A Numerical Optimization Approach to SwitchingSurface Design for Switching Linear Parameter-VaryingControlbyMoein JavadianB.Sc, Sharif University of Technology, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMasters of Applied ScienceinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)The University of British ColumbiaFebruary 2014c? Moein Javadian, 2014AbstractThis thesis proposes an algorithm to design switching surfaces for the switchinglinear parameter-varying (LPV) controller with hysteresis switching. The switchingsurfaces are sought for to optimize the bound of the closed-loop L2-gain performance.An optimization problem is formulated with respect to parameters characterizingLyapunov matrix variables, local controller matrix variables, and locations of theswitching surfaces. Since the problem turns out to be non-convex in terms of thesecharacterizing parameters, a numerical algorithm is given to guarantee the decreaseof the cost function value after each iteration, which consists of two steps: directionselection and line search. A hybrid method which is a combination of the steepestdescent method and Newton?s method is employed in the direction selection step todecide the orientation of proceeding. A numerical algorithm is used to computethe most appropriate length of the proceeding along the selected direction whichgenerates the most decrease in the cost function.To demonstrate the efficiency and usefulness of the proposed algorithm, it will beapplied to three examples in control applications: a tracking problem for a mass-spring-damper system, a vibration suppression problem for a magnetically-actuatediioptical image stabilizer, and an air-fuel-ratio control problem for automotive en-gines. In these examples, it will be shown that the proposed optimization approachto the design of the switching surfaces and the switching LPV controller is superiorto heuristic approaches in closed-loop performances, at the price of higher compu-tational costs. Additionally, it will be shown that the algorithm can be applied tothe general n-parameters case.iiiPrefaceThis thesis is an original intellectual property of the author, Moein Javadian,working under the supervision of Dr. Ryozo Nagamune. This work, which presentsa numerical optimization algorithm to switching surface design for switching linearparameter-varying control, has been completed in the Control Engineering Labora-tory at the University of British Columbia.A version of results in Chapters 2 and 3 has been accepted for publication in The2014 American Control Conference. Results in Chapter 4 and 5 will be submittedfor publication.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 LPV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 LPV-Based Control . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Switching LPV Control . . . . . . . . . . . . . . . . . . . . . 41.2.4 Smooth Switching LPV Control . . . . . . . . . . . . . . . . . 61.2.5 Switching LPV Control with Optimized Switching Surfaces . 7v1.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 92 An Optimization Approach to Switching LPV Control Design . 102.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Description of an LPV Plant . . . . . . . . . . . . . . . . . . 112.1.2 Description of a Switching LPV Controller with HysteresisSwitching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Switching Surface Design Problem . . . . . . . . . . . . . . . 142.2 Review of LPV Switching Controller Design . . . . . . . . . . . . . . 162.3 An Optimization Approach to Switching Surface Design . . . . . . . 192.3.1 The Descent Algorithm?s Main Structure . . . . . . . . . . . 192.3.2 Direction Selection Computations . . . . . . . . . . . . . . . . 222.3.3 Line Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Plant Parameter Variation . . . . . . . . . . . . . . . . . . . . 293.2 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 A Generalized Plant . . . . . . . . . . . . . . . . . . . . . . . 313.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Case One: 2 Switching Surface Variables . . . . . . . . . . . . 333.3.2 Case Two: n Switching Surface Variables . . . . . . . . . . . 36vi3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Application in Magnetically-Actuated Optical Image Stabilizer 434.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.1 Physical System . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.2 Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.3 Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.4 Plant Parameter Variation . . . . . . . . . . . . . . . . . . . . 514.2 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . 524.2.2 A Generalized Plant . . . . . . . . . . . . . . . . . . . . . . . 534.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Application in Spark Ignition Internal Combustion Engines . . . 615.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1.1 Plant Parameters Variation . . . . . . . . . . . . . . . . . . . 635.1.2 Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.1 Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . 665.2.2 A Generalized Plant . . . . . . . . . . . . . . . . . . . . . . . 675.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3.1 Partition Design Assumptions . . . . . . . . . . . . . . . . . . 705.3.2 Optimized Switching LPV Controller Design . . . . . . . . . 715.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74vii5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendix A Plant Model Derivations of Spark Ignition InternalCombustion Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Appendix B A Feedforward Term Added to the Control Input inAir-Fuel Ratio Control of Internal Combustion Engines . . . . . . 99viiiList of Tables3.1 Results of applying different numerical methods to optimize the switch-ing surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Assumptions on the switching surfaces . . . . . . . . . . . . . . . . . 374.1 System parameters [37]. . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Results of applying different numerical methods to optimize the switch-ing surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Results of applying different numerical methods for the DS step ofthe descent algorithm to optimize the switching surface . . . . . . . 74ixList of Figures1.1 The overlapping region and switching surfaces employed in hysteresislogic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Feedback connection between an LPV plant and an LPV controller. 132.2 An example of a two-dimensional parameter trajectory with four sub-regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Parameterizations of switching surfaces (dashed lines) in terms of thecenters (dotted lines) and the widths of the overlapping regions forthe case when N1 = 2 and N2 = 3. . . . . . . . . . . . . . . . . . . . 152.4 The optimization flowchart. . . . . . . . . . . . . . . . . . . . . . . . 213.1 Mass-spring-damper schematic. . . . . . . . . . . . . . . . . . . . . . 293.2 A feedback structure for controller design. . . . . . . . . . . . . . . . 323.3 Functionality of ? with respect to pS = (c1,1, c2,1) obtained by thefull-search method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Comparison of step responses between the optimal controller and aheuristic one with m = 60. . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Plant variation region with 5 divisions on each axis. . . . . . . . . . 383.6 The initial and final performance level for different number of switch-ing surface variables while applying the descent algorithm. . . . . . . 39x3.7 Computational time of convergence of the descent algorithm for dif-ferent number of switching surface variables, having used the hybridmethod and the steepest descent method for the DS step. . . . . . . . 403.8 Computational time of convergence of the descent algorithm for dif-ferent number of switching surface variables, having used the steepestdescent method for the DS step. . . . . . . . . . . . . . . . . . . . . . 404.1 Physical layout of the lens-shifting image stabilizer [37]. . . . . . . . 454.2 Block diagram of the plant and actuator. . . . . . . . . . . . . . . . 464.3 Block diagram of the closed-loop system. . . . . . . . . . . . . . . . . 474.4 Actual control input computation. . . . . . . . . . . . . . . . . . . . 504.5 A feedback structure for controller design. . . . . . . . . . . . . . . . 534.6 Functionality of ? with respect to pS = (c1,1, c2,1) obtained by thefull-search method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Convergence path of the descent algorithm having employed the hy-brid method or the steepest descent method. . . . . . . . . . . . . . . 564.8 Error envelope comparison between the cases having an optimal con-troller and a heuristic one. . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Plant variation region. . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Block diagram of the closed-loop system. . . . . . . . . . . . . . . . . 675.3 Generalized plant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4 The plant variation region with a partition. . . . . . . . . . . . . . . 715.5 Comparison of equivalent air-fuel ratio reference tracking betweencontrollers with the optimized switching surface and the heuristic one. 75xi5.6 Error comparison of equivalent air-fuel ratio reference tracking, aver-aged in every 10 seconds time interval, between controllers with theoptimized switching surface and the heuristic one. . . . . . . . . . . 765.7 Air-flow trajectory and the lower and upper bounds on its variations. 775.8 Engine speed trajectory and the lower and upper bounds on its vari-ations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.9 The operating point of the engine in the plant variation region. . . . 785.10 Air-flow variation rate trajectory and the lower and upper bounds onits variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.11 Engine Speed variation rate trajectory and the lower and upper boundson its variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.12 Comparison of switching variable on air-flow axis between controllerswith the optimized switching surface and the heuristic one. . . . . . 815.13 Comparison of switching variable on engine speed axis between con-trollers with the optimized switching surface and the heuristic one. . 815.14 Comparison of control input to the plant between controllers with theoptimized switching surface and the heuristic one. . . . . . . . . . . 82B.1 The block diagram of the control system with a feedforward controller.100xiiAcknowledgementsForemost, I would like to express my deep gratitude to my supervisor and in-structor, Dr. Ryozo Nagamune, for his enlightening guidance throughout the entireresearch process. He has been a tremendous mentor for me with the useful commentsand remarks. His assiduity towards research will inspire me for my lifetime. Further-more, I would like to thank the rest of my thesis committee, Dr. Farrokh Sassaniand Dr. Mu Chiao, for their great academic support of my thesis. My gratitudealso goes to the members of Control Engineering Laboratory at the University ofBritish Columbia, especially Mr. Masih Hanifzadegan for his insightful commentsabout my thesis. I would also like to thank my friends, Mr. Mahdi Nematimehr,Mr. Sina Amini Niaki, Mr. Ashkan Babaei, Mr. Amirhossein Hadi Hossein Abadi,and Mr. Reza Nickmanesh for their supports and encouragements.Moein JavadianThe University of British ColumbiaFebruary 2014xiiiThis thesis is dedicated to my Parents,for their priceless support and endless lovexivChapter 1Introduction1.1 MotivationA controller design procedure usually involves trial and error. Various charac-teristics of a controller to be designed are selected heuristically and required tobe modified multiple times. The controller structure, a state-space order of thecontroller, some tuning parameters of a specific type of controller, and weighingparameters used in some sort of control design methods are just a few examples ofsuch characteristics. An appropriate selection of these characteristics to maintaina stability and a satisfactory performance is always a challenging task in the con-trol design procedure which requires trial and error. To facilitate such procedureand to obtain the best controller in the search region of the heuristically-selectedparameters, it is usually desired to automate the parameters selection.Similarly, in switching linear parameter-varying (LPV) control design methodwhich is used in a wide range of control applications, many premises are madebeforehand and different parameters are heuristically selected. Nevertheless, optimal1selections instead of heuristic selections can be extremely beneficial. Therefore, asystematic approach to switching LPV control design method is sought, so as toshortcut the design process and to further improve the performance of the closed-loop system containing a switching LPV controller.1.2 Literature ReviewThe genesis of switching LPV control design method through the literature isdiscussed in this section. Before addressing this topic, LPV models and LPV controldesign method are explained and related research endeavours are mentioned.1.2.1 LPV ModelLPV models are known to be useful in representing nonlinear systems with varyingoperating conditions, as well as time-varying systems with measurable parameters.Different methods used to express nonlinear plants as LPV systems can be foundin [17] and [39]. In LPV models, one or more of the plant state-space matrices arefunctions of a vector, ?, called the scheduling parameter, which is assumed to bemeasurable in real-time. The scheduling parameter changes within bounds, calledthe plant variation region. Change in the vector ? causes variation not only in theinput and output of the plant but also in the dynamics of the linear-representedsystem expressed by the state-space matrices.1.2.2 LPV-Based ControlFor LPV systems, it is desired to design gain-scheduled LPV controllers ratherthan time-invariant ones, since LPV controllers can relatively increase the stabilitymargin and improve the performance of the system. Like any other LPV systems, the2dynamics of LPV controllers depend on the scheduling parameter, and the controlinput is updated by measuring the scheduling parameter in real-time. The gain-scheduling design techniques are first introduced in [4] where a single Lyapunovfunction defined over the entire plant variation region is used to formulate the LPVsynthesis condition as a linear matrix inequality (LMI) optimization problem. Theexistence of a single LPV controller, the state-space matrices of which satisfy theformulated LMI, guarantees