Static Output Feedback Stabilisation with H Performance for a Class of Plants

Static Output Feedback Stabilisation with H Performance for a Class of Plants E. Prempain and I. Postlethwaite Control and Instrumentation Research, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK, E-mail: ep26@sun.engg.le.ac.uk Abstract In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of Linear Matrix Inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H performance objectives. This results in a sub-optimal static H control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example. Notation R m n : real m n matrices I n : n n identity matrix M > 0: M is symmetric and positive definite, i.e. x T Mx > 0 for all non zero vectors x A + ( ) T := A + A T 1 Introduction In this paper, attention is focused on the design of a stabilising static output feedback gain matrix. Given the plant matrices (A, B, C) the problem is to find, if one exists, a static output feedback law such that the closed-loop system is asymptotically stable. Usually, the determination of a full order output Preprint submitted to Elsevier Preprint 15 January 2001

feedback reduces to the solution of two convex problems (a state feedback and a Kalman filter). However, the synthesis of a static output feedback gain or a controller of fixed order is more difficult. The reason is that the separation principle does not hold in such cases. A comprehensive survey on static output feedback is given in (Syrmos et al. [11]). The existence of a static output feedback controller is shown to be equivalent to the existence of a positive definite matrix Q satisfying simultaneously two Lyapunov inequalities [8], [9], where one inequality involves Q and the other Q 1. As a result, the determination of such a Q > 0 leads to solving a non-linear optimisation matrix problem [1], [9]. An algorithm (called min/max algorithm) has been developed for the numerical determination of Q [6]. This iterative algorithm involves, at each step, the determination of a solution to a particular LMI system. In [9], [7], the formulation has been extended to get suboptimal performance with respect to a given quadratic index. While attractive and able to handle general problems, the min/max algorithm is not guaranteed to find a solution, even if one exists. Various algorithms have been proposed to address the static output feedback design problem e.g. [4] [2]. But as with the min/max algorithm they are not guaranteed to find a solution. In this paper, necessary and sufficient conditions for the existence of a static output feedback law are provided. These conditions depend on a specific congruence transformation which is in fact a function of a rectangular matrix N. The conditions become a Linear Matrix Inequality whenever the structure of the Lyapunov function is fixed (i.e when N is fixed). Conditions on the congruence transformation matrix are also given. A strong motivation for using structured Lyapunov functions is to make the problem affine once the structure is fixed. One obvious advantage, for the class of plants considered (which are always static-output-feedback stabilizable), is to exploit all the remaining degrees of freedom in the Lyapunov function to optimize an H cost. The particular class of plants considered is the class of minimum phase plants with CB full row rank for which the matrix N is given algebraically. It follows, for this class of plants, that we can compute systematically and non-iteratively a sub-optimal static H output feedback control law. The effectiveness of the method is demonstrated on a numerical example. 2 Preliminaries Consider a linear time invariant system G described by state-space equations ẋ = Ax + Bu G : (1) y = Cx 2

where A R n n, u R n u is the control input, y R n y is the measured output. The pairs (A, B) and (A, C) are assumed to be, respectively, stabilizable and detectable, and we assume that C and B are full rank. Our aim is to compute a static output feedback law u = Ky that ensures the stability of the closed-loop system A cl = A + BKC. The closed-loop system matrix can be rewritten as A cl = A + B[K 0]T (2) where T T = [C T TC T ] is square and non-singular. Such a T C can always be chosen as an orthogonal basis of the null-space of C. The closed-loop is stable, is equivalent to A T = T A cl T 1 = A C + B C KC C stable (3) with A C = T AT 1, B C = T B and C C = CT 1 = [I ny 0]. (4) Clearly, the static output feedback problem can be viewed as the determination of a state-feedback gain F with the structural constraint F = [K 0]. Since the controller has to stabilize the closed-loop, the closed-loop system must admit a symmetric Lyapunov function Q > 0 such that A T Q + QA T T < 0 or equivalently L := A C Q + QA T C + B C KC C Q + QC T CK T B T C < 0 (5) Before stating the problem, we recall the following well-known lemmas [5], [3], [8]: Lemma 1 Consider the following LMI in the variable X, BXC + (BXC) T + Ω < 0. (6) This LMI has a solution X if and only if N T BΩN B < 0 and N T C ΩN C < 0 (7) where N B and N C denote bases of the null spaces of B T and C, respectively. 3

