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Multiple objective control with applications to teleoperation Hu, Zhongzhi 1996

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Multiple Objective Control with Applications to Teleoperation By Zhongzhi Hu B.Sc, Beijing Institute of Technology, Beijing, China, 1983. M.Sc, Beijing Institute of Technology, Beijing, China, 1986. M.A.Sc, University of British Columbia, Vancouver, Canada, 1992. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to th^/required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1996 © Zhongzhi Hu, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Zy'nee^Mj The University of British Columbia Vancouver, Canada Date V*c- 4y/9?6 DE-6 (2/88) Abstract Control system design inevitably involves tradeoffs among different and even con-flicting performance and robustness specifications. This thesis deals with some multiple objective control system design problems and with applications to teleoperation systems. First, the multiple objective linear-quadratic optimal control problem is solved. By using duality theory, this minimax problem is transformed into a convex optimization problem . In particular, the infinite time problem is shown to be an optimization problem involving linear matrix inequalities. Second, the multiple objective Hoo control problem for SISO systems is studied. Nonsmooth analysis is used to characterize optimality conditions for this problem. Under these conditions, either all-pass properties or the optimal performance values are obtained. Third, numerical solutions of the general multiple objective control system design problem by either convex optimization or non-convex optimization are presented. First, the convex optimization design procedure is described, the effectiveness of the cutting-plane based solver is demonstrated, and some computational issues are discussed. Then a non-convex optimization design procedure is proposed, in which an approximation to the free transfer function in the Q-parametrization is proposed. It is shown, by design examples, that it has the advantage of directly producing low-order controllers. Last, the robust controller design problem for teleoperation systems is investigated. First, a two-port ideal teleoperation model is proposed. It is shown that the model scales both positions and forces, yet is stable when terminated by any strictly passive hand and environment impedances. Then a transparency measure is proposed to be defined as the Hoo-distance to the ideal teleoperator model. Using a four channel control structure, the controller design problem is formulated as a multiobjective optimization problem maximizing transparency subject to robust stability for all passive environments. This problem is shown to be convex in the design parameters for a fixed hand impedance. To demonstrate the design procedure, this thesis treats the design of a controller for a simple one degree-of-freedom system model of a motion-scaling teleoperation system. Both simulations and experiments have been carried out to show the effectiveness of the proposed controller design methodology. i i i Table of Contents Abstract ii List of Tables viii List of Figures ix Notation xiii Acknowledgments . . . xv Chapter 1 Introduction '.. 1 1.1 Historical Overview 1 1.2 Literature Review 4 1.3 Thesis Outline . 8 Chapter 2 Multiple Objective Control Problems 12 2.1 Introduction 12 2.2 Problem Formulation 14 2.2.1 Parametrization of stabilizing controllers 15 2.2.2 Problem statement 16 2.3 Analytical Solutions 18 2.4 Numerical Solutions 20 2.5 Discussion 21 Chapter 3 Multiple Objective LQ Control Problem 23 3.1 Introduction . 23 3.2 Formulation of the Problem 24 3.3 Linear Quadratic Problem 25 3.4 Solution via Convex Optimization 27 iv 3.5 Solution via Linear Matrix Inequalities 31 3.6 Concluding Remarks 36 Chapter 4 Multiple Objective Control Problem 37 4.1 Introduction 37 4.2 Analytical Results via Nonsmooth Analysis 38 4.2.1 Some preliminaries 38 4.2.2 Main results . 40 4.3 Numerical Solution via LMI 49 4.4 Concluding Remarks .55 Chapter 5 Numerical Solutions of Multiple Objective Control Problems 56 5.1 Introduction 56 5.2 Solution via Convex Optimization 57 5.2.1 Linear approximation of Q 57 5.2.2 Convex optimization via sequential quadratic program methods 58 5.2.3 Convex optimization via cutting-plane algorithms 58 5.2.4 Design Examples 59 5.2.5 Computational aspects 70 5.3 Solution via Non-convex Optimization 71 5.3.1 Nonlinear approximation of Q 71 5.3.2 Computational Aspects 72 5.3.3 Numerical examples . . . . . . . 75 5.4 Concluding remarks .80 V Chapter 6 Teleoperation Controller Design Problem 82 6.1 Introduction 82 6.2 Background on Stability and Passivity 84 6.3 An Ideal Teleoperator 87 6.4 Robust Controller Design 93 6.4.1 Teleoperation controller structure 93 i 6.4.2 Performance measures and stability constraints . . . . . . 94 6.4.3 Controller design problem formulations 99 6.4.4 Numerical solutions 100 6.5 Design Example 101 6.5.1 Identification of the hand impedance 102 6.5.2 Tradeoff between performance and stability robustness . 103 6.5.3 Simulation results 104 6.5.4 Experimental results 113 6.6 Concluding Remarks 120 Chapter 7 Conclusions 122 7.1 Contributions 122 7.2 Future Work 124 References . . • 125 Appendix A Cutting-plane Algorithms 135 A.1 Elements of Convex Analysis 135 A.2 KCP Algorithm 135 A.3 EMCP Algorithm 138 Appendix B Linear Matrix Inequalities 140 B.1 Linear Matrix Inequalities 140 B.2 Software for Solving LMI Problems 141 vii List of Tables Table 3.1 Example 3.1: Solution of a multiple objective LQ control problem 31 Table 4.1 Example 4.2: Solution results 54 Table 5.1 Example 5.1: Results via constr, KCP and EMCP . . . . . 62 Table 5.2 Example 5.2: Results via minimax, KCP and EMCP . . . 66 Table 5.3 Example 5.4: Solution properties .76 Table 5.4 Example 5.4: Results via convex and non-convex optimization 77 Table 5.5 Example 5.5: solution properties . 78 Table 5.6 Example 5.5: Results via convex and non-convex optimization "- 80 Table 6.1 Simulation results: transmitted impedances to the hand .112 Table 6.2 Experimental results: transmitted impedances to the hand . 1 1 9 viii List of Figures Figure 2.1 Example 2.1: A feedback system with multiplicative plant perturbation 13 Figure 2.2 General feedback systems 14 Figure 3.1 Example 3.2: The level curves of the overall objective: (a) the mesh surface, and (b) the contocdur 34 Figure 3.2 Example 3.2: The level curves of the multiplier A i : (a) the mesh surface, and (b) the contour . 3 5 Figure 3.3 Example 3.2: The level curves of the multiplier A 2 : (a) the mesh surface, and (b) the contour 35 Figure 4.1 Example 4.2: Weighted objective functions \Ti(ju)\ (dotted line) and \T2(ju)\ (solid line) when vV = 14 54 Figure 5.1 Example 5.1: A feedback system with additive plant perturbation \ . 60 Figure 5.2 Example 5.1: Robust stability constraint 63 Figure 5.3 Example 5.2: A unity feedback control system ...' 64 Figure 5.4 Example 5.2: Weighted objective functions |Ti(ju)|(dotted line) and |T2(jw)|(solid line) associated with Figure 5.5 Example 5.2: Sensitivity function Ter(solid line) and its bounding function h (dotted line) associated with Q™x . . 68 Figure 5.6 Example 5.2: Robustness function Tur(solid line) and its bounding function associated with Qllxh(dotted line) . . . 68 ix Figure 5.7 Example 5.2: Weighted objective functions iT^ju)|(dotted line) and \T2(ju)|(solid line) associated with QU • • • 69 Figure 5.8 Example 5.2: Number of Constraints in KCP and EMCP . 70 Figure 5.9 Example 5.5: Weighted objective functions iT^ju)|(dotted line) and |T2(ju;)|(solid line) using parameter Q2ncvx from (5.45) 79 Figure 6.1 General Teleoperation System 83 Figure 6.2 2n-port representation of a teleoperation system 83 Figure 6.3 An n-port network : 85 Figure 6.4 A representation of an LTI one-port network, Y, coupled to Z . 8 6 Figure 6.5 Two-port representation of ideal scaled teleoperation . . . 88 Figure 6.6 Another representation of ideal scaled teleoperation . . . . 90 Figure 6.7 A four channel control structure. 93 Figure 6.8 General feedback system 94 Figure 6.9 Performance vs distance to passivity for three values of force and motion scaling 104 Figure 6.10 Maximum singular value frequency responses of Y/;# (solid line) and YT:H (circle) 105 Figure 6.11 Nyquist plot of Yte 106 Figure 6.12 Design example: simulation diagram of a motion-scaling system . 1 0 8 Figure 6.13 Simulation results: motion scaling and force scaling with a soft environment E = 2s + 200 109 X Figure 6.14 Simulation results: motion scaling and force scaling with a stiff environment E = 5s + 500 110 Figure 6.15 Simulation results: motion scaling and force scaling with a time-varying environment, which switches at 3, 6, 9, and 12 seconds 111 Figure 6.16 Simulation results: transmitted impedances to the hand, Zth (solid line) and Zth (dash-dotted line), with a soft environment E = 2s + 200 112 Figure 6.17 Simulation results: transmitted impedance to the hand, Zth (solid line) and Zih (dashdot line), with a stiff environment E = 5s + 500 113 Figure 6.18 Design example: experiment diagram of a motion scaling system 114 Figure 6.19 Design example: experimental setup 115 Figure 6.20 Experimental results: motion scaling and force scaling with a soft environment E = 2s + 200 116 Figure 6.21 Experimental results: motion scaling and force scaling with a stiff environment E - 5s + 500 117 Figure 6.22 Experimental results: motion scaling and force scaling with a time-varying environment, which switches at 3, 6, 9, and 12 seconds 118 Figure 6.23 Experimental results: transmitted impedances to the hand, Zth (solid line) and Zth (dashdot line), with a soft environment E = 2s + 200 119 xi Figure 6.24 Experimental results: transmitted impedance to the hand, Zth (solid line) and Zth (dashdot line), with a stiff environment E = 5s + 500 120 xii Notation Notation Meaning / : X —> Y A function from the set X into the set Y . ^ Equals by definition. • End of proof. d<j)(x) The subdifferential of the function <f> at the point x. m { l ,2, . . . ,m}. R The real numbers. R + The nonnegative real numbers. R m The vector space of m-component real vectors. C The complex numbers. C + The open right half plane, { ^ C : Res > 0). C m The vector space of m-component complex vectors. C m x n The vector space of mx n complex matrices. D The open unit disc, {z e C : \z\ < 1). Re{(.)} The real part of (.). L i Lebesgue space of integrable functions. L 2 Lebesgue space of square-integrable functions. LQO Lebesgue space of essentially bounded functions. H 2 Hardy space of square-integrable functions. HQO Hardy space of essentially bounded functions. prefix R Real rational subset of, e.g., RHoo. ||.||2 Norm on £2-H-lloo Norm on S1- Orthogonal complement of subspace S. S m The unit simplex in R m : A f m |^ sm = | / x e / r :^->o,y:^ = i j . xiii A > 0 The nxn complex symmetric matrix A is positive semidefinite, i.e., z*Az > 0, for all z G C n . A > 0 The nxn complex symmetric matrix A is positive definite, i.e., z*Az > 0, for all nonzero z G C n . AT Transpose of matrix A. A* Complex-conjugate transpose of matrix A. A 1 ! 2 A symmetric square root of a matrix A = A* > 0, i.e., All2Axl2 = A. A(-sf. The infimum of a real-valued function on a set. The supremum of a real-valued function on a set. The maximum singular value of a matrix A, equal to the square root of the largest eigenvalue of A* A. [A, B, C, D] The transfer matrix corresponding to the state-space equations x = Ax + Bu, y — Cx + Du, i.e., . [A, B,C,D] = D + C(sl - A)~lB. A~(s) inf sup 6(A) xiv Acknowledgments I would like to express my deep gratitude to my supervisor, Professor Tim Salcudean, for suggesting the subject of this thesis, for his invaluable support and inspiring guidance throughout this research. I would also like to extend my appreciation to my co-supervisor, Professor Phillip Loewen, for his many insightful suggestions and helpful discussions. I am grateful to many of past and present colleagues, for some technical help and discussions, and for their encouragement. Last, but no means least, I would like to thank my wife Lei Jin for her love, support and encouragement. This thesis is dedicated to my sons, Jinwei and Youja, to whom I owe the most. XV Chapter 1 Introduction In this chapter we first give a brief historical overview of control systems design, describing why we have to deal with multiple objective control problems, we then survey the literature on some particular multiple objective control problems, and finally we present the thesis outline. §1.1 Historical Overview From the ancient waterclock to today's space probes and automated manufacturing plants, control systems have played a key role in technological and scientific development. The ultimate goal in control system design is to achieve optimal performance while maintaining stability, under perhaps substantial system uncertainty and under design constraints imposed by the technology. Quite often, performance and stability are two different and even conflicting features. Therefore most of the practical controller design problems are multiple objective in nature. Control systems are designed and analyzed using mathematical models, arrived at by physical laws or perhaps from measured data. Mathematical models inevitably give an imperfect description of physical systems, and, in any case, the parameters involved in such a description are often subject to variation and uncertainty. Feedback can be used to reduce the effect of this uncertainty on the system behavior. Indeed, the ability of feedback to counteract the effects of uncertainty is generally the primary motivation for its use. However feedback come with several potential disadvantages. In particular, large gains in a feedback system usually cause the system to become unstable. Therefore determining just how much feedback can safely be applied is a key issue in control system design. 1 Chapter 1, Introduction Early efforts to use feedback ran into exactly this difficulty, and it motivated much of the early development of the so called classical control theory. Classical design methodologies dealt with linear time-invariant scalar systems and proceeded by shaping the gain and phase of the open loop transfer function to modify the feedback properties of the closed loop system. Several techniques for synthesizing feedback control systems that meet explicit specifications on disturbance attenuation and that have low sensitivity to large parameter variations are presented in Horowitz's book [1]. These techniques are based on root-locus and frequency response shaping, but are limited to single-input single-output systems. Furthermore, they are highly dependent on the specific structure of the problem and demand a large amount of the designer's time because they are not easily implemented on computers. The linear-quadratic-Gaussian (LQG) approach dominated the field in the 1960's and the early 1970's [2]. If the dynamic model is assumed to be exact and the disturbances are assumed to be white noise Gaussian processes, then the control law that minimizes the expected value of a quadratic form in the errors is called an L Q G controller. However, there is no guarantee of robustness for LQG-controllers [3]. For example, a small perturbation of the plant dynamics or the occurrence of control disturbances may produce an unstable closed-loop system. This is because the L Q G design method takes no account of either uncertainties in the dynamic model or of departures from uniformity in the power spectrum of the disturbance. Both factors are, of course, inevitable in any practical problem. A good control system design procedure should therefore take account of the uncer-tainty inherent in a mathematical model, and quantify the effect of feedback on uncer-tainty. This motivated people to seek quantitative measures for the size of uncertainty. A suitable measure that can incorporate both signal gain and robustness to uncertainty is the 2 Chapter 1. Introduction Hoo-norra. The synthesis of controllers using Hoc optimization methods was initiated in 1981 by Zames [4]. It represents one of the major recent advances in control system design. Put simply, the problem is to design a controller that achieves internal stability and minimizes the Hoo-norm of a closed-loop transfer matrix. Zames's work dealt with some of the basic questions of classical control theory, and immediately attracted a great deal of attention. It was soon extended to a wide range of sensible robust controller syn-thesis problems, in particular when it was recognized that the approach allows robustness to be incorporated more directly than other methods. The intensive research activity that. followed has produced valuable results in both theory and practice [5, 6, 7, 8, 9, 10]. Despite its success and consequent popularity, HQO control theory has some disadvan-tages, which were quickly recognized in control community. One serious disadvantage is that the theory can only cope with the problem of minimizing the Hoo-norm of a single transfer matrix. Therefore any problem with several and perhaps competing performance and robustness specifications must be restated in terms of just one objective in the fre-quency domain by ,the use of weighting functions. Selecting weighting functions that capture the essence of each specification, in the absence of a systematic procedure, may turn out to be quite hard and time consuming, and worse still the conservatism introduced by this approximation may render the final results useless. It should be noted that the same problem also occurs to the L Q G design method discussed before in selecting the weighting matrices. In practice, control system design almost inevitably involves tradeoffs among com-peting objectives. It is often the case that the controller is required to enhance several different performance and robustness criteria, and all of these cannot be improved simul-taneously. For example, it is intuitively clear that to obtain a greater stability margin, it is likely that the performance of the control systems needs to be compromised. Therefore it 3 . Chapter 1. Introduction-is, desired to formulate the controller synthesis problem using multiple design objectives rather than to cast the problem in terms of a single cost function. § 1 . 2 Literature Review The research in multiple objective control problems is very rich and broad. The literature review will only centre on • multiple objective linear quadratic (LQ) control problem, • multiple objective HQQ control problem, • numerical solutions of multiple objective control problems, and • controller design for teleoperation systems. Multiple objective L Q control problem Much attention has been paid to the L Q control in control theory, largely due to its wide applications and its mathematical elegance and tractability. The use of multiple objectives is well justified by many real-world control problems, for example, in aircraft control systems design [11, 12], in the control of space structures [13], and in industrial process control [14, 15,16]. Various multiobjective L Q control formulations and schemes have thus been motivated and investigated. Mainly, there are two research directions: 1) characterization of the set of noninferior solutions (e.g., [14, 17, 18]), and 2) search for a specific noninferior solution, for example, the minimax solution [19, 20], the ideal point [21], and the hierachical ordering [22]. For the first direction, the selection of the final implementing control is left unsolved. For the second direction, technical difficulties arise in the search process due to the involved dependency of the matrix Riccati differential equations (RDE) or algebraic Riccati equations (ARE) on a weighting coefficient vector. However, recently, the convex duality and optimization method has emerged as a powerful 4 Chapter 1. Introduction tool to solve the multiobjective L Q G problem [23, 24] and the non-convex multiobjective linear quadratic regulator problem for the infinite time horizon case [24]. Multiple objective H ^ control problem As we discussed in the last section, the H^o control theory plays an essential role in designing robustly stable control systems (see, e.g., [4, 6, 7, 25] and references therein). However, control system design is constrained not only by stability robustness but by other performance criteria as well. As shown in [26, 23], many design specifications can be quantified using closed-loop frequency responses. Therefore, the control system designer needs to find a controller such that several closed-loop frequency responses are simultaneously small in an appropriate sense. This problem may be cast as a multiple objective Hoo control problem, also referred to as a multi-disc H^o problem. In complete generality, the multiple objective Hoo control problem has only been solved numerically by convex optimization [26, 23] and general analytical solutions are not available. However, some qualitative properties of the optimum for related problems have been reported in [27, 28, 29, 30, 31]. Holohan and Safonov [29, 30] studied the nominal loop shaping problem (NLSP) for SISO systems, which they transformed into a two-disc Hoo problem, and then used function space duality theory to obtain bounds and an all-pass property. For the case where the plant is both stable and minimum phase, they determined the optimal achievable performance. Zames and Owen [28, 31] studied another form of the two-disc Hoo problem for both SISO and MIMO systems, which arose from the optimal robust disturbance attenuation problem (ORDAP). Their solutions have also been shown to satisfy a flatness or all-pass condition. Numerical solutions of multiple objective control problems Despite great progress in control theory and enormous advances in computing power, the design of a LTI controller which meets multiple and conflicting performance and 5 Chapter 1. Introduction robustness specifications can still be quite a challenge. Only for a few very special cases are there analytic methods for finding the exact form of the trade-offs among different specifications [32, 29]. However, in the last decade, optimization-based design methods [33, 34, 35, 36, 26, 23, 24] have emerged as powerful tools in multiple objective control system design. Most numerical methods proceed by solving approximate problems. These can be categorized as the non-convex optimization approach, such as the approximate scalarization [37], the U-parametrization [38, 39], and the iterative Hoo optimization [40], semi-infinite optimization [41], and the convex optimization approach, such as the Q-parameter design [36, 26, 23, 24]. The advantage of the non-convex approach is that it normally yields low-order controllers when it succeeds. Its major drawback is that it is not guaranteed to find a solution if one exists, nor is it guaranteed to find global optimum of the objective function. The advantage of convex optimization approach is that it always finds a solution if one exists. However, the parameter space is usually very large, which may produce two problems in the design process. First, it easily runs into computational problems, and therefore an efficient optimization solver is demanded in the design. Second, it generally yields high order controllers, which must then be judiciously reduced in order to be made feasible in practice. A practical example: teleoperation control Teleoperation has found wide applications in space exploration, waste management and undersea exploration, by extending an operator's sensing and manipulation capabili-ties to remote and hazardous locations. Recently, teleoperation has started to encompass the extension of such capabilities through barriers of scale, allowing human involvement at scales much smaller or much larger than possible directly. Examples include manip-ulation of cells, microsurgery and electronic assembly. Much research effort has been devoted to this kind of scaled teleoperation (see, e.g., [42, 43, 44], and references therein). 