Linear Matrix Inequalities in Control

Transcription

1 C. W. Scherer a S. Weiland b Linear Matrix Inequalities in Control a Mathematical Systems Theory, University of Stuttgart, Pfaffenwaldring 57, 7569 Stuttgart/ Germany Carsten.scherer@mathematik.uni-stuttgart.de b Department of Electrical Engineering, Eindhoven University of Technology P.O. Box 513, 56 MB Eindhoven, The Netherlands Postprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 7a 7569 Stuttgart publications@simtech.uni-stuttgart.de

2 Linear Matrix Inequalities in Control Carsten Scherer Siep Weiland Mathematical Systems Theory Department of Electrical Engineering University of Stuttgart Eindhoven University of Technology Pfaffenwaldring 57 P.O. Box Stuttgart 56 MB Eindhoven Germany The Netherlands 1 Introduction Optimization problems and control problems are highly intertwined. If a control configuration has been decided upon, controller parameters or control input signals can be interpreted as decision variables of an optimization problem that reflects the desired specifications and constraints of the controlled system. In recent years, linear matrix inequalities (LMI s) have emerged as a powerful tool for approaching control problems that appear difficult if not impossible to solve in an analytic fashion. Although the history of linear matrix inequalities goes back to the fourties, with a major emphasis on their role in control in the sixties through the work of Kalman, Yakubovich, Popov and Willems, only during the last decades have powerful numerical interior point techniques been developed to solve LMI s in a practical and efficient manner (Nesterov, Nemirovskii). Today, several commercial and non-commercial software packages are available that allow for simple coding of general LMI problems. For example, Yalmip [12] is a very flexible and non-commercial toolbox for defining and solving advanced optimization problems. Boosted by the availability of fast LMI solvers, research in robust control theory has experienced a significant paradigm shift. Instead of arriving at an analytical solution of an optimal control problem and implementing such a solution in software so as to synthesize optimal controllers, today a substantial body of research is devoted to reformulating a control problem to the question of whether a specific linear matrix inequality is solvable or, alternatively, to optimizing functionals over linear matrix inequality constraints. It is the purpose of this chapter to highlight the main role and use of LMI s in solving a large variety of control and estimation problems. Notation R and C denote the fields of real and complex numbers. Sets of real and complex matrices of dimension m n are denoted by R m n and C m n. A matrix A C m n is Hermitian if it is square (m = n) and if A = A where the star denotes taking complex conjugate transpose. For real matrices this amounts to saying that A = A in which case A is said to be symmetric. The vector spaces of all n n Hermitian and symmetric matrices will be denoted by H n and S n, respectively, and we will omit the superscript n if the dimension is not relevant for the context. A Hermitian or symmetric matrix A is called negative definite, negative semi-definite, positive definite or positive semi-definite if x Ax <,, > or for all non-zero complex vectors x C n. We will denote this by A, A, A and A, and extend this notation to expressions A B to mean that A B for any A, B H n. A congruence transformation 1

3 of a square matrix M is a mapping M T MT with some nonsingular T. The operator col( ) stacks its arguments in a vector, as in col(a, b) = ( a b ) where a, b are vectors or matrices with the same number of columns. L n 2 denotes the space of all signals x : [, ) R n with finite energy x L2 := x(t) 2 dt where. is the Euclidean vector norm. We refer to the appendix for a brief summary on notions of convex sets and convex functions. 2 Linear matrix inequalities and convexity A Linear Matrix Inequality (LMI) is a constraint of the form F (x) := F + x 1 F x n F n (1) where F, F 1,..., F n are given real symmetric matrices and where x = col(x 1,..., x n ) is a vector of unknown real scalar decision variables. The inequality F (x) means that x should render the symmetric matrix F (x) negative definite, i.e., the maximum eigenvalue of F (x) should be negative. The LMI (1) gives rise to two types of questions: (a) The LMI feasibility problem amounts to testing whether there exist real variables x 1,..., x n such that (1) holds. (b) The LMI optimization problem amounts to minimizing a cost function c(x) = c 1 x c n x n over all x 1,..., x n that satisfy the constraint (1). Classical linear programs easily fit in this formalism, but also quadratic programs and some instances of convex quadratically constrained quadratic programs can be reformulated in this setting. In fact, the LMI optimization problem is a natural generalization of a linear program in which inequalities are defined by the cone of positive definite matrices. In most control applications, LMI s arise with matrix variables rather than vector variables. This means that we consider inequalities of the more general form F (X) in which X is a matrix that belongs to an arbitrary finite dimensional vector space X of matrices and where F : X S m is an affine function. We recall that affine functions assume the form F (x) = F + T (x) where F is fixed and where T is a linear map. Affine functions are, therefore, linear mappings plus some offset. For this reason, a linear matrix inequality is actually better called an affine matrix inequality, but the world has decided to accept this erroneous nomenclature. Example 1 A simple example of an LMI in the matrix valued unknown X = X is F (X) = A X + XA + Q where A and Q = Q are given square real matrices. This is a special case of (1) by expanding X as X = n k=1 x kx k where X k = Xk, k = 1,..., n, is a basis of the set of symmetric matrices X. Indeed, F is affine and F (X) = F ( n k=1 x kx k ) = Q + n k=1 x k(a X k + X k A) which is of the form (1) with F = Q and F k = A X k + X k A. 2

