REGLERTEKNIK AUTOMATIC CONTROL LINKÖPING

Transcription

1 QC Synthesis based on nertia Constraints Anders Helmerssonn Department of Electrical Engineering Linkoping University, SE Linkoping, Sweden www: LiTH-SY-R-1994 January 12, 1998 REGLERTEKNK AUTOMATC CONTROL LNKÖPNG Technical reports from the Automatic Control group in Linkoping are available as UNX-compressed Postscript les by anonymous ftp at the address (ftp.control.isy.liu.se).

2 QC Synthesis based on nertia Constraints Anders Helmersson Department of Electrical Engineering Linkoping University SE Linkoping, Sweden www: Submitted to Automatica December 19, 1997 Abstract ntegral quadratic constraints (QCs) can be used for proving stability of systems with uncertainties and nonlinearities. Similarly, QCs can also be used for controller synthesis. Necessary and sucient conditions for the existence of such acontroller is derived. These conditions include linear matrix inequalities (LMs) and matrix inertia specifying the number of negative eigenvalues of a matrix. n general, these conditions are nonconvex. Connections to bilinear matrix inequalities and LMs with rank constraints are also given. Keywords: controller synthesis, matrix inertia, linear matrix inequalities, integral quadratic constraints. 1 ntroduction Linear matrix inequalities (LMs) have been used during the last ten years for analysis and synthesis of robust control systems. The reason for this emerging interest is twofold. First, analysis and synthesis problems can be formulated as LMs. Secondly, ecient numerical solvers have been developed and are now available. One important feature of the LM is that it denes a convex problem, for which the local solution (minimum) is also a global one. However, some important problems, such as model reduction and synthesis of reduced order-controllers, cannot be formulated as pure LMs. nstead, nonconvex elements, such as rank constraints or bilinear matrix inequalities (BMs) must be included. n this paper we propose a new formulation based on the inertia of symmetric matrices, that is, the numbers of positive, negative and zero eigenvalues. The LM formulation can be derived from H 1 and analysis. A more general analysis setting, based on integral quadratic constraints (QCs) was This work was supported by the Swedish National Board for ndustrial and Technical Development (NUTEK), which is gratefully acknowledged 1

3 originally introduced by Yakubovich and later rened by Megretski and Rantzer [9]. n an QC setting (after applying the Kalman-Yakubovich-Popov lemma) the synthesis problem can be formulated as an algebraic problem. Find a K such that A + BKC A + BKC < (1) holds for some P ; ; ;Q (2) dened to belong to a given convex set. The synthesis problem considers the condition on, A, B and C for the existence of such a solution K. Specically, if A, B and C are given we want to search for such a. Assuming that is nonsingular, two LMs can be derived: one in and the other in ;1. This is in general not a convex problem. However, employing the inherent structure of some important problems, suchash 1 and gain-scheduling synthesis, convexity can be recovered and the existence of a controller K can be formulated as a (convex) LM problem. For the general synthesis problem, there seems to be no convex characterization of the existence of a controller K. One important class of problems that the convexity is violated in is synthesis of controller with a specied (low) order and model reduction. One way to solve this type of problem is by iterative projection methods [] using bilinear inequalities (BMs) is another approach [5, 4, 3, 13]. Previous QC synthesis results [1, 14, 15] require P and Q in (2) to be positive denite or at least positive semidenite. These requirements are relaxed in this paper to inertia constraints on only. n some QC problems, the deniteness on P and Q must be relaxed in order to not produce too conservative results, see for instance [7]. n this paper we elaborate on general conditions for the existence of a controller K. Two conditions emerge: one LM and one inertia constraint. The latter gives a constraint onthenumber of negative eigenvalues of a matrix that depends anely on. Section 2 gives a brief introduction of integral quadratic constraints (QCs). n section 3 some basic facts on inertia of matrices are given. The main synthesis results are stated and proved in section 4. Conclusions are given in section Notations Here A denotes the (complex conjugate) transpose A y is the pseudo-inverse n denotes a unitary matrix of size n n (A) and (A) are the number of negative andpositive eigenvalues of A A? denotes any fullrank matrix such that ker A range A?, where ker A is the null space of A and range A is the range or image of A. Note that A? exists only if A has linearly dependent rows and that A? A. For a matrix M11 M M 12 M 22 M 21 2

