as a result of practical validity constraint, and for ADT switching owingto equality constraints on some of Lyapunov variables at the switching surfaces. A numerical exampledemonstrated the advantages of the proposed controller over a non-switching LPV controller consideringmeasurement uncertainties and an switching LPV controller ignoring measurement uncertainties. It was alsoillustrated that ADT switching may give better performance than hysteresis switching when the ADT is large.25Chapter 3Optimal Switching Surface Design forState-Feedback SLPV Control3.1 IntroductionIn conventional SLPV controller design, SSs are generally prespecified [6, 7, 27]. However, different SSsmay lead to significantly different system performance. Even though trial and error can be used to searchsatisfactory SSs, optimization-based methods are more desired for this problem. For SS design for switchingLPV control, a gradient-based optimization algorithm was proposed in [35]. However, the computationalcomplexity of this algorithm is high, because the problem of controller design under fixed SSs has to besolved several times at each step in order to numerically compute the gradient and Hessian of the objectivefunction.In this chapter, we consider optimal SS design for state-feedback SLPV control of LPV plants withpolynomial parameter-dependence. The main contribution of this chapter is a computationally efficientalgorithm for simultaneous design of a state-feedback SLPV controller and optimal SSs. Specifically, wefirst derive synthesis conditions in the form of affine LMIs for state-feedback SLPV controllers using slackvariable approach [15, 46]. Then, by incorporating SSs in the SLPV controller design, the simultaneousdesign problem is formulated as an optimization problem involving bilinear matrix inequalities (BMIs) withrespect to controller variables and SS variables. An algorithm is then proposed to solve this BMI problemby sequentially updating controller variables and SS while fixing the other. The main advantage of ourapproach over previous approaches [35, 47] is that our approach only involves solving optimization problemswith LMIs, which can be performed efficiently by off-the-shelf algorithms. Besides, the computationalcomplexity does not increase significantly with increasing number of SS variables. A numerical example isprovided to show the effectiveness of our approach.3.2 Preliminaries3.2.1 Problem FormulationIn this section, we consider the following LPV generalized plant:G(θ) :x˙ = A(θ)x+B1(θ)w +B2(θ)u,z = C(θ)x+D1(θ)w +D2(θ)u,y = x,(3.1)263.2. Preliminarieswhere θ = [θ1, . . . , θs]T is the vector of scheduling parameters which is measurable in real time, x ∈ Rnis the state vector, u ∈ Rnu is the control input vector, w ∈ Rnw is the vector of exogenous inputs, y isthe measured output vector which is assumed to be x, z ∈ Rnz is the vector of output signals related to theperformance of the control system. The matrices in (3.1) are assumed to be polynomially dependent on θ andto have compatible dimensions. Additionally, the parameter vector θ and its rate of variation θ˙ are assumedto be bounded as follows:θ ∈ Θ := {θ ∈ Rs : θi ≤ θi ≤ θi, i ∈ Zs},θ˙ ∈ Ω := {θ˙ ∈ Rs : ωi ≤ θ˙i ≤ ω¯i, i ∈ Zs}.(3.2)We consider hysteresis switching rule and hyper-rectangular subsets for the parameter set Θ obtained bypartitioning each Θi, i.e. the projection of Θ onto the ith axis. For hysteresis switching, adjacent subsetshave overlapped regions. Assuming that Θi is partitioned into Ni intervals, each subset can be described as{Θ(j)}j∈J , whereJ := {j = (j1, . . . , js), ji ∈ ZNi , i ∈ Zs}is the set of indices for all the subsets. Note that Θ =⋃j∈J Θ(j). The subsets are separated by a familyof SSs. Let S(p,q) denote the SS for controller switching from a controller for Θ(p) to that for Θ(q). Forthe hyper-rectangular subsets, SSs can be characterized by SS variables, which are the pairs of centers andwidths of overlapped regions between adjacent subsets, denoted by{(c(ni)i , w(ni)i ), ni ∈ ZNi−1, i ∈ Zs}. (3.3)An example of rectangular subsets and SSs for two-dimensional parameter sets is given in Fig. 3.1. Inthis case, Θ1 and Θ2 are partitioned into three and two intervals, respectively, which results in six subsets.Switching surfaces between adjacent subsets in this case are line segments. For instance, the SSs when θmoves from Θ(1,1) to Θ(2,1), and from Θ(2,1) to Θ(1,1), are line segments represented by(c(1)1 + w(1)1 )× [θ2, c(1)2 + w(1)2 ]and(c(1)1 − w(1)1 )× [θ2, c(1)2 + w(1)2 ],respectively. Note that SSs will become points for s = 1 and hyperplanes for s >= 2.Here, we would like to design a state-feedback SLPV controller K(θ) consisting of a family of localcontrollers with the formu = K(j)(θ)x, j ∈ J, (3.4)where K(j)(θ) takes charge of a specific parameter subset Θ(j). Controllers are switched according tohysteresis switching rule when the parameter trajectory hits one of the SSs. A switching signal σ is used todenote the active subset and generated as follows [6] for hysteresis switching. Let σ(0) = j if θ(0) ∈ Θ(j).For any t > 0, if σ(t−) = j and θ(t) ∈ Θ(j), keep σ(t) = j. On the other hand, if σ(t−) = j andθ(t) ∈ Θ(p) \ Θ(j), then let σ(t) = p. Under SLPV control, the closed-loop system becomes a switched273.2. PreliminariesFigure 3.1: Rectangular subsets and SSs for a two-dimensional parameter set (N1 = 3, N2 = 2)uFigure 3.2: Block diagram of a state-feedback SLPV controller K(θ) for an LPV plant G(θ)LPV system, depicted in Fig. 3.2, and described asGσcl(θ) :{x˙cl =Aσcl(θ)xcl +Bσcl(θ)w,z =Cσcl(θ)xcl +Dσcl(θ)w,(3.5)where xcl = x ∈ Rn and [Aσcl(θ) Bσcl(θ)Cσcl(θ) Dσcl(θ)]=[A(θ) +B2(θ)Kσ(θ) B1(θ)C(θ) +D2(θ)Kσ(θ)D1(θ)]. (3.6)As is mentioned above, SSs design has the potential to further improve the performance of an SLPVcontroller. One general objective of LPV controller design is to minimize the L2 gain of the closed-loopsystem for tracking error minimization, disturbance rejection, and control input reduction. Thus the problemcan be formulated as follows:Problem 3.1. Given an LPV plant (3.1) with constraints (3.2) and the number of subsets Ni for each axisθi, design SSs (3.3) and an SLPV controller (3.4) such that, for all trajectories of (θ, θ˙) satisfying (3.2), theclosed-loop system (3.5) is exponentially stable and the L2 gain of the closed-loop system is minimized, i.e.min γ subject to ||z||2 < γ||w||2. (3.7)283.2. Preliminaries3.2.2 Basic LemmaThe following lemma gives the condition for synthesizing a state-feedback SLPV controller.Lemma 3.1. Suppose that there exist a family of positive number γ(j) and parameter-dependent matricesX(j)(θ) and K(j)(θ) of compatible dimensions such that the following inequality conditions hold: for allj ∈ J ,X(j)(θ) > 0,∀θ ∈ ver(Θ(j)), (3.8)〈A(θ)X(j)(θ) +B2(θ)K(j)(θ)〉− X˙(j)(θ) ? ?BT1 (θ) −γ(j)Inw ?C(θ)X(j)(θ) +D2(θ)K(j)(θ) D1(θ) −γ(j)Inz < 0, ∀(θ, θ˙) ∈ ver(Θ(j))× ver(Ω),(3.9)and for all adjacent subsets Θ(p) and Θ(q),X(p)(θ) ≤ X(q)(θ), ∀θ ∈ S(p,q). (3.10)Then, the closed-loop LPV system (3.5) is exponentially stabilized by the SLPV controller (3.4) withK(j)(θ) = K(j)(θ)X(j)(θ)−1 (3.11)over entire parameter set and its performance ||z||2 < γ||w||2 is achieved with γ = maxj∈J{γ(j)}.Proof. This lemma results from can be easily derived by combining the synthesis condition for a state-feedback SLPV controller [22] and the conditions for a montonically decreasing Lyapunov function to en-force stability under arbitrary switching [6]. Specifically, for all (θ, θ˙) ∈ Θ(j) × Ω, the exponential stabilityand performance of the closed-loop system, i.e. ||z||2 < γ(j)||w||2, are guaranteed by (3.8) and (3.9) ac-cording to the analysis conditions in [22] and the state-feedback law (3.4). Moreover, for each subset Θ(j),a candidate of Lyapunov function for the closed-loop system isV (j)(xcl) = xTclX(j)(θ)−1xcl, (3.12)Now define a piecewise-continuous function asV σ(xcl) = V(j)(xcl), σ = j. (3.13)Equation (3.10) guarantees that V σ(xcl) is monotonically decreasing even under switching between anyadjacent subsets. Following the proof for Theorem 1 in [6], exponential stability and performance of theclosed-loop system for all (θ, θ˙) ∈ Θ × Ω can be guaranteed. Moreover, V σ(xcl) is a switched Lyapunovfunction for all admissible trajectories of θ.293.3. A Simultaneous Design Approach3.3 A Simultaneous Design Approach3.3.1 State-Feedback SLPV Synthesis using Slack VariablesIn this subsection, for specified SSs, we propose a method for design of state-feedback SLPV controllers byextension of the result in [15]. Slack variable approach is used to solve polynomially parameter-dependentLMIs, as is in [15]. Based on the method, an algorithm for simultaneous design of the controller and SSswill be proposed in the next subsection.Some notations are introduced as in [15]. Define the vector of non-negative power series for each pa-rameter θi up to mi asθ˘[mi]i := [θ0i θ1i . . . θmii ]T ∈ Rαi , (3.14)where αi := mi + 1. Defineθ˘ := θ˘[m1]1 ⊗ . . .⊗ θ˘[ms]s ∈ Rα, (3.15)where α := α(1, k) :=∏si=1 αi. Additionally, let α(k + 1, k) = α(1, 0) = 1 by definition. Using abovedefinitions, parameter-dependent state-space matrix A(θ) in (3.1) can be represented asA(θ) = (θ˘ ⊗ In)T Aˆ, Aˆ ∈ Rnα×n, (3.16)whileB1(θ), B2(θ), C(θ), D1(θ) andD2(θ) can be analogously described by Bˆ1, Bˆ2, Cˆ, Dˆ1 and Dˆ2 respec-tively. Please refer [15] for details. The following matrices are defined as well: where pii := αmi/αi.Hereafter, e denotes [1 01,α−1]T .By combining Lemma 3.1 and the slack variable approach for solving parameter-dependent LMIs pre-sented in [15], we get the following lemma for design of state-feedback SLPV controller using parametricallyaffine LMIs. The proof is straightforward and omitted for brevity.Lemma 3.2. Suppose that there exist a family of positive number γ(j) and affinely parameter-dependentmatrices X(j)(θ) and K(j)(θ) defined asX(j)(θ) = X(j)0 +k∑i=1θiX(j)i , K(j)(θ) = K(j)0 (θ) +k∑i=1θiK(j)i , (3.17)where X(j)i ∈ Sn, K(j)i ∈ Rnu×n, and M (j)i ∈ R(npii+nzpi)×(nα+nw+nzα) such that the following inequalityconditions hold: for all j ∈ J ,X(j)(θ) > 0, ∀θ ∈ ver(Θ(j)), (3.18) F1 ? ?Bˆ1T −γ(j)Inw ?F2 Dˆ1 −γ(j)(eeT ⊗ Inz)+〈 s∑i=1Ψi(θi)⊗ In 00 00 Ψi(θi)⊗ InzM (j)i〉> 0,∀(θ, θ˙) ∈ ver(Θ(j))× ver(Ω), (3.19)303.3. A Simultaneous Design ApproachwhereF1 =〈Aˆ(eT ⊗X(j)(θ))+ Bˆ2(eT ⊗K(j)(θ))〉−s∑i=1θ˙i((eeT )⊗X(j)i),F2 = Cˆ(eT ⊗X(j)(θ))+ Dˆ2(eT ⊗K(j)(θ)),and for all adjacent subsets Θ(p) and Θ(q),X(p)(θ) ≤ X(q)(θ), ∀θ ∈ S(p,q). (3.20)Then, the closed-loop LPV system (3.5) is exponentially stabilized by the SLPV controller (3.4) withK(j)(θ) = K(j)(θ)X(j)(θ)−1 (3.21)over entire parameter set and its performance ||z||2 < γ||w||2 is achieved withγ = maxj∈Jγ(j). (3.22)3.3.2 A Sequential Design AlgorithmLemma 3.2 presented in Section 3.3.1 can be directly used for design of a state-feedback SLPV controllergiven SSs. For simultaneous design of the controller and SSs, the LMIs involved in the theorem need tobe reformed so that both the controller variables and SS variables are included as optimization variables.Hereafter, letc(0)i − w(0)i = θi, c(Ni)i + w(Ni)i = θifor i ∈ Zs by definition. Note that subset vertices ver(Θ(j)) can be described using the SS variables asver(Θ(j)) = {c(j1−1)1 − w(j1−1)1 , c(j1)1 + w(j1)1 } × . . .× {c(js−1)s − w(js−1)s , c(js)s + w(js)s }, (3.23)By generalizing the example for a two-dimensional parameter set shown in Fig. 3.1, SSs for k-dimensionalparameter subsets Θ(p) and Θ(q) can be represented using SS variables asS(p,q)= [c(j1−1)1 − w(j1−1)1 , c(j1)1 + w(j1)1 ]× . . .× (c(ji)i + w(ji)i )× . . .×[c(js−1)s − w(js−1)s , c(js)s + w(js)s ](3.24)andS(q,p) = [c(j1−1)1 − w(j1−1)1 , c(j1)1 + w(j1)1 ]× . . .× (c(ji)i − w(ji)i )× . . .×[c(js−1)s − w(js−1)s , c(js)s + w(js)s ](3.25)313.3. A Simultaneous Design Approachrespectively, forp = (j1, . . . , ji−1, ji, ji+1 . . . , js),q = (j1, . . . , ji−1, ji + 1, ji+1 . . . , js),for jm ∈ ZNm , m ∈ Zs \ {i}.To consider the bound of parameter set and ensure no overlapped regions between any three or moresubsets (see Fig. 3.1), we need to add the following constraints for each i ∈ Zs:c(ni)i + w(ni)i ≤ c(ni+1)i − w(ni+1)i , ni ∈ ZNi−1,θi ≤ c(1)i − w(1)i ,θi ≥ c(Ni−1)i + w(Ni−1)i .(3.26)Note that (3.18) to (3.20) are respectively affine with respect to θ and θ˙. Thus, if they are satisfied at thevertices of related parameter sets, they also hold for all the elements in the sets. However, when (c(ni)i , w(ni)i )are added as optimization variables, (3.18) to (3.20) are no longer jointly convex with respect to controllervariables X(j)i , K(j)i , slack variables M (j)i and SS variables (c(ni)i , w(ni)i ). Actually they become BMIs.However, they reduce to LMIs when either controller variables (including slack variables) or SS variablesare fixed. In fact, when optimizing SS variables, we do not need to fix all the controller variables in order tomaintain convexity of optimization. Instead, we can still get LMIs when SSs variables as well as X(j)0 andK(j)0 are treated as optimization variables. Thus, we can iteratively optimize controller variables while fixingswitch surface variables and optimize SS variables while fixing part of controller variables. Based on thisobservation, Algorithm 3.1 is proposed for iteratively reducing the γ value. Note that the initially selectedSS variables should yield a feasible SLPV controller in Step 1 to make the iterations continue. If there is afeasible LPV controller for the entire parameter variation set, then we can always obtain an SLPV controllerfrom solving the optimization problem in Step 1. In case a feasible LPV controller cannot be found, thendifferent SS variables should be tried until some are found to yield a feasible SLPV controller in Step 1.Due to the non-convexity of the problem, our algorithm cannot guarantee to find the global or even localoptimum, and the result depends on initially selected SS variables. If the result is not satisfactory, we shouldchange the initial SS variables and try the algorithm again. We cannot provide a complete guideline forthe selection of the initial SSs up to now. However, from the numerical example to be presented in nextsection, choosing a large initial value for w(ni)i (corresponding to initial large overlapped areas betweenadjacent subsets) gives good results in terms of fast convergence. This is probably because with large initialoverlapping, the SS conditions (3.20) and the constraints on SSVs (3.26) can still be satisfied even withdrastic change of the SSVs from one iteration to another at the beginning of the iterations. On the otherhand, by solving a number of examples, we have found that relaxing and optimizing X(j)0 and K(j)0 togetherwith SS variables greatly increase the possibility for the algorithm to converge to an optimum.We claim that our method is more efficient than the method proposed in [35]. Our method involves onlysolving LMIs, which can be efficiently solved using off-the-shelf algorithms and software [48, 49]. Thecomputational complexity does not increase too much with increasing number of scheduling parameters and323.4. A Numerical ExampleAlgorithm 3.1 A Sequential Design Algorithm for Optimal SSsInitializationInitialize the SS variables {c(ni)i , w(ni)i , ni ∈ ZNi−1, i ∈ Zs} under the constraints (3.26).Iteration Repeat the following two steps until the termination criteria are met.Step 1: Update controller variablesFix the SS variables and solve an optimization problem over controller variables, i.e.minX(j)i ,K(j)i ,M(j)iγ (3.27)subject to (3.18) to (3.20) and (3.22). If this problem is infeasible, then go back to the initializa-tion step to re-initialize the SS variables.Step 2: Update SS variablesFix part of controller variables obtained in Step 1, i.e. X(j)i ,K(j)i ,M (j)i (i 6= 0), solve an opti-mization problem over SS variables:minc(ni)i ,w(ni)i ,X(j)0 ,K(j)0γ (3.28)subject to (3.18) to (3.20), (3.22) and (3.26) with (3.23)–(3.25).TerminationIf the number of iteration surpasses a predefined maximum iteration number, or if the decrease in γvalue in Step 2 is less than a prescribed small value , then terminate. Otherwise, iterate Step 1 andStep 2.subsets because all the SS variables or controller variables can be updated simultaneously through solvingLMIs. On the other hand, the controller is just redesigned once in each iteration step. However, the methodin [35] needs to numerically calculate the gradient and Hessian of the objective function to each scalaroptimization variable through perturbing the variable, which involves much more computation. Moreover,our method does not need gridding of the parameter sets [14], which further saves time.3.4 A Numerical ExampleConsider an LPV system with the following state matrices borrowed from [15]:A(θ) =[25.9− 60θ 120− 40θ 34− 64θ], B1 =[−0.03−0.47],B2 =[32], C =[1 10 0], D1 =[00], D2 =[01].333.4. A Numerical Examplewhere θ ∈ Θ = [0 1] and θ˙ ∈ Ω = [−0.1 0.1]. Note that Ω in our example is slightly different from thatin [15].All the computations involved were conducted in Matlab 2013b on a laptop with Intel Core i5-3317UCPU and 8 GB RAM running Win7 64-bit OS. We first tried a non-switching LPV controller, i.e. N = 1,and the optimal γ value was obtained as 0.4742. Then we divided Ω into two subsets. When we heuristicallydivided Ω into Ω(1) = [0 0.6] and Ω(2) = [0.4 1], i.e. (c1, w1) = (0.5, 0.1), we got an optimal γ value of0.4742, which is the same up to four decimal points as that given by the LPV controller. To obtain the globaloptimal SSs described by (c1, w1) and the corresponding γ value, we first implemented a brute-force searchover the set of{(c1, w1) : 0 ≤ c1 ≤ 1, 0 ≤ w1 ≤ 0.5},with the step of 0.01 and 0.005 for c1 and w1 respectively. The search process took 3749 seconds and the γvalues for different value of c1 and w1 are shown in Fig. 3.3. The optimal γ value was obtained as 0.3716 at(c1, w1) = (0.3, 0.035). Notice that the optimal point found is “optimal” up to the step size used.Figure 3.3: Switching surfaces and γ value obtained using brute-force searchNext, we tried the proposed algorithm with different initial values of the SS variables. To implementthe algorithm, we used CVX, a software package for convex optimization problems [49]. The constant forterminating the iterations and the maximum number of iterations were set to be 10−8 and 200, respectively.It turns out that the number of iterations for all the cases reaches the maximum number of iterations, exceptthe second one, for which the algorithm terminated after 180 iterations. The results are shown in Table 3.1.Even though we cannot guarantee the convergence of the algorithm to the global optimum or even localoptimum, the results are close to the optimal value achieved by the brute-force search. If we just considerthree decimal points for γ value, then all the initial values in Table 3.1 led to the “globally optimal” γ valuewith much less computational time compared to brute-force search. But the global optimum covergencecould also be because that there is no more than one local optima for this example, which can be seen fromthe brute-force search results in Fig. 3.3.343.4. A Numerical ExampleWe also tried fixing all the controller variables, i.e. including X(j)0 and K(j)0 , when optimizing SSvariables, and found that not all the cases converge to the “optimal” γ value. Thus optimizing X(j)0 andK(j)0 together with the SS variables enhances the possibility of the algorithm to converge to the “optimal”value. The γ value with respect to iterations under different initial values of the SS variables is shownTable 3.1: Optimization results for 2-subset caseInitial Condition Optimal Result Computation(c1, w1) γ (c1, w1) γ Time (s)(0.5,0.1) 0.4742 (0.291,0.023) 0.3720 496(0.5,0.4) 0.4742 (0.288,0.022) 0.3717 441(0.5,0.01) 0.4742 (0.294,0.022) 0.3724 522(0.6,0.1) 0.4742 (0.293,0.024) 0.3720 494(0.4,0.1) 0.4346 (0.289,0.023) 0.3717 497Brute-force search (0.3,0.035) 0.3716 5443in Fig. 3.4, which demonstrates that the γ value gets very close to the optimal value after 75 iterationsfor all the cases tested. Therefore, computational time can be further reduced if we relax the terminationtolerance . In addition, we can see that setting the initial value of w1 to a larger value (e.g. 0.4 and 0.1),corresponding to a larger initial overlapped area between adjacent subsets, led to faster convergence. Asmentioned in Section 3.3.2, this is probably because with large initial overlapping, the SS conditions (3.20)and the constraints on SSVs (3.26) can still be satisfied even with drastic change of the SSVs from oneiteration to another at the beginning of the iterations.0 50 100 150 2000.360.380.40.420.440.460.48Iteration numberγ value Initial (c1,w1)=(0.5,0.1)Initial (c1,w1)=(0.5,0.4)Initial (c1,w1)=(0.5,0.01)Initial (c1,w1)=(0.6,0.1)Initial (c1,w1)=(0.4,0.1)Figure 3.4: γ value VS iterationsFinally, we divided Θ into four subsets. When we heuristically chose the SS variables to bec1 = (c(1)1 , c(2)1 , c(3)1 ) = (0.25, 0.5, 0.75),w1 = (w(1)1 , w(2)1 , w(3)1 ) = (0.1, 0.1, 0.1),the optimal γ value obtained is 0.4148. Now there are six SS variables, and brute-force-search will takenotoriously long time. Thus we tried our proposed algorithm with different initial values of the SS variables353.5. Conclusionand the maximum number of iterations was still set to be 200. The result is shown Table 3.2. It can beseen that all the initial values lead to the “same” optimal γ value up to four decimal points. In addition, thecomputational time did not increase too much even though the number of SS variables increased from twoto six, compared to the two-subset case.Table 3.2: Optimization results for 4-subset caseInitial Conditions Optimal (Final) Results Comput.c1, w1 γ c1, w1 γ Time (s)(0.25,0.5,0.75),(0.1,0.1,0.1)0.4148(0.014,0.140,0.342),(0.0019,0.0034,0.0290)0.3616 767(0.3,0.5,0.7),(0.05,0.05,0.05)0.3759(0.018,0.153,0.347),(0.0014,0.0040,0.0330)0.3616 774(0.2,0.45,0.8),(0.1,0.1,0.1)0.4148(0.029,0.180,0.356),(0.0014,0.0037,0.0342)0.3616 722Heuristic selection(0.25,0.5,0.75),(0.1,0.1,0.1)0.4148 —3.5 ConclusionIn this chapter, we proposed an approach for simultaneous design of state-feedback SLPV controllers andSSs for LPV plants with polynomial parameter dependence. Using slack variable approach and Lyapunovvariables with affine parameter dependence, we formulated the simultaneous design problem into an opti-mization problem involving BMIs. An algorithm was proposed for solving the BMIs by iteratively fixingeither SS variables or part of controller variables while optimizing the other. An numerical example demon-strated the effectiveness of our approach.36Chapter 4SLPV Control with Improved LocalPerformance and Optimized SwitchingSurfaces4.1 IntroductionThe performance of an LPV control system is usually characterized by the upper bound of the L2-gain ofthe closed-loop system. To the best of the authors’ knowledge, in all the previous work on SLPV control ex-cept [50], the SLPV controllers were designed by solving an optimization problem minimizing the maximumof the local L2-gain bounds for all the subsets. We call this design approach as the traditional approach inthis thesis. It has been observed in our experiences that the traditional approach tends to yield almost iden-tical L2-gain bounds in all the subsets (as demonstrated by the examples in Section 4.2.3 and Section 4.5),which are often only marginally smaller than the L2-gain bound yielded by a non-switching LPV controller.This means that the performance improvement brought by the traditional SLPV control over the LPV controlin all the subsets for these plants is marginal, despite the much higher computational cost in designing theSLPV controllers.On the other hand, in most of previous work on SLPV, the SSs are prespecified in SLPV controllerdesign. However, SSs may play an important role in optimizing the performance of an SLPV controller.It was shown in [51] that different SSs may cause significant difference in the performance of an SLPVcontroller. Although trial and error and brute-force search can be used to search for satisfying SSs, it ispreferred to adopt optimization-based methods that are capable of finding the optimal SSs in an effectiveway. For SS optimization, a descent method was presented in [35]. However, the computational cost of thismethod is high due to numerical evaluation of gradient and Hessian of the objective function. In addition, themethod is a local search method that is sensitive to the attraction of local optima. In [51], the problem of SSdesign for state-feedback SLPV control was formulated as a bilinear matrix inequality (BMI) problem andsolved by a sequential optimization algorithm, in which the SS variables and controller variables are fixedand updated alternately. However, the method is just for state-feedback case, and it cannot guarantee to finda global or even local optimum.To address the problem of unnoticeable performance improvement over the LPV control with the tradi-tional SLPV control in all the subsets, this chapter presents a novel method for designing SLPV controllersbased on a new cost function. The new cost function considers the local performance in all subsets, as op-posed to only the worst-case performance used in the traditional SLPV design approach. An optimization374.2. Review of SLPV controlproblem is formulated to optimize the new cost function while the worst-case performance is bounded witha tuning parameter. By adjusting the tuning parameter, one can possibly improve the local performance insome subsets significantly at the price of slight sacrifice of the worst-case performance. SS optimizationbased on the particle swarm optimization (PSO) is then presented to further improve the performance of anSLPV controller. The advantages of the proposed SS optimization algorithm include its easiness to imple-ment requiring no gradient information over the work in [35], dealing with output-feedback case over thework in [51], and great potential to find the global minimum over both. Combination of the new design ap-proach and SS optimization enables design of SLPV controllers that yield improved performance in a largeproportion of the whole scheduling parameter set.This chapter is organized as follows. Section 4.2 gives a brief review of SLPV control and presents amotivating numerical example. Section 4.3 presents the main approach for designing SLPV controllers withimproved local performance including the novel cost function, while PSO-based SS optimization is presentedin Section 4.4. The numerical example introduced in Section 4.2.3 is then revisited in both Section 4.3 andSection 4.4. The proposed methods are applied to an air-fuel control problem in an automotive engine inSection 4.5. 