4 Of course, one is led to wonder what the practical interest in studying constraints of the special form (1) might be. There are a number of answers to this question. Firstly, (1) defines a convex constraint on the decision variable. This means that the solution set S := {x F (x) } of the LMI F (x) is convex. We refer the reader to the appendix for a brief overview of notions and results on convex sets and convex functions. Although the convex constraint F (x) on x may seem rather special, many convex sets can be represented in this way and, in fact, have more attractive properties than general convex sets. Secondly, the solution set of every finite set of LMI s F 1 (x),..., F k (x) can again be represented as one single LMI F (x) by setting F (x) = diag(f 1 (x),..., F k (x)). Hence, multiple LMI constraints on a decision variable x is again an LMI constraint. Thirdly, and very importantly, the partitioned LMI ( ) F 11 (x) F 12 (x) F (x) = F 21 (x) F 22 (x) is equivalent to { F 11 (x) S 22 (x) := F 22 (x) F 21 (x)f 11 (x) 1 F 12 (x) and at the same time equivalent to { S 11 (x) := F 11 (x) F 12 (x)f 22 (x) 1 F 21 (x) F 22 (x) Since F is affine, this equivalence means that also specific types of quadratic and rational inequalities can be reformulated as (1). The expressions S 11 (x) and S 22 (x) are called Schur complements of F 22 (x) and F 11 (x) in F (x). The above equivalences follow from the fact that congruence transformations of symmetric matrices leave the number of positive and negative eigenvalues invariant. In particular, M if and only if T MT ( for any nonsingular ) matrix T. For example, the first follows by taking M = F (x) and T =. I F 11 (x) 1 F 12 (x) I 3 Numerical solutions of LMI s Optimization problems over symmetric semi-definite matrix variables belong to the realm of semi-definite programming or semi-definite optimization. Although we mainly focus on control problems here, semi-definite programs occur in combinatorial optimization, polynomial optimization, topology optimization, etc. In the last few decades this research field has witnessed incredible breakthroughs in numerical tools, commercial and non-commercial software developments and fast solution algorithms. In particular, the introduction of powerful interior point methods allow us to effectively decide about the feasibility of semi-definite programs and to determine their solutions. This section aims to indicate briefly the main ideas behind interior point solvers of convex optimization programs. Let the solution set S := {x R n F (x) } of the LMI (1) be the domain of a convex function f : S R which we wish to minimize. That is, we consider the convex optimization problem v opt = inf f(x). (2) x S The idea behind interior point methods is to solve this constrained optimization problem by a sequence of unconstrained optimization problems. For this purpose, a barrier function φ is introduced. This is a function φ : R n R which is required to 3

5 (a) be strictly convex on the interior of S and (b) approach + along any sequence of points {x n } n=1 in the interior of S that converges to a boundary point of S. Given a barrier function φ, the constraint optimization problem is replaced by the unconstrained optimization problem to minimize the functional f t (x) := f(x) + tφ(x) (3) where t > is a penalty parameter. The main idea is to determine a curve t x(t) that associates with any t > a minimizer x(t) of f t. The minimum of f t is attained in the interior of S. Subsequently, the behavior of this mapping is considered as the penalty parameter t decreases to zero. In almost all interior point methods, the unconstrained optimization problem is solved with the classical Newton-Raphson iteration technique to approximately determine a local minimizer of f t. Since f t is strictly convex on R n, every local minimizer of f t is guaranteed to be the unique global minimizer. Under mild assumptions and for a suitably defined sequence of penalty parameters t n, t n as n, the sequence x(t n ) will converge to a point x. That is, the limit x := lim n x(t n ) exists and v opt = f(x ). If, in addition, x belongs to the interior of S then it is an optimal solution to (2); otherwise an almost optimal solution of (2) can be deduced from the sequence x(t n ). Interior point methods can be applied to either of the two LMI problems defined in the previous section. If we consider the feasibility problem associated with the LMI F (x) then (f does not play a role and) one candidate barrier function is the logarithmic function { log det( F (x) 1 ) if x S, φ(x) := otherwise. If S is bounded and non-empty, φ will be strictly convex. Convexity of φ implies the existence of a unique x opt such that φ(x opt ) is the global minimum of φ. The point x opt belongs to S and is called the analytic center of the feasibility set S. The LMI optimization problem to minimize c(x) subject to the LMI F (x) can be viewed as a feasibility problem for the LMI ( ) c(x) t G t (x) := F (x) where t > t := inf x S c(x) is a penalty parameter. Using the same barrier function yields the unconstrained optimization problem to minimize g t (x) := log det( G t (x) 1 ) = log 1 t c(x) + log det( F (x) 1 ) for a sequence of decreasing positive values of t. Due to the strict convexity of g t the minimizer x(t) of g t is unique for all t > t. Since closed form expressions for the gradient and Hessian of g t can be obtained, a Newton iteration is an efficient numerical method to find minimizers of g t. Currently much research is devoted to further exploiting the structure of LMI s in order to tailor dedicated solvers for specific semi-definite programs. 4 Stability characterizations with LMI s Around 189, Aleksandr Mikhailovich Lyapunov made a systematic study of the local expansion and contraction properties of motions of dynamical systems around an attractor. He worked 4

6 out the idea that an invariant set of a differential equation attracts all nearby solutions if one can find a function that is bounded from below and decreases along all solutions outside the invariant set. Such functions are called Lyapunov functions and they have been used to prove stability of equilibria of differential equations ever since. Consider the differential equation ẋ(t) = f(x(t), t) (4) with finite dimensional state space X = R n and where f : X R X is assumed to be sufficiently smooth so as to guarantee existence and uniqueness of the solution that satisfies the initial condition x(t ) = x X. The differential equation (4) is time-invariant if solutions satisfy x t +τ,x (t + τ) = x t,x (t) for any τ R with t and t + τ in the interval of existence. A point x X is called an equilibrium or fixed point of the differential equation if x t,x (t) = x satisfies (4) (which implies that f(x, t) = for all t t ). A wealth of stability concepts associated with fixed points of differential equations exists. Here, we focus on just one. The equilibrium point x of (4) is called exponentially stable if for all t R, positive numbers α, β and δ exist (all possibly depending on t ) such that x x δ = x t,x (t) x β x x e α(t t ) for all t t. (5) If α, β and δ do not depend on t then x is said to be uniformly exponentially stable. If δ is arbitrary, x is called globally exponentially stable. Hence, a fixed point is exponentially stable if all solutions of the differential equation that initiate nearby x converge to x with an exponential rate α >. The following result is standard and relates exponential stability of linear time-invariant differential equations to LMI feasibility. Theorem 2 Let A R n n. The following statements are equivalent. (a) The origin is an exponentially stable equilibrium point of ẋ = Ax. (b) All eigenvalues λ(a) of A belong to C := {s C R(s) < } (That is, A is Hurwitz). (c) The linear matrix inequalities A X + XA and X are feasible. Any solution X of the LMI s in item (c) defines the quadratic function V (x) := x Xx that serves as a Lyapunov function for the equilibrium point x = of the differential equation ẋ = Ax. Indeed, V achieves its minimum at x = and its derivative is d dt x(t) Xx(t) = ẋ(t) Xx(t) + x(t) Xẋ(t) = x(t) [A X + XA]x(t) and hence strictly decreasing by item (c). Rather straightforward arguments lead to (5) where δ is arbitrary, β = λ max (X)/λ min (X) and α > is any number for which A X +XA+2αX. For many applications in control and engineering one may be interested in characterizing eigenvalue locations of A in more general stability regions than C. For a real symmetric matrix P S 2m, the set of complex numbers { } I I L P := s C P si si 5

