Robust Stability and Stabilization of Discrete-time Infinite ... - IEEE Xplore

Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

TuC11.3

Robust Stability and Stabilization of Discrete-time Infinite Markov Jump Linear Systems Marcos G. Todorov and Marcelo D. Fragoso Abstract— This paper addresses the problems of robust stability and stabilization of discrete-time linear systems with Markov jump parameters taking values in a countably infinite set. We consider the problem of robustness against complex multiperturbations, which extends the setting currently encountered in the literature. By means of the introduction of blockdiagonal scaling techniques, we show how less conservative robust stability margins and robust controllers can be obtained by the solution of linear matrix inequality problems. The effectiveness of the main results is illustrated with a numerical example.

I. I NTRODUCTION In recent years, a great deal of interest has been devoted to linear systems whose dynamics is subjected to abrupt changes. Perhaps this is due to the fact that, while being an extension of a central topic in dynamic systems theory – that of linear systems – such mathematical models are powerful enough to capture a kind of phenomenon that is observed in an enormous quantity of processes in nature and industry. A particular class of such systems are those driven by a discrete-state Markov process, commonly known in the literature as Markov jump linear systems (MJLS). The importance of MJLS is well-illustrated by the great variety of applications which have been devised through the last four decades or so, comprising solar-power stations, robotics, communication systems, economics, and many other fields. We refer to [1] for a detailed discussion and list of references. Among the great variety of open problems in the MJLS literature, our interest here lies on those of robust stability and robust stabilization in discrete time. A sample of previous work on this subject can be found, for instance, in [2], [3], [4], [5]. In this paper, different from previous approaches, we study the robust stability of the discrete-time MJLS x(k + 1) = Aθ(k) x(k) with respect to multiperturbations of the form M    m m Bθ(k) ∆m (1) x(k + 1) = Aθ(k) + θ(k) Cθ(k) x(k), m=1

in which θ = {θ(k), k = 0, 1, 2, . . .} is a Markov chain taking values in a countably infinite state space. As indicated in, e.g., references [6], [7], this generalization of the usual norm-bounded uncertainty setting (which, in the MJLS case, This work was partially supported by the Brazilian National Research Council-CNPq, under the Grants No. 301740/2007-0, 470527/2007-2 and 141363/2007-0, and by FAPERJ under the Grants No. E-26/100.579/2007 and E-26/100.437/2009. The authors are with the National Laboratory for Scientific Computing - LNCC/CNPq, Av. Get´ulio Vargas 333, Petr´opolis, Rio de Janeiro, CEP 25651-070, Brazil. E-mail: [email protected] and [email protected].

978-3-9524173-9-3 © Copyright EUCA 2009

is of the form Ai  Ai + Bi ∆i Ci , i ∈ S) is suitable to describe a much broader class of problems. Besides, we assume that for each mode of operation θ(k) = i, the mth uncertainty block ∆m i can have either a full, diagonal, or scalar structure. Our approach is based on a version of the small-gain theorem, combined with scaling techniques which are also new in this context. One distinguishing aspect of the present work is that, different from the scaling optimization technique proposed, e.g., in [8], [9], [10], [5], here the scaling parameters are treated as additional decision variables in linear matrix inequalities, both in the robust stability and stabilization problems. Since scaling optimization is, in general, rather difficult to solve (in fact, a non-convex problem), we believe our approach is more suitable for computational purposes. Finally, it is important to mention that the results presented here remain original even in the usual full-block single perturbation setting (which was treated, for instance, in [5]). This paper is organized as follows. In sections II and III we provide the bare essentials of notation to be adopted in the rest of the paper, as well as the basic model and some preliminaries. In section IV we study the problem of robust stability of MJLS, followed by the robust control problem in V. We finish the paper with a numerical example, in section VI. II. N OTATION Let  ·  denote the euclidean norm in the complex nspace Cn . We write M(Cm , Cn ) as the Banach space of all n-by-m complex matrices, equipped with the standard induced matrix norm, ·. We define S := {1, 2, . . .} (unless otherwise stated) and denote the Kronecker product between two given complex matrices M and N by M ⊗ N (see, e.g., [11]). We also denote the transpose and conjugate transpose of such M by M ′ and M ∗ , respectively. Let us introduce the infinite dimensional Banach space Hm,n sup of all matrices of the form H = (H1 , H2 , . . .) with Hi ∈ M(Cm , Cn ) for every i ∈ S := {1, 2, . . .}, such that Hsup := supi∈S Hi  < ∞. We further write Hnsup in ˜ n+ place of Hn,n sup , and define Hsup as the set composed by all matrices H = (H1 , H2 , . . .) ∈ Hnsup such that Hi∗ = Hi ≥ εIn for all i ∈ S and some ε > 0 independent of i (here In stands for the n × n identity matrix). Accordingly, ˜ n− whenever −L ∈ H ˜ n+ . For short, we write that L ∈ H sup sup whenever there is ε > 0 such that Hi ≥ εI for all i ∈ S and supi∈S Hi  < ∞, we write that Hi ≫ 0 for all i ∈ S, the analogous for negative Li ≪ 0. We shall write