the stability of the closed-loop system. Also, the per-formance of the system is optimized by minimization of the bound of the closed-loopL2-gain performance. This problem is a convex optimization which can be solvedusing available LMI techniques.Although it is required in [4] that the state-space matrices of the plant are affinefunctions of the scheduling parameter, this necessity is removed in further develop-ment of the method in [3] by finitely sampling the plant variation region. In thisapproach, LMIs generated by formulating the problem for each sampled point aresolved together to compute the optimal LPV controller state-space matrices. How-ever, this controller satisfying LMI conditions for all the sampled points are requiredto be analyzed for other points in the plant variation region, so that the stability andperformance for the whole region are checked. If the controller is not satisfactory,more sampling in the plant variation region is required for control design. In addi-tion to the aforementioned development of LPV controller design in [3], unlike [4],a limit can be enforced for the variation rate of the scheduling parameter resultingin a less conservative controller.A novel control design approach is also proposed in [45] in which parameter-dependent Lyapunov functions are used to design LPV controllers for linear frac-3tional transformation (LFT) systems with guaranteed closed-loop stability and per-formance. Finally, control of LPV systems has been thoroughly and comprehensivelystudied in [30].The aforementioned methods have been successfully applied to real engineeringproblems, e.g., automotive engines [13, 36], aircraft systems [28], and active magneticbearing systems [23], to name a few.1.2.3 Switching LPV ControlIn the design of LPV controllers, or any other type of robust controllers, as therange of plant variations increases, it is often the case that we have to compro-mise the overall control performance, or that the stabilizing controller cannot befound, due to the conservatism inherent to robust controller design methods. Withlarge plant variations, one approach to the design of a satisfactory controller is theswitching LPV control system design in [24]. In this approach, the region of pa-rameter variations is divided into a given set of subregions, and for each subregion,one LPV controller is designed so that it performs well while the parameter varieswithin the subregion. The LPV controller switches to another LPV controller whenthe parameter moves into another subregion, with performance guarantee. Theseswitched systems are considered as a combination of continuous systems and discreteswitching events which is described in [20].Stability analysis of switched systems has come a long way primarily studiedin [10], [7], [35], and [20]. As shown in [6] and [20], the existence of a commonLyapunov function for a group of stable LTI systems guarantees stability of switchedsystems for any switching arrangements. Nevertheless, satisfying such a stability4condition results in a too conservative controller when a specific switching logic isapplied. However, for a particular class of switching signals, multiple Lyapunovfunctions can be employed for stability analysis. Multiple Lyapunov functions canbe defined either as piecewise continuous ones studied in [34], [44], [18], and [38], oras discontinuous ones described in [48] and [7]. Nevertheless, most of these researchhave considered each subsystem having LTI dynamics. In references [21], [22], and[33], this field of research has been further developed to the stability analysis ofswitched LPV systems.Switching logic plays an important role in the stability characteristics of switchedsystems. Diverse switching rules can be found in the literature. In references [44],[18], and [38], the state-space is partitioned and a construction method of conicswitching logic is developed for the stability of a family of LTI systems. This methodis further improved in [32] where a hysteresis switching logic is employed for conicsets. For this logic, it is assumed that every two adjacent sets have an overlapping re-gion engendering two switching surfaces in between as shown in Fig. 1.1. The activecontroller is switched when one of the switching surfaces is hit. A min-projectionstrategy is the basis for another type of state-dependent switching logic in [27] and[35]. Also, switching with average dwell time studied in [33] is an alternative methodbased on which switches occur in an adequately slow manner, so that the effects oftransient responses after switches are suppressed and stability is guaranteed. In[24], both hysteresis switching logic and switching with average dwell time are em-ployed to develop the switching LPV control design of LPV systems using multipleparameter-dependent Lyapunov functions.5Figure 1.1: The overlapping region and switching surfaces employed in hysteresislogic.The switching LPV control design method has been successfully applied to differ-ent applications, such as ball-screw drive servo control [15], aircraft systems control[25], and air-fuel ratio control in automotive engines [36].1.2.4 Smooth Switching LPV ControlA deficiency inherent to general switching LPV controllers described in [24] is thatthe smooth transition is not guaranteed while a switch event occurs. In other words,a jump in the control input is generally expected since the state-space matrices ofthe active controller would be changed abruptly when controller switching occurs.This nonsmooth transition might lead to signal saturations or damage the systemin some cases. In [9] and [8], a smooth switching LPV control design method hasbeen proposed which can be applied to the plants with one-dimensional parametervariation regions. Besides, state feedback is required in order to use this method.An alternative method is developed in [14] which can be applied to control design forsystems having output feedback and one or two-dimensional plant variation region.61.2.5 Switching LPV Control with Optimized Switching SurfacesAnother shortcoming of the method in [24] is that the set of subregions are as-sumed to be given a priori. Although the subregions may be selected by trial anderror, optimization-based methods will be desirable for finding satisfactory subre-gions systematically. Methods have been developed to determine the subregions au-tomatically based on the nonsmooth optimization technique and applied to variousapplications in [46, 5, 47]. In these methods, the subregions are designed sequen-tially such that the performance of the system over the entire plant variation regionis optimized. However, this optimization does not guarantee achieving the globaloptimum.1.3 Research ObjectivesAs described above, various works have been done in regard to switching LPVcontrol design to address stability analysis as well as smoothness of the transient re-sponse while switching the controller occurs. In addition, research has been recentlydone to design the set of subregions in the plant variation region rather than toprespecify such subregions set. Nevertheless, the nature of the problem is such thatdifferent approaches to optimized partition design are expectable, and more room toimprove the performance of the system is still remained to be explored. This thesisaddresses the partition design problem in a different manner from how [46, 5, 47]do and presents an alternative method to optimized switching LPV control design.In this approach:? An algorithm is proposed to design both subregions of the parameter variation7region and LPV controllers associated with the subregions, leading to theswitching LPV controller with hysteresis switching.? The proposed method optimizes the size of subregions such that the perfor-mance of the system over the entire plant variation region is optimized. How-ever, the number of subregions as well as the shape of each subregion is stillprespecified.? The subregions are characterized by two kinds of parameters, i.e., the centresand the widths of overlapping regions between adjacent subregions.? To design a switching LPV controller that optimizes the bound of the closed-loop L2-gain performance, an optimization problem is formulated by meansof these characterizing parameters, as well as matrix variables for switchingLPV controller design.? Since the problem is non-convex, we will provide an iterative algorithm toguarantee the decrease of the cost function value after each iteration, whichconsists of a direction selection step and line search step.? A hybrid method is proposed for the direction selection step which is a com-bination of the steepest descent method and Newton?s method explained, forexample, in [26].After proposing the algorithm, it is applied to three control applications includ-ing LPV systems: a tracking problem for a mass-spring-damper system, a vibrationsuppression problem for a magnetically-actuated optical image stabilizer, and anair-fuel ratio control problem for automotive engines. Simulation results of imple-menting the controllers are also provided and compared with that of implementing8non-optimized controllers.1.4 Organization of ThesisThe rest of this thesis is organized as follows. The numerical iterative descentalgorithm to optimize the switching surfaces in switching LPV control design ispresented in Chapter 2. Before the algorithm is described, formulation of an LPVplant is provided, and switching LPV control design method with fixed partition isreviewed. Then, switching surface design method is formulated in the form of anoptimization problem, to solve of which the proposed algorithm is employed.In Chapter 3, the proposed algorithm is applied to a numerical example whichis control of a simple mass-spring-damper system. Diverse aspects of applying thenumerical algorithm are investigated, and both strengths and weaknesses of theproposed optimization approach are explained.In Chapters 4 and 5, the numerical algorithm is applied to design switchingLPV controllers with optimized switching surfaces for two control applications:magnetically-actuated optical image stabilization in cameras and air-fuel ratio con-trol in spark ignition internal combustion engines. For each application, controldesign results are provided, and simulation results of applying controllers with opti-mized switching surfaces are compared with that of applying controllers with heuris-tic switching surfaces.Conclusion of the thesis and a summary of results are provided in Chapter 6. Thecontributions of this research are also briefly discussed and possible future worksare mentioned.9Chapter 2An Optimization Approach toSwitching LPV Control DesignDesign objectives in parameter-varying control problems consist of guaranteeingstability and improving performance over the entire plant variation region. Perfor-mance of a control system can be characterized by, e.g., speed of response, trackingerror, control input magnitude, disturbance rejection, and noise rejection. All theseperformance components using some weighting functions, introduced later in thisthesis, can be quantified to closed-loop L2-gain performance of the system, calledperformance level ?. It can be shown that there is an inverse relation between per-formance of the system and the performance level ?. Namely, the less ? it is, thebetter performance the system has. Moreover, the existence of ? itself, which corre-lates to existence of a set of common Lyapunov functions, guarantees the stability.In other words, the control design objective is to design a controller so that a ?-valueis found to have a stable system, and its value is minimized to have a satisfactoryperformance.10To further minimize ?-value and improve the performance, a numerical algorithmis proposed in this section to optimize switching surfaces in switching LPV controldesign. The formulation of switching LPV control design method with optimizedswitching surfaces is introduced before presenting the algorithm.2.1 Problem Formulation2.1.1 Description of an LPV PlantConsider the following continuous-time LPV plant:??????x?(t)z(t)y(t)??????= G(?(t))??????x(t)w(t)u(t)??????, (2.1)where the vector x is the state, z is the controlled variable, y is the measurement,w is the exogenous signal, and u is the control input. A matrix-valued function Gis defined for a parameter vector ? ? Rn? byG(?) :=??????A(?) B1(?) B2(?)C1(?) D11(?) D12(?)C2(?) D21(?) 0??????, (2.2)where the matrices in (2.2) have dimensions which are compatible with the lengthsof vectors in (2.1).Hereafter, to simplify the notation and the explanation of our proposed method,we will assume that the dimension of the parameter vector ? is two, i.e., n? = 2.However, all the results are valid even for cases n? > 2. The parameter vector? which varies in time is assumed to be measurable in real-time, to be used for11the gain-scheduling control purpose, and to move within a given normalized squareregion and given rates of variations as ?(t) ? ? and ??(t) ? ?, where? :={? ? R2 : ?k ? [?1, 1] , k = 1, 2}, (2.3)? :={? ? R2 : ?k ? [?i, ?i] , k = 1, 2}, (2.4)To guarantee the existence of the output feedback stabilizing controller, we alsoassume the stabilizability and detectability of the plant for each ? meeting theabove conditions.2.1.2 Description of a Switching LPV Controller with HysteresisSwitchingAs shown in Figure 2.1, the LPV plant in (2.1) is connected with a switching LPVcontroller expressed by???x?K(t)u(t)??? = K(?(t))???xK(t)y(t)??? , (2.5)where the vector xK is the controller state, and the system matrix of the controller,involving a piecewise-constant switching signal ?, is given byK(?(t)) :=???A(?(t))K (?(t)) B(?(t))K (?(t))C(?(t))K (?(t)) D(?(t))K (?(t))??? . (2.6)The switching signal ? specifies one active controller among a set of controllers ateach time instant. The switching rule in this thesis is the hysteresis switching rulepresented in [25, 24], and will be briefly reviewed with a simple example next.12Figure 2.1: Feedback connection between an LPV plant and an LPV controller.Suppose that the normalized square region (2.3) is divided into four overlappingrectangular subregions (see Figure 2.2):?(1,1) :={? ? R2 : ?1 ? [?0.1, 1] , ?2 ? [?1, 0.1]},?(1,2) :={? ? R2 : ?1 ? [?0.1, 1] , ?2 ? [?0.1, 1]},?(2,1) :={? ? R2 : ?1 ? [?1, 0.1] , ?2 ? [?1, 0.1]},?(2,2) :={? ? R2 : ?1 ? [?1, 0.1] , ?2 ? [?1, 0.1]},and that system matrices of the LPV controller for each subregion ?(i,j), i = 1, 2,j = 1, 2, are given byK(i,j)(?) :=???A(i,j)K (?) B(i,j)K (?)C(i,j)K (?) D(i,j)K (?)