6 Chapter 1. Introduction One of the major issues in teleoperation is the controller design. The teleoperation controller should be designed with the goal of achieving the best possible performance, normally termed as transparency, while maintaining stability when coupled to uncertain environments, in the possible presence of time delays, disturbances, and measurement noise. Therefore, the teleoperation controller design problem involves a compromise between performance and robust stability. It is challenging mainly due to the large uncertainties of operator and environment impedances that have to be accommodated and due to communication delays. The conventional teleoperation control schemes such as position-position and position-force schemes provide poor transparency, even at low frequencies, and poor stability properties [45]. Recent work in teleoperation controller design has focused more on stability or/and performance, and could be categorized as stability-optimized scheme [46, 47, 48], tansparency-optimized scheme [45] or some combinations of the two [49, 50]. Passivity theory is the basis for the modifications of basic position-position and position-force schemes presented in [46, 47] to deal with time delays. Robust control ideas based on small gain theory motivate the force-force scheme proposed in [51]. Transparency performance of these teleoperation systems are not presented. Some experimental testing of the passivity concept in [46] reveals that the stability guarantee comes at the expense of the reduced stiffness, resulting in poor transparency [52]. The problem of achieving performance is also difficult, in large part because perfor-mance specifications are also likely to change with the environment. However, various control schemes have been concentrated on performance. Objectives based on specifying network theory hybrid parameters are discussed in [53, 54]; the network hybrid parameter design problem in [55] is formulated in terms of a transparency objective, and suggests a 7 Chapter 1. Introduction position-position approach; a four-channel control structure has been suggested to achieve transparency in [45] and ideal response of teleoperation in [48]; Hoo-optimization theory has been used to best shape the interested closed-loop responses in [56, 44] and to shape the relationships between forces and positions at both ends of the teleoperator in [57]. However none of the above work has explicitly incorporated robust stability to large changes in hand and environment variation into the controller design. Both robust stability and performance are treated in [50] and [49] respectively. In [50], a combined Hoo-optimization and /j-synthesis framework are used to design a teleoperator which is stable for a pre-specified time delay and fixed operator and environment impedances while optimizing performance specifications. Concepts of passivity, impedance control and Hoo-optimization theory are used in [49] to formulate the controller design as a semi-infinite optimization problem. Two criteria, "transparency distance", and "passivity distance" [58] are employed. Unfortunately, both resulting designs are not convex in design parameters, and therefore the limit of the achievable performance and the exact trade-offs between stability and performance can not be obtained. § 1.3 Thesis Outline This thesis is concerned with control system design with multiple objectives for linear time-invariant (LTI) systems only. We first study two particular multiple objective control problems, then we develop numerical solutions to a general multiple objective control system design, and finally, in a practical problem of multiple objective control system design, we develop robust controller methodologies for teleoperation systems. The thesis consists of seven chapters and two appendices. An outline of the thesis is given as follows. 8 Chapter 1. Introduction Chapter 2: Multiple Objective Control Problems. We formulate the multiple objec-tive control problem as an optimization problem, which in most cases is convex but nondifferentiable, and review previous analytical and numerical work on this problem. Chapter 3: Multiple Objective LQ Control Problem. We solve the multiple objec-tive LQ optimal control problem, where the functional to be minimized is the maximum of several quadratic performance indices. By using duality theory, the minimax prob-lem is transformed into a convex optimization problem. Particularly for the infinite time horizon case, it can be reduced into a convex optimization problem only involving a lin-ear matrix inequality (LMI). These two optimization problems can be efficiently solved numerically by cutting-plane based solvers or LMI based solvers. The solution to the multiple objective L Q control problem shows that it depends on the initial states and is an open-loop control optimal only for the particular initial conditions. Chapter 4: Multiple Objective Hoo Control Problem. We study the multiple objec-tive Hoo control problem for SISO systems, in which the functional to be minimized is the maximum of several Hoo-norm performance indices. The problem is convex but nondifferentiable. Nonsmooth analysis is used to derive optimality conditions for this problem. Under these conditions, either all-pass properties or the optimal performance values can be obtained. Since the optimal solution for this problem might be infinite order in some situations, to get a realizable controller, i.e., a real-rational, proper and stable controller, it is suggested that only suboptimal solutions be pursued. One approx-imate solution approach based on LMIs is presented, in which Hoo-norm constraints are transformed into LMIs by using the Bounded Real Lemma. Chapter 5: Numerical Solutions of Multiple Objective Control Problems. We present convex and non-convex optimization based solutions to the general multi-9 Chapter 1. Introduction pie objective control system design problem. We first describe the convex optimization design procedure, demonstrate the effectiveness of a cutting-plane based solver, and discuss some computational issues arising in the design process. Then we propose a non-convex optimization design procedure, in which an approximation for the free transfer function in the Q-parametrization is proposed. Since the starting guess is vital for non-convex optimization, three schemes of updating the starting guess are proposed to improve the computational efficiency. We show by several design examples that this approach has the advantage of directly producing low-order controllers. Chapter 6: Teleoperation Controller Design Problem. We investigate a practical multiple objective control system design problem — the construction of robust controllers for teleoperation systems. These are inherently MIMO systems with large uncertainties. A controller design approach for teleoperation systems that optimizes performance sub-ject to robust stability for all passive environments is proposed. With the four channel control structure, the controller design problem can be formulated as a multiple objective optimization problem, which is shown to be convex using the Youla parametrization. The robust controller was obtained by solving the optimization problem with the cutting-plane based solver described in Chapter 5. To demonstrate the design procedure, we design a controller for a simple one degree-of-freedom (DOF) system model of a force-reflecting and motion-scaling teleoperation system. Simulations and experiments were carried out to show the effectiveness of the proposed design methodology. Chapter 7: Conclusions. In the last chapter we summarize the contributions of this thesis and suggest some future research. Appendix A: Cutting-plane Algorithms. In this appendix, we first describe some of the basic tools of convex analysis, and then summarize Kelley's cutting plane algorithm 10 Chapter 1. Introduction for solving convex optimization problems, and its refinements by Elzinga and Moore. Appendix B: Linear Matrix Inequalities. In this appendix, we give a short introduc-tion to linear matrix inequalities, and some software for solving linear matrix inequality problems. 11 Chapter 2 Multiple Objective Control Problems We first define multiple objective control problems, then briefly review some existing analytical and numerical solutions, and finally offer some discussion and remarks. § 2.1 Introduction Control system design is most often a study of tradeoffs between different and even conflicting performance and robustness specifications. Most of these specifications can be quantified using closed loop frequency responses. For instance, questions such as how large the Controlled signals (e.g., tracking errors, weighted control inputs, etc.) can get in response to a given class of exogenous inputs (e.g., load disturbances, command inputs, sensor noises, etc.) and how much modeling uncertainty can be tolerated before the closed loop system becomes unstable may be precisely quantified using a closed loop frequency response. Therefore the control problem becomes that of finding a controller such that some desired measures of closed loop transfer matrixes are below specified levels. This is the class of multiobjective control problems that we shall consider in this thesis. Before we formally define this problem, we shall give an example illustrated in Figure 2.1 to show that multiple objective control problems arise naturally in control systems design. 12 Chapter 2. Multiple Objective Control Problems r t^e — u I+AWi t yp Figure 2.1. Example 2.1: A feedback system with multiplicative plant perturbation Example 2.1: Robust Control Problem. The controlled plant is modeled as (I + AW\)P, where P denotes the nominal plant transfer matrix, A the modeling uncertainty, and W\ a known weighting function that reflects size and directional properties of the uncertainty at each frequency. The objective is to design a con-troller K such that the closed loop system is internally stable for every plant in P = {(/ + A W i ) P : HAII^ < 1} and the power of the nominal (i.e., A = 0) plant output yp is below some pre-specified level in response to an exogenous signal d which is either white noise or an unknown signal with bounded power. More specifically, after introducing a suitable frequency dependent scaling W2 of the controlled output yp, the control problem is to find a controller K that (i). internally stabilizes every plant in P , or equivalently [59, 60], WXPK(I + PK)'1 < 1; and (ii). keeps either the H 2 - norm or the Hoo-norm of the weighted closed loop transfer matrix from d to yp below some pre-specified level, i.e., W2(I + PK) or W2(I + PK)~1 < 1 < 1. The combination of (i) and (ii) is a multiobjective control problem. 13 Chapter 2.. Multiple Objective Control Problems § 2.2 Problem Formulation For general control design problems, we use the basic configuration of the feed-back systems as shown in Figure 2.2, where G is the generalized plant, which has absorbed all weighing functions, with two sets of inputs: the exogenous inputs w = (wT wT ... w„w)T, which may consist of disturbances and commands, and the control inputs u. The plant G also has two sets of outputs: the measured outputs y and the con-trolled outputs z = (zT zT ...zTz) . Note that the input channels wi and output channels ZJ may be vector-valued. K is the controller to be designed. It is assumed that G and K are real-rational and proper. Under the decomposition G G\\ G\2 G21 G22 (2.1) the equations corresponding to Figure 2.2 take the form z = G\\w + G12U, y = G2iw + G22u, (2.2) u = Ky. We will see later that this setup is very convenient for analyzing some specific properties of the closed loop system or designing the controller K such that the closed loop system is stable and the output signal z is specified, i.e., some performance is satisfied. w > • • G • • • ^ ^ -—• ^ U y K Figure 2.2. General feedback systems 14 Chapter 2. Multiple Objective Control Problems § 2.2.1 Parametrization of stabilizing controllers The parametrization of all internally stabilizing controllers was first introduced by Youla et al. [61], which is often referred to as the Youla-parametrization or the Q-parametrization. The central idea is to parametrize all controllers in terms of a free stable transfer matrix Q such that closed loop stability is guaranteed for any choice of Q G R H Q O and the closed loop transfer matrices of interest are affine in the. function Q e R H O Q . A more recent version of the parametrization using coprime factorization due to Desoer et al. [62] is summarized in the following theorem. Theorem 2.1 [62]: Let G 2 2 = N2M~l = M~XN2 (2.3); be a right coprime factorization (rcf) and a left coprime factorization (lcf) of G22 over R H Q O , respectively. Then (a). The set of all (proper-real-rational) transfer matrices K stabilizing G is parametrized by K = (Y2- M2Q)(X2 - N2Q)~\ det(X 2 - N2Q)(joo) ? 0, • = (X2 - QNi) _ 1 (f2 - QM2), det (X2 - QN2) (joo) ^ 0, ^ for Q e RHoo, where X2,Y2,X2,Y2 e R H ^ satisfy 'M2 Y2 I 0 ' - JV 2 M2 _ N2 X2_ 0 I (2.5) (b). With K given by (2.4) the transfer matrix Tzw from w to z equals Ti + T2QT3, where T\ = G\\ + G\2M2Y2G2\, (2.6) T2 = -G12M2, T3 = M2G2l. Under the hypotheses above, Ti e RHoo, (i = 1,2,3). 15 Chapter 2. Multiple Objective Control Problems Remarks: (i) . For each proper real-rational matrix G22, eight RHoo-matrices satisfying the equa-tions in (2.3) and (2.5) can be explicitly constructed by using state-space formulas as shown in [63, 64, 59, 7]. Note that there are many choices of finding the rcf and lef of £22-(ii) . As a special case, suppose G22 is already stable, i.e., G22 £ RHoo- Then in (2.5) we may take N2 = N2 = G22 X2 = M2 = I, X2 = M2 = I (2.7) F 2 = 0 ,F 2 = 0, . in which case the formulas in (2.4) and (2.6) become simply K = -Q(I - G22QT1 = -(/ - QG22)~1Q, and (2.8) Tzw = Gn - G12QG21, Q € RHoo. (2.9) (iii) . An important point to note is that the closed-loop transfer matrix is simply an affine function of the controller parameter Q G R H ^ . § 2.2.2 Problem statement The multiple objective control problem is defined as follows: given m closed-loop transfer matrices Tk (&G m), each relating some input wi(i 6 nw) to some output zj{j ^ "Hz) m Figure 2.2, find a controller K such that K internally stabilizes G and minimizes the maximum of the various norms of transfer matrices Tjt (k G m); that is, solve the following optimization problem: ( p 2 I ) : ^ J S S . , ™ ! ™ — . . * e m ) . (2.10) where, for each k, ||.||normfc denotes one kind of norms in 11,112,1100, etc. 16 Chapter 2. Multiple Objective Control Problems Applying the Q-parametrization of stabilizing controllers shown in Theorem 2.1 reveals that the I*, (k e ra ) affine matrix functions in Q E R H ^ [7, 65, 23], i.e., Tk = Ti,jfc + T2)kQT3jk, k E m, (2.11) where T i ^ , ^ , * and T^^, k E m, are known RHoo-matrices, which can be obtained directly from the plant data G. So problem (P21) is equivalent to the following convex optimization problem: ( P 2 ' 2 ) : Q ^ H ^ ^ i ^ 1 ' * + T 2 ' J t g T 3 ^ l l - r m f c ^ ^ (2.12) More generally, we can formulate the problem directly from the design specifications. As shown in [65, 23], a controller design problem can be cast as a set of functional inequalities that must be satisfied. A flexible approach is to divide the performance specifications into two classes: <t>i{Q) < ai,i E n, and (2.13) ii>j(Q)<bj,jem, (2.14) where ai E R+(« E ra), bj E R+(j E ra), and <£,•(•.) : RHoo—> R, i E n and V>.?(') : RHoo—> R ? i ^ 221, are convex but perhaps nondifferentiable. The inequalities (2.13) are soft constraints or specifications about which the designer is flexible; and the inequalities (2.14) are hard constraints or specifications that must be met. Thus the design of control systems can also be carried out by solving the following convex optimization problem: ( p 2 ' 3 ) : nSk {<KQ)MQ)<i>h (2-15) where <j){Q) = max{<^(Q) — ai,i E 21} and i>(Q) = ma,x{tpj(Q) — bj,j E m}. 17 Chapter 2. Multiple Objective Control Problems Example 2.2: We consider the same problem as that in Example 2.1 using the frame-work in Figure 2.2. To get the generalized plant G we first define w — (Wl ^ — \w2 ( d ) ' * = (ll) = ( w 2 y e ) ' y " r " y " a n d u = / ^ T h e n G 0 Wi W]P jy2 -w2 -W2P 1 -i -p (2.16) Suppose, for simplicity, that the nominal plant P is stable, real rational and proper. Using the Q-parametrization in (2.8), we have TZ1W1 = WiPK(I + PK)'1 = VPiPQ, and (2.17) TZ2Wa = W2(I + PK)-1 = W2(I + PQ). (2.18) Therefore the robust control problem discussed in Example 2.1 can be formulated as convex optimization problems in the form of ( P 2 2 ) : ( p 2 A ) • 7 = n c rmn max {||W.PQW^ \\W2(I + PQ)\\2}, or (2.19) (P2'5) •• -7= min maX{||W1Pg||0 O,||W2(/ + PQ)|| „,}.• (2.20) C^feK-Jtloo Clearly, if we have 7 < 1, then we can get a solution to the robust control problem. § 2.3 Analytical Solutions There are no analytical solutions to the above formulated multiple objective control problems in (2.12) and (2.15). Only a few had been obtained for some simpler related problems. For example, control problems with multiple objectives in the H 2 sense has been handled by assigning a cost functional to each specific design objective and by merging all the objectives into a "global" cost functional by a weighted sum [17]. The resulting optimal control law, which is seldom optimal from any single design objective 18 Chapter 2. Multiple Objective Control Problems point of view, brings the global functional to a minimum. A well known example for this merging of objectives is the standard linear quadratic regulator problem [66], where the weighted sum of the L.2-norms of the controlled output and the control input is optimized. The advantage of the weighted sum approach is two-fold: first, the control law that results is Pareto-optimal [17]. Second, the weighted sum approach enhances the tractability of the design problem since it reduces to the intensively researched optimization problem of a single cost functional. The major design drawback to this approach is that there is no apparent way of determining the summation weights in advance. These weights control the trade-offs that are made between conflicting objectives, and their choice involves a judicious intuition in a "cut and try" sequence. The latter intuition is lost quickly when the number of weights to be determined increases. Holohan and Safonov [29, 30] studied the two-objective control problem ( P 2 5 ) as in (2.20), for SISO systems. By using the concept of the Cartesian product of vector spaces, they converted this problem into a geometrical one: find that vector belonging to a certain subspace lying closest to another given vector. Function space duality theory is used to develop the duality relations for the problem. Bounds are established for the general case. For the case where the plant is both stable and minimum phase, the optimal achievable performance (i.e., the numerical value of the minimum in (2.20)) is explicitly determined. However, the optimal controller has not been obtained in closed form. Recently, Zames and Owen [31] [67] studied the dual form of this problem but for MIMO systems using Banach space duality theory. Again, only a flatness or all-pass condition is obtained. To get the controller, numerical methods are still required. Several analytical results [32, 68, 69, 70] have also been obtained for solving the following so-called mixed H^/Hoo problem: min i l i r ^ y i l T ^ l l ^ l } , (2.21) stabilizing K 19 Chapter 2. Multiple Objective Control Problems where TZiWi,i = 1,2, denote the closed loop transfer matrix from the exogenous input w{ to the controlled output z,. Bernstein and Haddad [32] studied a special case of (2.21). With the restriction w = w\ = w2, they obtained necessary conditions for optimality of controllers of a pre-specified (full or reduced) order. Doyle et al [68] and Zhou et al. [71] considered the dual of the Bernstein and Haddad type problem. They obtained necessary and sufficient conditions for the existence of an optimal controller (see also [72]). These conditions were given in terms of coupled nonlinear matrix equations of the Riccati type. Khargonekar and Rotea [70] have shown that in the state feedback case one can come arbitrarily close to the optimal mixed H2 [Hoo performance measure using constant gain state feedback and the output feedback problem can be reduced to a state feedback problem; moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of real matrices. Unfortunately, as we can see, all these results still demand numerical methods. § 2.4 Numerical Solutions As discussed before, the Q-parametrization of stabilizing controllers allows the multiple objective design problem to be formulated as a convex optimization problem in Hoo such as (2.12) or (2.15). This problem is infinite-dimensional in the sense that Q cannot in general be specified by a finite number of parameters. However it can be approximated by a finite-dimensional convex optimization in the following manner: fix N basis functions Qi (i G jV) in RHoo, and then choose where x = [xi, • • •, x^] is a vector of new design parameters. This approximation of Q(s) is known as the "Ritz approximation" [23]. It can be shown, by using Weierstrass' Approximation Theorem [73] as done in [65, 23], that any Q e RHoo can be uniformly N (2.22) i = l A ' 20 Chapter 2. Multiple Objective Control Problems approximated arbitrarily close by the Ritz approximation in the form of (2.22). However, since there is no systematic way available to choose the basis functions, a large number of parameters is normally needed to obtain a satisfactory approximation. By using the Ritz approximation, problem ( P 2 3 ) in (2.15) reduces into a convex optimization problem in the finite-dimensional vector space R ^ : min {/(x)|flf(x)<0}, (2.23) x£R,N where /(•) : R ^ -> R and g(-) : RN -> R are convex but perhaps nondifferentiable. For example, (2.23) approximates (2.15) when f(x) = max |^jfc(Q(a)) : Q{s) = J ^ S i Q i O o J ; and g(x) = max{rf>k(Q(s)):Q(s) = y2xiQi(s)\. kem { U J There are several simple but powerful algorithms specifically designed for the finite-dimensional convex optimization problem (2.23), e.g., the cutting-plane method [74, 75], and the ellipsoid method [76, 77]. These algorithms require only the ability to compute the function value and a single subgradient for / and g at any given point x. Therefore they are easy to implement. A key feature of these algorithms is that they maintain converging upper and lower bounds on the desired minimum value, and thus can compute this quantity to a guaranteed accuracy. There are also some more efficient methods such as interior-point methods [78]. They have been successfully applied to general convex optimization problems, especially in problems involving linear matrix inequalities (LMIs) [24]. § 2.5 Discussion Despite some progress in the multiple objective control problems, they have turned out to be very difficult and seem unlikely to have a complete analytical solution. Even 21 Chapter 2. Multiple Objective Control Problems though there exist some analytical solutions to some particular problems, they still demand efficient numerical algorithms. The convex optimization approach offers interesting possibilities in the design of multiple objective control systems. It enables us to solve control problems with complex specifications, including hard constraints on various loop response parameters, in the frequency domain, the time domain and some operator norms. The main limitation of this approach is that it generates high order controllers. However, the advent of cheap, high performance, digital processors has substantially reduced the relevance of controller order. Also, even if a controller designed by this approach is not implemented, knowledge that particular loop shaping specifications can or cannot be achieved is very valuable information to the designer. The designer then knows exactly how much performance is given up by using a lower order controller, obtained by some other design method. 