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that such Li ≪ 0 on N (Ji ) whenever there is ε > 0 such that Li ≤ −εI on N (Ji ) for all i ∈ S, where N (·) stands for the null space of a given complex matrix. Finally, we denote by diag(Ni ) an infinite-sized matrix with block diagonal entries N1 , N2 , . . ., and all the other entries equal to zero, and by diag(N 1 , . . . , N M ) the matrix with block diagonal entries N 1 , . . . , N M . Also, col(x1 , x2 , . . .) represents a column vector with real entries x1 , x2 , . . ., and c(H, L) := L∗ HL. We use the notations N = {0, 1, 2, . . .} and N∗ = {1, 2, . . .}. Concerning the random objects, we fix a complete probability space (Ω, F, P) carrying a filtration Fk ⊆ F, k ∈ N. Additionally, we denote by E(·) the mathematical expectation and define Ln2 (Ω) as the space of all second order random variables (Ω, F) → Cn .

Bearing (i)–(iii) in mind, for each ρ > 0 we define the sets   Di (ρ) = ∆i = (∆1i , . . . , ∆M (7) i ) ; ∆i  < ρ ,

in which ∆i  := max1≤m≤M ∆m i , which comprises all admissible disturbances of norm not larger than ρ > 0 which can affect the mode i ∈ S. Finally, we define the set of all such disturbances on system (3) as   D(ρ) = ∆ = (∆1 , ∆2 , . . .) ; ∆i ∈ Di (ρ), i ∈ S . (8) A. Scaling

In this paper, a leading role will be played by the scaling parameters {Sim , i ∈ S, 1 ≤ m ≤ M }, which we define with the aid of the following relations: ∆m =⇒ Sim = sm i is full, as in (4) i m ∆i is diagonal, as in (5) =⇒ Sim = Lm i ∆m =⇒ Sim = Mim , i is scalar, as in (6)

III. M ODEL SETTING AND PRELIMINARIES Consider in the stochastic basis (Ω, F, Fk , P) a homogeneous Markov process θ = {θ(k), k ∈ N}, with state space S ⊆ {1, 2, . . .}, such that   (2) P θ(k + 1) = j|θ(k) = i = pij  where pij ≥ 0 for i, j ∈ S, and j∈S pij = 1 for all i ∈ S. The initial condition θ0 : Ω → S is assumed to be a random variable with fixed distribution π0 in Ln2 (Ω). With θ defined in this way consider the following system, ⎧ M    ⎪ ⎨x(k + 1) = A m m + Bθ(k) ∆m θ(k) θ(k) Cθ(k) x(k) (3) m=1 ⎪ ⎩ n k ∈ N, x(0) = x0 ∈ L2 (Ω), ∈ M(C

bm i

n

, C ),

Cim

n

∈ M(C , C

cm i

1 ≤ m ≤ M,

),

1 M for given {b1i , . . . , bM i , ci , . . . , ci } ⊂ {1, 2, . . . , n}, i ∈ S. We assume that supi∈S max1≤m≤M Bim  < ∞, the analogous for {Cim , i ∈ S, 1 ≤ m ≤ M }. In this paper it is assumed that, for each m ∈ {1, . . . , M } and i ∈ S, the uncertainty ∆m i has either a full-block, diagonal, or scalar structure. That is, for each pair (m, i) ∈ {1, . . . , M } × S, one (and only one) of the following is true: m cm i , Cbi ) is a full block, such as (i) ∆m i ∈ M(C ⎛ δm (1,1) ··· δm (1,cm ) ⎞ i

⎝ ∆m i = dm i

i

.. .

..

δim (bm i ,1)

.

...

i

.. .

m δim (bm i ,ci )

dm i

⎠

(4)

m m (ii) ∆m , C ), when dm i ∈ M(C i := bi = ci , is a diagonal block, such as   δm (1) 0 i

∆m i

..