24.2 Review of SLPV control4.2.1 Plant DescriptionWe consider a generalized LPV plant in the following form:x˙ = A(θ)x+B1(θ)w +B2(θ)u,z = C1(θ)x+D11(θ)w +D12(θ)u,y = C2(θ)x+D21(θ)w,(4.1)where θ := [θ1, · · · , θs]T is the vector of scheduling parameters that are measurable online, x(t) ∈ Rn isthe state vector, u(t) ∈ Rnu is the control input vector, w(t) ∈ Rnw is the vector of exogenous inputs,y(t) ∈ Rny is the vector of measured outputs, and z(t) ∈ Rnz is the vector of outputs to be minimized.The matrices in (4.1) have compatible dimensions. In addition, the vector θ and its rate of variation θ˙ aresupposed to be bounded by hyper-rectangular sets asθ ∈ Θ, Θ := {θ ∈ Rs : θi ≤ θi ≤ θi, i ∈ Zs},θ˙ ∈ Ω, Ω := {θ˙ ∈ Rs : νi ≤ θ˙i ≤ νi, i ∈ Zs}.(4.2)4.2.2 SLPV Control and Switching SurfacesThe hysteresis switching rule and hyper-rectangular subsets for SLPV control are adopted in this chapter.The hyper-rectangular subsets for Θ are obtained by partitioning Θi, where Θi is projection of Θ onto thei-th coordinate. For hysteresis switching, to make sure that adjacent subsets have overlapped regions with2The data and Matlab codes used in this paper can be downloaded at http://cel.mech.ubc.ca/software-files/.384.2. Review of SLPV controlnon-empty interior, Θi is divided intoNi intervals with overlapped segments between any adjacent intervals.In this way, the set Θ will be partitioned into subsets {Θ(j)}j∈ZN where N = Πsi=1Ni and Θ = ∪j∈ZNΘ(j).The subsets are separated by a family of SSs (SSs). Let S(j,k) denote the SS specifying the one-directionalmove of θ from Θ(j) to Θ(k). See Figure 4.1 for an illustration of subsets and SSs.Figure 4.1: Subsets and SSs (s = 2, N1 = N2 = 2)For the LPV plant (4.1), we are interested in designing an SLPV controller consisting of a family offull-order output-feedback LPV controllers in the form of[x˙Ku]=[A(j)K (θ, θ˙) B(j)K (θ, θ˙)C(j)K (θ, θ˙) D(j)K (θ, θ˙)]︸ ︷︷ ︸:=K(j)(θ,θ˙)[xKy], j ∈ ZN , (4.3)where xK(t) ∈ Rn is the controller state vector, and a local LPV controller K(j)(θ, θ˙) ∈ R(n+nu)×(n+ny)with state-space matrices A(j)k , B(j)k , C(j)k and D(j)k takes charge of the parameter subset Θ(j). A switchingsignal, σ designating the active local controller, is generated based on θ as follows. Let σ(0) = j if θ(0) ∈Θ(j) (pick an arbitrary j if there exist multiple js). For any t > 0, if σ(t−) = j (where σ(t−) is the leftlimit of σ at time t) and θ(t) ∈ Θ(j), keep σ(t) = j. On the other hand, if σ(t−) = j, θ(t−) ∈ S(j,k), andθ(t) ∈ Θ(k) \Θ(j), then let σ(t) = k.The LPV plant (4.1) and the SLPV controller (4.3) constitute a closed-loop SLPV system. The perfor-mance of a closed-loop SLPV system is usually evaluated with the L2-gain bounds. We defines the localL2-gain bound as follows.Definition 4.1. The L2-gain of the closed-loop system with (4.1) and (4.3) is locally upper bounded by γ(j)if for σ(t) ≡ j and zero initial state conditions, the following inequality holds:T∫0z(t)T z(t)dt ≤ (γ(j))2T∫0w(t)Tw(t)dt, ∀T ≥ 0. (4.4)On the other hand, the worst-case L2-gain bound in this chapter refers to the maximum of local L2-gain bounds in all the subsets. In the traditional approach [6], given an LPV plant (4.1) with θ satisfying(4.2) and the parameter subsets {Θ(j)}j∈ZN , an SLPV controller (4.3) is designed by solving the followingoptimization problem.394.2. Review of SLPV controlProblem 4.1.min maxj∈ZNγ(j) (4.5)subject to1. constraints to guarantee exponential stability and local upper-bound γ(j) for any j ∈ ZN , and2. constraints to guarantee exponential stability under switching between any two adjacent subsets.The traditional SLPV control focuses on optimizing the worst-case L2-gain bound over all subsets in(4.5), and often leads to similar local performance, which will be demonstrated using a numerical examplenext. This example will highlight the potential drawback of the traditional SLPV controller design, and mo-tivate the work in the present chapter. Hereafter, (local or worst-case) L2-gain bound is used interchangeablywith (local or worst-case) γ-value and (local or worst-case) performance level.4.2.3 A Motivating ExampleLet us consider a numerical example borrowed from [52] with the following state-space matrices:A(θ) B1 B2(θ)C1 D11 D12C2 D21 =−4 3 5 1 00 7 −5 −2 160.1 −2 −3 1 −101 1 0 0 10 1 0 2+ θ1 0 1 0 12 0 −5 0 −52 5 1.5 0 3.50 0 0 0 00 0 0 0 , (4.6)where Θ := [−1, 1] and Ω := [−1, 1].We first designed an LPV controller for the plant (4.6) by using the method of [42]. The resultingL2-gainbound was 17.96, which is depicted with black dash-dotted line in Fig. 4.2, and denoted as γ∗.Next, we divided Θ into [−1, 0.1] and [−0.1, 1], and designed an SLPV controller. When we utilized thetraditional design approach by solving Problem 4.1 (the details of which will be explained later), the resultinglocal L2-gain bounds in the two subsets were almost the same and approximately 17.06, as indicated inFigure 4.2(a) with magenta dot lines. This traditional SLPV controller (denoted as SPLPV 0(a) in 4.2(a))gives an improvement of approximately 5.0% in terms of the worst-case L2-gain bound compared to theLPV controller.Despite the worst-case L2-gain bound improvement achieved by the SLPV controller above, it turns outthat there exist SLPV controllers which could yield smaller local L2-gain bound at the slight sacrifice of theworst-case L2-gain bound. For instance, by applying the method proposed in this chapter, we could get anSLPV controller (denoted as SLPV 1(a)) that yielded only 2.0% worse performance in Subset 2 but 7.2%better performance in Subset 1, compared to SLPV 0(a), as illustrated by the blue dashed line in Figure 4.2(a).Since SSs may have an effect on the performance of an SLPV controller, we also tried different SSs todivide Θ into [−1, 0.7] and [0.5, 1]. The associated traditional and proposed SLPV controllers are denotedas SLPV 0(b) and SLPV 1(b), and their performance is displayed in Figure 4.2(b). The local L2-gain boundsyielded by SLPV 0(b) were again almost identical to each other, and were around 16.35. Compared to404.3. SLPV Control with Improved Local Performance-1 -0.5 0 0.5 115161718-value LPVSLPV 0(a)SLPV 1(a)(a) Results for Θ(1) = [−1, 0.1], Θ(2) = [−0.1, 1]-1 -0.5 0 0.5 115161718-valueLPVSLPV 0(b)SLPV 1(b)(b) Results for Θ(1) = [−1, 0.7], Θ(2) = [0.5, 1]Figure 4.2: Performance of different SLPV controllers. SLPV 0(x): traditional SLPV controller, SLPV 1(x):proposed SLPV controller with improved local performanceSLPV 0(b), the performance given by SLPV 1(b) in Subset 2 (a smaller subset) was 2% worse while theperformance in Subset 1 (a much bigger subset) was improved by 4.3%.Both Figure 4.2(a) and Figure 4.2(b) indicate that it is possible to achieve significantly smaller local L2-gain bounds without much sacrifice of the worst-case L2-gain bound. Small local L2-gain bounds in somesubsets corresponds to good local performance in these subsets, and therefore, are desired. The improvementof local L2-gain bounds is even more valuable if the improvement is realized in subsets with large arearatios such as the case shown in Figure 4.2(b). The above analysis motivates our work on design of SLPVcontrollers with improved local performance.On the other hand, by comparing the performance of SLPV 0(a) and SLPV 0(b), and the performance ofSLPV 1(a) and SLPV 1(b), one could confirm that SSs do have an effect on the performance of the SLPVcontrollers. This motivates our work on design of SLPV controllers with optimized SSs.4.3 SLPV Control with Improved Local PerformanceIn this section, we first propose a new criterion to quantitatively evaluate the performance of an SLPVcontroller. Then an approach for designing an SLPV controller is presented by solving an optimizationproblem to minimize this cost function. The numerical example introduced in Section 4.2.3 is revisitedafterwards.4.3.1 A New Criterion to Evaluate the Performance of an SLPV ControllerBefore formally presenting the cost function, let us first try to intuitively compare the performance of twoSLPV controllers K1 and K2 in several cases. The local performance levels in two subsets for four differentcases are shown in Figure 4.3. Obviously, in Case 1, K1 is better because the SSs are the same and the localperformance levels given byK1 is smaller than that given byK2 in any subset. In Case 2, although the worst-case performance level yielded by the two controllers are the same, which indicates the same performancein the traditional SLPV design approach, it is reasonable to regard K1 better as the local performance levelgiven by K1 in Subset 1 is smaller. In Case 3, the local performance level yielded by the two controllers in414.3. SLPV Control with Improved Local Performanceany subset is the same. However, since the area ratio of the subset corresponding to smaller local performancelevel, associated with K1, is larger than that associated with K2, it makes more sense to conclude that K1 isbetter. The last case may be difficult to evaluate by pure intuitive reasoning. For K1, although the worst-caseperformance level obtained in Subset 2 is higher, the area ratio of Subset 2 is small and the local performancelevel in Subset 1 is much smaller than the local performance level in any subset of K2. Although K2 isregarded better in the traditional SLPV design approach, there may exist some situations for which K1 isdesired over K2.Case 1: Same SSs and different local performance Case 2: Same SSs and same worst-case performance Case 3: Different SSs and same local performance Case 4: Different SSs and different local performance Figure 4.3: Comparison of the performance of two SLPV controllers (s = 1, N = 2)The analysis above indicates that, to evaluate the performance of an SLPV controller, it may be morereasonable to consider the performance levels in all the subsets instead of only the worst-case performancelevel. A weighted summation of the local L2-gain bounds is a natural choice of the cost function to realizesuch purpose. The weight for the local L2-gain bound in a local subset should reflect the significance of thissubset relative to other subsets. Although there can be different ways to define the significance of a subset,in this chapter, we associate the significance of a subset with the active-time percentage of this subset amongall subsets. If a priori information of the active-time percentage of all the subsets is available, we can utilizethis information to select the weights. On the other hand, without such information, we make the followingreasonable assumption.(A0) The active-time percentage of a subset is proportional to its area ratio among all subsets.424.3. SLPV Control with Improved Local PerformanceUnder assumption (A0), we propose to evaluate the performance of an SLPV controller byJ(ΓK) :=N∑j=1w(j)γ(j) (4.7)where ΓK :={(γ(j),Θ(j)) : j ∈ ZN}, w(j) is the weight for the subset j and defined to be the area ratio ofthe subset j, i.e.w(j) := s(j)r /N∑k=1s(k)r (4.8)with s(j)r being the area of the subset j. Obviously,∑Nj=1w(j) = 1, even though there is an overlappedarea between any two adjacent subsets. Furthermore, the overlapped area between two adjacent subsetscontributes to the weights for both of them. Therefore, if there are more than two subsets and the localperformance level in both of the two adjacent subsets are relatively good compared to the performance levelin other subsets, then a large overlapped area between these two subsets is more desired than a small oneaccording to (4.7). Thus when using the (4.7) in SS optimization, large overlapped areas may be yieldedbetween adjacent subsets taht correspond to good local performance. This phenomenon will be furtherexplained in Section 4.5.2.We recently noticed that the performance criterion (4.7) was already mentioned in [7]; however, the wayto select the weights was not discussed in [7]. Moreover, the use of (4.7) for SLPV controller design wasnot tested in any previous work. Based on the the definition (4.7), we say that an SLPV controller K1 hasimproved local performance over K2, if J(ΓK1) < J(ΓK2).4.3.2 Conditions for SLPV SynthesisHereafter, the dependence of plant matrices and other variables on θ and θ˙ is dropped for brevity.By extending the LPV existence conditions in [42, Theorem 5.3], we can easily obtain the existenceconditions of an SLPV controller. For this, the following two assumptions are made for the plant matrices:(A1) The matrices in (4.1) are affine with respect to θ; for example,A(θ) = A0 +s∑i=1θiAi (4.9)(A2) The matrices [BT2 (θ) DT12(θ)], [C2(θ) D21(θ)] have full row-rank over Θ.Here, Assumption (A1) is needed for applying the multi-convexity method in [42] to solve polynomiallyparameter-dependent LMIs involved in SLPV control analysis and synthesis. Assumption (A2) is neededfor designing LPV and SLPV controllers based on projected solvability conditions [14, Theorem 2.2]. IfAssumption (A2) is not satisfied for some LPV plants, one can use basic characterization [14, Theorem 2.1]instead and multi-convexity method in [42] to obtain corresponding conditions.434.3. SLPV Control with Improved Local PerformanceLemma 4.1. Under the assumptions (A1), (A2) and (4.2) and given the SSs, there exists an SLPV controllerwith the hysteresis switching rule ensuring exponential stability and an L2-gain bound γ := maxj∈ZN γ(j)for the closed-loop system for any θ(·) ∈ Θ and θ˙(·) ∈ Ω, whenever there exist parameter-dependentmatrices X(j)(θ) and Y (j)(θ) in the form ofX(j)(θ) := X(j)0 +s∑i=1θiX(j)i , Y(j)(θ) := Y(j)0 +s∑i=1θiY(j)i , (4.10)scalars σ(j)1 and σ(j)2 and non-negative scalars λ(j)1 , . . . , λ(j)s , µ(j)1 , . . . , µ(j)s such that (4.11) to (4.15) andone of (4.16) and (4.17) hold.[Subset conditions]X˙(j) + 〈X(j)A〉 ? ?BT1 X(j) −γ(j)I ?C1 D11 −γ(j)I− σ(j)1CT2DT210[C2D21 0] < −( s∑i=1θ2i λ(j)i )I,∀(θ, θ˙) ∈ verΘ(j) × verΩ, (4.11)−Y˙(j) + 〈AY (j)〉 ? ?C1Y(j) −γ(j)I ?BT1 DT11 −γ(j)I− σ(j)2 B2D120[BT2 DT12 0] < −( s∑i=1θ2i µ(j)i )I,∀(θ, θ˙) ∈ verΘ(j) × verΩ, (4.12)[X(j) II Y (j)]> 0, ∀θ ∈ Θ(j), (4.13)[〈X(j)i Ai〉 ?BT1,iX(j)i 0]− σ(j)1[CT2,iDT21,i] [C2,i D21,i]≥ −λ(j)i I, (4.14)[〈AiY (j)i 〉 ?C1,iY(j)i 0]− σ(j)2[B2,iD12,i] [BT2,i DT12,i]≥ −µ(j)i I, (4.15)[Switching Surface Conditions]Y (j) ≤ Y (k), ∀θ ∈ verS(j,k), (4.16a)X(j) − Y (j)−1 ≥ X(k) − Y (k)−1, ∀θ ∈ S(j,k). (4.16b)444.3. SLPV Control with Improved Local PerformanceX(j) ≥ X(k), ∀θ ∈ verS(j,k), (4.17a)Y (j) −X(j)−1 ≤ Y (k) −X(k)−1, ∀θ ∈ S(j,k). (4.17b)for all adjacent Θ(j) and Θ(k).The controller construction, i.e. obtaining the state-space matrices of the SLPV controller (4.3) can bereadily realized by following the controller construction of an LPV controller explained in [42], and is givenin the Appendix for completeness.Before proving this lemma, we would like to point out that the existence conditions for an SLPV con-troller in this lemma are a combination of the LPV existence conditions from [42, Theorem 5.3] (correspond-ing to the subset conditions (4.11) to (4.15)) and the SS conditions from [6] for guaranteeing monotonicallydecreasing Lyapunov function. One difference between subset conditions in Lemma 4.1 and the conditionsin [42, Theorem 5.3] is that, σ(j)1 and σ(j)2 are used for (4.11) and (4.14), and (4.12) and (4.15), respectively,to reduce the potential conservatism from using a common σ(j) for each subset j for the corresponding fourequations in [42].Proof. According to [42, Theorem 5.3], the conditions (4.11) to (4.15) ensures exponential stability and alocal L2-gain bound γ(j) for σ(t) ≡ j. According to [42, Theorem 5.2] and [42, Theorem 5.3], the quadraticfunctionV (j)(xcl, θ) := xTclP(j)(θ)xcl, (4.18)is a Lyapunov function that satisfies the bounded real lemma conditions with a local L2-gain bound forσ(t) ≡ j. Furthermore, P (j) and its inverse can be parameterized asP (j) =[X(j) N (j)N (j)T?], P (j)−1=[Y (j) M (j)M (j)T?], (4.19)where ‘?’ denotes irrelevant matrix blocks. Define a parameter-dependent piecewise-continuous functionV (σ)(xcl, θ) asV (σ)(xcl, θ) := xTclP(σ)(θ)xcl, (4.20)where P (j) is from (4.18). The inequality (4.16) is equivalent to[Y (j)−100 X(j) − Y (j)−1]≥[Y (k)−100 X(k) − Y (k)−1],for all θ ∈ S(j,k), which is further equivalent toP (j) ≥ P (k), ∀θ ∈ S(j,k), (4.21)through congruence transformation, when N (j) and M (j) in (4.19) are chosen to be X(j) − Y (j)−1 and454.3. SLPV Control with Improved Local Performance−Y (j), respectively. Similarly, if we chooseN (j) = −X(j) andM (j) = Y (j)−X(j)−1, the condition (4.17)leads toP (j)−1 ≤ P (k)−1, ∀θ ∈ S(j,k), (4.22)which is again equivalent to (4.21) considering that P (j) is positive definite for all j ∈ ZN . In summary,any one of the conditions (4.16) and (4.17) guarantees that the function (4.20) is monotonically decreasingat SSs. Following the proof of Theorem 1 in [6], it is straightforward to show that the SLPV controller withthe hysteresis switching rule ensures exponential stability and an L2-gain bound γ := maxj∈ZN γ(j) for alladmissible trajectories of θ(·).Remark 4.1. The conditions in Lemma 4.1 contain only a finite number of LMIs. They are the sufficientconditions for the parameter-dependent LMIs (PDLMIs) generally used for SLPV analysis and synthesis [6],which actually involve an infinite number of LMIs. These sufficient conditions result from applying multi-convexity method [42] to those PDLMIs. The multi-convexity method is more desirable than the griddingmethod for solving the PDLMIs because it is computationally more effective and provides performanceguarantee for the entire parameter set. But it should also be aware of that the multi-convexity method canonly be applied to plants with polynomial parameter dependence.Note that the constraints (4.16) and (4.17) are nonconvex. To convexify them, a simple yet conservativetrick proposed in [6] is to impose Y (j) = Y (k) in (4.16) and X(j) = X(k) in (4.17). The resulting sufficientconditions for (4.16) and (4.17) are given respectively as (4.23) and (4.24).{Y (j) = Y (k),X(j) ≥ X(k), ∀θ ∈ verS(j,k), (4.23){X(j) = X(k),Y (j) ≤ Y (k), ∀θ ∈ verS(j,k). (4.24)4.3.3 Design of SLPV Controllers with Improved Local PerformanceThe SLPV controllers are usually designed by solving an optimization problem to optimize the performanceof the closed-loop system. In the traditional approach, the worst-case L2-gain bound of the closed-loopsystem in all the subsets is used as the cost function [6, 7, 27, 53]. With the existence conditions stated inLemma 4.1, the traditional SLPV synthesis approach can be summarized as:Approach 4.1. Traditional SLPV SynthesisGiven an LPV plant (4.1) with θ satisfying (4.2) and the parameter subsets {Θ(j)}i∈ZN , design an SLPVcontroller by solving the optimization problemminX(j),Y (j),γ(j),σ(j)1 ,σ(j)2 ,λ(j)i ,µ(j)imaxj∈ZNγ(j) (4.25)subject to (4.11) to (4.15), (4.23) and (4.24).464.3. SLPV Control with Improved Local PerformanceThe proposed SLPV controllers are also designed by solving optimization problems whereas the objec-tive of the optimization problems is to minimize the cost function (4.7). To guarantee a worst-case perfor-mance level γ¯ in all the subsets, we introduce the following constraints:γ(j) ≤ γ¯, ∀j ∈ ZN . (4.26)Setting γ¯ to +∞ means that no hard constraint is put on the worst-case L2-gain bound.The proposed SLPV controller design approach is summarized as:Approach 4.2. Proposed SLPV Synthesis Under Fixed SSsGiven an LPV plant (4.1) with θ satisfying (4.2) and the parameter subsets {Θ(j)}i∈ZN }, design an SLPVcontroller by solving the optimization problemminX(j),Y (j),γ(j),σ(j)1 ,σ(j)2 ,λ(j)i ,µ(j)iJ(Γ) (4.27)subject to (4.11) to (4.15), (4.23), (4.24) and (4.26).The bound γ¯ can be selected based on the performance given by an LPV controller or a traditional SLPVcontroller. For instance, if we set the γ¯ equal to the L2-gain bound yielded by an LPV controller (denotedas γ∗), it is ensured that the worst-case performance level given by the proposed SLPV control is not worsethan the LPV control. On the other hand, we can also relax the bound γ∗ by a small percentage, i.e. settingγ¯ equal to (1 + α)γ∗, where α is the relaxation percentage, in order to obtain large improvements of localperformance in some subsets. In addition, one can use the results from the traditional SLPV control to setthe value of γ¯. The following lemma states the relation between the traditional approach and the proposedapproach. The proof is neglected since the statement is straightforward.Lemma 4.2. Assume the worst-case L2-gain bound yielded by a traditional SLPV controller is γ˜. With thesetting ofγ¯ := γ˜, (4.28)there is always a feasible SLPV controller from Approach 4.2.Lemma 4.2 indicates that we can always design an SLPV controller using Approach 4.2 that yields thesame worst-case performance as a traditional SLPV controller, while potentially giving better local perfor-mance in some subsets, because of inclusion of the local performance in the cost function (4.7). This isverified by the second last row in Table 4.1 for the numerical example introduced in Section 4.2.3 and thefourth last row in Table 4.4 for the air-fuel ratio control example in Section 4.5, although the local perfor-mance improvement is small due to the tight bound γ¯. The upper bound γ¯ reflects the trade-off betweenthe local performance and the worst-case performance. A loose bound often leads to aggressive local per-formance in some subsets while a tight bound tends to make the local performance close to the worst-caseperformance.474.4. SLPV Control with Optimized Switching Surfaces4.3.4 Revisit of the Motivating ExampleHere, more tests on the numerical example used in Section 4.2.3 were conducted. The L2-gain bounds givenby an LPV controller, traditional SLPV controllers and proposed SLPV controllers under different scenariosare summarized in Table 4.1. Note that pi = {−0.1, 0.1} for two-subset case, pi = {−0.4, −0.3, 0.2, 0.3}for three-subset case. When designing SLPV controllers, both the SS conditions (4.23) and (4.24) were triedand (4.24) always gave less conservative results. Therefore, only the results using (4.24) are presented. Onecan see that the traditional SLPV synthesis approach always led to very close local L2-gain bounds in all thesubsets.For the proposed SLPV controller design, different values for γ¯ based on the results of LPV control andtraditional SLPV control were tested. The proposed SLPV controllers gave different results under differentγ¯. When we set γ¯ equal to the worst-case L2-gain bound yielded by traditional SLPV controllers, we eitherrecovered the L2-gain bounds for all the subsets given by the traditional controller (two subset case), orimproved the local performance in some subsets (Subset 2 in three subset case). When we relaxed the boundγ¯ by a small amount, the local L2-gain bounds were improved in all except one subsets. The performance ofthe proposed SLPV controller shown in Figure 4.2(a), i.e. SLPV 1(a) corresponds to γ¯ = 1.02γ0a.We also designed an LPV controller and traditional SLPV controllers using the a common σ(j) (i.e.