Robust Stability and Control of Uncertain Discrete-Time Markov JumpLinear Systems

Carlos E. de Souza

Abstract— This paper deals with robust stability and con-trol of uncertain discrete-time linear systems with Markovianjumping parameters. Systems with polytopic-type parameteruncertainty in either the state-space model matrices, or inthe transition probability matrix of the Markov process, areconsidered. This paper develops new methods of robust stabilityanalysis and robust stabilization in the mean square sensewhich are dependent on the system uncertainty. The designof both mode-dependent and mode-independent control lawsis addressed. The proposed methods are given in terms oflinear matrix inequalities. Illustrative numerical examples areprovided to demonstrate the effectiveness of the derived results.

I. INTRODUCTION

The study of linear systems with Markovian jumpingparameters has been attracting an increasing attention overthe past decade. This class of systems, referred to as Markovjump linear (MJL) systems, is very appropriate to modelplants whose structure is subject to random abrupt changesdue to, for instance, random component failures, abruptenvironment disturbance, changes of the operating point ofa linearized model of a nonlinear system, etc. A number ofcontrol problems related to discrete-time MJL systems hasbeen analysed by several authors; see, for instance, [1]-[3],[5]-[16] and the references therein. In particular, with regardto uncertain discrete-time MJL systems, robust stability ofsystems with norm-bounded uncertainty has been studied in,for instance, [2] and [3], whereas robust control schemeshave been proposed in, e.g. [2], [3], [8] and [15]. A commonfeature of the existing methods of robust stability and controlis that they are based on Lyapunov functions which are inde-pendent of the system uncertainty. These methods, referred toas uncertainty-independent, are numerically appealing, how-ever, since a common Lyapunov function is used to ensurestability for all admissible uncertainties, they can be quiteconservative. A way of reducing this conservatism, which isthe object of this paper, is to incorporate in the Lyapunovfunction some dependence on the system uncertainty.

This paper is concerned with robust stability and controlof discrete-time linear systems with Markovian jumping pa-rameters and subject to polytopic-type parameter uncertaintyin either the matrices of the system state-space model, orin the transition probability matrix of the Markov process.

This work was supported in part by “Conselho Nacional de Desenvolvi-mento Cientıfico e Tecnologico - CNPq”, Brazil, under PRONEX grantNo. 0331.00/00 and CNPq grants No. 472920/03-0/APQ and 30.2317/02-3/PQ

C.E. de Souza is with the Department of Systems and Control, Labo-ratorio Nacional de Computacao Cientıfica (LNCC/MCT), Av. GetulioVargas 333, 25651-075 Petropolis, RJ, Brazil csouza@lncc.br

Attention is focused on developing new methods of robuststability analysis and robust stabilization in the mean squaresense which are dependent on the system uncertainty. The de-sign of both mode-dependent and mode-independent robustlystabilizing controllers is addressed, where the latter is aimedat systems in which the Markov chain is not accessible.The proposed methods are given in terms of linear matrixinequalities (LMIs) and have the advantage that they areless conservative than the existing uncertainty-independentmethods of robust stability analysis and robust stabilization.The potentials of the proposed results are demonstrated vianumerical examples.

Notation. Throughout the paper the superscript “T”stands for matrix transposition, R

n and Rn×m denote the

n-dimensional Euclidean space and the set of n × m realmatrices, respectively, and In is the n×n identity matrix. Fora real matrix S, HerS denotes S + ST and S > 0 meansthat S is symmetric and positive definite. For a symmetricblock matrix, the symbol denotes the transpose of theblocks outside the main diagonal block, and E[ · ] stands formathematical expectation.

II. PROBLEM STATEMENT

Fix an underlying probability space (Ω,F , IP ) and con-sider the MJL system (S):

x(k+1) = A(θk)x(k)+B(θk)u(k) (1)

where x(k)∈Rn is the state, u(k)∈R

nu is the control input,and θk is a discrete-time homogeneous Markov chain withfinite state-space Ξ = 1, . . . , σ and stationary transitionprobability matrix Λ = [λij ], where

λij:=IP θk+1 =j | θk = i.The set Ξ comprises the operation modes of system (S) andfor each possible value of θk = i, i ∈ Ξ, we denote thematrices associated with the “i-th mode” by

Ai = A(θk = i), Bi = B(θk = i)

where Ai and Bi, i = 1, . . . , σ are known real constantmatrices of appropriate dimensions.