=

0

m

.

(5)

δim (dm i )

m

di m m (iii) ∆m , Cdi ), when dm i ∈ M(C i := bi = ci , is a scalar block, such as m . ∆m i = δi Idm i

(6)

(9)

is a real scalar, Lm is a diagonal matrix in where sm i m i dm di m m i M(C , C ), in which case dm = bm i i = ci , and Mi ∈ dm dm i i M(C , C ) is a full matrix. Notice that, whenever ∆m i is full, as in (4), the second diagonal block of (18) reads m m m m as −sm i Ibi . Otherwise, we trivially have −Si Ibi = −Si . We also define the arrays Si = (Si1 , . . . , SiM ),

S = (S1 , S2 , . . .).

(10)

Always bearing (9) in mind, we restrict our attention to uniformly positive scalings (that is, ones such that there is ε > 0 for which (Sim )∗ = Sim ≥ εI for all i ∈ S, and m ∈ {1, . . . , M }) which satisfy supi∈S max1≤m≤M Sim  < ∞. We denote the set of all such S as S.

where A = (A1 , A2 , . . .) ∈ Hnsup , M ∈ N∗ , and Bim

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IV. T HE ROBUST STABILITY PROBLEM In this paper we consider the following notion of robust stability for system (3), which extends a widely adopted terminology in the literature of MJLS. Definition 1: System (3) is said to be robustly stochastically stable (robustly SS) in D(ρ) if, for every ∆ ∈ D(ρ), we have ∞  E[x(k)2 ] < ∞ (11) k=0

satisfied for any initial condition x0 ∈ Ln2 (Ω) and initial distribution π0 .  One important fact to be observed at this point is that, by M M defining ˆbi = m=1 bm ˆi = m=1 cm i , c i , together with the ˆ matrices Bi ∈ M(Cbi , Cn ), Ci ∈ M(Cn , Ccˆi ), ⎡ 1⎤ Ci   Bi = Bi1 ... BiM , (12) Ci = ⎣ ... ⎦ , CiM

it is immediate that system (3) can be rewritten    x(k + 1) = Aθ(k) + Bθ(k) ∆θ(k) Cθ(k) x(k) x(0) = x0 ∈ Ln2 (Ω), k ∈ N, ˆ

(13)

cˆi bi with ∆i := diag(∆1i , . . . , ∆M i ) ∈ M(C , C ), for i ∈ S. An auxiliary result, which we prove next, is that the existence

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Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

˜ n+ such that, for a given ρ > 0, of X = (X1 , X2 , . . .) ∈ H sup the following system of infinitely coupled LMIs,   Li (X) − Xi + ρ2 Ci∗ Ci A∗i Ei (X)Bi ≪ 0 (14) Bi∗ Ei (X)Ai Bi∗ Ei (X)Bi − Iˆbi is satisfied, is a sufficient condition for the robust stability ˆ ˆ ,b of system (13) with respect to all ∆ = (∆1 , ∆2 , . . .) ∈ Hcsup such that ∆sup < ρ, where by definition   ˆ ˆ ,b Hcsup = (H1 , H2 , . . .); Hi ∈ M(Cci , Cbi ), sup Hi  < ∞ i∈S

(15) ˆ := (ˆb1 , ˆb2 , . . .), and c ˆ := (ˆ with b c1 , cˆ2 , . . .). Here and throughout the paper we employ the notation  pij Xj , Li (X) := A∗i Ei (X)Ai . (16) Ei (X) := j∈S

Lemma 1: Given ρ > 0, system (13) is SS for any ∆ = ˆ ˆ ,b (∆1 , ∆2 , . . .) ∈ Hcsup such that ∆sup < ρ, with 0 < ρ ≤ ρˇ, where   ρˇ := sup ρ > 0 such that (14) is feasible . (17) Proof: See the appendix. A distinguishing feature of the present work is that we aim at reducing the conservatism of Lemma 1 by means of scaling techniques, in a more general framework than that currently available in the literature of MJLS. Essentially, we replace (14) by the scaled LMI problem   A∗i Ei (X)Bi Li (X) − Xi + ρ2 Ci (S) ≪ 0, Bi∗ Ei (X)Ai Bi∗ Ei (X)Bi − Hib (S) (18) in which, for each i ∈ S, Ci (S) :=

M 

(Cim )∗ Sim Cim ,

(19)

m=1

 A∗i Ei (X)Bi = A∗i Ei (X)Bi1 ⎡

⎢ Bi∗ Ei (X)Bi = ⎣

(Bi1 )∗ Ei (X)Bi1 . . .