σ(j)1 = σ(j)2 = σ(j)) for each subset j. Note that different σ values were still used for different subsets. Theresulting {γ(j)} value for the LPV controller, traditional SLPV controller in two-subset case, and traditionalSLPV controller in three-subset case are 42.57, {39.10, 39.10} and {36.29, 36.29, 36.29}, respectively.A comparison of these results with those in Table 4.1 clearly shows that using σ(j)1 and σ(j)2 yields lessconservative results than using a common σ(j).Table 4.1: Performance of different SLPV controllersWorst-caseBoundCostFunction{γ(j)}LPV 17.96 := γ∗ 17.96TraditionalSLPV2-subset case 17.06 := γ0a {17.06, 17.06}3-subset case 16.53 := γ0b {16.53, 16.45, 16.53}ProposedSLPV2-subset caseγ¯ = γ0a 17.06 {17.06, 17.06}γ¯ = 1.02γ0a 16.61 {15.83, 17.40}γ¯ = γ∗ 16.28 {14.62, 17.96}γ¯ = +∞ 16.28 {13.39, 18.16}3-subset caseγ¯ = γ0b 16.08 {16.53, 15.81, 16.53}γ¯ = 1.02γ0b 15.95 {15.79,15.09,16.86}4.4 SLPV Control with Optimized Switching SurfacesAs mentioned in Introduction and validated in Section 4.2.3, SSs may have a crucial effect on the perfor-mance of an SLPV controller. SS optimization can further improve the performance of an SLPV controller byeither improving the local performance in some subsets, or increasing the area ratio of subsets corresponding484.4. SLPV Control with Optimized Switching Surfacesto good local performance, or both.4.4.1 A Switching Surface Design ProblemFor hyper-rectangular subsets, SSs can be characterized by SS variables (SSVs), which are the set of all thevariables that determine the partition of the intervals Θi for all i ∈ Zs, denoted aspi := {pii,1, pii,2, . . . , pii,2(Ni−1) : i ∈ Zs}. (4.29)Remember that Ni is the number of intervals Θi is divided into. For instance, in Figure 4.1, S(1,2) is thesegment between points (pi1,2, θ2) and (pi1,2, pi2,2) while S(2,1) is the segment between points (pi1,1, θ2) and(pi1,1, pi2,2).To enforce overlapped regions between adjacent subsets and the bounds of parameter variation (4.2), thefollowing constraints should be imposed on SSVs:pii,1 − θi ≥ pii,θi − pii,2(Ni−1) ≥ pii,pii,ni+1 − pii,ni ≥ pii, ∀ni ∈ Z2Ni−3,∀i ∈ Zs, (4.30)where pii is a prespecified minimum distance between adjacent SSs as well as between SSs and the edges ofΘi for θi.Simultaneous design of the SSs and an SLPV controller will result in a nonconvex problem involvingnonlinear matrix inequalities, which is difficult to solve. However, the problem of SLPV controller designunder fixed SSs, i.e. the problem in Approach 4.1 or Approach 4.2 is a convex problem that can be effi-ciently solved using the off-the-shelf solvers such as LMI Lab[45]. Hereafter, for brevity, we focus on SSoptimization for the proposed SLPV controllers. However, the idea and the algorithm for SS optimization tobe presented can be readily applied to traditional SLPV controllers without much extra effort. For proposedSLPV controller design, by treating (4.7) as the cost function and the SLPV controller design under fixed SSas evaluation of the cost function value, the SS optimization problem can be formulated asProblem 4.2 (SS Optimization for the Proposed SLPV Control). Given an LPV plant (4.1) with θ satisfying(4.2) and the number of partitions Ni for all i ∈ Zs, design the optimal SSs for the proposed SLPV controlby solving the following optimization problemminpiJ(Γ) subject to (4.30), (4.31)where Γ is from Approach 4.2 under the SSs specified by pi.4.4.2 Switching Surface Optimization Based on Particle Swarm OptimizationProblem 4.2 is an optimization problem for which the analytic expression of the gradient is unavailable.Therefore classical gradient-based optimization methods cannot be directly applied unless the gradient isnumerically computed, as done in [35]. However, the numerical evaluation of gradient is computationally494.4. SLPV Control with Optimized Switching Surfacesexpensive. In addition, the gradient-based methods are local search methods and may give results thatstrongly depend on the initial conditions considering that the SS optimization problem may be nonconvex.In this chapter, particle swarm optimization (PSO) is selected to solve Problem 4.2 because it is easy toimplement without needing the gradient information, computationally efficient [54], and is a global searchmethod that has great potential to find the global optimum.PSO is a population-based stochastic optimization algorithm inspired by the social behavior of biologicpopulations developed by Kennedy and Eberhart [55]. PSO has been shown to successfully find the globaloptima of numerous highly nonlinear and non-differentiable problems under appropriately selected param-eters. There are a variety in implementation of the PSO. The canonical version of PSO is adopted here forSS optimization and will be explained shortly. For faster convergence, the gbest topology is selected, whichmeans that each particle will be influenced by the best member in the entire population [56].Denoting the dimensionality of the search space as D, each particle will contain three D-dimensionalvectors, i.e. the current position xi, the previous best position pi and the velocity vi. Denote the position ofthe best particle in the swarm as pg. The canonical PSO updates the position and velocity of each particle ineach iteration asvi ← wvi + U(φ1) (pi − xi) + U(φ2) (pg − xi)xi ← xi + vi(4.32)where denotes element-wise multiplication and U(φi) represents a vector of random numbers uniformlydistributed in [0, φi]. The parameter w is the inertia weight, and φ1 and φ2 are acceleration coefficients. Incanonical PSO, the parameters in (4.32) are set as w = 0.7298, φ1 = φ2 = 1.49619, which were suggestedin [56]. The population size, i.e. the number of particles, is usually set between 10 and 100. The particlesin the canonical PSO are guaranteed to converge [56]. By applying the canonical PSO to Problem 4.2, weobtain the following algorithm for optimal SS design.Algorithm 4.1 Optimal SS design utilizing PSOInitialization Set the value of γ¯. Initialize a population of particles with random positions and velocitiessubject to the constraints (4.30).Iteration Repeat the following steps until the termination criterion is met:Step 1 For each particle with the position xi , evaluate the cost function in Problem 4.2 using Ap-proach 4.2 under the SSs specified by pi := xi.Step 2 Compare the particle’ cost function with its previous best value Ji,best. If the current value issmaller than Ji,best, then set Ji,best equal to the current value and pi equal to the current positionxi.Step 3 Identify the best particle with the entire population and assign its index to g. Denote the costfunction of the global best as Jgbest. Update the velocity and position of the particle accordingto (4.32).Termination Terminate the algorithm if the number of iterations reaches the prespecified maximum numberof iterations, or if the decrease of Jgbest is less than some small tolerance ε during the last m¯ iterations,where m¯ is a prespecified integer.504.4. SLPV Control with Optimized Switching Surfaces4.4.3 Revisit of the Motivating Example for SS optimizationTo test Algorithm 1, Algorithm 4.1 was applied to search for the optimal SSs. All the computations involvedin this chapter were conducted in Matlab 2015b on a PC with Intel i5-3470 CPU and 16 GB RAM runningWin7 64-bit OS. LMI Lab[45] was used to solve all the involved LMIs. The population size and the maxi-mum number of iterations for PSO were set to 10 and 50, respectively. The iteration was terminated whenthe decrease of the cost function of the global best is less than ε = 10−5 in the last m¯ = 8 iterations. Forthis numerical example, pi1 in (4.30) was set to be 0.01.We first tried to optimize the SSs in traditional SLPV controller design using an analogous version ofAlgorithm 4.1. The optimal SSs obtained is listed in Table 4.2, and the local L2-gain bounds yielded bythe traditional SLPV controller corresponding to the optimal SSs, denoted as SLPV 0(c), are illustrated inFigure 4.4. Compared to the traditional SLPV controller with the heuristic SSs pi = {−0.1, 0.1}, i.e. SLPV0(a), SLPV 0(c) improved the performance by 19.6%.Next, we optimized SSs for the proposed SLPV controller design using Algorithm 4.1 with γ¯ set equalto γ∗, i.e. the L2-gain bound given by the LPV controller. The algorithm was terminated after 44 and50 iterations, which took 178.4s and 359.5s, for two subset and three subset cases, respectively. The costfunction and SSVs corresponding to the global best particle in every iteration for the two subset case is shownin Figure 4.5. The results are summarized in Table 4.2. The performance of the proposed SLPV controllercorresponding to the optimal SSs for two subset case (denoted as SLPV 1(c) is also depicted in Figure 4.4.Compared to the proposed SLPV controller with heuristic SSs pi = {−0.1, 0.1}, i.e. SLPV 1(a), SLPV 1(c)improved the performance in Subset 1 and Subset 2 by 17.0% and 14.3%, respectively. Furthermore, thearea ratio of Subset 1 that corresponds to better performance among the two subsets was also increased from0.5 to 0.9. On the other hand, compared to SLPV 0(c), SLPV 1(c) improved the performance in Subset 1 ofSLPV 1(c) by 2.8% while worsened the performance in Subset 2 by 8.8%. The relatively high sacrifice ofthe worst-case performance level is due to using the γ∗ for worst-case performance bound. If we enforcedγ¯ equal to 1.02γ0c when implementing Algorithm 4.1, where γ0c is the worst-case L2-gain bound fromSLPV 0(c), different optimal SS were obtained, and the performance of the corresponding SLPV controller(denoted as SLPV 1(d)) is given in both Table 4.2 and Figure 4.4.Table 4.2: Performance of proposed SLPV controllers with optimized SSs (γ∗ and γ0b are defined in Ta-ble 4.1.)Worst-CaseBoundpioptCostFunction{γ(j)}TraditionalSLPV2-subset case {0.801, 0.812} 13.72 := γ0c {13.72,13.72}ProposedSLPV2-subset caseγ¯ = γ∗ {0.800, 0.810} 13.31 {13.14,14.92}γ¯ = 1.02γ0c {0.810, 0.820} 13.42 {13.36,13.99}3-subset case γ¯ = γ0b{0.487, 0.498,0.810, 0.822}12.40 {11.79, 13.24, 15.68}The global best particle and the associated cost function in PSO iterations for optimizing SSs for theproposed SLPV controller with γ¯ := γ∗ are displayed in Figure 4.5. To verify whether Algorithm 4.1 found514.5. Application to Air-Fuel Ratio Control of an Automotive Engine-1 -0.5 0 0.5 112141618-valueLPVSLPV 0(c)SLPV 1(c)SLPV 1(d)Figure 4.4: Performance of SLPV controllers with optimized SSs. SLPV 0(c): traditional SLPV controllerwith optimized SSs, SLPV 1(c): proposed SLPV controller with improved local performance and optimizedSSs under the setting of γ¯ = γ∗, SLPV 1(d): proposed SLPV controller with improved local performanceand optimized SSs under the setting of γ¯ = 1.02γ0c.0 10 20 30 40 50Iteration13141516Cost function0 10 20 30 40 50Iteration0.20.40.60.81SSVs 1,1 1,2Figure 4.5: Global best in each iteration of Algorithm 1 for the proposed SLPV controller with γ¯ = γ∗the global optimum, a brute-force search with the step of 0.01 for both pi1,1 and pi1,2 was also implementedfor the two subset case with γ¯ := γ∗. The result is shown in Figure 4.6. The optimum from the brute-forcesearch was the same as the one found by Algorithm 4.1, which indicates that Algorithm 4.1 found the globaloptimum for this special case.4.5 Application to Air-Fuel Ratio Control of an Automotive EngineCurrently internal combustion engines usually employ a three-way catalytic converter (TWC) to reduce toxicgases and pollutants in exhaust gases. Precise control of air-fuel ratio is crucial for emission reduction asTWC operates effectively only when the air-fuel ratio is extremely close to stoichiometric. In this section,we apply the proposed methods to the air-fuel ratio control of an spark ignition engine in automobiles.524.5. Application to Air-Fuel Ratio Control of an Automotive Engine12-1141Cost function161,101,21801-114151617Figure 4.6: Brute-force search result4.5.1 Modeling of the Air-Fuel Ratio DynamicsFor controller design purpose, an equivalence ratio φ, which is actually fuel-air ratio, is usually utilized anddefined asφ := Rstoichm˙fuelm˙air(4.33)where m˙fuel and m˙air are the fuel and air mass flow rate, respectively, andRstoich is stoichiometric ratio whichis approximately 14.7 for gasoline fuels. The air-fuel ratio dynamics describes the dynamic relation betweenthe air mass flow rate m˙air (i.e. the input) and the equivalence ratio φ (i.e. the output), and was modeledusing a first-order plus dead time (FOPDT) model (also adopted in [9]):L(φ)L(m˙fuel) =gsτ + 1e−sT , (4.34)where L(·) is the Laplace transform operator. In the model (4.34), the steady-state gain g, the time constantτ , and the pure delay T are assumed to depend on the (time-varying) air mass flow and the engine speed asfollows:g =Rstoichm˙air, τ =120(ncyl − 1)Ne · ncyl , T =120Ne+cm˙air, (4.35)where Ne is the engine speed in revolutions per minute (RPM), ncyl (= 4) is the number of cylinders, andc(= 5.33× 10−2) is a constant that depends on the exhaust line geometry.To apply the approach presented in previous sections, the pure delay term in the FOPDT model (4.34)was approximated using a Padé approximation to yield a rational plant model. For simplicity, a first-orderPadé approximation was utilized that led to a state-space model:[Ap BpCp Dp]:= −1/τ 4g/T −g0 −2/T 11/τ 0 0 . (4.36)534.5. Application to Air-Fuel Ratio Control of an Automotive Engine4.5.2 Controller Design and Computation ResultsThe feedback system structure for controller design is shown in Figure 4.7. The exogenous inputs includethe reference r and the disturbance d applied on the output. The performance channels include the weightedtracking error ze and the weighted control input zu. Note that the first order term (2 + 0.6s)/s is inserted+‐LPV/SLPV controllerCompen-sation gainPlantFigure 4.7: Controller synthesis structurefor both shaping the sensitivity function and ensuring zero steady-state error. The weighting functions areselected as We = 1 and Wu = 10.2.In the model (4.36), g and τ depend on m˙air and Ne, respectively, and T depends on both of m˙air andNe. However, the plant matrix Ap is not affine with respect to either m˙air and Ne, or their inverse. To obtainan affine LPV plant and to simplify the controller synthesis, we placed a parameter-dependent gain (whichis equal to m˙air) after the output of the LPV/SLPV controllers to cancel the time-varying steady-state gainof the system (4.35) (see Figure 4.7). In this case, the steady-state gain of the plant can be considered asconstant and equal to Rstoich for controller design purpose. Then, by setting the scheduling parameters asθ1 :=1T=1120/Ne + c/m˙air, θ2 := Ne, (4.37)we obtain an affine LPV plant with the following matrices: A(θ) B1 B2C1(θ) D11 D12C2(θ) D21 0 =−θ2/90 58.8θ1 0 0 0 −14.70 −2θ1 0 0 0 1−θ2/45 0 0 −2 2 0−θ2/150 0 1 −0.6 0.6 00 0 0 0 0 10−θ2/150 0 1 −0.6 0.6 0.It is assumed that the engine speed varies between 800 and 6000 RPM with the rate of variation between±2000 RPM/s, while the air mass flow varies between 10% and 100% with the rate of variation between544.5. Application to Air-Fuel Ratio Control of an Automotive EngineTable 4.3: Different partitions of the parameter set and associated SSsPartition of Θ Heuristically Selected SSs piN1 = 2, N2 = 1 pi = {7.5, 8}N1 = 2, N2 = 2 pi = {7.5, 8, 3000, 3500}N1 = 3, N2 = 1 pi = {5.5, 6, 9.5, 10}N1 = 3, N2 = 2 pi = {5.5, 6, 9.5, 10, 3000, 3500}±100%/s. Utilizing (4.37), the range of scheduling parameters and their rate of variation is obtained asθ1 ∈ Θ1 = [1.464 13.642], θ˙1 ∈ Ω1 = [−17.46 17.46],θ2 ∈ Θ2 = [800 6000], θ˙2 ∈ Ω2 = [−2000 2000].For the SLPV controller design, we need to create the subsets by dividing Θi. We tested different partition ofΘ with heuristically selected SSs, as listed in Table 4.3, where Ni is the number of intervals Θi was dividedinto (see Section 4.2.2).We first tried to design the traditional LPV controllers and the results are shown in Table 4.4. One cansee that for all the tested cases, the traditional SLPV design approach always yielded almost the same localperformance levels in all the subsets, which had only marginal improvement (less than 1% percent) comparedto the LPV controller. It was also found that dividing Θ2 does not bring much improvement compared to theintact Θ2 case. Due to space limit, only the results corresponding to N1 = 3, N2 = 1 yielded by proposedSLPV controllers are listed in Table 4.4. By adjusting the worst-case performance bound γ¯, it is possibleto acquire large improvements in some subsets without much sacrifice of the worst-case performance. Forinstance, when setting γ¯ = 1.02γ∗, the performance improvement of 27.2% and 6.2% were obtained inSubset 2 and Subset 3, respectively, while the performance in Subset 1 was only 2% worse than that ofthe LPV controller. Even when we set γ¯ equal to γ0, i.e. the worst-case performance level yielded bythe traditional SLPV controller, we still obtained some improvement of performance in Subset 2. This isconsistent with the statement right after Lemma 4.2.Table 4.4: Performance of different SLPV controllers with heuristically selected SSsWorst-CaseBoundCostFunction{γ(j)}LPV 3.394 := γ∗ 3.394TraditionalSLPVN1 = 2, N2 = 1 3.383 {3.383, 3.383}N1 = 1, N2 = 2 3.394 {3.394, 3.394}N1 = 2, N2 = 2 3.38 {3.38, 3.38, 3.38, 3.38}N1 = 3,N2 = 1 3.37 := γ0 {3.37, 3.37, 3.37}N1 = 3, N2 = 2 3.36 {3.36, 3.36, 3.36, 3.36}ProposedSLPVN1 = 3, N2 = 1γ¯ = γ0 3.36 {3.37, 3.33, 3.37}γ¯ = γ∗ 3.12 {3.39, 2.78, 3.39}γ¯ = 1.02γ∗ 3.04 {3.46, 2.47, 3.19}γ¯ = +∞ 2.66 {3.93, 1.98, 2.00}554.5. Application to Air-Fuel Ratio Control of an Automotive EngineFor SS optimization based on Algorithm 4.1 for the proposed SLPV controllers, the case of N1 =3, N2 = 1 and γ¯ = 1.02γ∗ was utilized. The computational time was roughly 2 hours. The populationsize and the maximum number of iterations for PSO were set to 10 and 50, respectively. The terminationparameters ε and m¯ were set as ε = 10−5 and m¯ = 8. The result is shown in both Table 4.5 and Figure 4.8,where the meaning of the legends in Figure 4.8 is as follows.SLPV 0: A traditional SLPV controller with heuristic SSs pi = {5.5, 6, 9.5, 10},SLPV 1: An proposed SLPV controller with heuristic SSs pi = {5.5, 6, 9.5, 10} and γ¯ = 1.02γ∗,SLPV 2: An proposed SLPV controller with optimized SSs and γ¯ = 1.02γ∗, given in Table 4.2.Compared with the SLPV 1, SLPV 2 improved the performance in Subsets 2 and 3 and also increased thearea ratio of these two subsets. Furthermore, optimizing the SSs yielded a large overlapped area betweenSubset 2 and Subset 3 because of the relatively good performance in these two subsets compared to that inSubset 1. This is consistent with our statement at the end of the second last paragraph in Section 4.3.1. Notethat what we really want to achieve from SS optimization is to maximize the ratio of the operating regionthat corresponds to good performance. This region should be the union (rather than the summation) of thesubsets corresponding to good performance, i.e. Subsets 2 and 3 for this example. Therefore, there is someslight mismatch between what we want to achieve in SS optimization and the weights defined in (4.8). Thismismatch can be alleviated by adding some constraints on SSVs similar to (4.30) to limit the overlapped areabetween any two adjacent subsets or using different weights that do not depend on the overlapped area.We also tested optimizing the SSs in traditional SLPV controller design which, as shown in Table 4.5,yielded almost 4% improvement in the worst-case L2-gain bound compared to the heuristic SS case.Table 4.5: Performance of SLPV controllers with optimized SSs (N1 = 3, N2 = 1)ControllerWorst-CaseBoundpioptCostFunction{γ(j)}Traditional SLPV {1.88, 4.34, 4.75, 9.92} 3.24 {3.24, 3.24, 3.24}Proposed SLPV γ¯ = 1.02γ∗ {3.12, 4.63, 4.80, 10.40} 2.68 {3.46, 2.42, 2.61}4.5.3 Simulation ResultsTime-domain simulations were performed to inspect the performance of different controllers, where a con-stant reference signal r = 1 and a square-wave disturbance d with the amplitude of 0.1 and the period of20s were applied. Figure 4.9 shows a realistic profile of air mass flow and engine speed, which includesvarious operation conditions of the engine such as idling, engine braking and high load. Figure 4.10 showsthe trajectory of the engine operating point in the m˙air-N plane and the scheduling parameter (θ1-θ2) plane.564.5. Application to Air-Fuel Ratio Control of an Automotive Engine0 2 4 6 8 10 12 14Scheduling parameter 3122.533.54. value LPVSLPV 0SLPV 1SLPV 2Figure 4.8: L2-gain bounds comparison for the air-fuel ratio control example0 10 20 30 40 50 60Time (s)050100Air flow (%)0 10 20 30 40 50 60Time (s)0200040006000Engine Speed (RPM)Figure 4.9: Profiles of air flow and engine speed574.5. Application to Air-Fuel Ratio Control of an Automotive Engine0 50 100Air flow (%)123456Engine Speed (#1000 RPM)(a) In m˙air-N plane2 4 6 8 10 12 1431 (s-1)1234563 2 (#1000 RPM)(b) In θ1-θ2 planeFigure 4.10: Trajectory of the engine operating point. The red dash-dot lines in the right figure denote theoptimized SSs while the blue dashed lines denote the bounds of θ1For comparison purpose, we also designed a H∞ controller using a linear time-invariant plant obtainedby fixingN = 3000, m˙air = 50%. The disturbance rejection performance of different controllers is shown inFigure 4.12 where the switching signals of the SLPV controllers is given in Figure 4.11. The H∞ controllercould not even stabilize the feedback system at the initial phase of the simulation. The SLPV controller withoptimized SSs performed similarly to the LPV controller and the SLPV controller with heuristic SSs, whenits local controller 1 or 2 was active, but performed much better than the other two when its local controller 3was active. The root-mean-squared values of the track error (RMSE) given byH∞, LPV, SLPV1 and SLPV 2are 0.2515, 0.1214, 0.1183 and 0.1169, respectively, which again demonstrated that the SLPV controller withoptimized SSs gave the best performance. Note that the drop in the equivalence ratio at around 28 secondyielded by SLPV 2 was because of the jump in the control signal, which could be potentially alleviated byconsidering smooth-switching requirement in the SLPV controller design[27].0 10 20 30 40 50 60Time (s)0.811.2Equivalence ratioH1 LPV SLPV 1 SLPV 20 10 20 30 40 50 60Time (s)123Switching signalSLPV 1SLPV 2Figure 4.11: Switching signals584.6. Conclusion0 10 20 30 40 50 60Time (s)0.811.2Equivalence ratioH1 LPV SLPV 1 SLPV 20 10 20 30 40 50 60Time (s)123Switching signalSLPV 1SLPV 2(a) Full scale20 30 40 50 60Time (s)0.90.951.051.1Equivalence ratio0 10 20 30 40 50 60Time (s)123Switching signalSLPV 1SLPV 2(b) Zoomed figure for 30-60 secondsFigure 4.12: Disturbance rejection performance4.6 ConclusionBased on the observation that a traditional SLPV controller often yields similar local performance in all thesubsets, this chapter presented a novel method to design SLPV controllers that could yield improved localperformance without much sacrifice of the worst-case performance. The design method utilized a new costfunction considering the local performance in all the subsets, and a tuning parameter reflecting the trade-off between the local performance and the worst-case performance. An algorithm based on particle swarmoptimization was also proposed to optimize the switching surfaces to further improve the performance ofSLPV controllers. The efficacy of the proposed approaches were validated on both a numerical example anda realistic example of air-fuel ratio control of an automotive engine.59Chapter 5Application 1: Miniaturized Optical ImageStabilizers5.1 IntroductionNowadays high-quality digital cameras have become one of the main attractions for smart phones and tablets.Although the image quality has been dramatically improved by increasing pixels, image blur due to involun-tary hand-shake while taking