true, and only partial information through the measured output is available. .... tation, we establish a new bounded real lemma for the closed-loop sy...

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Mixed H2 /H∞ Control of Discrete-Time Markovian Jump Systems via Static Output-Feedback Controllers ? Zhan Shu ∗ , James Lam ∗ , Junlin Xiong ∗∗ ∗

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. (E-mail: [email protected]) ∗∗ School of Information Technology and Electrical Engineering, University of New South Wales, Canberra ACT 2600, Australia. Abstract: This paper investigates the static output-feedback mixed H2 /H∞ control problem of discrete-time Markovian jump systems from a novel perspective. Unlike traditional methods, the closed-loop system is represented as an augmented form, in which input and gain-output matrices are decoupled. By virtue of the augmented representation, new characterizations on stochastic stability and H2 /H∞ performance of the closed-loop system are established in terms of matrix inequalities. Based on these, a sufficient condition with redundant matrices for the existence of the mode-dependent controller is proposed, and an iteration algorithm is given to solve the condition. An extension to the mode-independent case is provided as well. Keywords: Iteration, linear matrix inequality (LMI), Markovian jump systems, mixed H2 /H∞ control, static output-feedback. 1. INTRODUCTION Discrete-time Markovian jump linear systems (DMJLSs), modeled by a set of discrete-time linear systems with transitions among the models determined by a Markov chain taking values in a finite set, have appealed to a lot of researchers in the control community. This is due to their widespread applications to model various practical processes that experience abrupt changes in their structure and parameters, possibly caused by phenomena such as component failures or repairs, sudden environmental disturbances, changing subsystem interconnections. Stability of DMJLSs has been investigated thoroughly in Costa and Fragoso [1993], and the equivalence of different second moment stability has been established in Ji et al. [1991]. The linear quadratic optimal control problem for DMJLSs has been studied in Chizeck et al. [1986] and Costa and Fragoso [1995], and the filtering problem has been considered in Costa and Marques [2000]. Some results on the H2 and H∞ control problems are available in Costa and Marques [1998], Seiler and Sengupta [2003] and references therein. As for robust stability analysis, we refer readers to de Souza [2006] and references therein. More details on DMJLSs can be found in Costa et al. [2005]. In the literature mentioned above, it is often assumed that the system state is completely accessible to the controller. However, in practice, this assumption may not be always true, and only partial information through the measured output is available. Therefore, it is necessary to consider the more practical case that the system state is partially accessible, i.e., the output-feedback case. Although Costa ? This work was supported in part by RGC HKU 7029/05P.

978-1-1234-7890-2/08/$20.00 © 2008 IFAC

et al. [1997] proposed a non-convex cutting-plane algorithm based on the output structural constraint approach by Geromel et al. [1993] to solve the static output-feedback H2 control problem of DMJLSs, it is not easy to apply due to its nonlinearity and complexity. Apart from this work, there are very few results on output-feedback control of DMJLSs. This motivates us to seek an effective and easyto-use approach for output-feedback control of DMJLSs. In this paper, we investigate the mixed H2 /H∞ control problem of DMJLSs via static output-feedback controllers from a new point of view. The closed-loop system is represent as an augmented form with algebraic constraints. By virtue of the augmented representation, new characterizations on stochastic stability and H2 /H∞ performance of the closed-loop system are established in terms of matrix inequalities. Two advantages of our characterizations lie in the decoupling of the input matrix and the gain-output matrix, which enables us to parameterize the controller matrix by free matrix, and the separation of the Lyapunov matrix and the system matrix, which avoids imposing any constraint on the Lyapunov matrix when the controller matrix is parameterized. Based on these, a sufficient condition with redundant matrix variables for the existence of the mode-dependent controller is proposed, and an iterative algorithm is given to solve the condition. An extension to the mode-independent case is provided as well. When Markovian jumps disappear, the obtained results are also applicable to the usual deterministic discrete-time linear systems. Notation: Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y ) means that the matrix X − Y is positive-semidefinite