22 Chapter 3 Multiple Objective LQ Control Problem A minimax approach, based on convex duality and optimization, is presented for solving the multiple objective L Q optimal control problem. For the infinite time horizon case, the minimax solution of this problem can also be obtained by solving linear matrix inequalities (LMIs). § 3.1 Introduction In this chapter, we shall pursue the minimax solution to the multiple objective linear-quadratic optimal control problem, where the functional to be minimized is the maximum of several quadratic performance indices. In the previous work done in [19, 20, 79], technical difficulties arise in the search process due to the involved dependency of the matrix Riccati differential equations (RDE) or algebraic Riccati equations (ARE) on a weighting coefficient vector. We will show that by using convex duality and optimization method the search process can be efficiently completed. Although the method used here is in essence the same as in [23, 24], the problems to be solved are in different formulations. More importantly, the minimax solution for the finite time horizon problem has not been obtained before by convex optimization. The organization of this chapter is structured as follows. In the following section, the formulation of the problem is presented. In Section 3, basic results of linear quadratic theory are reviewed; they are instrumental in determining the solution of the problem considered. In Section 4, the multiobjective linear quadratic optimal control problem is solved by convex duality and optimization. In Section 5, the minimax solution for the infinite time horizon case is reduced to solving linear matrix inequalities. Some concluding remarks are given in the final section. 23 Chapter 3. Multiple Objective LQ Control Problem § 3.2 Formulation of the Problem Consider the LTI system described by x = Ax + Bu, x(0) — XQ, (3.1) Ql °1 \u 0 R? \ lx. (3.2) where a; is an ra-dimensional state vector, u is a p-dimensional control input, and zi, i 6 m, are the exogenous outputs. For any u, the m performance indices are defined by Assume that (A,B) is controllable; that \Q?,AJ, i e m, are observable; and that Qi > 0, Ri > 0, and F{ > 0. These assumptions ensure that each of the objectives Ji, i £ m, has a finite minimum value, and indeed this minimum is attained by a linear feedback control law that stabilizes the system [80]. The convexity of each objective Ji, i G m, is also assured under these assumptions. This model may also be used for determining the feedback strategy in a control problem where m distinct situations may arise, each case being managed by selecting the control minimizing a corresponding performance index. If it is not known beforehand which of the equally likely situations will arise, a reasonable choice for the overall performance index, J(u), in the control problem is the maximum of the m performance indices. Therefore the multiple objective linear-quadratic control problem (MOLQP) is formulated as a minimax problem: (3.3) o where tf > 0 is the time horizon. min max Ji(u) u jgm subject to x = Ax + Bu, x(0) = XQ. (3.4) 24 Chapter 3. Multiple Objective LQ Control Problem § 3.3 Linear Quadratic Problem Before proceeding with the solution of the minimax problem MOLQP described above, we shall recall briefly the solution to the linear quadratic problem. The finite time horizon problem is to choose u e Z/2[0, tf) so as to minimize a single performance index J« '(«) = If + / ( + R*u dt,0<t< tf, (3.5) subject to the dynamic constraint (3.1), whereas the infinite time horizon problem is to find u £ L,2[0,oo) which minimizes |2^ J°°(U) /( 0 1 2 I Q2x + R2u dt. (3.6) We summarize the well-known results as follows. Lemma 3.1 [66, 80, 81]: Let R > 0, Q > 0, and F > 0. Then the minimum value of J T F in (3.5) subject to (3.1) is xTP(0)xo, where P(0) is the solution to the matrix Riccati differential equation (RDE) P(t) + P(t)A + ATP(t)- P(t)BR-lBTP(t) + Q = 0, P(tf) = F. (3.7) The matrix P(t) exists for all t < tf. The associated optimal control is given by the linear feedback law u (t) = -RT1BTP{t)x(t), t <E [0,tf]. (3.8) Lemma 3.2 [66, 80, 81]: Let R > 0 and Q > 0. Suppose that (A, B) is controllable and (Q%,A^ is observable. Then the minimum value of J°° in (3.6) subject to (3.1) equals XTPXQ, and is achieved by choosing u*(t) = -R~lBTPx(t), t> 0, (3.9) where P is the unique positive definite solution of the algebraic Riccati equation (ARE) PA + A'P- PBR-'B1 P + Q = 0. 25 (3.10) Chapter 3. Multiple Objective LQ Control Problem Lemma 3.3 [82, 24]: Assume that (A, B) is controllable. Let J* = inf {J°°(u), x = Ax + Bu, x(0) = x0}. «ei2[o,o6) v ' . ' J (3.11) J* is bounded if and only if there exists a real symmetric matrix P ~ PT such that (3.12) ATP + PA + Q PB BTP R > 0. When J* is bounded, its numerical value can be obtained by solving the following optimization problem with choice of variable P = PT { max < XQ PXQ\ ATP + PA + Q PB BTP R > 0 and the corresponding optimal control is u (t) = -R-1BTP*x(t),t> 0, where P* is the solution in (3.13). (3.13) (3.14) Remarks: (i) . The positive-definiteness condition in (3.12) is called a linear matrix inequality (LMI) in the parameter P. The LMIs can be formulated as convex optimization problems that are amenable to computer solution. Recently the efficient solution of LMIs has attracted considerable interest among researchers [24]. A short introduction to LMI and the software for solving LMI problems is given in Appendix B. (ii) . The LMI formulation of LQ problem in Lemma 3.3 can be derived from Theorem 3 of [82] by Willems. The details of the connection between Lemma 3.2 and Lemma 3.3 can be found in [24]. 26 Chapter 3. Multiple Objective LQ Control Problem § 3.4 Solution via Convex Optimization Before we go further note that MOLQP is not in a form that can be handled by the usual methods for LQ design. However, MOLQP can be transformed into a convex optimization problem, which can be easily solved by some numerical methods, such as the cutting-plane algorithm [23, 75] in Appendix A. m A f m 1 Define J A ( « ) = £ A,-Ji(u), where A e S m = \ A e R m : Ai > 0, £ A,- = 1 \, a i=i I !=i J unit simplex in R m . Then as proven in [19, 83], m i n m a x Ji{u) = m i n m a x J\(u) U i € m " A e £ m (3 15) = m a x m i n J\(u). The first equality comes from the elementary scalarization result for finding the maximum element of an m-vector [19, 83]; and the second equality is obtained by interchanging the order of the min and max operations, which can be justified because J\(u) is linear in A and convex in u [84]. Define V(A) = m i n { J A H : x = Ax + Bu, x(0) = xQ}. (3.16) u Since V is the minimum of a family of linear functions of A, it is concave. We can thus pose the MOLQP as a convex optimization problem, which maximizes over the compact, convex, finite-dimensional set E m , the concave function V, i.e., m a x V(X) (3.17) and it can be solved by using the cutting-plane algorithm in Appendix A. To use the cutting-plane algorithm, we handle the equality constraint Ai + A 2 + ... + A T O = 1 (3.18) 27 Chapter 3. Multiple Objective LQ Control Problem by letting be our optimization variables and setting (3.19) A m = l - A i A m _ i . (3.20) The constraints on the optimization variables are then Ai > 0, • i (3.21) 1 - Ai A m _ ! > 0. From (3.16), evaluation of the objective function V(\) at a given vector A 6 S m requires the solution of a linear quadratic problem. According to Lemma 3.1, V(X) = xTPx(0)x0, (3.22) where -PA(O) is the solution of a single Riccati differential equation p\(t) + Px(t)A + ATPx(t)-Px(t)BR^BTPx(t) + Qx = 0, Px(tf)=Fx, (3.23) where, m m m FX = Y, ^ > 0, Qx = ^ >0, Rx = J2 > °- (3-24) i=l t'=l i'=l The minimizer in the definition of V(X) for any evaluation point A is given by ux(t) = -R-lBTPxx(t), t> 0. A subgradient #0bj 6 - f t" 1 - 1 for the objective V(A) is easily obtained (3.25) 9ohi = J\{ux) - Jm(ux) (3.26) Jm-\(uX) - Jm(uX), Before evaluating <70bj, we first present the following lemma. 28 Chapter 3. Multiple Objective LQ Control Problem Lemma 3.4: IJZX ,K\ = —R\1BTP\. With the control in (3.25), for each i G m, Ji is given by Ji = x?Wi(0)xo (3.27) where Wi = Wf satisfies Wi(t) + {A + BKx)TWi(t) + Wt(t)(A + BKX) + KTR^X + Qi = 0, Wi(tf) = Ft. (3.28) Proof: Define 2 t f Hi(t) = xT(t)Wi(t)x(t)=\\F?x(tf)j + J (jQ*x(t) + dt, (3.29) then we have J , = #;(0) = xTWi(0)x0 and Wi(t/) = F,-, Differentiating (3.29) with respect to t and using the dynamic constraint (3.1) and the control (3.25), we can obtain (3.28). • From Lemma 3.4, J 8 (i G m) can be obtained by solving the differential Riccati equation (3.28) for each i G m. With the obtained objective values Ji (i G m)> the subgradient #0bj can be easily evaluated according to (3.26). Example 3.1: We solve the following problem taken from [20]: J* - minmax{J i , i G 3}, u i subject to x = Ax + Bu, x(0) = XQ, where, for each i G 3, + 2 QU + dt. (3.30) (3.31) 29 Chapter 3. Multiple Objective LQ Control Problem and " 3 1 0 " "1 0 " ' 2 " A = - 1 2 0 0 1 ,xo = 2 (3.32) 1 0 2 1 - 1 - 1 "1 0 0" "3 0 0" Fx = 0 0 0 ,Qi = 0 0 0 ,Ri — 0 0 0 0 0 0_ "0 0 0" "0 0 0" F2 = 0 1 0 ,Q2 ='• 0 2 0 0 0 0 0 0 0 "0 0 0" "0 0. 0" F3 = 0 0 0 ,Qs = 0 0 0 , R3 — 0 0 2 0 0 1 (3.33) 1 0 0 1 1 0 0 1 1 0 0 1 In Example 3.1, we first transformed this three-objective min-max problem into max-min problem as in (3.17), then we used the cutting-plane algorithm written in Matlab to automatically adjust the value of A in order to find the minimax solution. Since P A a n d W» (* = 1 , 2 , 3 ) are symmetric, for each matrix six elements need to be computed as functions of time. For each given value of A, at each iteration of the cutting-plane algorithm, we first solved the 6th-order differential equation given in (3.23) for P A to evaluate the objective function V(A), and then we solved three 6th-order differential equations given in (3.28) for W , (i = 1,2,3) to evaluate the subgradient of V(X). A l l these differential equations were solved by using the 4th and 5th order Runge-Kutta formulas. There are only two optimization variables Ai and X2 in this convex optimization problem. To start the cutting-plane algorithm, the initial values for A were chosen as Ai = 0.3 and X2 = 0.5; and the objective tolerance e0bj and constraint tolerance eC On were eQbj = 1 0 - 1 0 and e c o n = 1 0 - 6 . After 32 iterations with 146.6000 seconds of Sparc 5 cpu time, the algorithm terminated and yielded the optimal objective value V(A*) = 31.3989 and the corresponding maximizers A^  = 0.2839 and A^  = 0.4627. The minimax control u* = u\* can be computed by using (3:25). Table 3.1 presents the iteration process with only first four steps and last four steps. We can see that 30 Chapter 3. Multiple Objective LQ Control Problem J\, J2 and J 3 are conflicting objectives and approximately approach the same value at the last iteration. Iteration Step Ai A2 «M«A) MO V(X) 1 0.3000 0.5000 29.8709 29.1578 38.1917 31.1785 2 0.0000 0.0000 309.0146 18.8556 83193 8.3193 3 0.0967 0.0000 46.1859 49.5584 14.7746 17.8111 4 0.0667 0.4740 114.5509 21.7482 14.5768 24.6394 i j • 29 0.2838 0.4633 31.4319 31.3847 31.3878 31.3974 30 0.2834 0.4630 31.3990 31.3985 31.3985 31.3988 31 0.2838 0.4628 31.3994 31.3988 31.3985 31.3989 32 0.2839 0.4627 31.3989 31.3989 31.3989 31.3989 Table 3.1. Example 3.1: Solution of a multiple objective LQ control problem § 3.5 Solution via Linear Matrix Inequalities When tf —* 00 and x(tf) —> 0, MOLQP becomes ' 0 0 / (H mm max 2 1 Rfu 1 \dt (3.34) subject to x= Ax+Bu, x(0)= XQ. Since the infinite time horizon problem in the single-objective L Q design is a special H2 norm minimization problem as shown in [85], the problem considered here is a special case of ( -P 2 1 ) defined in the previous chapter. As done for the finite time horizon case in the last section, this problem can be solved by the cutting-plane algorithm. The only difference here is that the objective function V(\) is evaluated by using Lemma 3.2 31 Chapter 3. Multiple Objective LQ Control Problem instead of Lemma 3.1, and the subgradient of V(\) is computed by solving AREs in (3.10) instead of integrating RDEs in (3.28) at each iteration. In this section, we present an alternative solution approach based on Lemma 3.3. This is done by avoiding the Riccati equation approach altogether and reformulating the infinite time horizon problem as an LMI-based convex optimization problem, which is summarized as the following Theorem. Theorem 3.4: Let Ri > 0 and Qi > 0 for i em. For any A 6 E m , define Q\ = m m ' / 1 \ AjQj, and R\ = ^ A ^ . Suppose also that (A,B) is controllable and [Qf, Aj,i e i=l t = l V / 221, is observable. The solution to the infinite time horizon problem in (3.34) can be obtained by solving the following LMI-based optimization problem with choice of variables P = PT and A XQPXQ subject to A The corresponding minimax control is computed by max ATP + PA + Q\ PB' BTP RX . i > 0 , Ai A m _ i >0 . > 0 , (3.35) u -R-lBTP*x, (3.36) where P* is the solution to (3.35). Proof: The solution of the infinite time problem, by using the scalarization [19, 83] and invoking the saddle-point theorem [84], is given by i2\ m a x m m m J / (jQix 2 I + R\u dt (3.37) subject to x= Ax+Bu, x(0)= XQ. Note that for any fixed A € S m , since Qx > 0 and Rx > 0, the integral inside the min is always bounded below by zero. Therefore the minimum over u of the weighted 32 Chapter 3. Multiple Objective LQ Control Problem linear quadratic problem, according to Lemma 3.3, is computed by solving the LMI in P = PT in the form of (3.13). < Example 3.2: We consider the following minimax control problem: J* — min max {Ji, i £ 3), u i subject to x — Ax + Bu, x(0) = XQ. where, for each i' £ 3, (3.38) oo + and Qi = 2 1 1 1 , 0-2 = 1 - 1 - 1 3 Rlu , g 3 dt, 1 1.5 (3.39) 1.5 3 Ri = 2, i? 2 = 1, i?3 = 1, A 0 1 0 0 and B According to Theorem 3.4, we can easily formulate this problem as an LMI-based optimization problem in the form of (3.35). Totally there are 5 optimization variables: the entries pi 1, P12, P22 oi P , and the multipliers A i , A 2 . An LMI-based P11 P12 P12 P22 . solver, sdpsol, developed at Stanford [86], was used for the optimization. The relative tolerance was chosen as e r ei = 1 0 - 6 . The solutions for two different initial states were obtained as follows: For the initial state xj = [5 2]', sdpsol terminated after 11 iterations. It generated [3.0806 1.1680' 1.1680 2.4166 the following results: J* - 55.0208, P , Ai = 0.1680, and A2 = 0.8320. The minimax control, given by (3.36), is u* = [-1.000 -2.0690 ]x. With this control, we can compute the three individual objectives: J\ = 55.0208, J 2 = 55.0208, and J3 = 23.7708. Clearly, J\ and J 2 are conflicting objectives and J3 is not active in the optimization. 33 Chapter 3. Multiple Objective LQ Control Problem For initial state XQ = [—2 5]', sdpsol terminated after 11 iterations. It generated [2.9077 1.3407' 1.3407 2.5892 the following results: J* = 24.7723, P = , A x = 0.3407, and A2 0.6593. The minimax control, given by (3.36), is u* = [-1.000 -1.9310 }x. With this control, we can compute the three individual objectives: J\ = 24.7723, J 2 = 24.7723, and J 3 = 19.7720. Again, J\ and J 2 are identified as active and J3 is not active in the optimization. Clearly, the solution to the M O L Q P is dependent on the initial condition. To better look at this, by using sdpsol, we solved Example 3.2 for some sampling initial conditions satisfying —10 < xo < 10. Fig. 3.1 shows the level curves of the overall performance with respect to the initial conditions. Also, in this example, J3 was automatically identified being non-active, and J\ and J 2 active. From the level curves of the multipliers with respect to the initial conditions as shown in Fig. 3.2 and Fig. 3.3, J\ and J 2 .are conflicting each other on the initial conditions that the multipliers are positive but less than unity. (a) Figure 3.1. Example 3.2: The level curves of the overall objective: (a) the mesh surface, and (b) the contocdur 34 Chapter 3. Multiple Objective LQ Control Problem (a) X0(1) Figure 3.2. Example 3.2: The level curves of the multiplier A i : (a) the mesh surface, and (b) the contour Chapter 3. Multiple Objective LQ Control Problem § 3.6 Concluding Remarks In this chapter, we used convex optimization to obtain the minimax solution of MOLQP. The minimax solution is identified in a two-step iterative scheme. In the first step, the weighted linear quadratic problem is solved and the set of noninferior solutions is generated. Then, in the second step, the weighting coefficients are updated to approach the minimax solution. These two steps are repeated under the control of the cutting-plane method until the desired accuracy is attained. Two examples show that the cutting-plane-based solver is quite effective and can automatically identify the active objectives without the prior knowledge of the conflicting situations. For the infinite time horizon case, the minimax solution can also be obtained in a single step by solving an LMI-based convex optimization problem. Since the LMI approach adjusts the matrix P and the weighting coefficients A simultaneously instead of alternating between the two, it is more direct than the discussed two-step scheme. From the numerical examples and from (3.35), the solution to MOLQP depends on the initial states and is open loop control optimal only for the particular initial conditions. This is different from the solution of the single-objective LQ optimal control problem, where one can always get a feedback control law. Since a number of researchers have solved the single-objective Hoo control problem by using L Q optimal game theory [87, 88, 89], one possible direction for future research is to explore the connections between MOLQP and the multiple objective Hoo control problem and to investigate the possibility of finally solving the multiple objective H^ control problem. 