photos is still an issue.Technologies in cameras to alleviate the hand-shake induced image blur are called image stabilization.The two categories of image stabilization techniques are electronic image stabilization (EIS) and opticalimage stabilization. EIS is cost-effective and easy to implement as it only relies on digital image process-ing [57, 58]; however, EIS often leads to degraded image quality due to image scaling and image processingartifacts [59]. Cameras with the optical image stabilization, on the other hand, are more expensive as theyneed hardware components, named as the optical image stabilizer (OIS), to stabilize the image projectedon the image sensor before the sensor converts the image into digital information. Despite the higher cost,OIS can provide superior performance compared to EIS, and therefore it is popular among single-lens reflex(SLR) and point-and-shoot cameras [59].As of December 2017, there has been an increasing tendency to popularize OIS’s in mobile platforms,such as Apple iPhone 8 and Samsung Galaxy S8. There are mainly four mechanisms for OIS„ i.e. CCD-shifting [60, 61], lens-shifting [62], module-tilting [59] and lens-tilting [63]. Among these four mechanisms,lens-shifting and lens-tilting are most appropriate for mobile applications because of their easiness for minia-turization. In [64], a concept of miniaturized magnetically-actuated lens-tilting OIS based on micro-electro-mechanical-system (MEMS) technology was proposed, and the concept was validated with large-scale pro-totypes in [65]. The device uses four folded beams to support a lens platform (LP) and four moving-magnetactuators that can actuate the LP in three degrees of freedom (DOFs). The proposed OIS employs a feedbackcontrol structure, where the LP is tilted to track reference angles under an OIS controller in order to mitigatethe image blur. The lens-tilting tracking performance yielded by the feedback controller directly determinesthe image quality.A number of controllers have been proposed to control OIS systems, such as lead-lag compensator [66],fuzzy proportional-integral-derivative (PID) controller [67], adaptive PID controller [68], gain-schedulinglead-lag compensator [60], and sliding mode controller [69]. All the OIS systems controlled by those con-trollers employed voice coil motor (VCM) actuators, and most of the controllers were mainly for dealingwith the nonlinearity associated with the VCM actuators such as hysteresis and friction. In contrast, the OIS605.1. Introductionsystem proposed in [64] that is targeted in this chapter employs a moving-magnet actuator instead of a VCMactuator, and the nonlinearity can be approximately canceled using the estimated relation between air gapand magnetic force. Thus, the aforementioned work is not suitable for controlling the OIS system presentedin this chapter. In addition, none of the above methods consider dynamics variation among different productsdue to product variations, and thus cannot provide stability guarantee for all the OIS products.A couple of efforts have been made to control the lens-tilting OIS proposed in [64]. A state-feedbackcontroller was designed and validated on a 1-DOF large-scale prototype in [64]. However, without con-sideration of unavoidable product variations [36, 37] in micro-scale devices, the performance or even thestability of the feedback system cannot be guaranteed for all the mass-produced OIS products. It was vali-dated in [65, 70] on multiple 3-DOF prototypes that lead-lag compensator, LQR and H∞ controllers wereunable to guarantee robust stability due to ignorance of product variability, while robust controllers such asthe µ controller [71, 72] and the robustH∞ controller [73] have an ability to offer consistent and satisfactoryresults for all the prototypes. Despite the successful application of robust control to OIS, there is still a roomfor reducing the conservatism inherent to robust controllers. This is because some of the parameters regardedas uncertain in robust controller design can actually be identified after the manufacturing, and then employedfor controller adaptation.The contribution of this chapter is to propose a method to design multiple parameter-dependent robust(MPDR) controllers for the mass-produced OIS’s, which can reduce the conservatism of the robust con-trollers previously presented in [65]. The designed controllers are parameterized by the natural frequenciesof the LP that represent the most crucial product variations in fabrication of the LPs. Considering the dif-ficulty in accurately estimating the natural frequencies due to the extremely low damping of the devices,the natural frequencies are assumed to be estimated with uncertainties. In addition, to characterize the un-balanced forces as a result of unavoidable errors in actuator fabrication and installation, an uncertain gainparameter is introduced in modeling of the OIS’s. The controllers are designed to be robust in the sense thatboth estimation uncertainties of the natural frequencies and the uncertain gain parameter for the unbalancedforces are explicitly taken into account in controller design. To design the MPDR controllers, the approachto the SLPV controller design under uncertain scheduling parameters presented in Chapter 2 is adopted. Theadvantages of the MPDR controllers over an existing µ-synthesis robust controller and a classical controllerconsisting of a lead-lag compensator and multiple notch filters are experimentally verified on large-scale3-DOF OIS prototypes.This chapter is organized as follows: Section 5.2 gives a brief introduction of the conceptual miniatur-ized OIS and large-scale prototypes, as well as the control objectives and challenges. Section 5.3 presentsthe feedback control structure employed in the OIS. Mathematical modeling of the OIS’s is presented inSection 5.4. Section 5.5 explains the implementation and design of the MPDR controllers. The designedcontrollers are experimentally validated on large-scale prototypes in Section 5.6. Details of the controllerdesign are given in Appendix C.3615.2. Conceptual Miniaturized OIS and Large-Scale Prototypes5.2 Conceptual Miniaturized OIS and Large-Scale Prototypes5.2.1 Miniaturized Lens-Tilting OISThe mechanical layout of the miniaturized lens-tilting OIS proposed in [64] is shown in Fig. 5.1. The deviceconsists of a flexible lens platform (LP) and four moving magnet actuators. The LP includes a plate supportedby four folded beams, a lens installed at the center of the plate, and four permanent magnets attached to theplate underneath. The magnets and four air-core electromagnetic coils below them constitute the actuatorsthat can provide 3-DOF actuation of the LP, which are translation along z-axis, pitch (i.e. rotation aboutx-axis) and yaw (i.e. rotation about y-axis)3. Among the three DOFs, pitch and yaw are utilized for imagestabilization while the translational DOF is for autofocus.Flexible lens platform10 mm5 mmFolded beamMeasurement pointsAB10 mmA’ B’O(a) Isometric view1234Flexible lens platformLensPermanent magnet Coil10 mm10 mm5 mmFolded beam(b) Front viewFigure 5.1: Mechanical layout of a miniaturized OISLike most other OIS’s, the proposed OIS employs a feedback control mechanism illustrated in Fig. 5.2.A gyro-sensor is used to detect the hand-induced movement of the camera body while taking photos, whichwill be used to calculate desired lens-tilting angles in order to restore the optical path to the image sensor. TheOIS controller, utilizing the error between the desired and measured tilting angles, determines the currentsapplied to the magnetic actuators, which actuate the LP to track the reference angles. This chapter assumesthat the desired lens-tilting angles are prespecified, while the generation of these angles needs the knowledgeof optical imaging theory and is beyond the scope of this chapter.For feedback control, the rotation angle of the LP needs to be measured in real time. Due to the flexibilityof the LP, angle sensors may not be applicable to directly measure the rotation angle. A feasible sensingmechanism is to measure the z-direction displacements of two points on the LP for each rotational DOF,and convert them to the rotation angles of the LP. For instance, the displacements of points A and A’ (Band B’) in Fig. 5.1a can be measured for reconstruction of the pitch (yaw) angle. One type of the potentialsensors for measuring the displacement is hall effect sensor, which has actually been widely used in OISsystems (see [59, 74] for example). Note that in the ideal case when there is no translational motion in thecenter of the LP during pitch and yaw actuation, the displacement of one point A (B) is enough to accurately3For cameras, yaw, pitch and roll are conventionally defined as presented here. See, e.g., [59].625.2. Conceptual Miniaturized OIS and Large-Scale PrototypesOIS controller—Lens platformGyro sensorReferenceCalculatorHand-induced camera movementDesiredlens-tilting angles Magnetic actuatorsForcesMeasuredlens-tilting anglesSensorCurrentsErrorFigure 5.2: Block diagram of the OIS systemreconstruct the pitch (yaw) angle. This idealization may not be valid in practice due to the flexibility of theLP and imperfection in fabrication of the OIS devices; thus measuring the displacement of two points foreach rotational DOF should be selected when the sensor cost is not a concern. However, if the magnitudeof the translational motion is small relative to that of the rotational motion, the single-point measurementfor each rotational DOF may still provide rather accurate estimation of the rotation angle and be preferredbecause of the sensor cost saving.5.2.2 Control Objectives for OIS Control SystemThere are three control objectives in designing the feedback controller. Namely, for all the OIS products, thecontroller should:(1) minimize the tracking error for desired tilt angle signals below 10 Hz,(2) use minimal control input, and(3) have minimal order.The first objective is for mitigating the image blur caused by vibrations within frequencies of human handtrembles as much as possible, while the second objective is for the prolonged battery life. The third ob-jective is for the reduced computational cost so that the controller is implementable on inexpensive micro-controllers.5.2.3 OIS Control ChallengesAlthough the proposed OIS is compact and easy for miniaturization, its control is challenging. There aremainly three challenges involved.The first challenge is related to the noticeable product variations of dynamics, due to the uncertainty infabrication of small-scale devices. According to [37] and [36], the tolerances in MEMS devices as a resultof fabrication uncertainties can be as high as ±10% of the nominal value. As verified by the finite-element-analysis results for miniature devices in [70] and the experimental results on large-scale prototypes in [65],variations in beam width (see Fig. 5.3) will cause stiffness change of the LPs, and thus the variation of theirnatural frequencies.635.2. Conceptual Miniaturized OIS and Large-Scale PrototypesFor each OIS, the frequency response can be measured and utilized to identify the natural frequencies,which can be further adopted for controller adjustment. However, due to the extremely low damping of theOIS system, it is difficult to accurately estimate the natural frequencies. This will cause the second challenge,namely, the uncertainties of the identified natural frequencies.The third challenge is the translational force (denoted by the red arrow in z-direction in Fig. 5.3) as theresultant force of the unbalanced forces from the four actuators, which may excite the translational mode ofthe LP and lead to performance degradation of the OIS control system. The force unbalance may result fromuncertainties in fabrication and installation of the electromagnetic actuators.Figure 5.3: Demonstration of beam width variation and unbalanced forces (δb and δF,i represent the devia-tions from the nominal beam width b and nominal force output F , respectively.)As opposed to the previous work, these three challenges are explicitly taken into account in the proposedmodeling and controller design.5.2.4 Large-Scale OIS PrototypesSince the fabrication of miniaturized OIS’s involves non-trivial issues which require careful thoughts, e.g.,in material selections and fabrication methods, the large-scale prototypes developed in [65] were utilizedfor experimental validations in this chapter. Testing of the proposed methods in micro-scale experiments isleft as a future work. Although the focus will be put on the large-scale OIS prototypes hereafter, since theaforementioned challenges in control of the miniaturized devices are mimicked in the large-scale prototypes,it is expected that the proposed modeling and control techniques will be valid and similar conclusions willhold for miniaturized OIS’s.One large-scale prototype is shown in Fig. 5.4a. The LP made of 17-7 stainless steel was fabricated usinga water-jet cutter, and the LP and coils were installed on an aluminum base with screws for adjusting theheight of coils. To mimic the beam dimension variation in the micro-scale devices, we produced five LPswith different beam widths, as shown in Fig. 5.4b. The designed values of the beam widths of LP 1–LP 5differ from the nominal value of 1.5 mm by −10%, −5%, 0%, +5%, and +10%, respectively. Variationsin fabrication and installation of magnetic actuators are also inevitable in the prototypes due to the non-uniformity of hand-made coils and misalignment of permanent magnets. Parameter values of the magneticactuators in the prototypes can be found in [65], and they are also given in Table C.1 in C.1 for interestedreaders.645.3. Feedback Control StructureMagnetsCoilsBaseLens Platform (LP)60 mmDifferent Beam WidthLP 1 LP 2LP 3LP 4 LP 5(a) Components of one prototypeMagnetsCoilsBaseLens Platform (LP)60 mmDifferent B am WidthLP 1 LP 2LP 3LP 4 LP 5(b) Five LPs with different beam widthsFigure 5.4: Large-scale 3-DOF prototypes5.3 Feedback Control StructureTo compensate for pitch and yaw vibrations of the LP, the lens needs to be tilt around x- and y-axis, re-spectively. For an ideal case with perfectly symmetric LP and folded beams, and perfectly manufacturedand aligned magnetic actuators, finite element analysis (FEA) confirms that an actuation to one of the tworotational DOFs will not affect the other one. In addition, experimental results in [65] indicate that, evenfor non-ideal cases where the imperfection of fabrication and installation exists, the influence of one DOFactuation on the other DOF is negligible. Consequently, the pitch and yaw control can be considered to bedecoupled, and thus each controller can be designed for each of two rotational DOFs independently.Figure 5.5 depicts the feedback control structure for the OIS, where φrx and φry denote the desiredpitch and yaw angles, while φx and φy are the angles calculated using the displacement measurement inz-direction. Pitch and yaw controllers determine the desired torques Trx and Try respectively based on thecorresponding tracking errors. The currents are calculated from the desired torques via a current calculator,and applied to the four magnetic actuators to generate the actual torques Tx and Ty. The current calculatorutilizes the relation between the torque (Trx and Try) and the magnetic force generated by each actuator, aswell as the relation among magnetic force, air gap and current for each actuator. This relation was identifiedfor large-scale prototypes using a software package, Finite Element Method Magnetics (FEMM) [75], asexplained in C.1. As pitch and yaw control can be decoupled, and as the two control problems are essentiallythe same, only the modeling and control for pitch motion will be explained henceforth.As mentioned before, the displacement measurement is necessary for reconstruction of the rotation an-gles. For the large-scale prototypes, the single-point measurement strategy was utilized mainly because thelaser Doppler vibrometer (LDV) sensor as a part of our experimental setup (see Fig. 5.9 in Section 5.6) canonly measure the displacement of a single point at a time. Due to this limit, all of the experiments presentedin this chapter are based on the single-point measurement. However, an experiment presented in C.2 verified655.4. Modeling of Large-Scale OIS PrototypesLens platformOIS controllerpitch controlleryaw controller__Current calculatorMagneticactuatorsFigure 5.5: Block diagram of the OIS controllerthat, during pitch actuation, the magnitude of the translational motion is rather small compared to that of therotational motion. Therefore, the single-point measurement technique can provide rather accurate estimationof the the rotation angle for feedback use with negligible influence on the image quality. Actually, as men-tioned before, the single-point measurement is even preferable from an economic viewpoint due to reductionof the sensor cost.5.4 Modeling of Large-Scale OIS PrototypesTo derive a mathematical model of the large-scale OIS prototypes, the frequency response of all the proto-types was measured with a dynamic signal analyzer, and is shown in Fig. 5.6a, where the input is the torquecommand Trx and the output is the LP angle φx. Note that φx was calculated based on the displacementmeasurement of a point on the LP in z-direction.As observed in Fig. 5.6a, the flexible LP has several natural frequencies, each associated with one vi-bration mode. The first two modes (below 200 Hz) correspond to two DOFs of the LP, i.e. translation andpitch, while higher frequency modes (above 200 Hz) are from the folded beams, which have infinitely manyvibration modes as continuous systems. Note that the mode corresponding to the yaw motion did not appearin the frequency response data since only the pitch motion was intentionally excited.Ideally, the commanded torque Trx should not generate the z-direction force in the center point of theLP. This may not be true in reality due to the imperfection of the fabrication and installation of the magneticactuators (the third challenge in Section 5.2.3), as well as the estimation error of the relation among the airgap, the current, and the magnetic force. The directional force will induce a translational motion in z direc-tion. This undesired translational motion is negligible from the perspective of rotation angle reconstructionas verified by the experiment in C.2. Nevertheless, considering the gain and phase distortion caused bythe translational motion as seen in Fig. 5.6a, it is beneficial to explicitly consider the translational motionin both modeling and controller design for guarantee of the robust stability. In fact, even with a robust µ-synthesis controller that was designed based on an uncertain model consisting of only the second mode (i.e.,a model which will appear later in (5.1) but with i = 2), the closed-loop system often became unstable in theexperimental validation.665.4. Modeling of Large-Scale OIS Prototypes101 102 103-50050100LP 1LP 2LP 3LP 4LP 5101 102 103Frequency (Hz)-400-300-200-1000 LP 1LP 2LP 3LP 4LP 5(a) Measured frequency response of different LPs101 102 103-50050100101 102 103Frequency (Hz)-400-300-200-1000(b) Frequency response of sampled modelsFigure 5.6: A comparison between measured frequency response and frequency response of the model set(5.4)To consider the translational motion in the modeling and controller design, the fact was utilized that thetranslational motion contributes to the tilting angle measurement φx, due to the way of angle measurementexplained in Sections 5.2.1 and 5.3. The level of this contribution may depend on various factors, such asthe attitude of the mobile phones, desired tilting angles, and imperfection level of actuator fabrication andinstallation. Thus, this contribution is modeled as a time-varying gain uncertainty.On the other hand, due to the imperfection and estimation error related to the actuators, the commandedtorque Trx may not be equal to the generated torques Tx. It is possible to express the discrepancy betweenTrx and Tx as an uncertain gain for the pitch mode in the mathematical modeling. However, extensive ex-periments indicated that the potential discrepancy between Trx and Tx has much less crucial influence onthe control system performance, compared to the gain uncertainty for the translational mode and measure-ment uncertainties of the natural frequencies. Therefore, to simplify the modeling and controller design, it isassumed that Trx ≈ Tx.5.4.1 Transfer Function Model for a Single OISTo simplify the pitch controller design, a simple model that captures only the first two modes is adopted.Higher frequency modes above 200 Hz will be dealt with via a band-stop filter, for mainly two reasons.First, the higher frequency modes are difficult to model as accurately as the lower frequency ones. Second,applying a band-stop filter on the frequency range above 200 Hz has negligible influence on the trackingperformance below 10 Hz. The details for design of the band-stop filter are given in C.1.A model structure to capture the first two modes of the dynamics from Trx to φx was proposed in [65] asP (s) := e−τds2∑i=1Kiω2is2 + 2ζiωis+ ω2i, (5.1)675.4. Modeling of Large-Scale OIS Prototypeswhere τd represents the delay time in the system, mainly from the delay in the driving circuits, and Ki,ωi and ζi are the DC gain, natural frequency, and damping ratio of the LP, respectively, corresponding tothe i-th mode (i = 1: translation, i = 2: pitch). Note that, although the output is pitch angle, the model(5.1) includes the translational mode. This is because the translational motion of LP as a result of the forceunbalance contributes to the pitch angle calculated based on the measured translational displacement of onepoint on the LP.5.4.2 Model Set for Multiple OIS’sOne can see in Fig. 5.6a the variation of the frequency response of LP 1–LP 5. Specifically, the variationmainly appears in the natural frequencies corresponding to the first and second modes as well as the low-frequency gain.The primary sources of the dynamics variations are beam dimension (including width, thickness andlength) variations, and variations in fabrication and installation of magnetic actuators. The former willchange the stiffness of LP and subsequently the natural frequencies ω1 and ω2 in (5.1), while the latter willlead to different levels of force unbalance, characterized by different K1 values in (5.1). Therefore, thedynamics variation due to product variations can be represented in terms of the following parameter vector:θ := [ω1 ω2 K1]T . (5.2)The lower and upper bounds of θ can be estimated from the frequency response data, and denoted by the setΘ :=θ :ω1 ∈ [ω1, ω1]ω2 ∈ [ω2, ω2]K1 ∈[K1,K1] . (5.3)5.4.3 Identified ParametersThe identified parameter values for the large-scale OIS prototypes, with the model structure (5.1) and theparameter set (5.3), are listed in Table 5.1, where the range of varying parameters are selected to cover allthe five plants.Table 5.1: Identified parameter valuesParameter Value Unit[K1,K1] [−0.2, 0.2] —K2 4.113 —[ω1, ω1] [61, 71] Hz[ω2 ω2] [95 108] Hzζ1 5× 10−3 —ζ2 5× 10−4 —τd 0.0011 sec685.4. Modeling of Large-Scale OIS PrototypesTo verify the identified parameters in Table 5.1, the frequency response of five linear time-invariantmodels obtained by randomly sampling the parameter values in (5.3) are plotted in Fig. 5.6b. A comparisonbetween Fig. 5.6a and Fig. 5.6b, shows that the first and second modes of the real system are well capturedin the model.5.4.4 State-Space Model RepresentationFor subsequent controller design, the transfer function model (5.1) needs to be transformed to a finite-dimensional state-space model. With the parameter vector (5.2) and first-order Padé approximation of thetime-delay term e−τds in (5.1), the model (5.1) can be converted into a state-space model parameterized byθ asP (θ) :[x˙Pφx]=[AP (θ) BPCP (θ) 0][xPTrx], (5.4)where xP ∈ R5 is the state vector, andAP (θ) :=0 ω1 0 0 0−ω1 −2ζ1ω1 0 0 4/τd0 0 0 ω2 00 0 −ω2 −2ζ2ω2 4/τd0 0 0 0 −2/τd ,BP :=[0 −1 0 −1 1]T,CP (θ) :=[K1ω1 0 K2ω2 0 0].5.4.5 Parameter UncertaintiesFor each OIS, the value of ωi can be estimated by inspecting the measured frequency response and locatingthe frequencies corresponding to peaks in the frequency response. However, the low damping of the OISplants will lead to sharp peaks in the frequency response, which makes the natural frequencies difficult toidentify accurately due to the limited resolution