(BiM )∗ Ei (X)Bi1

... ... .

.

.

...

A∗i Ei (X)BiM , (20) (Bi1 )∗ Ei (X)BiM . . .

(BiM )∗ Ei (X)BiM

Hib (S) := diag(Si1 Ib1i , . . . , SiM IbM ), i

⎤ ⎥ ⎦

(21)

(22)

with the scaling parameters {Sim , i ∈ S, 1 ≤ m ≤ M } defined as in section III-A. The precise result goes as follows. Theorem 1: Given ρ > 0, system (3) is SS for any ∆ = (∆1 , ∆2 , . . .) ∈ D(ρ), ∆i = (∆1i , . . . , ∆M i ), such that 0 < ρ ≤ ρˆ, where   ρˆ := sup ρ > 0 such that (18) is feasible . (23) Moreover, ρˇ ≤ ρˆ. Proof: See the appendix.

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A. On the calculation of ρˆ In the sequel we propose an algorithm for the optimization of ρ, by means of a bisectional procedure. Algorithm 1: (robust stability). The maximal robustness margin ρˆ in (23) can be computed with arbitrary precision ε > 0, as follows: S1 : Find 0 ≤ ρmin < ρmax such that the LMI problem (18) is feasible for ρ = ρmin and unfeasible for ρ = ρmax , respectively. ֒→ If such ρmax can’t be found, then stop: system (3) is always SS for this class of perturbations, and ρˆ → ∞. S2 : Let ρ ← (ρmin + ρmax )/2 and check whether (18) is feasible. ֒→ If so, then let ρmin ← ρ; ֒→ otherwise, let ρmax ← ρ; S3 : Repeat S2 until (ρmax − ρmin )/2 < ε. S4 : Return ρ ≈ ρˆ. Remark 1: In the finite case, the characterization presented in, e.g., (14) and (18), fits in with the usual LMI framework. In the general situation studied here, however, one is faced with infinitely coupled LMIs, whose computational solvability still represents a challenge. In order to overcome this problem, approximate numerical solutions could be sought for particular applications. Also, it is to be expected that analytical solutions could be found in some cases, which is currently being taken into consideration. For an alternative approach, see [12].  V. ROBUST STABILIZATION In this section we consider the problem of robust stabilization of the following controlled version of system (3),   M m m x(k + 1) = Aθ(k) + m=1 Bθ(k) ∆m C θ(k) θ(k) x(k) (24) k ∈ N, + Gθ(k) u(k),

with x(0) = x0 ∈ Ln2 (Ω), by means of a static state-feedback control law (x, θ) → u, such as u(k) = Fθ(k) x(k),

(25)

u ,n Hnsup

where G = (G1 , G2 , . . .) ∈ describes how the control signal u = {u(k), k ∈ N} in Cnu is to be inserted on u the system, and F = (F1 , F2 , . . .) ∈ Hn,n defines the sup controller. Substituting (25) into (24) we obtain the corresponding closed-loop system of the form   M m m x(k + 1) = Aˆθ(k) + m=1 Bθ(k) ∆m θ(k) Cθ(k) x(k) (26)

where Aˆi := Ai + Gi Fi , for each i ∈ S. Hence, as a direct consequence of Theorem 1 we get that a given controller u F ∈ Hn,n sup guarantees the robust SS of system (26) for all ∆ ∈ D(ρ) whenever   Aˆ∗i Ei (X)Bi Lˆi (X) − Xi + ρ2 Ci (S) ˆ ∗ Ei (X)Aˆi ˆ ∗ Ei (X)B ˆi − Hb (S) ≪ 0, B B i i i (27)

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˜ n+ and S ∈ S, where Lˆi (X) := is satisfied for some X ∈ H sup Aˆ∗i Ei (X)Aˆi . Bearing this in mind, we state the main result of this section as follows. Theorem 2: Given ρ > 0, suppose there are Y = ˜ n+ , W = (W1 , W2 , . . .) ∈ Hn,nu , and a (Y1 , Y2 , . . .) ∈ H sup sup scaling Z ∈ S such that, for all i ∈ S, ⎤ ⎡ −Yi ρYi Ci∗ p′i ⊗ Mi (Y, W )∗ ⎦ ≪ 0, ⎣ ρCi Yi −Hic (Z) 0 pi ⊗ Mi (Y, W ) 0 Qi ⊗ Gi (Z) − D(Y ) (28) in which Mi (Y, W ) = Ai Yi + Gi Wi ,  Yi Ci∗ = Yi (Ci1 )∗ . . . Yi (CiM )∗ , Hic (Z) = diag(Zi1 Ic1i , . . . , ZiM IcM ) i Gi (Z) =