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(respectively, positive-definite). I is the identity matrix with appropriate dimension, and the superscript “T ” represents the transpose. |·| denotes the Euclidean norm for vectors and k·k denotes the spectral norm for matrices. E {·} stands for the mathematical expectation with respect to some probability measure. l2 refers to the space of mean square infinite vector sequences with r summable nP o ∞ 2 norm |f |2 = E . The symbol # is used k=0 |f (k)| to denote a matrix which can be inferred by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. PRELIMINARIES Consider the following class of discrete-time stochastic systems, denoted T : x(k + 1) = A(r (k))x(k) + B (r (k)) u (k) +Bw (r (k)) w (k) (1) (T ) : z (k) = C(r (k))x(k) + D (r (k)) u (k) y (k) = Cy (r (k))x(k) where x(k) ∈ Rn , u(k) ∈ Rl , y (k) ∈ Rm , z(k) ∈ Rp , w(k) ∈ Rq are the system state, control input, measured output, regulated output to be controlled and exogenous noise process, respectively, and A(r(k)) ∈ Rn×n , B(r(k)) ∈ Rn×l , Bw (r(k)) ∈ Rn×q , C(r(k)) ∈ Rp×n , D(r(k)) ∈ Rp×l , Cy (r(k)) ∈ Rm×n are the system matrices of the stochastic jumping process {r(k), k > 0}; the parameter r (k) represents a discrete-time, discrete-state Markov chain taking values in a finite set S = {1, 2, . . . , s} with transition probabilities Pr {r (k + 1) = j |r (k) = i } = πij , (2) Ps where πij ≥ 0, and for any i ∈ S, j=1 πij = 1. The processes w (k) and r(k) are mutually independent. To simplify the notation, M (r (k)) and MN (r (k)) will be denoted by Mr(k) and MN r(k) , respectively, and, for a set ˆ i denotes Ps πij Mj . of matrices Mi , i ∈ S, M j=1 Definition 1. (1) System T is said to be stochastically stable if, when ˜ (x0 , r0 ) > w(k) ≡ 0, u (k) ≡ 0, there exists a scalar M 0, for x (0) = x0 and r (0) = r0 , such that ( ν ) X 2 ˜ (x0 , r0 ). lim E |x(k)| x0 , r0 ≤ M ν→∞ k=0

(2) Assume that system T is stochastically stable. The H∞ norm of system T with x (0) ≡ 0 and u (k) ≡ 0, denoted as kT k∞ , is defined as kT k∞ , |z| supr0 ∈S sup06=w∈l2 |w|2 . 2 (3) The H2 norm of system T with x (0) ≡ 0 and 2 u (k) ≡ 0, denoted as kT k2 , is defined as kT k2 , Pq Ps 2 i=1 j=1 |zi,j |2 , where zi,j represents the output sequence generated by (1), i.e., (z (0) , z (1) , . . .), when (a) the input sequence is w = (w (0) , w (1) , . . .), where w (0) = ei , the unit vector formed by one at the ith position and zero elsewhere, and w (k) = 0, for k > 0. (b) r (0) = r (1) = j.

The static output-feedback controller under consideration is of the form u (k) = Kr(k) y(k). (3) When static mode-dependent controller (3) is applied to (1), the closed-loop system becomes x(k + 1) = Aclr(k) x(k) + Bwr(k) w (t) , (4) (Tcl ) : z (k) = Cclr(k) x (t) , where Aclr(k) = Ar(k) + Br(k) Kr(k) Cyr(k) , Cclr(k) = Cr(k) + Dr(k) Kr(k) Cyr(k) . Our goal is to design a controller in (3) such that system Tcl is stochastically stable and satisfies kTcl k∞ < γ∞ , kTcl k2 < γ2 , where γ∞ > 0 and γ2 > 0 are prescribed scalars. Since Kr(k) is embedded in the middle of two matrices, it is hard to parameterize it by matrix variables. Hence, our fundamental idea is to extract Kr(k) from the middle of two matrices. To this end, we view the input u (k) as a state T component and choose xT (k) uT (k) as a new state variable. Then the closed-loop system can be re-written as the following augmented form: Eξ (k + 1) = Ar(k) ξ (k) + Bwr(k) w (k) , z (k) = Cr(k) ξ (k) , where x (k) I 0 ξ (k) = , E= , u (k) 0 0 Ar(k) Br(k) Bwr(k) Ar(k) = , Br(k) = , Kr(k) Cyr(k) −I 0 Cr(k) = Cr(k) Dr(k) . An advantage of this augmented representation lies in the separation of Br(k) and Kr(k) Cyr(k) , which enables us to parameterize Kr(k) by matrix variables. It is noted that T if we choose xT (k) y T (k) as a new state variable, we may also obtain a similar augmented representation, which we call dual augmented representation. In this paper, we do not intend to present any results on dual augmented representation, due to the page length consideration, and further discussion on this issue will appear in our future work. In addition, many dynamic outputfeedback synthesis problems can be reformulated as a static output-feedback control design involving augmented plants. Therefore, the approach presented in this paper is applicable to the dynamic output-feedback case as well. We end this section by giving several lemmas, which will be useful in the sequel. Lemma 1. (Seiler and Sengupta [2003]). Assuming system Tcl is weakly controllable 1 , it is stochastically stable with kTcl k∞ < γ∞ if and only if there exist matrices Pi > 0 such that, for each i ∈ S, T Pi 0 Acli Bwi Acli Bwi Pˆi 0 − < 0. 2 Ccli 0 Ccli 0 0 γ∞ I 0 I 1