of measurement devices. Hence, the actual parameter ωi isassumed to be estimated asωˆi = ωi + δi, i = 1, 2, (5.5)where δi represents the estimation uncertainty of ωi. The bounds of δi are estimated for subsequent controllerdesign, and are presumed to satisfyδ1 ∈ [δ1 δ1], δ2 ∈ [δ2 δ2]. (5.6)Similarly, in order to obtain the value of K1, parameter identification is needed, in which identificationerror of K1 is inevitable. Furthermore, the level of force unbalance may change with the attitude of mobiledevices due to the influence of gravity, as well as the magnitude of tilting angles, which will lead to time-varying K1 value even for a fixed OIS product. Because of the identification error and its time-varying695.5. Multiple Parameter-Dependent Robust Control of OIS’sproperty , the parameter K1 will be treated as an uncertain time-varying parameter. The nominal value ofK1, denoted by Kˆ1, is set to be zero, corresponding to the perfectly manufactured case where the forceunbalance and the resulting translation of the LP during the LP tilting do not exist. The value Kˆ1 will beneeded for controller reconstruction (see (C.8) for details). For later use, the measured value of the vector θis denoted asθˆ := [ωˆ1 ωˆ2 Kˆ1]T , Kˆ1 = 0. (5.7)To sum up, there are two kinds of uncertainties considered in the following controller design. One is thetime-invariant uncertainties δi, i = 1, 2 in (5.6), whereas the other corresponds to the time-varying uncertainparameter K1 in (5.3). The controller to be designed is required to be robust against both of these two kindsof uncertainties.5.5 Multiple Parameter-Dependent Robust Control of OIS’sTo accomplish the control objectives in Section 5.2.2 for the model represented by (5.4) with the inaccurateestimation ωˆi in (5.5) and the time-varying uncertain gain K1 in (5.3), the authors propose to design themultiple parameter-dependent robust (MPDR) controllers. To be more specific, we divide the parameter setΘ intoN subsets {Θ(j)}Nj=1 and designN -number of parameter-dependent robust controllers {K(j)(θˆ)}Nj=1,where the local controller K(j) takes care of the subset Θ(j). Before presenting the details of the MPDRcontrollers design, it is necessary to first explain how to implement such controllers.5.5.1 Implementation SchemeEach local controller K(j)(θˆ) taking care of the subset Θ(j) is parameterized by the estimated parametervector θˆ. For each specific product with an estimated θˆ, the corresponding local controllerK(j)(θˆ) is actuallya linear time-invariant (LTI) controller, and is designed to be robust against the time-invariant uncertaintiesδi in (5.6) and the time-varying uncertainty of K1 in (5.3).After designing the MPDR controllers, the process for determining an LTI robust controller Kk for eachOIS product k is illustrated in Figure 5.7, and described as follows. First, estimate the vector θˆ in (5.7) byinspecting the natural frequencies ωi, i = 1, 2. Second, identify the subset Θ(j) which the estimated vectorθˆ belongs to. Finally, based on the identified index j, select the corresponding controller parameterizationK(j)(·) and determine the LTI robust controller as Kk := K(j)(θˆ).5.5.2 MPDR Controllers DesignDivision of parameter setFor design of the MPDR controllers, first divide the parameter set Θ into N -number of hyper-rectangularsubsets by partitioning the range interval of ωi into Ni sub-intervals for i = 1, 2. Obviously, N = N1 ×N2.The number Ni (and thus N ) were determined heuristically. Roughly speaking, the larger N is, the betterthe controllers’ performance will be, but the larger the computational cost for obtaining the controllers willbe as well. To be more precise, the performance of the MPDR controllers depends on not only N but also705.5. Multiple Parameter-Dependent Robust Control of OIS’s.−Controller determinationParameter estimationp…….Figure 5.7: Implementation of the MPDR controllers for a specific product for pitch controlthe partition of Θ. Optimal partition of Θ is not pursued in this chapter, but was investigated in Chapters 3and 4 and the paper [35]. The algorithm based on particle swarm optimization proposed in Chapter 4 canbe applied to optimize the partitioning of Θ to further improve the performance of the MPDR controllers.However, since optimal partition is not the focus of this chapter, it is omitted for the sake of brevity.Formulation of a generalized LPV plantGiven the partition of the parameter set, the method for designing an SLPV controller under inexact mea-surement of scheduling parameters presented in Chapter 2 will be utilized to design the MPDR controllers,by regarding the model P (θ) and the estimated vector θˆ as an LPV system and an inexactly measured gain-scheduling parameter vector, respectively. For applying the SLPV controller design method, a generalizedLPV plant is needed, which is obtained as follows.For pitch controller design, the block diagram in Fig. 5.7, in combination with weighting functions Weand Wu for e and u, is transformed into a block diagram depicted in Fig. 5.8. A dynamic function for Wewas selected in the form ofWe(s) =s/MH + ωbs+ ωbML, (5.8)to minimize the tracking error below 10 Hz. In (5.8), MH , ML and ωb are respectively the high-frequencygain, the low-frequency gain and the crossover frequency of W−1e [76]. A static gain was used for Wu forsimplicity to avoid control input saturation. In Fig. 5.7, G(θ) is the generalized LPV plant:G(θ) :x˙ = A(θ)x+B1(θ)w +B2u,z = C1(θ)x+D11(θ)w +D12(θ)u,y = C2x+D21(θ)w,(5.9)where θ is given in (5.2), the control input u := Trx is the desired pitch torque (assumed to be equal to the715.5. Multiple Parameter-Dependent Robust Control of OIS’sactual torque Tx), w := φrx is the desired pitch angle, y := e˜ is the tracking error, and z := [ze, zu]T =[Wee,Wuu]T ∈ R2 is the vector of output signals to be minimized. The matrices in (5.9) have compatibledimensions. The vector x ∈ Rn is the state vector with n := 5 + nWe + nWu + nf = 7, where nWe andnWu (= 0) are the number of states in We and Wu, whereas nf (= 1) is the number of states in the filterf(s) := 10.001s+1 for filtering the error signal in order to lead to constant (i.e. parameter-independent) B2and C2. An assumption of constantB2 and C2 is needed for reconstruction of controller matrices (see (C.8)).For the details of how filtering leads to constant B2 and C2, see [14, p. 1255]). Since the cutoff frequencyof the filter is much higher than the bandwidth of the system to be controlled, it has negligible effects on theperformance of the closed-loop system.−Figure 5.8: A synthesis structure with the generalized plant and MPDR controllersDesign of the MPDR controllers utilizing the SLPV controller design methodFor the LPV system (5.9), the MPDR controllers to be designed are a family of full-order output-feedbackcontrollers in the form of [x˙Ku]=[A(j)K (θˆ) B(j)K (θˆ)C(j)K (θˆ) D(j)K (θˆ)]︸ ︷︷ ︸=:K(j)(θˆ)[xKy], j ∈ ZN , (5.10)where xK ∈ Rn is the controller state vector, and K(j)(θˆ) ∈ R(n+1)×(n+1) is a local controller parameteri-zation in charge of the subset Θ(j).With the generalized LPV plant (5.9), the method for SLPV controller design under inexactly measuredscheduling parameters in Chapter 2 can be readily trimmed to design the MPDR controllers by consideringthe fact that ωi (i = 1, 2) are time-invariant and the assumption that Kˆ1 is equal to 0. The details are givenin C.3 for interested readers. Specifically, the state-space matrices of the controller (5.10) are determined byequations (C.8) and (C.9) in C.3.725.6. Experimental Validation5.6 Experimental ValidationIn this section, the proposed MPDR controllers are experimentally validated on the large-scale prototypes.5.6.1 Experimental SetupThe experimental setup is shown in Fig. 5.9. A dSPACE DS1003 platform is used for implementation of allthe control algorithms and runs at a sampling frequency of 2 kHz. The DC power supply is for supplyingpower to the amplifier, which converts the control voltages from dSPACE controller to the currents appliedto the coils of the magnetic actuators. The displacement of one point on the LP is measured using a laserDoppler vibrometer (LDV) and converted to the rotation angle of the LP for feedback use.dSPACEController AmplifierLaser Doppler Vibrometer (LDV)DC Power SupplyOIS Prototype Figure 5.9: Experimental setup5.6.2 Assumptions for Controller DesignFor controller design and validation, it was assumed that ω1 was accurately measured, and ω2 was mea-sured with an uncertainty bounded by 1 Hz, which was estimated based on the frequency resolution of themeasuring equipment near the measured natural frequency. Accordingly, the parameters in (5.6) were set tobeδ1 = δ1 = 0, δ2 = −δ2 = 2pi · 1. (5.11)As mentioned in Section 5.5.2, for MPDR controller design, Θ is divided into N -number of hyper-rectangular subsets by partitioning the range interval of ωi, i = 1, 2. To balance the computational costand the controller performance, the values N1 = N2 = 2 were selected. Different partitioning of the rangeinterval of ω1 and ω2 were manually tested, and the one yielding the smallest cost function in design of theMPDR controllers (see C.3) was selected. The resulting subsets are shown in Fig. 5.10.For minimizing the tracking error below 10 Hz without causing control input saturation, the weightparameters were tuned and finally fixed to the values listed in Table 5.2.735.6. Experimental Validationw2 (rad/s) 21r · 108 -------e(2) e(4)21r · 103 -------e(1) e(3)21r · 95 -------,-21r · 61 21r · 65 21r · 71 w1 (rad/s)Figure 5.10: Subsets from partitioning the set of the natural frequenciesTable 5.2: Weights parametersParameter MH ML ωb WuValue 3 0.028 750 rad/s 0.1155.6.3 Designed controllersFor comparison purpose, efforts were made to design five controllers, denoted by Kcla, Kµ, KS , KM andKM24. Properties of each controller are summarized in Table 5.3, and explained next.Table 5.3: Designed controllersControllerParameterdependentRobustfor δ2Robustfor K1Kcla No – –Kµ No – YesKS Yes (single) Yes YesKM Yes (multiple) Yes YesKM2 Yes (multiple) No Yes• Kcla is a classical controller consisting of a lead-lag compensator and notch filters, designed for thenominal plant LP 3, presented in [65] with detailed parameters. In other words, the uncertainties ofω1, ω2 and K1 are not explicitly considered in designing Kcla.• Kµ is a µ-synthesis robust controller designed by treating all of ω1, ω2 and K1 as time-invariantuncertain parameters, also presented in [65].• KS is a single parameter-dependent robust controller, which can be considered as a specific case ofKM in the sense that N = 1, i.e. the natural frequency set in Fig. 5.10 is not partitioned.• KM are MPDR controllers proposed in this chapter, which consider the uncertainties of the parametersω1, ω2 and K1. However, instead of treating all the parameters as uncertain parameters like Kµ, it4The controller parameters along with the frequency response data and the Matlab codes can be downloaded at http://cel.mech.ubc.ca/software-files/.745.6. Experimental Validation100 101 102 103 104Frequency (Hz)-100-80-60-40-20020Magnitude (dB)Figure 5.11: Frequency response of different controllers including the band-stop filter for LP 3assumes ω1 and ω2 are measurable, and additionally considers the uncertainty δ2 in measurement ofω2.• KM2 are MPDR controllers similar to KM except that exact measurement of ω2 is assumed in designof KM2. In other words, δ2 = δ2 = 0 is enforced in design of KM2 via the proposed method.In the attempt to design KS , the authors were not able to find a feasible solution for the correspondingoptimization problem presented in C.3, due to the relatively large product variation shown in Fig. 5.10. Thisinfeasibility issue demonstrated the necessity of MPDR controllers.As mentioned before, a band-stop filter will be used to suppress the magnitude of the high frequencymodes of the LP over 200 Hz, which are not considered in the mathematical modeling and robust controllerdesign. Details of the band-stop filter are given in C.1.2. All of the aforementioned controllers except Kclaare connected with the band-stop filter in series for experimental verification. Hereafter, we use Kµ, KS ,KM and KM2 to denote the series-connected version of the aforementioned corresponding controllers andthe fourth-order band-stop filter.5.6.4 Frequency-Domain AnalysisThe frequency response of the aforementioned controllers for LP 3 (nominal plant) are given in Fig. 5.11.It can be seen that, by virtue of parameter dependency, KM achieves a higher low-frequency gain than Kµ,which corresponds to the better tracking performance. Another parameter-dependent robust controller KM2is even more aggressive than KM due to ignorance of measurement uncertainties δ2. However, it will beshown in the next sub-section that KM2 cannot stabilize the closed-loop system for any of the five plants.5.6.5 Robust Stability and Time-Domain Tracking PerformanceTo compare the performance of the designed controllers, both the stability and the tracking performance wereexperimentally tested for all the five plants. For the test, a summation of sinusoidal signals with frequenciesranging from 3 Hz to 10 Hz was used as the reference angle. The stability test results for different plants and755.6. Experimental Validationcontrollers are shown in Table 5.4, whereXand×mean that the closed-loop system was stable and unstable,respectively. This table also exhibits the order of designed controllers. Note that all the controllers listed inTable 5.4 contain the fourth-order band-stop filer. From Table 5.4, one can see that only Kµ and KM couldstabilize all the plants; even KM2 failed to stabilize any of the plants due to disregard of the measurementuncertainty of ω2.Table 5.4: Closed-loop stability and controller orderKcla Kµ KM KM2Order 10 17 11 11LP 1 × X X ×LP 2 × X X ×LP 3 X X X ×LP 4 X X X ×LP 5 × X X ×The tracking performance and control effort are quantitatively evaluated using root-mean-squared (RMS)value of tracking error and integration of absolute current value over time, respectively. The comparison ofthese two properties between different controllers is shown in Fig. 5.12. Compared with Kµ, the controllerKM offers over 27% improvement in the tracking performance at the price of 5%–12% increase in controleffort. Moreover, the order of KM is much lower than that of Kµ, which is desired for practical implemen-tation on mobile platforms.LP 1 LP 2 LP 3 LP 4 LP 500.050.1degRMS value of tracking error K7 KM KclaLP 1 LP 2 LP 3 LP 4 LP 500.10.20.3mA"hIntegral of absolute current value over timeK7 KM KclaFigure 5.12: Quantitative comparison of different controllers. Top: tracking performance, bottom: controlinputFinally, the tracking performance of three stabilizing controllers for three of the five plants is plottedin Fig. 5.13. One can observe that the proposed MPDR controllers KM always yields smaller trackingerror than the classical controller Kcla and the conventional robust controller Kµ. On the other hand, theincreased oscillations associated with Kcla indicated that the closed-loop system corresponding to LP 1cannot be stabilized by Kcla.765.6. Experimental Validation0 0.2 0.4 0.6 0.8 1Time (s)-101Rot. angle(deg)0 0.2 0.4 0.6 0.8 1Time (s)-0.200.2Error (deg)(a) LP 30 0.2 0.4 0.6 0.8 1Time (s)-4-2024Rot. angle(deg)0 0.2 0.4 0.6 0.8 1Time (s)-101Error (deg)(b) LP 10 0.2 0.4 0.6 0.8 1Time (s)-101Rot. angle(deg)0 0.2 0.4 0.6 0.8 1Time (s)-0.4-0.200.2Error (deg)(c) LP 5Figure 5.13: Tracking performance of different controllers. Reference (solid blue, only in the top sub-figure),Kcla (dash-dot black), Kµ (solid green), KM (dash-dot red)775.7. Conclusion5.7 ConclusionIn this chapter, by applying the approach proposed in Chapter 2, we designed the so-called multiple parameter-dependent robust (MPDR) controllers for mass-produced miniaturized optical image stabilizers (OIS’s). Thedynamics of batch-fabricated OIS’s with inevitable product variations was represented by a set of linearmodels, parameterized by two product-dependent natural frequencies and one uncertain gain. Due to thedifficulty in accurately measuring the natural frequencies for each OIS product, they were assumed to bemeasured with errors. The proposed controllers were designed to be parameter-dependent on the measurednatural frequencies, as well as to be robust against both estimation errors of the natural frequencies andthe gain uncertainty. By using the measured natural frequencies for gain scheduling, the proposed MPDRcontrollers achieved noticeably better performance than a conventional robust controller, which was experi-mentally verified on large-scale prototypes. Experimental results also indicated that explicit consideration ofthe measurement uncertainties of the scheduling parameters is crucial for robust stability guarantee for thisapplication.It is worth mentioning that implementation of the proposed controllers necessitates measurement of thefrequency response for each OIS product. This is the trade-off for the improved performance. Our futurework includes experimental validation of the proposed controller on mass-produced micro-scale devices withmore realistic product variations.78Chapter 6Application 2: Floating Offshore WindTurbine6.1 IntroductionWind power has been the world’s fastest-growing renewable energy source in recent years [38], and has greatpotential to replace non-renewable fossil fuels and nuclear power, since it is indigenous, inexhaustible andnon-polluting [39]. One of the recent trends in the wind energy industry is to place large turbines offshore,where the wind is stronger and steadier and vast open space is available for building large wind farms [77].From the economic perspective, research has shown that the average offshore wind speed offshore can be90% greater than onshore [78], allowing more power capture; reduced spatial variation of wind speed de-creases structural loads, reducing the maintenance cost and extending a turbine’s lifespan. To further improvethe profitability of offshore wind farms, it is desirable to place them far away from the coastline in deep water,where the water depth is larger than 30 m [39]. While offshore turbines can be fixed to the seabed underneathlike the onshore ones, floating platforms are needed for deep-water turbines as fixed bottom structures areeither practically infeasible or uneconomical as the water depth continues to increase [79]. Fig 6.1 presentsdifferent types of offshore wind turbine foundations and the associated water depth. For economic advan-tages, the capacity of future floating wind turbines are expected to be 5 MW or larger [77, 80]. For offshoreturbine development, a 5 MW baseline wind turbine was proposed by the National Renewable Energy Labo-ratory (NREL) in Colorado [81] and is so far the most widely used turbine system in the literature. Hereafterwe denote this wind turbine as NREL 5 MW baseline wind turbine. Several floating platforms for offshorewind turbines have been proposed or adapted from the oil and gas industry including the tension-leg platform(TLP), semi-submersible platform, and spar buoy platform.Research and development in offshore wind turbines need dynamic models to predict the behaviors ofthe turbine system in various scenarios. Different models have been proposed for floating offshore windturbines with a range of modeling fidelity and mathematical complexity. The most well-known model isa nonlinear model developed at NREL [82] and implemented in the open-source software named Fatigue,Aerodynamics, Structures, and Turbulence (FAST) [83]. FAST is capable of modeling the physical dynamicswith high accuracy for both onshore and offshore wind turbines but its high complexity makes it not suitablefor controller design. A number of control-oriented models have been proposed as well. A simple 2-Dcontrol-oriented model consisting of only 7 states was proposed by Betti [84] for a 5 MW turbine on atension-leg platform. The model is developed based on the assumption that the turbine moves only in a 2-Dplane that is aligned with the wind direction. This model was validated by comparison with FAST and was796.1. Introduction20 Deep Water - The next step for offshore wind energyFIgURE 11 oFFShoRE WIND FoUNDATIoNS Source: Principle Power2011 was the best year on record for deep offshore development with two floating substructures tested, SeaTwirl and SWAY, in addition to the grid connected Windfloat project.Currently, offshore wind farms have been using three main types of deep offshore foundations, adapted from the offshore oil and gas industry:• Spar Buoy: a very large cylindrical buoy stabilises the wind turbine using ballast. The centre of gravity is much lower in the water than the centre of buoy-ancy. Whereas the lower parts of the structure are heavy, the upper parts are usually empty elements near the surface, raising the centre of buoyancy. The Hywind concept consists of this slender, ballast-sta-bilised cylinder structure.