??? . (2.7)In Figure 2.2, the switching surfaces are indicated with dashed lines. With the hys-teresis switching rule, the controller switches when the varying parameter ? crossesthe switching surfaces which are also the boundary of the subregion from which ?moves into an adjacent region. For an example of a trajectory of ? provided in13Figure 2.2: An example of a two-dimensional parameter trajectory with four subre-gions.Figure 2.2, the switching signal ? in (2.6) changes its value at points marked with?x? as (1, 1)? (1, 2)? (2, 2)? (1, 2)? (1, 1)? (2, 1).2.1.3 Switching Surface Design ProblemWe will tackle the following design problem in this chapter. Given an LPV plant(2.1)-(2.4), and the number of divisions in each axis of the square region ?, denotedby N1 and N2, design both the subregions?(i,j), i = 1, . . . , N1, j = 1, . . . , N2, (2.8)and the associated LPV controllersK(i,j), i = 1, . . . , N1, j = 1, . . . , N2, (2.9)such that the bound ? of the closed-loop L2-gain from w to z is minimized, namely,min ? subject to ?z?2 < ? ?w?2 , (2.10)14for any trajectory of ? with (2.3) and (2.4), and with the hysteresis switching LPVcontroller (2.6).We remark that the problem formulated above is different from the one in [24],in that the subregions ?(i,j) are prespecified in that paper, while they are variablesto be sought in our formulation.If we consider only regularly aligned rectangular subregions, as shown in Fig. 2.2,then the design problem of subregions is equivalent to that of switching surfaces.The switching surfaces for each coordinate ?k, k = 1, 2, are characterized by thepairs of centers and widths, denoted by {(ck,nk , wk,nk) : nk = 1, . . . , Nk ? 1}, of theoverlapping regions. This characterization is illustrated in Figure 2.3. We will utilizethese characterizing parameters to solve the formulated problem.Figure 2.3: Parameterizations of switching surfaces (dashed lines) in terms of thecenters (dotted lines) and the widths of the overlapping regions for the case whenN1 = 2 and N2 = 3.152.2 Review of LPV Switching Controller DesignHere, we review the LPV switching controller design method developed in [24],and set the notation to propose an algorithm for switching surface design in thenext section.The L2-gain performance in (5.21) holds as far as the vector ? changes withinthe region ? and within the region of rate of variations ?, if there exist a constantmatrix Y ? and matrices which depend on ? by means of scalar-valued functions hm,m = 1, . . . ,M ?, expressed asA?(i,j)K (?) := A?(i,j)0 +?Mm=1 hm(?)A?(i,j)m ,B?(i,j)K (?) := B?(i,j)0 +?Mm=1 hm(?)B?(i,j)m ,C?(i,j)K (?) := C?(i,j)0 +?Mm=1 hm(?)C?(i,j)m ,D?(i,j)K (?) := D?(i,j)0 +?Mm=1 hm(?)D?(i,j)m ,X(i,j)(?) := X(i,j)0 +?Mm=1 hm(?)X(i,j)m ,i = 1, . . . , N1, j = 1, . . . , N2,(2.11)that have appropriate matrix dimensions? and fulfill the following matrix inequal-ities. (The dependency of the matrices in (2.12)?(2.14) on ? and ?? is omitted forbrevity.)?For practical validity, we have to assume that either X or Y is a constant matrix. Here, weassume Y to be constant, but formula are analogous for a constant X. See [3, 36].?The functions hm, m = 1, . . . ,M , are chosen based on the functionality of the plant G(?) in(2.2). See [3, Section IV].?See [42] for the dimensions of these matrices.161. For each vector ? ? ?(i,j) and each vector ?? ? ?,??????????M11 ? ? ?M21 M22 ? ?M31 M32 ??I ?M41 M42 M43 ??I??????????< 0, (2.12)????X(i,j) II Y??? < 0, (2.13)where the entry ? represents a matrix which makes the whole matrix symmet-ric, and block matrices are defined byM11 := X?(i,j) +X(i,j)A+ B?(i,j)K C2 + (?),M21 := [A?(i,j)K ]T +A+B2D?(i,j)K C2,M22 := ?Y? +AY +B2C?(i,j)K + (?),M31 := (X(i,j)B1 + B?(i,j)K D21)T ,M32 := (B1 +B2D?(i,j)K D21)T ,M41 := C1 +D12D?(i,j)K C2,M42 := C1Y +D12C?(i,j)K ,M43 := D11 +D12D?(i,j)K D21.2. For each vector ? on each switching surface associated with the case when ?leaves the subregion ?(i,j) and enters its adjacent subregion ?(k,`),X(i,j) ?X(k,`) < 0. (2.14)Once matrices in (2.11) satisfying the matrix inequalities (2.12)?(2.14) are found,the controller parameters for a subregion ?(i,j) are obtained, with N (i,j) := Y ?1 ?17X(i,j), byA(i,j)K :=[N (i,j)]?1 [A?(i,j)K ?X(i,j){A?B2D?(i,j)K C2}Y?B?(i,j)K C2Y ?X(i,j)B2C?(i,j)K]Y,B(i,j)K :=[N (i,j)]?1 [B?(i,j)K ?X(i,j)B2D?(i,j)K],C(i,j)K :=[C?(i,j)K ? D?(i,j)K C2Y]Y T ,D(i,j)K := D?(i,j)K (?),For controller design, matrix inequalities (2.12)?(2.14) must hold at infinitelymany points ?, and extreme points of ??, i.e., ?k and ?k in (2.4), due to the linearityof the matrix inequality (2.12) with respect to ??. In order to make the designpractically feasible, a common technique is the gridding of the parameter region,leading to a finite number of matrix inequalities.We can gather all of the inequalities into one matrix inequality, expressed in theform of M(?,pK ,pS) < 0. Here, the matrix M is a block diagonal matrix consistingof left-hand sides of the inequalities (2.12)?(2.14) evaluated at gridded points, andpK and pS are vectors of optimization variables related to, respectively, the con-troller design (A?(i,j)K , B?(i,j)K , C?(i,j)K , D?(i,j)K , X(i,j), Y , i = 1, . . . , N1, j = 1, . . . , N2) andthe switching surface design ({(ck,nk , wk,nk) : nk = 1, . . . , Nk ? 1}, k = 1, 2).The optimization problemmin ? subject to M(?,pK ,pS) < 0 (2.15)is non-convex in general, due to the coupling between optimization variables pK andpS in the matrix M . However, with a fixed vector pS = p?S , this problem reduces18tomin ? subject to M(?,pK ,p?S) < 0, (2.16)which is exactly same as that considered in [24], and thus solvable via numericallyefficient convex optimization techniques. By solving (2.16), the corresponding min-imizers p?K and ?? are obtained. One of the contribution of this thesis is to proposean algorithm for the adjustment of the vector p?S , i.e., the switching surface, so asto further improve the L2-gain bound ??.2.3 An Optimization Approach to Switching SurfaceDesignAn optimization algorithm presented in this thesis utilizes numerical methods tosuggest such p?S , so that the performance level ?? is minimized. The iterative descentalgorithm is explained in this section. For ease of explanation of the algorithm, weintroduce a function f which relates ?? to p?S as follows:?? = f(p?S). (2.17)Note that the explicit form of f is not available. However, for any input to thefunction f , it is possible to find the output by solving a convex optimization problem(2.16).2.3.1 The Descent Algorithm?s Main StructureThe main structure of the algorithm can be seen in Fig. 2.4 and described asfollows:191. Initialize a switching surfaces vectorpS,old := pS,init, (2.18)and compute?old := f(pS,old), (2.19)by solving the minimization problem (2.16) at given pS,old.2. Find a new vectorpS,new := pS,old + ?pS , (2.20)and compute?new := f(pS,new), (2.21)by solving the minimization problem (2.16) at given pS,new. In (2.20), we have?pS := ? d, (2.22)where d is a direction vector numerically obtained by applying the idea behindthe steepest descent method and Newton?s method explained in [26]. Also, in(2.22) is numerically set by line search. In the next subsections, the directionselection and line search steps will be described.3. Test whether the ?-value has decreased by comparing ?new with the ?-valuein the previous iteration, ?old. Assume ? is a sufficiently small positive value.? If ?old ? ?new < ?, then terminate the algorithm. The optimizers are?? := ?old and p?S := pS,old.? Otherwise, set ?old := ?new, pS,old := pS,new, and go back to Step 2.20Figure 2.4: The optimization flowchart.The algorithm is terminated at Step 3 and the optimizers, ?? and p?S , are obtained.Meanwhile, the optimized controller parameter vector denoted by p?K is computedby solving the optimization problem (2.16) at given p?S . Having obtained all thedesign parameters, controller reconstruction is done as explained in Section 2.2.Regarding the algorithm presented above, we give two remarks. First, the pro-posed algorithm guarantees that the ?-value decreases every time when the algo-rithm comes to Step 2. In other words, the algorithm is a descent algorithm. Sec-ondly, since the problem being solved is non-convex, the final solution obtained bythis algorithm may depend on the initial selection of the switching surfaces in Step 1,which can be heuristic. If the final value of ? is not satisfactory, we have to either21change the initial switching surfaces, or increase the number of divisions N1 and N2on each axis.2.3.2 Direction Selection ComputationsThere are different numerical methods for the direction selection (DS) step, two ofwhich are the steepest descent method and Newton?s method used in this thesis [26].The steepest descent method utilizes a first order approximation of the cost functionto select a descending direction. The order of approximation of this method bringsabout a low convergence rate. However, selection of a descending direction is guar-anteed. On the other hand, Newton?s method employs a second order approximationwhich makes convergence rate be comparably faster. The disadvantage inherent toNewton?s method is that it needs Hessian matrix of the objective function to bepositive definite to ensure a descending direction [26].A hybrid method is employed in this thesis to use the advantages of both meth-ods while covering the disadvantages of each. It starts with Newton?s method andcheck eigenvalues of Hessian matrix to see whether they are positive or not. If theHessian matrix is positive definite, the direction suggested by Newton?s method isadopted. Otherwise, the method is switched to the steepest descent method, so thatthe descending direction is guaranteed. In the next chapters of this thesis, the hybridmethod is applied for DS step of the algorithm. However, the results of applyingeither of above methods exclusively are also presented and compared with that ofthe hybrid method.In the following, computation steps of both the steepest descent method and New-ton?s method are described.22DS Based on the Steepest Descent MethodBased on the steepest descent method, we do first order approximation to decidethe direction to proceed. Therefore, it is assumed thatd = ??f(pS)T , (2.23)where ?f(pS) is the gradient of the function f evaluated at pS . We have?f(pS) =[?f?pS1?f?pS2...?f?pSn], (2.24)where pSi is the ith element and n is the number of elements in the vector pS . Forobtaining the ith element of the gradient vector ?f(pS), first we need to perturbthe ith element of pS in forward and backward while other elements are set to befixed, i.e.,pS,b := [pS1 pS2 ... pSi ? h ... pSn ],pS,f := [pS1 pS2 ... pSi + h ... pSn ], (2.25)where indices b and f stand for backward step and forward step around pS , respec-tively. Then we compute?f?pSi=f(pS,f )? f(pS,b)2h, (2.26)by solving the minimization problem (2.16) at given pS,f and pS,b. We iterate abovecomputations for every element of pS to obtain all the elements of ?f(pS).DS Based on Newton?s MethodAs it is explained before, Newton?s method utilizes second order approximationfor setting the direction, d. Based on this method it is assumed thatd = ?H(pS)?1?f(pS)T . (2.27)23where ?f(pS) is the gradient of the function f and computed same as what is shownabove. Also, H(pS) is the Hessian of the function f evaluated at pS . Each elementof H(pS) is expressed byHij(pS) =?2f?pSi?pSj, i, j = 1, ..., n, (2.28)where pSi is the ith element of pS and n is the number of elements in the vector pS .For computing Hij(pS), we need to perturb the first-order partial derivative of f .First assumepS,bb := [pS1 pS2 ... pSi ? h ... pSj ? h ... pSn ],pS,bf := [pS1 pS2 ... pSi ? h ... pSj + h ... pSn ],pS,fb := [pS1 pS2 ... pSi + h ... pSj ? h ... pSn ],pS,ff := [pS1 pS2 ... pSi + h ... pSj + h ... pSn ].Then we compute?2f?pSi?pSj=(f(pS,ff )?f(pS,fb)2h )? (f(pS,bf )?f(pS,bb)2h )2h, (2.29)by solving the minimization problem (2.16) at given pS,ff , pS,fb, pS,bf , and pS,bb.We need to iterate above computations for obtaining every element of H(pS).2.3.3 Line SearchThere are different methods for doing line search, such as the bisection techniqueand Armijo?s method [26]. However, applying these methods on this particular prob-lem brings about additional charge to total computational time of the algorithm.In other words, although searching along the previously selected direction to findthe best point on the line is reasonable at the first glance, it requires solving the24minimization problem 2.16 numerously. Therefore, using a technique with less com-putation for the line search is beneficial. For that matter, an alternative algorithmis proposed to set the step size as described below:1. Initialize := ?, (2.30)where ? is a sufficiently large positive pre-specified value, so that sufficientlylarge step size to proceed is guaranteed in the first iteration.2. Compute?pS := ? d, (2.31)using d obtained by the DS done before. Then set a candidate for the newswitching surface vector:pS,cand := pS,old + ?pS . (2.32)Next, Compute?cand := f(pS,cand), (2.33)3. Test whether the ?-value has sufficiently decreased by comparing ?cand with?old. Assume ?? and ? are sufficiently small positive values.? If ?old??cand?old < ?? or ? ?, then terminate the line search algorithm. isthe desired step size.? Otherwise, choose a smaller -value by setting :=2, (2.34)and go to step 2.25In other words, we generally expect that ?-value will decrease because of thedirection that has been chosen. Therefore, if the sufficient decrease in ?-value, i.e.,??, is not the case, the step size is too large that the first-order approximation is notaccurate enough. That is why we need to choose smaller to make smaller step size.Also, if the -value, which has been initialized with an adequately large value, hasbecome too small, without resulting in sufficient decrease in the ?