System Tcl is said to be weakly controllable with respect to w (k) if for every initial state/mode, (x0 , r0 ), and any final state/mode, (xf , rf ), there exists a finite time tc and an input w (k) such that Pr[x(tc ) = xf , r(tc ) = rf ] > 0.

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T T Lemma 2. (Costa et al. [1997], Costa and Marques [1998]). Acli Pˆ1i Acli − P1i + Ccli Ccli ATcli Pˆ1i Bwi Ψ = i T ˆ 2 T ˆ Assuming that system Tcl is stochastically stable, if Pi > 0 Bwi P1i Acli −γ∞ I + Bwi P1i Bwi is the unique solution of the following equations: T Acli Bwi Acli Bwi Pˆ1i 0 T ATcli Pˆi Acli − Pi + Ccli Ccli = 0, i ∈ S, = C 0 Ccli 0 0 I cli P s 2 T Pi Bwi . then kTcl k2 = i=1 trace Bwi P1i 0 − 2 Lemma 3. If system Tcl is stochastically stable, then, for 0 γ∞ I any Qi ∈ Rn×n , i ∈ S, there exists a unique solution < 0. Pi ∈ Rn×n , i ∈ S, such that Pi − ATcli Pˆi Acli = Qi . Moreover, if Q1i ≥ Q2i ≥ 0 (> 0, respectively) and Therefore, according to Lemma 1, system Tcl is stochastiP1i − ATcli Pˆ1i Acli = Q1i , P2i − ATcli Pˆ2i Acli = Q2i , then cally stable with kTcl k∞ < γ∞ . P1i ≥ P2i ≥ 0 (> 0, respectively). (Necessity) If system Tcl is stochastically stable with kTcl k∞ < γ∞ , then according to Lemma 1, there exist Lemma 3 is an analogue in the real number field of matrices P > 0 such that 1i Proposition 6 in Costa and Fragoso [1993]. Its proof can Ψi < 0. be conducted in a similar way, and thus omitted here. Now set P4i to be any positive definite matrices, P2i = 0, Q4i = Pˆ4i , and αi > 0 to be sufficiently large scalars such 3. NEW CHARACTERIZATIONS ON STOCHASTIC that STABILITY AND H2 /H∞ PERFORMANCE T ˆ T ˆ −zTi Ψ−1 i zi + Bi P1i Bi + Di Di − (2αi − 1) P4i < 0, (7) 3.1 Stochastic Stability and H∞ Performance (Bounded where T T Real Lemma) Acli Pˆ1i Bi + Ccli Di zi = T ˆ Bwi P1i Bi On the basis of the proposed augmented system represen- Then, directly manipulating together with (6), (7), and tation, we establish a new bounded real lemma for the Schur complement equivalence yields that closed-loop system in the following theorem. Theorem 1. Assuming that system Tcl is weakly controlΨi zi −T −T −1 lable, it is stochastically stable with kTcl k∞ < γ∞ , if and BiT Pˆ1i Bi + DiT Di T2−1 T1i Ω∞i = T1i T2 zTi T T only if there exist P1i = P1i , P4i = P4i , P2i , Q4i = QT4i , ˆ − (2αi − 1) P4i and scalars αi > 0 such that, for each i ∈ S, < 0. Ω∞i = ATi Pˆi Ai − E∞ Pi E∞ + Qi Li + LTi QTi < 0, (5) This completes the proof. where Remark 1. When the assumption of weak controllability is not satisfied, the necessity of Theorem 1 may be lost, Ai Bwi E 0 but the sufficiency still holds. For the subsequent synthesis, Ai = , Li = [ Ai 0 ] , E∞ = , Ci 0 0 γ∞ I we only need to use the sufficiency of Theorem 1, since the T T controller matrices are unknown before they are computed. 0 −αi Cyi KiT Q4i P1i P2i 0 . Pi = P2i P4i 0 > 0, Qi = 0 αi Q4i In the following theorem, we provide an equivalent char0 0 I 0 0 acterization of the bounded real lemma, which will play a key role in the subsequent controller synthesis. Proof: (Sufficiency) Define two nonsingular transformation T Theorem 2. (5) holds if and only if there exist P1i = P1i , matrices as follows: T T " # " # P = P , P , Q = Q , H , G , (ν = 1, 2, . . . , 6), and 4i 2i 4i νi νi 4i 4i I 0 0 I 0 0 scalars αi > 0 such that, for each i ∈ S, T1i = Ki Cyi I 0 , T2 = 0 0 I . Hi Ai + ATi HiT − E∞ Pi E∞ 0 0 I 0 I 0 T T Ai Gi − Hi < 0, +Qi Li + LTi QTi T Pre- and post-multiplying (5) by T2T T1i and its transpose T T ˆ G A − H P − G − G i i i i i i yields that (8) where Ai , Pi , Qi , Li , and E∞ are defined as in Theorem T T T2 T1i Ω∞i T1i T2 1, and T " # " # ATcli Pˆ1i Acli − P1i + Ccli Ccli ATcli Pˆ1i Bwi H1i 0 H2i G1i 0 G2i T ˆ 2 T ˆ = Bwi P1i Acli −γ∞ I + Bwi P1i Bwi Hi = H3i 0 H4i , Gi = G3i Q4i G4i . BiT Pˆ1i Acli − Pˆ2i Acli + DiT Ccli BiT Pˆ1i Bwi − Pˆ2i Bwi H5i 0 H6i G5i 0 G6i T ˆ T ˆT T Acli P1i Bi − Acli P2i + Ccli Di Proof: (Sufficiency) By pre- and post-multiplying (8) by T ˆ T ˆT and its transpose, we obtain (5) immediately. I ATi T Bwi P1i Bi T− BTwi P2i Bi Pˆ1i Bi − Bi Pˆ2i − Pˆ2i Bi (Necessity) If there exist P1i > 0, P2i = 0, P4i > 0, +Pˆ4i + DiT Di − 2αi Q4i Q4i = Pˆ4i , and sufficiently large αi > 0 such that < 0, (6) (5) holds, then, by simple manipulating and Schur comwhich implies that plement equivalence, we can obtain that (8) holds with