• Tension Leg Platform: a very buoyant structure is semi submerged. Tensioned mooring lines are at-tached to it and anchored on the seabed to add buoyancy and stability. • Semi-submersible: combining the main principles of the two previous designs, a semi submerged struc-ture is added to reach the necessary stability. Wind-Float uses this technology.The table 1 outlines the deep offshore wind designs and projects developed in Europe, Japan and the US: 2.1 State of the artThe concept of a floating wind turbine has existed since the early 1970s, but the industry only started researching it in the mid-1990s. In 2008, Blue H technologies installed the first test floating wind turbine off the Italian coast. The turbine had a rated capacity of 80 kW and after a year of test-ing and data collection it was decommissioned. A year later the Poseidon 37 project followed, a 37m-wide wave energy plant and floating wind turbine foundation tested at DONG’s offshore wind farm at Onsevig. In 2009, Statoil installed the world’s first large scale grid connected floating wind turbine, Hywind, in Nor-way, with a 2.3 MW Siemens turbine. The second large scale floating system, WindFloat, developed by Principle Power in partnership with EDP and Repsol, was installed off the Portuguese coast in 2011. Equipped with a 2 MW Vestas wind turbine, the installation started producing energy in 2012.chapter 2: the introduction of deep offshore designsMonopile0-30m, 1-2 MWJacket/Tripod25-50m, 2-5 MWFloating Structures>50m, 5-10 MWFloating Structures>120m, 5-10 MWTlP Semi-Sub SparFigure 6.1: Different types of foundations for offshore wind turbines (Source: Principle Power). TLP:tension-leg platformshown to be able to capture the major dynamics. To more thoroughly capture the dynamics, a 3-D control-oriented model was proposed for a 5MW wind turbine on a semi-submersible platform [40]. A rigid-bodyassumption for the wind turbine and its floating foundation and first principles was based to derive the model.The resulting nonlinear model consists of 15 states and is able to capture the major dynamics of the turbinesystem in the 3-D space with satisfactory accuracy as validated by FAST.In floating offshore wind turbine systems, feedback control plays a significant role in improving turbineperformance [77], and has been investigated by many researchers. In [81], a baseline controller was proposedfor the NREL 5 MW baseline wind turbine. The baseline controller consists of a generator torque controllerand a blade pitch controller that work independently. The generator torque controller tries to maximize powercapture below the rated operating point while above the rated operating point the generator torque is inverselyproportional to the generator speed so that the generator power is held constant. The blade pitch controllerregulates the generator speed using a gain-scheduled proportional-integral (GSPI) controller above ratedoperating point. Although the baseline controller may work well for onshore and fixed-foundation offshorewind turbines, it is not adequate for floating offshore wind turbines. This is because the baseline controllerdid not account for platform oscillation and can easily induce negative damping for the platform pitch [85],leading to large fatigue loads on the tower due to amplifying platform pitch motion.Control techniques that try to reject the wind disturbance and mitigate the platform oscillation for float-ing offshore wind turbines were also extensively investigated. In [86], a combined feedforward-feedbackcontrol strategy was proposed for generator speed regulation and platform motion suppression above ratedoperating point, where the feedforward part is based on measurement of the wind disturbance using Lidar(light detection and ranging) technology. A periodic state-space controller utilizing individual blade pitch(IBP) control was proposed in [87] to improve power and platform pitch regulation above rated wind speed.According to the authors, by independently controlling the blade pitch angle of each blade to adjust the806.1. Introductionthrust force applied to each blade, a restoring moment can be generated to effectively reduce platform pitch-ing. In [88], disturbance accommodating control using IBP was proposed to reject the wind disturbances,where a disturbance estimator is designed to estimate the wind disturbance assuming a uniform step changein the wind velocity.There are also some attempts to reject the wave disturbance in controller design. In [89], a referencemodel-based predictive control was presented to reject the wave disturbance. In the proposed method, aclosed-loop system consisting of a floating wind turbine model ignoring the wave disturbance and a basiccontroller is set as the reference model, which is used to generate the reference state trajectory. A modelpredictive controller (MPC) was designed to control the blade pitch to follow the reference trajectory. Inthis way, the influence of the incident waves is reduced in the sense that, the actual behavior of the turbinesystem is controlled to track the behavior of the same system as if there is no wave disturbing it. In [84],a H∞ controller was proposed to reject both wind and wave disturbance for control of the NREL 5 MWbaseline wind turbine on a tension-leg platform. The controller was designed based on a 2-D control-orientednonlinear model developed in the same work, which was able to produce wind and wave disturbance matriceswhen this model is linearized. Weighting functions were added to reject the wind and wave disturbance inspecific frequency ranges where the power of wind and wave disturbances are dominant. The designedH∞ controller was compared with the baseline controller in FAST. However, although the designed H∞controller achieved a big improvement in generator speed regulation, the performance in power and platformpitching regulation did degrade compared to the baseline controller.Most of the aforementioned work on floating wind turbine control focuses on control of the turbinesystem at an operating point using an LTI model, which is obtained through linearization of a nonlinear model(either FAST or control-oriented models) at this point. Control of the floating wind turbine in the wholeoperating range was also explored. Bagherieh and Nagamune [90] proposed GS linear-quadratic-regulator(LQR) and LPV controllers for the NREL 5 MW wind turbine on a barge platform above rated operatingpoint. The GS LOR controllers were designed utilizing a family of LTI models from FAST linearizationwhile an LPV model was first obtained by interpolating those LTI models and used for LPV controllerdesign . Simulations in FAST verified that the designed controllers especially the GS LPV controller withstate feedback yielded substantial improvement over the baseline controller in both power regulation andplatform pitch motion reduction. Despite the promising performance displayed in this work, there is somespace for improvement. First, the designed controllers are not validated using wind profiles covering thewhole range of above rated wind speed. Second, the designed LPV controllers use wind speed as schedulingparameters; however, the inevitable measurement inaccuracy of the wind speed is not investigated.This chapter presents SLPV control of the NREL 5 MW wind turbine on a semi-submersible platformabove rated wind speed with explicit consideration of the plant dynamics variations with respect to the vary-ing wind speed. By linearizing a control-oriented 3-D nonlinear model developed in [40] at a series ofoperating points above rated wind speed, a family of LTI models are obtained and used to directly designSLPV controllers. SSs are also optimized in SLPV controller design utilizing the algorithm developed inChapter 4. The uncertainty in measurement of the wind speed is also investigated in both simulations andcontroller design. The SLPV controllers are validated in medium-fidelity simulations based on FAST using816.2. System Descriptionrealistic wind profiles covering the whole range of above rated wind speed. Simulation results demonstratethe superior performance of the SLPV controllers in power regulation and platform motion mitigation, com-pared to other controllers including the baseline controller, LPV controllers, and an H∞ controller.6.2 System Description6.2.1 5 MW Baseline Wind Turbine SystemThe reference wind turbine system used in this chapter is the NREL 5 MW baseline wind turbine with asemi-submersible floating platform [81, 91]. Hereafter, we use WTS to denote the system consisting of thethe wind turbine and the floating platform. The layout and dimensions of this WTS are illustrated in Fig. 6.2.The floating platform consists of three columns, each of which is fastened to the seabed by a mooring line.The 3-blade rotor is connected to the generator through a gearbox inside the nacelle box. The control inputsof the system include the collective blade pitch angle (β), the generator torque (Tg) and the nacelle yawangle (γn). The blade pitch angle is the rotation of a blade along its longitudinal axis and is used to adjust theaerodynamic power and thrust force, the generator torque controls the generator speed and extracted power,while the nacelle yaw angle determines the relative angle of the rotor plane to the wind direction.Water Level~70~1500Wind DirectionBuoyancy ColumnsMooring LinesNacelleβXYFront View Top ViewFigure 6.2: The baseline WTS with a semi-submersible floating platformThe WTS operates in three main regions depending on the wind speed v. In Region I, the WTS isswitched off to either save the operational cost (when v < vcut-in) or to prevent from structural overload(when v > vcut-out ). In Region II where vcut-in ≤ v < vrated, the control objective is to maximize the powercapture. In Region III where vrated ≤ v ≤ vcut-out, the captured power should be regulated at the rated value.Table 6.1 lists the properties of the baseline wind turbine.826.3. Dynamics ModelingVariable Symbol ValueCut-in wind speed vcut-in 3 m/sRated wind speed vrated 11.4 m/sCut-out wind speed vcut-out 25 m/sRated electrical power Prated 5 MWRated generator speed ωg,rated 1173.7 rpmElectrical generator efficiency ηg 94.4%Gearbox ratio NGR 97Table 6.1: Properties of the NREL 5 MW baseline WTS6.2.2 Control Problem DefinitionAmong the different regions, the control problem in Region III is the most critical as negative damping ofthe platform pitch can easily arise, due to the fact that the thrust force may decrease as the wind speedincreases under a standard pitch controller [85]. Negative damping will lead to amplifying platform pitchingand increasing structural loads, and thus should be avoided. Due to its challenging property, we focus oncontrol of the WTS in Region III. Therefore, the problem to be solved can be summarized as follows.Problem 6.1. Design an SLPV controller for Region III of the reference WTS using the measured windspeed with potential measurement uncertainty such that for the whole region above rated wind speed,• the captured power and the generator speed are regulated at their rated value, and• the platform rotations especially the pitch motion are minimized.6.3 Dynamics Modeling6.3.1 FAST ModelFatigue, Aerodynamics, Structures, and Turbulence (FAST) is an open-source, highly accurate and complexnon-linear simulator for wind turbines developed at NREL in Colorado [83]. This software is capable ofmodeling the dynamics of two and three-bladed horizontal-axis wind turbines, placed both onshore and off-shore including the ones on floating platforms. FAST has been validated for several onshore wind turbinesby Germanischer Lloyd, the world’s largest renewable energy consultancy. The FAST model is developedbased on a mixture of fundamental physics law and empirical relationships, and the modeling details aredocumented in [82]. FAST calculates the rotor-wake effects and the aerodynamic loads on each blade us-ing full field in-flow data. As a result of the interaction between the platform and the surrounding water,hydrodynamic loads are calculated over the entire platform structure including the supporting beams. Thehydrodynamic loads considered include hydrostatic, radiation, diffraction and viscous forces. Forces fromthe mooring lines are computed using static-equilibrium assumption while the effect of wave and currentdisturbance to the mooring lines is ignored. Other components and factors that are also modeled include theactuators associated with the control and the flexibility of all the blades, the tower and the drive-train. TheFAST model consists of 22 DOFs and 44 systems sates when all DOFs are activated.836.3. Dynamics ModelingAlthough the nonlinear FAST model is too complex to be used for controller design, it can generatelinearized models at static equilibrium points. These linearized modes have been utilized in wind turbinecontroller design, e.g. in [90]. However, to date, FAST cannot generate wave disturbance matrix whenlinearization is made. This means that FAST cannot be used for design of advanced control techniques thattake into account the wave disturbance rejection. Moreover, for gain-scheduling controller design in thischapter, model linearization will be implemented many times during the controller design process. A simplemodel with fast linearization process is preferred. Therefore, FAST will only be used in simulations forvalidation and comparison purpose in this chapter. For controller design, we will use a simplified control-oriented model as will be explained next.6.3.2 Simplified Nonlinear ModelA 3-D control-oriented nonlinear model was proposed in [40] for floating offshore wind turbines. This modelassumes that the wind turbine and the platform behaves as a rigid body. In other words, the flexibility of thetower structures and blades is neglected in deriving the model. The model includes 15 states, defined asx := [xp, yp, zp, θx, θy, θz, x˙p, y˙p, z˙p, θ˙x, θ˙y, θ˙z, ωr, ωg,∆θr]T , (6.1)among which are 3 translational coordinates (xp, yp, zp), 3 rotational coordinates (θx, θy, θz) and their deriva-tives. In addition, rotor speed (ωr), generator speed (ωg) and the rotor-side equivalent elastic torsional de-formation of the drive-train (∆θr), i.e. the difference between the rotor azimuth angle θr and the generatorazimuth angle θg divided by the gearbox ratio NGR, are also included to describe the dynamics of the drive-train. The model presented in [40] is a nonlinear state-space model in the form ofx˙ = f(x, u, v, w), (6.2)where x is the state vector, u is the vector of control inputs [β, Tg, γn]T , v is the wind disturbance vector con-sisting of uniform wind speeds in three axes denoted as vx, vy and vz , respectively, w is the wave disturbancevector consisting of wave velocity, acceleration and dynamic pressures and f is a nonlinear vector-valuedfunction relating the states to the control inputs and the disturbances.6.3.3 Linearized ModelsSolving Equilibrium PointsAlthough the nonlinear model in [40] is much simpler than FAST model, it is still not suitable for controllerdesign because it involves look-up tables to calculate the aerodynamic load and implicit equations to modelthe mooring lines. A feasible way is to linearize the nonlinear model at static equilibrium points and usethe linearized models for controller design. We use the superscript 0 to denote the value of a variable atan equilibrium point. For instance, x0 denotes the value of the state vector at an equilibrium point. InRegion III, one of the control objectives is to regulate the generator speed and power at their rated value.Therefore, ω0g = ωg,rated = 1173.7 rpm, P0 = Prated = 5 MW . For linearization, we assume that the water846.3. Dynamics Modelingis still, and thus there is no wave disturbance, which means w0 = 0. Under this assumption, given a winddisturbance vector v0, the equilibrium point can be obtained by solvingf(x0, u0, v0, w0) = 0. (6.3)Note that most of the states can be pre-determined before solving (6.3). More specifically, the values of thesix states involving derivatives (i.e. x˙0p, y˙0p, z˙0p , θ˙0x, θ˙0y, θ˙0z ) should be zero at a static equilibrium point. Therotor speed w0r is determined by ω0r = ω0g/NGR. In terms of the control inputs, the generator torque T0g canbe determined by T 0g = P0/(ηgω0g) = 43.094 kNm. Since the nacelle yaw angle is controlled to be alignedwith the wind direction, γ0n can also be determined by the wind direction that can be calculated given thewind speed vector v0.For simplicity, we assume that the wind always blows in the x-axis, i.e. vy = vz = 0 always hold. Inthis way, the dynamics of WTS is uniquely determined by vx and thus vx is the only scheduling parameterwhen applying the LPV technique. Without this assumption, then two scheduling parameters, either vx andvy or the magnitude of the wind speed and the wind direction should be employed to fully characterize thedynamics of the WTS.To summarize, given a wind disturbance vector v0, the equilibrium point (x0, u0, v0, w0) can be deter-mined by pre-selection and solving (6.3) numerically, e.g. using Newton’s method.State-Space Model DerivationAfter obtaining an equilibrium point (x0, u0, v0, w0), we can compute a linearized modelδx˙ = Aδx+Bδu+Bvδv +Bwδw, (6.4)where δx := x−x0, δu := u−u0, δv := v−v0, δw := w−w0, and the state space matricesA, B, Bv, Bware the partial derivatives of the non-linear function f , defined to beA =∂f∂x∣∣∣(x0,u0,v0,w0), B =∂f∂u∣∣∣(x0,u0,v0,w0),Bv =∂f∂v∣∣∣(x0,u0,v0,w0), Bw =∂f∂w∣∣∣(x0,u0,v0,w0). (6.5)The partial derivatives can be approximated by finite difference approximations.Up to now, we have obtained general linearized models with 15 states and three independent controlinputs. Later, considering the specific generator torque controller, we will derive a model that has the bladepitch angle as the sole control input. The derived model will be used for blade pitch controller design.856.4. Controller Design6.4 Controller Design6.4.1 Nacelle Yaw Angle ControlIt is assumed that a wind vane is used to measure the wind direction. The nacelle yaw angle γn shouldbe controlled to match the wind direction. Because of the huge inertia of the WTS and the fact that thewind direction does not alter quickly, the nacelle yaw angle should change very slowly. Thus, a simpleproportional, proportional-integral or bang-bang controller should be enough for control of the nacelle yawangle. Furthermore, in this chapter, it is assumed that wind always blow in x-axis; therefore, the nacelle yawangle γn is constantly equal to zero by assumption.6.4.2 Generator Torque ControlAs mentioned before, the generated power should be regulated to the rated value in Region III. To achievethis, a baseline generator torque controller was proposed in [81] and takes the formTg =Pratedηgωg. (6.6)This controller was adopted for generator torque control in this chapter.6.4.3 Model Re-Derivation and SimplificationBlade pitch control is crucial in regulating the generator speed (thus the generator power) and the platformmotion in Region III. Due to employing the baseline generator-torque controller (6.6), the generator torqueis automatically determined by the measured generator speed. Thus, we need to re-derive a plant model withthe blade pitch angle as the sole control input, and use this model to design the blade pitch controller.Considering that δu = [δβ, δTg, δγn] and δγn = 0 (since γn ≡ 0 by assumption), (6.4) can be rewrittenasδx˙ = Aδx+B(:, 1)δβ +B(:, 2)δTg +Bvδv +Bwδw. (6.7)From (6.6), the differential δTg can be expressed asδTg = − Pratedηg(ω0g)2δωg, (6.8)Plugging (6.8) into (6.7) and noticing that δωg is the 14th element of δx, we haveδx˙ = Aδx+B(:, 1)δβ − Pratedηg(ω0g)2B(:, 2)δωg +Bvδv +Bwδw,= A˜δx+ B˜δβ +Bvδv +Bwδw, (6.9)where B˜ := B(:, 1) and A˜ is equal to A with its 14th column replaced with A(:, 14)− Peηgω2g0B(:, 2).The frequency response of the model (6.9) from δβ and δvx (note that vy ≡ vz ≡ 0 by assumption) toδωg and δθy under different wind speed is given in Fig. 6.3. The dynamics variations from varying wind866.4. Controller Designspeed is clearly illustrated in this figure.10-4 10-2 100 102Frequency (rad/s)-100-50050Singular Values (dB)From /v to /!gv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(a) From wind speed to generator speed10-4 10-2 100 102Frequency (rad/s)-150-100-500Singular Values (dB)From v to yv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(b) From wind speed to platform pitch angle10-4 10-2 100 102Frequency (rad/s)-80-60-40-20020406080Singular Values (dB)From /u to /!gv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(c) From blade pitch angle to generator speed10-4 10-2 100 102Frequency (rad/s)-150-100-50050Singular Values (dB)From /u to /3yv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(d) From blade pitch angle to platform pitch angleFigure 6.3: Open-loop frequency response of the 15th order model (6.9). Note that in this figure, δu := δβ,δv := δvxTo simplify the controller design and implementation, we simplified the linearized model (6.9) by drop-ping the six states related to the translational motion of the WTS, i.e. xp, yp, zp, x˙p, y˙p and z˙p, as they arenot directly related to our control objectives. Furthermore, we ignored the wave disturbance term for mainlytwo reasons. Firstly, we designed twoH∞ controllers, of which one considered the wave disturbance and theother ignored the wave disturbance. Comparison of these two controllers showed that considering wave dis-turbance in the controller design did not help much reduce the platform motion caused by wave disturbance.The details of the comparison are given in Appendix D. Secondly, our computational results indicated thatconsidering the wave disturbance tends to make the LMI-based on LPV/SLPV controller design sensitive tonumerical issues. Based on the above two simplifications, a simplified linearized model is obtained asδ ˙ˆx = Aˆδxˆ+ Bˆδβ + Bˆvδv, (6.10)876.4. Controller Designwhereδxˆ := [δθx, δθy, δθz, δθ˙x, δθ˙y, δθ˙z, δωr, δωg, δ(∆θr)]T , (6.11)Aˆ, Bˆ and Bˆv are obtained from A˜, B˜ and Bv and by removing the rows and columns corresponding to the 6dropped states. The frequency response of the simplified linearized models corresponding to different windspeeds is shown in Fig. 6.4. Although the simplified models (6.10) are noticeably different from the full-order models (6.9) in frequency response, controllers designed based on them do work well as validated bymedium-fidelity simulations which will be shown later. The simplified model (6.10) is used for subsequentcontroller design.10-4 10-2 100 102Frequency (rad/s)-100-50050Singular Values (dB)From /v to /!gv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(a) From wind speed to generator speed10-4 10-2 100 102Frequency (rad/s)-200-150-100-500Singular Values (dB)From v to yv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(b) From wind speed to platform pitch angle10-4 10-2 100 102Frequency (rad/s)-80-60-40-20020406080Singular Values (dB)From u to gv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(c) From blade pitch angle to generator speed10-4 10-2 100 102Frequency (rad/s)-150-100-50050Singular Values (dB)From /u to /3yv = 11.4 m/sv = 12 m/sv = 13 m/sv = 16 m/sv = 18 m/sv = 23 m/sv = 25 m/s(d) From blade pitch angle to platform pitch angleFigure 6.4: Open-loop frequency response of the 9th order model (6.10). Note that in this figure, δu := δβ,δv := δvxIn addition, our frequency domain analysis and time-domain simulation results show that consideringwave disturbance in controller design does not help much in rejecting the wave disturbance for platformmotion reduction. Therefore, the wave disturbance is ignored in design of all the LPV and SLPV controllers,i.e.886.4. Controller Design6.4.4 Blade Pitch ControlGeneralized Plant and Weighting FunctionsAmong the 9 states in the simplified model (6.11), the generator speed can be measured by a speed sensor,the six states related to the rotational motion of the platform can either be measured using gyroscopes orestimated using a state observer. The rotor speed and the torsional deflection probably have to be estimated.In this chapter, we assume that all the 9 states in (6.11) are available for feedback use, either through puresensor measurement, or through the combination of sensor measurement and a reduced-order state observer.With the 9 states available, we are interested in designing state-feedbackH∞, LPV and SLPV controllers forthe blade pitch control and evaluate their performance. The synthesis structure of the blade pitch controlleris shown in Fig. 6.5. In terms of weighting function selection, we tried both dynamic and constant weightingAll the statesState-feedback LPV or SLPV controller States from weighting functions Figure 6.5: Synthesis structure of the blade pitch controller. The input δw does not exist when the wavedisturbance is ignored in controller designfunctions for δωg and [δθx, δθy, δθz]T and were not able to obtain noticeable performance improvement byusing dynamic weights that consider the frequency range of the wind and wave disturbances, as did in [84].Thus static weights were finally selected for those performance channels. For the weight Wu, we selected adynamic function in the form ofWu(s) =(s+ ωbc/k√Muk√ε1s+ ωbc)k, (6.12)where Mu, ε1, and ωbc are the DC gain, high-frequency gain and control bandwidth of W−1u , respectively,and k is a positive integer to adjust the rolloff speed [92]. Utilizing the fact that the main frequency com-ponents of both the wind and wave disturbance are below 1 Hz (= 6.28 rad/s), the parameters of the896.4. Controller Designweighting functions were tuned and finally selected to beWg = 0.1, Wr = diag{0.01, 0.01, 0.01}, (6.13)Mu = 10, wbc = 3, ε1 = 0.001. (6.14)Circumventing Estimation of an LPV ModelFor designing the LPV or SLPV controllers, a general procedure is to first obtain an LPV model and thenformulate and solve parameter-dependent LMIs (PDLMIs). The LPV models may be obtained using a quasi-LPV approach or a Jacobian linearization approach [5]. The Jacobian linearization approach may directlygive a linearized model parameterized by the scheduling parameters, i.e. an LPV model, for nonlinearsystems with analytic forms. However, for complex nonlinear systems that do not have analytic forms, lineartime-invariant (LTI) models usually have to be obtained first by linearization at a series of equilibrium pointsand an LPV model is then estimated by interpolating these LTI models. This approach was adopted by [90]for gain-scheduling control of a floating offshore wind turbine.However, estimating an LPV model is not an inevitable step in designing an LPV or SLPV controller.LPV and SLPV controllers are designed by solving PDLMIs involving the plant matrices, Lyapunov vari-ables and controller variables, all of which may be parameterized by the scheduling parameters. See [14],[6] and Chapters 2 to 4 of this thesis for examples of such PDLMIs. Notice that when we use the griddingmethod [14, Section IV] to solve the PDLMIs involved, we actually just need the plant matrices evaluatedat the gridding points to construct a number of LMIs from the PDLMIs. The plant matrices at a specificpoint can be given by the linearized model at this point. Thus constructing an LPV model is unnecessary forconstructing those LMIs. Actually, there is another benefit of directly using linearized models instead of anLPV model for this application. If we first construct an LPV model from linearized models, we need to re-evaluate the LPV model at a gridding point, says P, for formulating the LMIs at this point. The re-evaluationmay yield plant matrices that are different from those given by the LTI model from linearization at this point,while the latter are supposed to be exact. This discrepancy is due to the inevitable error in estimating theLPV model.Based on the analysis above, we will directly use linearized models and the gridding method to solve thePDLMIs for LPV or SLPV controller design. Whenever a gridding point is determined where the PDLMIsshould be