36 Chapter 4 Multiple Objective Control Problem In this Chapter, the solution to the multiple objective Hoo control problem for SISO systems is pursued. Optimality conditions, and, in special cases, either all-pass properties or optimal performance values are obtained via nonsmooth analysis. One numerical solution approach based on linear matrix inequality (LMI) formulation is also presented. § 4.1 Introduction In this chapter, we study the multi-disc Hoo problem only for SISO systems via nonsmooth analysis [90]. This is a special case of (2.10) presented in Chapter 2 and can be rewritten as the following optimization problem ( P 4 - 1 ) : min max {||al<? - 6 ^ } , (4.1) where, for convenience, a,, 6, and q are used, respectively, to denote T^T i^ , —T^i, and Q in (2.10). The problems N L S P [29, 30] and O R D A P [28, 31] mentioned in Chapter 2 could be transformed into some special forms of the general problem considered here. For example, if we only consider specifications on robust stability and on nominal disturbance attenuation, then the problem ( P 4 1 ) in (4.1) reduces into N L S P in the form of ( P 2 5 ) in (2.20); if we only consider specifications on robust stability and on robust disturbance attenuation, then the problem ( P 4 1 ) in (4.1) reduces into ORDAP. While the results obtained here are similar, to the results of Holohan and Safonov [29, 30], our problem is more general and as we can see from the following sections, nonsmooth analysis also provides a more straightforward solution. This chapter is structured as follows. In the following section, using nonsmooth analysis [90], optimality conditions, and, in special cases, either all-pass properties or 37 Chapter 4. Multiple Objective Control Problem optimal performance values are obtained. In Section 3, an LMI-based solution approach is presented and illustrated by a numerical example. We conclude with some discussions in Section 4. § 4.2 Analytical Results via Nonsmooth Analysis Note that problem (P41) in (4.1) is convex and nondifferentiable. Nonsmooth analysis [90] provides a powerful tool for problems with this structure. In this section, we first give some preliminaries about nonsmooth and functional analysis, and then present the main results. § 4.2.1 Some preliminaries We start with a characterization of subgradients for a finite max-functions. Lemma 4.1 [90]: Let Y be a Banach space, and U c Y be an open convex set. Suppose fi : U —> R, i £ JV, is a finite collection of functions each of which is finite-valued and convex on U . Define f(y) = max{fi(y):ieN}, (4.2) and denote the set of "active" indices by I(y) = {i e N_ : /,-(j/) = f(y)}. Then the subdifferential of the function / obeys df(y) = co{dfi{y),ieI(y)} = I Yi X& : & e dfi{y),\i >0,Y Xi = 1 In particular, at any point y where each fi is differentiable, we may take & = V/,-(y) in (4.3). Here is an extension of the simpler case above to a max-function with infinite index set. 38 Chapter 4. Multiple Objective H T O Control Problem Lemma 4.2 [90]: Let Y be a Banach space and fe be a family of functions on Y parametrized by 0 G T , where T is a compact topological space. Assume that: (i) . for each y £ Y , fe(y) is continuous as a function of 6, and {fe(y) : # G T} is bounded; (ii) . for each 9 G T , fe(y) is convex as a function of y, and is Lipschitz in y. Define a function / : Y —» R via f(y) = m*x{fe{y)}. (4.4) Then the subdifferential of / is given by df(y) = df9(y)ii(dB) : pt G P[S(y)]|. (4.5) Here, S(y) = {0 G T : /#(?/) = f{y)} denotes the "active set" and for any subset S of T , P[S] signifies the collection of probability Radon measures supported on S. The following two lemmas in functional analysis are essential to the derivation of the all-pass property in the solution of the multi-disc problem. Lemma 4.3 [91]: If v is a Borel measure on T , where T = [0,2ir], such that / e^n6du = T 0 for n > 0, then v is absolutely continuous with respect to Lebesgue measure and there exists / in H i such that dv = fd6. If / is a nonzero function in H i , then the set {0 G T: f(e}6)= 0} has measure zero. Lemma 4.4 [73]: Suppose n is a positive measure on a cr-algebra in a Banach space, and g G Li(/x). If d\ = gdu, then d\X\ = \g\du. 39 Chapter 4. Multiple Objective Hoo Control Problem § 4.2.2 Main results To use nonsmooth analysis, we reduce the space RHoo over which the minimization is performed in (4.1) to the Banach algebra Ao of RHoo functions that are continuous on the unit circle [92]. Therefore we shall next solve the following problem: ( P 4 2 ) : minmaxdla^-feilU}. (4:6) In this thesis, the systems are normally represented by their transfer functions, which are functions of the Laplace transform variable s G C+ (open right half plane) or functions of the Z-tansform variable z G D (open unit disc). Here the solution to (P 4 ' 2 ) will be derived using functions defined on the unit disc. Any function F G Ao defined on C + can be represented in terms of function / G Ao(D), and vice versa, e.g., f(z) = P(y^f) and F(s) = /(f^y). The conformal map between C + and D preserves all the important properties of F(s) as a bounded analytic function, e.g., f(z) is a bounded analytic function on D and I i n » = sup | J W I = sup | / ( ^ ) | = ||/||oo- (4.7) weR 0G[O,27r] 1 v n Let /(•) : Ao —> R be given by f(q) = ma,x{\\aiq-bi\\00} max = max max = max fo{q), max{ |a i 9 -& t | ( e^)} | naax j\d{q — b{ \ (e-?^ } (4.8) where f$ is a family of functions on Ao (a Banach space) parametrized by 9 G T = [0,27r] (a compact space), and is given by fe{q) = max {|a t9 - 6t| (e^) }. (4.9) 40 Chapter 4. Multiple Objective H T O Control Problem Clearly (P 4 - 2 ) in (4.6) is equivalent to ( P 4 - 3 ) : min/(g), (4.10) ?€A0 and the optimality condition for ( P 4 3 ) is Oedf(q). (4.11) The trivial case in which some g 6 Ao obeys aiq — 68 — 0 for all i G rahave a solution in Ao is excluded in the discussion below. To use the optimality condition (4.11), we need to compute the subdifferential of / by using Lemma 4.1. Before doing this, we check each assumption in Lemma 4.1. First, the continuity of function f$(q) and the boundedness of {fe(q) : 6 G T} for each q G Ao are obvious since a,g — bi (i G ra) are proper and stable transfer functions. Second, the convexity of /<?(<?) can be easily established as follows: Vgi,<?2 G Ao, A G [0,1], fe(Xqi + (1 - X)q2) = max{|a,(Agi + (1 - X)q2) - bi\} < m&x{X\aiqx - 6,-| + (1 - A)|a^ 2 ~ k\} (4.12) = X\aMq\-hM\-\-(\ - X)\aMq2-bM\ < Amax{|aigi - 6,1} + (1 - A)max{|a,g2 - k\} = A/*((?i) + (1 - X)fe(q2), here, M is the index corresponding to the maximum of m functions X\aiq\ — bl\ + (1 - X)\aiq2 — bi\,i G m . To verify that fo(q) is Lipschitz in q, we first show that for 41 Chapter 4. Multiple Objective H w Control Problem each i G m , and 9 G T , gf(q) = | a t g — 6 i | ( e j e ) is Lipschitz: Vr/i, <72 € A 0 , rf(?l)"rf(92)| = k ? l - & i | ( e J * ) - | a ; g 2 - M ( e J < ? ) | (4.13) < Kdiqx - bi) - (a,iq2 - (eJ'*) = \ai\(e>')\q1-q3\(e>9). Since each is Lipschitz with constant H a i U ^ , the next result shows that f$ = m a x {gf,i G in} is Lipschitz with constant m a x { H o i H ^ , i € m}. Lemma 4.5 If 'g\, ...,gm are functions on a linear space with respective Lipschitz constants Ki,...,Km> then the max-function / — m a x { < ? i , z G ra} is Lipschitz with constant K = m a x {if,, i £ m}. Proof: For any points p, q, • f(q) = max{gi(q)} < max {gi(p) + Ki\q - p\} (4.14) < m a x {gi(p)} + m a x {K{}\q - p\ = f(p) + K\q - p\. Thus f(q) - f(p) < K\q - p\. But since q and p are arbitrary, we can exchange them to obtain \f(q) — f(p)\ < K\q — p\ as required. • Now we proceed to compute the subdifferential of /. By using Lemma 4.2, for any direction h € Ao , we have < df(q),h>= lj< df9,h>du(9) : pL G P[S(q)}V (4.15) 42 Chapter 4. Multiple Objective Control Problem where S(q) = { O e T : f(q) = fe(q)}. From ( 4 . 1 5 ) , we need to compute the subdifferential of fe(q)- Before doing this by Lemma 4 . 2 , we notice that fe(q) = max {\aiq — 6t|(e J I'*)}, and clearly , for each fixed i G m, \aiq — 6,-|(eJ'*) is finite-valued and convex in q G Ao. Therefore the assumptions in Lemma 4 . 1 are satisfied. Next we compute the directional directive of g\ for each i G rn.i & G T , as follows: V/i G Ao,t G R, 9i(q'+th)-gei(q) <Vg°i(q),h> = ]im i2|«i/t|2 + 2*Re{(oig - &i)*a,-ft} t^o *(|a,-(g + ifc) - 6i| + |a,-g - 6,-|) Re{(a,'9 - fej)*a,7i} (4.16) Finally, from Lemma 4 . 1 , the subdifferential of f$(q), for any direction h G Ao , is given by < df0(q),h>= ( 4 . 1 7 ) Therefore, from the optimality condition ( 4 . 1 1 ) and the subdifferentials of f(q) and f$(q), respectively, in ( 4 . 1 5 ) and ( 4 . 1 7 ) , we have q solves (P 4 - 3 ) => o G om 3p(6) eP[S(q)] such that V / i G A o , ( 4 - 1 8) 0 -Hi ai(aiq — bi)*ai Wiq- bi\ (ej*) am-Since for a fixed h, (—jh) also satisfies ( 4 . 1 8 ) , we have cti(a,iq — bi)*ai 0 (e*) ( 4 . 1 9 ) - &i| The following theorem gives the all-pass properties of the solution, and, in special cases, the optimal performance value. 43 Chapter 4. Multiple Objective Control Problem Theorem 4.1: Assume that the simultaneous equations — bt; = 0 (i G m) have no solution q in Ao- If the multi-disc Hoo problem (-P4'2) (m > 2) in (4.6) has a solution q G Ao , then either (i) max { \aiq — 6,-| (e^), i G 221} — const. ,0 G T , or (ii) some 0 G S(if) obeys condition (4.28) below. In this case, if a,-(eJ'*) 7^  0 for all i G 221, V0 G T, then there exists £ S m such that TO , i'=i Proof: We show first that the following hypothesis implies conclusion (i): cti(aiq — bi)*a,i i=i \a{q- bi\ Choose h = e>ne,n > 0. Then (4.19) becomes a>i(a,iq — &i)*a,-(f) ± 0,ye G s(q). (4.21) /tit, K<7 - &i| (>) <W) 0,n > 0. (4.22) From Lemma 4.3, 3/(e j e) G Hi such that 'ai(aiq- UfaA ( -e\ Since // is a positive measure, and by the assumption, ^ tt;(aig—&i)*a. |a,g—6;| T, / is a nonzero function in Hi . Therefore the support of / is supp(/) = supp(|/|) = T. From Lemma 4.4, since du > 0 and d0 > 0, (4.23) becomes cti(aiq- bi)*ai (4.23) (e>*) / 0, V0 G (4.24) E du = l/l <*0. (4.25) 44 Chapter 4. Multiple Objective Hoo Control Problem Therefore supp(/x) = supp(|/|) = T . But we have (j, <E P[S(q)] in (4.18), so T = S(5) = {0 € T : f(q) = fe(q)}. (4.26) (4.27) Hence conclusion (i) of Theorem 4.1 holds. In this case, the "all-pass" property from single-objective Hoo control theory extends to the multiobjective case. It remains to treat the possibility that (4.21) fails, i.e., that some 0 6 S(q) obeys ~ai(aiq - 6i)*a,' E i=i o-iq — b{ 0. (4.28) Under this condition we prove conclusion (ii). Indeed, since each ai(eJe) ^ 0, V0 e T , by hypothesis, and V0 G S(<f), |a,<7- 6,-|^e^ = \akq — bk\(e^ for alH,fc e m, from (4.28), (4.29) g [ a r f ( ? - ^ ) ] ( ^ = o , which immediately gives the desired result in (4.20) by choosing |2 i2 (4.30) i=i When conclusion (i) of Theorem 4.1 holds, we can get the infinite-order property for the optimal solution summarized in Corollary 4.1. This corollary suggests that in some situations, the optimal controller may not be realizable, so that only approximate solutions can be pursued. This justifies the use of approximate numerical methods, such as the LMI-based optimization presented in the following section. To obtain Corollary 4.1, we first prove the following lemma. 45 Chapter 4. Multiple Objective Hoo Control Problem Lemma 4.6: Let T(s) = —2 ™ 1 ^ — - (n > m > 0 , (4.31) Dnsn + A,-!*'*- 1 + ... + D0 be a proper real-rational transfer function. If \T(joj)\ = const., V w £ U c [0, oo), where the measure of the set U is not zero, then indeed |T(ju;)| = const., Vu; e [0, oo). Proof: Let hN(w) = Nm(joj)m.+ Nm^(ju)m-1 + ... + No and hD(w) = Dn(jto)n + Dn_\(ju)n 1 + ... + D 0 (4.32) (4.33) These can be rewritten as, respectively, 2m hN(u) = YhN,iu\ (4.34) i=0 2n and hD(u>) = hoju3, (4.35) j=0 where the real coefficients h^^ (i e 0 U 2m) and hpj (j G 0 U 2n) can be obtained from (4.32) and (4.33). We denote c as a constant and positive real number. Then, by assumption, we have 2m l r - H I = TTTA = Tn = c, Vu; e U , i'=0 (4.36) i.e., 2n 2m i=0 c Y hD,iu3 ~ Y h^,iujl = 0, Vu> G U , i=0 . 2n or y^ckuk - 0, Vu; £ U , (4.37) (4.38) it=o 46 Chapter 4. Multiple Objective Control Problem where, cjfc(fc£0U2n) are the real coefficients, which can be expressed by c, hNii(ieOU 2m), and hDj (j £ 0 U 2n). Taking any 2n + 1 distinct points {u-i,W2, •••,W2n+i} C U , we have 2n Y^ckuf = (U £ 2n + 1. (4.39) it=0 If we took c/t (fe G OU 2n) as unknown variables, solving (4.39) would yield ck = 0,k £ 0 U 2 n . (4.40) This implies that 2n ^ c ^ f c = 0,Vcu£[0,oo), (4.41) it=o i.e., 2n 2m cY,hDtiJ - £ fcjv.iw'" = 0, Vu; £ [0, oo),. (4.42) i=0 t'=0 or |T(ju>)|2 = c,Vu>£[0,oo). (4.43) Corollary 4.1: Assume that the multi-disc Hoo problem (m > 2) has a solution g £ Ao. If max{|a,<f — £ m} = 7 ,V0 £ T , and 3i E m such that the measure of the set {6 £ T : 7 = |a,-9 — bi\(e?e)} C T is not zero, then q is of infinite order. Proof: First notice that ai(i £ m) and bi(i £ m) are given real-rational proper transfer functions of finite order. Then the corollary can be proven by contradiction using Lemma 4.6. 47 Chapter 4. Multiple Objective H m Control Problem From (ii) of Theorem 4.1, the explicit expression of the optimal performance value, denoted by 7 , can be obtained by solving the following equations in 7 and (i e m): Wiq- k\ = 7 , ( J £ m), (4.44) i—l m X)ft = l , # > 0 , t e m . (4.46) t'=l This could be done by using symbolic math software, such as Maple [93]. An example follows Example 4.1: Under the assumption of (4.28), obtain the optimal performance value for two-disc problem: (P 4 - 4 ) : 7 = min max {\\aiq - b^}. (4.47) Solution: To solve this problem, we use Maple command solve(eqns,vars), where eqns := {equations in (4.42 — 4.44)}, and vars : = B 2 , 7 } . Executing the com-mand yielded the following results: 7 . |ai « 2 | ' M + « 2 | ' Ctl&2 - a 2 & i M + |«21 (4.48) Combining (i) of Theorem 4.1 and (4.48), the results for two-disc problem are summarized as the following corollary. 48 Chapter 4. Multiple Objective Control Problem Corollary 4.2: Assume that all of a,g — bi = 0, i-= 1,2, have no solution in Ao. If the two-disc HQO problem ( P 4 4 ) has a solution q G Ao , then either (i) max{|aiif— 6,-|(eJI'*),i € 2} = const. ,0 £ T , or (ii) some 0 G S(if) obeys condition (4.28). In this case, if a i (e j e ) ^ 0 for all i G 772, V0 G T , then the optimal performance is given by ax 62 — G2&1 (4.49) We can use this corollary to obtain the optimal performance value for N L S P as in (2.20), studied by Holohan and Safonov in [29, 30]. To do this, we rewrite N L S P here as follows: ( p 4 ' 5 ) : 7 2 = ^{\\wipQ\LA\W2(i + PQ)\\ OQ}. (4.50) Clearly, (P 4 - 5 ) is a special form of two-disc problem ( P 4 4 ) in (4.47) with a\ = WiP, h — 0, and a 2 = W2P, fe2 = — W2, where W\ and Wi are two known outer functions. Using (4.49), we obtain the same results as in [29, 30]: 72 = w1w2 |Wi|+|W a | § 4.3 Numerical Solution via LMI Note that the solution to problem ( P 4 1 ) in (4.1) can be obtained by solving (P 4 - 6 ) : ' min 7 subject to \\aiq — biW^ < 7 , i G m. 1 The Hoo-norm constraints in (4.51) can be transformed into LMIs by using the following version of the Bounded Real Lemma, which may be derived via a bilinear transformation of the generalized Positive Real Lemma in [94, 95]. 49 Chapter 4. Multiple Objective Control Problem Lemma 4.7 [24]: Consider a transfer function T(s) with a state-space realization T(s) = [A, B,C, D]. The following statements are equivalent: (1) The LQO—norm of T(s) satisfies the constraint \\T(s)\\Lao < 7 ; (4.52) < 0. (4.53) (2) There exists an symmetric matrix P = PT such that the following LMI holds: \ATP + PA PB CT ' BTP - 7 / DT C D - 7 / To transform the Hoo-norm constraints in (4.51) into LMIs, we first approximate q(s) in (4.51) as a linear combination of known basis functions, qi(s), as in (2.22) N where x = (xi, • • •, x^) is the vector of new design parameters, and the basis functions, qi(s), can be chosen as proper and stable transfer functions, for example, the all-pass functions, s — p s +p i-1 ,p>Q. (4.55) With N chosen sufficiently large, the Hoo-norm performance in (4.51) can be approx-imated arbitrarily closely [23]. By using the parametrization of q(s) given in (4.54), for each i£ m. ai(s)q(s) - bi(s) ai(s)qi(s) - f j x1 i ; ai(s)q2(s) ai(s)qN(s) . -his) [xT l][Ai,Bi,Ci,Di] A{, Bi, X^Ci, Di (4.56) 50 Chapter 4. Multiple Objective Control Problem where XT = [xT 1], and [Ai,Bi,d,Di] = d(sl — Ai) xBi + Di is the state-space realization of the transfer function matrix ai(s)q2(s) (4.57) a,i(s)qN+1(s) . -bi(s) J We can now apply Lemma 4.6 to obtain this section's main result. Theorem 4.2: If q(s) is given by the N-th order approximation in (4.54), then the solu-tion to ( P 4 6 ) in (4.51) can be obtained by solving the following LMI-based optimization Clearly, (P 4 ' 7 ) is a finite-dimensional convex problem and efficient numerical al-gorithms exist for its solution [23, 24]. Here, to illustrate the solution procedure, we approximately solve a two-disc problem, previously called a robust control problem in Example 2.1 and Example 2.2, by using the LMI-based solver sdpsol [86]. Example 4.2: Find a stabilizing controller K such that 7 < 1, where (4.58) 7 = max{||r 1(^)|| 0 0,||r 2(^)|| 0 0}, where T1 = Wl(;I + PK) - 1 (4.59) T 2 = W2PK(I + PK)-\ The plant and the weighting functions are s - 1 2 ' 11s -= l s + 1 ( s + 6 ) 2 (4.60) 3 s + 2 (s2 •+ 2s + 37)' 51 Chapter 4. Multiple Objective Control Problem Solution: Notice that the plant here is not stable. Therefore the solution of this problem proceeds by first obtaining a stable coprime factorization of P as in Theorem 2.1. We take N = — ; M = ; s + 6' 5 + 6 ' (4.61) X = - 0 . 8 ; a n d F = - 1 . 8 . In terms of the Q-parametrization, we cast this problem as the following convex opti-mization problem: (P 4 - 8 ) : 7 * = min max{||a i g - b ^ , \\a2q - b2\\00} (4.62) qt ltx loo with ai = -WiMN- h = -WiMX; (4.63) a2 = -W2NM; b2 = —W2NY. In this example, we chose all-pass functions as the basis functions as in (4.55) with p — 1. The number of the design parameters was first chosen as N = 5, and then was increased one by one until we obtained a satisfactory solution. From (4.57), the corresponding state-space realization data, [Ai,P>i,Ci,Di],i = 1,2, were generated. This realization was high-order. Since sdpsol can only efficiently handle low-order systems, a model-reduction procedure was used in this step. Then, by using Theorem 4.2, problem ( P 4 8 ) was transformed into an LMI-based optimization problem in the form of (4.58). Finally, the LMI-based solver sdpsol was used to solve the optimization problem. Key features of this procedure are tabulated in Table 4.1, which shows the number of terms used to parametrize q(s), N; the obtained performance value, 7 ^ ; the number of variables, ravar; the number of constraints, n c o n ; and the cpu time (in Sparc 5 seconds), £ c p u , needed in the computation. For example, at N = 14, [Ai, Bi, d, Di], i = 1,2, were both obtained as 15-th order systems, and therefore the LMI-based optimization problem in the form of (4.58) had 255 variables: P i = P f e R 1 5 x 1 5 , P 2 = P 2 T e R 1 5 x 1 5 , x 6 R 1 5 and 7 € R, and 578 constraints. With 206 seconds of Sparc 5 cpu time, sdpsol 52 Chapter 4. Multiple Objective Hoo Control Problem x = and 714 = 0.9977. (4.64) terminated after 21 iterations and yielded the following results 0.8102 -0.3325 -0.1207 0.0596 0.1769 0.2224 0.2062 0.1504 0.0808 0.0197 -0.0194 -0.0332 -0.0274 -0.0128 The corresponding controller to this solution can be obtained according to (2.4). With this controller, the frequency responses of two weighted objectives in (4.59) are shown in Fig. 4.1. The graph clearly shows that the designed controller meets the required specifications: 714 = 0.9977 < 1. With an increase of the order of the approximated q(s), we can expect that max {|TI(JOJ)| , |T2(jo;)|} approaches "flatness" and the two objective functions are shaped over mutually exclusive frequency ranges. This would be consistent with all-pass property for the optimal solution described by (i) of Theorem 4.1 in the previous section. 53 Chapter 4. Multiple Objective Hoo Control Problem N 7iV ^var ^con ^cpu ( s ) 5 1.3358 48 128 3.0215 6 1.2019 63 162 4.8477 7 1.1047 80 200 9.1406 8 1.0381 99 242 14.1154 9 1.0215 120 288 24.5263 10 1.0204 143 338 40.1288 11 1.0184 168 392 61.1081 12 1.0100 195 450 94.3341 13 1.0007 224 512 135.7785 14 0.9977 255 . 578 206.1405 15 0.9973 288 648 284.4093 Table 4.1 Example 4.2: Solution results "I 1 1 -„,-r^ , , 1 1 1 •' ' '•"•n • ' 0.8 | 0 . 6 0.4 0.2 n 10~2 10"1 . 