This paper addresses the problems of robust stability anal-ysis and robust stabilization of system (S) in the presenceof parameter uncertainty and two cases will be considered:

Case 1: The matrices Ai and Bi, i = 1, . . . , σ areuncertain, but belong to given matrix polytopes Πi, i =

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MC2.5

0-7803-9354-6/05/$20.00 ©2005 IEEE 434

1, . . . , σ described by:

Πi :=[

Ai Bi

] ∣∣ [Ai Bi

]=

υi∑r=1

αir

[Air Bir

];

αir≥ 0,

υi∑r=1

αir =1

(2)

where υi is the number of vertices of Πi, and Air and Bir

are given matrices.Case 2: The transition probability matrix Λ = [λij ] is

unknown, but belongs to a given polytope, namely Λ ∈ ΠΛ,where ΠΛ is a polytope with vertices Λi, i=1, . . . , υ, i.e.

ΠΛ :=

Λ |Λ=

υ∑r=1

αrΛr; αr ≥ 0,υ∑

r=1

αr =1

(3)

where Λr = [λ(r)ij ], i, j = 1, . . . , σ, r = 1, . . . , υ are given

transition probability matrices. Note that the convex hull oftransition probability matrices is also a transition probabilitymatrix.

The scalars αir and αi will be referred to as the systemuncertain parameters for the Cases 1 and 2, respectively

The notion of stochastic stability for system (S) used inthis paper is in the mean square sense. In the sequel we recallthe concept of mean square stability and a related result forperfectly known MJL systems.

Definition 2.1: System (S) with known matrices Λ andAi, i=1, . . . , σ is mean square stable (MSS), if the solutionto the stochastic difference equation:

x(k + 1) = A(θk)x(k)

is such that E[ ‖x(k)‖2 ] → 0, as k → ∞ for any finite initialcondition x(0) ∈ R

n and θ0 ∈ Ξ.

Lemma 2.1 ([6]): System (S) with known matrices Λ andAi, i=1, . . . , σ is MSS if and only if there exist matricesPi >0, i=1, . . . , σ satisfying the inequalities:

Pi − Ai

σ∑j=1

λjiPjATi > 0, i = 1, . . . , σ. (4)

In relation to the uncertain systems of Cases 1 and 2, weintroduce the following definitions:

Definition 2.2: System (S) with uncertain Ai ∈ Πi, i =1, . . . , σ (respectively, Λ ∈ ΠΛ) is robustly mean squarestable if (1) is MSS for every Ai ∈ Πi, i = 1, . . . , σ(respectively, Λ ∈ ΠΛ).

Definition 2.3: System (S) with either uncertain Ai∈Πi,i=1, . . . , σ, or Λ ∈ ΠΛ, is robustly mean square stabilizableif there exists a mode-dependent control law

u(k) = K(θk)x(k); K(θk) = Ki, when θk = i (5)

such that the closed-loop system (Sc):

x(k + 1) = [A(θk) + B(θk)K(θk)]x(k) (6)

is robustly mean square stable.

Definition 2.4: System (S) with either uncertain Ai∈Πi,i = 1, . . . , σ, or Λ ∈ ΠΛ, is robustly strongly mean squarestabilizable if there exists a mode-independent control law

u(k) = Kx(k) (7)

such that the closed-loop system (Scs):

x(k + 1) = [A(θk) + B(θk)K]x(k) (8)

is robustly mean square stable.Remark 2.1: Note that the notion of robust strong mean

square stabilization is aimed at MJL systems in which thejumping parameter θk is not accessible and thus cannot beused in the control law.

