Linear Matrix Inequalities Linear Matrix Inequalities (LMIs) and LMi techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. Three factors make LMI techniques a variety of design specifications and constraints can be expressed as lmis Once formulated in terms of LMIs, a problem can be solved exactly by efficient convex optimization algorithms(the "LMI solvers") While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. This makes LMI-based design a valuable alternative to classical"analytical"methods See [9] for a good introduction to LMI concepts. The LMI Control Toolbox is designed as an easy and progressive gateway to the new and fast-growing field of lmis For users mainly interested in applying lmi techniques to control design, the LMI Control Toolbox features a variety of high-level tools for the analysis and design of multivariable feedback loops(see Chapters 2 through 7) For users who occasionally need to solve LMi problems, the"LMI Editor"and the tutorial introduction to LMi concepts and lMI solvers provide for quick For more experienced LMI users, the"LMI Lab"(Chapter 8)offers a rich, flexible, and fully programmable environment to develop customized LMI-based tools The LMI Control Toolbox implements state-of-the-art interior-point LMI solvers. While these solvers are significantly faster than classical convex optimization algorithms, it should be kept in mind that the complexity of LMI computations remains higher than that of solving, say, a Riccati equation. For instance, problems with a thousand design variables typically take over an hour on today s workstations. However research on LMi optimization is still very active and substantial speed-ups can be expected in the future. Thanks to its efficient"structured"representation of LMIs, the LMI Control Toolbox is geared to making the most out of such improvements 1-2
1 Introduction 1-2 Linear Matrix Inequalities Linear Matrix Inequalities (LMIs) and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. Three factors make LMI techniques appealing: • A variety of design specifications and constraints can be expressed as LMIs. • Once formulated in terms of LMIs, a problem can be solved exactly by efficient convex optimization algorithms (the “LMI solvers”). • While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. This makes LMI-based design a valuable alternative to classical “analytical” methods. See [9] for a good introduction to LMI concepts. The LMI Control Toolbox is designed as an easy and progressive gateway to the new and fast-growing field of LMIs: • For users mainly interested in applying LMI techniques to control design, the LMI Control Toolbox features a variety of high-level tools for the analysis and design of multivariable feedback loops (see Chapters 2 through 7). • For users who occasionally need to solve LMI problems, the “LMI Editor” and the tutorial introduction to LMI concepts and LMI solvers provide for quick and easy problem solving. • For more experienced LMI users, the “LMI Lab” (Chapter 8) offers a rich, flexible, and fully programmable environment to develop customized LMI-based tools. The LMI Control Toolbox implements state-of-the-art interior-point LMI solvers. While these solvers are significantly faster than classical convex optimization algorithms, it should be kept in mind that the complexity of LMI computations remains higher than that of solving, say, a Riccati equation. For instance, problems with a thousand design variables typically take over an hour on today's workstations. However, research on LMI optimization is still very active and substantial speed-ups can be expected in the future. Thanks to its efficient “structured” representation of LMIs, the LMI Control Toolbox is geared to making the most out of such improvements
Toolbox features Toolbox features The LMI Control Toolbox serves two purposes Provide state-of-the-art tools for the LMi-based analysis and design of robust control system Offer a flexible and user-friendly environment to specify and solve general LMI problems(the LMI Lab) The control design tools can be used without a priori knowledge about LMis or LMI solvers. These are dedicated tools covering the following applications of LMI techniques Specification and manipulation of uncertain dynamical systems (linear-time invariant, polytopic, parameter-dependent, etc. Robustness analysis. various measures of robust stability and performance are implemented, including quadratic stability, techniques based on parameter-dependent Lyapunov functions, analysis, and Popov analysis Multi-model/multi-objective state-feedback design Synthesis of output-feedback Ho controllers via Riccati-and LMI-based techniques, including mixed H2/Hoo synthesis with regional pole placement constraints Synthesis of robust gain-scheduled controllers for time-varying parameter-dependent systems For users interested in developing their own applications, the lmi lab provides a general-purpose and fully programmable environment to specify and solve virtually any LMI problem. Note that the scope of this facility is by no means restricted to control-oriented applications
