Proceedings of 2008 CACS International Automatic Control Conference National Cheng Kung University, Tainan, Taiwan, Nov. 21-23, 2008

Improved H∞ Control for Discrete-Time Polytopic Systems Chia-Po Wei

Li Lee

Department of Electrical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan Email: [email protected]

Department of Electrical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan Email: [email protected]

Abstract—In this paper, we consider H∞ performance analysis and synthesis of discrete-time polytopic systems. A sufficient LMI condition is proposed to ensure the H∞ performance of a polytopic system with the help of a parameter-dependent Lyapunov function depending linearly on the parameters. Our result contains a related result by de Oliveira et al. and quadratic H∞ performance as special cases by putting additional constraints on LMI variables. Based on the analysis result, a state-feedback controller is designed such that the closed-loop system meets the H∞ performance. Finally, the effectiveness of the proposed method is illustrated by a numerical example. Keywords—H∞ control, Polytopic systems, Robust performance synthesis, Linear matrix inequality.

I. I NTRODUCTION It is well known that stability and performance analysis of linear systems affected by real structured parametric uncertainty are widely studied difficult problems in robust control. Quadratic stability has been the most popular approach for robust stability analysis (Ch 5.1 of [1]). But it is known that quadratic stability is conservative. There has been much work done on reducing the conservatism by introducing parameter-dependent Lyapunov functions, see [2] and references therein. LMI conditions of increasing precision for robust stability analysis, tending to necessity as the degree of the polynomial increases, have been proposed in [3] for affine uncertain continuous-time systems and in [4] for continuous-time and discrete-time polytopic systems. H∞ performance analysis for both discrete-time and continuous-time polytopic systems is addressed in [5], [6]. The tightness of the upper bound to the H∞ performance of a polytopic system is established in [7]. Although robust stability and performance analysis have been well studied in the literature, this is not the case for controller design. Designing robust controllers via parameter-dependent Lyapunov functions (PDLFs) is difficult since this often leads to nonconvex optimization problems. It is found that introducing additional variables to separate product terms of system matrices and the Lyapunov matrix is beneficial for controller design via PDLFs [8], [9]. The stabilization problem via state feedback for discrete-time polytopic systems is addressed in [8]. [9] This work was supported by National Science Council of Taiwan, R.O.C., under grant no. NSC 96-2221-E-110-087-MY2.

extends the result of [8] to robust H∞ performance problems. Corresponding results for continuous-time polytopic systems are addressed in [10], [11]. In this paper, we consider H∞ performance analysis and synthesis of discrete-time polytopic systems. A sufficient LMI condition is proposed to ensure the H∞ performance of a polytopic system. It is well known that once a problem is formulated as LMIs, then it can be solved efficiently via interior-point methods [1]. Our result contains Theorem 4 of [9] and quadratic H∞ performance as special cases by putting additional constraints on LMI variables. Based on the analysis result, a state-feedback controller is designed such that the closed-loop system meets the H∞ performance. The following notations are used in the sequel. The set of n × m real matrices is denoted by Rn×m . The set of n × n symmetric matrices is denoted by Sn . For a matrix A, its transpose is denoted as AT . For a square matrix A, sym (A) stands for the symmetric matrix A + AT . For a symmetric matrix A and a matrix B, (◦)T AB means B T AB. II. P RELIMINARIES Consider the linear discrete-time system x(k + 1) = Ax(k) + Bw(k) z(k) = Cx(k) + Dw(k)

(1)

where x is the state, w is the disturbance input, z is the control output, and A, B, C, D are given matrices. Denote the transfer function from w to z by G(z) = C(zI − A)−1 B + D. The H∞ norm of a stable transfer function G is

kG(z)k∞ = sup σ ¯ G ejω ω∈[0,2π]

where σ ¯ (·) represents the maximal singular value of a matrix. The H∞ performance problem for system (1) is to determine whether system (1) is stable and kG(z)k∞ < γ for a given positive number γ. The next lemma provides a necessary and sufficient LMI condition for the H∞ performance problem for system (1). Lemma 1 (Lemma 7.1.2 of [12]). Consider system (1) and let γ be a positive number. Then the following statements are equivalent. 1) The matrix A is stable and kG(z)k∞ < γ.