This paper develops novel LMI conditions for:

(i) robust mean square stability,(ii) robust mean square stabilization,(iii) robust strong mean square stabilization

of uncertain systems (S) in the Cases 1 and 2.Lemma 2.1 could be used to test the robust mean square

stability of system (S), however, as (4) is now required tobe satisfied for every Ai ∈ Πi, i = 1, . . . , σ (or, Λ ∈ ΠΛ),this will involve testing the feasibility of an infinite numberof LMIs. A simple remedy to turn this problem into afinite number of LMIs is to impose that (4) be satisfiedwith fixed, and uncertainty-independent, matrices Pi, i =1, . . . , σ for all Ai ∈ Πi, i = 1, . . . , σ (or, Λ ∈ ΠΛ). Thisuncertainty-independent approach has the advantage that (4)needs to be tested only at the vertices of the uncertaintypolytope, namely, robust mean square stability is ensured ifthe following LMIs are feasible:

In Case 1:[

Pi AirPTΛi

PTΛiATir PTΛi

]> 0, i = 1, . . . , σ, r = 1, . . . , υi

(9)where

P =[P1 · · · Pσ

]T, Λi =

[λ1iIn · · · λσiIn

]T.

In Case 2:[

Pi AiPTΛ(r)i

PTΛ(r)i AT

i PTΛ(r)i

]> 0, i = 1, . . . , σ, r = 1, . . . , υ

(10)where

Λ(r)i =

[λ

(r)1i In · · · λ

(r)σi In

]T.

However, since the matrices Pi, i=1, . . . , σ in (9) and (10)are independent of the system uncertainty, the latter robuststability conditions can be quite conservative as it will beillustrated via examples in Section V. To overcome this draw-back, this paper proposes new methods of robust stabilityanalysis and robust stabilization of MJL systems which havethe advantages that they are uncertainty-dependent and givenin terms of LMIs.

We conclude this section by recalling a version of Finsler’slemma.

435

Lemma 2.2 ([4]): Given matrices Ψi = ΨTi ∈ R

n×n andHi ∈ R

m×n, i = 1, . . . , κ, then

xTi Ψixi > 0, ∀xi ∈ R

n : Hixi = 0, xi =0;i = 1, . . . , κ

(11)

if and only if there exist matrices Li ∈ Rn×m, i = 1, . . . , κ,

such that

Ψi + LiHi + HTi LT

i > 0, i = 1, . . . , κ. (12)

Note that conditions (12) remain sufficient for (11) to holdeven when arbitrary constraints are imposed to the scalingmatrices Li, including setting Li =L, i=1, . . . , κ.

III. ROBUST STABILITY ANALYSIS

First, we present a novel necessary and sufficient con-ditions for MSS of MJL systems without parameter un-certainty. These new MSS conditions will be fundamentalin the derivation of the robust mean square stability andstabilization results proposed in this paper.

Theorem 3.1: Consider system (S) with known matricesΛ and Ai, i = 1, . . . , σ. Then the following conditions areequivalent:(a) System (S) is MSS;(b) There exist matrices Gi and Xi ∈ R

n×n, i = 1, . . . , σsatisfying the following LMIs:

[Xi AiGi

GTi A

Ti HerGi− 0.5ΛT

i X

]> 0, i = 1, . . . , σ (13)

where

X =[X1 · · · Xσ

]T, Λi =

[λ1iIn · · · λσiIn

]T. (14)

(c) There exist matrices Gi, Xi ∈ Rn×n, Mi ∈ R

n×nσ andNi ∈ R

nσ×nσ, i=1, . . . , σ satisfying the following LMIs:⎡⎢⎣

Xi

GTi A

Ti HerGi+MiΛi

0 0.5X+MTi +NiΛi Ni+NT

i

⎤⎥⎦> 0, i = 1, . . . , σ

(15)where the matrices X and Λi are as in (14).

Proof . (a)⇒ (b): By Lemma 2.1 and applying Schur’scomplement to (4), there exist matrices Pi, i = 1, . . . , σsatisfying the LMIs:

[Pi AiPi

PiATi Pi

]> 0, i = 1, . . . , σ (16)

where

Pi =σ∑

j=1

λjiPj .

Then, it results from (16) that (13) holds with Gi = Pi andXi = Pi.

(b)⇒ (c): From (13), it follows that there exist sufficientlysmall scalars αi > 0, i = 1, . . . , σ such that the followinginequalities hold for i = 1, . . . , σ:

[Xi AiGi

GTi A

Ti HerGi− 0.5ΛT

i X]− αi

2

[0 00 ΛT

i Λi

]> 0

or equivalently,⎡⎣

Xi AiGi 0GT

i ATi HerGi− 0.5X TΛi αiΛT

i

0 αiΛi 2αiI

⎤⎦> 0, i = 1, . . . , σ

(17)Then, it follows from (17) that (15) is satisfied with the

same matrices Xi and Gi as above, Mi = −0.5X T andNi = αiI .