M 

(29)

Bim Zim (Bim )∗ ,

m=1

D(Y ) = diag(Y1 , Y2 , . . .), 1/2

1/2

Qi = pi p′i .

pi = col(pi1 , pi2 , . . .), Then, the static state-feedback controller −1 u(k) = Wθ(k) Yθ(k) x(k),

k∈N

(30)

ensures the robust stochastic stability of system (3) for any ∆ ∈ D(ρ). Bearing (12) in mind, let BiZ := Bi Hib (Z) =  1 Proof: 1 Bi Zi . . . BiM ZiM . From fundamental properties of the Kronecker product (see [11]), we have   Qi ⊗ Gi (Z) = (pi p′i ) ⊗ Bi Hib (Z)Bi∗   = (pi p′i ) ⊗ BiZ Hib (Z)−1 (BiZ )∗ = (pi ⊗ BiZ )(1 ⊗ Hib (Z)−1 )(pi ⊗ BiZ )∗ = (pi ⊗ BiZ )(Hib (Z))−1 (pi ⊗ BiZ )∗ ,

so that, letting Mi := Mi (Y, W ), (28) can be easily rewritten as ⎡ ⎤ −Yi ρYi Ci∗ p′i ⊗ M∗i ⎦ ⎣ ρCi Yi −Hic (Z) 0 pi ⊗ Mi 0 −D(Y ) ⎡ ⎤ ⎡ ⎤∗ 0 0   −1 ⎦ −Hib (Z) ⎣ ⎦ ≪ 0, 0 0 −⎣ Z Z pi ⊗ Bi pi ⊗ Bi

or, as it is immediately verified, as

Υ∗i Mi Υi ≪ 0, where

⎡ I ⎢0 Υi := ⎢ ⎣0 0

and ⎡

−Yi ⎢ 0 Mi := ⎢ ⎣ ρCi Yi pi ⊗ Mi

0 0 I 0

0 −Hib (Z) 0 pi ⊗ BiZ

(31)

Notice that, since c(I, Υi ) ≥ I ≫ 0, a consequence of (31) is that Mi ≪ 0, for all i ∈ S. Besides, by introducing the matrices ⎡ ⎤ (pi ⊗ Mi )∗ Vi := ⎣ (pi ⊗ BiZ )∗ ⎦ , (32) 0    I we have, after applying c ·, D(Y )−1 V∗i ≫ 0 on Mi ≪ 0, that ⎡ ⎤ −Yi 0 ρYi Ci∗ ⎣ 0 ⎦ + V∗i D(Y )−1 Vi ≪ 0. (33) −Hib (Z) 0 c ρCi Yi 0 −Hi (Z)

Next, let Fi := Wi Yi−1 and Aˆi := Ai + Gi Fi for each i ∈ S. Then, due to the fact that Mi = Ai Yi +Gi Wi = Aˆi Yi , (33) may be rewritten ⎤ ⎤ ⎧⎡ ⎡ 0 ρCi∗ Yi 0 0 ⎨ −Xi ⎦ ⎣ 0 I 0⎦ ⎣ 0 −Hib (Z) 0 ⎩ 0 0 I ρCi 0 −Hic (Z) ⎤⎫ ⎡ ⎤ ⎡ ζi (Aˆi , Aˆi ) ζi (Aˆi , BiZ ) 0 ⎬ Yi 0 0 + ⎣ζi (BiZ , Aˆi ) ζi (BiZ , BiZ ) 0⎦ ⎣ 0 I 0⎦ ≪ 0, ⎭ 0 0 I 0 0 0 (34) where Xi := Yi−1 , and

ζi (M, N ) := {pi ⊗ M }∗ D(X){pi ⊗ N } &X =

√ √ [ pi1 M ∗ pi2 M ∗ ... ]

=



j∈S

0 (pi ⊗ Bi )∗ ⎥ ⎥, ⎦ 0 I ⎤ ρYi Ci∗ (pi ⊗ Mi )∗ 0 (pi ⊗ BiZ )∗ ⎥ ⎥. c ⎦ −Hi (Z) 0 0 −D(Y )

1

' & √pi1 N ' √

X2

..

.

pi2 N

.. .

pij M ∗ Xj N = M ∗ Ei (X)N.