Lemma 2 (Finsler Lemma) The following inequality N T BΩN B < 0 (8) holds if and only if there exists σ R such that Ω σbb T < 0 (9) 3 Structured Lyapunov functions Without any lose of generality let Q = Q T R n n, Q := Q 1 Q 1 N (10) N T Q 1 Q 2 + N T Q 1 N with Q 1 = Q T 1 R ny ny, Q 2 = Q T 2 R (n ny) (n ny), N R ny (n ny). Q can be rewritten as Q = T N Q d T T N, Q d := diag(q 1, Q 2 ) (11) where T N is a non-singular matrix defined as T N := I n y 0 (12) N T I n ny Note that Q > 0 is equivalent to Q d > 0 for any matrix N. Theorem 1 The system (1) is stabilizable by a static output feedback if and only if there exist Q d > 0, Y R n u n y and N R n y (n n y ) such that L M := M + M T < 0 (13) with M := (TN 1 A C T N )Q d + (TN 1 B C )Y C C (14) Moreover when (13) is feasible, a stabilising static output feedback is K = Y Q 1 1. 4

Proof Using (11), (5) can be rewritten as A C T N Q d T T N + B C KC C T N Q d T T N + ( ) T < 0 (15) With the congruence transformation TN 1 C C T N = C C, we get applied to (15) and using the identity TN 1 A C T N Q d + TN 1 B C KQ 1 C C + ( ) T < 0 (16) which is exactly (13). This completes the proof. Remark. This theorem shows that when N is fixed, that is when the Lyapunov matrix Q = Q T has a special structure, then the Lyapunov inequality L M becomes an LMI in the variables Q d > 0 and Y. Thus, when N is fixed, the synthesis of K becomes tractable by LMI optimisation. However, the matrix N cannot be chosen arbitrarily for a given problem. The next theorem provides conditions which must be satisfied by the matrix N. Theorem 2 Let A C and B C be conformably partitioned with the symmetric matrix Q defined in (10) i.e. A C := A C 11 A C21 A C12 A C22, B C := B C 1 B C2, (17) A C11 R ny ny, B C1 R ny nu. The plant G is stabilizable by a static output feedback only if (A C22 N T A C12 )Q 2 + ( ) T < 0 (18) and for some real σ (A C11 + A C12 N T )Q 1 + ( ) T σb C1 B T C 1 < 0. (19) Proof. Using the projection conditions (lemma 1), the LMI system (13) is feasible if and only if there exist Q d > 0 and N R n y (n n y ) such that N1 T ΩN 1 < 0 (20) and N T 2 ΩN 2 < 0 (21) 5

with Ω := TN 1 A C T N Q d + (TN 1 A C T N Q d ) T (22) and where N 1 and N 2 denote any orthonormal bases of the null space of B T CT T N and C C respectively. Necessity of (18). A possible choice for N 2 is N 2 = [0, identities I n ny ] T. Using the N T 2 T 1 N = [ N T, I n ny ] and T N Q d N 2 = [0, Q 2 ] T, (23) the condition (21) reduces to finding a Lyapunov matrix Q 2 > 0 so that L 2 = [ N T, I n ny ] A C 11 A C21 A C12 A C22 0 + ( ) T Q 2 = (A C22 N T A C12 )Q 2 + ( ) T < 0, (24) but inequality (24) is satisfied if and only if A C22 N T A C12 proves the necessity of (18). is stable. This Necessity of (19). Inequality (20) is feasible if there exist Q d σ R so that > 0, N and Φ := Ω σtn 1 B C BCT T N T < 0 (25) By substituting the expression of Ω given in (22) into (25) and by post and pre multiplying (25) by C C = [I ny, 0] and C T C respectively, we get the following inequality (A C11 + A C12 N T )Q 1 + ( ) T σb C1 B T C 1 < 0; (26) which means that the pair (A C11 +A C12 N T, B C1 ) is stabilizable. This completes the proof. 4 A special class of plants We will consider the class of minimum phase plants plant with CB full row rank. We will see that the plants in that class are stabilizable by static output 6

feedback. Moreover, we will provide a choice for N in terms of the plant matrices. With such an N, the problem of finding a static output feedback reduces to solving LMI (13). Lemma 3 Assume G is square with CB full row rank and a minimal realisation, then the eigenvalues of à C22 := A C22 B C2 B T C 1 EA C12 with E := (B C1 B T C 1 ) 1 are the transmission zeros of the plant G. Proof. The transmission zeros of G can be defined as the complex numbers z such that A C zi B C (27) C C 0 loses rank. Or equivalently, using the partitioned state-space representation of G, when A C11 zi A C12 B C1 A C21 A C21 zi B C2 I ny 0 0 (28) loses rank. Permuting the first and the third column of (28), it is clear that the transmission zeros of G are the complex numbers z such that B C2 A C12 S 12 := B C 1 (29) A C22 zi loses rank. Now, since CB is full row rank, the following transformation is non-singular T E := EBT C 1 0 (30) B C2 EBC T 1 I and we get T E S 12 := I ( ) (31) 0 ÃC 22 zi 7