concretized, we linearize and simplify the nonlinear model (6.2) to get an LTI model (6.10). Thestate-space matrices of this model will be used to formulate the LMIs at this point. We notice that a similaridea was adopted in [93, Section 6.1].Settings for LPV/SLPV Controller DesignThe scheduling parameter is the wind speed in x-axis, i.e. vx. Note that the range of wind speed in Region IIIis [11.4 25] m/s. However, it was found in simulations that considering the speed range of [11.4 12] m/swould produce overly aggressive blade pitch controllers so that the connected saturation block was easilyactivated. As a result, the performance of the system degraded significantly. This is probably because inthe low speed range, generator speed variation is very sensitive to wind speed variation, as shown by the906.4. Controller Designfrequency response plot in Fig. 6.4. Subsequently, the controller will use large control inputs to reject thewind disturbance. After conducting extensive simulations, we found that considering the wind speed rangeof [12 25] m/s will produce a controller that works well not only for the speed range of [12 25] m/s, butalso for the speed range of [11.4 12]m/s, although there is no theoretical guarantee for the latter. Therefore,we set θ ∈ Θ := [12 25] m/s. The rate of variation of the wind speed is assumed to between −1 m/s2 and1 m/s2, i.e. θ˙ ∈ Ω := [−1 1] m/s2.The synthesis conditions in Lemma 3.1 were used to design an SLPV controller without consideringthe measurement uncertainties of wind speed. Design of an SLPV controller considering the measurementuncertainties will be addressed later. The parameter-dependent matrices K(j)(θ) and X(j)(θ) (K(θ) andX(θ)) for SLPV (LPV) in Lemma 3.1 were selected to have an affine form. The interval for gridding theparameter variation set was chosen to be 1 m/s. We used Yalmip [94] and Sedumi [95] to solve all theLMI problem involved in this chapter. We first tried to design an LPV controller, which yielded a γ value of0.2298.Switching Surface Optimization for SLPV ControlFor improving the performance of SLPV controllers, the SSs were optimized using an analogous algorithmof Algorithm 4.1 for state-feedback LPV control. This algorithm is still based on particle swarm optimiza-tion (PSO), and the cost function is the weighted average of the local L2-gain bound in all the subsets, asexpressed by (4.7). The parameters in implementing the algorithm were selected as N = 2, pi1 = 2 m/s,γ¯ = 1.05γ0, where pi1 is the minimum distance between adjacent SSs as well as between edges of Θ andSSs, and γ0 = 0.2298 was the L2-gain bound yielded by the LPV controller. The population size and themaximum number of iterations for PSO were set to 10 and 50, respectively. The termination parameters εand m¯ were set to be 10−5 and 8, respectively. We ran the algorithm in MATLAB 2016b on a PC with Inteli5-3470 CPU and 16 GB RAM running Win7 64-bit OS. The algorithm terminated after 11 iterations, whichtook 31.4 mins. The cost function and SSVs at each iteration is shown in Fig. 6.6. The optimal values ofSSVs are obtained as piopt = {14, 16}.916.5. Simulation Results0 2 4 6 8 10 120.120.140.160.180.2Cost function global bestworst of local bestsmean of local bests0 2 4 6 8 10 12Iteration14151617SSVs 1 2Figure 6.6: Cost function and SSVs in PSO iterations. A “local best” refers to the best position of anindividual particleWith the optimal SSs piopt, a state-feedback SLPV controller was finally designed by solving an opti-mization problem minimizing the cost function (4.7).Considering Measurement Uncertainty of Wind SpeedConsidering that the wind speed may not be accurately measurable in reality, we now investigate how theperformance of the SLPV controllers is influenced by the uncertainty in wind speed measurement. Wedesigned an SLPV controller assuming a measurement uncertainty bounded by 2 m/s, i.e. δ := θˆ − θ ∈[−2, 2]m/s, where θˆ is the measured wind speed. The synthesis conditions for designing such state-feedbackSLPV controller considering the measurement uncertainty are given in Lemma E.1. Different from previouscases, we utilized second-order polynomial form for all the decision matrices K(j)(θˆ) and X(j)(θˆ) to reducethe conservatism due to consideration of the measurement uncertainty.6.5 Simulation Results6.5.1 Simulation SettingsIn order to validate the proposed SLPV controllers, simulations were conducted in FAST with all the DOFsactivated. For comparison, we also utilized or designed several other controllers for blade pitch control inaddition to the LPV controller. They are926.5. Simulation Results• A baseline controller presented in [81] which is a gain-scheduled PI controller, and• A H∞ controller designed using an LTI model in the form of (6.10) from linearization at wind speedequal to 18 m/s.Notice that the baseline controller works in the pitch-to-feather mode that always gives a non-negative bladepitch angle, while all other controllers employ the pitch-to-stall mode to help avoid the occurrence of negativedamping for platform pitch motion, thus always giving non-positive blade pitch angles. Both the bladepitch controller and the generator controller use the measured generator speed for feedback. To avoid high-frequency excitation of the WTS, a first-order low-pass filter proposed in [81] was also adopted here for thegenerator speed measurement and the filtered generator speed was used for all the controllers. The low-passfilter has a cutoff frequency of 0.25Hz. To enforce the control inputs to satisfy the physical constraints of theactuators in the WTS, saturation and rate-limitation were applied to the controller outputs before input to theWTS. The saturation and rate-limit parameters are determined based on those defined in [81] and are listedin Table 6.2. Furthermore, all these controllers work together with the same generator torque controller (6.6).The step size for all the simulations was chosen to be 0.0125 seconds.Control Input Saturation Rate-Limitβ[0 90] (deg) for baseline controller[−60 0] (deg) for others [−8 8] (deg/s)Tg [4 4.5] (kN ·m) [−1.5 1.5] (kN ·m/s)Table 6.2: Saturation and rate-limit on control inputs of the WTS6.5.2 Results Without Measurement Uncertainty of Wind SpeedConstant Mean Wind SpeedWe first evaluate the performance of the controllers using a realistic wind profile with a constant mean windspeed of 18 m/s, which was generated by TurbSim. The profiles of the wind disturbance in x-axis and thewave height are illustrated in Fig. 6.7.936.5. Simulation Results0 100 200 300 400 500 600Time (s)102030Wind speed (m/s)OriginalFiltered0 100 200 300 400 500 600Time (s)-505Wave height (m)Figure 6.7: Profiles of wind with a constant mean speed and wave. Top: wind speed in x axisTo implement the LPV and SLPV controllers, a moving-average filter with a window length of 200points was used to smooth the wind speed and the filtered wind speed (shown in Fig. 6.7) was utilized asthe scheduling parameter. The performance of different controllers in terms of platform pitch reduction,generator speed regulation and power regulation are shown in Fig. 6.8. Among all the tested controllers, thebaseline controller yields the worst performance in all the evaluated criteria, while the other three controllersgive similar performance in platform pitch suppression. However, in terms of generator speed and powerregulation, the SLPV controller outperforms both the H∞ controller, which is further better than the LPVcontroller.946.5. Simulation Results0 100 200 300 400 500 600-505y (deg)Baseline H LPV SLPV0 100 200 300 400 500 6001000110012001300g (rpm)Baseline H LPV SLPV0 100 200 300 400 500 600Time (s)4.64.855.2Power (MW)Baseline H LPV SLPVFigure 6.8: Performance of different controllers in regulating the platform pitch, generator speed and powerunder the wind and wave profiles shown in Fig. 6.7Figure 6.9 illustrates the control inputs of different controllers, which again verifies the advantages ofthe SLPV controller over all other controllers from the relatively small deviation of the blade pitch angle andgenerator torque.956.5. Simulation Results0 100 200 300 400 500 600-1001020 (deg)Baseline H LPV SLPV0 100 200 300 400 500 600Time (s)44.24.44.6T g (Nm)104(a) Control inputs0 100 200 300 400 500 600Time (s)11.52Switching signal (b) Switching signal of the SLPV controllerFigure 6.9: Control inputs and switching signal (of the SLPV controller) under the wind and wave profilesshown in Fig. 6.7To quantitatively evaluate the performance of various controllers, we calculated the root-mean-squarederror (RMSE) of different variables, defined to beN∑i=1(xi − x¯)2/N, (6.15)where x represents one of the variables ωg, θy, Pe, β and Tg, xi denotes the value of x at the i-th samplinginterval, and x¯ represents the desired value (for ωg, Pe and Tg) or mean value (for θy and β) of x. In addition,we divided the RMSE value of a variable under different controllers by the value under the baseline controllerfor normalization. The normalized RMSE value is listed in Fig. 6.10, which again validates the advantage of966.5. Simulation Resultsthe SLPV controller. More specifically, the SLPV controller achieved an improvement of 77% in generatorspeed regulation, 69% in platform pitch reduction, and 72% in power regulation compared to the baselinecontroller.0.230.31 0.280.650.270.610.360.310.461.00 1.00 1.00g y Power00.20.40.60.811.2Normalized RMSEBaseline H LPV SLPV(a) Performance0.130.220.330.680.210.331.00 1.00Tg00.20.40.60.811.2Normalized RMSEBaseline H LPV SLPV(b) Control inputsFigure 6.10: Normalized root-mean-squared error yielded by different controllers under the wind and waveprofiles shown in Fig. 6.7. Performance of the baseline controller was utilized as the basis for normalizationVarying Mean Wind SpeedWe have simulated the controllers using a wind profile with a constant mean wind speed. However, the meanwind speed may vary in Region III. To verify the performance of the controllers under significantly varyingwind speed, we repeated the above simulations but with a different wind profile shown in Fig. 6.11. Thiswind profile was created by concatenating the wind profiles with a constant wind speed of 12 m/s, 18 m/s and25 m/s, respectively. The resulting wind profile completely covers the wind speed range of [11.4 25] m/sin Region III. The same moving-average filter was utilized to smooth the wind speed trajectory for gain-scheduling use. The performance and the control inputs of the different controllers are shown in Figs. 6.12to 6.14. This time, the baseline controller failed to stabilize the WTS in the low wind-speed phase; thus,part of the trajectories corresponding to this controller was outside the scope of the figures presented. Allother three controllers are still very similar in platform motion regulation. However, the SLPV controller stillperforms the best in generator speed and power regulation. Between the H∞ and LPV controller, althoughthe H∞ controller outperforms the LPV controller in the low and median wind-speed phase, its performanceis worse in the final phase when the wind speed is high. In Fig. 6.14, the superior performance of the SLPVcontroller is also demonstrated through the relatively small deviation in control inputs.976.5. Simulation Results0 100 200 300 400 500 600Time (s)102030Wind speed (m/s)OriginalFiltered0 100 200 300 400 500 600Time (s)-505Wave height (m)Figure 6.11: Profiles of wind with a varying mean speed and wave. Top: wind speed in x axis0 100 200 300 400 500 6001000110012001300g (rpm)Baseline H LPV SLPV0 100 200 300 400 500 600Time (s)4.555.5Power (MW)Figure 6.12: Performance of different controllers in generator-speed and power regulation under the windand wave profiles shown in Fig. 6.11986.5. Simulation Results0 100 200 300 400 500 600-1-0.500.51x (deg)Baseline H LPV SLPV0 100 200 300 400 500 600-50510y (deg)Baseline H LPV SLPV0 100 200 300 400 500 600Time (s)-2024z (deg)Baseline H LPV SLPVFigure 6.13: Performance of different controllers in platform motion regulation under the wind and waveprofiles shown in Fig. 6.11996.5. Simulation Results0 100 200 300 400 500 600-20-10010 (deg)Baseline H LPV SLPV0 100 200 300 400 500 600Time (s)44.24.44.6T g (Nm)104(a) Control inputs0 100 200 300 400 500 600Time (s)11.52Switching signal (b) Switching signal of the SLPV controllerFigure 6.14: Control inputs and switching signal (of the SLPV controller) under the wind and wave profilesshown in Fig. 6.11Quantitative comparison of the different controllers using the normalized RMSE is shown in Fig. 6.15,which again validates the advantage of the SLPV controller over all other controllers. More specially, theSLPV controller achieved an improvement of 57% in generator speed regulation, 1% in platform pitch re-duction, and 62% in power regulation compared to H∞ controller which ignores the dynamics variation dueto the varying wind speed.1006.5. Simulation Results0.43 0.99 0.381.20 0.99 1.121.00 1.00 1.0017.802.9023.57g y Power051015202530Normalized RMSEBaseline H LPV SLPV(a) Performance0.610.511.431.261.00 1.002.231.96Tg00.511.522.53Normalized RMSEBaseline H LPV SLPV(b) Control inputsFigure 6.15: Normalized root-mean-squared error yielded by different controllers under the wind and waveprofiles shown in Fig. 6.11. Performance of the H∞ controller was utilized as the basis for normalization6.5.3 Results Under Measurement Uncertainty of Wind SpeedNow we evaluate the performance of different controllers when only inexact measurement of the wind speedis available for gain scheduling. The wind profile with a varying mean wind speed used in Section 6.5.2 wasstill utilized, while a square wave with an amplitude of 4 m/s and a period of 200 second was added to theactual wind profile to mimic the measurement uncertainty, as shown in Fig. 6.16.0 100 200 300 400 500 600Time (s)1015202530Wind speed (m/s) ActualMeasured0 100 200 300 400 500 600Time (s)-505Wave height (m)Figure 6.16: Actual and measured (filtered) wind speedThe time-domain simulation results are shown in Figs. 6.17 to 6.19 while the quantitative comparison isgiven in Fig. 6.20. Note that the performance of theH∞ controller was also exhibited for comparison. For thetested scenario, the SLPV controller ignoring the measurement uncertainty was still able to stabilize the WTSsystem although there is no theoretical guarantee. Furthermore, it even yielded even better performance thanthe SLPV controller considering measurement uncertainty. This is because considering the measurementuncertainty in the SLPV controller design tends to generate a more robust yet more conservative controller.Even with the inherent conservatism, the SLPV considering uncertainty still outperforms the H∞ controller,yielding an improvement of 33% in generator speed regulation, 4% in platform pitch reduction and 38%in power regulation. Considering that the performance degradation of SLPV control due to consideringthe measurement uncertainty is not so huge, it is still worth utilizing the SLPV controller considering themeasurement uncertainty as it provides theoretical guarantee.1016.5. Simulation Results0 100 200 300 400 500 600105011001150120012501300g (rpm)H SLPV ignoring SLPV considering 0 100 200 300 400 500 600Time (s)4.855.2Power (MW)Figure 6.17: Performance of different controllers in generator-speed and power regulation under measure-ment uncertainty of wind speed0 100 200 300 400 500 600-0.500.51x (deg)H SLPV ignoring SLPV considering 0 100 200 300 400 500 6002468y (deg)H SLPV ignoring SLPV considering 0 100 200 300 400 500 600Time (s)-2024z (deg)H SLPV ignoring SLPV considering Figure 6.18: Performance of different controllers in platform motion regulation under measurement uncer-tainty of wind speed1026.5. Simulation Results0 100 200 300 400 500 600-15-10-50 (deg)H SLPV ignoring SLPV considering 0 100 200 300 400 500 600Time (s)44.24.44.6T g (Nm)104(a) Control inputs0 100 200 300 400 500 600Time (s)11.52Switching signal (b) Switching signals of the SLPV controllersFigure 6.19: Control inputs and switching signals of different controllers. Note that the switching signals ofthe two SLPV controllers were the same due to use of the same SSs1036.6. Conclusion0.770.960.620.671.010.511.00 1.00 1.00g y Power00.20.40.60.811.2Normalized RMSEH SLPV ignoring SLPV considering (a) Performance0.980.940.750.811.00 1.00Tg00.20.40.60.811.2Normalized RMSEH SLPV ignoring SLPV considering (b) Control inputsFigure 6.20: Normalized root-mean-squared error yielded by different controllers under measurement un-certainty of wind speed. Performance of the H∞ controller was utilized as the basis for normalization.6.6 ConclusionIn this chapter, we considered control of a baseline 5 MW offshore wind turbine on a semi-submersibleplatform for both power regulation and platform motion reduction in the whole Region III. Linearized modelswere obtained through linearization of a control-oriented nonlinear model and were utilized for synthesis ofH∞, LPV and SLPV controllers. For SLPV controller design, SSs were optimized for improved performanceand the measurement uncertainty of the wind speed was also considered. The designed controllers anda baseline controller previously proposed were compared with each other in medium-fidelity simulationsbased on FAST using realistic wind profiles covering the whole Region III. Simulation results showed that thebaseline controller could not stabilize the wind turbine system in the whole region, while all other controllerscould. The SLPV controller ignoring the measurement uncertainty achieved better results than both theH∞ controller and the LPV controller. The SLPV controller considering the measurement uncertainty ofwind speed is slightly more conservative than the SLPV controller ignoring the measurement uncertainty.However, since the former provides theoretical guarantee of robust stability and performance, and was stillmuch better than the H∞ controller, it was recommended by the author.Current work is based on the assumption that the nine states concerning the platform rotational motionand drive-train dynamics are all available for feedback use. Our future work includes designing an LPVstate-observer to estimate the states that are unlike to be measurable and testing the performance of ourproposed controllers with the observer.104Chapter 7Conclusion7.1 Summary of ContributionsThis thesis made three contributions to the switching LPV (SLPV) control theory. The first one is an ap-proach to designing SLPV controllers with guaranteed stability and performance even when the schedulingparameters cannot be exactly measured. The second one is to develop algorithms to optimize the switchingsurfaces (SSs) to further improve the performance of an SLPV controller. The last one is an approach todesigning SLPV controllers that could significantly improve the local performance in some subsets withoutmuch sacrifice of the worst-case performance. Utilizing the theoretical methods developed in this thesis, twopractical problems were also addressed. One is control of miniaturized optical image stabilizers for mobileapplications with product variations. The other one is control of a floating offshore wind turbine for powerquality improvement and platform motion reduction above rated wind speed.Specifically, for SLPV control using uncertain scheduling parameters, admissible regions of actual andmeasured scheduling parameters for all local controllers were obtained according to the switching rulesbased on measured scheduling parameters. Convex conditions on SSs were acquired for hysteresis switchingas a result of practical validity constraint, and for average-dwell-time (ADT) switching owing to equalityconstraints on some of Lyapunov variables at SSs. A numerical example demonstrated the advantages ofthe proposed controller over a non-switching LPV controller considering measurement uncertainties and anSLPV controller ignoring measurement uncertainties. It was also illustrated that the ADT switching maygive better performance than hysteresis switching when the average dwell time is large.For LPV plants with polynomial parameter-dependence, the problem of simultaneous design of a state-feedback SLPV controller and SSs was formulated as an optimization problem involving bilinear matrixinequalities (BMIs). A sequential algorithm was then proposed to solve the BMI problem by alternatelyupdating SS variables and controller variables while fixing the other. A numerical example demonstratedthe effectiveness of the algorithm although there is no theoretical guarantee of finding the global optimumor even a local optimum. For output-feedback case, an algorithm based on particle swarm optimization(PSO) was proposed, in which each evaluation of the cost function is achieved by solving a problem ofSLPV controller design under fixed SSs. The efficacy of this algorithm was validated by both a numericalexample and a realistic example of air fuel ratio control of an automotive engine. Compared to the sequentialalgorithm, the PSO-based algorithm is more general in the sense that it can be applied to both state-feedbackand output-feedback cases. It is also a global search algorithm while the sequential algorithm can only do alocal search. On the other hand, the sequential algorithm is computationally more efficient.For SLPV control with improved local performance, a new cost function considering the local perfor-1057.2. Limitations and Future Workmance in all the subsets was proposed, as opposed to considering only the worst-case performance overall subsets in the traditional SLPV controller design. A tuning parameter was introduced to take the trade-off between improving the local performance and optimizing the worst-case performance. The efficacy ofthe proposed approach was validated on both a numerical example and a realistic example of air-fuel ratiocontrol of an automotive engine.By applying the approach to SLPV control using uncertain scheduling parameters, multiple parameter-dependent robust (MPDR) controllers were designed for control of mass-produced miniaturized optical im-age stabilizers (OIS’s) to deal with the product variations and uncertainties caused by inaccurate parameterestimation and imperfect actuator manufacturing. By using the natural frequencies of the lens platform asgain-scheduling parameters, the MPDR controllers achieved noticeably better performance than a conven-tional robust controller, which was experimentally verified on large-scale prototypes. Experimental resultsalso indicated that explicit consideration of the measurement uncertainties of the scheduling parameters iscrucial for robust stability guarantee for this application.This thesis also dealt with control of a baseline 5 MW offshore wind turbine on a semi-submersibleplatform for both power regulation and platform motion mitigation in the entire region above rated operat-ing point. Linearized models were derived based on a control-oriented nonlinear model and were utilizedfor synthesis of H∞, LPV and SLPV controllers. For SLPV controller design, SSs were optimized for im-proved performance, and uncertainty in measurement of the wind speed was also considered. The designedcontrollers and a baseline controller previously proposed were compared with each other in medium-fidelitysimulations based on FAST using realistic wind profiles. Simulation results show that the baseline controllerwas not able to stabilize the wind turbine system in the entire region above rated operating point, while allother controllers were. The SLPV controller ignoring the measurement uncertainty achieved better resultsthan both the H∞ controller and the LPV controller. The SLPV controller considering the measurementuncertainty of wind speed is slightly more conservative than the SLPV controller ignoring the measurementuncertainty. However, since the former provides a theoretical guarantee of robust stability and performance,and yielded much better performance than the H∞ controller, it was recommended by the author.7.2 Limitations and Future WorkThe limitations of the research results presented in this thesis and future work are summarized as below.