Toolbox Features 1-3 Toolbox Features The LMI Control Toolbox serves two purposes: • Provide state-of-the-art tools for the LMI-based analysis and design of robust control systems • Offer a flexible and user-friendly environment to specify and solve general LMI problems (the LMI Lab) The control design tools can be used without a priori knowledge about LMIs or LMI solvers. These are dedicated tools covering the following applications of LMI techniques: • Specification and manipulation of uncertain dynamical systems (linear-time invariant, polytopic, parameter-dependent, etc.) • Robustness analysis. Various measures of robust stability and performance are implemented, including quadratic stability, techniques based on parameter-dependent Lyapunov functions, analysis, and Popov analysis. • Multi-model/multi-objective state-feedback design • Synthesis of output-feedback H∞ controllers via Riccati- and LMI-based techniques, including mixed H2 /H∞ synthesis with regional pole placement constraints • Loop-shaping design • Synthesis of robust gain-scheduled controllers for time-varying parameter-dependent systems For users interested in developing their own applications, the LMI Lab provides a general-purpose and fully programmable environment to specify and solve virtually any LMI problem. Note that the scope of this facility is by no means restricted to control-oriented applications
LMIs and lmi problems A linear matrix inequality (Lmi) is any constraint of the form A(r):=A0+x1A1+.+xNAN<O where xn) is a vector of unknown scalars(the decision or optimization variables A0,……, AN are given symmetric matrices .<0 stands for "negative definite, "i.e., the largest eigenvalue of A(x)is negative Note that the constraints A(x)>0 and A(x)<B(x)are special cases of(1-1)since they can be rewritten as-A(x)<0 and A(x)-b(x)<0, respectively The LMI(1-1)is a convex constraint on x since A(y)<0 and a(z)<0 imply that A(2)<0 Its solution set, called the feasible set, is a convex subset of r Finding a solution x to(1-1), if any, is a convex optimization problem Convexity has an important consequence: even though( 1-1)has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of lMi constraints can be regarded as a single lmi since A1(x)<0 is equivalent to A(x): =diag(A1(),, AK(x))<0 AK(x)<0 re diag(A(),., Ax(x)) denotes the block-diagonal matrix with A,(x),., Ak(r)on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity 1-4
1 Introduction 1-4 LMIs and LMI Problems A linear matrix inequality (LMI) is any constraint of the form (1-1) where • x = (x1, . . . , xN) is a vector of unknown scalars (the decision or optimization variables) • A0, . . . , AN are given symmetric matrices • < 0 stands for “negative definite,” i.e., the largest eigenvalue of A(x) is negative Note that the constraints A(x) > 0 and A(x) < B(x) are special cases of (1-1) since they can be rewritten as –A(x) < 0 and A(x) – B(x) < 0, respectively. The LMI (1-1) is a convex constraint on x since A(y) < 0 and A(z) < 0 imply that . As a result, • Its solution set, called the feasible set, is a convex subset of RN • Finding a solution x to (1-1), if any, is a convex optimization problem. Convexity has an important consequence: even though (1-1) has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of LMI constraints can be regarded as a single LMI since where diag (A1(x), . . . , AK(x)) denotes the block-diagonal matrix with A1(x), . . . , AK(x) on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity. A x( ) := A0 x1A1 … xNAN + ++ < 0 A y z + 2 ----------- < 0 A1( ) x < 0 . . . AK( ) x < 0 is equivalent to A x( ) := diag(A1( ) x ,… AK , ( )) x < 0
LMIs and LMI Problems In most control applications, LMIs do not naturally arise in the canonical form (1-1), but rather in the form LOX where L()and R() are affine functions of some structured matrix variables X1 Xn. A simple example is the lyapunov inequality AX+XA<0 where the unknown X is a symmetric matrix Defining x,..., xN as the independent scalar entries of X, this LMi could be rewritten in the form(1-1) Yet it is more convenient and efficient to describe it in its natural form(1-2) which is the approach taken in the LMi Lab The three generic LMI Problems Finding a solution x to the lmi system A(x)<0 (1-3) is called the feasibility problem. Minimizing a convex objective under LMI constraints is also a convex problem. In particular, the linear objective minimization problem Minimize c x subject to A(x)<0 plays an important role in LMI-based design. Finally, the generalized eigenvalue minimization problem A(x)<λB(x) Minimize n subject toB(x)>0 C(x)<0 is quasi-convex and can be solved by similar techniques. It owes its name to the fact that is related to the largest generalized eigenvalue of the pencil (A(x),B(x) Many control problems and design specifications have LMI formulations [ 9] This is especially true for Lyapunov-based analysis and design, but also for 1-5