4 f - i v + - G(s) w? i + e Figure 1: Basic feedback conguration. and a matrix Q with compatible dimension the lower fractional transformation (LFT) is dened as M?Q F l (M Q) M 11 + M 12 Q( ; M 22 Q) ;1 M 21 : The set of rational stable transfer functions is denoted by RH 1 and L 2 denotes the Lebesgue space of signals with bounded energy L 2e denotes the extended Lebesgue space of signals with bounded energy over a nite interval [ T]. 2 ntegral Quadratic Constraints The integral quadratic constraints (QCs) have been proposed for robustness analysis [9]. The QC forms a stability criterion for the interconnection of a stable system G 2RH 1 and a bounded causal operator, see gure 1. ( v Gw + f (3) w v + e: Wesaythat the interconnection of G and is well-posed if the map (v w)! (e f) dened by (3) has a causal inverse on L 2e. The interconnection is stable if, in addition, the inverse is bounded, that is, if there exists a constant C such that Z T (jv(t)j 2 + jw(t)j 2 )dt C Z T (jf(t)j 2 + je(t)j 2 )dt for any T and for any solution of (3). Depending on the particular application, various versions of QCs are available. Two signals w 2L 2 [ 1) andv 2L 2 [ 1) are said to satisfy the QC dened by,if Z 1 ;1 ^v(j!) ^v(j!) (j!) d! (4) ^w(j!) ^w(j!) where absolute integrability isassumed. Here ^v(j!) and ^w(j!) represent the harmonic spectrum of the signals v and w at the frequency!. n principle, :jr! C can be any measurable Hermitian-valued function. n most applications, however, it is sucient to use rational functions that are bounded on the imaginary axis. A time-domain form of (4) is Z 1 (x (t) v(t) w(t))dt (5) 3

5 where is a quadratic form, and x is dened by _x (t) A x (t)+b v v(t)+b w w(t) x () where A is a Hurwitz matrix. The main theorem from [9] goes as follows Theorem 1 ([9]) Let G 2RH 1 and let be abounded causal operator. Assume that: i) for every 2 [ 1], the interconnection of G and is well-posed ii) for every 2 [ 1], the QC dened by is satised by iii) there exists> such that G(j!) G(j!) (j!) Then the feedback interconnection of G and is stable. ; 8! 2 R: () Note that if the upper left corner, 11 (j!), of is positive semidenite for all! 2 R then satises (4). f further the lower right corner, 22 (j!), is negative semidenite for all! 2 R, then any convex combination of 's satisfying (4) also satises the QC. Thus, 11 and 22 imply that satises (4) for 2 [ 1] if and only if does so. This simplies assumption ii). The search for multipliers,, can be carried out as a convex optimization problem by parametrizing (j!) X i x i i (j!) where x i are positive real parameters and i is a set of basis multipliers. Usually, i and G are proper rational functions with no poles on the imaginary axis, so that we can rewrite G(j!) G(j!) i (j!) D + C(j! ; A) ;1 B M i D + C(j! ; A) ;1 B n this formulation the matrices A, B, C and D depend on G and, while M i depends on i only. Thus, M is independent ofg. By applying the Kalman-Yakubovich-Popov lemma [17, 18, 12], the search for x i, can be implemented using linear matrix inequalities (LMs). Then () is equivalent to the existence of P P such that PA+ A P PB C D C D B + M < P : holds, where M11 M X M 12 M M i x i M i : 4