Thus, (34) yields & −Xi + Aˆ∗i Ei (X)Aˆi (BiZ )∗ Ei (X)Aˆi ρCi

Aˆ∗i Ei (X)BiZ ζi (BiZ , BiZ ) − Hib (Z) 0

'

ρCi∗ ≪ 0. 0 Hic (Z)

(35) Finally, let us define the inverse scaling S = (S , S , . 1 2 . .),  with Si := (Zi1 )−1 , . . . , (ZiM )−1 . Then, it is immediate that Hib (Z)Hib (S) = Hib (S)Hib (Z) = Iˆbi , the analogous holding for c, cˆi in place of b, ˆbi , from which application of the congruence transformation ⎡ ⎤∗ ⎡ ⎤ I 0 I 0   ⎣ 0 Hib (S)⎦ · ⎣ 0 Hib (S)⎦ (36) c c ρHi (S)Ci 0 ρHi (S)Ci 0 on (35) leads us to  2

−Xi + Lˆi (X) + ρ Ci∗ Hic (S)Ci Bi∗ Ei (X)Aˆi

⎤

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Aˆ∗i Ei (X)Bi ≪ 0, Bi∗ Ei (X)Bi − Hib (S)

(37) with Lˆi (X) := Aˆ∗i Ei (X)Aˆi , which is just (27). To be more precise, this follows from the fact that ⎛ ⎡ ⎤⎞ I 0 0 Hib (S)⎦⎠ c ⎝I, ⎣ c ρHi (S)Ci 0   2 ∗ c I + ρ Ci Hi (S)2 Ci 0 ≫ 0, = 0 Hib (S)2

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Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

if we keep in mind that Sim = (Zim )−1 ≥ εI for some ε > 0, regardless of i ∈ S and m ∈ {1, . . . , M }. In the following section we present a simple example of application of the preceding results, in the finite case. VI. N UMERICAL EXAMPLE Suppose the discrete-time system x(k + 1) = Ax(k),

A=

 0.25 0

−2 0.25

,

(38)

is perturbed according to a Markov chain {θ(k), k ∈ N} ⊂ {1, 2, 3}in (Ω, F, P), with fixed transition probabilities P = 1 111 ˜ 111 3 1 1 1 , giving rise to the MJLS x(k + 1) = Aθ(k) x(k), with     A˜1 = A, A˜2 = A + α0 00 , A˜3 = A + βγ γδ , (39)

in which α, β, γ and δ are uncertain parameters. Equivalently, notice we can write that       β 0 A˜2 = A + 10 α [ 1 0 ] , A˜3 = A + + γ 01 10 , 0 δ

or, put in the form (3), with M = 2 and     B11 = 00 , C11 = [ 0 0 ] , B12 = 00 , C12 = [ 0 0 ] ,     B21 = 10 , C21 = [ 1 0 ] , B22 = 00 , C22 = [ 0 0 ] ,       B31 = C31 = 10 01 , B32 = 10 01 , C32 = 01 10 .

In this case, in accordance to (9), the scaling parameters may be chosen as in Fig. 1.

i=1 i=2 i=3 Fig. 1.

m=1

m=2

scalar scalar 2-by-2, diagonal

scalar scalar 2-by-2, full

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In this case, after 23 iterations of an analogous procedure to that of Algorithm 1, we have from Theorem 2 that the controller gains   F1 = −0.2500 2.1104 , F2 = −0.2500 2.1104 ,  F3 = 0.0265 −0.4886

guarantee the robust stability of the system at hand whenever max{|α|, |β|, |γ|, |δ|} < 0.4774. If scaling is not employed, the analogous procedure yields that stability of the controlled system is guaranteed for just as long as max{|α|, |β|, |γ|, |δ|} < 0.3613, with the controller   F1 = −0.2500 2.0796 , F2 = −0.2500 2.0796 ,  F3 = 0.0335 −0.5515

which again is a more conservative margin. This illustrates the effectiveness of the proposed results. VII. C ONCLUDING REMARKS In this paper we addressed the robust stability and stabilization of discrete-time infinite MJLS, in face of block-diagonal structured uncertainty. By introducing blockdiagonal scaling techniques, which are new in this context, we have shown how less conservative results may be obtained. Besides, the approach presented here does not rely on the optimization of scaling, which is a non-convex problem. Instead, a bisectional procedure is proposed in order to achieve maximal robustness. We have restricted ourselves to the scenario of complex perturbations, which is the usual setting when norm-bounded uncertainty is considered. In this framework, a general and open problem in robust control theory consists on how to extend the results available to the complex case to the real uncertainty setting (see [7]). This will be the subject of further research. A PPENDIX

Structure of the scaling parameters Sim and Zim .