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Hνi = 0, (ν = 1, 2, . . . , 6) , G2i = G3i = G4i = G5i = 0, G1i = Pˆ1i , G6i = I. Remark 2. The major merit of the equivalent characterization is the separation of the Lyapunov matrices P1i and the system matrices, which avoids imposing any constraint on the Lyapunov matrices when Ki is parameterized. In addition, by following the idea proposed in de Oliveira et al. [1999], redundant matrices Hνi and Gνi are introduced to reduce the conservatism and to improve the solvability of the iterative calculation to be presented later. Remark 3. It should be pointed out that, without loss of generality, the matrices P4i and Q4i in Theorems 1 and 2 can be set to be mode-independent, i.e., P41 = P42 = . . . = P4s and Q41 = Q42 = . . . = Q4s , and the corresponding conditions are still necessary and sufficient. In view of this feature, it is easy to design a modeindependent controller for the case that the jump variable r(k) is not available without imposing any restriction on the Lyapunov matrices P1i , which may cause excessive conservatism.

Hence, the stochastic stability of system Tcl follows from (11) immediately. On one hand, it follows from (11) that there must exist some Fi ≥ 0 such that T ˆ 1i Acl = Ccli X1i − ATcli X Ccli + FiT Fi . (13) i On the other hand, by Lemma 2, s X 2 T kTcl k2 = trace Bwi Si Bwi , i=1

where T Si − ATcli Sˆi Acli = Ccli Ccli . (14) Therefore, from (13), (14) and Lemma 3, we obtain that X1i ≥ Si > 0, and thus

2

kTcl k2 =

s X

s X T T trace Bwi Si Bwi ≤ trace Bwi X1i Bwi

i=1

<

s X

i=1

trace (Λi ) < γ22 ,

i=1

where (12) is used.