-value in between,the algorithm is terminated as well.2.4 SummaryThe numerical algorithm was developed and explained in this chapter. Formula-tion of an LPV plant, as well as a switching LPV controller with hysteresis switchinglogic was first introduced. It was shown how the plant variation region could be di-vided into a set of overlapping subregions by means of a set of switching surfaces.The switching LPV controller design problem with fixed switching surfaces wasthen reviewed. The closed-loop L2-gain performance of the system was set as a costfunction to be minimized in this problem. The resulting minimizers would be theswitching LPV controller parameters. Afterwards, optimization of the switchingsurfaces was formulated as optimization of centres and widths of the overlappingregions. This optimization problem, which is non-convex in general, was rewrittenin the form of minimization of a cost function, the explicit form of which is not avail-able. However, obtaining the function at each point, i.e., a specific set of switchingsurfaces, is possible by solving a convex optimization problem. This feature waswidely used in the proposed algorithm to numerically minimize the cost function.26Next, the main structure of the optimization algorithm was provided. This al-gorithm is iterative and descent in which a decrease in the cost function in eachiteration is guaranteed. However, a local minimum point is obtained depending onthe initial condition, and finding the global minimum point is not guaranteed. Eachiteration consists of two steps: a direction selection (DS) step and a line search step.In DS step, the orientation of proceedings to the next point in the variables spaceis selected by means of the hybrid method which is a combination of the steepestdescent method and Newton?s method. The idea behind the hybrid method is to usethe relatively better accuracy of Newton?s method while preserving a guaranteeddescent direction obtained by the steepest descent method. The first and secondorder approximations of the cost function corresponding respectively to the steepestdescent method and Newton?s method are computed numerically by solving a seriesof convex optimization problems. In the line search step, the size of the proceed-ing along the proposed direction in the DS step is determined using an iterativealgorithm. These two steps are repeated until sufficiently large decrease in the costfunction is no more the case, resulting in termination of the optimization algorithm.27Chapter 3A Numerical ExampleThe proposed algorithm in Chapter 2 is applied to design a switching LPV con-troller with optimized switching surfaces for a mass-spring-damper system. It willbe also shown how computationally it costs to design such a controller via the de-scent algorithm rather than a full search method, described later. Moreover, a timedomain response of the closed-loop having the optimized switching LPV controlleris compared with that having a switching LPV controller with a trivial partition.283.1 Problem Statement3.1.1 Plant ModelConsider a mass-spring-damper system shown in Fig. 3.1 and represented in astate-space form asP :???????????????????????x?(t) =???0 1? km ?bm???? ?? ?Apx(t) +???01m???? ?? ?Bpu(t),y(t) =[1 0]? ?? ?Cpx(t).(3.1)Here, scalars m, k, and b are respectively mass, spring, and damping coefficients,x is a state vector consisting of the position and the velocity of the mass, u is theinput force, and y is the position measurement.3.1.2 Plant Parameter VariationThe dynamics of this system is subjected to change due to change in the massand stiffness. To cover different scenarios, we consider two different sources of plantvariation. It is assumed that the mass changes due to product variation, meaningFigure 3.1: Mass-spring-damper schematic.29that it remains constant for each product during the operation of the system. Therange of product variation of the mass is set as follows:m ? [20, 100]. (3.2)Besides, it is assumed that the spring is nonlinear, and the stiffness-value dependson the length of the spring, so it varies while the system operates. As explained in[41], the force-displacement relations for a hard spring are given byF = c1y + c2y3, c1, c2 > 0. (3.3)Then, the stiffness-displacement equation is obtained by dividing above equation byy as follows:k = c1 + c2y2. (3.4)In this example c1 and c2-values are given byc1 = 10, c2 = 50. (3.5)The displacement y is assumed to be within the range of [?1.34 1.34]. Therefore,the allowable range of the stiffness is obtained ask ? [10, 100], (3.6)where k = 10 and k = 100 correspond to y = 0 and y = ?1.34, respectively. Also,the maximum stiffness variation rate is assumed to be 100 standard unit per secondeither increasing or decreasing. Moreover, the damping coefficient b is assumed tobe fixed and equal to 5. Having defined the plant parameters, we introduce a gain-scheduling parameter ? which is a function of the varying parameters and given by? =????1?2??? =???1/mk??? , (3.7)30where ?1 is equal to the inverse of the mass for controller reconstruction purposessince the mass appears in the denominator in (3.1). The variation range and varia-tion?s rate range can be expressed by? :={? ? R2 : ?1 ? [1/100, 1/20] , ?2 ? [10, 100]},? :={? ? R2 : ?1 = 0, ?2 ? [?100, 100]}.(3.8)3.2 Control Objectives3.2.1 Weighting FunctionsFor the system P , we form a feedback structure shown in Fig. 3.2, where r is thereference signal, e, ew, u, and uw are respectively the error, the weighted error, thecontrol input, and the weighted control input, We and Wu are weighting functions,and K(?) is a switching LPV controller. In this example, we assume that Wu is ascalar constant gain, while We is a dynamical system with a state-space realizationasWe :?????x?e(t) = WeAxe(t) +WeBe(t),ew(t) = WeCxe(t) +WeDe(t),(3.9)where the vector xe is the state vector of the system We, and matrices in (3.9) havecompatible dimensions.3.2.2 A Generalized PlantThe controller K(?) is designed such that the L2-gain from r to ew and uw isto be minimized. This L2-gain minimization corresponds to reference tracking andcontrol input energy minimization. For this purpose, we obtain the input-output31Figure 3.2: A feedback structure for controller design.relation for the generalized plant G(?) in Fig. 2.1 from Fig. 3.2 as????????????????x?x?e??????ewuw???e?????????????= G(?)?????????????xxe???ru??????????, (3.10)where the matrix-valued function G(?) is given byG(?) =?????????????Ap 0 0 BpWeBCp WeA WeB 0?WeDCp WeC WeD 00 0 0 Wu?Cp 0 1 0?????????????. (3.11)In addition, we select the weighting functions as???WeA WeBWeC WeD??? =????0.0499 3.23.12 0.05??? , Wu = 0.01. (3.12)323.3 Controller DesignIn this section, control design has been done for two cases to properly addressdifferent aspects of the numerical algorithm. In the first case, 2 switching surfacevariables are assumed to be designed. Different methods used to find the optimizedpartition are applied, and simulation results of implementing switching LPV con-trollers with optimized switching surfaces and heuristic ones are compared to eachother. For the second case, applicability of the proposed algorithm to the n-variablescase is investigated. The relations between the number of switching surface vari-ables and both the performance level and computational time of convergence arediscussed.3.3.1 Case One: 2 Switching Surface VariablesFor the LPV generalized plant G(?), we design a switching LPV controller withN1 = 2 and N2 = 2, i.e., four subregions, including the center lines of the twooverlapping regions c1,1 and c2,1. For simplicity, the width vector w of the overlap-ping regions are fixed to be 0.1 times the variation range of each gain-schedulingparameter. The results of applying different methods to obtain the switching LPVcontroller with optimized switching surfaces are explained below and summarizedin Table 3.1.First, we applied the full-search method to find the optimal pS = (c1,1, c2,1) bygridding the pS-space. To do the full search, we designed the switching LPV con-troller that minimizes ?-value for each gridded point by solving (2.16). The result ofthe full-search method is shown in Fig. 3.3. As it can be observed in this figure, theminimum performance level ?? = 10.06 is achieved at p?S = (0.038, 23). Although33Figure 3.3: Functionality of ? with respect to pS = (c1,1, c2,1) obtained by thefull-search method.it is guaranteed that this method finds the optimal pS , up to the resolution ofthe gridding, its main disadvantage is that the computational complexity increasesexponentially as the dimension of ? does.Secondly, we applied the descent algorithm proposed in Section 2.3. Having em-ployed the hybrid method for the DS step and having started at ps,init = (0.03, 55),the middle point of the plant variation region, the algorithm converged to the samepair of (??,p?s) with much less computational time in comparison to the full-searchmethod. The advantages of this method will be more remarkable if plant has twovarying parameters and more divisions are desired in each axis. Although dependingon the initial point, the algorithm could have converged to the other local minimumpoint, one could try different initial points to find the global minimum while stillspending much less time compared to the full search method.34Table 3.1: Results of applying different numerical methods to optimize the switchingsurfaceMethod Initial Point Final PointComp.Time (min)Full Search N/A?? = 10.06p?s = (0.038, 23)228Descent Algorithm(Hybrid)?init = 10.24ps,init = (0.03, 55)?? = 10.06p?s = (0.038, 23)9Descent Algorithm(Steepest Descent)?init = 10.24ps,init = (0.03, 55)?? = 10.06p?s = (0.038, 23)15Descent Algorithm(Newton?s)?init = 10.24ps,init = (0.03, 55)N/A N/AThirdly, both the steepest descent method and Newton?s method were used singlyfor the DS step, instead of the hybrid method. It was seen that having the sameinitial point, applying Newton?s method did not lead to convergence to any localminimum point, since Hessian matrix is negative definite around the initial pointtrying to maximize the cost function. On the other hand, applying the steepestdescent method singly for the DS step led to convergence to (??,p?s). However, thesteepest descent method utilizing first order approximation took one and half timeslonger than the hybrid method to converge, since the latter one employs second orderapproximation at some iterations in which Hessian matrix is positive definite.In order to meaningfully compare the computational time of applying differentmethods, computations were done with the same number of gridded points of theplant variation region, equal to 2500, as well as the same 3.20 GHz CPU computerwith 32.0 GB RAM and MATLAB version 8.1.0.604 (R2013a).35Simulation ResultsIt is important to note that improvement of the worst case L2-gain, ?, doesnot necessarily correlate to improvement of the time domain simulation results ofa specific plant parameter trajectory. However, advancement of ? could increasethe possibility of advancement in time domain responses for trajectories that areoften the case in each particular example. Therefore, it is inherent to the controllerdesign procedure to go back and forth and readjust the switching surfaces as wellas weighting functions, so that improvement in time domain responses for requiredtrajectories is guaranteed.Nevertheless, for the system explained above, improvement of the step response fordifferent mass-values in the product variation range specified in (3.2) was observed.In this section, the step responses of two switching LPV controllers for just a samplemass-value are compared to each other in Fig. 3.4. First controller is the optimalone, the switching surfaces of which are placed at p?S = (0.038, 23). The other oneis designed with a heuristic partition where the switching surfaces are placed atthe middle of each plant variation region?s axis, pS = (0.03, 55). As can be seenin Fig. 3.4, the optimized controller generates step response with smaller overshootand less oscillations.3.3.2 Case Two: n Switching Surface VariablesIn the second case study, more than two number of switching surface variables arerequired to be designed. This work was done by making different assumptions on thenumber of divisions on each axis of the plant variation region. These assumptionsare summarized in Table 3.2. As seen in this table, the widths of the overlapping36Figure 3.4: Comparison of step responses between the optimal controller and aheuristic one with m = 60.regions have been assumed to be fixed for simplicity. However, varying width couldhave also been assumed. For the case that number of variables is eight, a schematiclayout of the plant variation region with assumed switching surfaces are shown inFig. 3.5.Table 3.2: Assumptions on the switching surfacesN1 N2 Width Number of Variables2 2 Fixed 22 3 Fixed 33 3 Fixed 43 4 Fixed 54 4 Fixed 65 4 Fixed 75 5 Fixed 837Figure 3.5: Plant variation region with 5 divisions on each axis.A switching LPV controller with optimized switching surfaces is designed foreach set of assumptions in Table 3.2 by applying the descent algorithm. Employingboth the hybrid method and steepest descent method have resulted in the same finalswitching surfaces p?s and performance level ??. However, applying Newton?s methodsingly for the DS step of the algorithm does not lead to convergence to any localminimum point. The initial and final performance levels are shown in Fig. 3.6. Itcan be seen that ?-value decreases when the number of switching surface variablesincreases?an observation which coincides with our expectations.Fig. 3.7 shows the time of convergence for each case. It can be seen that thesteepest descent method results in smaller computational time for high number ofvariables than the hybrid method, different from two-variables case. The computa-tional time of applying the hybrid method increases dramatically by increasing thenumber of variables since Hessian matrix is not positive definite and useless in most382 3 4 5 6 7 89.89.91010.110.210.3Performance LevelNumber of Switching Surface Variables Initial Performance Level, ?initFinal Performance Level, ?