By running Algorithm 1 with ρmin = 0, ρmax = 10, and precision ε = 10−6 , we found, after 23 iterations, that the above system is stable whenever max{|α|, |β|, |γ|, |δ|} < 0.3092, which corresponds to the maximal ρ for which (18) is feasible. Without scaling, this margin drops to 0.2155. For comparison, by performing a random search (bearing in mind the stability criteria of [1, Chapter 3]), we found that the stochastic stability of the above uncertain system is lost when α = 0.3046 − 0.1301i, β = 0.3195 + 0.0874i, γ = −0.3214 − 0.0804i, δ = 0.3229 + 0.0739i, √ where i := −1, for which max{|α|, |β|, |γ|, |δ|} = 0.3313. Suppose now that the control u(k) = Fθ(k) x(k) is applied to this system, with input matrices     G1 = G2 = 10 , G3 = 11 .

Before proceeding to the proof of Lemma 1 we need the following auxiliary result. ˜ n+ , let Proposition 1: Given X = (X1 , X2 , . . .) ∈ H sup F (k) = Ex(k), Xθ(k) x(k), k ∈ N. Then, for any ∈ N∗ , F ( ) − F (0) =

ℓ−1 

Ex(k), Tθ(k) (X)x(k),

(40)

k=0

 ∗    Li (X)−Xi A∗ I i Ei (X)Bi in which Ti (X) := ∆iICi , ∗ ∗ B E (X)A B E (X)B ∆ C i i i i i i i i  ∗ with Ei (X) := j∈S pij Xj , and Li (X) := Ai Ei (X)Ai for each i ∈ S. Proof: To begin with, notice that ( )* E x(k + 1), Xθ(k+1) x(k + 1) * x(k), θ(k) + , *  = A˜θ(k) x(k), E Xθ(k+1) * θ(k) A˜θ(k) x(k) = x(k), A˜∗θ(k) Eθ(k) (X)A˜θ(k) x(k)

where A˜i := Ai +Bi ∆i Ci . Thus, bearing in mind the relation

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F ( ) − F (0) =

ℓ−1 

k=0

F (k + 1) − F (k),

(41)

M. G. Todorov and M. D. Fragoso: Robust Stability and Stabilization of Discrete-Time Ininite Markov Jump Linear Systems

together with the easily verifiable fact that ( ) F (k + 1) − F (k) = E x(k), A˜∗ Eθ(k) (X)A˜θ(k) x(k) θ(k)

− Ex(k), Xθ(k) x(k) = Ex(k), Tθ(k) (X)x(k),

we immediately have (40). The proof of Lemma 1 relies on a study of the following cost functional. Jρℓ (∆)

ℓ−1    E ρ2 Cθ(k) x(k)2 − ∆θ(k) Cθ(k) x(k)2 . = k=0

(42) Proof of Lemma 1. Assuming that ∆sup < ρ, we have that the cost in (42) satisfies Jρℓ (∆) ≥ (ρ2 − ∆sup )2

ℓ−1 

k=0

Then, from an easy calculation, we have that (18) is equivalent to   A∗i Ei (X)BiS Li (X) − Xi + ρ2 (CiS )∗ CiS ≪0 (BiS )∗ Ei (X)Ai (BiS )∗ Ei (X)BiS − Iˆbi (46) for all i ∈ S. Hence, due to Lemma 1, we have the robust SS of the system S S ∆θ(k) Cθ(k) )x(k), x(k + 1) = (Aθ(k) + Bθ(k)

To see why this yields robust SS of system (3), simply notice that, from (9), we have that ∆i ≡ diag(∆1i , . . . ∆M i ) satisfies

E{Cθ(k) x(k)2 } ≥ 0. (43)

BiS ∆i CiS

so that, since ℓ−1 

k=0

Jρℓ (∆)

E x(k), c



Ex(k), Tθ(k) (X)x(k),

R EFERENCES 1

in (42) is equal to

   , ∆θ(k)ICθ(k) x(k),

∗ ρ2 Cθ(k) Cθ(k) 0 0 −I

we immediately have

+c



ℓ−1 +  E x(k), Tθ(k)

k=0

∗ ρ2 Cθ(k) Cθ(k) 0 0 −I

   , , ∆θ(k)ICθ(k) x(k)