7 is called an LMI region. Important stability regions such as half-planes C stab 1 := {s R(s) < α}, circles C stab 2 := {s s < r} or conic sectors C stab 3 := {s R(s) tan(θ) < I(s) } can be represented by LMI regions L P1, L P2 and L P3, respectively, by taking sin(θ) cos(θ) 2α 1 r 2 P 1 =, P 2 =, P 3 = cos(θ) sin(θ) 1 1 sin(θ) cos(θ). cos(θ) sin(θ) LMI regions include sets bounded by circles, ellipses, strips, parabolas and hyperbolas. Since any finite intersection of LMI regions is again an LMI region, one can virtually approximate any convex region in C as long as it is symmetric with respect to the real axis. To present the main result of this section, we recall the definition of the Kronecker product A B of two matrices A C m n, B C k l, which is the mk nl matrix A 11 B... A 1n B A B =... A m1 B... A mn B The following result as originating from [4] is an interesting and elegant generalization of the stability characterization in Theorem 2. Theorem 3 Let A R n n. The following statements are equivalent. (a) All eigenvalues of A are contained in the LMI region { } I Q S I s C si S. R si (b) There exists X = X such that ( ) I X Q X S I X and A I X S. X R A I The condition in item (b) is an LMI in X. The result of Theorem 2 is recovered by taking Q =, S = 1 and R =. With Q = 1, S = and R = 1 the LMI region corresponds to the open unit disc; hence A has all eigenvalues within the open unit disc iff there exists X such that A XA X. In turn, this LMI test is equivalent to saying that the discrete time system x(k + 1) = Ax(k) is exponentially stable. Example 4 Consider the problem to find a stabilizing feedback law u = F x that simultaneously stabilizes the continuous-time systems ẋ = A k x + B k u where k = 1,..., 4 and A 1 =, B 1 =, A 2 =, B 2 =, A 3 =, B 3 =, A 4 =, B 4 =

8 By Theorem 2, the equivalent problem is to find X k and F such that (A k + B k F ) X k + X k (A k + B k F ) for k = 1,..., 4. Since both X k and F are unknown, this is not an LMI constraint. However, assuming X 1 = = X 4 = X, a congruence transformation with the matrix Y := X 1 transforms the five matrix inequalities to Y, A k Y + Y A k + B km + M B k, k = 1,..., 4 where we set M = F Y. These are LMI s in Y and M. When implemented with the given matrices, this set of LMI s turns out to be feasible and any solution defines a feedback F = MY ( 1 that solves the ) stabilization problem. One of these feedbacks is computed to be F = An analogous synthesis strategy can be applied for the simultaneous pole placement problem which amounts to finding F such that eigenvalues of A k + B k F belong to an LMI region L P for all k. Its solution is then an application of Theorem 3. 5 Performance characterizations with LMI s In this section we consider a linear system ẋ = Ax + Bd, e = Cx + Dd, x() = (6) in which d is viewed as an undesired external disturbance and e is an error output. Many control synthesis problems can be translated into a question of disturbance attenuation: the controller should reduce the effect of the disturbance d onto the error e as much as possible. In this section we discuss techniques how to quantify or analyze the effect of d onto e for the system (6) whose transfer function matrix is given by T (s) = C(sI A) 1 B + D. 5.1 H 2 -Performance Let us assume for (6) that A is Hurwitz and D =. If the number of inputs is m then B = (b 1,..., b m ) has m columns. If the system is excited with a unit impulse in the ν-th input, it responds with the output trajectory z ν (t) = Ce At b ν. The energy of this output trajectory equals ( ) [Ce At b ν ] [Ce At b ν ] dt = b ν e A t C Ce At dt b ν = b νy b ν (7) with Y being the observability Gramian, the unique solution of the Lyapunov equation A Y + Y A + C C =. (8) If we add the output energies for impulsive inputs in all input components we obtain m ν=1 z ν (t) z ν (t) dt = m Trace(b ν b νy ) = Trace(BB Y ) = Trace(B Y B) ν=1 where Trace denotes the sum of the diagonal elements of any matrix. In view of the explicit formula for the observability Gramian as used in (7) combined with Parseval s theorem, we infer that Trace(B Y B) actually equals T 2 H 2 := 1 2π Trace [C(iωI A) 1 B] [C(iωI A) 1 B] dω. (9) 7