(c)⇒ (a): It will be shown that if the inequalities of (15)hold, then conditions (4) of Lemma 2.1 are satisfied.

First, note that (4) is equivalent to:

ηTi Υiηi > 0, ηi = Φix,∀ 0 = x ∈ R

n; i = 1, . . . , σ (18)

where

Υi =

⎡⎣

Pi 0 00 0 −0.5PT

0 −0.5P 0

⎤⎦, Φi =

⎡⎣

I

ATi

ΛiATi

⎤⎦,

P =[P1 · · · Pσ

]T.

Considering that

ηi : ηi =Φix, ∀x∈ Rn, x =0

=

ηi : Hiηi =0, ηi =0

where

Hi =[

ATi −I 00 Λi −I

],

it follows that (18) is also equivalent to

ηTi Υiηi > 0, ∀ ηi = 0 : Hiηi = 0, i = 1, . . . , σ. (19)

By Lemma 2.2, (19) holds iff the following inequalitiesare feasible for some matrix Li of appropriate dimensions:

Υi + LiHi + HTi LT

i > 0, i = 1, . . . , σ. (20)

Next, setting

Li =[

0 −Gi 00 MT

i −NTi

]T

for some matrices Gi,Mi and Ni of appropriate dimensions,it can be readily verified that (20) becomes identical to (15).Hence, one concludes that (15) ensures the mean squarestability of system (S). ∇∇∇

Note that Theorem 3.1 is equivalent to Lemma 2.1.However, due to the extra degrees of freedom introducedby the variables Gi in conditions (13) and Gi, Mi and Ni inconditions (15), Theorem 3.1 has the advantage that it can beeasily extended to solve the problems of robust mean squarestability and stabilization of MJL systems with polytopic-type uncertainty in either the matrices Ai, i=1, . . . , σ, or Λ,including the design of a mode-independent state-feedbackcontrol law. The introduction of the matrices Gi, Mi and Ni

allows for the derivation of uncertainty-dependent conditionsof robust mean square stability, robust mean square stabi-lization and robust strong mean square stabilization based onLMIs. In particular, the matrices Mi and Ni are introduced to

436

handle MJL systems with a polytopic-type uncertain matrixΛ.

In the sequel, robust mean square stability conditions forsystem (S) in the Cases 1 and 2 will be developed based onTheorem 3.1 (b) and (c), respectively.

First, consider the Case 1 for uncertain matrices Ai ∈Πi, i = 1, . . . , σ. An uncertainty-dependent approach isproposed, where the matrix Xi of (13) depends affinely onthe uncertain parameter vector αi =[αi1, . . . , αiυi

]T , whereαir are the non-negative scalars of (2), namely, Xi is givenby:

Xi(αi) =υi∑

r=1

αirX(r)i , i=1, . . . , σ (21)

where X(r)i are symmetric positive definite matrices.

Theorem 3.2: System (S) with the matrix Ai belonging tothe polytope Πi, i=1, . . . , σ is robustly mean square stableif there exist matrices Gi and X

(r)i ∈ R

n×n, i = 1, . . . , σ,r=1, . . . , υi satisfying the following LMIs:

[X

(r)i AirGi

GTi A

Tir Her

Gi− 0.5ΛT

i X [πir]

]> 0, i = 1, . . . , σ,

r = 1, . . . , υi, ∀πir ∈ Iir (22)

where Λi are as in (14), Iir denotes the set of σ-tuples

Iir =

πir|πir = [ p1, . . . pi−1, r, pi+1, . . . , pσ]T ,

ps = 1, . . . , υs, ∀ s ∈ 1, . . . , σ, s = i

and where to each πir is associated the matrix

X [πir ] =[X

(p1)1 , · · ·, X

(pi−1)

i−1 , X(r)i , X

(pi+1)

i+1 , · · ·, X(pσ)σ

]T

. (23)

Proof . Considering (2) and since the LMIs of (22) areaffine in the matrices Air and X