6 f - i v + - G(s) w? i + e - K Figure 2: Feedback conguration with a controller K. Note that this can also be written as 2 4 C D P M 11 M 12 P M 12 M C D < : (7) n a more general setting, we mayalsoletm be dened as a convex set specied by an LM. For instance, we may add constraints such that 11 (j!) > and 22 (j!) <, for all! 2 R, see [9, 7] for examples. 2.1 Controller Synthesis n (7), A, B, C and D depend on G and, while M depends on only, that is, M is independent ofg. We may let G or, equivalently, A, B, C and D depend on some controller, see gure 2. We assume that they are parametrized as a linear fractional transformation (LFT). t is no loss of generality to assume that the controller is represented as a static matrix dynamics can be included by augmenting G. Thus, C D ~A ~ B ~C ~ D?K ~ A + ~ BK( ; ~ DK) ~ C: f we assume that D ~, the matrices A, B, C and D depend anely on K. f ~D,we replace K with K ; D ~? K ~ K( ~ + D ~ K) ~ ;1 : Then, C D ~A ~ B ~C ~ D?K ~A ~ B ~C? ~ K ~ A + ~ B ~ K ~ C which depends anely on K. ~ The modied problem is equivalent to the original one as long as + D ~ K ~ is nonsingular. Thus, we have arrived at a the following matrix inequality problem. Determine if there exists a controller, K,suchthat ~ ~A + B ~ K ~ C ~ ~A + B ~ K ~ C ~ < (8) 5

7 holds. f such a controller exists, nd one such controller or, if possible, nd the set of all controllers that satisfy (8). n this paper we will focus on the existence conditions. n order to simplify the notation, we will rewrite (8) as (A + BKC) (A + BKC) < where A ~A ~B B C ~ C and K ~ K: n both of this two formulations it is assumed that has a given structure, for instance 2 4 P M 11 M 12 P M 12 M 22 where P P and M M are convex sets. 3 Matrix nertia The conditions for having a solution to the synthesis problem will be based on the inertia of matrices. The inertia of a matrix is dened as the numbers of negative, zero and positive eigenvalues. We will denote the number of negative eigenvalues of a (square) matrix A by (A) and the numberofpositive eigenvalues by (A) (;A). n the sequel we will only consider the inertia of hermitian matrices. One important fact (a theorem by Sylvester and Jacobi) of the inertia of an hermitian matrix is that it is unaected by any congruence transformation, see for instance [1]. A congruence transformation of a matrix P P is T PT where T is any nonsingular (square) matrix. Thus, (P )(T PT) Lemma 1 The truncation 11 of a hermitian matrix () ( 11 ). Proof: First assume that 11 is nonsingular. Then ; ; ; ; h i satises ; 12 ; and consequently () ( 11 )+( 22 ; 12 ; ) ( 11 ). f 11 is singular then we can modify the problem without aecting () nor ( 11 ),by adding " to 11, where "> is suciently small. For a given, wecanchoose " to be less than the minimum of the absolute values of the negative eigenvalues of 11 and. 2 Note that the trick of modifying a singular matrix, say, without modifying () will be used for derivation of some results in the sequel. Such a modication (9)

8 does aect the inertia since it modies the number of zero eigenvalues. However, since we here only consider the number of negative (or positive) eigenvalues this operation is legal. The following lemma connects a certain structure of inertia conditions to LMs. Lemma 2 Let X 2 R nn. Then if and only if B? XB? <. B B X n (1) Proof: For any suciently small ">, (1) is equivalent to "m B "m B X X ; " ;1 BB (X ; " ;1 BB )nn which inturnisequivalent tox<" ;1 BB for any suciently small ">, or equivalently, using Finsler's theorem, see for instance [11, 8],B? XB? <. 2 Note that in this case (1) is an equality, since(x) n for any matrix X X 2 R nn. 3.1 Reformulations Conditions on the inertia of a matrix can be seen as an extension to the linear matrix inequalities (LMs). For instance, if P P 2 R nn. Then (P ) n, or (P ) n, is equivalent to P <. Other conditions on the inertia can be translated into LMs with rank constraints. Lemma 3 Let P P 2 R (n+m)(n+m). The following three statements are equivalent: (i) (P ) n (ii) There exists a Q Q with rank Q m such that P <Q (iii) There exists a U 2 R (n+m)n such that U PU <. Proof: (i) ) (ii) Diagonalizing P using a congruent transformation yields a matrix with its eigenvalues along its diagonal. t is clear that there are no more than m non-negative eigenvalues and there are at least n negative eigenvalues. Thus, it is clear that we can choose a Q such that rank Q m. (ii) ) (iii) Choose U as a full rank matrix spanning the nullspace of Q. Note that U hasatleastn columns truncate it, if necessary, to exactly n columns. Then, U (P ; Q)U U PU ; <. (iii) ) (i) t is clear that we can nd a full rank matrix V such that U V becomes nonsingular. Using lemma 1 it follows that (P ) (U PU)n. 2 LMs with rank constraints also emerge in synthesis of reduced-order controllers and model reduction. n general, these problems are hard to solve, since they are not convex. Several methods have been proposed for this class of problems, for instance projection methods [], inversion of analytic centering [2], and bilinear matrix inequalities (BMs) [5, 4, 3, 13]. 7