9 Note that T H2 is the co-called H 2 -norm of the transfer matrix T, the name of which is motivated by the theory of Hardy-spaces in pure mathematics; this relation is not relevant for our purposes. We have actually derived the impulse-response performance interpretation of the H 2 -norm. Moreover this interpretation also provides a concise link to classical LQ theory, since the impulse response z ν ( ) is identical to the output response if the system s state is initialized as x() = b ν. Let us briefly touch upon the stochastic interpretation of the H 2 -norm. If w is white noise with unity covariance, the asymptotic variance of the output process of (6) satisfies ( lim t E[z(t) z(t)] = Trace lim t CE[x(t)x(t) ]C ) = Trace(CX C ) where X is the system s controllability Gramian, the unique solution of AX + X A + BB =. (1) Since T (s) = B (si A ) 1 C, we can conclude that the asymptotic output variance is equal to T H2. Due to (9) this is the same as T H2, which leads to a dual version for computing T H2. Let us summarize our findings as follows: If D = and X and Y are the system s controllability and observability Gramians satisfying (1) and (8) respectively, then T 2 H 2 = Trace(CX C ) = Trace(B Y B) is the sum of the energies of the output of (6) for impulsive inputs in each input channel, and it also equals the asymptotic output variance if the input is white noise with unity covariance. Note that H 2 -norms can be easily determined by solving linear equations and computing traces. Since these explicit formulas are inadequate for applying the synthesis procedure as developed in the next section, let us provide a genuine characterization of a bound on the H 2 -norm by LMI s. It is important to stress that this formulation actually combines an LMI characterization of system stability with a bound on system performance. Theorem 5 A is Hurwitz and T H2 < γ iff D = and there exist X = X and W = W with ( ) A X + XA XB X C B, and Trace(W ) < γ. (11) X γi C W Sketch of proof of if. For the general manipulation of performance specifications it is an instructive illustration of how to show that the feasibility of the system of LMI s (11) implies T H2 < γ. Indeed, by taking Schur complements we infer that (11) leads to A X + XA + 1 γ XBB X, X, CX 1 C W, Trace(W ) < γ. The first inequality implies A X + XA. Together with X this implies (Theorem 2) that A is Hurwitz. On the other hand, ˆX = γx 1 satisfies A ˆX + ˆX A + BB and Trace(C ˆX C ) < Trace(γW ) < γ 2. Combining the latter inequalities with (1), we get A( ˆX X )+( ˆX X )A which implies, using the stability of A, that ˆX X. Consequently, also Trace(CX C ) < γ 2 which yields that T H2 < γ. 8

10 5.2 H -Performance A different way of quantifying the effect of the disturbance d on the output e in (6) is in terms of the so-called energy gain e L2 T H := sup. < d L2 < d L2 The norm reflects the worst amplification of disturbances on outputs if measuring the sizes of the input and output signals in terms of their L 2 -norm or energy. Dissipativity theory [19] provides a direct path towards an LMI characterization of an upper bound T H < γ. Theorem 6 A is Hurwitz and T H < γ iff there exists X with I X I I γ 2 I I +. (12) A B X A B C D I C D Sketch of proof of if. Let us assume that (12) holds. Then the left-upper block of (12) just reads as A X + XA + C C which implies A X + XA. Therefore X guarantees that A is Hurwitz. Since the inequality (12) is strict, it continues to hold if we replace γ 2 by (γ ɛ) 2 for some suitably small ɛ >. Let us choose any d( ) with < d L2 < and let e( ) be the output of (6). If we right-multiply the perturbed version of (12) with col(x(t), d(t)) and left-multiply with its transpose, and if we exploit the relations in (6), we obtain ( ) x(t) X x(t) d(t) (γ ɛ) 2 I d(t) + = ẋ(t) X ẋ(t) e(t) I e(t) By integration on [, T ] and with x() = we infer x(t ) Xx(T ) + T = d dt x(t) Xx(t) + e(t) e(t) (γ ɛ) 2 d(t) d(t). T e(t) e(t) dt (γ ɛ) 2 d(t) d(t) dt for all T. Since both x( ) and ẋ( ) are of finite energy, x(t ) for T. After taking the limit T we hence get e 2 L 2 (γ ɛ) 2 d 2 L 2 or e L2 / d L2 γ ɛ. Since this holds for all < d L2 < we finally arrive at T H < γ. We started from the formulation of the LMI s (12) which gives the most insight for a dissipationbased proof of these results. In the literature, more common equivalent representations of the LMI (12) are ( A X + XA + C C XB + C D B X + D C D D γ 2 I ) or (Schur) A X + XA XB C B X γ 2 I D. (13) C D I Equivalently, A X+XA+C C+(XB+C D)(γ 2 I D D) 1 (B X+D C) and D D γ 2 I which touches upon the relation to Riccati inequalities and equations. 9

11 5.3 The Kalman-Yakubovich-Popov (KYP) Lemma We have established the link of (12) to the time-domain energy-gain by dissipation arguments. The relation of this LMI to the frequency-domain is the subject of the celebrated KYP lemma. Algebraic arguments proceed as follows. If (12) holds with X = X then A X +XA implies that A has no eigenvalues on the imaginary axis. For ω R let us observe that ( ) ( ) (iωi A) 1 B I X I (iωi A) 1 B = I A B X A B I = [(iωi A) 1 B] I X I (iωi A) 1 B = iωi X iωi = [(iωi A) 1 B] [iωx iωx](iωi A) 1 B =. Therefore feasibility of (12) implies, the validity of the frequency-domain inequality I γ 2 I I for all ω R { }, T (iω) I T (iω) which follows for ω = from the right-lower block of (12). Note that this inequality translates into T (iω) T (iω) γ 2 I or σ max (T (iω)) < γ for all ω R { }. In turn, this reveals the relation of the L 2 -gain bound to the peak of the largest singular value of the system s frequency response, which is simply the classical L -norm for transfer matrices without poles on the imaginary axis, or the H -norm for transfer matrices whose poles are all contained in the open left-half plane. The following result captures a generalization of the strict version of the KYP lemma for an arbitrary symmetric matrix P and without involving any sign-constraint on X. Theorem 7 In the case that A has no eigenvalues on the imaginary axis then I I P for all ω R { } (14) T (iω) T (iω) iff there exists some X = X with I X I I I + P. (15) A B X A B C D C D The survey article [1] provides a nice historical account of the development around the KYP lemma with many references, also to the Russian literature, and discusses further versions of this result even without any a priori hypotheses. In particular [2] is a rich source for the link of the KYP lemma to semi-definite programming duality and the related control theoretic interpretations. 5.4 Variants The books [3, 7] contain a whole variety of concrete variations on the theme of formulating performance specifications with LMI s. Let us provide a sample. 1