(r)i , then multiplying (22)

by appropriate scalars αjk of (2) and summing up theseinequalities, it can be easily verified that (13) holds for everyAi∈ Πi, i=1, . . . , σ and with a matrix Xi as in (21). Thus,system (S) is robustly mean square stable. ∇∇∇

Next, the case of an uncertain transition probability matrixΛ ∈ ΠΛ will be treated. In parallel with Case 1, Theo-rem 3.1 (c) is applied to ensure robust mean square stability,where the matrix Xi of (15) is set to be affinely dependenton the uncertain parameter vector α=[α1, . . . , αυ]T , whereαi are the non-negative scalars of (3). Specifically, the matrixXi is of the form:

Xi(α) =υ∑

r=1

αrX(r)i , i=1, . . . , σ (24)

where X(r)i are symmetric positive definite matrices. In the

above setting, the following robust mean square stabilityresult is obtained from Theorem 3.1 (c).

Theorem 3.3: System (S) with an uncertain transitionprobability matrix Λ belonging to the polytope ΠΛ is robustlymean square stable if there exist matrices Gi, X

(r)i ∈ R

n×n,

Mi ∈ Rn×nσ and Ni ∈ R

nσ×nσ , i=1, . . . , σ, r=1, . . . , υsatisfying the following LMIs:

⎡⎢⎣

X(r)i

GTi A

Ti HerGi+MiΛ

(r)i

0 0.5X [πir ]+MTi +NiΛ

(r)i Ni+NT

i

⎤⎥⎦> 0,

i = 1, . . . , σ, r = 1, . . . , υ, ∀πir ∈ Jir (25)

where

Λ(r)i =

[λ

(r)1i In · · · λ

(r)σi In

]T, (26)

Jir denotes the set of σ-tuples

Jir =πir|πir = [ p1, . . . pi−1, r, pi+1, . . . , pσ]T ,

ps = 1, . . . , υ, ∀ s ∈ 1, . . . , σ, s = i

and where to each πir is associated the matrix X [πir] asdefined in (23).

Proof . It is similar to that of Theorem 3.2. ∇∇∇Remark 3.1: Theorems 3.2 and 3.3 present novel condi-

tions of robust mean square stability for discrete-time MJLsystems with polytopic-type uncertainty in the state and tran-sition probability matrices, respectively. The derived resultshave the feature that they incorporate information on thesystem uncertainty. Similarly to the case of systems withoutjumps, it turns out that the conditions of Theorems 3.2 and3.3 are less conservative than the uncertainty-independentrobust stability conditions of (9) and (10).

IV. ROBUST STABILIZATION

This section deals with the robust stabilization of uncertaindiscrete-time MJL systems. The first two theorems presentuncertainty-dependent conditions for robust mean squarestabilization based on Theorems 3.2 and 3.3.

Theorem 4.1: System (S) with uncertain matrices Ai andBi belonging to the polytope Πi, i = 1, . . . , σ is robustlymean square stabilizable if there exist matrices Gi, X

(r)i ∈

Rn×n and Yi ∈ R

m×n, i=1, . . . , σ, r=1, . . . , υi satisfyingthe LMIs:⎡

⎣ X(r)i

GTi A

Tir+ Y T

i BTir Her

Gi− 0.5ΛT

i X [πir]

⎤⎦> 0,

i = 1, . . . , σ, r = 1, . . . , υi, ∀πir ∈ Iir (27)

where Λi, Iir and X [πir] are as in Theorem 3.2. Moreover,u(k) = K(θk)x(k), where

K(θk) = YiG−1i , when θk = i (28)

is a stabilizing control law.

Proof . First, note that (27) implies the nonsingularity ofthe matrix Gi. Next, applying Theorem 3.2 to the closed-loopsystem (Sc) of (8) and setting Yi = KiGi, we get the LMIsof (27) and the controller gain of (28) follows immediately.