*Figure 3.6: The initial and final performance level for different number of switchingsurface variables while applying the descent algorithm.of iterations. Therefore, computing second order approximation of the functiondoes not help the algorithm to speed up converging, while it charges the algorithmin terms of computational time.Convergence computational time of applying the steepest descent method has beenreplotted singly in Fig. 3.8, so that the trend can be seen more clearly. Lookingat these three figures demonstrates that increasing number of switching surfacevariables can burden the design process computationally, and it should be evaluatedthat if such computation is beneficial. In other words, it is important to maintain abalance between the computational time and improvement in the performance levelby choosing an appropriate number of switching surface variables which can be aninteresting topic for future research in this field.392 3 4 5 6 7 80200040006000800010000Number of Switching Surface VariablesComputational Time (min) The Steepest Descent MethodThe Hybrid MethodFigure 3.7: Computational time of convergence of the descent algorithm for differ-ent number of switching surface variables, having used the hybrid method and thesteepest descent method for the DS step.2 3 4 5 6 7 8050100150200250Computational Time (min)Number of Switching Surface VariablesFigure 3.8: Computational time of convergence of the descent algorithm for differentnumber of switching surface variables, having used the steepest descent method forthe DS step.403.4 SummaryThe proposed numerical algorithm in this thesis was then applied to a numericalexample in this chapter. A simple mass-spring-damper system was used as a plantfor which controllers were designed. It was assumed that the plant?s mass is subjectto product variation, and the stiffness-value of the spring changes during the oper-ation of the system as a result of nonlinearity in the spring. These two parametersform a plant variation region with specified variation range and variation rate range.This problem was assumed as reference tracking one in which the position of themass is required to track a reference signal by means of applying a force generatedby a controller.For controller design, two cases were studied. First, two divisions on each axis ofthe plant variation region were assumed, and the widths of the overlapping regionswere set to be fixed. Therefore, 2 switching surface variables, i.e., centres of theoverlapping regions, are sought to be designed. Results of applying different meth-ods to find the switching LPV controller with optimized switching surfaces wereobtained in this chapter. It was shown that applying the proposed algorithm inChapter 2 is superior than using the full search method in terms of computationalcost. Moreover, it was seen that using Newton?s method singly for the DS stepof the descent algorithm does not lead to convergence to a local minimum pointsince Hessian matrix of the cost function at some iterations are not positive definite.However, the algorithm perfectly worked while the hybrid method or the steepestdescent method were used for the DS step of the descent algorithm. Afterwards,closed-loop performance of the switching LPV controller with optimized switchingsurfaces and that with heuristic switching surfaces were juxtaposed to each other.41Although advancement in bounds of the closed-loop L2-gain performance is onlyguaranteed, it was shown time-domain performance of the system with a specificplant parameters trajectory is more likely to be improved if optimized switchingsurfaces are employed.In the second case, the control design problem with more than 2 switching sur-face variables were studied by dividing the plant variation region into more than4 subregions. Final performance levels having different number of variables wereobtained and compared to each other. Also, computational time of convergenceof the descent algorithm for the different cases were plotted. It was seen that inhigh number of variables, the computational time increases dramatically, while thefinal performance level is not remarkably decreased. Therefore, cost benefit analysissuggests that lots of divisions on the axes of the plant variation region is not alwaysfavourable.42Chapter 4Application inMagnetically-Actuated OpticalImage StabilizerImage blurriness as a result of involuntary hand movement has always been anissue for the users while taking photos. Especially at low-light conditions, thisphenomenon is intensified because the shutter speed is slow. In photography, theeffective length of time when a camera?s shutter is open is called shutter speed. Inorder to avoid this annoying blurriness and stabilize the image, there are differentmethods widely used in industry which are categorized into software-based andhardware-based technologies.Software-based image stabilization methods are mainly applied for lower-end cam-eras [19], while hardware-based technologies are used for higher-end ones. Althoughhardware-based mechanisms are more effective in stabilizing the images, most mo-43bile cameras are not equipped with them because of size constraints. However, ahardware-based method is designed recently for miniaturized applications in whicha lens-shifting mechanism is used to stabilize the image [37]. The idea is that whenthe whole camera including the micro lens moves due to hand shake, a controllersystem tries to compensate for the hand movement and move back the lens to itsright position in order to restore the image in a good quality with no blurriness.This mechanism utilizing a magnetic actuator is used for not only compensation forundesired vibrations but also auto focusing purposes.This device called optical image stabilizer (OIS) is built in micro scale. Pro-duction variation in micro-scale devices is considerable and causes huge effect onthe dynamics of the system from one product to another. In micro-scale OISs, thechange in dynamics of the plant is caused by variation in the equivalent mass andstiffness of the system as a result of imprecise cutting. This variation in dynamics isconsidered in this chapter for designing a switching LPV controller to stabilize theimage.4.1 Problem Statement4.1.1 Physical SystemThe physical system described in [37] consists of a sufficiently flexible platform atthe center of which a micro lens is placed. See Fig. 4.1. Four magnets are attachedto the platform. After some air gap, there are four shielded electromagnetic coilsunder each magnet. Depending on the currents in the electromagnetic coils anddepending on the air gap between each magnet and coil, four vertical forces will beapplied to the whole platform. Due to the flexibility and elasticity of the platform,44it will work like a spring at each side connected to the chassis of the camera. Thesuperposition of the forces applied to the platform will generate some movements,namely vertical motion along the z axis and rotational motions about the x and they axes. More technically, the plant including a platform, a lens, and magnets hasoutputs of (z, ?x, ?y) measured using strain gauge sensors, and inputs of (F, Tx, Ty).The set of the output defines the position and attitude of the whole platform withthree degrees of freedom in the space, and the set of the input consists of the netvertical force and torques around the x and the y axes. These torques are generatedbecause of the eccentricity of four forces applied by interaction of four pairs of coiland magnet.Figure 4.1: Physical layout of the lens-shifting image stabilizer [37].The inputs to the plant are the outputs of an actuator consisting of coils, magnets,and some electric circuits. The inputs to the actuator are four currents going throughthe coils, and the air gap between each pair of coil and magnet. This air gap can bedetermined by vertical and angular positions of the platform which are the outputsof the plant, so the air gap can not be manipulated directly. Therefore, the fourcurrents are the only inputs to the actuator which can be determined by a controller.45A typical block diagram of the plant and the actuator can be seen in Fig. 4.2.Figure 4.2: Block diagram of the plant and actuator.4.1.2 Control ProblemNow that the plant, actuator, and sensor are explained, it is desired to find a con-troller which inputs such currents to the actuator that the outputs of the plant, i.e.,(z, ?x, ?y), track a reference signal which is obtained based on the hand movement.First, the entire hand movement, including voluntary and involuntary ones, are mea-sured using some gyroscopes or accelerometers. Then the measured movements arepassed through a high pass filter in order to ignore zero frequency disturbance, sincewe do not want to reject intentional hand movements. In other words, by applyinga control system we intend to reject the involuntary hand shakes which have highfrequencies. Therefore, we pass a part of the signal which has such frequencies andreject the rest. Afterwards, the oscillatory signal is multiplied by ?1, and eventually,the reference signal is built. This multiplication is done because we would like totrack negative involuntary hand shake in order to compensate for that. The blockdiagram of the closed-loop system can be seen in Fig. 4.3.46Figure 4.3: Block diagram of the closed-loop system.Now, if the controller enforces the output of the plant to track the obtainedreference signal perfectly, the overall movement of the micro lens will be almost zeroand the image will be clear and in good quality. Moreover, there is a bound for thecurrents going through the coils which should be considered in the control designprocess. Additionally, due to the product variation, the dynamics of the plantchanges from one device to another, and designing a single controller for all thedevices might not be satisfying. Therefore, an optimized switching LPV controlleris designed in this chapter to achieve the design objectives explained above.4.1.3 Plant ModelAfter applying Newton?s 2nd law and doing realization, the state space form ofthe plant dynamics is obtained in [37] which has the formP :?????x?p = Apxp +BpuFy = Cpxp +DpuF, (4.1)47wherexp =?????????????????zz??x??x?y??y?????????????????, uF =??????FTxTy??????, y =??????z?x?y??????, (4.2)andAp =?????????????????0 1 0 0 0 0?keqMeq?beqMeq0 0 0 00 0 0 1 0 00 0 ?kxIx?bxIx0 00 0 0 0 0 10 0 0 0 ?kyIy?byIy?????????????????, Bp =?????????????????0 0 01Meq0 00 0 00 1Ix 00 0 00 0 1Iy?????????????????,Cp =??????1 0 0 0 0 00 0 1 0 0 00 0 0 0 1 0??????, Dp = 0 (4.3)In this state space form, Meq is the equivalent mass of the platform, F is the totalvertical force, keq is the equivalent translational stiffness and beq is the equivalentdamping coefficient. Also, Ix and Iy are the mass moments of inertia, Tx and Ty arethe equivalent torques, kx and ky are the equivalent rotational stiffness, and bx andby are the rotational damping constants about each axis.It can be seen that the matrices AP and BP are block diagonal which provides uswith the situation to deal with each degree of freedom independently in controller48design. Therefore, the model can be broken down into three subsystems for con-troller design purposes, and each controller is designed for each subsystem. Thisapproach to the problem needs to assume the force vector uF as the control input tothe plant, while the actual control input for implementation purposes is the currentsgoing through the coils. To deal with this issue, we need to have the input-outputequation of the actuator. The simplified version of this equation obtained in [37]can be expressed as??????FTxTy??????= M ???????????i1i2i3i4??????????. (4.4)We haveM = ???????1 1 1 1?R ?R R R?R R ?R R??????, (4.5)where ? and R are constants shown in Table 4.1. Therefore, the force vector uF isconsidered as the control input to be obtained by designing the controller, and theactual control input, i.e., the currents to the coils, is computed as follows:??????????i1i2i3i4??????????= MT (MMT )?1??????FTxTy??????. (4.6)See Fig. 4.4. Note that if we dealt with the currents as the control input in thecontrol design process, the combination of the plant and the actuator as the new-defined plant would not have block diagonal Ap matrix and would not be brokendown into three subsystems.49Figure 4.4: Actual control input computation.Table 4.1 shows all the nominal values of the parameters of the plant and actuatorwhich are obtained in [37].Table 4.1: System parameters [37].Parameter Value UnitMeq 1.49? 10?4 kgkeq 89.09 N.m?1beq 2.30? 10?3 N.m?1.s?1Ix = Iy 9.40? 10?10 kg.m2kx = ky 2.40? 10?3 N.rad?1bx = by 3.89? 10?8 N.rad?1.s?1? 0.105 N.A?1R 2.5? 10?3 mIn the rest of this chapter, we assume one of the subsystems, i.e., the one whichmodels rotation of the platform around x-axis, as the plant for which we are goingto design controller. The state space form of this system can be given byPx :?????x?px = Apxxpx +BpxTx?x = Cpxxpx +DpxTx, (4.7)50wherexpx =????x??x??? , (4.8)andApx =???0 1?kxIx?bxIx??? , Bpx =???01Ix??? ,Cpx =[1 0], Dpx = 0 . (4.9)4.1.4 Plant Parameter VariationAs it is explained before, the change in dynamics of the system in (4.7) is causedby product variation which leads to variation in the equivalent mass moments ofinertia, Ix and the equivalent rotational stiffness kx of the system as a result ofimprecise cutting. These two parameters, Ix and kx, form a plant variation regionin which the bounds of the variation are defined based on manufacturing tolerances.These manufacturing tolerances are assumed to be 75% of the nominal parametervalues, I?x and k?x. Moreover, once the product is manufactured, it is assumed thatthe parameters of the plant are not varying during the operation of the system. Theplant variation vector ? is defined as? =????1?2??? =???1/Ixkx??? , (4.10)where ?1 is assumed to be equal to the inverse of the plant?s moment of inertiadue to controller reconstruction purposes. Also, the plant variation region ? and51variation rate region ?r are expressed as? :=?????? ? R2 :?1 ? [0.6079, 4.2553] 1/(g.mm2),?2 ? [0.6000, 4.2000] g.mm2/ms2?????,? :={? ? R2 : ?1 = ?2 = 0}.(4.11)4.2 Control ObjectivesAs mentioned in the previous sections, this is a reference tracking control problem.Moreover, there is a constraint on the control input to be applied. To formulate theproblem in an appropriate manner so that these objectives are sought, weightingfunctions and the generalized plant are introduced in this section.4.2.1 Weighting FunctionsThe block diagram of the control problem can be seen in Fig. 4.5 where rx isthe reference signal for rotation tracking around x-axis, ex, e?x, Tx, and T?x arerespectively the error, the weighted error, the control input, and the weighted controlinput. Also, K(?) is a switching LPV controller to be designed. We is a weightingfunction expressed byWe :?????x?e(t) = WeAxe(t) +WeBe(t),ew(t) = WeCxe(t) +WeDe(t),(4.12)where xe is the state vector of the system and???WeA WeBWeC WeD??? =????0.000866 810.83 0.5??? . (4.13)Moreover, Wu is assumed to be a static gain given byWu = 0.01. (4.14)52The selection procedure of a weighting function has been explained in [11]. Theprocedure for the current example can be briefly expressed in two steps. First, theshape of the weighting function in frequency-domain is selected such that it coversthe FRF measurements of the hand movements. By this selection, the weightingfunction penalizes the performance level more around frequencies that have rela-tively high amplitudes. Next, parameters representing exact form of the functionsare selected by trial and error such that the final performance of the system issatisfactory.Figure 4.5: A feedback structure for controller design.4.2.2 A Generalized PlantHaving defined the weighting functions, we can introduce the control objective asto minimize the error and the control input by minimization of the L2-gain from rto e?x and T?x. Based on this control objective, the generalized plant in (2.1) relating53the inputs and outputs as????????????????x?pxx?e??????e?xT?x???e?????????????= G(?)