12 Generalized H 2 -performance. If replacing Trace(W ) < γ by W γi in Theorem 5, the formulated LMI s characterize that the system gain from finite energy input signals to the peak value of the output signal is bounded by γ: e L sup < d L2 < d L2 < γ with e L := sup e(t). t This criterion allows to impose time-uniform bounds on the error variable under the assumption that the disturbance has bounded energy. Quadratic performance. Our discussion of L 2 -gain performance opens the path towards the following generalization. With a symmetric weighting matrix P, quadratic performance with index P is achieved if an ɛ > exists such that the following integral quadratic constraint on the performance channel d e of the stable system (6) is satisfied: d(t) d(t) P dt ɛ d 2 L e(t) e(t) 2 for all d L 2. This time-domain specification directly translates into the frequency-domain inequality (14) and, due to Theorem 7, into feasibility of the LMI (15). If the right-lower block of P is positive semi-definite, it is elementary to see that stability of A is guaranteed by imposing the additional positivity constraint X. Discrete-time. All described results have counterparts for discrete-time systems x(t + 1) = Ax(t) + Bd(t), e(t) = Cx(t) + Dd(t), x() =, t =, 1, 2,.... If A has all its eigenvalues in the unit disc (Schur stability) then quadratic performance holds, by definition, if there exists some ɛ > with d(t) d(t) P ɛ d(t) d(t). e(t) e(t) t= This is equivalent to the frequency-domain inequality I I P for all z C, z = 1 T (z) T (z) on the unit circle, which is, in turn, equivalent to the existence of X = X satisfying the LMI I X I I I + P. A B X A B C D C D If the right-lower block of P is positive semi-definite, Schur stability of A is once again guaranteed by X. We stress the pleasing parallel structure of continuous-time and discrete-time performance formulations, which is even more striking for synthesis since the proposed general procedure applies to both domains without the need for any adaptation. t= 6 Optimal Performance Synthesis Let us now consider a system ẋ = Ax + B d + Bu e = C x + D d + Eu y = Cx + F d (16) 11

13 e System d y Controller u Figure 1: Generalized Plant Configuration where, in addition to the disturbance d and the error e, the signal u is a control input and y is a measured output. In feedback synthesis, the goal is to determine a controller ẋ c = A c x c + B c y u = C c x c + D c y (17) which feeds the measurements y back to u such that the controlled system, the interconnection of (16) and (17), is internally stable and satisfies a desired performance property (Figure 1). Note that the controlled system with state ξ = col(x, x c ) is easily seen to be described by ξ = Aξ + Bd e = Cξ + Dd with A B := C D C + ED c C EC c A + BD c C BC c B c C A c B + BD c F B c F D + ED c F (18). In view of the generalized plant framework [2, 18] it is essential to understand that this innocent problem formulation comprises surprisingly many specific configurations as they are needed in one- or two-degrees of freedom controller synthesis for reference tracking and disturbance attenuation. In this section emphasis is put on an H -norm bound as a measure of performance. 6.1 State-Feedback Synthesis The simplest control law is state-feedback u = D c x with a gain D c, which leads to the controlled system ẋ = (A + BD c )x + B d (19) e = (C + ED c )x + D d. Using Theorem 6 and with the LMI expressed as in (13), this controller stabilizes (19) and renders the H -norm of the transfer matrix d e smaller than γ iff there exists some X with X and (A + BD c ) X + X(A + BD c ) XB (C + ED c ) B X γ2 I D (C + ED c ) D I. (2) Recall that the first inequality guarantees stability while the second captures performance. We observe that, in synthesis, we need to search for both D c and X in order to satisfy (2). The performance inequality is, with some abuse of notation, a so-called bilinear matrix inequality problem since the left-hand side is affine in X (for fixed D c ) and affine in D c (for fixed X). Actually, a large variety of design problems in control can be easily seen to admit this structure. Unfortunately, however, bilinear matrix inequalities are as hard to handle as general nonlinear programs. Fortunately, for the problem at hand there is a surprisingly simple remedy. 12

14 There exists a celebrated procedure which actually turns the non-convex bilinear matrix inequality (2) into a convex problem of linear matrix inequalities. This is achieved by applying a nonlinear change of variables (D c, X) (M, Y ) and a congruence transformation to (2). Indeed if we transform (2) by congruence with X 1 and diag(x 1, I, I) respectively, and if we introduce the new variables Y := X 1 and M := D c X 1, (21) we arrive at Y and (AY + BM) + (AY + BM) B (C Y + EM) B γ 2 I D (C Y + EM) D I. (22) Obviously the constraints (22) constitute linear matrix inequalities in (M, Y ) whose feasibility can be verified. If (M, Y ) is a solution of (22), we can solve (21) for (D c, X) as X = Y 1 and D c = MY 1 and perform a congruence transformation of (22) with Y 1 and diag(y 1, I, I) which leads back to (2). This proves, with X being a certificate, that D c is indeed stabilizing and achieves the desired performance specification. On the other hand, if the inequalities (22) are not feasible, it is assured that no state-feedback gain can exist for which both these properties are satisfied. 6.2 Output Feedback Synthesis A general output feedback controller leads to the controlled system (18). It achieves stability of A and C(sI A) 1 B + D H < γ iff there exists some X with A X + X A X B C X and B X γ 2 I D. (23) C D I Due to the affine dependence of (A, B, C, D) on the controller matrices (A c, B c, C c, D c ), this involves again a bilinear matrix inequality. It is pleasing that one can identify, again, a convexifying controller parameter transformation [13, 17]. If we partition X U X = U and X 1 Y V = V (24) according to A (where X and Y share their dimension with A and the s denote matrices that are irrelevant for our purposes) reads as K L XAY U XB A c B c V = +. (25) M N I C c D c CY I We view this as a transformation (X, A c, B c, C c, D c ) (X, Y, K, L, M, N) =: v. This definition is motivated by the following easily verified relations: Y I with Y := V we have Y Y I X Y = =: X(v) and (26) I X Y (X A)Y Y AY + BM A + BNC B + BNF (X B) A(v) B(v) = K XA + LC XB + LF =:. (27) CY D C(v) D(v) C Y + EM C + ENC D + ENF 13