Proceedings of 2008 CACS International Automatic Control Conference National Cheng Kung University, Tainan, Taiwan, Nov. 21-23, 2008

2) There exists a positive definite matrix P such that     T  A B P 0 A B P 0 − < 0. (2) C D 0 I C D 0 γ2I To facilitate robust performance analysis and synthesis in the next section, we will introduce additional variables to (2) by means of the elimination lemma (p. 32 of [1] or Theorem 2.3.12 of [12]). Lemma 2 (Elimination Lemma). Let Υ ∈ Sn , U ∈ Rq×n , and V ∈ Rp×n be given. Denote by NU and NV any matrices whose columns form bases of the null spaces of U and V , respectively. There exists a matrix X ∈ Rp×q such that

if and only if the following two conditions hold NUT ΥNU < 0 or U T U > 0, NVT ΥNV < 0 or V T V > 0. holds if and only if P and matrices F , (.)T (.)T −I D

 (.)T (.)T   < 0. (.)T  −γ 2 I (3)

Proof: Inequality (3) can be written as    −I    A    I 0 0 0     Ψ + sym   0  F L 0 I 0 0  < 0 C

where

By Lemma  0  0 (◦)T Ψ  I 0



P 0 Ψ= 0 0

0 −P BT 0

0 B −I D

Lemma 3 provides an equivalent condition for the LMI condition in Lemma 1. Two important features of this lemma are the separation between the Lyapunov matrix P and system matrices A, B, C, and D, and the introduction of two additional variables F and L. As we shall see in the next section, the first feature enables one to introduce parameter-dependent Lyapunov functions easily, while the second feature allows more freedom in designing robust controllers. III. M AIN R ESULTS Consider the uncertain linear discrete-time system

Υ + U T X T V + V T XU < 0

Lemma 3. Statement 2) of Lemma 1 there exists a positive definite matrix L such that  P − F − FT (.)T T T  AF − LT L A + AL − P   0 BT CF CL

which, by Schur complement with respect to the (3,3)block, is equivalent to (2).

 0 0  . DT  −γ 2 I

2, the last inequality holds if and only if    T A CT 0 0  0 0 0  < 0.  < 0 and (◦)T Ψ  I  0 0 I 0 0 I 0 I (4) Next, we’ll prove the two inequalities in (4) hold if and only if (2) holds. The first inequality in (4) is   −I DT < 0 or DDT − γ 2 I < 0 D −γ 2 I which is the (2,2)-block of (2). The second inequality in (4) is   AP AT − P AP C T B  CP AT CP C T − γ 2 I D  < 0 BT DT −I

x(k + 1) = A(α)x(k) + B(α)w(k) + Bu (α)u(k) (5) z(k) = C(α)x(k) + D(α)w(k) + Du (α)u(k) where x is the state, u is the control input, w is the disturbance input, z is the control output, and the system matrices are given by     X N Ai Bi Bui A(α) B(α) Bu (α) αi = Ci Di Dui C(α) D(α) Du (α) i=1 (6) where Ai ∈ Rn×n , Bi ∈ Rn×q , Bui ∈ Rn×p , Ci ∈ Rm×n , Di ∈ Rm×q , Dui ∈ Rm×p are given, i = 1, . . . , N , and α ∈ RN belongs to the unit simplex ) ( N X N αi = 1 . Γ = α ∈ R : αi ≥ 0, i=1

For the unforced system (5) (u = 0), denote the transfer function from w to z by G(z, α) = C(α)(zI − A(α))−1 B(α) + D(α). In view of Lemma 1 and Lemma 3, we immediately have the following result. Lemma 4. Consider system (5) with u = 0. 1) The matrix A(α) is stable and kG(z, α)k∞ < γ for all α ∈ Γ. 2) There exist matrix-valued functions P (α) : RN → Sn , F (α) : RN → Rn×n , and L(α) : RN → Rn×n such that   Π11 (α) (.)T (.)T (.)T  Π21 (α) Π22 (α) (.)T (.)T   <0 T  0 B (α) −I (.)T  C(α)F (α) C(α)L(α) D(α) −γ 2 I (7) for all α ∈ Γ, where Π11 (α) = P (α) − F (α) − F T (α), Π21 (α) = A(α)F (α) − LT (α), Π22 (α) = LT (α)AT (α) + A(α)L(α) − P (α).