?????????????xpxxe???rxTx??????????, (4.15)is obtained from Fig. 4.5 as follows:G(?) =?????????????Apx 0 0 BpxWeBCpx WeA WeB 0?WeDDpx WeC WeD 00 0 0 Wu?Cpx 0 1 0?????????????. (4.16)4.3 Controller DesignUsing different methods explained in Chapter 2, switching LPV controllers withN1 = 2 and N2 = 2 are designed for the LPV generalized plant described above. Likethe numerical example in Chapter 3, having two divisions in each axis leads to fournumber of subregions in the plant variation region. The width of the overlappingregions are assumed to be zero since this is a time invariant system, and it doesnot need to have hysteresis switching logic. In other words, no switching occursduring the operation of the system. Having made the assumptions on the number ofsubregions and the width of the overlapping regions, we introduce pS = (c1,1, c2,1)as the switching surface variable vector.54Like what was done for the numerical example in Chapter 3, first, the full searchmethod was applied by gridding the pS-space to obtain the minimum point. It canbe seen in Fig. 4.6 that the optimal ?? = 6.76 was obtained at p?S = (2.00, 1.95).Figure 4.6: Functionality of ? with respect to pS = (c1,1, c2,1) obtained by thefull-search method.Secondly, the descent algorithm developed in Chapter 3 employing the hybridmethod for the DS step was applied. Starting from pS,init = (2.43, 2.40), i.e., themiddle point of the plant variation region, the algorithm converged to ?? = 6.82and p?S = (2.18, 2.00), as shown in Fig. 4.7. It can be seen that the final pointobtained by applying the descent algorithm is close to the minimum point obtainedby the full search method. However, it is not the exact same point. The reason forthat is because of the discontinuity in the first derivative of the function f in (2.17)which can be observed in Fig. 4.6. In other words, the algorithm gets stuck in adiscontinuous-first-derivative point where the numerically-computed derivatives do55not suggest necessarily a descending direction. Thus, the algorithm terminates atthat point which is not the local minimum point.In fact, the descent algorithm perfectly works with an objective function f thefirst derivative of which is continuous, and it exactly converges to the minimumpoint. Nevertheless, since the algorithm is descent, it guarantees decrease in thecost function and finding a point with smaller performance level ? compared to thatof achieved by a trivial choice of the switching surface. Moreover, computationalcost of using the descent algorithm over the full search method is remarkably small.See Table 4.2.Figure 4.7: Convergence path of the descent algorithm having employed the hybridmethod or the steepest descent method.Thirdly, we used the steepest descent method and Newton?s method singly forthe DS step of the descent algorithm. It was seen that Newton?s method was not56Table 4.2: Results of applying different numerical methods to optimize the switchingsurfaceMethod Initial Point Final PointComp.Time (min)Full Search N/A?? = 6.76p?s = (2.00, 1.95)114Descent Algorithm(Hybrid)?init = 6.96ps,init = (2.43, 2.40)?? = 6.82p?s = (2.18, 2.00)5Descent Algorithm(Steepest Descent?init = 6.96ps,init = (2.43, 2.40)?? = 6.82p?s = (2.18, 2.00)3Descent Algorithm(Newton?s)?init = 6.96ps,init = (2.43, 2.40)N/A N/Aapplicable since Hessian matrix of the objective function f is negative definite allthe time, as the convexity of the cost function, seen in Fig. 4.6, demonstrates suchresult. Due to the same reason, applying the steepest descent method led to the sameconvergence path as applying the hybrid method did, as shown in Fig. 4.7. In otherwords, second order approximation of the objective function was never used in anyiteration of the algorithm employing the hybrid method since Hessian matrix of fwas always negative definite. However, computational time of applying the steepestdescent method for the DS step was smaller than that of applying the hybrid methodbecause at each iteration, Hessian matrix of f should be computed additionally inthe hybrid method. See Table 4.2.Also, in order to meaningfully compare the time, computations were done withthe same number of gridded points of the plant variation region, equal to 2500,as well as the same 3.20 GHz CPU computer with 32.0 GB RAM and MATLABversion 8.1.0.604 (R2013a).574.4 Simulation ResultsSimulation results of implementing two switching LPV controllers are shown inFig. 4.8. One of the controllers has the optimized switching surface obtained byapplying the descent algorithm, and the other one has a heuristic switching surfaceassumed to be the one placed at the middle of the plant variation region. Thereference signal to be tracked is a sinusoidal signal with the frequency of 20 Hzand the amplitude of 10?3 rad which is the working condition of the actual system.The error signal which is the difference between the attitude of the platform ?x andthe reference signal r is plotted in the figure below. The simulation was done fordifferent plants covering the whole product variation region. As a result, two errorsignal envelopes each corresponding to one of the switching LPV controllers wereobtained.As seen in this figure, the worst case error has been improved by applying theoptimized switching LPV controller. However, for many other plants in the productvariation region, the amplitude of the error signal has increased compared to thecase having a heuristic switching surface. This is not surprising since as explainedbefore, improvement of the performance level ? guarantees improvement of just theworst case L2-gain from r to e?x and T?x.4.5 SummaryIn this chapter, a magnetically-actuated optical image stabilizer was used as acontrol application. The micro-scale image stabilizer system studied in this thesisuses a hardware-based mechanism to suppress vibrations produced by involuntary58Figure 4.8: Error envelope comparison between the cases having an optimal con-troller and a heuristic one.hand-movements, so that the image becomes clear with no blurriness. After re-viewing the mechanical system and the operation of the image stabilizer, it wasshown how the plant is modelled as a mass-spring-damper system. The mass andstiffness-value of a system in micro-scale forms are subject to product variations.Therefore, the plant variation region was defined for this system as a result of suchproduct variations. The control objective in this control design problem is that theorientation of the platform containing a lens at the centre tracks a reference signal,which is the negative of the hand-movements.For designing a switching LPV controller, the plant variation region was dividedinto 4 rectangular subregions with fixed widths of the overlapping regions. Thus, 2switching surface variables, i.e., the centres of the overlapping regions, were remainedto be designed. The switching LPV controller with optimized switching surfaces was59designed using the descent algorithm presented in Chapter 2. Different methods forthe DS step of the algorithm were employed and convergence computational time ofapplying each of them was computed and compared with one another. Also, the fullsearch method was applied to validate that the final point obtained by the descentalgorithm is actually a local minimum or sufficiently close to a local minimum.Besides, it was shown how fast is the algorithm compared to the full search method.Finally, simulation results of implementing two switching LPV controllers, onewith the optimized switching surfaces and the other one with heuristic switchingsurfaces, were juxtaposed together. For each controller, simulation was done bysampling the product variation region, and an envelope of response was obtained. Byusing the switching LPV controller with optimized switching surfaces, improvementin the closed-loop performance of the worst case product was observed. Nevertheless,many products showed relatively worse responses. This phenomenon coincides withour expectation that applying switching LPV controller with optimized switchingsurfaces can only improve the worst case L2-gain performance of the closed-loopwhich relates to the worst case product.60Chapter 5Application in Spark IgnitionInternal Combustion EnginesOne of the major concerns of using Internal Combustion Engines (ICE) is toreduce hazardous emissions to health and environment. One of the technologies usedin modern automotive engines for that matter is Three-Way Catalytic Converter(TWC). The TWC is located in the exhaust line of the engine, and through somechemical reactions, improves the quality of exhaust gas. Some typical reactions inTWC are reducing nitrogen oxides to nitrogen and oxygen, oxidizing hydrocarbonsto carbon dioxide and water, and oxidizing carbon monoxide to carbon dioxide [16].In general, TWC can be assumed as a buffer to small fluctuations of air and fuelflow. This happens by storing oxygen on the catalytic surface when excess oxygenexists in exhaust gas and releasing oxygen when deficiency in oxygen occurs.As explained in [36], TWC requires at least two conditions to work properly:First, it is important that oxygen content on the catalytic layer should be as near as61possible to half of its capacity, and second, exhaust gas chemical composition shouldbe near stoichiometric composition. Therefore, it is important to have air-fuel ratioas close as possible to stoichiometric air-fuel ratio. When the driver pushes thethrottle pedal, the throttle valve is opened and allows more air into the engine?scylinder. To compensate for the excess amount of air, Engine Control Unit (ECU)changes the duration of the fuel injection pulses.In modern control systems, Universal Exhaust Gas Oxygen (UEGO) sensor hasbeen used to compensate for drawbacks in conventional control systems in whichstatic maps and feedforward controllers are used to determine the amount of injectedfuel. In modern systems, air-fuel ratio in the exhaust gas is measured and based onthe measurement, a feedback is sent to ECU [2]. Simultaneous use of feedback andfeedforward controller can perform fast and accurate air-fuel ratio control in engine.In the normally aspirated, port injected, Spark Ignition (SI) Internal Combustion(IC) engine, air flow is controlled by driver using throttle pedal. In intake manifold,air flow is divided to separate stream for each cylinder. Afterwards, fuel is injectedto each stream and air-fuel mixture enters cylinders. After combustion processin each cylinder, exhaust gases are combined in exhaust manifold and go into theTWC. Measurement of air-fuel ratio takes place before and after TWC using UEGOsensors.There are three remarks about the dynamics of an IC engine: In the first place,there is no rational transfer function for an IC engine since its discrete strokes causea pure delay in the system. In the second place, since the engine dynamics is nottime invariant, the pure delay could varies notably with various parameters such as62engine speed and air flow. Finally, as far as the air-fuel ratio is a fraction of the ICengine?s inputs, i.e., air and fuel, the system could only be expressed in a nonlinearform instead of a simple linear transfer function.Different control design methods have been used so far in order to control air-fuelratio in SI IC engines, e.g., PI control [1], H? Robust Control [29], Kalman Filter[31, 43] Adaptive Control [12, 40, 49, 50], LPV Control [51, 52], and switching LPVcontrol [36].In [36], the strengths and weaknesses of applying each control design methodmentioned above are discussed, and a switching LPV controller with a heuristicselection of switching surfaces is designed. In current research, an extension to theresearch done in [36] is presented by designing a switching LPV controller withoptimized switching surfaces using the descent algorithm.5.1 Problem Statement5.1.1 Plant Parameters VariationAs explained before, the dynamics of the IC engine change due to change inengine speed N and air flow m?air. As shown in Fig. 5.1, these two parametersform a plant variation region in which the engine speed is assumed to change from800 rpm to 6000 rpm because of the limits on the operation of the engine. Also, theair flow is assumed to change from 10% to 100%. The red dashed line in Fig. 5.1shows an example of the varying parameters trajectory in the plant variation region.Additionally, the maximum rate of change of the varying parameters are assumed tobe 6000 rmp/s and 100% per second for the engine speed and air flow, respectively.63Figure 5.1: Plant variation region.For controller reconstruction purposes, the scheduling parameter vector, ? is definedas? =????1?2??? =???1/m?air1/N??? . (5.1)We also have? :={? ? R2 : ?1 ?[1100% ,110%], ?2 ?[16000 rpm ,1800 rpm]},? :={? ? R2 : ?1 ?[?100%/s(10%)2 ,100%/s(10%)2], ?2 ?[?6000 rpm/s(800 rpm)2 ,6000 rpm/s(800 rpm)2]},(5.2)where ? and ? are the plant variation region and variation rate region, respectively.In the next subsection, the functionality of the transfer function model and a statespace form of the IC engine with respect to the scheduling parameter ? will beshown.645.1.2 Plant ModelBelow is the transfer function obtained in [36] to model the IC engine:?m?fuel=g(?)sT (?) + 1?6? 2s?(?)6 + 4s?(?) + (s?(?))2, (5.3)where the input to the system is the fuel mass flow m?fuel which is a control input tomanipulate the equivalent air-fuel ratio, denoted by ?, as the output of the system.Also, as modelled in [36] and shown in Appendix A, we haveg(?) =14.7m?air, (5.4)T (?) ?90N, (5.5)?(?) =180N+5.33m?air, (5.6)where g, T , and ? are parameters of the modelled transfer function and depend onthe varying parameters, m?air and N . After realization, a state space form of theplant is given byP (?) :?????x?p = Ap(?)xp +Bp(?)m?fuel? = Cp(?)xp +Dpm?fuel(5.7)where x ? R3 is the state vector of the plant andAp(?) =???????1T (?)6?(?)2?2?(?)0 0 10 ?6?(?)2?4?(?)??????, Bp(?) =??????001??????,Cp(?) =[g(?)T (?) 0 0], Dp = 0. (5.8)65In the next section we will show how to design a controller for the above varyingplant.5.2 Control ObjectivesIn this problem, we would like to design an optimized switching LPV controller,K(?), such that disturbance d is rejected, and the air-fuel ratio ? tracks a referencesignal which is not necessarily stoichiometric. In fact, it may happen that duringwarm up or while a correction in oxygen storage level of TWC is needed, the air-fuel ratio mixture is required to be rich or lean. So, if the air-fuel ratio tracks thisreference well, the overall performance of the IC engine will be improved. Since thisis a reference tracking problem, an integrator is used to help removing tracking error.In addition, there is a limitation for the control input, i.e., m?fuel, which should beconsidered in the controller design process. In order to achieve the control objectivesdescribed above, wighting functions and a generalized plant are introduced in thefollowing.5.2.1 Weighting FunctionsA schematic structure of the closed-loop system is shown in Fig. 5.2. In order tomake the performance channels, ew and uw, two weighting functions We and Wuare employed for the error signal e and control input u, respectively, as shown inFig. 