10° ' " 101 102 Frequency Figure 4.1. Example 4.2: Weighted objective functions Ti(ju)\ (dotted line) and |T2(icc;)| (solid line) when 7Y = 14 54 Chapter 4. Multiple Objective Control Problem § 4.4 Concluding Remarks Nonsmooth analysis has been shown to be a useful tool in the study of multiple objective Hoo control problems. Optimality conditions, and, in special cases, either all-x pass properties or optimal performance values were obtained. It was inferred from the all-pass property that the optimal controller in some situations is not of finite order. To get a realizable controller, only approximate solutions can be obtained. A numerical solution approach in terms of state-space LMIs has been proposed to obtain the approximate solutions: By re-parametrization of the free parameter q, the multiple objective Hoo control problem in frequency domain is transformed into a finite-dimensional convex optimization problem. This problem can be solved by many efficient algorithms. A robust control design problem, cast as a two-disc problem, has been solved to illustrate the solution procedure and to show the effectiveness of the proposed technique. rv 55 Chapter 5 Numerical Solutions of Multiple Objective Control Problems In this Chapter, convex and non-convex optimization-based solutions to the general multiple objective control system design problem are presented. The solution procedures are illustrated, and some computational issues arising in the design process are discussed in terms of design examples. § 5.1 Introduction As discussed in Chapter 2, parametrizing all stabilizing linear controllers in terms of a stable transfer function Q allows many specifications to be formulated as convex constraints on the free design parameter Q. In most cases, Q is approximated by a linear combination of given stable basis functions, so that convex constraints on Q become convex constraints on finite dimensional vector of design parameters. The advantage of the convex optimization approach is that it always finds a solution if one exists. However, the parameter space is usually very large, which may produce two problems. First, computational problems increases with the dimension of the unknown, and therefore an efficient optimization solver is demanded in the design. Second, it generally yields high order controllers, which must then be judiciously reduced in order to be made feasible in practice. In this Chapter, by solving two multiple objective control problems, we show that the cutting-plane based solver with constraint dropping scheme is quite efficient in convex optimization based control system design, and discuss some computational issues in the design process. Then we present an approximate numerical solution of the 56 Chapter 5. Numerical Solutions of Multiple Objective Control Problems multiple-objective control system design problem for SISO systems by using a nonlinear approximation of the free design parameter Q. We show through two numerical examples that low order controllers can be guaranteed by this new solution procedure. This chapter is organized as follows. In the following section, by solving a mixed H 2/Hoo control problem and a two-objective Hoo control problem, we describe the convex optimization based solution procedure, demonstrate the effectiveness of the cutting-plane based solver, and discuss some computational issues arising in the design process. In Section 3, the solution via non-convex optimization, which has the advantage of directly producing low-order controllers, is presented, and the solution procedure and the computational issues are also illustrated and discussed by solving the same design examples. We conclude with some remarks in the last section. § 5.2 Solution via Convex Optimization § 5.2.1 Linear approximation of Q As shown in Chapter 2, the multiple objective control system design problem can be formulated as a convex optimization problem in the form ( P 2 2 ) or ( P 2 3 ) with unknown Q e RHoo- These two problems are infinite dimensional and therefore an approximation of Q G RHoo is needed to obtain numerical solutions. To preserve the convexity property, as proposed in [36, 26], Q can be parametrized as a linear combination of basis functions: N where the Qi are chosen stable transfer functions and Xi,X2,and XJJ are N real-valued matrices of appropriate dimension, which are the new design parameters. With the approximation of Q by Q^vx, the infinite-dimensional convex program ( P 2 3 ) becomes finite-dimensional convex optimization program for the unknown X : ( P 5 - 1 ) : r m n { 0 ( g L ( X ) ) | ^ ( Q L w ) < 0 } . (5.2) 57 Chapter 5. Numerical Solutions of Multiple Objective Control Problems One important property of a convex optimization program is that every local solution is actually a global solution, so there is no danger of getting "stuck" at a local minimum. Another important feature of the parametrization is that even for such straightforward choices as Qi(s) = (f^y)^ \ any desired Q in RHoo can be approximated arbitrarily well by some Q^VX(X) provided N is sufficiently large [26, 23]. Complementing this advantage, however, is the problem that adequate approximations often require large dimensions N, and correspondingly heavy computation and high-order controllers. Until now, little work has been done on how to systematically approximate the free design parameter Q to reduce the size of the parameter space and therefore the computational complexity, and the order of the designed controller. § 5.2.2 Convex optimization via sequential quadratic program methods Many algorithms for convex optimization have been devised, and we could not hope to survey them in this thesis. For a good survey, please refer to [23, 24]. We first tried the sequential quadratic program (SQP) based solvers, such as constr and minimax, in the Optimization Toolbox for Use with Matlab [96]. These solvers are descent methods, which require the computation of a descent direction or even steepest descent direction for the function at a point. For smooth problems, these descent methods offer fast convergence, e.g., quadratic. For nondifferential optimization problems, which are often the case for the controller design problems, these solvers often exhibits slow convergence near the solution due to truncation error in the gradient calculation by finite difference approximation. § 5.2.3 Convex optimization via cutting-plane algorithms The cutting-plane method was chosen here simply due to its easy implementation. The cutting-plane method is an iterative algorithm. It is attractive since the subproblem to be solved at each iteration is a simple linear program that changes only sightly from 58 Chapter 5. Numerical Solutions of Multiple Objective Control Problems one iteration to the next and no line searches are required. It has simple stopping criteria that guarantee the optimum has been found to a known accuracy. We have developed two numerical solvers, K C P and E M C P , for convex optimization problems in Matlab. K C P is based on Kelley's original cutting-plane algorithm [74]. One criticism of K C P is that the number of constraints in the linear programs at each iteration increases by one with each iteration. E M C P is based on Elzinga and Moore's cutting-plane algorithm [75]. In contrast to K C P , it has rules for dropping old constraints in the optimization process and therefore the linear programs solved in each iteration do not grow as rapidly as the algorithm proceeds. We will illustrate this later by an example. For details of these two cutting-plane algorithms, please refer to Appendix A. § 5.2.4 Design Examples To illustrate the solution procedure by using the SQP based solvers and the cutting-plane based solvers, we solve two design examples: one is a mixed H 2/Hoo control problem [38], and the other is a two-objective HQO control problem, taken from [40]. Example 5.1: As shown in Fig. 5.1, consider a unstable nominal plant with additive perturbation bounded by \AP(ju)\<—±—,\/u, (5.4) W2(ju) where W2 = 20 is constant. The design objectives are to guarantee robust stability for perturbed plants and to minimize the H 2 -norm of weighted sensitivity function <f>(K)= Wiil + PK)'1 with weighting function (5.5) W1{s) = ^—: (5.6) 5 + 1 59 Chapter 5. Numerical Solutions of Multiple Objective Control Problems r + e K u P+AP + yp Figure 5.1. Example 5.1: A feedback system with additive plant perturbation Solution: From Hoo control theory [59, 7], a sufficient condition for robust stability is W2K(I + PK)'1 < 1. (5.7) Therefore, the controller design problem can be posed as a mixed H 2/Hoo optimization problem (5.8) ( P 5 2 ) : min U{K)\<p{K) < 0}, stabilizing K where cp(K) = W2K(I + PK)-1 - 1. (5.9) Since the nominal plant is unstable, the solution to ( P 5 2 ) proceeds by first obtaining a stable coprime factorization of the nominal plant P = NM~X as in Theorem 2.1. We take N = X = s-1 s2 + 3s + 2 s2'+ 75 - 26 M s2-s-2 s2 + 35 + 2' , Y and = -485 - 4 8 (5.10) s 2 + 3 s + 2 ' ... 5 2 + 35 + 2' Second, by using the Q-parametrization, we have W^I + PK)-1 = Thl+T1}2Q, and W^K(I + PK)-1 = T2,1+T2aQi (5.11) 60 Chapter 5. Numerical Solutions of Multiple Objective Control Problems where Ti i = WiMX, Ti 2 = -WiMN, (5.12) r 2,i = w 2 My, r 2 l 2 = - F F 2 M M . Therefore we can express (P 5 - 2 ) as an infinite-dimensional convex optimization problem (P 5 - 3 ) : min { « ) < 0 } , (5.13) where W ) = l|ri,i + r 1 ) 2g|| 2, (5.14) ^(Q) = -+- 1-N Then, by approximating Q as YjXiQi m the form of (5.1), ( P 5 3 ) is approximated i=i by the finite-dimensional convex optimization program (P 5 - 4 ) : 7 i V = min {<f>(x)\<p{x) < 0}. (5.15) Here we chose basis functions in (5.1) as all-pass functions: •/ _ \ N-i Q^(S)={J^) ,i£K- (5-16) First we solved ( P 5 4 ) by using the SQPbased solver constr in Matlab Optimization Toolbox, with both objective tolerance and constraint tolerance as 1 0 - 6 . To use constr and other developed solvers in this thesis, we make one naive approximation. We simply approximate semi-infinite constraints and objectives by discretization, e.g., replacing an Hoo-norm constraint by a very large number of single frequency constraints log-spaced in a specified frequency range. Then we solved ( P 5 4 ) by using the cutting-plane based solvers K C P and E M C P , with the stopping criteria for K C P as e0bj = 1 0 - 6 and e c o n = 1 0 - 6 , and the stopping criterion for E M C P as e = 1 0 - 6 . . To solve ( P 5 4 ) using either K C P or E M C P , we need to evaluate the objective function, <f>(x), the constraint function, y>(x), and to compute a subgradient of each. For a given x, the evaluations of <f>(x) and ip(x) can 61 Chapter 5. Numerical Solutions of Multiple Objective Control Problems be easily done by computing the H 2^norm of T M + T1)2Qgx(x) and the H oo—norm of T2,i + T2)2Qfvx(x), respectively. According to [23], one subgradient of tp(x) is given by <f>sub(x) = (<t>lub(x)' > W h e r e ' oo —oo Now denote Q the frequency at which the Hoo-norm of T2)i + T2t2Q^vx(x) is achieved. Then, according to [23], one subgradient of <p(x) is given by y>Sub{x) = T (vlub(X)'---'iPsub(X)) . W h e r e Ke{(T2il + T2t2Q^vx(x)yT2t2QA ^ { X ) = . \ \ T , i + T M x ) l L ^ ^ A l l computations were done in a Sparc 5 Sun Workstation. We solved ( P 5 4 ) for three different choices of basis functions in the form of (5.16) with p = 0.5,1, and 5. We took iV = 20. The obtained objective, 720, and the cpu time needed in the computation, tcpu, are summarized in Table 5.1 for three different solvers. p 720 tcpu (s) constr KCP EMCP constr KCP E M C P 0.5 2.6015 2.6009 2.6010 4530.65 2768.81 1559.98 1 2.5998 2.5994 2.5994 3378.89 2305.45 1238.15 5 2.5993 2.5990 2.5990 1265.45 687.28 302.32 Table 5.1 Example 5.1: Results via constr, K C P and E M C P From Table 5.1, it can be observed that 1) the three different basis functions produce different convergence speeds; and among them, the all-pass function with p = 5 yields the best result in terms of objective value; and 2) E M C P is the most efficient solver among minimax, K C P and E M C P , in terms of the computation time. Taking the solution 62 Chapter 5. Numerical Solutions of Multiple Objective Control Problems corresponding to p = 5 from E M C P , we obtained the approximated Q in (P 5 - 4 ) as QlvX(x). Since further increasing the number of parameters does not reduce the objective value significantly (< 1 0 - 6 ) , we can consider this solution as a nearly optimal one. With QlvX(x), we obtained a 25th-order stabilizing controller according to (2.4). The frequency response of the resulting weighted robust stability function is depicted in Fig. 5.2, which shows that the obtained controller satisfies the robust stability constraint, given by (5.7). 0.2r 10 10 10" 10 Frequency 10 10' 10 Figure 5.2. Example 5.1: Robust stability constraint Example 5.2: Consider an unstable plant with transfer function - 3 + 10 P(s) = s2 -0.5s + 1' It is required to design a controller K(s) which will stabilize the plant in a negative unity feedback configuration (Fig. 5.3), such that the sensitivity function Ter = (I + PK)-1 and the inverse additive stability margin TUT = K(I + PK)-1 satisfy \Ter(M\ < \h(ju)\, Vu,, (5.20) 63 Chapter 5. Numerical Solutions of Multiple Objective Control Problems and \Tur(ju)\ < \l2{jw)\, Vw, where the bounding functions li(s) (i = 1,2) are given by 'a + 0.01\ and h(s) = 2 h(s) = 10 5 + 4.5 5 + 2 5 + 1 0 (5.21) (5.22) (5.23) r + e K> + Figure 5.3. Example 5.2: A unity feedback control system Solution: If we define weight functions W\ and W2 as W\ = i j - 1 and W2 = l2x respectively, then the closed-loop system can be represented by Fig. 2.2, as shown in Chapter 2, with 2 i = Wi(I + PK)~l and T2 = W2K(I + PK)'1. The design specifications given in (5.20) and (5.21) will be met if and only if max {11^(^)11^,11^(^)11^} < 1. stabilizing K (5.24) Therefore the controller design problem can be cast as a two-objective Hoo optimization problem: ( P 5 - 5 ) : 7 = .min m a x { W T ^ K ) ^ \\T2(K)\U. (5.25) stabilizing K If the solution 7 < 1, then the minimizing controller K meets the design specifications. 64 Chapter 5. Numerical Solutions of Multiple Objective Control Problems Since the plant is unstable, the solution of problem (P 5 - 5 ) proceeds by first obtaining a stable right coprime factorization of P = NM~l as in Theorem 2.1. We take N = -* + 1 0 M = s 2 - ° - 5 s + l s 2 + 3s + 2' s 2 + 3s + 2 ' ( 5 2 6 ) v s2 + 6.5s + 16.5 , -1 .253 + 1.25 X — —r-r , and Y = —T . s 2 + 3s + 2 ' 3 2 + 3s + 2 In terms of the Q-parametrization of stabilizing controllers, the set of achievable closed-loop transfer functions Ti (i = 1,2) can be parametrized as {Ti,i + Tij2Q, Q G RHoo}, (5.27) where T^i and 2^2 (i = 1,2) are the following stable transfer functions: Ti i = I f j M I , Ti 2 = - W i M / V , and (5.28) T 2 l i = w 2 M y , T 2 ) 2 = -W2MM. Thus the original problem ( P 5 5 ) is equivalent to the following infinite-dimensional convex optimization program: N-By parametrizing Q using ^ ^ ( x ) = ^ in (5.1), we approximate ( P 5 6 ) by a finite-dimensional convex optimization program (P 5 - 7 ) : l N = m i l l i m a x { | | r , - i + r i- 2g^e(a:)| }. (5.30) Here we chose the basis functions in (5.1) as the following all-pass functions: Qi{s)={i^y i>ieK- (53i) Both the SQP based solver minimax in Matlab and the developed cutting-plane based solvers K C P and E M C P were used to solve (P 5 7 ) , . To use K C P and E M C P , we need to 65 Chapter 5. Numerical Solutions of Multiple Objective Control Problems evaluate the objective function, <f>(x) = max {||Ti(a;)HQQ, H ^ M X ) ^ } , ^ to compute one of its subgradients. For a given x, <j>(x) can be evaluated by computing two Hoo-norms: . IITitx)!!^ and || T2(«) 11 oo-Denote fc the "active" index for which ^(x) = WTkW^k = 1,2, and Q the frequency at which Hoo-norm of 7* is achieved. Then, according to [23], one subgradient of <j)(x) is given by (f>sub(x) = (<^u6(x), • • •, <f>guh(x)) , where cj>iub = -^(Tiuo^kiU^QiU^ym^i e N. (5.32) A l l computations were done in a Sun Sparc 5 workstation, we solved ( P 5 7 ) for three different choices of basis functions in the form of (5.31) with p = 0.5,1,5. We took iV = 30. The stopping criterion for minimax was taken as 1 0 - 6 , and the stopping criteria for KCP as e o b j = 10~ 6 and eCOn = 10~ 6 , and for EMCP as e = 1 0 - 6 . The obtained objective, 730, and the cpu time needed in the computation, i c p u , are shown in Table 5.2. p 730 tcpu (s) minimax KCP EMCP minimax KCP E M C P 0.5 1.0269 1.0257 1.0217 819.03 1266.72 460.95 1 0.9715 0.9713 0.9711 767.77 988.85 352.43 5 0.9741 0.9735 0.9724 | 1456.45 1139.43 390.40 Table 5.2 Example 5.2: Results via minimax, KCP and EMCP From Table 5.2, we can see again that different basis functions produce different results. Among these three choices of the basis functions, p = 1 yields the best solution to (P 5 - 6 ) in terms of the objective value. Again, EMCP appears to be the most efficient solver among minimax, KCP and EMCP, in terms of the computation time. Taking the solution from EMCP, we obtained the approximated Q in ( P 5 6 ) as Ql®x(x,s). With Ql®x(x, s) from EMCP, a 36th-order stabilizing controller K(x,s) was obtained according to (2.4). The frequency responses of objective functions TI(K^) and TAK) 66 Chapter 5. Numerical Solutions of Multiple Objective Control Problems are shown in Fig. 5.4, the unweighted sensitivity function Ter(^K^ together with its bounding function l\ and the robustness indicator Tur (^K^j together with its bounding function I2 are depicted respectively in Fig. 5.5 and Fig. 5.6. The graphs show that the designed controller meets the required specifications given in (5.20) and (5.21). Therefore QlyX(x,s) gives a reasonable solution to this example. Frequency Figure 5.4. Example 5.2: Weighted objective functions \TX (ju;) I (dotted line) and |T3(jw)|(solid line) associated with Q 67 Chapter 5. Numerical Solutions of Multiple Objective Control Problems 10' 10- 3 ' " . i — i — — i 10"3 10"2 10"' 10° 101 102 103 Frequency Figure 5.5. Example 5.2: Sensitivity function Ter(solid line) and its bounding function l\ (dotted line) associated with ( J 3 ° x 10' Frequency Figure 5.6. Example 5.2: Robustness function TUT(solid line) and its bounding function associated with Qzclxh (dotted line) 68 Chapter 5. Numerical Solutions of Multiple Objective Control Problems From Theorem 4.1, however, the above solution is not an optimal one since max{|Ti(ju;)|, |T2(ju;)|} in Fig. 5.4 is not flat. To get a better solution, we should increase the number of parameters in Q^vx until there is no significant improvement of using E M C P with p = 1 and N = 50 in (5.31), and e = 1 0 - 6 . The objective value for the approximation to this problem is 0.9666. Increasing N in Q^vx does not significantly reduce the objective value (< 1 0 - 6 ) ; and the resulting max{|Tj(ja;)|, |Ti(ja>)|} is al-most flat as shown in Fig. 5.7. This gives some confidence that the solution is nearly optimal. The controller K obtained from QHX is 56th-order, which has a 51st-order minimal realization. the solution and max {|Ti(ja;)|, |T2(ju;)|} is nearly flat. For example, we solved ( P 5 6 ) 0.2 y 0.8 h o.4 y 10' ,-2 10' ,-1 10* Figure 5.7. Example 5.2: Weighted objective functions {T^ju)|(dotted line) and |T2(jo;)|(solid line) associated with Q 50 cvx 69 Chapter 5. Numerical Solutions of Multiple Objective Control Problems § 5.2.5 Computational aspects Solving the above two design examples clearly demonstrates that EMCP is the most efficient solver among constr, minimax, KCP and EMCP. The slow convergence to the solution in the SQP based solvers is due to the truncation error in the gradient calculation by finite difference approximation. EMCP is always faster KCP since EMCP drops old constraints so that the linear programs solved at each iteration does not grow rapidly in size as the algorithm proceeds. In contrast, KCP does not drop old constraints and therefore suffers from a size problem. For example, Fig. 5.8 shows the number of constraints in linear programs versus the number of elapsed iterations when KCP and EMCP were used to solve Example 5.2 with p — 1 and iV = 30. As we can see clearly from Fig. 5.8, the size of the linear program in KCP grows linearly with the number of iterations, while the size of the linear program in EMCP turn out to be almost constant after 20 iterations. We can expect that with an increase of the size of the optimization problem the advantage of EMCP over KCP will be more apparent. 250 > 200 ! 150 • 100 50 + KCP I O EMCfi "'4 ,ooooooooojooooooooo°iooooo<>oooO|00oooooo 20 40 60 80 Number of Iterations 120 140 Figure 5.8. Example 5.2: Number of Constraints in KCP and EMCP 70 Chapter 5. Numerical Solutions of Multiple Objective Control Problems Both control design examples show that the convergence of the optimization process depends on the choice of the basis functions in the linear approximation of Q. Therefore a good choice of the basis functions should reduce the computational complexity and maybe the order of the designed controller. Unfortunately, no systematic methods currently exist on how to choose good basis functions. The two examples also show that taking all-pass functions as the basis functions usually generates very high-order controllers, and a model reduction procedure is needed in order to implement an obtained controller. It should be remarked here that the LMI-optimization based method proposed in Chapter 4 had been also tried to solve the two-disc Hoo control problem in Example 5.2. Unfortunately due to the large size of the formulated LMIs the LMI-based solver sdpsol failed to produce any results. § 5.3 Solution via Non-convex Optimization § 5.3.1 Nonlinear approximation of Q In this section, we only work on SISO systems. As shown in the last section, in convex optimization-based controller design, except for some special cases, we do not know how to choose good basis functions in linear approximation of Q, and usually very high-order controllers are generated. To circumvent this, we propose to sacrifice convexity and parametrize Q(s) 6 RHoo as a rational and proper transfer function in the following form: Here, N(> 0) is chosen to give a desired degree of accuracy for the minimal value. For N = 0, we define Qncvx(x) = x b a function of only one parameter. For N > 1, 71 2N+1 E xis .2N+l-i (5.33) Chapter 5. Numerical Solutions of Multiple Objective Control Problems x = (XI,...,X2N+I)T £ B?N+1 and y = (3/11,3/12,-,ym,yN2)T e R 2 A r are two design parameter vectors. Comparing the parametrization in (5.33) with that in (5.1) reveals that Q*tvx is one special form of QnCVX. Indeed, if y in Q„cvx is specified, i.e., the poles of QnCVX 3 X 0 fixed in the left half plane, then the form of QnCVX becomes the form of Q^vx. Therefore any transfer function in R H ^ can also be approximated arbitrarily well by the form of (5.33), and if the orders of QnCVX and Q^vx are equal, then QnCVX should approximate Q at least as well as with Q^vx. To see this, note that we can always choose the solution of Qfvx in the convex program as the initial guess of Q^cvx in the non-convex program by an appropriate mapping and then make further improvements by adjusting parameters of Qncvx m m e non-convex program. More importantly, the form of (5.33) allows the direct placement of 2N poles (some real, some in complex-conjugate pairs) at arbitrary points in the left half plane, in contrast to the fixed poles of Q*?vx in convex optimization. This is the key to limiting controller complexity: our method usually produces controllers of lower order than those produced by other methods, especially the convex optimization method. § 5.3.2 Computational Aspects With the transfer function Q parametrized as Q^cvx(x,y,s) in (5.33), the infinite-dimensional convex program ( P 2 3 ) is approximated by a finite-dimensional non-convex optimization program: (P 5 - 9 ) : lucvx = x g R 2 ™ * e ^ (5-34) There are some specifically well-developed algorithms for solving these, e.g., [35, 97], which have been shown to be quite successful. In our computations, we use the SQP based solvers, such as constr and minimax, in the Optimization Toolbox for Use 72 Chapter 5. Numerical Solutions of Multiple Objective Control Problems with Matlab [96]. To use these solvers, we make one naive approximation. We simply approximate semi-infinite constraints and objectives by discretization, e.g., replacing an Hoo-norm constraint by a very large number of single frequency constraints log-spaced in a specified frequency range. No doubt great improvements in performance would result from the use of sophisticated methods for semi-infinite optimization programs such as those described in [35, 97], Even if our method is not guaranteed to find the global minimum of the objective function, computational examples show that an acceptable suboptimal solution can always be obtained with a litde common sense. Since most non-convex optimization problems benefit from good starting guesses at the solution, here, we propose three schemes of updating the starting guesses to improve the execution efficiency. Scheme 1: Note that we can write N _ ^(^ Qncvxi.xiV) — jjN^y^ ' (5.35) where NQ(X) and Dq(y) are the numerator and denominator of QnCVX{x,y). Clearly NQ(X) is convex in x. Based on this observation, we can use a convex optimization-based solver, such as E M C P , to update the starting points in the optimization process. For a fixed N, the steps to obtain a better starting guess for solving non-convex problem (P 5 - 9 ) are the following: 1. Solve (P 5 - 9 ) by using a SQP based solver, and obtain a solution (x*cvx,y*cvx). 