The next theorem proposes a sufficient LMI condition to ensure that the unforced system (5) satisfies the robust H∞ performance.

Proceedings of 2008 CACS International Automatic Control Conference National Cheng Kung University, Tainan, Taiwan, Nov. 21-23, 2008

Theorem 1. Consider system (5) with u = 0. If there exist symmetric positive definite matrices Pi , i = 1, 2, . . . , N , and matrices F and L such that   Pi − F − F T (.)T (.)T (.)T  Ai F − LT LT ATi + Ai L − Pi (.)T (.)T  <0   0 BiT −I (.)T  Ci F Ci L Di −γ 2 I (8) for i = 1, 2, . . . , N , then A(α) is stable and kG(α, z)k∞ < γ for all α ∈ Γ. Proof: Multiplying (7) by αi and sum we obtain  P (α) − F − F T (.)T (.)T (.)T  A(α)F − LT Φ(α) (.)T (.)T   < 0 (9)  T  0 B(α) −I (.)T  C(α)F C(α)L D(α) −γ 2 I PN where P (α) = i=1 αi Pi and 

T

T

Φ(α) = L A(α) + A(α)L − P (α).

The proof is then finished in view of Lemma 4. Note that (8) holds for i = 1, 2, . . . , N if and only if (9) holds for all α ∈ Γ. The condition in Theorem 1 is sufficient because P (α) in (9) is a linear function of α and F and L in (9) are constant matrices. In Lemma 4, P (α), F (α), and L(α) can be any functions of α. However, in view of Lemma 1 and Lemma 3, we see that Theorem 1 is necessary when system (5) has only one vertex. Next, we show Theorem 1 is less conservative than Theorem 4 of [9]. To do this, post- and pre-multiplying (8) by   0 I 0 0 −I 0 0 0    0 0 I 0 0 0 0 −I

and its transpose, respectively, we obtain  T T L Ai + Ai L − Pi (.)T (.)T T T  −F T ATi + L P − F − F (.) i  T  −Bi 0 −I Ci L −Ci F −Di

 (.)T (.)T   < 0. (.)T  −γ 2 I (10) If the LMI condition in Theorem 4 of [9] holds, then (10) holds with L = 0. In other words, Theorem 1 reduces to Theorem 4 of [9] by letting L = 0. Theorem 1 also contains quadratic H∞ performance (Corollary 2 of [9]) as a particular case. If the quadratic H∞ performance test in Corollary 2 of [9] holds, then (10) holds with L = 0 and F = Pi = P , i = 1, 2, . . . , N . The closed-loop system for system (5) with a statefeedback controller u = Kx is x(k + 1) = Acl (α)x(k) + B(α)w(k) z(k) = Ccl (α)x(k) + D(α)w(k) where Acl (α) = A(α) + Bu (α)K and Ccl (α) = C(α) + Du (α)K. Define Gcl (α, z) = Ccl (α)(zI − Acl (α))−1 B(α) + D(α).