5.2. The weighting functions are realized into the state space forms as follows:We :?????x?e = WeAxe +WeBeew = WeCxe +WeDe, (5.9)66Figure 5.2: Block diagram of the closed-loop system.andWu :?????x?u = WuAxu +WuBuuw = WuCxu +WuDu, (5.10)where xe and xu are the state vectors. The weighting functions used in this thesisare first order with same parameters set in [36] and given by???WeA WeBWeC WeD??? =????502.5 10?55.02? 105 10?5??? , (5.11)and???WuA WuBWuC WuD??? =????9.95e? 105 1?9.949e? 105 1??? . (5.12)5.2.2 A Generalized PlantHaving set the weighting functions, we define a generalized plant which is used incontroller design process. The generalized plant with varying parameters in Fig. 5.367has the state space realization as follows:G(?) :???????????x? = A(?)x+B1(?)w +B2(?)uz = C1(?)x+D11(?)w +D12(?)uy = C2(?)x+D21(?)w +D22(?)u, (5.13)wherex =??????????xpxexuxi??????????, (5.14)is the state vector of the generalized plant. xi is the state of the integrator addedto the model. This integrator is actually a part of the controller. However, it isaggregated to the generalized plant for control design purposes. y, the output of theintegrator, is sent to a controller to be designed, and u is the control input to thegeneralized plant.Figure 5.3: Generalized plant.68Also, we havew =???dr??? , z =???ewuw??? , (5.15)which are the exogenous input and the performance channels vector, respectively. dis the disturbance added to the output of the IC engine plant, and r is the referencesignal to be tracked. After doing algebraic calculations, the matrices in (5.13) areobtained as follows:A(?) =??????????Ap(?) 0 0 0?WeBCp(?) WeA 0 00 0 WuA 0?Cp(?) 0 0 0??????????B1(?) =??????????0 0?WeB WeB0 0?1 1??????????B2(?) =??????????Bp(?)0WuB0??????????C1(?) =???0 WeC 0 00 0 WuC 0??? C2(?) =[0 0 0 1]D11(?) =????WeD WeD0 0??? D12(?) =???0WuD???D21(?) =[0 0]D22(?) = 0.(5.16)As it can be seen, the matrices of the generalized plant are functions of ? in generaldue to the functionality of the IC engine model with respect to ?.695.3 Controller Design5.3.1 Partition Design AssumptionsAs explained in Chapter 2, in order to design a switching LPV controller, weneed to design a partition for the plant variation region as well as a couple ofcontrollers each suitable for a subregion. To design the partition, we need to makesome assumptions beforehand.? The number of divisions on each axis of the plant variation region in Fig. 5.1is assumed to be 2, which means that we have 4 subregions in total.? The shape of each subregion is taken to be rectangular.? The width of the overlapping region between each two subregions are assumedto be fixed and is given bywor =???w1w2??? = 0.1 ? L, (5.17)where L is a vector containing the variation length of the gain-schedulingparameter ?.Fig. 5.4 shows an example of a partition on the plant variation region with theabove assumptions. Having made these assumptions, we would like to design thesubregions?(i,j), i = 1, 2, j = 1, 2, (5.18)which means designing the centre position of the overlapping regions. Therefore,the switching surface vector pS in (2.15) is given bypS =???c1,1c2,1??? , (5.19)70where c1 and c2 are parameters to be designed.Figure 5.4: The plant variation region with a partition.5.3.2 Optimized Switching LPV Controller DesignAs well as designing subregions, we would like to design LPV controllersK(i,j), i = 1, 2, j = 1, 2, (5.20)each of which is associated with a subregion in (5.18), in order to minimize thebound ? of the closed-loop L2-gain from the input w to the performance channel z,i.e.,min ? subject to ?z?2 < ? ?w?2 , (5.21)for any possible trajectory of ? within the region ? and the variation rate ?? withinthe region ? set in (5.2). Each LPV controller in (5.20) has the formK(i,j)(?) :?????x?K = A(i,j)K (?)xK +B(i,j)K (?)yu = C(i,j)K (?)xK +D(i,j)K (?)y, i = 1, 2, j = 1, 2. (5.22)71As explained in Chapter 2, the controller parameters are reconstructed using thefollowing equations:D(i,j)K (?) := D?(i,j)K (?),C(i,j)K (?) :=[C?(i,j)K (?)?D(i,j)K (?)C2(?)Y]MT ,B(i,j)K (?) :=[N (i,j)(?)]?1 [B?(i,j)K (?)?X(i,j)(?)B2(?)D(i,j)K (?)],A(i,j)K (?) :=[N (i,j)(?)]?1 [A?(i,j)K (?)?N(i,j)(?)B(i,j)K (?)C2(?)Y?X(i,j)(?)B2(?)C(i,j)K (?)MT?X(i,j)(?){A(?) +B2(?)D(i,j)K (?)C2(?)}Y]/MT ,i = 1, 2, j = 1, 2,in which Y is assumed to be fixed, and A?(i,j)K (?), B?(i,j)K (?), C?(i,j)K (?), D?(i,j)K (?), andX(i,j) are all assumed to be linear functions of the scheduling parameter ?, expressedasA?(i,j)K (?) := A?(i,j)0 +1m?fuelA?(i,j)1 +1N A?(i,j)2 ,B?(i,j)K (?) := A?(i,j)0 +1m?fuelB?(i,j)1 +1N B?(i,j)2 ,C?(i,j)K (?) := C?(i,j)0 +1m?fuelC?(i,j)1 +1N C?(i,j)2 ,D?(i,j)K (?) := D?(i,j)0 +1m?fuelD?(i,j)1 +1N D?(i,j)2 ,X(i,j)(?) := X(i,j)0 +1m?fuelX(i,j)1 +1NX(i,j)2 ,i = 1, 2, j = 1, 2,(5.23)The elements of these matrices are stacked together in the controller parametersvector pK and subjected to be designed.Now that the variables pK and pS are introduced for the IC engine problem, theoptimization problem (2.15) repeated asmin ? subject to M(?,pK ,pS) < 0, (5.24)72and transformed to the minimization problem of function f in (2.17) repeated as? = f(pS), (5.25)is numerically solved using the descent algorithm presented in Section 2.3. First,the hybrid method was employed for the DS step of the algorithm. Starting frompS,init = (5.50? 10?2, 7.08? 10?4), i.e., the middle point of the plant variation set?, and corresponding ?init = 3.30, the algorithm converged to p?S,init = (9.09 ?10?2, 3.50? 10?4) and ?? = 0.818.Secondly, both the steepest descent method and Newton?s method were appliedsingly for the DS step. Having used the steepest descent method and started fromthe middle point of the plant variation set, the algorithm converged to the samepoint as above but in shorter computational time. The reason for such result is thatin most of iterations, Hessian matrix of f was not positive definite, so that the secondorder approximation of f was not used, while calculation of Hessian matrix in eachiteration charged the computational cost of the algorithm. For the same reason,applying Newton?s method singly for the DS step did not lead to convergence to anypoint. The results mentioned above have been summarized in Table 5.1. In order tomeaningfully compare the time, computations were done with the same number ofgridded points of the plant variation region, equal to 2500, as well as the same 3.20GHz CPU computer with 32.0 GB RAM and MATLAB version 8.1.0.604 (R2013a).Different from what was done for the numerical example in Chapter 3 and the OISexample in Chapter 4, the full search method was not applied to find the optimizedswitching surface for the IC engine example. This is because of the fact that applyingthe full search method would take weeks to find the optimized point for the IC73Table 5.1: Results of applying different numerical methods for the DS step of thedescent algorithm to optimize the switching surfaceMethod Initial Point Final PointComp.Time(min)Hybrid?init = 3.30ps,init =[5.50? 10?27.08? 10?4]?? = 0.818ps,init =[9.09? 10?23.50? 10?4]259SteepestDescent?init = 3.30ps,init =[5.50? 10?27.08? 10?4]?? = 0.818ps,init =[9.09? 10?23.50? 10?4]148Newton?s?init = 3.30ps,init =[5.50? 10?27.08? 10?4]N/A N/Aengine problem which is much longer than what took for other examples. Thereason for such computational time difference is that the generalized plant in thecurrent example has 6 states which is double the number of states of the generalizedplants in previous examples. Therefore, computation of ? for each gridded pointwould take remarkably longer time. This evidence elucidates and emphasizes theadvantage of using the descent algorithm over the full search method, which is notapplicable to some examples.5.4 Simulation ResultsTwo switching LPV controllers have been designed and implemented in the sys-tem. One of the controllers has the optimized switching surfaces and the other onehas the switching surfaces heuristically placed at the middle of the plant variationregion, such that the whole region is divided into 4 equal subregions. In the simula-74tion, a feedforward term is also added to the system to modify the control input tothe plant so as to improve the tracking performance of the system. The logic behindsuch modification has been explained in Appendix B. In Fig. 5.5, reference trackingsimulations of two controllers with a square-wave reference signal are juxtaposedtogether. As seen in this figure, the optimized controller produced a response withsmaller overshoots and undershoots and improved settling time. It was consequentlyseen that the absolute value of the area between the equivalent air-fuel ratio and thereference signal, i.e., the error, was decreased by 17%. To clearly show the differencebetween the performance of two controllers, the error signal averaged in every 10seconds time interval has been shown in Fig. 5.6.Figure 5.5: Comparison of equivalent air-fuel ratio reference tracking between con-trollers with the optimized switching surface and the heuristic one.75Figure 5.6: Error comparison of equivalent air-fuel ratio reference tracking, averagedin every 10 seconds time interval, between controllers with the optimized switchingsurface and the heuristic one.76Fig. 5.7 and Fig. 5.8 show realistic profiles of the air flow and engine speed in[36] representing operating conditions of the engine. Also, assumed upper boundsand lower bounds on their variations are shown. As seen in these figures, the profileof the air flow and engine speed are limited to these bounds as expected. Fig. 5.9represents the resulting trajectory of the air flow and engine speed within the plantvariation region.Figure 5.7: Air-flow trajectory and the lower and upper bounds on its variations.77Figure 5.8: Engine speed trajectory and the lower and upper bounds on its varia-tions.Figure 5.9: The operating point of the engine in the plant variation region.78The variation rates of these two parameters and their bounds are plotted inFig. 5.10 and Fig. 5.11. As shown in these figures, the derivatives of the air flow andengine speed are in the assumed variation rate region ?. Otherwise, the stability ofswitching LPV controllers obtained based on these variation rate bounds would notnecessarily be guaranteed.Figure 5.10: Air-flow variation rate trajectory and the lower and upper bounds onits variations.79Figure 5.11: Engine Speed variation rate trajectory and the lower and upper boundson its variations.80Fig. 5.12 and Fig. 5.13 show the resulting changes in the switching variables onthe air flow axis and the engine speed axis, respectively. The combination of thesetwo switching signals determines the active controller in the Fig. 5.4 at each time.Figure 5.12: Comparison of switching variable on air-flow axis between controllerswith the optimized switching surface and the heuristic one.Figure 5.13: Comparison of switching variable on engine speed axis between con-trollers with the optimized switching surface and the heuristic one.81As explained before, the fuel flow is the control input to the plant. The twoprofiles of the fuel flow generated by two controllers have been shown in Fig. 5.14.Both signals are within the allowable region. As seen in this figure, little differencein the control input can make a huge impact in the performance of the system. Thetransient responses of the control input at the times of switching events are not largeenough to be observed in this figure. Albeit abrupt changes in the control inputsignal still exist, they are negligible compared to the absolute value of the controlinput. Moreover, a low pass filter has been employed just after the controller forperformance improvement purposes which has faded instant variations in the controlinput plotted here.Figure 5.14: Comparison of control input to the plant between controllers with theoptimized switching surface and the heuristic one.825.5 SummaryIn this chapter, the proposed algorithm was applied to air-fuel ratio control inspark ignition internal combustion engines. At the beginning of this chapter, theimportance of air-fuel ratio control in the engines was discussed, and different controlapproaches have been used so far in the literature were mentioned. By reviewingthe modelling of an IC engine, it was shown that the dynamic of the system dependson the engine speed and air flow which form a 2 dimensional plant variation region.For switching LPV controller design, the plant variation region was divided into 4rectangular subregions with fixed widths of the overlapping regions, like what wasdone for the image stabilizer example. As a result, the centres of the overlappingregions are the switching surface variables. Reference tracking and disturbancerejection are design objectives in this control design problem.The switching LPV controller with optimized switching surfaces was designedusing the descent algorithm. For the DS step of the algorithm, different methodswere employed and corresponding results were compared. It was observed thatthe algorithm with the steepest descent method converges faster than that withthe hybrid method, and applying Newton?s method led not to converge to a localminimum point. At the end of the chapter, simulation results of implementing theswitching LPV controller with optimized switching surfaces were compared withthat with heuristic switching surfaces. A realistic plant parameters trajectory wasused and an arbitrary reference signal was input to the closed-loop. It was shownthat the optimized controller could result in better tracking response in terms of theerror, overshoot, and settling time.83Chapter 6ConclusionIn conclusion, applying the descent algorithm in this thesis to design switchingsurfaces for switching LPV control has been shown to be beneficial for some reasons.First of all, the computational time of getting the solution by applying the proposedmethod is remarkably smaller than that of applying the full search method. Sec-ondly, it was shown that implementing switching LPV controllers with optimizedswitching surfaces improves time-domain simulation performances in different con-trol applications. However, performance improvement of the worst-case can be onlyguaranteed by employing optimized switching surfaces.In this chapter, a summary of what was explained in this thesis, as well as thecontributions of my work and some future works in this area has been provided.6.1 SummaryIn this thesis, a numerical algorithm was developed