3.2 H2 Performance

(Necessity) If system Tcl is stochastically stable kTcl k2 < γ2 , then there exist Z1i > 0 and Si > 0 Likewise, we first propose a new condition for H2 perfor- that mance, and then give an equivalent characterization. ATcli Zˆ1i Acli − Z1i < 0 Theorem 3. System Tcl is stochastically stable with kTcl k2 < s X T T 2 T γ2 if and only if there exist X1i = X1i , X4i = X4i , X2i , kTcl k2 = trace Bwi Si Bwi < γ22 , T W4i = W4i , Λi = ΛTi , and scalars βi > 0 such that, for i=1 each i ∈ S, where Si satisfying (14). Now, define s X trace (Λi ) < γ22 , (9) X1i = Si + εZ1i , i=1

with such (15) (16)

T Λi = Bwi X1i Bwi + δI,

T T Ω2i = Iup ATi Xˆi Ai Iup + Idn ATi Xi Ai Idn

−E2i Xi E2i + Wi Li + LTi WiT < 0, (10) where Ai and Li are defined as in Theorem 1, and T T 0 −βi Cyi KiT W4i X1i X2i 0 , Xi = X2i X4i 0 > 0, Wi = 0 βi W4i 0 0 I 0 0 E 0 I 0 0 0 Iup = n+l , Idn = , E2i = 1/2 . 0 0 0 Iq 0 Λi

where ε > 0 and δ > 0 are sufficiently small numbers such that

Proof: (Sufficiency) By pre- and post-multiplying (10) by T T2T T1i and its transpose, we obtain that

Then, it follows that

T Bwi X1i Bwi

< Λi .

s X T T trace Bwi trace Bwi Z1i Bwi Si Bwi + ε

i=1

+

i=1

s X

trace (δI)

i=1

< γ22 .

T Bwi X1i Bwi < Λi ,

T T2T T1i Ω2i T1i T2 T ˆ T Acli X1i Acli − X1i + Ccli Ccli 0 T = 0 Bwi X1i Bwi − Λi ˆ 1i Acli − X ˆ 2i Acli + DiT Ccli BiT X 0 T ˆ T ˆT T Acli X1i Bi − Acli X2i + Ccli Di 0 T T ˆ 1i Bi − BiT X ˆ 2i ˆ 2i Bi Bi X −X ˆ 4i + DT Di − 2βi W4i +X i < 0, which implies that

ˆ 1i Acl − X1i + C T Ccli < 0, ATcli X cli i

s X

s X

trace (Λi ) < γ22 .

(17)

i=1

Meanwhile, from (14) and (15), we have that T ˆ 1i Acl − X1i + Ccli ATcli X Ccli i = ε ATcli Zˆ1i Acli − Z1i

< 0.

(11) (12)

(18)

Combining (17)–(18), and following the same line as used in the proof of Theorem 1, we obtain that (9) and (10) hold. This completes the proof. Theorem 4. (9) and (10) hold if and only if there exist T T T X1i = X1i , X4i = X4i , X2i , W4i = W4i , Uνi , Vνi , (ν = 1, 2, . . . , 6), and scalars βi > 0 such that, for each i ∈ S,

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s X

trace (Λi ) < γ22 ,

(19)

i=1 T Vi Ai Iup + Iup ATi ViT T T Iup ATi UiT − Vi +Idn ATi Xi Ai Idn < 0, −E2i Xi E2i +Wi Li + LTi WiT T T Ui Ai Iup − Vi Xˆi − Ui − Ui (20) where Ai , Xi , Wi , Li , and E2i are defined as in Theorem 3, and " " # # V1i 0 V2i U1i 0 U2i Vi = V3i 0 V4i , Ui = U3i W4i U4i . V5i 0 V6i U5i 0 U6i