15 Although looking intricate, the key is the affine dependence of the blocks X(v), A(v), B(v), C(v) and D(v) on the new variables v. If Y is non-singular, a congruence transformation with Y and diag(y, I, I) on (23) leads to Y X Y and Y (A X)Y + Y (XA)Y Y (X B) Y C (B X )Y γ 2 I D CY D I, (28) which is, in turn, nothing but the following LMI in v: A(v) + A(v) B(v) C(v) X(v) and B(v) γ 2 I D(v). (29) C(v) D(v) I This brings us to the following result which is true without any hypothesis on Y. Theorem 8 There exists a controller (17) which stabilizes (18) and renders the H -norm of the transfer matrix d e smaller than γ iff the LMI s (29) are feasible. The actual design of a controller proceeds as follows. Find a solution of the LMI s (29); due to the first inequality in (29) the matrix I XY is non-singular (Schur); therefore one can find square and non-singular matrices U and V with I XY = UV ; then we can solve (25) for (A c, B c, C c, D c ). This controller does the job since we can define the non-singular matrix Y and solve for X in (26), which implies that (27) is valid; then (29) is nothing but (28); since Y is non-singular, this transforms into (23) by congruence. In summary, we have reduced the design problem to find a stabilizing controller that establishes a bound on the H -norm of the closed loop system by an equivalent problem that amounts to checking the feasibility of the LMI s (29). Since γ 2 enters these constraints in an affine fashion, one can minimize γ 2 subject to the feasibility of these LMI s in order to directly compute the optimal H -attenuation level that is achievable by stabilizing controllers. 6.3 General Synthesis Procedure Although we discussed H -synthesis in quite some detail, we can extract the following generic procedure for moving from analysis to synthesis inequalities for a whole variety of other performance specifications that can be expressed by LMI s. Rewrite the analysis inequalities Find a formal congruence transformation involving Y which leads to inequalities in terms of the blocks Y X Y, Y (X A)Y, Y (X B), CY and D. Then the synthesis inequalities are obtained by the substitution ( ) ( ) Y Y (X A)Y Y (X B) A(v) B(v) X Y X(v), C D C(v) D(v) with (27) for the variables v := (X, Y, K, L, M, N). For state-feedback synthesis one can apply the very same procedure with the formulas X(v) = Y, A(v) = AY +BM, B(v) = B, C(v) = C Y +EM, D(v) = D for v = (M, Y ). 14

16 The controller construction is independent from the particular analysis inequalities and remains, both for state-feedback and output-feedback synthesis, unaltered. This procedure applies to H 2 -synthesis as well as to the variants of the LMI analysis specification discussed in Section 5.4. As an illustration, the discrete-time system x(t + 1) = Ax(t) + Bd(t), e(t) = Cx(t) + Dd(t) is Schur-stable and its l 2 -gain is bounded by γ iff there exists some X with X, ( I A B ) ( X Since the second inequality can also be expressed as X, ) I I γ 2 I I +. X A B C D I C D ( X γ 2 I ) ( A B C D ) ( X I ) ( A B a Schur complement argument reveals that these analysis inequalities can be written as X A X C γ 2 I B X D X A X B X C D I C D and thus admit the precise format in order to apply the general synthesis procedure. Based on the generic dualization and elimination results described in [16], it is often possible to eliminate matrix variables that only appear in one of these synthesis inequalities, with the benefit of reducing computational complexity. For example, one can eliminate all matrices K, L, M, N from the H -synthesis inequalities in order to arrive at the inequalities as proposed in [8, 11]. Finally, in many practical problems the disturbance and error signals are partitioned as d = col(d 1,..., d p ) and e = col(e 1,..., e q ) in order to impose multiple individual performance requirements on some of the channels d ν e µ, possibly with different norms. The general controller synthesis procedure described in this section is applicable for this kind of multi-objective control problems provided that one is willing to allow some level of conservatism by expressing the combined desired specifications with one and the same matrix X in the closed-loop system. This variant of multi-objective control, which is addressed as a Lyapunov-shaping paradigm in [17] in more detail, even allows the incorporation of pole-placement requirements on A in terms of general LMI regions. We refer to [17] for an instructive controller design example. The introduction of slack-variables offers a possibility to reduce conservatism as shown for discrete-time systems in [5], while an LMI solution of the general multi-objective control problem based on the Youla-Kucera parameterization of all stabilizing controllers is discussed in [15] and its references. ), 6.4 Observer and Estimator Synthesis Estimation problems are related to the configuration in Figure 2. Based on measurements y, the goal is to determine an estimator whose output approximates the system output z (which can be equal to the state or comprise components of d) as closely as possible, despite the corruption of the system by the disturbance d. If d is white noise with unity covariance, minimization of the asymptotic variance of e amounts to minimizing the H 2 -norm of d e which results in the classical Kalman filter. Alternatively, using the energy-gain of d e as a performance indicator amounts to considering the H -estimation problem. These and many other variants 15

17 e u + Estimator z y System d Figure 2: Estimator Synthesis of estimation problems can be rephrased as an output-feedback synthesis problem for a system as described by (16) with B = and E = I (which means that u does not excite the system dynamics). Let us stress that in this reformulation u admits the interpretation of the estimator output and e is the estimation error. If the estimator admits the structure of an observer with a to-be-designed gain D c, ẋ c = Ax c + D c (Cx c y), u = C x c, (3) then the dynamics of the state-error ξ = x x c is described by ξ = (A + D c C)ξ + (B + D c F )d, e = C ξ + D d and defines the transfer matrix d e. This representation is dual (transposed) to what we considered for state-feedback synthesis, and only slight modifications are required in order to determine the LMI s for synthesizing observer gains D c that stabilize the error dynamics and achieve e.g. an H - or H 2 -norm bound γ on d e. Let us now assume that A in (16) is stable. Instead of assuming a particular structure, we can then try to find a general estimator (17) with stable A c such that a desired performance level for d e is achieved. In view of (18) and the fact that B =, stability of A c now boils down to stability of A, and the very same output-feedback synthesis procedure can be followed in order to design optimal estimators by LMI s. However, it is interesting to observe that the structural property B = allows a simplification of the convexifying controller parameter transformation which will be essential for robust estimator synthesis [9]. Indeed, starting from (24) and with Z = Y 1 let us define K L U A c B c V Z I I := and Y := M N I C c D c I V. (31) Z After a direct computation we get Y Z Z X Y = =: X(v) and (32) Z X Y (X A)Y Y ZA ZA ZB (X B) A(v) B(v) = XA+LC +K XA+LC XB +LF =: CY D C(v) D(v) C +ENC +EM C +ENC D +ENF with an affine dependence on X, Z, K, L, M, N. Therefore the general synthesis procedure in Section 6.3 does apply, with the only modification being the use of these new substitution formulas and by recalling the relation Z = Y 1 for the actual design of the estimator. 16