Based on Theorem 1, the next theorem proposes a sufficient condition to design a state-feedback controller u = Kx such that the closed-loop system of (5) satisfies the robust H∞ performance. Theorem 2. Consider system(5). If there exist symmetric positive definite matrices Pi , i = 1, 2, . . . , N , matrices F and Y , and a scalar τ such that   T Pi − F − F T (.)T (.)T (.)   Aˆ − τ F T sym τ Aˆi − Pi (.)T (.)T    i <0   0 BiT −I (.)T  Cˆi τ Cˆi Di −γ 2 I (11) for i = 1, 2, . . . , N , where Aˆi = Ai F + Bui Y,

Cˆi = Ci F + Dui Y

then there exists a state-feedback controller u = Kx with K = Y F −1 such that Acl (α) is stable and kGcl (α, z)k∞ < γ for all α ∈ Γ. Proof: The (1,1)-block of (11) reads Pi −F −F T < 0 which implies −F − F T < 0 since Pi is positive definite. Hence, F is nonsingular. Substituting Y = KF into (11), we obtain   Pi − F − F T (.)T
(.)T (.)T  A¯i F − τ F T sym τ A¯i F − Pi (.)T (.)T  <0   0 BiT −I (.)T  C¯i F τ C¯i F Di −γ 2 I where

A¯i = Ai + Bui K,

C¯i = Ci + Dui K.

The proof is then finished in view of Theorem 1 by letting L = τF . Note that in Theorem 1, Ai F and Ai L appear in (2,1)block and (2,2)-block of (10), respectively. Because both F and L are LMI variables, an application of Theorem 1 with simple changes of variables to state-feedback controller design will lead to a nonconvex optimization problem. A simple remedy is to let F and L be the same LMI variable, but this introduces conservatism. As seen in the proof of Theorem 2, our approach is to let L = τ F where τ is treated as a tuning parameter. Although our approach still introduces conservatism, it is alway better than letting L = F . To write the sufficient condition in Theorem 2 as LMIs, the value of τ has to be specified before solving LMIs. The optimal H∞ performance γ varies with the choice of τ . How to find a scalar τ minimizing γ results in a one-dimensional optimization problem which can be solved by various algorithms such as the golden section search method (p.293 of [13]), the parabolic interpolation (p. 299 of [13]), the Nelder-Mead simplex method (p. 162 of [14]), etc. The last method is implemented by fminsearch from the Optimization Toolbox of Matlab.

Proceedings of 2008 CACS International Automatic Control Conference National Cheng Kung University, Tainan, Taiwan, Nov. 21-23, 2008 30

quadratic performance. three methods are  K1 = − 9.2137  K2 = − 8.9932  K3 = − 7.9443

20

Magnitude (dB)

10

0

−10

−20

−30

The controllers obtained by the  0.4710 1.0241 1.0197  0.6415 0.9043 0.8854  0.6643 0.7373 0.7136

respectively. Let Gcl (z, δ) denote the closed-loop transfer function for Σ1 with u = K3 x. Since the H∞ norm of Gcl (z, δ) is less than 23.7545 for all admissible δ, we have
Gcl ejω , δ < 23.7545. sup ω∈[0,2π], δ∈δ

−40 −4 10

Fig. 1.

−3

10

−2

−1

10 10 Frequency (rad/sec)

0

10

1

10

Bode plots of Gcl (z, δ) for 125 samples of admissible δ’s

Hence, the peak values of Bode plots of Gcl (z, δ) for all admissible δ must be less than 20 log(23.7545) = 27.5149. The Bode plots of Gcl (z, δ) for 125 samples of admissible δ’s are shown in Fig. 1. V. C ONCLUSION

Note that the aforementioned algorithms do not need to calculate the gradient of γ with respect to τ , which is difficult to obtain analytically. IV. N UMERICAL E XAMPLE In this section, we use an example to demonstrate the effectiveness of the result proposed in the previous section. The nominal model of this example is borrowed from [15]. Consider the following uncertain discrete-time system, denoted by Σ1 ,   1.45 0.2 0 0  0 0.4 0 0.2   x(k) x(k + 1) =   1 0.2 1.1 0.75  0 −1 0 0.4 + δ1     0.6 0 0  1 + δ2     + 0.6 w(k) +  0  u(k) 0 1 + δ3   z(k) = 1 0 0 1 x(k) + w(k) + u(k)