∇∇∇

437

Theorem 4.2: System (S) with an uncertain transitionprobability matrix Λ belonging to the polytope ΠΛ is robustlymean square stabilizable if there exist matrices Gi, X

(r)i ∈

Rn×n, Mi∈R

n×nσ , Ni∈Rnσ×nσ and Yi∈R

m×n, i=1, . . . ,σ, r=1, . . . , υ satisfying the LMIs:⎡⎢⎣

X(r)i

GTi A

Ti +Y T

i BTi HerGi+MiΛ

(r)i

0 0.5X [πir ]+MTi +NiΛ

(r)i Ni+NT

i

⎤⎥⎦> 0,

i = 1, . . . , σ, r = 1, . . . , υ, ∀πir ∈ Jir (29)

where Λ(r)i , Jir and X [πir] are as in Theorem 3.3. Moreover,

u(k) = K(θk)x(k), where

K(θk) = YiG−1i , when θk = i (30)

is a stabilizing control law.

Proof . It will be first shown that (29) ensures that thematrix Gi is nonsingular. To this end, note that using similararguments as in the proof of Theorem 3.2, (29) implies that

⎡⎣

Xi

GTi A

Ti +Y T

i BTi HerGi+MiΛi

0 0.5X+MTi +NiΛi Ni+NT

i

⎤⎦> 0

(31)

where the matrix Xi is given by (24), and X and Λi are asin (14). Then, pre- and post multiplying (31) by [ 0 I −ΛT

i ]and its transpose, respectively, we get

Gi + GTi − ΛT

iX > 0

which implies that Gi + GTi > 0, and thus the matrix Gi is

nonsingular. Next, applying Theorem 3.3 to system (Sc) andsetting Yi = KiGi, the result follows immediately. ∇∇∇

We shall now consider the problem of robust strong meansquare stabilization where the control law is required to beindependent of the jumping parameter θk. The design of suchrobust controllers can be also accomplished via Theorems 3.2and 3.3 by imposing the constraints Gi = G, i = 1, . . . , σ.These results are presented in the next two theorems, whoseproofs are similar to that of Theorems 4.1 and 4.2, exceptthat now (Sc), Gi, Ki and Yi are replaced by (Ssc), G, Kand Y , respectively.

Theorem 4.3: System (S) with uncertain matrices Ai andBi belonging to the polytope Πi, i = 1, . . . , σ is robustlystrongly mean square stabilizable if there exist matrices G∈R

n×n, Y ∈ Rm×n and X

(r)i ∈ R

n×n, i = 1, . . . , σ, r =1, . . . , υi satisfying the LMIs:

⎡⎣ X

(r)i

GTATir+ Y TBT

ir Her

G − 0.5ΛTi X [πir]

⎤⎦> 0,

i = 1, . . . , σ, r = 1, . . . , υi, ∀πir ∈ Iir (32)

where Λi, Iir and X [πir] are as in Theorem 3.2. Moreover,

u(k) = Kx(k), K = Y G−1 (33)

is a stabilizing control law.

Theorem 4.4: System (S) with an uncertain matrix Λbelonging to the polytope ΠΛ is robustly strongly meansquare stabilizable if there exist matrices G ∈ R

n×n, Y ∈R

m×n, X(r)i ∈ R

n×n, Mi ∈ Rn×nσ and Ni ∈ R

nσ×nσ ,i=1, . . . , σ, r=1, . . . , υ satisfying the LMIs:⎡⎢⎣

X(r)i

GTATi +Y TBT

i HerG +MiΛ(r)i

0 0.5X [πir ]+MTi +NiΛ

(r)i Ni+NT

i

⎤⎥⎦> 0,

i = 1, . . . , σ, r = 1, . . . , υ, ∀πir ∈ Jir (34)

where Λ(r)i , Jir and X [πir] are as in Theorem 3.3. Moreover,

u(k) = Kx(k), K = Y G−1 (35)

is a stabilizing control law.Remark 4.1: Theorem 4.3 and 4.4 provide new LMI meth-

ods for the design of mode-independent robustly stabilizingstate-feedback controllers. This is in contrast with existingrobust control designs for uncertain discrete-time MJL sys-tems, such as those of [2], [3], [8] and [15], which requirethe accessibility of the jumping parameter.

V. EXAMPLES

Example 1: This example deals with the robust meansquare stability of an uncertain MJL system with 2 operatingmodes described by:

Λ =[

0.1 0.90.15 0.85

], A1 =

[1 0.7

−0.4 0.82

],

whereas the matrix A2 is uncertain but belongs to a 2-verticespolytope Π2 with:

A21 =[

1 0.5−0.3 0.8

], A22 =

[1 0.50 0.4

].