LMIs and LMI Problems 1-5 In most control applications, LMIs do not naturally arise in the canonical form (1-1), but rather in the form L(X1, . . . , Xn) < R(X1, . . . , Xn) where L(.) and R(.) are affine functions of some structured matrix variables X1, . . . , Xn. A simple example is the Lyapunov inequality (1-2) where the unknown X is a symmetric matrix. Defining x1, . . . , xN as the independent scalar entries of X, this LMI could be rewritten in the form (1-1). Yet it is more convenient and efficient to describe it in its natural form (1-2), which is the approach taken in the LMI Lab. The Three Generic LMI Problems Finding a solution x to the LMI system (1-3) is called the feasibility problem. Minimizing a convex objective under LMI constraints is also a convex problem. In particular, the linear objective minimization problem (1-4) plays an important role in LMI-based design. Finally, the generalized eigenvalue minimization problem (1-5) is quasi-convex and can be solved by similar techniques. It owes its name to the fact that is related to the largest generalized eigenvalue of the pencil (A(x),B(x)). Many control problems and design specifications have LMI formulations [9]. This is especially true for Lyapunov-based analysis and design, but also for ATX XA + < 0 A x( ) < 0 Minimize c T x subject to A x( ) < 0 Minimize λ subject to A x() λ < B x( ) B x( ) > 0 C x( ) < 0
optimal LQG control, Hoo control, covariance control, etc. Further applications of LMIs arise in estimation, identification, optimal design, structural design [6, 7, matrix scaling problems, and so on. The main strength of LMi formu- lations is the ability to combine various design constraints or objectives in a numerically tractable manner A nonexhaustive list of problems addressed by LMI techniques includes the following Robust stability of systems with lti uncertainty (u-analysis)[24, 21, 27 Robust stability in the face of sector-bounded nonlinearities(Popov criterion) [22,28,13,16 Quadratic stability of differential inclusions [15, 8 Lyapunov stability of parameter-dependent systems [12] Input/state/output properties of lTI systems (invariant ellipsoids, decay rate, etc. )9] Multi-model/multi-objective state feedback design [4, 17, 3, 9, 10 Robust pole pla Optimal lQG control [9] Robust H control [11, 14 thesis[17,23,10,18] Design of robust gain-scheduled controllers 5, 2 Control of stochastic systems [9] Weighted interpolation problems [9] To hint at the principles underlying LMI design, let,s review the LMi formulations of a few typical design objectives Stability. the stability of the dynamical system Ax is equivalent to the feasibility of Find P= pt such that ATP+PA<OP>L. 1-6
1 Introduction 1-6 optimal LQG control, H∞ control, covariance control, etc. Further applications of LMIs arise in estimation, identification, optimal design, structural design [6, 7], matrix scaling problems, and so on. The main strength of LMI formulations is the ability to combine various design constraints or objectives in a numerically tractable manner. A nonexhaustive list of problems addressed by LMI techniques includes the following: • Robust stability of systems with LTI uncertainty (µ-analysis) [24, 21, 27] • Robust stability in the face of sector-bounded nonlinearities (Popov criterion) [22, 28, 13, 16] • Quadratic stability of differential inclusions [15, 8] • Lyapunov stability of parameter-dependent systems [12] • Input/state/output properties of LTI systems (invariant ellipsoids, decay rate, etc.) [9] • Multi-model/multi-objective state feedback design [4, 17, 3, 9, 10] • Robust pole placement • Optimal LQG control [9] • Robust H∞ control [11, 14] • Multi-objective H∞ synthesis [17, 23, 10, 18] • Design of robust gain-scheduled controllers [5, 2] • Control of stochastic systems [9] • Weighted interpolation problems [9] To hint at the principles underlying LMI design, let’s review the LMI formulations of a few typical design objectives. Stability. the stability of the dynamical system is equivalent to the feasibility of Find P = PT such that AT P + P A < 0, P > I. x · = Ax