to design switching surfacesfor switching LPV controllers. Time-varying systems with measurable parameters,84as well as nonlinear systems, represented in the form of linear models with varyingoperating conditions, were considered for controller designing. It was shown howstability and performance of the system can benefit from switching LPV controllerswith optimized switching surfaces rather than that with heuristic ones. The algo-rithm proposed was applied to three examples in control applications including LPVsystems: a tracking problem for a mass-spring-damper system, a vibration suppres-sion problem for a magnetically-actuated optical image stabilizer, and an air-fuelratio control problem for automotive engines.The numerical algorithm to switching LPV control design with optimized switch-ing surfaces was developed and explained in Chapter 2 of the thesis. It was shownthat optimization of the switching surfaces can be formulated as optimization ofcentres and widths of the overlapping regions. The cost function to be minimizedis the closed-loop L2-gain performance of the system, ?. The proposed algorithmis iterative and descent in which a decrease in the cost function in each iterationis guaranteed. However, a local minimum point is obtained depending on the ini-tial condition, and finding the global minimum point is not guaranteed. A hybridmethod which is a combination of the steepest descent method and Newton?s methodis employed in this algorithm.The proposed numerical algorithm in this thesis was then applied to a numer-ical example in Chapter 3. A simple mass-spring-damper system with parametervariations was used as a plant for which controllers were designed. Switching LPVcontrollers with different premises on the number of subregions were designed forthis system. It was shown that applying the proposed algorithm in Chapter 2 issuperior than using the full search method in terms of computational cost. Also,85by juxtaposing closed-loop performance of the switching LPV controller with opti-mized switching surfaces and that with heuristic switching surfaces, it was shownthat optimized switching LPV controller has the better performance.In Chapter 4, a magnetically-actuated optical image stabilizer was used as a con-trol application in which the mass and stiffness-value are subject to product varia-tions. Two switching LPV controllers were designed for this system: the controllerwith optimized switching surfaces and the one with heuristic switching surfaces.Simulation results of implementing two switching LPV controllers showed that theclosed-loop performance of the worst case product will be improved if the opti-mized controller is employed. Nevertheless, many products showed relatively worseresponses with the optimized controller. This phenomenon coincides with our expec-tation that applying switching LPV controller with optimized switching surfaces canonly improve the worst case L2-gain performance of the closed-loop which relates tothe worst case product.In Chapter 5, the proposed algorithm was applied to another control applicationwhich is air-fuel ratio control in spark ignition internal combustion engines. Thedynamic of the system depends on the engine speed and air flow which form a 2-dimensional plant variation region. The switching LPV controller with optimizedswitching surfaces was designed using the descent algorithm. Then, simulation re-sults of implementing such controller were compared with that of implementing aswitching LPV controller having heuristic switching surfaces. A realistic plant pa-rameters trajectory was used, and an arbitrary reference signal was input to theclosed-loop. It was shown that the optimized controller could result in better track-ing response in terms of the error, overshoot, and settling time.866.2 ContributionsThe main contributions of this thesis can be outlined as follows:? A numerical algorithm was developed to optimize switching surfaces in switch-ing LPV control design. By applying this algorithm, decrease in the closed-loop L2-gain performance from initial selection of the switching surfaces to theconvergent point is guaranteed.? A hybrid method which is the combination of the steepest descent method andNewton?s method was proposed to be used in the descent algorithm.? A numerical example was developed, and the proposed algorithm was appliedto design a switching LPV controller with optimized switching surfaces. Thepotential of applying the descent algorithm to design switching surfaces withany number of variables was successfully tested in this example.? The proposed algorithm was applied to design switching surfaces for a switch-ing LPV controller used in a magnetically-actuated optical image stabilizersystem with production variation, and improvement in the worst case refer-ence tracking error was achieved.? For air-fuel ratio control in spark ignition internal combustion engine, a switch-ing LPV control with optimized switching surfaces was designed by applyingthe descent algorithm and improvement in the closed-loop performance of thesystem was obtained.876.3 Future WorkOptimization of switching surfaces in switching LPV control design is an effectiveway to improve the performance of the systems, the dynamics of which vary duringthe operation of them. What is done in this research has been addressed thisoptimization problem in a particular manner with a diverse set of premises. To putit in a nutshell, there are still lots of room for research and study in this field. Someresearch topics can be categorized as follows:? Combine the proposed descent algorithm with the genetic algorithm in orderto achieve global optimum: It is important to have an algorithm which doesnot depend on the initial point, and the global minimum point is possible tobe obtained, so that further improvement of the performance can be achieved.? Optimize the number of divisions on each axis of the plant variation region,so that a balance between computational time and improvement in the perfor-mance level is sought: Intuitively, increase in the number of subregions leadsto improvement in the performance level. However, in high number of subre-gions, the computational cost increases dramatically. Thus, it is important tomaintain a balance between these two factors in controller design.? Optimize shapes of subregions: In current research the shape of the subregionsare assumed to be rectangular. 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In Proceedingsof 1994 33rd IEEE Conference on Decision and Control, volume 4, pages 3492?3497, 1994.[45] Fen Wu and Ke Dong. Gain-scheduling control of LFT systems usingparameter-dependent Lyapunov functions. Automatica, 42(1):39?50, January2006.[46] Ehsan Azadi Yazdi and Ryozo Nagamune. Multiple robust H? controller designusing the nonsmooth optimization method. International Journal of Robust andNonlinear Control, 20:1197?1312, 2010.95[47] Ehsan Azadi Yazdi, Mohammad Sepasi, Farrokh Sassani, and Ryozo Naga-mune. Automated multiple robust track-following control system design in harddisk drives. IEEE Transactions on Control Systems Technology, 19(4):920?928,2011.[48] Hui Ye, Anthony N. Michel, and Ling Hou. Stability theory for hybrid dynam-ical systems. IEEE Transactions on Automatic Control, 43(4):461?474, April1998.[49] Yildiray Yildiz, Anuradha M. Annaswamy, Diana Yanakiev, and Ilya Kol-manovsky. Adaptive air fuel ratio control for internal combustion engines. InProceedings of 2008 American Control Conference, Seattle, Washington, June2008.[50] Yildiray Yildiz, Anuradha M. Annaswamy, Diana Yanakiev, and Ilya Kol-manovsky. Spark ignition engine fuel-to-air ratio control: An adaptive controlapproach. Control Engineering Practice, 18(12):1369?1378, 2010.[51] Feng Zhang, Karolos M. Grigoriadis, Matthew A. Franchek, and Imad H.Makki. Transient lean burn air-fuel ratio control using input shaping methodcombined with linear parameter-varying control. In Proceedings of 2006 Amer-ican Control Conference, Minneapolis, Minesota, June 2006.[52] Feng Zhang, Karolos M. Grigoriadis, Matthew A. Franchek, and Imad H.Makki. Linear parameter-varying lean burn air-fuel ratio control for a sparkignition engine. Journal of Dynamic Systems, Measurement, and Control,129(4):404?414, 2007.96Appendix APlant Model Derivations ofSpark Ignition InternalCombustion EnginesAs explained in [36], a parameter dependent first-order plus dead time (FOPDT)model is used in this thesis for modelling the IC engine. This model is simple enoughto design controller for and accurate enough to represent the overall dynamics of theIC engine. The input to the system is the fuel mass flow m?fuel which is a controlinput to manipulate the air-fuel ratio(AFR), denoted by ?, as the output of thesystem. The FOPDT model is?m?fuel=gsT + 1e?s?, (A.1)where g is the steady state gain of the system, T is the time constant of the first-order term, and ? is the pure delay. These parameters are functions of N andm?air, the varying parameters setting the dynamics of the plant. The mentioned97functionalities explained in [36] can be seen in below:g(?) =Rstm?air, (A.2)where Rst is the stoichiometric mass ratio of air over fuel which is almost 14.7. Itcan be seen that steady state gain g is just function of m?air, not N . Also, we haveT (?) ? 120 ? (1?1ncyl) ?1N, (A.3)where ncyl is the number of cylinders in the engine which is equal to 4 in our case,and?(?) =180N+5.33m?air, (A.4)shows that the pure delay is function of both the engine speed and the air flow [36].The delay in FOPDT model is estimated by a Pade approximation with a second-order denominator and first-order numerator. Then, the plant transfer function isapproximated by?m?fuel=g(?)sT (?) + 1?6? 2s?(?)6 + 4s?(?) + (s?(?))2. (A.5)98Appendix BA Feedforward Term Added tothe Control Input in Air-FuelRatio Control of InternalCombustion EnginesAfter designing switching LPV controllers for air-fuel ratio control in internalcombustion engines, it has been seen that the simulation performance of the systemwill be improved if a feedforward term is added to the output of the switching LPVcontroller to make the control input to the engine. In other words, m?fuel-commandto the engine has two components: a base-fuelling term obtained by a feedforwardcontroller and a correction term which is the output of the switching LPV controller,as shown in Fig. B.1.To set the base-fuelling term, we make three assumptions as follows:99Figure B.1: The block diagram of the control system with a feedforward controller.? The output of the switching LPV controller in Fig. B.1 is set to be zero,and the feedforward controller is used exclusively in the system to control theair-fuel ratio.? There is no delay in the system.? By applying the feedforward controller, the output of the plant, i.e., the air-fuel ratio ?, will be equal to 1.Having set these assumptions, we havem?fuel = m?fuel forward, (B.1)and the transfer function of the engine block in Fig. B.1 with no delay is thenexpressed by?m?fuel=g(?)sT (?) + 1. (B.2)This system can be realized in a state-space form asP (?) :?????x? = ?1T (?) ? x+ m?fuel? = g(?)T (?) ? x,(B.3)100in which x ? R is the state vector of the plant. By taking derivative of the outputequation, we have?? =g?(?)T (?)? g(?)T? (?)T 2(?)? x+g(?)T (?)? x?. (B.4)As mentioned before, we would like to input m?fuel such that the output of the plantis equal to 1. Therefore? ? 1, (B.5)and?? ? 0. (B.6)By substituting (B.5) in the output equation in (B.3), we havex =T (?)g(?). (B.7)Now, by plugging (B.7), (B.6), and (B.3) in (B.4), the equation can be expressedby0 =g?(?)T (?)? g(?)T? (?)T 2(?)?T (?)g(?)+g(?)T (?)? (?1g(?)+ m?fuel). (B.8)By simplification, m?fuel for the base-fuelling is obtained as follows:m?fuel =1g(?)+g(?)T? (?)? g?(?)T (?)g2(?). (B.9)From (B.1), the relation between input and output of the feedforward controller inFig. B.1 can be written asm?fuel forward =1g(?)+g(?)T? (?)? g?(?)T (?)g2(?), (B.10)where the functionalities of g and T with respect to the varying parameter ? areavailable. In all the simulations of Chapter 5, this feedforward term has been addedto the output of the designed switching LPV controllers, so as to build up the finalcontrol input to the plant.101
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A numerical optimization approach to switching surface design for switching linear parameter-varying… Javadian, Moein 2014
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Title | A numerical optimization approach to switching surface design for switching linear parameter-varying control |
Creator |
Javadian, Moein |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | This thesis proposes an algorithm to design switching surfaces for the switching linear parameter-varying (LPV) controller with hysteresis switching. The switching surfaces are sought for to optimize the bound of the closed-loop L2-gain performance. An optimization problem is formulated with respect to parameters characterizing Lyapunov matrix variables, local controller matrix variables, and locations of the switching surfaces. Since the problem turns out to be non-convex in terms of these characterizing parameters, a numerical algorithm is given to guarantee the decrease of the cost function value after each iteration, which consists of two steps: direction selection and line search. A hybrid method which is a combination of the steepest descent method and Newton's method is employed in the direction selection step to decide the orientation of proceeding. A numerical algorithm is used to compute the most appropriate length of the proceeding along the selected direction which generates the most decrease in the cost function. To demonstrate the efficiency and usefulness of the proposed algorithm, it will be applied to three examples in control applications: a tracking problem for a mass-spring-damper system, a vibration suppression problem for a magnetically-actuated optical image stabilizer, and an air-fuel-ratio control problem for automotive engines. In these examples, it will be shown that the proposed optimization approach to the design of the switching surfaces and the switching LPV controller is superior to heuristic approaches in closed-loop performances, at the price of higher computational costs. Additionally, it will be shown that the algorithm can be applied to the general n-parameters case. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-02-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0072159 |
URI | http://hdl.handle.net/2429/46064 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2014-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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