≤ Ex(0), Xθ(0) x(0) − ε

ℓ−1 

E[x(k)2 ]

(44)

k=0

for some ε > 0, due to the hypothesis that (14) holds true. Finally, Cauchy-Schwarz inequality implies that lim

ℓ→∞

ℓ−1 

k=0

E[x(k)2 ] ≤

1 Xsup E[x(0)2 ] < ∞, (45) ε

from which robust SS is guaranteed for any such ∆. Proof of Theorem 1. Given S = (S1 , S2 , . . .) ∈ S, Si = (Si1 , . . . , SiM ), let us define ⎡ 1 1/2 1 ⎤ Ci (Si )   S S 1 1 −1/2 M M −1/2 . ⎦. ⎣ ... Bi (Si ) , Ci = Bi = Bi (Si ) .. (SiM )1/2 CiM

1 Here

m Bim (Sim )−1/2 (Sim )1/2 ∆m i Ci

for each i ∈ S. This, together with (23), allows us to conclude that system (3) is SS for any ∆ ∈ D(ρ) such that 0 < ρ ≤ ρˇ = ρˇ(I) ≤ ρˆ, which yields the desired result.

k=0

0 ≤ Ex(0), Xθ(0) x(0) +

m 1/2 m Bim (Sim )−1/2 ∆m Ci i (Si )

m=1 M 

m=1

Ex(k), Tθ(k) (X)x(k) ℓ−1 

M 

= Bi ∆i Ci ,

k=0

≤ Ex(0), Xθ(0) x(0) +

= =

0 = Ex(0), Xθ(0) x(0) − Ex( ), Xθ(ℓ) x( ) +

k ∈ N (47)

guaranteed, in particular, for any ∆ = (∆1 , ∆2 , . . .) ∈ Hc,b sup , ∆i = diag(∆1i , . . . ∆M ), such that 0 < ρ ≤ ρ ˇ (S), where i   ρˇ(S) := sup ρ > 0 such that (46) is feasible in X . (48)

˜ n+ , we have from (1) that, for any ∈ N, Also, since X ∈ H sup ℓ−1 

TuC11.3

c(M, N ) := N ∗ M N , as defined in section II.

[1] O. L. V. Costa, M. D. Fragoso, and R. P. Marques, Discrete-Time Markov Jump Linear Systems, ser. Probability and Its Applications. New York: Springer-Verlag, 2005. [2] E. K. Boukas and Z. K. Liu, “Robust stability and H∞ -control of discrete-time jump linear systems with time-delay: An LMI approach,” Trans. ASME J. Dyna. Syst. Meas. Control, vol. 125, no. 2, pp. 271– 277, June 2003. [3] C. E. de Souza, “Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,” IEEE Trans. Automat. Control, vol. 51, no. 5, pp. 836–841, May 2006. [4] M. Karan, P. Shi, and C. Y. Kaya, “Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems,” Automatica, vol. 42, pp. 2159–2168, 2006. [5] A. El Bouhtouri and K. El Hadri, “Robust stabilization of discrete-time jump linear systems with multiplicative noise,” IMA J. Math. Control Inform., vol. 23, no. 4, pp. 447–462, Dec. 2006. [6] D. Hinrichsen and A. J. Pritchard, “Real and complex stability radii: a survey,” in Control of Uncertain Systems, ser. Progress in System and Control Theory, D. Hinrichsen and B. M˚artensson, Ed. Basel: Springer-Verlag, 1990, vol. 6, pp. 119–162. [7] ——, Mathematical systems theory I: modelling, state space analysis, stability and robustness, ser. Texts in applied mathematics. New York: Springer-Verlag, 2005, vol. 48. [8] ——, “Stability radii of systems with stochastic uncertainty and their optimization by output feedback,” SIAM J. Control Optim., vol. 34, pp. 1972–1998, 1996. [9] ——, “Stochastic H ∞ ,” SIAM J. Control Optim., vol. 36, no. 5, pp. 1504–1538, Sept. 1998. [10] A. El Bouhtouri and K. El Hadri, “Robust stabilization of jump linear systems with multiplicative noise,” IMA J. Math. Control Inform., vol. 20, no. 1, pp. 1–19, Mar. 2003. [11] J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst., vol. 25, no. 9, pp. 772–781, Sept. 1978. [12] J. Lee and G. E. Dullerud, “Uniform stabilization of discrete-time switched and markovian jump linear systems,” Automatica, vol. 42, no. 2, pp. 205–218, 2006.

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