2. Fix Dq(y) by setting y = y l c v x , in ( P 5 9 ) and hence form a convex program (P510) : levx = x m i n + i { ^ ( ^ ( ^ / ^ ( ^ . ^ ^ ( ^ ( x ) / ^ ^ , ) ) < o}. (5.36) 3. Solve (p 5 - 1 0 ) by using the convex program solver E M C P , and obtain a solution x 4. Take (x*vx,ylcvx) a s a n e w starting guess for problem ( P 5 9 ) . . . * cvx-73 Chapter 5. Numerical Solutions of Multiple Objective Control Problems This solution procedure for ( P 5 , 9 ) , using the combination of a SQP-based solver and a convex optimization-based solver, can be simply summarized as following: define ( P 5 9 ) in (5.34) in terms ofQ%cvx = N$ (x)/D%(y) in (5.33) for some N; e7 <— stopping tolerrance; k « - 0; (XQ,?/O) <— any initial guess for (P 5 ; 9 ) ; repeat { obtain a solution (x^.vx,yncVX) t 0 (P 5 ' 9 ) and'jkcvx, using a QP based solver; define ( P 5 1 0 ) in (5.36) by setting y = y*kcvs in ( P 5 9 ) ; obtain a solution x*kx to ( p 5 1 0 ) and ^ v x , using a cutting-plane based solver; take {x*kx, y^cvx) a s a n e w initial guess for ( P 5 9 ) ; k «- k + 1; } Until "fncvx ~ levx < e 7 -In this solution procedure, since ( p 5 1 0 ) is convex, we have that 7 ^ < 7 * + ^ < levx ^ Incvx- The two sequences {"fnCV-x} and {"ikvx\ are thus monotonic nonincreasing. Clearly, both sequences are bounded below (e.g., by zero) and hence they are convergent to a same value. Scheme 2: Solutions from lower order problems can generally be used as starting points for higher order problems by using an appropriate mapping. For example, if we denote a solution to ( P 5 9 ) when N = 1 as Q}ncvx, then we can take (s2 + as + b) . (s2 + a s + fe) Qncvx (5-37) as an initial guess for ( P 5 9 ) when N = 2, where a and b are some positive real numbers. Scheme 3: The solutions for higher order problems, after using model reduction tech-niques, may also be used as starting guesses for lower order problems. For example, 74 Chapter 5. Numerical Solutions of Multiple Objective Control Problems if we denote a solution to (P 5 - 9 ) when iV = 2 as Ql,cvx, using the balance-truncation model reduction method such as balmr in Matlab, we can take its second-order model as an initial guess for ( P 5 9 ) when N — 1. § 5.3.3 Numerical examples Initial experimentation with the proposed solution of the multiple objective control system design problem for SISO systems is certainly encouraging. Here we solve two control design problems given in the last section to illustrate the presented design procedure and to demonstrate its effectiveness. Example 5.4: Solve the mixed H 2 / H [ X ; control problem: ( P 5 ' 3 ) : o ^ S {11^ 1,1+ Ti ,2g | | 2 | | |r 2 ) 1+r 2 , 2g|| o o<l}, (5.38) where the data T^i, Ti)2,i = 1,2, are the same as those in Example 5.1. Solution: By approximating Q as QnCVX in the form of (5.33), ( P 5 3 ) is approximated as a finite-dimensional non-convex optimization program (P5-n):lN = mm{\\Tn + T12Qlvx(x,y) | T 2 1 + T22QNncvx - 1 < o } . (5.39) xiV y II 2 oo ) Then using the SQP based solver constr in Matlab Optimization Toolbox and the convex optimization-based solver E M C P developed in the previous section, with Scheme 1 and Scheme 2 for updating initial guesses, we solved ( P 5 1 1 ) for N = 0,1, and 2, in a Sparc 5 Sun Workstation. The results are summarized in Table 5.3. Now we show that using the starting-guess-updating Scheme 3, we can obtain a better lower-order approximation of Q(s). We did model reduction for Q2ncvx in Table 5.3, using Matlab command balmr, and obtained its second-order model as following : n n — r s * 2 + 2.5731s + 6.5601 2 a 0 0 0 0 ^ + 2 .30H 3 + 7 . 1 0 2 5 - ( 5 - 4 0 ) 75 Chapter 5. Numerical Solutions of Multiple Objective Control Problems We took this as a new guess for Qncvx ( P 5 1 1 ) . Invoking Scheme 1 again yielded a new solution for ( P 5 1 1 ) : Q i - 20 0000 3 2 + 2 - 5 4 7 8 3 + 6 - 4 0 4 1 (5 41) LJncvx ZU.WW g 2 + + g 9 5 1 3 . ( .41) With this Qncvx we obtained a 4th-order stabilizing controller (s + l)(s + 0.7584)(s2 + 2.18945 + 5.1094) K M = 2 0 - ° 0 0 0 ( , I 0.7724)(s - 13.6993)(^ + 2.2025, + 4.9762) ( 5 ' 4 2 ) and the corresponding minimal objective value 71 = 2.5994. Comparing with 71 in Table 5.3, clearly 71 < 71. N 0N IN tcpu (s) 0 20.000 2.6771 7.97 1 20.0000(s2+0.8453s+0.228l) s2+0.9282s+0.2493 2.6340 173.77 2 20.0000(s2+0.0980s+0.1017)(s2+2.5731.s+6.560l) (s2+0.0992s+0.1021)(s2+2.3014s-|-7.1025) 2.5993 592.78 Table 5.3 Example 5.4: Solution properties Compared with the results obtained from convex optimization given in Example 5.1, here the order of the obtained controller is much lower, while there is not much compromise in the objective value. This is shown in Table 5.4, where n0T& denotes the order of the obtained controller, and 7 denotes the corresponding objective value. 76 Chapter 5. Numerical Solutions of Multiple Objective Control Problems Method 7 ^ord Convex optimization 2.5990 25 non-convex optimization 2.5994 4 Table 5.4 Example 5.4: Results via convex and non-convex optimization This problem was first solved in [38] by the U-parametrization method. This method also solves a finite-dimensional non-convex program, aiming at getting a low-order controller. The solution given in [38] corresponds to our solution for N = 0. Both solutions yielded second-order controllers, but the U-parametrization method resulted larger objective value, 2.8616, which is 6.90% higher than that in this solution, 2.6771. Example 5.5: Solve the two-objective H^, control problem ( ^ 3 ) = 7 = . imn max 1 + Th2Q||oo}, (5.43) where the data T^\, T,,2, i = 1,2, are the same as those in Example 5.2. Solution: By approximating Q using the parametric expression Qncvx in the form of (5.33), we reduce ( P 5 3 ) to a finite-dimensional non-convex optimization program (P 5 - 1 2 ) : 1 N = minmax {||rt-!(S) + Tl>2(s)Qf!cvx(x,y,s)\\ }. (5.44) xiV tt£ Ml I loo J By using the SQP-based solver minimax in Matlab Optimization Toolbox and the convex optimization-based solver E M C P , with starting-guess-updating Scheme 1 and Scheme 2, we solved ( P 5 1 2 ) on a Sparc 5 Sun Workstation. We started from N = 1 and increased N to 3, at which we obtained a satisfactory solution, which is a 6th-order model. The results are shown in Table 5.5. From Table 5.5, at N = 3, = 0.9701 < 1 meets the required specifications. To get a lower-order solution than Qncvx(s), we first reduced it into a 4th-order model, by using starting-guess-updating scheme 3, and then took the result as a new ,77 Chapter 5. Numerical Solutions of Multiple Objective Control Problems initial guess for Q\cvx. Invoking Scheme 1 again, with minimax solver and E M C P , we also obtained a satisfactory solution, which is a 4th-order approximation of Q(s) by QlcvM = 9.7013 0 2 + 5.93085 + 10.1786) (s 2 + 0.66575 + 1.1339) (5.45) K(s) (.5 + 14.4398)(5 + 4.7052)(52 + 0.50065 + 0.9999)' With QnCVX, a 6th-ofder controller was obtained, according to (2.4), as follows: 9.7028(52 + 1.92995 + 0.9444) (52 + 0.32135 + 0.7136) (s2 + 3.97425 + 4.1408) (5 + 29.9390)(5 + 0.0100)(52 + 1.88785 + 0.9148)(52 + 4.01035 + 4.2807) (5.46) and the corresponding minimal objective value is 72 = 0.9707 < 72 = 1.0030. N 0 N IN tcpu (s) 1 11.7556(s+2.3575)(s+0.0004) (s+16.8085)(s+0.0004) 1.1755 126.30 2 10.0306s(s+2.2290)(52+0.8165s+l.5746) s(s+16.5203)(s2+0.6564s+1.2929) 1.0030 141.48 3 9.7013(s+0r09356)(s+0.00019)(s2+5.9308s+10.1790)(s2+0.6657s+1.1339' 0.9701 816.35 (s+0.09359)(s+0.00018)(s+14.4395)(s+4.7054)(s2+0.5005s+0.9999) Table 5.5 Example 5.5: solution properties With this controller, the frequency responses of objective functions Ti and T2 are shown in Fig. (5.9). The figure shows that the designed controller meets the required specifications given in (5.20) and (5.21), and max{|Ti(jw)|, |T2(ju;)|} is almost "flat". Therefore even the 4th-order QnCVX(s) gives a very reasonable solution to this example. 78 Chapter 5. Numerical Solutions of Multiple Objective Control Problems Frequency Figure 5.9. Example 5.5: Weighted objective functions [T^ju) | (dotted line) and |T3(ju;)| (solid line) using parameter Q2ncvx from (5.45) The controller design problem in this example was first solved in [40] by a nested iterative #oo optimization procedure. This method is computationally demanding and the order of the desired controller can increase very rapidly with the number of iterations. Therefore at each step, model reduction techniques had to be used. The best results reported in [40] is a seventh order controller and its corresponding objective value is 0.9729. Our method here produces better results even when N — 2 in terms of objective value and the controller order [98]. In contrast to the convex optimization approach, the key feature of our proposed method is that it directly yields lower order controllers without significantly compromising the objective value. This can be seen from Table 5.6, in which 7 denotes the objective value and nord denotes the order of the obtained controller. 79 Chapter 5. Numerical Solutions of Multiple Objective Control Problems Method 7 H-ord Convex optimization using 30 parameters 0.9711 36 Convex optimization using 50 parameters 0.9666 51 The iterative Hoo [47] 0.9729 7 Non-convex optimization 0.9707 6 Table 5.6 Example 5.5: Results via convex and non-convex optimization § 5.4 Concluding remarks While there are no known analytical solutions to general multiple objective control problems, many problems can be effectively solved by convex optimization. We have demonstrated by solving two robust control problems that the cutting-plane based solver E M C P is most efficient among constr, minimax, K C P , and E M C P . We also have shown that efficiency of the solution via convex optimization depends on the choice of basis functions in the Q-parametrization. However for most control design problems, high-order controllers generally result. Therefore a further study is needed on choosing good basis functions. To get a low-order controller, a non-convex optimization procedure is proposed for the multiple objective design for SISO systems. The key is a nonlinear approximation of the transfer function Q in the Q-parametrization in terms of controller complexity. To improve the solution efficiency, three initial-guess-updating schemes are proposed. Two examples illustrate the effectiveness of the proposed techniques. They seem to indicate 80 Chapter 5. Numerical Solutions of Multiple Objective Control Problems that the approximate solution is very close to at least a local minimum. In comparison to other methods, especially the convex optimization method, the design procedure presented here has the advantage of directly yielding low-order controllers. The extension of the non-convex optimization procedure to the MIMO systems is straightforward. However a large number of optimization parameters may cause some computaional problems: slow convergence and local optimum, as we experienced in the controller design- for teleoperation systems in the next chapter. 81 Chapter 6 Teleoperation Controller Design Problem In this chapter, first we formulate the robust controller design problem for teleoperation systems as a constrained multiple objective optimization problem, and then we demon-strate the effectiveness of the proposed methodology, with simulations and experiments, through a controller design example for a motion-scaling system. § 6.1 Introduction A typical teleoperation system consists of five interacting subsystems: human op-erator, master manipulator, controller, slave manipulator and environment as shown in Fig. 6.1. For simplicity and to a first approximation, the blocks in Fig. 6.1 are usually modeled as LTI systems, in which the dynamics of the force transmission and position responses can be mathematically characterized by a set of network functions. This allows the designer to draw upon well developed linear systems and network theory for anal-ysis and synthesis of the controller. For example, the teleoperation system in Fig. 6.1 can be modeled as an LTI 2n-port network illustrated in Fig. 6.2, in which the master, controller and slave are grouped into one block called the teleoperator M C S . Here, vm is the master velocity, vs the slave velocity, fh the force that the operator applies to the master, fe the force that the environment applies to the slave, and the operator hand and environment impedances are modeled, respectively, by Zh and Ze. Usually, continuous contact is assumed between the manipulator and its environment so v^ = vm and ve = vs. 82 Chapter 6. Teleoperation Controller Design Problem Operator Vh fh i 0=ffl Master L Controller Vs f. Slave Environment V Teleoperator J . s> Figure 6.1. General Teleoperation System Figure 6.2. 2n-port representation of a teleoperation system The teleoperation controller should be designed with the goal of achieving the best possible performance, normally termed as transparency, while maintaining stability when coupled to uncertain environments, in the possible presence of time delays, disturbances, and measurement noise. Therefore, the teleoperation controller design problem involves a compromise between performance and robust stability. It is challenging mainly due to the large uncertainties of operator and environment impedances that have to be accommodated and due to communication delays. The objective of this thesis work is to design a teleoperation controller that can achieve an appropriately defined measure of performance, i.e., transparency, while main-taining the system stability against any passive environment. First, by using the four-channel i/oo-control framework proposed by Yan and Salcudean in [44], we define two transparency measures. One is defined by the closed-loop frequency responses on the kinematic correspondence error and the force tracking error [99]. The other is defined 83 Chapter 6. Teleoperation Controller Design Problem as the admittance matrix gap between the designed teleoperator and a proposed ideal teleoperator [100]. Then the stability constraints are characterized by using the scattering matrix, for any passive operator and environment impedances, or the transmitted admit-tance to the environment, for a fixed operator impedance and any passive environments, of the designed teleoperator. After parametrizing all stabilizing controllers via the Youla parametrization, the controller design problems are cast as multiple objective optimiza-tion problems. For the case that the operator impedance is assumed to be known, the formulated controller design problems are shown to be convex. Therefore they are nu-merically solvable, and the limit of transparency achievable, thus the exact form of the tradeoffs between transparency and stability robustness can be obtained. This chapter is structured in the following way. In the next section, we review some basic passivity concepts and stability conditions for teleoperation systems. In Section 3, the ideal teleoperator is proposed for scaled teleoperation. In Section 4, the teleoperation controller design problems are formulated as multiple objective optimization problems by defining transparency measures and stability constraints. In Section 5, we demonstrate the effectiveness of the proposed robust controller design methodology by treating the design of a controller for a one-degree-of-freedom (1-DOF) force-reflecting and motion-scaling teleoperation system. The tradeoff curves between transparency and robustness are displayed, and the results of both simulations and experiments are presented. Some concluding remarks are included in the final section. § 6.2 Background on Stability and Passivity In this section, we review some .basic concepts of passivity and stability conditions for teleoperation systems. Those conditions will provide useful design constraints for the development of robust controllers for teleoperation. 84 Chapter 6. Teleoperation Controller Design Problem Definition 6.1 [95]: For a linear time-invariant (LTI) n-port network as shown in Fig. 6.3, the impedance matrix Z is defined as the map from v to / by / = Zv; the admittance matrix Y as the map from / to v by v = Yf; and the scattering matrix S as the map from the input wave c the equation b = Sa. a = (/ + v)/2 to the output wave b = (/ — v)/2, i.e., satisfying Figure 6.3. An n-port network These matrices are interrelated by S = (I-Y)(I + Y)-1 = {Z-I)(Z+.I)-1. (6.1) Theorem 6.1 [101]: (a) An LTI n-port as shown in Fig. 6.3 is strictly passive if and only if the matrix criterion below holds for some 6 > 0 : Y(]u) + Y*(}LO) >.8I,\/ui G R. (6.2) (b) An LTI n-port is passive if and only if criterion (6.2) holds for 6 = 0. Notice that if n = 1, condition (6.2) reduces to Re[Y(ju)} > 8/2. (6.3) Definition 6.2 [58]: The i/-index, also referred to as passivity distance, is defined as the distance of a stable LTI system to strict passivity. For the one-port network in Fig. 6.4, v = - inl: {Re[F(ja>)]}. (6.4) 85 Chapter 6. Teleoperation Controller Design Problem Theorem 6.2 [102]: Consider the network in Fig. 6.4, in which an LTI one-port with admittance Y is coupled to an impedance Z. This network will be stable for every strictly passive Z if and only if the one-port is itself passive, i.e., Re[F(ju;)] > 0, Vu; G R. (6.5) Figure 6.4. A representation of an LTI one-port network, Y, coupled to Z. Theorem 6.3 [103]: Consider the bilateral system shown in Fig. 6.2. This teleoperation system will be stable for every pair of strictly passive Zh and Ze if and only if the scattering matrix St V1 s 12 s 21 s 22 (6.6) of the 2n-port teleoperator M C S has no poles in the closed right-half-plane, and moreover satisfies ,11 ^12' sup {fiA(St(ju>))\ - sup inf a u w /?>o s „2l < 1. (6.7) [s21//3 s 22 Here, a denotes the maximum singular value, and denotes the structured singular value against the block structure A Sh 0 0 Se (6.8) where Sh and Se are the scattering matrices of strictly passive Zh and Ze, respectively. 86 Chapter 6. Teleoperation Controller Design Problem Remarks: (1) . The robustness criterion for the 2n-port teleoperation system in Theorem 6.3 is based on the finding that the system can be transformed into a structured uncertainty problem by using the scattering matrix [103], which can be addressed by considering the structured singular value [104]. Here we point out that this robustness criterion can also be derived from network theory on absolute stability [105, 95]. (2) . When the hand impedance is fixed, Fig. 6.2 reduces to an one-port network as in Fig. 6.4. Here we point out that the stability criterion for the one-port network in Theorem 6.2 can also be obtained by using the structured singular value of its scattering matrix. For the one-port network, the scattering matrix now becomes St = yipY, and the uncertainty structure is only one block A = Se, with H ^ H ^ < 1. As shown in [104], the coupled network is stable if and only if p.&(St) < 1, which is equivalent to in (6.5). 1+Y(ju) < l,Vw e R , i.e., Re[Y{jto)} > 0,Vw e R as shown § 6.3 An Ideal Teleoperator In this section, we propose an ideal teleoperator that has good transparency and robust stability. We shall see in the next section that this teleoperator can be used as a model-reference to aid the design of controllers. An ideal teleoperator is one that provides complete transparency of the man-machine interface such that the operator has the perception of working directly on the task environment. In order to achieve this, for non-scaled teleoperation, the effort and impedance of the operator port should be identical to the effort and impedance of the environment and vice versa [53, 45], i.e., vm = vs, fh = fe. This can be represented 87 Chapter 6. Teleoperation Controller Design Problem by an infinitely stiff and weightless mechanical connection between the master and the slave [106], which cannot be physically realized. For scaled teleoperation, an ideal teleoperator is proposed here and is modeled by a two-port network as illustrated in Fig. 6.5. Instead of eliminating the dynamics of the teleoperator and realizing the ideal response defined by vm = npvs, fh = riffe, where np and rif are constant scaling ratios of position and force respectively, a passive tool represented by impedance Zm is used by the operator, as similarly done in [107]. The end effector force is multiplied by nf and directly fed forward to the operator's hand so that the operator can get a feel of the task, and the motion command from the operator is divided by np. For micro-manipulations, nf,np > 1. There are instances in which rif,np — 1 (passive tool) or rif,np < 1, i.e., manipulation is at a large scale. Here Zth denotes the transmitted impedance to the hand, and Zte the transmitted impedance to the environment. / \ Figure 6.5. Two-port representation of ideal scaled teleoperation The dynamics of this MCS teleoperator block are given by: Vm = ym(fh + nffe), (6.9) vs - Vm/n. 88 Chapter 6. Teleoperation Controller Design Problem where ym = 1/Zm. Therefore, the ideal MCS teleoperator block can also be represented by the following admittance matrix Yi = (6.10) 1 Uf _l_ nf \Vm-- T i p Tip . When nf — np — 1, this corresponds to the case in which the operator manipulates the task environment direcUy with the assistance of a tool with impedance Zm. The important properties of this MCS teleoperator block are summarized in the following theorem. Theorem 6.4: The proposed ideal teleoperator has the following properties: (i). The transmitted impedance to the hand is Zth Zm H ~%e, and the transmitted impedance to the environment is Zte = —Zm-\ ~Zh. Uf Uf (ii) . If ym is passive, then Yj is passive if and only if n f { j u ) n * p ( j u ) = l , V w <E R . (iii) . n . (6.11) (6.12) (6.13) (a) . If ym is passive, then sup{/ iA (Sj ( H ) } = l (6.14) for any constant, positive and real scahngs nf and np. (b) . If ( H - ^ j ! / m is passive, then sup { ^ ( ^ ( j w ) ) } = 1 for any frequency dependent scahngs n f ( j u ) and np(juj) satisfying I Tin (6.15) Here, Si is the scattering matrix of the teleoperator MCS, and the block structure \Sh o is given by A 0 Se with WSHW^ < 1 and \\SE < 1. 