18 7 Polytopic Uncertainties and Robustness Analysis First principle models of physical systems are often represented by state-space descriptions in which the various components of the state represent well-defined physical quantities. Variations, perturbations or uncertainties in physical parameters lead to uncertainty in the model. Often, this uncertainty is reflected by variations in well distinguished parameters or coefficients in the model, while, in addition, the nature and/or range of the uncertain parameters may be known, or partially known. Since very small parameter variations may have a major impact on the dynamics of a system, it is of evident importance to analyse parametric uncertainties of dynamical systems. Suppose that δ = (δ 1,..., δ p ) is the vector which expresses the ensemble of all uncertain quantities in a given dynamical system. Then there are at least two distinct cases which are of independent interest: (a) Time-invariant parametric uncertainties: the vector δ is a fixed but unknown element of an uncertainty set δ R p. (b) Time-varying parametric uncertainties: the vector δ is an unknown time varying function δ : R R p whose values δ(t) belong to an uncertainty set δ R p, and possibly satisfy additional constraints. 7.1 Time-invariant parametric uncertainty Consider the uncertain time-invariant system defined by ẋ = A(δ)x (33) where A( ) is a continuous function of the real valued parameter vector δ = col(δ 1,..., δ p ) which is only known to be contained in an uncertainty set δ R p. The problem of robust stability amounts to characterizing whether the equilibrium point x = of (33) is exponentially stable for all parameters δ δ. With time-invariant uncertainties, (33) is robustly stable iff A(δ) is Hurwitz for all δ δ. Since δ generally consists of infinitely many points, the verification of this condition is rather troublesome from a computational point of view. The uncertain system (33) is called quadratically stable if there exists X = X such that X, A(δ) X + XA(δ) for all δ δ. (34) The importance of this definition becomes apparent after observing that V (x) := x Xx is a quadratic Lyapunov function for (33) which, by Theorem 2, implies that A(δ) is Hurwitz for all δ δ. Hence, quadratic stability implies the origin of (33) to be robust exponentially stable against time-invariant uncertainties δ δ. Unless δ has a finite number of points, (34) can not be verified easily. Therefore, the following result is of considerable interest. Theorem 9 If A(δ) is affine in δ and δ = co{δ 1,..., δ N } then (33) is quadratically stable if and only if there exists some X such that X, A(δ k ) X + XA(δ k ), k = 1,..., N. Hence, for polytopic uncertainty sets and affine parametric dependence in (33), quadratic stability can be numerically verified by a feasibility test for a finite number of LMI s only. The proof is an illustrative example of the use of convexity. It requires showing that F (δ) := A(δ) X + XA(δ) for all δ δ if F (δ k ) for k = 1,..., N. To see this, first observe that F ( ) is a convex function on δ whenever A( ) is affine. Second, any δ δ can be written as a convex combination of the points δ 1,..., δ N, say δ = N k=1 α kδ k with nonnegative coefficients α k that sum up to 1; F (δ) = F ( N k=1 α kδ k ) N k=1 α kf (δ k ). 17

19 7.2 Time-dependent parametric uncertainty Robust stability against time-varying uncertainties is generally a more demanding requirement than robust stability against time-invariant uncertainties. Consider the system ẋ(t) = A(δ(t))x(t) (35) which is affected by an uncertain parameter curve δ : R δ. Unlike the case with time-invariant uncertainties, robust stability of the origin is now not implied by the condition that A(δ) is Hurwitz for all δ δ. However, the uncertain system with time-varying parametric uncertainties is exponentially stable if there exists X such that (34) holds. Therefore, quadratic stability does, in fact, imply robust stability against arbitrary fast time-varying parametric uncertainties. This is a nice, but in general, conservative test if additional a priori information on the uncertainty is available. For example, the parameter curves δ( ) are often known to be continuously differentiable and constrained in terms of their values and their rate-of-variation as δ(t) δ, δ(t) ρ for all t R. (36) Less conservative robust stability tests can be inferred by postulating the existence of parameter dependent Lyapunov functions. A popular instance of such functions takes the form V (x, δ) := x X(δ)x and requires a search over matrix functions X(δ) = X(δ) with δ δ. For notational convenience, let us introduce, for a continuously differentiable matrix function X(δ), the derivative p X(δ, ρ) := k X(δ)ρ k, (δ, ρ) δ ρ (37) k=1 where k X( ) denotes the partial derivative of the function X( ) with respect to the k-th entry of δ and where ρ k is the k-th component of the vector ρ. (We stress that X(δ, ρ) is purely symbolic notation which is not to be confused with the partial derivative of X( ) itself.) The following result provides a sufficient condition for robust stability and actually covers many tests in the literature. The proof provides much insight into the understanding of stability arguments based on parameter dependent Lyapunov functions. Theorem 1 Suppose that δ and ρ are compact subsets of R p and suppose that X(δ) = X(δ) is a continuously differentiable matrix function that satisfies X(δ), X(δ, ρ) + A(δ) X(δ) + X(δ)A(δ) (38) for all δ δ and ρ ρ. Then the origin of (35) is exponentially stable for all time-varying parametric uncertainties that satisfy (36). Proof. Suppose that X(δ) satisfies (38). Continuity of X( ) and compactness of δ and ρ guarantee the existence of constants a, b, c > such that, for all δ δ and ρ ρ, ai X(δ) bi, X(δ, ρ) + A(δ) X(δ) + X(δ)A(δ) ci. (39) Let δ( ) and x( ) be a parameter curve and state trajectory that satisfy (36) and (35), respectively. With ξ(t) := x(t) X(δ(t))x(t) we clearly have ] ξ(t) = x(t) [ p k=1 k X(δ(t)) δ k (t) x(t) + x(t) [A(δ(t)) X(δ(t)) + X(δ(t))A(δ(t))]x(t). If we exploit (39) we get a x(t) 2 ξ(t) b x(t) 2 and ξ(t) c x(t) 2. This implies that ξ(t) c b ξ(t) and hence ξ(t) ξ(t ) exp( c b (t t )) for all t t. In turn, this leads to 18