Note that A2 can be also expressed in the norm-boundeduncertainty form A2 = A20 + L2δR2, where |δ| ≤ 1 and

A20 =[

1 0.5−0.15 0.6

], L2 =

[01

], R2 =

[0.15−0.2

]T

.

By Theorem 3.2, the above system is robustly meansquare stable. On the other hand, if we use the uncertainty-independent robust stability condition of (9), or any existingrobust stability condition, such as those of [2], [3] and [15],no conclusion can be drawn regarding the robust mean squarestability of the above system. This result demonstrates thesuperiority of the uncertainty-dependent condition for robustmean square stability of Theorem 3.2.

Example 2: Consider a 2-modes uncertain MJL systemwith state matrices

A1 =√

0.9, A2 =√

1.1

and an uncertain transition probability matrix Λ belongingto the polytope ΠΛ of (3) with 2 vertices:

Λ1 =[

0.9 0.10.3 0.7

], Λ2 =

[0.95 0.050.15 0.85

].

438

Applying Theorem 3.3 it follows that the above systemis robustly mean square stable. On the other hand, theuncertainty-independent robust stability condition of (10), aswell as any existing robust mean square stability condition,fail for the above system. Similarly to Example 1, this resultdemonstrates the superiority of the uncertainty-dependentcondition for robust mean square stability of Theorem 3.3.

Example 3: Consider the problem of robust stabilizationfor an uncertain MJL system with 2 operating modes de-scribed by:

Λ =[

0.1 0.90.15 0.85

], A1 =

[1 0.1

0.1 −1

], B1 =

[00

]

whereas [A2 B2 ] belongs to a 2-vertices polytope Π2 with:

A21 =[

1 0.5−0.3 0.8

], B21 =

[10

], A22 =

[1 0.50 0.6

], B22 =

[00

].

Note that A2 can be also expressed in the norm-boundeduncertainty form A2 = A20 + L2δR2, where |δ| ≤ 1 and

A20 =[

1 0.5−0.15 0.7

], L2 =

[01

], R2 =

[0.15−0.1

]T

.

Observe that as the above system is not MSS when A2

coincides with A22, then it is not robustly mean square stable.It turn out this system cannot be robustly stabilized by any

of the existing methods of robust mean square stabilization,such as those of [2], [3], [8] and [15]. However, by Theo-rem 4.1 it follows that the above system is robustly meansquare stabilizable and a stabilizing control law is given by:

u(k) = K(θk)x(k); K(θk) = Ki when θk = i

where K1 = [ 1.1420 0.0670 ] and K2 = [ 0 0 ].

Example 4: Consider the delay-free nominal system ofthe example in [2], namely, let the system (S) with threeoperating modes described by:

A1 =[

1 00 1.2

], A2 =

[1.13 00.16 0.478

],

A3 =[

0.3 0.130.16 1.14

], B1 =

[0.1 00.1 0

],

B2 =[

0.2 0.10 −0.1

], B3 =

[0 0.10 −0.1

].

The transition probability matrix Λ is unknown, but be-longs to polytope ΠΛ with 2 vertices:

Λ1 =

⎡⎣

0.15 0.85 00 0.45 0.550 0.2 0.8

⎤⎦, Λ2 =

⎡⎣

0.25 0.75 00 0.6 0.40 0.4 0.6

⎤⎦.

Applying Theorem 4.4, the above system is robustlystrongly mean square stabilizable and a stabilizing state-feedback gain is given by:

K =[ −5.6133 −3.5715

0.2906 5.8966

].

VI. CONCLUSIONS

This paper has addressed the problems of robust stabilityand robust stabilization of discrete-time Markov jump linearsystems subject to polytopic-type parameter uncertainty ineither the transition probability matrix of the Markov process,or in the matrices of the system state-space model for thepossible modes of operation. LMI methods of robust stabilityanalysis and robust stabilization in the mean square sensehave been developed. The design of both mode-dependentand mode-independent robustly stabilizing controllers hasbeen considered, where the latter is aimed at systems inwhich the Markov chain is not accessible. The proposedmethods incorporate information on the system uncertaintyand have the advantage that they are less conservativethan the existing uncertainty-independent methods of robuststability analysis and robust stabilization.

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