which clearly establishes that the zeros of G are also the eigenvalues of ÃC 22. This completes the proof. Theorem 3 The following assumptions are made: (i) CB full row rank (ii) ÃC 22 := A C22 B C2 B T C 1 (B C1 B T C 1 ) 1 A C12 stable. Then the system G is stabilizable by a static output feedback. Moreover N = (B C1 B T C 1 ) T B C1 B T C 2 can be used in Theorem 1 for the numerical determination of a stabilising static output feedback law K. Proof. Let us first partition Ω and Φ given in (22) and (25) conformably with A C i.e. Ω 11 = (A C11 + A C12 N T )Q 1 + ( ) T Ω 21 = ( N T A C11 N T A C12 N T + A C22 N T + A C21 )Q 1 + Q 2 A T C 12 (32) Ω 22 = (A C22 N T A C12 )Q 2 + ( ) T and Φ 11 = Ω 11 σb C1 BC T 1 Φ 21 = Ω 21 σrbc T 1 (33) Φ 22 = Ω 22 σrr T where R = B C2 N T B C1. Let N T = B C2 BC T 1 (B C1 BC T 1 ) 1. Note that RBC T 1 = 0. Since B C1 is full row rank, for any Q 1 > 0 there exists σ 0 > 0 such that Φ 11 < 0. Since ÃC 22 is stable there exists Q 2 > 0 such that Ω 22 < 0 which implies Φ 22 < 0 for any σ > 0. To prove that G is stabilizable by static output feedback, we have to prove that Φ < 0. Applying the Schur complement formula to Φ we get Φ 22 < Ω 21 Φ 1 11 Ω T 21 (34) But for any σ such that σ > σ 0 > 0, we have Φ 11 + σ B C1 B T C 1 = Ω 11 σ 0 B C1 B T C 1 < 0 with σ = σ σ 0, which in turn implies Φ 1 11 > (B C1 B T C 1 ) 1 /σ. 8

Then Φ < 0 if Φ 22 < Ω 21 (B C1 B T C 1 ) 1 Ω T 21/σ (35) But (35) obviously holds for an arbitrarily large σ. This completes the proof. Remark. Theorem 3 generalises the result presented in [10]. 5 Extension to H sub-optimal static output control Stability is the minimum requirement we can expect of a control law. Further performance objectives can be achieved via H optimisation. In this section, attention is given to the synthesis of a sub-optimal H output static control law. Let us consider a generalised plant governed by: ẋ = Ax + B w w + Bu G : z = C z x + D zw w + D z u y = Cx (36) Let T be a non-singular state coordinate transformation such that C C T = C, where C C := [I ny, 0]. It follows that G is also governed by ẋ = A C x + B wc w + B C u G : z = C zc x + D zw w + D z u y = C C x (37) where A C = T AT 1, B C = T B, B wc = T B w, C zc = C z T 1. Let T zw denote the closed-loop transfer function from w to z for the staticoutput feedback control law u = Ky. T zw is governed by x cl = A cl x cl + B cl w T zw : z = C cl x cl + D cl w (38) 9

where A cl := A C + B C KC C, B cl := B wc, C cl := C zc + D z KC C and D cl := D zw. From the real bounded lemma e.g [12], [3], [8], the closed-loop system is stable and the H -norm of T zw is smaller that γ if and only if there exists a symmetric matrix Q > 0 such that A cl Q + QA T cl QCcl T B cl Ω Q := C cl Q γi D cl < 0 (39) Bcl T Dcl T γi The LMI (39) can be transformed with Π := diag(tn T, I, I) to Π T Ω Q Π := TN 1 A cl T N Q d + (T 1 N A cl T N Q d ) T C cl T N Q d γi < 0 (40) BclT T N T Dcl T γi Using the identities C C T N = C C and KC C Q d = Y C C we get TN 1 A cl T N Q d = TN 1 A C T N Q d + B C Y C C and C cl T N Q d = C zc T N Q d + D z Y C C. Collecting these results we can state the following theorem: Theorem 4 The system G given in(36) is stabilizable by a static output feedback gain and the H -norm of the corresponding closed-loop system, T zw, is smaller than γ if and only if there exist a block-diagonal symmetric matrix Q d > 0, partitioned as in (11), and matrices N R n y (n n y ) and Y R n u n y such that TN 1 A C T N Q d + TN 1 B C Y C C + (TN 1 A Cy T N Q d + TN 1 B C Y C C ) T C zc T N Q d + D z Y C C γi < 0(41) BwcT T N T Dcl T γi where T is a non-singular matrix such that CT = C C and where A C = T AT 1, B C = T B, B wc = T B w, C zc = C z T 1. (41) is an LMI whenever N is fixed. As for the stability problem, the matrix N must satisfy the conditions of theorem 2. For a plant satisfying the assumptions of theorem 3, we can always compute a sub-optimal H static output feedback gain using (41) with the matrix N given in theorem 3. 10