9 3.2 Solving nertia nequalities We will here briey discuss how problems with inertia constraints can be solved numerically. We assume that there are constraints on the form (F (x)) n and C(x) >, where F and C are ane functions of x. We may maximize or minimize a linear combination of x, thatisc T x, subject to these constraints. Such an optimization could be based on (local) optimization subject to a barrier function. One choice is to use (x) log det C(x) +logj det F (x)j, which goestoinnity as the constraints are violated. This is similar to the barrier function used in algorithms for solving linear matrix inequalities, see for instance [1]. The analytic center and the analytic path both play important role in these algorithms. Both are the minimizers of (x) subject to F (x) > andc(x) >, in the latter also subject to c T x. t is possible to compute the the minimum of (x) even if F (x) is not positive denite. However, since the convexity islost, (x) may have several local minima, which have to be searched for. Also, the barrier function may divide the parameter space into several non-connected, non-convex regions, with the same number of negative eigenvalues, (F (x)). As the size of F increases the number of local minima is likely to increase and the complexity of the problem increases as well. n [2] it is shown that an LM problem with rank constraints is NP-hard. Consequently, according to lemma 3, problems with inertia constraints are NP-hard as well. Despite this fact, numerical algorithm searching for local minima as described above, could work well in many applications, especially if the search starts from an well-educated guess. 4 Synthesis We will here study the general synthesis problem: what are the conditions for the existence of K 2 R mp such that (A + BKC) (A + BKC) < holds. We start by looking at special case, which provides a simpler problem. Lemma 4 There exists a K 2 R mp such that < (11) K K if and only if () p. Proof: ()) Applying lemma 1 to P K K and using (11), yields () (P ) (P 11 )p. U1 (() Let U be a matrix spanning the eigenspace corresponding to U 2 the negative eigenvalues of. f U 1 2 R pp is non-singular then K U 2 U ;1 1 8

10 satises (11). f U 1 is singular add " to it, where "> is a suciently small number such that U1 + " U 2 U1 + " < U 2 still holds and use K U 2 (U 1 + ") ; Main Theorem We are now ready to state and prove the following new theorem. Theorem 2 Let A 2 R kn, B 2 R km and C 2 R pn. There exists a K 2 R mp such that holds if and only if (A + BKC) (A + BKC) < (12) C? A AC? < n: (13a) (13b) Proof: t is clear that (13a) is a necessary condition. Without loss of generality we may assume that C is full row rank. f not, we replace C in the original problem by a full rank matrix with the same nullspace and modify the size of K accordingly, so that the set of matrices generated by KC is unaected. By pre-multiplying (12) by C? and post-multiplying by its transpose, we infer that (13a) is a necessary condition. We absorb the dependency on A and B onto by rewriting (12) as (A + BKC) (A + BKC) KC P < KC KC where P. We transform (12) into an equivalent problem by performing a congruence transformation using C y C?, such that C C y C? p. Denote by C y ~A 1 The inequality (12) is equivalent to C? ~A 2 KC ~B ~A1 + BK ~ A ~ 2 P ~A1 + BK ~ A ~ 2 ( A1 ~ + BK) ~ P ( A1 ~ + BK) ~ ( A1 ~ + BK) ~ P A2 ~ ~A P ( ~ 2 A1 + BK) ~ A ~ P ~ < : (14) 2 A2 : 9