The proof can be conducted by following the same line as used in the proof of Theorem 2, and thus omitted here for brevity. 4. CONTROLLER SYNTHESIS We are now in a position to establish a sufficient condition for the existence of desired mode-dependent controllers. T T Theorem 5. If there exist P1i = P1i , P4i = P4i , P2i , Hνi , T T Gνi , X1i = X1i , X4i = X4i , X2i , Uνi , Vνi , (ν = 1, 2, . . . , 6), Mi , Li , Q4i = QT4i , Λi = ΛTi , and scalars αi > 0, βi > 0 such that, for each i ∈ S, T T P1i P2i X1i X2i > 0, > 0, (21) P2i P4i X2i X4i s X trace (Λi ) < γ22 , (22) i=1

ˆ Φ11i ˆ Φ 21i Φ ˆ 31i Φ∞i (αi , Mi ) = Φ ˆ 41i Φ ˆ 51i ˆ 61i Φ

# ˆ 22i Φ ˆ 32i Φ ˆ 42i Φ ˆ 52i Φ ˆ 62i Φ

# # ˆ Φ33i ˆ 43i Φ ˆ 53i Φ ˆ 63i Φ

# # # ˆ 44i Φ ˆ 54i Φ ˆ 64i Φ

# # # # ˆ 55i Φ ˆ 65i Φ

ˇ 11i Φ Φ ˇ 21i Φ ˇ 31i Φ2i (βi , Mi ) = Φ ˇ 41i ˇ 51i Φ ˇ 61i Φ

# ˇ 22i Φ ˇ 32i Φ ˇ 42i Φ ˇ 52i Φ ˇ 62i Φ

# # ˇ 33i Φ ˇ 43i Φ 0 ˇ 63i Φ

# # # ˇ 44i Φ ˇ 54i Φ ˇ 64i Φ

# # # # ˇ 55i Φ ˇ 65i Φ

# # # <0 # # ˆ 66i Φ (23) # # # <0 # # ˇ 66i Φ (24)

where T T ˆ 11i = H1i Ai + H2i Ci + ATi H1i Φ + CiT H2i − P1i T T 2αi MiT Q4i Mi − 2αi Cyi Li Mi − 2αi MiT Li Cyi ,

ˇ 11i = V1i Ai + V2i Ci + AT V T + C T V T − X1i Φ i 1i i 2i

ˆ 32i = H5i Bi + H6i Di + B T H T , Φ ˇ 32i = V5i Bi + V6i Di , Φ wi 3i T T T ˆ 33i = H5i Bwi + Bwi ˇ 33i = Bwi Φ H5i − γ∞ I, Φ X1i Bwi − Λi , T ˇ ˆ 41i = G1i Ai + G2i Ci − H1i Φ , Φ41i = U1i Ai + U2i Ci − V1iT , T ˇ ˆ 42i = G1i Bi + G2i Di − H2i Φ , Φ42i = U1i Bi + U2i Di − V2iT ,

ˆ 43i = G1i Bwi − H T , Φ ˇ 43i = −V T , Φ 3i 3i T ˇ T ˆ ˆ ˆ Φ44i = P1i − G1i − G1i , Φ44i = X1i − U1i − U1i , ˆ 51i = G3i Ai + G4i Ci + Li Cyi , Φ ˇ 51i = U3i Ai + U4i Ci + Li Cyi , Φ ˆ 52i = G3i Bi + G4i Di − Q4i , Φ ˇ 52i = U3i Bi + U4i Di − Q4i , Φ ˆ 53i = G3i Bwi , Φ ˆ 54i = Pˆ2i − G3i , Φ ˇ 54i = X ˆ 2i − U3i , Φ ˆ 55i = Pˆ4i − 2Q4i , Φ ˇ 55i = X ˆ 4i − 2Q4i , Φ T ˇ ˆ 61i = G5i Ai + G6i Ci − H4i Φ , Φ61i = U5i Ai + U6i Ci − V4iT , ˆ 62i = G5i Bi + G6i Di − H T , Φ ˇ 62i = U5i Bi + U6i Di − V T , Φ 5i 5i T ˇ T ˆ Φ63i = G5i Bwi − H6i , Φ63i = −V6i , T ˆ 64i = −G5i − GT2i , Φ ˇ 64i = −U5i − U2i Φ , T ˇ T ˆ Φ65i = −G4i , Φ65i = −U4i ,