• Throughout this thesis, the problem of discontinuous control signal during switching among localcontrollers was not addressed, simply because it is not the focus of this thesis. However, as mentionedin Section 1.2, severe discontinuity in control inputs may cause actuator saturation and mechanicaldamage, and thus is highly undesired. When implementing the SLPV controllers, the discontinuityin control inputs should be closely monitored in both simulations and experiments. If necessary, theSLPV controllers should be redesigned in order to reduce the discontinuity. One way to achieve thisis to impose smooth switching constraints in SLPV controller design as did in [8, 27]. In practice,we found that SSs often influenced the severity of the control input discontinuity. Thus different SSscan also be tried in order to make the switching more smooth. However, we do not have a systematic1067.2. Limitations and Future Workprocedure to guide this up to now.• For SLPV control with uncertain scheduling parameters, the proposed approach can only deal witheither additive uncertainty or multiplicative uncertainty, but not co-existence of them. The workin [32, 33] can be utilized to generalize the results in this thesis to design SLPV controllers that couldcope with co-existence of additive and multiplicative uncertainties in measurement of scheduling pa-rameters.• The synthesis conditions in Lemma 4.1 used in SLPV control with improved local performance andoptimized SSs presented in Chapter 4 were derived based on an assumption of affine parameter depen-dence for both the LPV plants and the Lyapunov variables. This assumption was needed for convertingthe PDLMIs into a finite number of LMIs utilizing the multi-convexity approach proposed in [42].While higher-order polynomial parameter dependence for the LPV plants and the Lyapunov variablecan still be handled by the multi-convexity approach, derivation of the corresponding LMI conditionsis painstaking. Furthermore, our computational experiences on the air-fuel ratio control problem indi-cated that the multi-convexity-based synthesis conditions in Lemma 4.1 are rather conservative, oftenleading to infeasible solutions. Therefore, we suggest employing the approach of LMI relaxation us-ing homogeneous polynomially parameter-dependent Lyapunov function [43] to solve those PDLMIsinvolved in SLPV controller synthesis. A software package is available for facilitating this [96]. Theapproach in [43] is less restrictive and less conservative compared to the multi-convexity approach.However, the synthesis conditions are not the main contributions of this thesis; replacing the synthesisconditions does not prevent applying the research results presented in this thesis.• For control of the optical image stabilizers, implementation of the proposed multiple parameter-dependent robust controllers necessitates measurement of the frequency response for each OIS prod-uct. 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The bounds for rate of variation of θˆi can be estimated from sensor measurements or through thebounds of θ˙i and δ˙mi if the latter is known.Comparing with additive uncertainties, the differences for dealing with multiplicative uncertainties arethe constraints on modified SSs for ADT switching and the expressions of admissible regions for localcontrollers, while the LMIs in Theorem 2.4 and Theorem 2.5 are still the same. Under multiplicative uncer-tainties, for ADT switching controller design, the conditions (2.9) need to be modified tomax(S(j,k)κ1 + δmκ,S(j,k)κ1 + δmκ)≤ Snomκ ≤ min(S(k,j)κ1 + δmκ,S(k,j)κ1 + δmκ). (A.1)By imposing the constraints (A.1), we can derive counterparts of Lemma 2.1 and Theorem 2.2 for multi-plicative uncertainties. The expression of admissible region Φ(j)i in (2.12) for additive uncertainties needs tobe replaced byΦ(j)i := {(θi, θˆi) : (1 + δmi )θi = θˆi ∈ Θ(j)i , δmi ∈ ∆mi , θi ∈ Θi}. (A.2)116Appendix BConstruction of an SLPV ControllerIn this appendix, we will review how to construct the SLPV controller from values of the optimization vari-ables X(j), Y (j) and γ(j) obtained by solving the optimization problems in Approach 4.1 or Approach 4.2.To simplify the controller construction, the following assumption is made.(A3) The matrices D12 and D21 are full-column and full-row rank, respectively.For LPV plants that does not satisfy this assumption, the controller could still be constructed by followingthe idea in [97]. With the assumption (A3), each local controller K(j) in (4.3) can be constructed in thefollowing steps [14]:1. Compute D(j)K that satisfiesσmax(D11 +D12D(j)K D21) ≤ γ(j) (B.1)and set Dcl := D11 +D12D(j)K D212. Compute Bˆ(j)K and Cˆ(j)K that satisfy the following linear matrix equations 0 D21 0DT21 −γ(j)I DTcl0 Dcl −γ(j)I[(Bˆ(j)K )T?]= − C2BT1 X(j)C1 +D12D(j)K C2 , 0 DT12 0D12 −γ(j)I Dcl0 DTcl −γ(j)I[Cˆ(j)K?]= − BT2C1Y(j)(B1 +B2D(j)K D21)T .3. ComputeAˆ(j)K =− (A+B2D(j)K C2)T +[(X(j)B1 + Bˆ(j)K D21)TC1 +D12D(j)K C2]T [−γ(j)I DTclDcl −γ(j)I]−1 [(B1 +B2D(j)K D21)TC1Y(j) +D12Cˆ(j)K .]4. Solve for N (j),M (j) from the following factorization problemI −X(j)Y (j) = N (j)(M (j))T117Appendix B. Construction of an SLPV Controller5. Compute A(j)K , B(j)K , C(j)K according toA(j)K =(N(j))−1(Aˆ(j)K +X(j)Y˙ (j) +N (j)(M˙ (j))T −X(j)(A−B2Dˆ(j)K C2)Y (j)−Bˆ(j)K C2Y (j) −X(j)B2Cˆ(j)K)(M (j))−T ,B(j)K =(N(j))−1(Bˆ(j)K −X(j)B2D(j)K),C(j)K =(Cˆ(j)K − Dˆ(j)K C2Y (j))(M (j))−T .118Appendix CParameters and Details of OIS ControllerDesignC.1 Parameters and Functions Used in ExperimentsC.1.1 Actuator ParametersThe parameters of magnetic actuators in the large-scale prototypes are listed in Table C.1.Table C.1: Parameters of magnetic actuatorsParameter Value UnitInside diameter 2.06 mmOutside diameter 14.1 ∼ 14.8 mmHeight 20.7 ∼ 21.8 mmWire gauge 28 AWGNumber of wire turns ∼ 960 -Magnet diameter 6.34 mmMagnet thickness 1.6 mmTo acquire the relation among the current, air-gap, and the output force of the magnetic actuators, asoftware package, Finite Element Method Magnetics (FEMM) [75], was used. It turned out that the magneticforce F is proportional to the applied current i and is nonlinear with respect to the air gap za. The relationwas approximated with a functionF (i, za) = i · f(za), (C.1)where f(za) is defined asf(za) := (27.13− 9.87za + 1.64z2a − 0.11z3a)/100.A good match between the FEMM data and the polynomial approximation (C.1) can be found in [65, Fig. 4].Utilizing (C.1) in the block ‘Current calculator’ in Fig. 3, the desired current for each magnetic actuator canbe inversely calculated given the desired torque and air gap.119C.1. Parameters and Functions Used in ExperimentsC.1.2 Band-Stop FilterAs mentioned earlier, the high frequency modes of the LP over 200 Hz were not included in the mathematicalmodeling and robust controller design, and instead, were handled with a band-stop filter. Specifically, thegain of the band-stop filter is used to notch out the high frequency modes between 200 and 300 Hz. Noticethat even higher frequency modes (above 300 Hz) were not addressed using the band-stop filter as they haveon effect on the stability of the system as observed in the experiments. The band-stop filter is designedusing Matlab Analog Filter Design tool, where the Chebyshev II type band-stop filter was chosen because itenables selection of the stop-band attenuation and does not have pass-band ripple that has an adverse effecton the low-frequency gain. Other tuning parameters include the filter order, and the lower/higher pass-bandedge frequency. These parameters have to be tuned so that the filter provides sufficient suppression of thehigh frequency modes between 200 and 300 Hz in order to guarantee the closed-loop stability for all theprototypes.The reasoning for selecting and tuning those parameters are explained as follows. A higher filter ordermakes it easy to get large stop-band attenuation without causing much phase change outside the stop-band,but is not desired from the perspective of implementation. Considering the trade-off, a filter order of two wasinitially selected.5 The lower and higher pass-band frequency were chosen to cover the frequency range ofthe high frequency modes between 200 and 300 Hz. The stop-band attenuation is a trade-off between largemagnitude suppression inside the stop-band and large phase change outside the stop-band.The tuning procedure is as follows. Step 1: tune the stop-band attenuation and lower/higher pass-bandedge frequency. Step 2: for each large-scale prototype, theoretically investigate whether the closed-loopsystem consisting of the plant described by the measured frequency response data, the robust µ-synthesiscontroller and the band-stop filter, is stable using Nyquist stability criterion. If so, terminate the tuningprocedure; if not, go to the Step 1 and repeat the process.After several iterations, a bandstop filter was obtained asKbs(s) = T1(s)T2(s),whereT1(s) :=s2 + (2pi · 249.2)2s2 + 2(0.0587)(2pi · 237)s+ (2pi · 237)2 ,T2(s) :=s2 + (2pi · 260.5)2s2 + 2(0.0587)(2pi · 274)s+ (2pi · 274)2 .Its frequency response is shown in Fig. C.1. The zeros of Kbs(s) are associated with the two notches inthe frequency response curve. Notice that the band-stop filter is included in all the controllers designed inSection 5.6 except Kcla.5In Matlab Analog Designer, increment of one in the filter order corresponds to increment of two in the order of the state-space(or transfer function) model corresponding to the filter. The minimum filter order for a band-stop filter is one; in this case, theband-stop filter is actually a notch filter.120C.2. Experimental Investigation of the Translational Motion under Pitch Actuation102 103Frequency (Hz)-150-100-500Magnitued (dB)Figure C.1: Frequency response of the band-stop filterC.2 Experimental Investigation of the Translational Motion under PitchActuationTo investigate the magnitude of the translational motion under pitch actuation, an open-loop experiment wasimplemented on the prototype with LP 3. By open-loop, the authors mean that control inputs ( (either Trx orthe currents i1 ∼ i4 in Fig. 5.5) were prespecified instead of being determined by the pitch controller. Theprocedure is explained as follows.First, a sinusoidal signal 0.4 sin(2pi · 1) as the desired torque Trx was inputted to the open-loop system.The displacement of point A with rOA = 12 mm (see Fig. 5.1a) was measured and converted into therotation angle φx, assuming there was no translational motion. Utilizing φx and Trx, the Current calculator(see Fig. 5.5) generated the currents i1 ∼ i4, the trajectories of which were recorded. Next, the sametrajectories of currents were applied to the coils while the displacement of point A’ that is symmetric to Awith respect to the LP center, was measured. The results are shown in Fig. C.2, where the time axes of thedisplacement measurements of A and A’ were adjusted to have meaningful comparisons. The mean value ofdisplacements of A and A’ was calculated to estimate the displacement of the LP center, i.e. the translationalmotion. One can see that the estimated displacement of the LP center was roughly below 0.005 mm, whichis rather small compared to the magnitude of the displacements of points A and A’, which were roughly0.25 mm (corresponding to an estimated rotation angle of 1.2 degree) under pitch actuation. Therefore, thesingle-point measurement can be regarded as a feasible way to reconstruct the rotation angle.C.3 Details of MPDR Controllers DesignFor design of MPDR controllers for pitch control, the method for designing SLPV controllers under inexactmeasurement of scheduling parameters presented in Chapter 2 was employed. The notations used in thissection are as follows. ZN denotes the integer set {1, 2, · · · , N}. I and 0 denote an identity matrix ofappropriate dimension and a zero matrix of appropriate dimension. 〈X〉 is the shorthand notation ofX+XT .For symmetric matrices M and N , M > N means that M −N is positive definite. In symmetric matrices,? denotes an block that is induced by symmetry.121C.3. Details of MPDR Controllers Design0 1 2 3 4 5 6-0.500.5Disp. of A & A' (mm)A A'0 1 2 3 4 5 6Time (s)-10-505Disp. of center (mm)#10-3Figure C.2: Displacement measurements at two symmetric points under the same currentsC.3.1 Admissible Regions for Local ControllersThe following content for admissible region determination is a direct application of Section 2.4.Same as an SLPV controller, the multiple robust controllers will be designed using parameter-dependentlinear matrix inequalities (PDLMIs) with respect to θ and θˆ, as the plant and controller depend on θ and θˆ,respectively. For formulating and solving the PDLMIs, one needs to determine the admissible region of θand θˆ for each local controller K(j), denoted by Φ(j). The region Φ(j) can be expressed as the Cartesianproduct:Φ(j) = Φ(j)ω1 × Φ(j)ω2 × Φ(j)K1, (C.2)where Φ(j)ω1 , Φ(j)ω2 and Φ(j)K1are the admissible regions of (ω1, ωˆ1), (ω2, ωˆ2) and (K1, Kˆ1), respectively. Fori = 1, 2, Φ(j)ωi includes all admissible pairs of (ωi, ωˆi) to make ωˆi belong to Θ(j)i , and is expressed byΦ(j)ωi = {(ωi, ωˆi) : ωˆi ∈ Θ(j)i , ωˆi = ωi + δi, δi ∈ [δi, δi], ωi ∈ [ωi, ωi]}, (C.3)where Θ(j)i denotes the projection of Θ(j) onto the i-th coordinate. The shape of Φ(j)ωi can be readilyillustrated as polytopes by following the line of Section 2.4. Analogously, the region Φ(j)K1 is given asΦ(j)K1= {(K1, Kˆ1) : Kˆ1 = 0, K1 ∈ [K1,K1]}.C.3.2 MPDR Controllers DesignThe following content for synthesis of MPDR controllers is a straightforward application of the SLPV con-troller design method in Section 2.5 to the specific OIS plant (5.9), where the scheduling parameters ωi andtheir ωˆi measurement are time-invariant. An SLPV controller is usually designed to minimize the worst-caseL2-gain bound, denoted as γ, which is the maximum of local L2-gain bounds γ(j) for j ∈ ZN , where N isthe number of subsets. The same cost function is used in design of the MPDR controllers design. Due to thetime-invariant property of ωˆi, there is no “switching” really happening in the implementation of the designedcontrollers for each OIS product. Thus, the conditions on SS that are necessary for the time-varying schedul-122C.3. Details of MPDR Controllers Designing parameter case, are not needed here. In other words, only the conditions for guaranteeing the stabilityand local L2-gain bound in each subset (i.e. (C.6) and (C.7)) are needed for design of the MPDR controllers.Following Section 2.5, the MPDR controllers can be designed through solving the following LMI prob-lem:min γ (C.4)s.t. γ(j) ≤ γ, j ∈ ZN (C.5)[X(j) InIn Y(j)]> 0, ∀θˆ ∈ Θ(j), j ∈ ZN (C.6)[M11 ?M21 M22]?[εX(j) (A(θ)−A(θˆ))TY (j) 0 0]−εIn < 0,∀(θ, θˆ) ∈ Φ(j), j ∈ ZN (C.7)M11 :=[〈AX(j) +B2Cˆ(j)K 〉 ?Aˆ(j)K + [A+B2Dˆ(j)K C2]T 〈Y (j)A+ Bˆ(j)K C2〉],M21 :=[[B1 +B2Dˆ(j)K D21]T [Y (j)B1 + Bˆ(j)K D21]TC1X(j) +D12Cˆ(j)K C1 +D12Dˆ(j)K C2],M22 :=[−γ(j)Inw ?D11 +D12Dˆ(j)K D21 −γ(j)Inz].The optimization variables include parameter-dependent and symmetric n×nmatricesX(j)(θˆ) and Y (j)(θˆ),parameter-dependent matrices Aˆ(j)K (θˆ), Bˆ(j)K (θˆ), Cˆ(j)K (θˆ) and Dˆ(j)K (θˆ) of compatible dimensions, and a posi-tive scalar ε. Note that in (C.6) and (C.7) and hereafter, the dependence of optimization variables on θˆ andof plant matrices on θ are dropped for brevity.PDLMIs in (C.6) and (C.7) lead to an infinite-dimensional and infinitely constrained problem. A standardapproach to convert the problem into a finite-dimensional problem is to mimic the parameter dependence ofthe plant (5.9) in the optimization variablesX(j), Y (j), Aˆ(j)K , · · · , Dˆ(j)K [14]. In this chapter, all the parameter-dependent matrices X(j), Y (j), Aˆ(j)K , · · · , Dˆ(j)K are selected to be affine with respect to measured schedulingparameters ωˆ1 and ωˆ2; for instance,X(j)(θˆ) = X(j)0 + ωˆ1X(j)1 + ωˆ2X(j)2 .The gridding technique [14] is applied to convert the PDLMIs into a finite number of LMIs that are solvedby the solver LMI Lab [45]. Since the matrix inequality (C.7) is affine with respect to K1, guarantee ofthe inequality at the vertices of Φ(j)K1 in (C.2) is necessary and sufficient. The scalar ε was determined bya line search conducted with 17 logarithmically equally-spaced points between 10−1 and 103. The resultedγopt = 14.92 was attained at ε = 177.8.123C.3. Details of MPDR Controllers DesignAfter computing the optimization variables X(j), Y (j), Aˆ(j)K , · · · , Dˆ(j)K by solving the aforementionedoptimization problem, the controller matrices of K(j) are reconstructed asA(j)K =N(j)−1(Aˆ(j)K − Y (j)(A(θˆ)−B2Dˆ(j)K C2)X(j)−Bˆ(j)K C2X(j) − Y (j)B2Cˆ(j)K)M (j)−T,B(j)K =N(j)−1(Bˆ(j)K − Y (j)B2D(j)K),C(j)K =(Cˆ(j)K −D(j)K C2X(j))M (j)−T,D(j)K = Dˆ(j)K ,(C.8)where M (j) and N (j) are from the factorizationN (j)M (j)T= In − Y (j)X(j). (C.9)Note that for the uncertain parameter K1, its nominal value Kˆ1(= 0) defined in (5.7) is used in calculatingA(θˆ).124Appendix DInfluence of Considering the WaveDisturbance in Controller DesignAlthough there is some work that tries to reject the wave disturbance by considering it in modeling andcontroller design, e.g. in [84], a comparison study still lacks to investigate whether explicitly consideringthe wave disturbance in controller design really help improve the performance, especially in regulating theplatform motion. To answer this question, we designed another H∞ controller (denoted as K2 hereafter)using a model that includes the wave disturbance term, in the form ofδ ˙ˆx = Aˆδxˆ+ Bˆδβ + Bˆvδv + Bˆwδw, (D.1)as opposed to the model (6.10). Note that the model (6.10) is used to design all the model-based controllers inChapter 6 including the H∞ controller, which is denoted as K1 hereafter. Similar to the matrices in (6.10),Bˆw is obtained from Bw and by removing the rows and columns corresponding to the 6 dropped states.The same weighting functions used in designing K1 were used here for designing K2. To compare theperformance of K1 and K2, we generated the singular value plots of the closed-loop systems composed ofthe plant (D.1) and K1 and K2, respectively, which are depicted in Fig. D.1. From the singular value plot ofthe frequency response from δw to δθy at the bottom left, one can see that considering the wave disturbancein controller design barely helped reduce the platform pitching induced by the wave. The bottom right sub-figure shows that the reduction of the generator speed variation due to the wave was improved by consideringthe wave in controller design. However, the improvement is not big.The two controllers were also compared in FAST simulations using the wind profile with a constant meanwind speed of 18 m/s that was used in Section 6.5.2. The results are shown in Fig. D.2. One can see that theplatform pitching almost did not change no matter whether the wave disturbance was considered or not incontroller design, which is consistent with the frequency-domain analysis result in Fig. D.1. The performanceof generator speed regulation was improved slightly as a result of considering the wave disturbance.125Appendix D. Influence of Considering the Wave Disturbance in Controller Design10-2 10-1 100 101 102-200-150-100-500From v to yFrequency (rad/s)Singular Values (dB)10-2 100 102-150-100-500From v to gFrequency (rad/s)Singular Values (dB)10-2 10-1 100 101 102-150-100-50050From w to yFrequency (rad/s)Singular Values (dB)10-2 100 102-300-200-1000100From w to gFrequency (rad/s)Singular Values (dB)Figure D.1: Singular value plots of the open-loop system and the closed-loop systems consisting of differentcontrollers. OL: open loop, CL: closed loop, K1: H∞ controller ignoring wave disturbance, K2: H∞controller considering wave disturbance126Appendix D. Influence of Considering the Wave Disturbance in Controller Design0 100 200 300 400 500 60034567y (deg)H ignoring wave H considering wave0 100 200 300 400 500 6001100115012001250g (rpm)H ignoring wave H considering wave0 100 200 300 400 500 600Time (s)4.84.955.15.2Power (MW) H ignoring wave H considering waveFigure D.2: Performance of controllers considering and ignoring the wave disturbance127Appendix EState-Feedback SLPV Control withUncertain Scheduling ParametersFor state-feedback controller design, we consider a generalized plant in the form ofG(θ) :x˙ = A(θ)x+B1(θ)w +B2(θ)u,z = C(θ)x+D1(θ)w +D2(θ)u,y = x,(E.1)where θ = [θ1, . . . , θs]T is the vector of scheduling parameters and is assumed to be measured as θˆ online.The meaning of all other notations is the same as defined in Chapter 2.For plant (E.1), we would like to design a state-feedback SLPV controller K(θˆ) in the form ofu = K(j)(θˆ)x, j ∈ ZN , (E.2)where K(j)(θˆ) is a local controller taking charge of the subset Θ(j) and depends on the vector of measuredscheduling parameters θˆ. Based on Lemma 2.3 and the state-feedback law (E.2), we can easily obtain thefollowing lemma for synthesis of state-feedback SLPV controllers with uncertain scheduling parameters.Lemma E.1. Suppose that there exist a family of positive number γ(j) and parameter-dependent matricesX(j)(θˆ) and K(j)(θˆ) of compatible dimensions such that the following inequality conditions hold: for allj ∈ J ,X(j)(θˆ) > 0, ∀θ ∈ ver(Θ(j)), (E.3)〈A(θ)X(j)(θˆ) +B2(θ)K(j)(θˆ)〉− X˙(j)(θˆ) ? ?BT1 (θ) −γ(j)Inw ?C(θ)X(j)(θˆ) +D2(θ)K(j)(θˆ) D1(θ) −γ(j)Inz < 0, ∀(θ, θˆ, ˙ˆθ) ∈ Φ(j) × ver(Ωθˆ),(E.4)and for all adjacent subsets Θ(j) and Θ(k),X(j)(θˆ) ≤ X(k)(θˆ), ∀θˆ ∈ S(j,k). (E.5)Then, the closed-loop LPV system (3.5) is exponentially stabilized by the SLPV controller (3.4) withK(j)(θˆ) = K(j)(θˆ)X(j)(θˆ)−1 (E.6)128Appendix E. State-Feedback SLPV Control with Uncertain Scheduling Parametersand its performance ||z||2 < γ||w||2 is achieved with γ = maxj∈ZN γ(j) for all admissible trajectories of θand θˆ.The proof of this lemma is omitted since it is straightforward. The notations Φ(j) and Ωθˆ are defined in(2.11) and (2.4) in Chapter 2, respectively.129
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Robust and optimal switching linear parameter-varying control Zhao, Pan 2018
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Title | Robust and optimal switching linear parameter-varying control |
Creator |
Zhao, Pan |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | Linear parameter-varying (LPV) control is a systematic way for gain-scheduling control of a nonlinear or time-varying system that has parameter-dependent dynamics variations in its operating region. However, when the dynamics variations are large, LPV control may give the conservative performance. One way to reduce the conservatism is switching LPV (SLPV) control, in which we partition the operating region into sub-regions, design one local LPV controller for each sub-region, and switch among those local controllers according to some switching rules. On the one hand, this thesis makes three theoretical contributions to the SLPV control theory. Firstly, this thesis proposes a new approach to designing SLPV controllers with guaranteed stability and performance even when the scheduling parameters cannot be exactly measured. Secondly, this thesis presents two algorithms to optimize the switching surfaces (SSs) that can further improve the performance of an SLPV controller. One algorithm is based on sequentially optimizing the SSs and the SLPV controller for the state-feedback case. The other one is based on particle swarm optimization and can be used for both state-feedback and output-feedback cases. Finally, this thesis introduces a novel approach to designing SLPV controllers that could yield significantly improved local performance in some sub-regions without much sacrifice of the worst-case performance. This is different from the traditional approach that often leads to similar performance in all the sub-regions. On the other hand, this thesis addresses two practical problems using the theoretic approaches developed in this thesis. One is control of miniaturized optical image stabilizers with product variations. Specifically, multiple parameter-dependent robust (MPDR) controllers are designed to adapt to the product variations, while being robust against the uncertainties in measurement of the scheduling parameters that characterize the dynamics variation. Experimental results validate the advantages of the proposed MPDR controllers over a conventional robust mu-synthesis controller. The other application is control of a floating offshore wind turbine on a semi-submersible platform. SLPV controllers are designed for regulating the power and the generator speed and reducing the platform motion. The superior performance of the SLPV controllers is demonstrated in high-fidelity simulations. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-09-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0371959 |
URI | http://hdl.handle.net/2429/67151 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2018-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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