20 x(t) 2 b a x(t ) 2 e c b (t t ) which is (5) for α = c/(2b) and β = b/a. The constraints (38) on X( ) define a purely algebraic test that do not involve the system- or parameter-trajectories. The test is not easy to apply directly because the matrix function X( ) needs to satisfy a partial differential LMI. By considering specific classes of matrix functions X( ), (38) can be converted and implemented with LMI solvers. One of these classes is the set of affine symmetric matrix functions X : δ S n. A few special instances are worth mentioning. If the parameters are time-invariant we have ρ = {} and (38) simplifies to the conditions X(δ) and A(δ) X(δ) + X(δ)A(δ). In that case, (38) is also necessary for robust stability. If parameters vary arbitrarily fast, ρ is unbounded and (38) is feasible only if the partial derivatives k X(δ) vanish identically. This means that X(δ) = X is not depending on δ and we recover the quadratic stability test. 7.3 Robust performance In the previous subsections we have shown how tests for robust stability against parametric uncertainties can be inferred from characterizations of nominal stability. The same generalization applies in order to obtain conditions for verifying robust performance. Consider the uncertain parameter depending system ẋ(t) = A(δ(t))x(t) + B(δ(t))d(t) e(t) = C(δ(t))x(t) + D(δ(t))d(t) where δ( ) is a continuously differentiable rate-bounded uncertainty that satisfies (36). In the case of H -performance, we quantified the effect of d on e in terms of the L 2 gain of the system. However, the output e of (4) not only depends on the input d but also on the uncertainty δ( ). Hence we say that the robust L 2 -gain is smaller than γ if for d = and for all parameter curves δ( ) with (36), x = is an exponentially stable equilibrium of the system (4); (4) for x() = it holds that sup δ( ) satisfies (36) e L2 sup < γ. < d L2 < d L2 With some abuse of terminology, robust L 2 -gain performance is often referred to as robust H -performance, but it is important to realize that frequency domain characterizations do not make sense when considering time-dependent parametric uncertainties in (4). Only with time-invariant parameter uncertainties (ρ = {}) does a robust L 2 -gain that is smaller than γ imply that T δ H < γ for all δ δ, where T δ is the transfer function associated with (4) for time-invariant parameters δ(t) = δ. The following result generalizes Theorem 6 to robust L 2 -gain performance. Theorem 11 Suppose there exists a continuously differentiable matrix function X(δ) = X(δ) such that X(δ) and ( ) X(δ, ρ) + A(δ) X(δ) + X(δ)A(δ) X(δ)B(δ) B(δ) + X(δ) ( ) ( ) I γ 2 I I + (41) C(δ) D(δ) I C(δ) D(δ) for all δ δ and ρ ρ. Then the uncertain system (4) has a robust L 2 -gain smaller than γ. 19

21 The proof of this result is analogous to that of Theorem 6. The main merit of Theorem 11 is that it converts robust L 2 -gain performance to an algebraic property. As in Theorem 1, the condition (41) requires the numerical search for a matrix function X( ) that needs to satisfy a partial differential linear matrix inequality. Moreover, the discussion about the extreme cases concerning time-invariant or arbitrary fast time-varying parameters in Section 7.2 remains valid for (41). Finally, let us remark that Theorem 11 extends, mutatis mutandis, to all other performance specifications that have been mentioned in Section 5. 8 Robust State-Feedback and Estimator Synthesis Consider the system ẋ(t) = A(δ(t))x(t) + B (δ(t))d(t) + B(δ(t))u(t) e(t) = C (δ(t))x(t) + D (δ(t))d(t) + E(δ(t))u(t) y(t) = C(δ(t))x(t) + F (δ(t))d(t) (42) whose describing matrices are affected by a time-dependent parametric uncertainty δ(t) R p. Let us also assume that the dependence of the system matrices is actually affine, and that δ(t) takes values that are, for t, confined to the polytope δ := co{δ 1,..., δ N } R p. (43) Robust controller synthesis deals with the problem of determining a feedback controller that processes measurements y to control inputs u, so as to guarantee robust stability and a desired robust performance specification on the mapping from the disturbance d to the output e of the controlled system. The robust state-feedback synthesis problem is a special case in which the whole state is assumed to be measurable (y = x in (42)) and the controller is a static feedback law u = D c x with some gain D c. In that case, the resulting closed-loop system is described by ẋ(t) = [A(δ(t)) + B(δ(t))D c ]x(t) + B (δ(t))d(t) e(t) = [C (δ(t)) + E(δ(t))D c ]x(t) + D (δ(t))d(t). If applying Theorem 11 for a parameter-independent X(δ) = X and exploiting affine parameterdependence as for Theorem 9, we infer that the robust L 2 -gain of the controlled system is smaller than γ if there exists X = X with X, [A(δ k )+B(δ k )D c ] X +X[A(δ k )+B(δ k )D c ] XB (δ k ) [C (δ k )+E(δ k )D c ] B (δ k ) X γ 2 I D (δ k ) C (δ k )+E(δ k )D c D (δ k ) I for all k = 1,..., N. (44) Literally following the nominal synthesis procedure of Section 6.1 with the convexifying transformation (21) we infer that there exist (D c, X) satisfying (44) iff there exist (M, Y ) satisfying Y, [A(δ k )Y +B(δ k )M] +[A(δ k )Y +B(δ k )M] B (δ k ) [C (δ k )Y +E(δ k )M] B (δ k ) γ 2 I D (δ k ) C (δ k )Y +E(δ k )M D (δ k ) I for all k = 1,..., N. 2