ˆ 66i = I − G6i − GT , Φ ˇ 66i = I − U6i − U T , Φ 6i 6i then, a mode-dependent control law u (k) = Q−1 (25) 4i Li y (k) exists, and makes the closed-loop system stochastically stable with kTcl k2 < γ2 and kTcl k∞ < γ∞ . Proof: It follows from Theorems 2 and 4 that a desired control law exists if (8), (19), and (20) hold. For the purpose of parameterization, Q4i in Theorem 2 and W4i in Theorem 4 can be set to be equal without loss of generality, i.e., Q4i = W4i , for i ∈ S. By expanding (8) and (20), and noting that T T −2Cyi KiT Q4i Ki Cyi ≤ −2 Cyi KiT QT4i Mi −2MiT (Q4i Ki Cyi ) + 2MiT Q4i Mi , we obtain that (8), (19) and (20) hold if (22)–(24) hold, where the parameterization Li = Q4i Ki is used. When αi , βi , and Mi are fixed, (23) and (24) become strict LMIs, which could be verified easily by conventional LMI solver. According to the proof of Theorems 1 and 3, the larger the αi and βi , the higher the reduction in conservatism of (23) and (24). If (21)–(24) do not hold for sufficiently large αi > 0 and βi > 0, it is plausible to conclude that a desired controller does not exist. As a matter of fact, when Mi = Q−1 4i Li Cyi , the left sides of (23) and (24) are monotonic decreasing matrix functions with (1) (2) respect to αi and βi , respectively, i.e., for αi > αi and (1) (2) βi > βi ,

T T 2βi MiT Q4i Mi − 2βi Cyi Li Mi − 2βi MiT Li Cyi ,

(1) (2) −1 Φ∞i αi , Q−1 4i Li Cyi ≤ Φ∞i αi , Q4i Li Cyi , (1) (2) −1 Φ2i βi , Q−1 L C ≤ Φ β , Q L C i yi 2i i yi . 4i i 4i

ˆ 21i = H3i Ai + H4i Ci + B T H T + DT H T + 2αi Li Cyi , Φ i 1i i 2i ˇ 21i = V3i Ai + V4i Ci + BiT V1iT + DiT V2iT + 2βi Li Cyi , Φ T T ˆ 22i = H3i Bi + H4i Di + BiT H3i Φ + DiT H4i − 2αi Q4i , T T T T ˇ 22i = V3i Bi + V4i Di + B V + D V − 2βi Q4i , Φ i

3i

i

4i

T T ˇ ˆ 31i = H5i Ai + H6i Ci + Bwi Φ H1i , Φ31i = V5i Ai + V6i Ci ,

Hence, we can set αi and βi to be large values. The remaining problem is how to select Mi . It can be seen from the proof of Theorem 3 that the left sides of (23) and (24), Φ∞i (αi , Mi ) and Φ2i (αi , Mi ), achieve their minima only

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17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

if Mi = Q−1 4i Li Cyi . Therefore, we adopt a simple iterative algorithm to solve the condition of Theorem 5. Algorithm: (1) Set ν = 1 and αi , βi to be sufficiently large values (for example, αi = βi = 104 , for each i ∈ S). Select initial (ν) values Mi , i = 1, 2, . . . , s, such that system T with (ν) u (k) = Mi x (k) (denoted as Tsf ) is stochastically stable with kTsf k2 < γ2 or kTsf k∞ < γ∞ . (ν) (2) For fixed αi , βi , and Mi , solve the following convex (ν) (ν) optimization problem with respect to Li , Q4i , (ν) (ν) (ν) (ν) (ν) (ν) P1i > 0, P4i > 0, P2i , X1i > 0, X4i > 0, X2i , (ν) (ν) (ν) (ν) Hτ i , Vτ i , Gτ i , Uτ i , (τ = 1, 2, . . . , 6). Minimize γ (ν) subject to, for each i ∈ S, (21), (22) and (ν) Φ∞i αi , Mi < γ (ν) I, (ν) Φ2i αi , Mi < γ (ν) I. (ν)