89 Chapter 6. Teleoperation Controller Design Problem (iv). If \ym(joj)\ —> oo, Vtu 6 R, then Si 1 nf + np rip — rif 2rifnp 2 rif — nr, (6.16) Proof: First, (i) is obvious. For example, the first part can be proven by transforming Fig. 6.5 into Fig. 6.6. riife + rif/ripZe Figure 6.6. Another representation of ideal scaled teleoperation Second, we prove (ii). For each u e R, the eigenvalues \l'2(u>) of the matrix Yf(jw) + Yj(ju) are given by Aj>2(u>) = Re{yn(ju)} ±J[Re{yn(jio)}}2 + nf(ju) n*p(joj) 2 ym(ju)\ , (6.17) where yn = {(1 + ^ )2 /™}- ^ n e D a r t c a n ^ e e a s u V obtained by plugging nf(j<jj)n*(ju) = 1 into (6.17). The "only if" part can be proven by contradiction, we just need to notice that if nf(ju)n*(ju>) ^ 1, then the inside of the square root in (6.17). is bigger than Re(yn(ju)), Vu> e R. This would imply that at least one of the eigenvalues Ay2(u;) of the matrix Yj(ju>) + Yi(ju) is negative. 90 Chapter 6. Teleoperation Controller Design Problem Third, we prove (iii). The scattering matrix of the ideal teleoperator, S1/, can be obtained, by using formula (6.1), as Si = (i+^>m+i 2nfy„ (6.18) ,11 „12 -21 „22 5 7 5 7 For each u e R, we compute fi&(Si(iu>)). Since Sj has two block structured uncer-tainties, fiA(SI)=Ma(SIJ), (6.19) where a denotes the maximum singular value and Si,p = To get the singular values of Sitp, we calculate the eigenvalues \%s(i = 1,2) of the matrix (6.20) A(/9)±WA2(/3) + [2Re{y B}]^- \yn\z + 1 i 2 |y„r+ 2Re{y» } + l (6.21) where ^(0) = 2 T2 + |?/n|2 + 4 ny nP } ) (6.22) + 1. Since a(Sitp) = max (y/Xl, V ^ f ) ' and from equations (6.21) and (6.22), we can see that minimizing a(Sij) is equivalent to minimizing A(/3), and the infimum in (6.19) 91 Chapter 6. Teleoperation Controller Design Problem is achieved when 0M- i P\n, 0, or B = 1 \nfnp\ Therefore, for any constant and positive scalings rif and np, we have 1 tiA(Si)=.'mf(T(SItp)=a[SItp[ i; and for any frequency dependent scalings rif (jus) and np(ju>) satisfying HL{ju) = R e { ^ ( i u ; ) ) , V u , G R , np I np ) we also have (6.23) (6.24) (6.25) / * A ( S J ) = cr Sitp 1. yj\nf~np\ Finally, (iv) can be obtained from (6.18) by letting \ym(ju>)\ —> oo,Vo; G R. (6.26) Remarks: (1) . Clearly, as shown in (i) of Theorem 6.4, with this ideal teleoperator, the operator feels a. scaled version of the environment impedance plus the tool impedance. Similarly, the impedance transmitted to the environment is a scaled version of the hand impedance plus a scaled version of the tool impedance. It wil l be assumed later that a teleoperator with this property shall be considered as having ideal transparency. A transparency measure will be introduced, based on the proposed teleoperator model, in the next section. (2) . Although this teleoperator is not generally passive (see Theorem 6.4, (ii)), it is robustly stable for any strictly passive operator and environment impedances, for any 92 Chapter 6. Teleoperation Controller Design Problem constant scalings (see Theorem 6.4, (iii).(a)) and even for some frequency-dependent searings (see Theorem 6.4, (iii).(b)), since it satisfies the stability criterion in (6.7). (3). From (iv) of Theorem 6.4, we can see that the scattering matrix of this teleoperator approaches that of teleoperation for the ideal response in [43] as the tool's dynamics disappear, i.e., vm —• npvs and fh —> nffe as \Zm(jio)\ —> 0, Vu; e R. § 6.4 Robust Controller Design In this section, we first define the performance measures and stability constraints for teleoperation systems, and formulate robust teleoperation control problems as multiob-jective optimization problems. § 6.4.1 Teleoperation controller structure f h a + Operator Environment tea fh + + f m Master X m X s Slave fe X X m = Pm ( fh + f m ) Controller X s = Ps ( fe + fs ) + + fs Figure 6.7. A four channel control structure. A general four channel structure in "admittance" form is used for controller design as shown in Fig. 6.7. Laplace transforms and transfer function notation are assumed throughout. For simplicity, we consider only the one-degree-of-freedom case. In Fig. 93 Chapter 6. Teleoperation Controller Design Problem 6.7, xm = vm/s is the master position, xs = vs/s the slave position, Pm the master plant, and Ps the slave plant. The hand force fh is decoupled into an active component fha and a passive component —Hxm, and similarly, the environment force fe is decoupled into fea and —Exs_ Note that since H and E are the maps from the positions to forces, the hand impedance, , and the environment impedance, Ze, are given by, respectively, Zh = \H and Ze = ^E. The master dynamics Pm can also be expressed as the impedance Zm = 7T5- or the admittance ym = = sPm. We can do the same with the slave too, i.e., Zs = -jjr or ys = = sPs. The controller K, which is a 2 by 4 matrix of real-rational transfer functions, takes force and position measurements from both master and slave, and generates the actuator driving forces fm on the master and fs on the slave. As shown in [45, 44], this four channel structure can provide sufficient freedom to shape various closed loop frequency responses of interest. We shall show later that this controller structure can also realize scaled teleoperation with good transparency while maintaining stability. In this work, for simplicity, we do not consider the master and slave modeling errors, the measurement noise, the control disturbances, and the time delays. § 6.4.2 Performance measures and stability constraints w G z K u y Figure. 6.8. General feedback system 94 Chapter 6. Teleoperation Controller Design Problem The model presented in Fig. 6.7 can be transformed into the basic configuration of the general feedback systems as shown in Chapter 2, which is re-plotted in Fig. 6.8. As discussed in Chapter 2, this setup can be used to define multiple performance specifications and stability constraints, and therefore to find a realizable controller K which stabilizes the augmented plant G. To define performance specifications and stability constraints, we first look at the general case, where both the hand and the environment impedances are unknown but strictly passive. Then we look at the special case where the hand impedance is fixed and the environment impedance is strictly passive. In the general case, we choose w = [fh fe ]T, u = [fm f3]T, and y = [fh fe xm xs]T. The output vector signal z should be selected to be able to characterize the performance specifications and stability constraints. There is a great deal of freedom in selecting the output vector z. To realize a scaled teleoperation, as an example, a possible set of signals is chosen as the following: (1) . z\ = Wf (fm — n//e), the force tracking error for some frequency range of interest. The master actuator force should track the environment force, scaled by a specified force-scaling ratio nf. Since it is more important at low frequency range, Wf is chosen to be a low-pass filter. (2) . z2 — Wp(xs — xm/np), the kinematic correspondence error . The slave motion should track the master motion scaled by a specified motion scaling ratio np. Again Wp is chosen to be a low-pass filter. (3). z z = , an unweighted output signal. This may be used to define transparency and robust stability constraints, as we shall see later. Then the generalized plant can be expressed as z 'WGzw WGZU W y. Gyu U (6.27) 95 Chapter 6. Teleoperation Controller Design Problem where, Gz 0 Pmj Hp sP 0 —rif Ps 0 sP, 1 0 Pmfrip Ps sPm 0 0 sPs (6.28) G yw ' 1 0 " ' 0 0 " 0 1 1 Gyu — 0 0 p 1 m 0 p 1 m 0 _ 0 Ps _ 0 Ps (6.29) Here, for notational convenience, W = diag(W/, Wp, 1, l ) . Let T i , T2 and be the maps from ti; to z\, z2, and 23 respectively, which are closed-loop transfer functions. Here we propose two performance measures. We will use the Hoo-norm to reflect the error sizes of transfer function responses. First we define performance specifications, directly according to the above setup. Clearly, to have a good force tracking and a good kinematic correspondence between the master and the slave, both ||Ti(iir)||00 and 1^2(^)11^ should be minimized. Therefore a performance measure, denoted by up, can be defined as /,p(ir) = max{||ri(/^)| | o o, | |r 2(/0||oo}- (6.30) proposed in the previous section as a fh fe to , and therefore it is the Second, we can use the ideal teleoperator M C S reference model. Note that Yp is the map from admittance matrix of the teleoperator to be designed. As shown in the Theorem 6.4 of last section, the ideal teleoperator provides good transparency. Therefore, to get good transparency, Yp should be designed to match the ideal model Yi in (6.10) as closely as possible. Hence, a transparency measure, denoted by p,p, can be defined as the gap between the teleoperator to be designed and the proposed ideal teleoperator, quantified by the following weighted Hoo-norm: f*T(K) = \\WT[YT(K)-YI}\\00, 96 (6.31) Chapter 6. Teleoperation Controller Design Problem where WT is a weighting matrix that reflects the frequency bands of interest. The smaller the value of ur, the better the transparency of the teleoperation system. To get the stability constraint for the teleoperator, as discussed before, we take the teleoperator as a two-port network. From its admittance matrix Yr, using (6.1), we have its scattering matrix ST = {Yr - I){YT + Therefore, to ensure stability of the two-port network coupled to any strictly hand and environment impedances, by using Theorem 6.3, the following absolute stability constraint should be satisfied: svLVuA(ST{K)(ju)):< l,Vw E R. (6.32) In the special case where the hand impedance is fixed and the environment impedance is strictly passive, the signals u, y, and z are selected as the same as those defined in the general case. Since we assumed that the hand impedance (= ^H) is fixed, we define w as w — the generalized p fha .fe ant becomes Let W = diag(W), Wp, 1, l ) and PH = Pm/(1 + PmH). Then z WGZW WGZU' W y. Gyy, U (6.33) where, Gz G yw 0 -nf 1 0 SPH Ps 0 1 G z u — SPH Ps 0 0 SPS _ 0 SPS \ - H P H 0 " '-HPH 0 " 0 1 . ) Gyu = 0 0 PH 0 PH 0 0 Ps 0 Ps (6.34) (6.35) The performance measure and the transparency measure are, respectively, in the same forms as in (6.30) and (6.31). However, to differentiate this special case from the general case discussed above, we denote them, respectively, as W { K ) = max {HTi^/pIL, \\T2,H{K)\\J (6.36) 97 Chapter 6. Teleoperation Controller Design Problem and WMK) = \\WT,H(YT,H(K)-YI,H)\\00- (637) The teleoperator, in this case, can be taken as a one-port network. Let Yu be the admittance transmitted to the environment, which is the map from fe to vs. According to Theorem 6.3, to maintain stability of the teleoperator coupled to any strictly passive environment impedance, the following stability constraint should be satisfied: inf {Re[Yte{K){ju)]} > 0, Vu; G R. (6.38) Remarks: (1) . As shown in [108, 109], other useful signals could be added in the output vector z. For example, z± = Wmfm. Wm is chosen to be high-pass. This could be used to account for master actuator saturation at high frequencies. (2) . From the definitions of up in (6.30), up in (6.31), and Yj in (6.10), it is easy to verify that up = 0 •<=>• up = 0. This equivalence, in the optimal case, shows that the perfect force tracking and kinematic correspondence would imply the good match between the designed teleoperator and the proposed one, and vice versa. Therefore, we can use either up or up as an optimization objective in the controller deign. (3) . In practice, the variation of the operator hand impedance is relatively small if compared with drastic changes of the environment impedance. Therefore, to design a less conservative controller for teleoperation, the stability constraint in (6.38) could be used. The advantage of using (6.38) is that it is convex in design parameters, as we wil l show next. 98 Chapter 6. Teleoperation Controller Design Problem § 6.4.3 Controller design problem formulations From the above discussion, it is clear that the teleoperation controller design involves tradeoffs between performance and robust stability. The general problem of designing a robustly stable controller for the scaled teleoperation system can be formulated as a multiobjective optimization problem as follows: ( P 6 - 1 ) : min aP{K) (6.39) stabilizing K sup HA{ST(K)(JLO)) < l,Vw G R, (6.40) or ( P 6 - 2 ) : min aT{K) (6.41) stabilizing K sup fiA(ST(K)(juj)) < l,Vu> G R. ' (6.42) For the special case where the hand impedance is fixed, the robust teleoperation controller design problem can be cast as stabilizing K inf {Re[Yte{K){ju)]} > -v, Vu; G R, (6.44) or ( P 6 - 4 ) : min UT,H(K) (6.45) stabilizing K inf {Re[Yte(K)(jco)]} > - i / , Vw G R, (6.46) 99 Chapter 6. Teleoperation Controller Design Problem here, the positive parameter v is used to ensure a given distance to passivity defined in (6.4) and determine the degree of conservatism of the design. As discussed in Chapter 2, the Youla parametrization of the stabilizing controllers K [7] makes the closed-loop transfer matrices Ti,T2,Yp Ti,#, T^jy, YT,H-> and YTE affine functions of Q € RH%£*. Since the scattering operator Sp is a bilinear transform of Yp it is not affine in Q £ RH^£A. Therefore p,p, up, //p,#, and p,ptjj are convex and Sp is non-convex in Q £ RH%£*. Hence, after the Youla-parametrization, the general teleoperation control design problems (P 6 - 1 ) in (6.39-6.40) and (P 6 - 2 ) in (6.41-6.42) are non-convex. Only in the special case where the hand impedance is fixed, can the controller design problems ( P 6 3 ) in (6.43-6.44) and ( P 6 4 ) in (6.45-6.46) be cast as convex optimization problems in Q e RH2x4: • oo (P 6 - 5 ) : mm uPtH(Q) (6.47) QeRH2^ inf {Re[Yte(Q)(ju)]} > -v,Vu £ R, (6.48) and (P 6 - 6 ) : mm ap,H(Q) (6.49) QeRH2^ inf {Re[Yte(Q)(ju)}} > -u, Vu; 6 R. (6.50) § 6.4.4 Numerical solutions In this thesis, we only solve problems ( P 6 5 ) and ( P 6 6 ) . We tried both the convex optimization procedure and the non-convex optimization procedure presented in Chapter 5. We found that the latter often produced slow convergence and local optimum due to the large number of parameters and the truncation error in gradient calculation by finite 100 Chapter 6. Teleoperation Controller Design Problem difference approximation. Even though the cutting-plane based solver E M C P produced high-order controllers, it was found more efficient and reliable. Therefore we will use the cutting-plane-based solver E M C P developed in Chapter 5, in a design example shown next. To produce a finite-dimensional approximation of ( P 6 5 ) and ( P 6 6 ) , as discussed in Chapter 2 and Chapter 5, Q G RH^* is approximated as a linear combination of fixed scalar stable basis functions Qi e RH^, as in N Q(XUX2,...,XN) = ^ X t g „ X% e R 2 * 4 , (6.51) where the N real-valued matrices Xi(i = 1,2, ...,iV) are the design .parameters. For / _ \N-i example, the basis functions can be chosen as all-pass functions Qi = ( f+£ J for some fixed p with positive real part, as in [26]. § 6.5 Design Example In this section, we present an example to show the trade-off between performance and stability robustness, and to demonstrate the effectiveness of the above developed methodology, through both simulations and experiments. The design problem considered here concerns a prototype telerobotic system for use in microsurgery experiments [108, 44]. In this system, two magnetically levitated wrists, each having a stator and a levitated floater, are used as the master and the slave. Both force and position need to be scaled down from the operator's hand to the task. The transfer functions mapping force to position for the master Pm(s) and slave Ps(s) are modeled, respectively, as P m ^ = 1.422*2 +42.663 +383.94' ( 6 - 5 2) and ftW= 0.035,'+ 1.0.+9.6- < 6 ' 5 3 ) 101 Chapter 6. Teleoperation Controller Design Problem The design here is only for motion along a single D O F (the vertical z-axis). As discussed in the previous section, this teleoperation controller can be obtained by solving either (P 6 - 5 ) in (6.47-6.48) or ( P 6 6 ) in (6.49-6.50), according to the designer's specifications. § 6.5.1 Identification of the hand impedance To use the proposed controller design method, we need to identify the operator's hand impedance by using the master wrist. A JR? force/torque sensor was mounted on the top of the master wrist. As suggested in [55, 57, 44], the operator's hand impedance can be modeled as a constant mass-spring-damper system, i.e., Zh(s) = rrihs + bh + or H(s) = sZh(s) = mhS2+b}ls-\-kil, and the operator hand force fh is considered to possess active exogenous component //,„ and passive feedback component —Hxm dependent on the hand impedance. In the identifying process, the operator was grasping the wrist's handle with his right hand without imposing any force on the wrist (fha = 0), and was following the wrist's motion. The master wrist was driven by a white noise signal. The experimental data of the hand force fh and the master wrist's position was collected. We took xm as the output signal and fh as the input signal. Then xm = J J ^ J A - Since it is assumed as a second-order system, the model ARX(2,2) was chosen to describe the relations between the two signals: xm(t) + aixm(t - Ts) + a2xm(t - 2TS) = hfh(t - Ts) + b2fh(t - 2TS), (6.54) where, a\,a2,b\ and b2 are unknown parameters to be identified, and Ts = 0.002 seconds, the sampling interval. By using the Identification Toolbox [110], we obtained the parameters of the model ARX(2,2). Based on this estimated model, the hand impedance model H(s), the map from xm to /ft, was obtained in frequency domain as H(s) = 0.26s2 + 26.23s + 170.83 (N/m). (6.55) We shall use this model in the following control system analysis and design. 102 Chapter 6. Teleoperation Controller Design Problem § 6.5.2 Tradeoff between performance and stability robustness The tradeoffs between performance and stability robustness can be displayed by solving either ( P 6 5 ) or (P 6 - 6 ) with different distances to passivity v. Here, by using the linear approximation of Q as in (6.51) with N = 20 and p .= 1, and using the developed EMCP solver, we solved the convex program ( P 6 5 ) with different v, and different force and motion scaling ratios rif = 5,10,20, and np = 5,10,20. The weighting functions in (P 6 - 5 ) were selected as Wt(s) = U f P,(s), uf = 6TT rad/sec, (6.56) S+LOf LO„ HJI(S) = , top = 57T rad/sec. (6.57) These reflect the frequency bandwidths of force tracking and kinematic correspondence, and are low-pass filters. For example, for the kinematic correspondence, we want the error to be low for frequencies below 57t rad/sec. The performance vs robust stability trade-off curve can be plotted as shown in Figure 6.9. As expected, the performance gets worse as the distance to passivity increases; also higher scaling leads to worse performance. We can also see the large increase in performance measure imposed by the passivity requirement. For example, in the case of np = rif = 5, with passivity constraint (v = 0), aptH = 0.0265, and without any passivity constraint (v = oo), UP%H = 0.0112. 103 Chapter 6. Teleoperation Controller Design Problem ~i r np= iu= 20 nP= nr= 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Distance to Passivity v Figure 6.9. Performance vs distance to passivity for three values of force and motion scaling § 6.5.3 Simulation results In the simulations presented here and the following experiments, we specified the force scaling ratio and the motion scaling ratio as nf = 50 and np = 10, respectively. The weighting matrix was selected as WH(s) = U>. y s +w- -7/2x2, wy = 57r rad/sec. (6.58) y (6.59) The robust controller was obtained by solving ( P 6 6 ) for v = 0 : m i l l 2 x 4 W'HiQ) inf {Re[Yte(Q)(ju)}} > 0, Vu, e R, with the linear approximation of Q in (6.51). We also took N = 20 and p = 1 in (6.51). The resulting controller is of high order and was reduced to an 8th-order system by using the balanced model reduction technique. With the obtained controller, the maximum 104 Chapter 6. Teleoperation Controller Design Problem singular value frequency responses of the admittance matrices YI,R and YT,H for the ideal teleoperator and the designed one, respectively, are shown in Fig. 6.10, and the Nyquist plot of the transmitted admittance to the environment, Yte, is depicted in Fig. 6.11. cn E 3 E Frequency (rad/sec) Figure 6.10. Maximum singular value frequency responses of Yjtjj (solid line) and YT,H (circle) 105 Chapter 6. Teleoperation Controller Design Problem 0.4 0 0.1 0.2 0.3 0.4 0.5 0 .6 ' 0.7 0.8 Real Axis Figure 6.11. Nyquist plot of Yte Clearly, Fig. 6.10 shows that the designed teleoperator is very close to the ideal one, and therefore should have good transparency. Fig. 6.11 shows that Yte is positive real and satisfies the stability criterion in (6.5), and therefore the scaling system is also robustly stable for any strictly passive environment. In order to illustrate the stability and the validity of the scaling system with the proposed controller design approach, both ; simulations and experiments were carried out. Fig. 6.12 shows the simulation diagram in Simulink [111] that we used. To be consistent with the experiments in the next section, the master was modeled in continuous-time, and the controller, the slave and the environment were modeled in discrete-time. The dynamics of the master and the slave were simulated as in (6.52) and (6.53) respectively. Three sets of simulations were performed with the following passive environment impedances (note that E = sZe): (1) . a soft impedance E = 2s + 200, (2) . a stiff impedance E = 5s + 500, and 106 Chapter 6. Teleoperation Controller Design Problem (3). a time-varying impedance, which simulates the free motion to a soft environment, to a hard environment, to a soft environment again, and back to the free motion again: E 0, 0 < * < 3 sec 55 + 300, 3 < t < 6 sec 105 + 1000, 6 < i < 9 s e c 25 + 100 9 < t < 1 2 s e c 0 1 2 < t < 1 5 s e c >. (6.60) Fig. 6.13, Fig. 6.14 and Fig. 6.15 show the motion-scaling and force-scaling results for each case. It can be observed that the system is always stable for the passive environments and both the motion and the force scahngs are realized. To look at transparency of the designed teleoperator, for the cases (1) and (2), we need to obtain the transmitted impedances to the hand, Zth, defined as the map from vm to fh in frequency domain. We shall use Matlab Identification Toolbox [110] to do this. During the course of simulations for case (1) and case (2), we collected the data of the hand force fh and the master position xm, as shown in Fig. 6.13 and Fig. 6.14. The system identification problem is to estimate a model of a system based on observed input-output data. We took xm as the output signal and fh as the input signal. We assumed that the two signals are related by a linear system, described by the model ARX(2,1): xm(t) + aixm(t - Ts) + a2xm(t - 2TS) = b ^ t - Ts), (6.61) where, a\, a2, and b\ are unknown parameters to be identified, and Ts = 0.002 seconds, the sampling interval. By using the Identification Toolbox, we identified the parameters of the model ARX(2,1). Based on this estimated model, the map Eth from xm to fh was obtained in frequency domain, and then transformed into the estimated transmitted impedances to the hand, Zth — ^Eth. The results are summarized in Table 6.1, where as defined before Zth = Zm + ^Ze is the ideal transmitted impedance to the hand. The Bode plots of the transmitted impedances Zth and-2^'are illustrated in Fig. 6.16 and 107 Chapter 6. Teleoperation Controller Design Problem Fig. 6.17, which clearly show very good transparency in the specified frequency range (< 57T rad/sec). [t.fh] From Workspace Forces rZEZH-1 To Workspace2 Mux fm/nHe + 1 ZJ Sum1 Force Error •*yy-Zero-Order Hold Master Controller r*r-i Slave Environment nf Zero-Order Hold! -+B&>-np 0-+ rvl Sum Position Error Mux np*xs/xm Positions To Workspacel -H time Clock To Workspace Figure 6.12. Design example: simulation diagram of a motion-scaling system 108 Chapter 6. Teleoperation Controller Design Problem (a)fh i 1 1 1 1 1 r I i i i i i i i i i I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) (b) xm (solid line) and 10*xs (dashdot line) i 1 1 1 1 1 1 1 r I i i i i i i i i i I 0 0.2 0.4 0.6 0.8 ,1 1.2 1.4 . .1.6 1.8 2 Time (second) (c) fm (solid line) and 50*fe (dashdot line) i i i i i i i i r I 1 1 1 1 I 1 ; I I I I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) Figure 6.13. Simulation results: motion scaling and force scaling with a soft environment E = 2s + 200. 