Design methods T zw T zw 2 optimal full-order (4th order) H regulator 10.08 1.054 K = [ 2.139, 4.347] T 14.65 2.99 min/max H 2 design: K = [ 1.6368, 4.9221] T 13.78 2.97 Table 1 Comparative results 6 Numerical example Let us consider the following state-space model of the longitudinal motion of a VTOL helicopter. This example has been used in [9] to illustrate the min/max algorithm. A minimal realisation of the augmented plant G is: 0.0366 0.0271 0.0188 0.4555 0.4422 0.1761 0.0482 1.0100 0.0024 4.0208 3.5446 7.5922 A := B := 0.1002 0.3681 0.7070 1.4200 5.5200 4.4900 0 0 1 0 0 0 [ ] C := 0 1 0 0, B w = I 4, C z = diag(i 2, 0 2 2 ), D zw = 0 4 4, D z = [0 2 2, I 2 ] T. In this example, it is easy to check that ÃC 22 is stable and CB = [3.5446 7.5922] is full row rank. Therefore we can solve (41) with N = (B C2 B T C 1 (B C1 B T C 1 ) 1 ) T = [ 0.0332, 0.764, 0] to get a sub-optimal H control law. The optimal value of γ for this problem is 10.08 when a full-order H regulator is considered. Solving (41) leads to the following sub-optimal H control law K = [ 2.139, 4.347] T for which γ = 14.65. The performance of the static output feedback gain K obtained here noniteratively, is close to the performance of the H 2 regulator designed in [9] which makes use of the non-linear and iterative optimisation min/max algorithm (see table 1). Also, we see that the static output feedback gain designed by our approach enforces quite good H attenuation when compared to the optimal full-order (4th order) H regulator. This example shows that solving the LMI sysytem (41) with the matrix N given in theorem 3 is indeed consevative but not by much in this case. 11

7 Conclusions In this paper, necessary and sufficient conditions for a given plant to be stabilisable by a static output feedback have been given. The conditions become a linear matrix inequality in the optimisation variables whenever a particular structure is assigned to a Lyapunov matrix. This structure is parametrized in terms of a rectangular matrix N. Necessary conditions on the matrix N have been given. For a special class of plants (the class of minimum phase plants with CB full row rank), LMI conditions are given to compute a static output feedback gain. Finally, the method has been extended to incorporate H performance. This results, for the class of plants considered, in a sub-optimal static H regulator found by non-iterative means. An example was given to illustrate the effectiveness of the method. Acknowledgements The authors would like to acknowledge financial support from the European Community and the UK Engineering and Physical Sciences Research Council. References [1] Packard A. and J. C. Doyle. The complex structured singular value. Automatica, 29(1):71 109, 1993. [2] R. E. Benton and D. Smith. Static output feedback stabilization with prescribed degree of stability. IEEE Trans. Aut. Cont., 43(10):1493 1496, October 1998. [3] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM. Studies in Applied Mathematics, 1994. [4] Y. Y Cao and Y. X. Sun. Static output feedback simultaneous stabilization: Ilmi approach. Int. Jour. of Cont, 70(5):803 814, 1998. [5] C. Davis, W. M. Kahan, and H. F. Weinberger. Norm-preserving dilations and their applications to optimal error bounds. SIAM J. Numerical Anal., 19(3):445 469, June 1982. [6] J. C. Geromel, P. L. D. Perez, and R. Souza. Output feedback stabilisation of uncertain systems through a min/max problem. IFAC World Congress, pages, 1993. [7] J. C. Geromel, R. Souza, and R. E. Skelton. Static output feedback controllers: Stability and convexity. IEEE trans. Aut. Cont., 43(1):, 1998. 12

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