If a γltz ≤ 0 is found during solving the convex optimization problem, then the system is outputfeedback stabilizable, and a controller law can be obtained as (25). STOP. (ν) (3) Denote γ∗ as the optimal value of γ (ν) . If (ν) (ν−1) γ∗ − γ∗ ≤ δ, where δ is a prescribed tolerance, then go to next step, (ν+1) else update Mi as −1 (ν) (ν+1) (ν) Li Cyi , Mi = Q4i and set ν = ν + 1, then go to Step 2. (4) A desired control law may not exist. STOP. Remark 4. The convergence of the iteration is not guaranteed. However, it can be shown easily that the sequence (ν) (ν) γ∗ is monotonic decreasing with respect to ν, i.e., γ∗ ≤ (ν−1) (ν) γ∗ . If γ∗ does not converge to a positive number, (ν) then, after a sufficiently large number of iterations, γ∗ will always be negative, which means that the system is output-feedback stabilizable. Therefore, the case that the iteration is non-convergent is trivial. (1)

Remark 5. Initial values Mi are H2 or H∞ statefeedback controller matrices, which can be found by existing approaches Ji et al. [1991], Costa et al. [1997]. It should be pointed out that the optimum of the converged value (∞) (1) γ∗ is affected by the initial values Mi , αi , and βi , and (1) the optimization of Mi , αi , and βi will be investigated in the future. By setting P4i , X4i , and Q4i to be mode-independent, as stated in Remark 3, we give a sufficient condition for the existence of mode-independent controllers. T Theorem 6. If there exist P1i = P1i , P4 , P2i , Hνi , Gνi , T X1i = X1i , X4 , X2i , Uνi , Vνi , (ν = 1, 2, . . . , 6), Mi , L, Q4 , and scalars αi > 0, βi > 0 such that, for each i ∈ S, (21)–(24) hold, then a mode-independent control law u (k) = Q−1 4 Ly (k) exists, and makes the closedloop system stochastically stable with kTcl k2 < γ2 and kTcl k∞ < γ∞ .

5. CONCLUSION The mixed H2 /H∞ control problem of discrete-time Markovian jump systems via static output-feedback controllers has been solved by employing an augmented system representation. New characterizations on stochastic stability and H2 /H∞ performance of the closed-loop system are established in terms of the new representation and the matrix inequality technique. Based on these new results, a sufficient condition with redundant matrix variables for the existence of the mode-dependent controller is proposed, and an iterative algorithm is given to solve the condition. An extension to the mode-independent case is provided as well. REFERENCES H. J. Chizeck, A. S. Willsky, and D. Castanon. Discretetime Markovian-jump linear quadratic optimal-control. Int J. Control, 43(1):213–231, January 1986. O. L. V. Costa and M. D. Fragoso. Stability results for discrete-time linear-systems with Markovian jumping parameters. Journal of Mathematical Analysis and Applications, 179(1):154–178, October 1993. O. L. V. Costa and M. D. Fragoso. Discrete-time LQoptimal control problems for infinite Markov jump parameter systems. IEEE Trans. Automatic Control, 40 (12):2076–2088, December 1995. O. L. V. Costa and R. P. Marques. Robust H2 -control for discrete-time Markovian jump linear systems. Int J. Control, 73(1):11–21, January 2000. O. L. V. Costa and R. P. Marques. Mixed H2 /H∞ -control of discrete-time Markovian jump linear systems. IEEE Trans. Automatic Control, 43(1):95–100, January 1998. O. L. V. Costa, J. B. R. Do Val, and J. C. Geromel. A convex programming approach to H2 control of discretetime Markovian jump linear systems. Int J. Control, 66 (4):557–579, March 1997. O. L. V. Costa, M. D. Fragoso, and R. P. Marques. Discrete-Time Markov Jump Linear Systems. Springer, London, 2005. M. C. de Oliveira, J. Bernussou, and J. C. Geromel. A new discrete-time robust stability condition. Syst. Control Lett., 37(4):261–265, July 1999. C. E. de Souza. Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems. IEEE Trans. Automatic Control, 51(5):836–841, May 2006. J. C. Geromel, P. L. D. Peres, and S. R. Souza. Convex analysis of output feedback structural constraints. In Proceedings of the 32th Conference on Decision and Control, volume 2, pages 1363–1364, December 1993. Y. Ji, H. J. Chizeck, X. Feng, and K. A. Loparo. Stability and control of discrete-time jump linear-systems. Control-Theory and Advanced Technology, 7(2):247–270, June 1991. P. Seiler and R. Sengupta. A bounded real lemma for jump systems. IEEE Trans. Automatic Control, 48(9): 1651–1654, September 2003.

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