109 Chapter 6. Teleoperation Controller Design Problem (a)fh (b) xm (solid line) and 10*xs (dashdot line) | 0.5 £ - 0 . 5 -1 - V \\ V\ \\ 1 1 1 1 1 L r v J /.' \ i •• \ • ' Y \\ i ' \ /' Ii X />• : \ \ U // li 1 1 • 1 1 1 • 0 0.2 0.4 0J 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) (c) fm (solid line) and 50*fe (dashdot line) n 1 1 1 1 1 1 1 r o o // if \ L/L \ '/ \ il \ if \ / J I I I I I I 1_ 0 0.2 0.4 . 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) Figure 6.14. Simulation results: motion scaling and force scaling with a stiff environment E = 5 s + 500. 110 Chapter 6. Teleoperation Controller Design Problem (a) hand force: fh CD O E E, c o in o E E c. o ' o Q_ 3 6 9 12 (b) motion scaling: xm (solid line) and 10*xs (dashdot line) 15 _ . . . . I J I 1 6 9 12 (c) motion tracking error: xm-10*xs CD O CD O 3 6 9 12 (d) force scaling: fm (solid line) and 50*fe (dashdot line) 6 9 (e) force tracking error: fm-50*fe 12 6 9 Time (second) 15 15 15 Figure 6.15. Simulation results: motion scaling and force scaling with a time-varying environment, which switches at 3, 6, 9, and 12 seconds 111 Chapter 6. Teleoperation Controller Design Problem E (= sZe) z^ 2s + 200 1.422s + 52.66 + 1.635 + 60.77+ 1 3 9 0 5 55 + 500 1.4225+67.66+ 1.765 + 5 8 . 9 7 + 2 ^ 1 Table 6!1 Simulation results: transmitted impedances to the hand (a) 1S0i •:, • • 0I ; ; i i i i i i i ; : : ; ; : ; : : : : : ; ; ; ; ; \ 10"3 10"2 ' 10"' 10° 10' 102 103 Frequency (rad/sec) (b) Frequency (rad/sec) Figure 6.16. Simulation results: transmitted impedances to the hand, Zth (solid .—. line) and Zth (dash-dotted line), with a soft environment E = 2s + 200. 112 Chapter 6. Teleoperation Controller Design Problem (a) Frequency (rad/sec) Figure 6.17. Simulation results: transmitted impedance to the hand, Zth (solid line) and Zth (dashdot line), with a stiff environment E = 5s + 500. § 6.5.4 Experimental results Experiments were performed with a UBC magnetically levitated (maglev) wrist [56] as the master manipulator, a slave manipulator model and a virtual environment, both of which were simulated in a SPARC 1-e processor under VxWorks, shown in Fig/ 6.18. The dynamics of the master and the slave were modeled as in (6.52) and (6.53) respectively. The virtual environments were taken as the same ones as in the simulations in the last section. Fig. 6.19 illustrates the hardware configuration of the experiment setup. It consists of a UBC maglev wrist, a current amplifier, a SPARC 1-e processor which resides on a V M E bus, as well as an analog-to-digital (A/D) board and a digital-to-analog (D/A) board. A J R 3 force/torque sensor was mounted on the top of the master wrist to measure the hand force. AUsoftware development is performed in a SPARC station host, which is networked to the SPARC 1-e processor. 113 Chapter 6. Teleoperation Controller Design Problem Figures 6.20, 6.21 and 6.22 presents the motion scaling and force seating results when the operator manipulates the three different virtual environments using the UBC wrist. It can be observed that the system is stable against the passive environments, and a fairly good motion and force scaling are realized as specified. As done in the simulations, the transmitted impedances to the hand, Z^, for the soft environment (case (1)) and the stiff environment (case (2)), were estimated by using the Identification Toolbox in Matlab with the collected experimental data. The obtained results are presented in Table 6.2. The good transparency of this scaling system in the specified frequency range can be also seen from the Bode plots of the transmitted impedances Zth and Zth illustrated in Fig. 6.23 and Fig. 6.24. SPARC 1-e VxWorks Slave Manipulator Model UBC Wrist Controller Virtual Environment Figure 6.18. Design example: experiment diagram of a motion scaling system 114 Chapter 6. Teleoperation Controller Design Problem CPU A/D D/A Host SPARCstation Ethernet < x I 5" Force/Position V M E bus U B C Wrist Current Amplifier Figure 6.19. Design example: experimental setup 115 Chapter 6. Teleoperation Controller Design Problem (a)fh (b) xm (solid line) and 10*xs (dashdot line) Ji Ii • 1 \ AN Ji \\ V i / \\ \ Ii i \ i Ji • v \ It Ji '/ \ \ A : \ /' ! \ E E o -5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) (c) fm (solid line) and 50*fe (dashdot line) </ \ ,7 A i A / / \ \ . / \ : \y : \// : \// 5h p - 5 h _i J i i i i i i_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) Figure 6.20. Experimental results: motion scaling and force scaling with a soft environment E = 2s + 200. 116 Chapter 6. Teleoperation Controller Design Problem (a)fh 41 1 1 1 : — i 1 r i : i i i i i i i i I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) (b) xm (solid line) and 10*xs (dashdot line) "co o 1 V 1' V / / \ '/ > 1 / / . \ \ \...//... \ \ ... \v \ \ 1/ Ii V\ : Ii \ \ 1/ \ ^; J" \ \ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) (c) fm (solid line) and 50*fe (dashdot line) 5 o LL -5 1 1 I 1 I l 1 1 ! if if • / . A il \ . \ il . . / / \ .. .\\... il if A il \\ / / \\ il \\ ll \ / \\ ij il \\ il i/ i i i i i i i i i I 1 I I I I I I I I I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (second) Figure 6.21. Experimental results: motion scaling and force scaling with a stiff environment E = 5s + 500. 117 Chapter 6. Teleoperation Controller Design Problem 5 0 - 5 (a) hand force: fh CD O E E. c g o 3 6 9 12 (b) motion scaling: xm (solid line) and 10*xs (dashdot line) 15 1 1 1 v y v V| \j \] y v y CD O 6 9 (c) motion tracking error: xm-10*xs 12 3 6 9 12 (d) force scaling: fm (solid line) and 50*fe (dashdot line) 6 9 (e) force tracking error: fm-50*fe 12 15 15 15 CD O - 5 b 0 6 9 Time (second) 12 15 Figure 6.22. Experimental results: motion scaling and force scaling with a time-varying environment, which switches at 3, 6, 9, and 12 seconds 118 Chapter 6. Teleoperation Controller Design Problem E (= sZe) Zth Zth 2s + 200 1,422s + 52.66 + 1.46s + 27.73 + i i ^ i 5s + 500 1.422s + 67.66 + S 1.65s + 13.75 + 2 9 9 4 8 Table 6.2 Experimental results: transmitted impedances to the hand (a) 1501 • ! • II • • Frequency (rad/sec) (b) > w i—i i i i i i i l i : ; : : : : :::t ; : : : : : : = = = : : : : : : : : 10" 3 10~2 ic f 1 10° 10* 10 2 103 Frequency (rad/sec) Figure 6.23. Experimental results: transmitted impedances to the hand, Zth (solid line) and Zth (dashdot line), with a soft environment E = 2s + 200. 119 Chapter 6. Teleoperation Controller Design Problem (a) 150 m 100 C ' a ° 50 0 10"3 10~2 10"1 10° 101 102 10 3 Frequency (rad/sec) (b) 100 50 o> u -100 10~3 10" 2 10"' . 10° 101 102 103 Frequency (rad/sec) Figure 6.24. Experimental results: transmitted impedance to the hand, Zth (solid line) and Zth (dashdot line), with a stiff environment E = 5s •+ 500. § 6.6 Concluding Remarks Teleoperation controller design is challenging in large part due to large uncertainties of the hand impedance and especially the environment impedance, and communication delays. In addition to this, a teleoperation system is inherently multi-input and multi-output even in the single DOF case, and involves tradeoffs between performance and stability robustness. In this chapter, an optimization-based robust controller design approach for scaled teleoperation systems has been proposed, to optimize transparency subject to robust stability for all passive environments, The transmitted admittance to the hand was used to define the stability criterion, which was shown to be easily incorporated into the controller design. With the four channel control structure and a proposed ideal teleoperator, using Q-parametrization, performance measure or transparency measure were defined, while controllers were obtained as solutions to multiobjective convex 120 Chapter 6. Teleoperation Controller Design Problem optimization problems. By defining distance to passivity as a robustness measure, the trade-offs between performance and robustness were also displayed. A design example of a motion seating and force reflecting system, with both simulations and experiments, demonstrates that such an approach leads to an effective controller design for scaled teleoperation. Even if this approach is for 1-DOF systems, it may extend to multi-DOF systems as well. For multi-DOF case, first the performance measure and transparency measure can be easily defined in the same framework as those in 1-DOF case, and then the stability criterion can be defined now as positive realness of the transmitted admittance matrix to the hand. However, this approach still needs the hand impedance model. In addition to this, this approach may lead to a conservative design when some understanding of environment structure exists. These issues still need investigation. 121 Chapter 7 Conclusions In this chapter, we first summarize the contributions of this thesis, and then propose some future research. § 7.1 Contributions This thesis has dealt with the multiple objective control problems analytically and numerically, and, particularly, as an application example, has studied the scaled teleop-eration control problem. The main contributions of this thesis are summarized as the following: (1) . Analytical work. (a). Solutions to the SISO version of multiple objective HQO control problem are characterized by using Nonsmooth Analysis. Either the "all-pass" property is es-tablished, or the optimal performance value is obtained under some assumptions. (2) . Numerical work. (a) . The solution to the multiple objective LQ control problem is obtained by using duality theory and convex optimization. It is shown that the solution depends on the initial states and is an open-loop control optimal only for the particular initial states. (b) . A cutting-plane based solver has been developed for solving convex optimization problems. It is demonstrated that this solver is quite efficient in multiple objective control system design. Some computational issues arising in the design process are also discussed. 122 Chapter 7. Conclusions (c) . An LMI-based solution in state-space to the multiple objective Hoo control problem is presented. (d) . An approximate numerical solution to the multiple objective control system design for SISO systems, which yields low-order controllers, is described. Three schemes for updating the initial guess in non-convex optimization are proposed to improve the computational efficiency. (3). Application work. (a) . A two-port ideal teleoperation model is proposed. It is shown that the model scales both positions and forces, yet is stable when terminated by any strictly passive hand and environment impedances. (b) . A transparency measure is proposed to be defined as the Hoo-distance to the ideal teleoperator model, and for a fixed hand impedance, a robust stability criterion, against any strictly environment impedances, is proposed to be positive realness of the transmitted admittance to the environment port. (c) . The robust control problem for teleoperation systems is formulated as a multiple objective optimization problem. By using the Youla-parametrization, for a fixed hand impedance, this problem has been shown to be convex in design parameters. (d) . The tradeoff between performance and stability robustness is displayed for different position and force seating ratios. (e) . A controller design example for a motion-scaling system designed for micro-surgery experiments is presented. Both simulations and experiments have been carried out to show the effectiveness of the proposed multiobjective optimization-based controller design methodology. 123 Chapter 7. Conclusions § 7.2 Future Work Multiple objective control design is quite a challenging problem, and wil l continue to attract attention in the control community, and to find more applications in practice. Some goals of future research stemming from this thesis work include: (1) . Study on the MIMO version of multiple objective Hoo control problem. One possible direction is to extend the results for SISO case to MIMO case by using Nonsmooth Analysis. Another possible direction is to explore the connection between the multiple objective LQ control problem and the multiple objective Hoo control problem, since a number of researchers have solved single objective Hoo control problem by using LQ game theory. (2) . 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The directional derivative of /(•) at x in the direction h, denoted by df(x, h) is defined by df (x, h) = l im f(x + th)-f(x) (A.1) t (ii). The subdifferential of /(•) at x is the set df(x) = { ( £ R " : f(z) > f(x) + iT{z - x), for all ze R"| . (A.2) Every vector in df(x) is called a subgradient of / at x. § A.2 KCP Algorithm We consider the following constrained optimization problem: ( P a l ) : <j>* = min {(f>(z) : ipi(z) < 0, • • • 1pm(z)< 0}, (A.3) 135 where both the objective function <f> and the constraint functions i/>; (i G m) are convex functions from R n to R. Define the constraint function ib(z) = max ipi(z), (AA) l<t<m so ( P a l ) in (A.3) is equivalent to (pa.2\ . ^ = m i n . ^ < Q j ( A 5 ) z€R n We will say that z is feasible if ib(z) < 0. Suppose we have computed function values and at least one subgradient at a sequence of points x\, • • •, xk for both the objective and the constraint functions: <t>{x\), • • •, (j>(xk), g1 G cty(xi), • • • ,-gk e d(t>(xk), (A.6) ^(xi ) , • • •,^(xjfc), / i l 6 • • • ,hk G d^(xk), where these points x2 need not be feasible. Then we can form piecewise linear lower bound functions for both the objective and the constraint: <t>k(z) = max (<j)(xi) + gj(z - xi)), . A r \ <A.7) The lower bound function ibk yields a polyhedral outer approximation to the feasible set < 0 } c { ^ ? ( « ) < 0 } , (A.8) since V^6 (*) < for all 2 G R".Thus we have the following lower bound on <j)* : f>Lk = min{4b(z)\1>2(z) < o}. (A.9) But Lk in (A.9) can be obtained by solving a linear program: Lk= min \cTw\Aw < b\ (A.10) weR n + 1 t J 136 where ' z' '0 ' w = , c = L. 1 ,A = r T 9i - 1 " r T 9l xi - <j>(xi) ' 9l - 1 9kxk ~ <f>(xk) ,6 = hi 0 hTx\ - if>(xi) -hi 0 . -hlxk ~ ^{xk). Thus, the K C P algorithm for solving the econ-relaxed problem (P°- 3) : min {<f>(z) : ib(z) < econ} (A. 11) (A. 12) is summarized as the follows: €0bj <— objective tolerance; tcon <— constraint tolerance; x\ ±-r- starting parameter vector; k^-0; repeat { k <- k + 1; compute <f>(xk) and any gk G d<j){xk); compute rb(xk) and any hk G dip(xk); solve LP in (A.10) to find its solution xk and Lk; xk+\ <- xk', } until (if>(xk) < econ and <f>(xk) - Lk < eobj). Theorem A.1 [51,10]: The sequence {xk} constructed by KCP Algorithm converges to a point that is feasible and has objective value within e0j,j of optimal for ( P a 3 ) . 137 § A.3 EMCP Algorithm We consider the problem of maximizing a linear function subject to m-concave constraints from R n to R: ( P - 4 ) : /* = m a x { c ^ : 9i(x) > 0, • • • ,gm(x) > o]. (A.14) First, note that the assumption of a linear objective function involves no loss of generality. For if the objective function f(x) is concave, then f(x) — y is concave, and (P a - 5 ) : max{/(x ) : gi(x) > 0, • • •,gm(x) > 0} (A.15) is equivalent to ( P a 6 ) : m a x { y : f(x) - y >0, gi(x) >0,---,gm(x) >0}. (A. 16) x£R" Second, note that ( P a l ) , (PAA) and (p a - 5 ) are equivalent optimization problems. Here we only consider problem ( P a 4 ) in (A.14). Let F = {x : gi{x) > 0, i G m}, let / be a lower bound on the optimal value of ( P a f ) , e a pre-specified objective tolerance. Suppose we have computed at least one subgradient at a sequence of points x\, • • •, xk for constraint functions: 9i(xi),--- ,gi{xk), h\ £ dgi(xi),---,hk e dgi(xk), (A.17) 9m(xi), • • -,gm(xk), hm e dgm{xi), • • •, hm e dgm(xk). Then the E M C P algorithm is summarized as follows: Step 0. Let LPQ be the linear program Let k = 1. max U : cTx - 8 > f\. (A.18) 138 Step 1. Let (xk,8k) D e a solution to the linear program LPk_\. If 8k < e, stop; xk is optimal. Otherwise go to Step 2. Step 2. Delete constraints from the linear program LPk-i according to the following deletion rules: Deletion rule I: Delete the objective constraint from LPk_i if e F. Deletion rule II: Delete a constraint from LPk-i if (a) , the constraint was generated at the Ith iteration, I < k, (b) . 8k < 08l, where /? is a fixed scalar, 0 < /3 < 1, and (c) . the constraint was not tight in LPk_\. Call the resulting program L P ^ _ r Step 3. ^ I f X k e j? a (jjoin to LP'h_^ the objective constraint cTx-8>cxk. (A. 19) (ii). If xk F, adjoin to LP'k_v the constraints: gr{xk)+ hkr(xk)(x - xk) - hkr(xk) £ > 0 , (A.20) 2 where, gr(xk) < 0, for r 6 m. Call the resulting program LPk, set k = k + 1 and return to Step 1. Theorem A.2 [30]: If / < /*, then under either deletion rule I or rule II (or both or neither), the sequence {xk} constructed by E M C P Algorithm converges to an optimal solution to ( P a 4 ) . 139 Appendix B Linear Matrix Inequalities In this appendix, we first give a short introduction to linear matrix inequalities, and then point out sources of software for solving linear matrix inequality problems. § B.1 Linear Matrix Inequalities A linear matrix inequality (LMI) is an inequality of the following form m F{x) = F0 + J2xiFi>0 (B.l) i=i where x £ R m is the variable and Fo,...,Fm £ ~£inxn are given symmetric matrices. The inequality symbol means that F is positive semidefinite. An L M I is a convex constraint on x, i.e., the set {x\F(x) > 0} is convex. The LMI (B.l) can represent a wide range of convex constraints on x. In particular, linear inequalities, (convex) quadratic inequalities, matrix norm inequalities, and constraints that arise in control theory, such as Lyapunov and convex quadratic matrix inequalities, can all be cast in the form of an LMI. For example, the Lyapunov inequality ATP + PA<0, (B.2) where A £ R n x n is given and P = PT is the variable, can be readily written out in the form of (B.l), as follows. Let P\,...,Pm be a basis for symmetric n x n matrices (m = n(n + l)/2). Then take Fo = 0, and Fi = -ATP{ - Pi A. We often encounter problems in which the variables are matrices as in (B.2). For these problems, we will not write out the L M I explicidy in the form F(x), but instead make clear which matrices are the variables. 140 Many problems in system and control theory can be formulated as optimization problems involving LMIs (see, e.g., [11]): T mm c x (B.3) s.t. F(x) > 0. The optimization problem in the form of (B.3) is called semidefinite programming. Semidefinite programs are convex optimization problems; conversely, many convex optimization problems can be expressed as semidefinite programs. The survey paper [98] gives an overview of semidefinite programming and applications. The formulation and applications of more general problems than (B.3) can be found in [99]. § B.2 Software for Solving LMI Problems Pascal Gahinet, Arkadi Nemirovsky, and Alan Laub, at Mathworks, developed the L M I Control Toolbox, which allows one to efficiently solve problems involving LMIs. This toolbox solves the problems using Nesterov and Nemirovsky's Interior-Point methods [71] and taking advantages of some of the problem structure (e.g., block structure, diagonal structure of some of the matrix variables). This toolbox was not available while the related LMI work of this thesis was being done. Vandenberghe and Boyd at Stanford developed a software package, called SP, for solving semidefinite programming. They used a primal-dual interior-point algorithm [97]. SP was written in C, and is available via anonymous ftp at isl.stanford.edu in pub/boyd/semidef_prog. E l Ghaoui at ENSTA in Paris (elghaoui@ensta.fr) also developed a software package, called LMItool , that acts as an interface for SP, and allows users to specify semidefinite programming problems interactively from within Matlab. It is especially well suited for problems with matrix variables. It is also available via anonymous ftp at ftp.ensta.fr in pub/elghaoui/lmitool. 141 Based on SP, Boyd and Wu at Stanford developed another software package, called sdpsol, as a parser/solver to simplify the specification and solution of semidefinite programs, sdpsol parses semidefinite programming problems expressed in the sdpsol language, solves them, and reports the results in a convenient way. It is available via anonymous ftp at isl.stanford.edu in pub/boyd/semidef_prog/sdpsol. 142 

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