where −0.1 ≤ δ1 ≤ 0.1, 0 ≤ δ2 ≤ 0.5, and 0 ≤ δ3 ≤ 0.5. Let δ = (δ1 , δ2 , δ3 ) and δ be the set of all admissible δ. The objective is to find a controller u = Kx which minimizes the H∞ norm γ of the transfer function from w to z in the face of uncertain parameters. Three methods are applied to solve this problem: quadratic H∞ performance, Theorem 6 of [9], and Theorem 2. The tuning parameter τ in Theorem 2 is determined by the procedure mentioned in the remark below Theorem 2. The simulation results are shown in the following table. Method quadratic performance de Oliveira Theorem 2

γ 112.1513 28.7799 23.7545 (τ = −0.49798)

The best result is given by Theorem 2 which represents an improvement in the H∞ performance of 78.82% over

We have proposed an improved characterization of H∞ performance for discrete-time polytopic systems. Thanks to the introduction of additional variables and the separation between the Lyapunov matrix and system matrices, H∞ control of discrete-time polytopic systems can be formulated as convex optimization problems with a finite number of LMIs. The proposed LMI conditions perform well when compared to the related results in the literature, as illustrated by the numerical example. R EFERENCES [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, 1994. [2] D. Peaucelle, Y. Ebihara, D. Arzelier, and T. Hagiwara, “General polynomial parameter-dependent Lyapunov functions for polytopic uncertain systems,” in International Symposium on Mathematical Theory of Networks and Systems, Kyoto, 2006. [3] P.-A. Bliman, “A convex approach to robust stability for linear systems with uncertain scalar parameters,” SIAM J. Control Optim., vol. 42, no. 6, pp. 2016–2042, 2004. [4] R.C.L.F. Oliveira and P.L.D. Peres, “Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. Automat. Contr., vol. 52, no. 7, pp. 1334–1340, 2007. [5] P.J. de Oliveira, R.C.L.F. Oliveira, V.J.S. Leite, V.F. Montagner, and P.L.D. Peres, “H∞ guaranteed cost computation by means of parameter-dependent Lyapunov functions,” Automatica, vol. 40, pp. 1053–1061, 2004. [6] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter dependent Lyapunov functions for robust H∞ performance analysis,” in 16th IFAC world congress on automatic control, Prague, Czech Republic, 2005. [7] G. Chesi, “Establishing tightness in robust H∞ analysis via homogeneous parameter-dependent Lyapunov functions,” Automatica, vol. 43, pp. 1992–1995, 2007. [8] M.C. de Oliveira, J. Bernussou, and J.C. Geromel, “A new discretetime robust stability condition,” Syst. Contr. Lett., vol. 37, pp. 261– 265, 1999. [9] M.C. de Oliveira, J.C. Geromel, and J. Bernussou, “Extended H2 and H∞ norm characterizations and controller parametrizations for discrete-time systems,” Int. J. Contr., vol. 75, no. 9, pp. 666– 679, 2002. [10] Y.-Y. Cao and Z. Lin, “A descriptor system approach to robust stability analysis and controller synthesis,” IEEE Trans. Automat. Contr., vol. 49, no. 11, pp. 2081–2084, 2004.

Proceedings of 2008 CACS International Automatic Control Conference National Cheng Kung University, Tainan, Taiwan, Nov. 21-23, 2008

[11] Y. He, M. Wu, and J.H. She, “Improved bounded-real-lemma representation and H∞ control of systems with polytopic uncertainties,” IEEE Trans. Circuits and Syst. II, vol. 52, no. 7, pp. 380–383, 2005. [12] R.E. Skelton, T. Iwasaki, and K.M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, New York: Taylor and Francis, 1997. [13] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, 1988. [14] D.P. Bertsekas, Nonlinear Programming. Athena Scientific, second edition, 1999. [15] L. Lu, L. Xie, and M. Fu, “Optimal control of networked systems with limited communication: A combined heuristic and convex optimization approach,” in Proc. 42th IEEE Conf. Decision Control, 2003, pp. 1194–1199.