11 Since (13a) or, equivalently, ~ A 2 P ~ A2 < holds, we rewrite (14) using the Schur complement as ( ~ A1 + ~ BK) P ( ~ A1 + ~ BK) ; ( ~ A1 + ~ BK) P ~ A2 ( ~ A 2P ~ A2 ) ;1 ~ A 2P ( ~ A1 + ~ BK) (~ A1 + ~ BK) Q( ~ A1 + ~ BK) < (15) where Q P ; P A2 ~ ( A ~ 2 P A2 ~ ) ;1 A ~ 2 P. Next, we transform (15) into ( A1 ~ + BK) ~ Q( A1 ~ + BK) ~ C y Q C y K K D QD < (1) K K where D C y : m Using lemma 4, we infer that (1) has a solution K if and only if (D QD) p. Observing that C? has n ; p rows, and consequently n ; p ( A ~ P ~ 2 A2 ) (C? A AC? ), it follows that n p +(n ; p) (D QD)+ ~A 2P A2 ~ D (P ; P ~ A2 ( ~ A 2 P ~ A2 ) ;1 ~ A 2 P )D A ~ ~A 2 P A2 ~ A ~ PD 2 D P A2 ~ D PD ~A2 D P ~A2 D C? C y (P ) C P? C y 2P ~ A2 where we have used a congruence transformation similar to the one in (9). 2 Note that the condition (13a), including C, is convex in, while the condition (13b), including B, is in general not convex. Condition (13b) tells us that the number of negative eigenvalues in must be greater than or equal to the numberofrows in A. We can also reformulate the inertia condition (13b) using lemma (3), as the existence of a full row rank matrix matrix U 2 R (n+m)p such that U U<: (17) This can be interpreted as an LM where U selects the appropriate subspace from. Note that this reformulation is not convex since U is not given, but must be searched for. The space spanned by U must at least contain the space spanned by C?. 1

12 4.2 The Standard QC Synthesis Case We will now reconsider the standard QC synthesis problem (8), for which we state and prove the following new theorem. Theorem 3 Let A 2 R kn, B 2 R km and C 2 R pn. Assume that 2 R (k+n)(k+n) is non-singular and has k positive and n negative eigenvalues. Then there exists a K 2 R mp such that A + BKC A + BKC < if and only if C? A A C? < B? ;A ;1 ;A B? > : (18a) (18b) Proof: We apply theorem 2: the rst condition (13a) gives (18a), and the second condition (13b) becomes n: (19) Using a congruence transformation, (19) is equivalent to n + k 2 4 ; ;1 We next use lemma 2, with 2 4 B A ; ;1 B? ;A + ( ;1 ) C A :? C A which yields (18b). 2 The inertia assumption on can relaxed to () k. Since (19) must hold, we infer that () n and () k. n previous synthesis results using QCs, the assumption on P ; ; ;Q is that P and Q are both positive (semi-)denite, see [1, 14,15]. n theorem 3 the only assumption on is its inertia, which is weaker than in previous results. n many applications we may apply the following lemma. 11

13 Lemma 5 Let P ; ; ;Q be a nonsingular matrix where P 2 R kk, Q 2 R nn and P Q. Then () k and () n. Proof: f P is nonsingular, that is P >, then P ; P ; ;Q ;Q ; ; P ;1 ; where denotes similarity by a congruence transformation. Consequently, () k and () (Q +; P ;1 ;) k, since is nonsingular and Q. f P is singular then we can modify it by adding " n,such that the inertia of is unchanged ( is nonsingular), where "> is a suciently small real number. 2 However, in some applications the assumption that P and Q are positive semidenite is too conservative, see for instance [7] 4.3 An example { Static feedback A simple problem concerns the stabilization of a linear time-invariant system, G(s) C(s ; A) ;1 B, using static feedback, K. The system with static feedback is strictly stable if A + BKC P A + BKC < P holds for some P P >. Applying theorem 2, the existence of a K for a given P is equivalent to C? (A P + PA)C? < P P where n is the dimension of A and P, that is the number of states. The last condition can be rewritten as A P + PA PB B P n n: (2) For rst and second order systems, the solution set of P is convex. For thirdorder systems this is generally not true. When B is a unit or a non-singular matrix, that is the full control, the matrix is non-singular and can be considered as a congruence transformation in (2). Thus, the number of negative eigenvalues is equal to the number of negative eigenvalues of P P 12

14 which is n if P is nonsingular. Thus (2) is always satised, and the only remaining condition for the existence of a stabilizing controller is C? (A P + PA)C? < which isconvex in P. The full control problem is dual to the more common full information problem, see for instance [19]. 5 Conclusions Synthesis based on integral quadratic constraints (QC) can be expressed as solving a quadratic inequality involving a parametrization of the controller. Necessary and sucient conditions for the existence of acontroller have been derived. The conditions comprise a linear matrix inequality (LM) and a matrix inertia constraint. n general the matrix inertia condition is not convex and the problem becomes numerically hard since several local minima must be searched for and inspected. The inertia constraint can be seen as an alternative to LMs with rank constraints and to bilinear matrix inequalities (BMs). t is hoped that this new formulation could lead to better insight into the synthesis problem in order to better understand the problem and its complexity. References [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix nequalities in System and Control Theory. SAM Studies in Applied Mathematics. SAM, [2] J. David. Algorithms for Analysis and Design of Robust Controllers. PhD thesis, Dept. of Electrical Engineering, K. U. Leuven, Leuven, Belgium, [3] K. Goh, J. H. Ly, L.Turan, and M. Safonov. K m -synthesis via bilinear matrix inequalities. n Proceedings of the 33rd Conference on Decision and Control, volume 3, pages 232{237, Lake Buena Vista, Florida, December [4] K. Goh, M. Safonov, and G. Papavassilopoulos. A global optimization approach for the BM problem. n Proceedings of the 33rd Conference on Decision and Control, volume 3, pages 29{214, Lake Buena Vista, Florida, December [5] K. Goh, L. Turan, M. Safonov, G. Papavassilopoulos, and J. Ly. Biane matrix inequality properties and computational methods. n Proceedings of the American Control Conference, volume 1, pages 85{855, Baltimore, Maryland, June [] K. Grigoriadis and R. Skelton. Fixed-order control design for LM control problems using alternating projection methods. n Proceedings of the 33rd Conference on Decision and Control, volume 3, pages 23{28, Lake Buena Vista, Florida, December

15 [7] A. Helmersson. An QC-based stability criterion for systems with slowly varying parameters. Technical Report LiTH-SY-R-1979, Linkoping University, Linkoping, Sweden, Submitted to the ACC 1998 conference. [8] T. wasaki and R. E. Skelton. All controllers for the general H 1 control problem: LM existence conditions and state space formulas. Automatica, 3:137{1317, August [9] A. Megretski and A. Rantzer. System analysis via integral quadratic constraints. EEE Transactions on Automatic Control, 42():819{83, June [1] R. Njio, C. Scherer, and S. Bennani. Application of LPV control with full block scalings for a high performance ight control system. n Selected Topics in dentication, Modelling and Control, pages 113{12. Delft University Press, Delft, Netherlands, December 199. [11]. R. Petersen and C. V. Hollot. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22:397{411, January 198. [12] A. Rantzer. A note on the Kalman-Yacubovich-Popov lemma. n Proceedings of the 3rd European Control Conference, volume 3, part 1, pages 1792{1795, Rome, taly, September [13] M. Safonov, K. Goh, and J. Ly. Control system synthesis via bilinear matrix inequalities. n Proceedings of the American Control Conference, volume 1, pages 45{49, Baltimore, Maryland, June [14] G. Scorletti and L. El Ghaoui. mproved linear matrix inequalities conditions for gain-scheduling. n EEE Proceedings of the 31st Conference on Decision and Control, volume 4, pages 32{331, New Orleans, Louisiana, December [15] G. Scorletti and L. El Ghaoui. mporved LM conditions for gain scheduling and related control problems. nternational Journal of Robust and Nonlinear Control, Accepted for publication. [1] G. W. Stewart and J. Sun. Matrix Perturbation Theory. Computer Science and Scientic Computing. Academic Press, 199. [17] J. C. Willems. The Analysis of Feedback Systems. MT Press, Cambridge, MA, [18] V. A. Yakubovich. A frequency theorem for the case in which the state and control spaces are Hilbert spaces with an application to some problems of optimal controls Part {. Sibirskii Mat. Zh., 15(3):39{8, English translation in Siberian Math. J. [19] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall,