Robust stability, H-2 analysis and stabilization of discrete …vargas/ap02Final.pdfTitle Robust...

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Robust stability, H 2 analysis and stabilization of discrete-time Markov jump linear systems with uncertain probability matrix R. C. L. F. Oliveira, A. N. Vargas and J. B. R. do Val and P. L. D. Peres Abstract The stability and the problem of H 2 guaranteed cost computation for discrete-time Markov jump linear systems (MJLS) are investigated, assuming that the transition probability matrix is not precisely known. It is generally difficult to estimate the exact transition matrix of the underlying Markov chain and the setting has an special interest for applications of MJLS. The exact matrix is assumed to belong to a polytopic domain made up by known probability matrices, and a sequence of linear matrix inequalities is proposed to verify the stability and to solve the H 2 guaranteed cost with increasing precision. These LMI problems are connected to homogeneous polynomially parameter-dependent Lyapunov matrix of increasing degree g. The mean square stability can be established by the method since the conditions that are sufficient, eventually turns to be also necessary, provided that the degree g is large enough. The H 2 guaranteed cost under mean square stability is also studied here, and an extension to cope with the problem of control design is also introduced. These conditions are only sufficient, but as the degree g increases, the conservativeness of the H 2 guaranteed costs is reduced. Both mode-dependent and mode-independent control laws are addressed, and numerical examples illustrate the results. I. I NTRODUCTION The Markov Jump Linear Systems (MJLS) comprise a class of hybrid systems with random switching structure that is appropriate to model a number of systems subject to abrupt variations in their structure. Among others, [1–7] can be cited as important contributions on this subject. Supported by the Brazilian agencies CNPq and FAPESP. R. C. L. F. Oliveira, J. B. R. do Val and and P. L. D. Peres are with the School of Electrical and Computer Engineering, Univer- sity of Campinas, CP 6101, 13081-970, Campinas, SP, Brazil. Emails: {ricfow,jbosco,peres}@dt.fee.unicamp.br A. N. Vargas is with the Electrotechnical Department, Federal University of Technology - Paran´ a, Av. Alberto Carazzai 1640, 86300-000, Corn´ elio Proc´ opio - PR - Brazil. Email: [email protected] http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete- time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

Transcript of Robust stability, H-2 analysis and stabilization of discrete …vargas/ap02Final.pdfTitle Robust...

  • Robust stability, H2 analysis and stabilization of

    discrete-time Markov jump linear systems with

    uncertain probability matrix

    R. C. L. F. Oliveira, A. N. Vargas and J. B. R. do Val and P. L. D. Peres

    Abstract

    The stability and the problem ofH2 guaranteed cost computation for discrete-time Markov jump

    linear systems (MJLS) are investigated, assuming that the transition probability matrix is not precisely

    known. It is generally difficult to estimate the exact transition matrix of the underlying Markov chain and

    the setting has an special interest for applications of MJLS. The exact matrix is assumed to belong to a

    polytopic domain made up by known probability matrices, anda sequence of linear matrix inequalities

    is proposed to verify the stability and to solve theH2 guaranteed cost with increasing precision. These

    LMI problems are connected to homogeneous polynomially parameter-dependent Lyapunov matrix of

    increasing degreeg. The mean square stability can be established by the method since the conditions

    that are sufficient, eventually turns to be also necessary, provided that the degreeg is large enough.

    The H2 guaranteed cost under mean square stability is also studiedhere, and an extension to cope

    with the problem of control design is also introduced. Theseconditions are only sufficient, but as the

    degreeg increases, the conservativeness of theH2 guaranteed costs is reduced. Both mode-dependent

    and mode-independent control laws are addressed, and numerical examples illustrate the results.

    I. I NTRODUCTION

    The Markov Jump Linear Systems (MJLS) comprise a class of hybrid systems with random

    switching structure that is appropriate to model a number ofsystems subject to abrupt variations

    in their structure. Among others, [1–7] can be cited as important contributions on this subject.

    Supported by the Brazilian agencies CNPq and FAPESP.

    R. C. L. F. Oliveira, J. B. R. do Val and and P. L. D. Peres are with the School of Electrical and Computer Engineering, Univer-

    sity of Campinas, CP 6101, 13081-970, Campinas, SP, Brazil. Emails:{ricfow,jbosco,peres}@dt.fee.unicamp.br

    A. N. Vargas is with the Electrotechnical Department, Federal Universityof Technology - Parańa, Av. Alberto Carazzai 1640,

    86300-000, Corńelio Proćopio - PR - Brazil. Email:[email protected]

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • On the wider context of stochastic systems there is a great number of applications such as

    finance, telecommunications, neural networks (see for instance [8–11]), for which the stochas-

    tic information is not available in the strict sense but important random phenomena is only

    known to be within certain ranges, or to belong to given sets.The paper investigates MJLS

    with uncertainties on the stochastic information, in particular, we focus the situation when the

    transition probability matrix of the Markov chain is not precisely known but it belongs to a

    certain polytope made up by known matrices. It is a hard task to get accurate estimates for the

    precise transition matrix in applications, and estimationerrors may prevent the use of MJLS

    to real-world problems [12]. Similar characterization waspreviously studied in [13–15] for the

    discrete and in [16] for the continuous-time MJLS. The main aim here is to verify the stochastic

    stability of this family of processes and to obtain an evaluation of the H2 guaranteed norm

    applied to the control synthesis. The stability assessmentor the norm evaluation can be made

    tighter as the method becomes more refined.

    The mean square stability (MSS) and a guaranteedH2-norm in the MJLS setting just described

    are coined in terms of parameter-dependent linear matrix inequalities (LMIs). The search for a

    feasible solution assuring that the set of LMIs with parameters in the unit simplex can be regarded

    as the analysis of positivity of a matrix-valued polynomial. Positivity analysis of polynomials is

    an issue that has gained renewed interest in this decade due to the emerging of efficient numerical

    algorithms based on semidefinite programming. By assuring that a polynomial matrix is positive

    definite, scalar sum-of-squares relaxations [17, 18], matrix-valued sum-of-squares relaxations

    [19–22], Gram matrix representation [23–25] and Pólya’s Theorem based relaxations [26–28]

    can be used to search for a parameter-dependent solution. See [28] for a comparison of these

    methods in the context of robust stability of linear systems.

    In this paper, following along the lines in [28], sequences of LMI relaxations are proposed in

    terms of the degreeg of the homogeneous polynomially parameter-dependent Lyapunov solutions

    and the leveld of the Ṕolya’s relaxation. The MSS characterization, or theH2-norm under MSS,

    tends to be exact as the degreeg increases, namely, the level of sufficiency in conditions decreases

    as the degreeg increases and the stability conditions turn to be also necessary forg large enough.

    Sufficient conditions for the existence of a state feedback control law assuring a guaranteedH2-

    norm of MSS with uncertain probability matrix are also presented. Although only sufficient, this

    approach generalizes the results presented in [29]. Examples illustrate that the increase ofg and

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • d provideH2-norm evaluations less conservative than others found in the literature.

    The organization of the paper is as follows: the definitions and preliminary results are given in

    Section II, including the model for the uncertainty in the probability matrix and the machinery to

    handle homogeneous matrix polynomials. Section III presents the robustH2 and stability analysis

    results. The robust state feedback control problem is addressed in Section IV, for both mode-

    independent and mode-dependent cases and also includes theH2 guaranteed norm minimization.

    Numerical examples for robust stability analysis and statefeedback control design are given in

    Section V and the final remarks appear in Section VI.

    II. PRELIMINARIES

    Let IRn denote then-th dimensional Euclidean space with the usual norm| · |, and letS :={1, . . . ,σ} be a finite set. The set of natural numbers is represented byN. Consider(Ω,F ,{Fk},P)the fundamental probability space. Let{θ(k);k≥ 0} be the discrete-time homogeneous Markovchain, withS as state space, havingΓ = [pi j ], for all i and all j in S , as the transition probability

    matrix, namely,

    pi j := Pr(θ(k+1) = j|θ(k) = i), ∀k≥ 0.

    The initial distribution of the Markov chain is given byµ = {µ1, . . . ,µσ} in such a way thatPr(θ(0) = i) = µi . Let ℓ2 denote the Hilbert space formed byz= {z(k);k ≥ 0}, second order,real-valued stochastic processes that are{Fk}-adapted and

    ‖z‖22 :=∞

    ∑k=0

    E[|z(k)|] < ∞,

    here, E[·] stands for the mathematical expectation.Consider the discrete-time MJLS defined by

    G0 =

    x(k+1) = Aθ(k)x(k)+Eθ(k)w(k)

    z(k) = Cθ(k)x(k), k≥ 0w∈ ℓ2, E[|x0|2] < ∞, θ0 ∼ µ,

    (1)

    wherex(k)∈ IRn is the system state,w(k)∈ IRm is the noisy input andz(k)∈ IRp is the output. Thefinite sets of matricesA= {Ai ∈ IRn×n, i ∈S }, E = {Ei ∈ IRn×m, i ∈S }, C = {Ci ∈ IRp×n, i ∈S }are given, and wheneverθ(k) = i, i ∈ S , one has that(Aθ(k),Eθ(k),Cθ(k)) = (Ai,Bi,Ci). Weassume throughout that {w(k)} is an i.i.d. random sequence of real numbers, with nullmean value and covariance matrix equals to the identity.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • One of the most important concepts of stability applied to MJLS is the mean square stability

    (MSS), which requires thatthe systemG0 with w(k) ≡ 0 satisfies

    E[|x(k)|] → 0 ask→ ∞ (2)

    for any initial condition x0 ∈ IRn,θ0 ∈ S . A necessary and sufficient condition for MSS isreproduced below.

    Lemma 1: ([4, 7]) SystemG0 is MSS if and only if there exist matrices 0< Pi = P′i ∈IRn×n, i = 1, . . . ,σ such that the following inequalities hold

    A′i( σ

    ∑j=1

    pi j Pj)

    Ai −Pi < 0, i = 1, . . . ,σ (3)

    The definition of theH2-norm for MJLS is introduced as follows.

    Definition 1: ([7, 29]) TheH2-norm for the systemG0 is given by

    G0 =m

    ∑s=1

    σ

    ∑i=1

    µi‖zs,i‖22,

    wherezs,i represents the outputz= {z(k);k≥ 0} when:(a) the input is given byw = {w(k);k≥ 0}, w(0) = es, w(k) = 0,k = 1,2, . . . andes∈ IRm is thes-th basis vectores = [0· · ·0 1 0· · ·0]′,(b) x0 = 0 andθ0 = i.

    The H2-norm of systemG0 can be calculated by means of an LMI based optimization

    procedure using the following lemma [29].

    Lemma 2: SystemG0 is MSS and‖G0‖22 < ρ if and only if there exist matrices 0< Pi =P′i ∈ IRn×n, i = 1, . . . ,σ such that the LMIs

    σ

    ∑j=1

    trace(CjPjC′j) < ρ,

    σ

    ∑j=1

    p ji(

    A jPjA′j + µ jE jE′j

    )

    −Pi < 0, (4)

    hold for eachi = 1, . . . ,σ or, equivalently, by duality, the LMIsσ

    ∑j=1

    µ j trace(

    E′j( σ

    ∑ℓ=1

    p jℓPℓ)E j)

    < ρ, A′i( σ

    ∑j=1

    pi j Pj)

    Ai −Pi +C′iCi < 0, (5)

    hold for eachi = 1, . . . ,σ .

    Minimizing the valueρ under the LMI constraints (4) (or (5)) produces the value of the

    H2-norm of systemG0.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • A. Uncertainties in the probability matrix

    Suppose now that the transition probabilityΓ = [pi j ] is not precisely known but belongs to a

    polytopic domain, i.e.Γ ∈ Γ(α), with

    Γ(α) = {Γ | Γ =N

    ∑r=1

    αrΓ(r), α ∈ ∆N},

    where∆N is the unit simplex given by

    ∆N ={

    α ∈ IRN :N

    ∑r=1

    αr = 1, αr ≥ 0, r = 1, . . . ,N}

    and Γ(r) , [p(r)i j ] are given probability matrices. Note that convex combinations of probability

    matrices are also probability matrices.

    The extension of Lemma 2 to the case ofH2-norm of MJLS with uncertainties in the

    probability matrix is presented in the sequel.

    Lemma 3: SystemG0 is robustly MSS for allΓ ∈ Γ(α) and ‖G0‖22 < ρ if and only ifthere exist parameter-dependent matrices 0< Pi(α) = Pi(α)′ ∈ IRn×n, i = 1, . . . ,σ such that theparameter-dependent inequalities

    L(α) ,σ

    ∑j=1

    trace(CjPj(α)C′j) < ρ, (6)

    Mi(α) ,σ

    ∑j=1

    p ji (α)(

    A jPj(α)A′j + µ jE jE′j)

    −Pi(α) < 0, (7)

    hold for eachi = 1, . . . ,σ and for allα ∈∆N or, equivalently, the parameter-dependent inequalitiesσ

    ∑j=1

    µ j trace(

    E′j( σ

    ∑ℓ=1

    p jℓ(α)Pℓ(α))E j)

    < ρ, (8)

    A′i( σ

    ∑j=1

    pi j (α)Pj(α))

    Ai −Pi(α)+C′iCi < 0, (9)

    hold for eachi = 1, . . . ,σ and for all α ∈ ∆N.The smallest valueρ such that the parameter-dependent LMIs (6–7) (or (8–9)) hold for all

    α ∈ ∆N is theH2 worst case norm of the systemG0 with uncertainties in the probability matrix.If only robust MSS is concerned, the inequalities (7) or (9) in Lemma 3 with voided matrices

    Ei or Ci, i = 1, . . . ,σ , respectively, can be alternatively employed.

    Lemma 3 assess theH2-norm of the systemG0 with uncertainties in the probability matrix by

    means of parameter-dependent matricesPi(α), i = 1, . . . ,σ depending generically onα. In other

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • words, for any possible fixed value ofα ∈ ∆N, matricesPi(α) must fulfill inequalities (6–7) (or(8–9)). This is clearly a infinite dimension problem based onLMIs (parameter-dependent LMIs),

    a known problem of hard solution [30].

    Recently, it has been proved that parameter-dependent LMIsdepending continuously upon

    scalar parameters lying in a compact set always admits a solution which is polynomial of

    certain degreeg on the parameters whenever one solution exists [31]. Specifically in the case

    of parameters lying in the unit simplex, the solution can be approximated by a homogeneous

    polynomial solution of sufficiently large degreeg, without loss of generality [32].

    Theorem 1: There exist parameter-dependent matrices 0< Pi(α)= Pi(α)′ ∈ IRn×n, i = 1, . . . ,σsatisfying inequalities (6–7) (or (8–9)) if and only if there exist homogeneous polynomially

    parameter-dependent matrices 0< Pi(α) = Pi(α)′ ∈ IRn×n, i = 1, . . . ,σ of sufficiently large degreeon α.

    Proof: See [32].

    Using the results of Theorem 1, the solvability of the parameter-dependent LMIs from Lemma 3

    can be cast equivalently as the search for a homogeneous polynomial solution of arbitrary

    degreeg Pi(α), i = 1, . . . ,σ . In this case, the problem relies on the positivity analysisof matrix-

    valued homogeneous polynomials of degreeg+ 1 due to the product between the uncertain

    probability matrix (affine onα) and the Lyapunov matrices (degreeg on α). The machinery

    necessary to handle homogeneous polynomials is presented in what follows.

    B. Homogeneous polynomial Lyapunov matrix

    The following definitions and operations are needed to manipulate expressions involving

    products and sums of homogeneous polynomials of arbitrary degree. A homogeneous matrix-

    polynomialP(α) of degreeg can be generally written as

    P(α) = ∑k∈K (g)

    αkPk, αk = αk11 αk22 · · ·α

    kNN , k = k1k2 · · ·kN (10)

    whereαk11 αk22 · · ·α

    kNN , α ∈ ∆N, ki ∈N, i = 1, . . . ,N are the monomials, andPk ∈ IRn×n, ∀k∈K (g)

    are matrix valued coefficients. By definition,K (g) is the set ofN-tuples obtained as all possible

    combinations of nonnegative integerski, i = 1, . . . ,N, such thatk1+k2+ . . .+kN = g. Since the

    number of vertices in the polytopeΓ(α) is equal toN, the number of elements inK (g) is given

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • by

    J(g) =(N+g−1)!g!(N−1)! (11)

    To give an example, for homogeneous polynomials of degreeg = 2 with N = 2 variables, the

    possible values of the partial degrees areK (2) = {02,11,20} (J(2) = 3), corresponding to thegeneric formP(α) = α22P02+ α1α2P11+ α

    21P20. Constant (zero-degree) matrices are obtained

    from (10) for g = 0.

    By definition, for N-tuplesk,k′ one writesk� k′ if ki ≥ k′i , i = 1, . . . ,N. Usual operations ofsummationk+k′ and subtractionk−k′ (wheneverk� k′) are defined componentwise. Consideralso the following definitions for theN-tuple ei and the coefficientπ(k)

    ei = 0· · ·0 1︸︷︷︸i−th

    0· · ·0, π(k) = (k1!)(k2!) · · ·(kN!)

    The i-th column (transposed) and the i-th row of each vertexr of the polytopic domainΓ(α)

    of uncertain transition probability matrices are respectively grouped as

    Γ(r)i =[

    p(r)1i In p(r)2i In · · · p

    (r)σ i In

    ]

    , Λ(r)i =[

    p(r)i1 In p(r)i2 In · · · p

    (r)iσ In

    ]

    When homogeneous polynomial Lyapunov matricesPi(α) of degreeg for eachi = 1, . . . ,σ are

    considered, the following coefficient matrices are used

    Xk =

    P1k

    P2k...

    Pσk

    , Yk =

    A1P1kA′1

    A2P2kA′2

    ...

    Aσ PσkA′σ

    , Z =

    µ1E1E′1µ2E2E′2

    ...

    µσ Eσ E′σ

    With these definitions and notations, it is possible to construct a systematic procedure to search

    for homogeneous polynomials solutions of arbitrary degreeto the parameter-dependent LMIs of

    Lemma 3, yielding MSS andH2 guaranteed costs for the MJLSG0. Thanks to some algebraic

    properties of positive homogeneous polynomials in the unitsimplex, it is also possible to assure

    the convergence of such procedure, presented next sectionsin terms of finite dimension LMI

    relaxations for robust stability analysis and stabilization with H2 guaranteed costs.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • III. ROBUST ANALYSIS RESULTS

    In this section, LMI conditions are given in term of the degree g of the homogeneous solutions

    and the leveld of Pólya’s relaxations.

    Theorem 2: SystemG0 is robustly MSS for allΓ ∈ Γ(α) and‖G0‖22 < ρ holds if and onlyif there exist symmetric matricesPik ∈ IRn×n, k ∈ K (g), i = 1, . . . ,σ a degreeg ∈ N and asufficiently large degreed ∈ N such that fori = 1, . . . ,σ , the following LMIs hold

    Sk = ∑k′∈K (d)

    k�k′

    d!π(k′)

    (

    Pik−k′)

    > 0, ∀k∈ K (g+d) (12)

    Tk = ∑k′∈K (d)

    k�k′

    d!π(k′)

    ∑j=1

    trace(CjPjk−k′C

    ′j

    )− g!

    π(k−k′)ρ)

    < 0, ∀k∈ K (g+d) (13)

    Uk = ∑k′∈K (d)

    k�k′

    ∑ℓ∈{1,...,N}

    kℓ>k′ℓ

    d!π(k′)

    (

    Γ(ℓ)i(

    Yk−k′−eℓ +g!(kℓ−k′ℓ)π(k−k′) Z

    )

    −Pik−k′−eℓ

    )

    < 0,

    ∀k∈ K (g+d+1) (14)Proof: From Theorem 1, one has that if there exists a positive definite solutionPi(α),

    i = 1, . . . ,N to (6–7), then there existsPi(α) which is a homogeneous polynomial solution of

    certain degreeg. In this case, matrixPi(α) and matricesL(α) andMi(α) given in (6–7) can be

    equivalently rewritten as

    (α1 + · · ·+αN)dPi(α) = ∑k∈K (g+d)

    αk11 αk22 · · ·α

    kNN Sk, i = 1, . . . ,σ (15)

    (α1 + · · ·+αN)dL(α) = ∑k∈K (g+d)

    αk11 αk22 · · ·α

    kNN Tk, (16)

    (α1 + · · ·+αN)dMi(α) = ∑k∈K (g+d+1)

    αk11 αk22 · · ·α

    kNN Uk, i = 1, . . . ,σ (17)

    Thus, if Sk > 0, Tk < 0, k ∈ K (g+d) andUk < 0, k ∈ K (g+d+1), i = 1, . . . ,σ then, (6–7)and Pi(α) > 0 hold ∀α ∈ ∆N, concluding the sufficiency of the proof. Necessity: if (6–7) andPi(α) > 0 hold ∀α ∈ ∆N, then from the extension of Pólya’s Theorem to the case of matrix-valued homogeneous polynomials [26–28] it is known that these homogeneous polynomials

    can be represented by homogeneous polynomials, possible ofhigher degrees, with the same

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • properties but with all coefficients positive (or negative)definite. For a given degreeg, if the

    homogeneous polynomial is positive for allα ∈ ∆N, then there exists a sufficiently larged suchthat all matrix coefficients are positive definite, implyingthat Sk > 0, Tk < 0, ∀k ∈ K (g+ d),andUk > 0 ∀k ∈ K (g+ d+ 1); Finally, note thatSk, Tk, andUk will render such coefficientssooner or latter with the increase ofd for a given degreeg.

    Note that the conditions (12–13) do not require the increaseof d when g = 1 since, to test

    the positivity of a homogeneous polynomial of degree one (affine on α), is necessary and

    sufficient to test the positivity of its coefficients. This aspect is always taken into account in the

    numerical implementations, decreasing the computationaleffort. A robust MSS condition can be

    immediately derived from the conditions of Theorem 2, as stated in next corollary.

    Corollary 1: SystemG0 is robustly MSS for allΓ∈Γ(α) if and only if there exist symmetricmatricesPik ∈ IRn×n, k∈K (g), i = 1, . . . ,σ a degreeg∈N and a sufficiently large degreed ∈Nsuch that (12) and (14) hold forEi = 0, i = 1, . . . ,σ , which rendersZ = 0.

    LMI relaxations with guaranteed convergence for the dual formulation of Lemma 3 can be

    derived similarly.

    Theorem 3: SystemG0 is robustly MSS for allΓ ∈ Γ(α) and the inequality‖G0‖22 < ρ holdsif and only if there exist symmetric matricesPik ∈ IRn×n, k∈ K (g), i = 1, . . . ,σ a degreeg∈ Nand a sufficiently large degreed ∈ N such that fori = 1, . . . ,σ , the following LMIs hold

    ∑k′∈K (d)

    k�k′

    d!π(k′)

    (

    Pik−k′)

    > 0, ∀k∈ K (g+d) (18)

    ∑k′∈K (d)

    k�k′

    ∑ℓ∈{1,...,N}

    kℓ>k′ℓ

    d!π(k′)

    ∑j=1

    µ j trace(

    E′jΛ(ℓ)j Xk−k′−eℓE j

    )

    − g!(kℓ−k′ℓ)

    π(k−k′) ρ)

    < 0,

    ∀k∈ K (g+d+1) (19)

    ∑k′∈K (d)

    k�k′

    ∑ℓ∈{1,...,N}

    kℓ>k′ℓ

    d!π(k′)

    (

    A′i(

    Λ(ℓ)i Xk−k′−eℓ)

    Ai −Pik−k′−eℓ +g!(kℓ−k′ℓ)π(k−k′) C

    ′iCi

    )

    < 0,

    ∀k∈ K (g+d+1) (20)

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • Proof: Similar to the proof of Theorem 2.

    The minimization ofρ under the LMI constraints of Theorems 2 or 3 yields an upper bound,

    i.e., a guaranteed cost for theH2 worst case norm of systemG0. Next lemma gives more details

    about the resultingρ as a function ofg (degree of the homogeneous polynomial Lyapunov

    matrices) andd (level of Ṕolya’s relaxations).

    Lemma 4: Considerρ(g,d) as the optimal solution of

    minρ

    such that(12–14) or (18–20) hold (21)

    for g andd given. Then

    (a) ρ(g+1,d) ≤ ρ(g,d);(b) ρ(g,d+1) ≤ ρ(g,d);

    Proof: Consider , for instance, minρ such that (12–14) hold. To prove (a), note that if there

    exists feasible solutionsPi(α), i = 1, . . . ,σ of degreeg for a givend to (12–14), then

    P∗i (α) = (α1 + · · ·+αN)Pi(α), i = 1, . . . ,σ

    are feasible solutions of degreeg+1 fulfilling the LMIs (12–14) for the samed, yielding at least

    an H2 guaranteed cost such thatρ(g+1,d) ≤ ρ(g,d). Item (b) is similar, that is, if there existsdegreeg solutionsP∗i (α), i = 1, . . . ,σ such thatSk > 0, Tk < 0, ∀k ∈ K (g+ d) and Uk < 0,∀k ∈ K (g+ d + 1), then these same solutions also verify (12–14) fork ∈ K (g+ d + 1) andk∈K (g+d+2), which can be generated in this case by positive combinations of the previousLMIs, assuring thus thatρ(g,d + 1) ≤ ρ(g+ d). The same properties apply to the conditions(12), (19–20).

    The results of Lemma 4 assure that the conditions of Theorems2 or 3 provide upper bounds

    ρ to theH2-norm. Wheng tends to infinity, the value ofρ tends to theH2 worst case norm.

    Note that for a given degreeg, the results of Theorems 2 and 3 may provide different upper

    bounds, but asg grows unbounded, the results will tend to the same optimal value ρ∗. Finally,

    a robust MSS condition can also be derived from conditions (18) and (20) of Theorem 3.

    Corollary 2: SystemG0 is robustly MSS for allΓ∈Γ(α) if and only if there exist symmetricmatricesPik ∈ IRn×n, k∈K (g), i = 1, . . . ,σ a degreeg∈N and a sufficiently large degreed ∈Nsuch that (18) and (20) hold withCi = 0, i = 1, . . . ,σ .

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • It is important to emphasize that the robust MSS conditions of Corollaries 1 and 2 are not

    equivalent for giveng andd, but eventually will produce the same results with the increase of

    both g andd; see an illustration in the numerical experiment section.

    IV. STATE FEEDBACK DESIGN RESULTS

    Let the systemG0 with control input be described by

    G =

    x(k+1) = Aθ(k)x(k)+Eθ(k)w(k)+Bθ(k)u(k)

    z(k) = Cθ(k)x(k)+Dθ(k)u(k), k≥ 0w∈ ℓ2, E(|x0|2) < ∞, θ0 ∼ µ

    (22)

    whereu(k) ∈ IRr is the control action. The finite sets of matricesB = {Bi ∈ IRn×r , i ∈ S } andD = {Di ∈ IRp×r , i ∈ S } are given.

    Using slack variables, as in the methods presented in [29, 33], state feedback design conditions

    for systemG with uncertainties in the probability matrix can be obtained.

    Theorem 4: If there exist symmetric matricesPik ∈ IRn×n, matricesWik ∈ IRp×p, Hi ∈ IRn×n,Zi ∈ IRr×n, k∈ K (g), i = 1, . . . ,σ , a degreeg∈ N and an integerd ∈ N such that the followingLMIs hold for eachi = 1, . . . ,σ ,

    Jk = ∑k′∈K (d)

    k�k′

    d!π(k′)

    ∑j=1

    trace(

    Wjk−k′)

    − g!π(k−k′)ρ

    )

    ≤ 0, ∀k∈ K (g+d) (23)

    Lk = ∑k′∈K (d)

    k�k′

    ∑ℓ∈{1,...,N}

    kℓ>k′ℓ

    d!π(k′)

    Wik−k′−eℓ

    λk,ℓ(CiHi +DiZi)

    ⋆ λk,ℓ(Hi +H ′i )−Γ(ℓ)i Xk−eℓ

    > 0,

    ∀k∈ K (g+d+1) (24)

    Mk = ∑k′∈K (d)

    k�k′

    ∑ℓ∈{1,...,N}

    kℓ>k′ℓ

    d!π(k′)

    Pik−k′−eℓ

    −λk,ℓµiEiE′i λk,ℓ(AiHi +BiZi)⋆ λk,ℓ(Hi +H ′i )−Γ

    (ℓ)i Xk−k′−eℓ

    > 0,

    ∀k∈ K (g+d+1) (25)

    with λk,ℓ = g!(kℓ−k′ℓ)/π(k−k′) thenKi = ZiH−1i is a mode-dependent mean square stabilizingstate feedback gain for system (22). Moreover,ρ is a guaranteed cost for theH2-norm of the

    closed-loop system.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • Proof: Multiplying the left-hand side of (23) byαk11 αk22 · · ·α

    kNN and summing up fork ∈

    K (g+d) and multiplying the left-hand side of (24–25) and summing upfor k∈K (g+d+1),with Âi = Ai +BiZiH−1i andĈi = Ci +DiZiH

    −1i , one has that

    ρ ≥σ

    ∑j=1

    trace(Wj(α)

    ), (26)

    Wi(α) ĈiHi

    ⋆ Hi +H ′i −σ

    ∑j=1

    p ji (α)Pj(α)

    > 0, (27)

    Pi(α)−µiEiE′i ÂiHi

    ⋆ Hi +H ′i −σ

    ∑j=1

    p ji (α)Pj(α)

    > 0 (28)

    Note that (28) impliesPi(α)−µiEiE′i > 0, thus assuringPi(α) > 0, i = 1, . . . ,σ . Multiplying (27)on the left by[I −Ĉi] and on the right by its transpose, and multiplying (28) on theleft by[I − Âi] and on the right by its transpose, one has, respectively, that

    Wi(α) > Ĉi

    ∑j=1

    p ji (α)Pj(α)

    )

    Ĉ′i , Pi(α)− Âi(

    σ

    ∑j=1

    p ji (α)Pj(α)

    )

    Â′i −µiEiE′i > 0 (29)

    The choiceP̄i(α) =(

    ∑σj=1 p ji (α)Pj(α))

    in (29) yields

    Wi(α) > ĈiP̄i(α)Ĉ′i , P̄i(α)−σ

    ∑j=1

    p ji (α)(Â j P̄j(α)Â′j −µ jE jE′j

    )> 0 (30)

    Taking into account (26),Ki = ZiH−1i is a mode-dependent mean square stabilizing state feedback

    gain with anH2 guaranteed cost given byρ.

    In contrast with the LMI relaxations presented in the analysis case, which are necessary (for

    a large enoughg andd) and sufficient, the conditions of Theorem 4 for state feedback control

    design are only sufficient. This restriction is intrinsically related to the use of the slack variables

    as proposed in [29, 33]. Despite the sufficiency, the increase ofg andd can improve the estimation

    of H2 guaranteed costs, as illustrated in the section of numerical experiments. In addition, if the

    states of the Markov chain are not available for observations, one can easily adapt the conditions

    of Theorem 4 to cope with robust state feedback design (i.e. mode independent control).

    Corollary 3: If the conditions of Theorem 4 are satisfied withHi, Zi replaced byH, Z,

    respectively, thenK = ZH−1 is a mode-independent mean square stabilizing gain for system (22).

    Moreover,ρ is a guaranteed cost for theH2-norm of the closed-loop system.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • It is noteworthy that, if the probability matrix is precisely known, the conditions in Theorem 4

    and Corollary 3 withg = 0, d = 0 retrieve the results presented in [29]. Theorem 4 can also be

    used to cope with the mean square stabilization problem, as presented in next corollary.

    Corollary 4: Let Ei = 0, i = 1, . . . ,σ . If there exist symmetric matricesPik ∈ IRn×n, matricesHi ∈ IRn×n, Zi ∈ IRr×n, k ∈ K (g), i = 1, . . . ,σ , a degreeg∈ N and an integerd ∈ N such that(25) holds fori = 1, . . . ,σ , thenKi = ZiH−1i is a mode-dependent mean square stabilizing state

    feedback gain for system (22).

    V. NUMERICAL EXAMPLES

    We first evaluate the performance of the proposed method in terms of the efficiency of the

    results produced by the LMI relaxations and the computational effort. The numberv of scalar

    variables, the numberr of LMI rows and the computational times are given in the examples

    presented in the sequel, which have been performed in a Pentium IV 2.6 GHz, 512 MB RAM,

    using SeDuMi [34] and YALMIP [35] within the Matlab environment.1

    Example 1

    This example is concerned with a MJLS (borrowed from [2, 29])for which the data are

    A1 =

    2 2

    3 1

    , A2 =

    1 0

    0.5 1

    , B1 =

    2

    1

    , B2 =

    0

    0

    , C1 =

    1 −11 1

    0 0

    , C2 =

    1 0

    0 1

    0 0

    ,

    E1 =

    0.5 0

    0 0.4

    , E2 =

    1 0

    0 0.8

    , D1 =

    0

    0

    1

    , D2 =

    0

    0

    1

    , µ =[

    0 1]

    Here, the probability matrix is not precisely known, affected by uncertainties in the form

    Γ(λ ,η) = A1 =

    0.9−λ 0.1+λ0.8−η 0.2+η

    , λ ∈ [0 0.5], η ∈ [0 0.5].

    yielding a polytopeΓ(α) of four vertices, constructed by taking the maximum and minimum

    values ofλ and η . The conditions of Theorems 4 and Corollary 3 are used to synthesize state

    1The numerical implementation of all theorems presented in the paper can be downloaded from www.dt.fee.unicamp.br/˜

    ricfow/robust.htm.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • feedback gains and the results are shown in Table I. The resulting H2 guaranteed costs indicate

    that the conditions of Theorems 4 and Corollary 3 cannot provide smaller values with the increase

    of g and d (within the precision considered). Forg = 2 andd = 1, the robust state feedback

    gain provided by Corollary 3 is given byK = [−1.1974 −0.9921] and the mode-dependentgains provided by Theorem 4 are given byK1 = [−1.1648 −0.9949], K2 = [0 0]. Becausethe conditions of Theorems 4 and Corollary 3 are only sufficient, the necessary and sufficient

    conditions of Theorems 2 and 3 were used to determine tighterbounds to theH2 guaranteed

    costs for the closed-loop systems. Using the controllers obtained in the mode-dependent case,

    Theorem 3 yields√ρ = 6.50 for g= 1 andd = 0 or d = 1. For the mode-independent controller,

    Theorem 3 yields√ρ = 7.96 for g = 1 andd = 0 or d = 1.

    Concerning only mean square stabilization by means of mode-dependent gains, the conditions

    of Corollary 4 can be compared with the approach of [15, Theorem 4.2], which is based on

    affine parameter-dependent Lyapunov matrices. Forg = 1, d = 0, Corollary 4 yields a feasible

    solution demandingv= 34, r = 80 and 0.09 s while [15, Theorem 4.2] requiresv= 82, r = 64 and

    0.10 s to provide a feasible solution. Figure 1 shows (in logarithmscale) the number of decision

    variablesv and the number of LMI rowsr as a function ofn = N = σ , n = σ = N = 1, . . . ,10,

    for [15, Theorem 4.2] ([15, T4.2]) and Corollary 4 (C4) withg = 1,d = 0 when the number

    of control inputs isr = 1. Although for this example both methods require comparable amount

    of computation, the growth of the number of variables when using [15, Theorem 4.2] is more

    prominent than the one associated to Corollary 4, as shown in Figure 1 (a), while the increase

    in the number of LMI rowsr is similar, see Figure 1 (b).

    Example 2

    Consider the MJLS defined in (1) with the state-space matricesgiven by

    A1 =

    0.6 0.4

    0.3 0.7

    , A2 =

    0.2 0.6

    0.6 0.5

    , A3 =

    0.3 0.3

    0.4 0.5

    , Ei =

    2 0

    0 1

    , Ci =

    1 0

    0 1

    , i = 1,2,3

    The uncertain transition probability matrix is given by

    Γ(α) = α1Γ(1) +α2Γ(2) +α3Γ(3), α1 +α2 +α3 = 1, αr ≥ 0, r = 1,2,3

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • 1 2 3 4 5 6 7 8 9 1010

    0

    101

    102

    103

    104

    105

    106

    n = σ = N

    v[15, T4.2]

    C4g=1

    (a)

    1 2 3 4 5 6 7 8 9 1010

    0

    101

    102

    103

    104

    105

    n = σ = N

    r

    [15, T4.2]

    C4g=1

    (b)

    Fig. 1. Values ofv (fig. (a)) andr (fig. (b)) demanded by [15, Theorem 4.2] ([15, T4.2], solid line) and Corollary 4 (C4,

    dashed line) forg = 1 andd = 0, using the dimensionsn = σ = N = 1, . . . ,10 andr = 1.

    TABLE I

    STATE FEEDBACK DESIGN FOREXAMPLE 1 USING THE CONDITIONS OFTHEOREMS4 (T4) AND COROLLARY 3 (C3);v IS

    THE NUMBER OF SCALAR VARIABLES, r IS THE NUMBER OFLMI ROWS; COMPUTATIONAL TIMES IN SECONDS.

    mode-dependent mode-independent

    Method√ρ v Time Method √ρ v Time r

    T4g=0,d=0 11.58 31 0.2 C3g=0,d=0 13.01 25 0.2 77

    T4g=1,d=0 7.89 85 0.5 C3g=1,d=0 9.16 79 0.5 200

    T4g=1,d=1 7.82 85 0.9 C3g=1,d=1 9.10 79 0.8 380

    T4g=1,d=2 7.80 85 1.6 C3g=1,d=2 9.09 79 1.4 650

    T4g=1,d=3 7.78 85 2.5 C3g=1,d=3 9.08 79 2.5 1028

    T4g=2,d=0 7.66 193 1.1 C3g=2,d=0 9.08 187 1.1 410

    T4g=2,d=0 7.66 193 2.1 C3g=2,d=1 9.08 187 1.9 730

    with

    Γ(1) =

    0.09 0.81 0.10

    0.27 0.32 0.41

    0.11 0.32 0.57

    , Γ(2) =

    0.74 0.04 0.22

    0.26 0.36 0.38

    0.71 0.02 0.27

    , Γ(3) =

    0.64 0.33 0.03

    0.17 0.36 0.47

    0.79 0.20 0.01

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • The aim is to computeH2 guaranteed costs for this system using the results of Theorems 2

    and 3 for various values ofg andd. Table II shows the guaranteed costs as well asv, r and the

    computational times. In both cases (Theorems 2 and 3), it seems that convergence was attained

    for g = 5, since the results do not present significant changes within the adopted precision (two

    decimal digits).

    Again, if only robust MSS is investigated, it is immediate toconclude that the conditions of

    Corollaries 1 and 2 are not equivalent. In fact, Corollary 1 with g = 0,d = 0 (v = 9, r = 24) is

    not able to assure robust MSS, requiring at leastg = 1,d = 0. On the other hand, Corollary 2

    assures robust MSS withg = 0,d = 0 (v = 9, r = 24). For comparison purposes, the robust MSS

    condition of [15, Theorem 3.3] (based on affine parameter-dependent Lyapunov matrices) was

    also applied (v = 183, r = 90) and no feasible solution was found. For this example, it can be

    noted that the use of parameter-independent (i.e. constant) slack matrices as in [15, Theorem

    3.3]) increases considerably the number of decision variables but conservativeness still remains.

    TABLE II

    H2 GUARANTEED COSTS FOREXAMPLE 2 USING THEOREMS2 AND 3; v IS THE NUMBER OF SCALAR VARIABLES, r IS THE

    NUMBER OF LMI ROWS; COMPUTATIONAL TIMES IN SECONDS.

    Method√ρ r Time Method √ρ r Time v

    T2g=0,d=0 – 25 0.1 T3g=0,d=0 5.56 27 0.1 10

    T2g=1,d=0 7.06 57 0.2 T3g=1,d=0 5.28 60 0.2 28

    T2g=1,d=4 6.79 315 0.7 T3g=1,d=4 5.25 322 0.7 28

    T2g=1,d=5 6.75 412 1.0 T3g=1,d=5 5.25 420 0.8 28

    T2g=3,d=0 5.25 160 0.3 T3g=3,d=0 5.25 165 0.3 91

    T2g=3,d=3 5.24 412 0.8 T3g=3,d=3 5.24 420 0.8 91

    T2g=3,d=4 5.24 522 1.2 T3g=3,d=4 5.24 531 1.1 91

    T2g=5,d=0 5.24 315 0.4 T3g=5,d=0 5.24 322 0.4 190

    T2g=5,d=1 5.24 412 0.7 T3g=5,d=1 5.24 420 0.9 190

    Example 3

    Consider a MJLS defined in (1) with the state-space matrices given by A1 = 0.999, A2 =

    0.986, A3 = 1.082, A4 = 0.931, Ei = 1, Ci = 1, i = 1, . . . ,σ . Also, set µ = [1 0 0 0] and

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • Γ(α) = α1Γ(1) +α2Γ(2), α1 +α2 = 1, αr ≥ 0, r = 1,2 with

    Γ(1) =

    0.16 0.74 0.01 0.09

    0.59 0.02 0.11 0.28

    0.02 0.15 0.55 0.28

    0.65 0.13 0.21 0.01

    , Γ(2) =

    0.13 0.48 0.27 0.12

    0.48 0.35 0.08 0.09

    0.43 0.01 0.26 0.30

    0.51 0.36 0.00 0.13

    The conditions of Theorems 2 and 3 were applied to provideH2 guaranteed costs for this

    system, and Table III show the results. Untild = 20, the conditions of Theorems 2 and 3

    were not able to detect robust MSS forg = 1, indicating that the robust MSS of this system

    cannot be assured by means of affine parameter-dependent Lyapunov matrices. Forg≥ 2, theproposed conditions converge rapidly. The robust MSS condition [15, Theorem 3.3], based on

    affine parameter-dependent Lyapunov matrices, was not ableto detect robust MSS (v= 92,r = 48,

    0.32 s). On the other hand, Corollary 2 assures MSS withg = 2,d = 0 (v = 12, r = 28, 0.05 s).

    This example illustrates that even for small dimension systems, Lyapunov matrices depending

    on higher degrees of the uncertain parameterα can be necessary to detect robust MSS and,

    consequently, to provideH2 guaranteed costs.

    TABLE III

    H2 GUARANTEED COSTS OBTAINED FOREXAMPLE 3 USING THEOREMS2 AND 3; v IS THE NUMBER OF SCALAR

    VARIABLES, r IS THE NUMBER OFLMI ROWS; COMPUTATIONAL TIMES IN SECONDS.

    Method√ρ r Time Method √ρ r Time v

    T2g=0,d=0 – 13 0.05 T3g=0,d=0 – 14 0.04 5

    T2g=1,d=20 – 102 0.11 T3g=1,d=20 – 123 0.17 9

    T2g=2,d=7 524.36 94 0.13 T3g=2,d=0 137.18 32 0.08 13

    T2g=3,d=0 126.81 40 0.08 T3g=3,d=0 126.81 41 0.08 17

    T2g=3,d=1 126.81 49 0.08 T3g=3,d=0 126.81 50 0.08 17

    Example 4

    Finally, consider the MJLS representing an economic system, with state-space model as in (1)

    and matrices given in Table IV. The input variableu represents the government expenditure and

    the state variablex2 represents the national income. More details about the economic system can

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • TABLE IV

    PARAMETERS OF THEMJLS OF EXAMPLE 4 (ECONOMIC SYSTEM)

    Operation modes

    Parameters i = 1 (normal) i = 2 (boom) i = 3 (slump)

    Ai

    0 1

    −2.5 3.2

    0 1

    −4.3 4.5

    0 1

    5.3 −5.2

    Bi

    0

    1

    0

    1

    0

    1

    Ci

    1.5477 −1.0976−1.0976 1.9145

    0 0

    3.1212 −0.5082−0.5082 2.7824

    0 0

    1.8385 −1.2728−1.2728 1.6971

    0 0

    Di

    0

    0

    1.6125

    0

    0

    1.0794

    0

    0

    1.0540

    be found in [14, 36]. The model considers three distint situations for the economy, associated

    to the operation modes:i = 1 (normal), i = 2 (boom) andi = 3 (slump). The dynamics of the

    MJLS follows a Markov chainθ(k).

    ConsideringEi = [1 1]′, i = 1,2,3 andµ = [0 0 1] the aim here is to design mode-dependent

    state-feedback controllers through the LMI relaxations ofTheorem 4, that is, to provide an input

    u which is proportional to the actual incoming but also takes into account the scenario of the

    economy. To appropriately model this uncertain situation,the probability matrix is not precisely

    known, but can assume arbitrary values (inside the unit simplex) whenever the system operates

    on modesi = 2 (boom) andi = 3 (slump). When the economy is on the normal model (i = 1),

    the probability to stay is 50% and to jump to modesi = 2 or i = 3 is 25%. Hence, the uncertain

    probability matrix is given by

    0.5 0.25 0.25

    β1 β2 β3λ1 λ2 λ3

    yielding Γ(r), r = 1, . . . ,9 (all combinations of the extreme points of 0≤ βi ≤ 1, 0≤ λi ≤1, i = 1,2,3). Using the results presented in Theorem 4, it is possible to synthesize mode-

    dependent controllers forg = 0,d = 0 (H2 guaranteed cost(√ρ = 11.9154)) and g = 1,d = 0

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • (H2 guaranteed cost(√ρ = 7.7269)). The use of a mode-dependent controller of degreeg = 1

    improved theH2 guaranteed cost in 35.15%. In this particular example, Corollary 3 fails to

    provide a mode-independent control gain that stabilizes the closed-loop system.

    To give an idea of the dynamical behavior of the economic system associated to two mode-

    dependent controllers (i.e.g= 0 andg= 1), Monte Carlo simulations have been performed using

    the transition probability matrix

    Γ =

    0.50 0.25 0.25

    0 1 0

    0 0 1

    for a total of 100 possible realizations of the Markov chain.The mean values of the national

    incomex2(k) and the control effortu(k) are shown in Figure 2, for the initial conditionx(0) =

    [1 1]′, without noise (i.e.w(k) = 0). This example illustrates the ability of the proposed approach

    to cope with models that represent real world problems, especially in the case where few

    information about the transition probability matrix is available.

    VI. CONCLUSION

    The paper studies the mean square stability and stabilization, theH2 guaranteed norm and

    control synthesis for Markov jump linear systems when the transition probability matrix is not

    precisely known. Theorems 2 and 3 and their corollaries present in dual forms the conditions for

    theH2-norm evaluation and the stability assessment in terms of LMIs. TheH2-control synthesis

    for mode-dependent and mode-independent feedback gains are, respectively, in Theorem 4 and

    its corollary.

    In the stability test, the conditions can be made more and more refined to attain a necessary

    and sufficient level, at expenses of larger and larger computational burden. This yields in the

    analysis case an LMI relaxation procedure which gets arbitrarily near to theH2 worst case norm.

    This in turn, allows one to cope with control design dealing with sufficient LMI conditions of

    increasing precision. Numerical test cases illustrate thepotential use of these methods, aimed at

    real-world problems.

    REFERENCES

    [1] Y. Ji and H. J. Chizeck, “Controllability and stabilizability and continuous-time jump linear-quadratic control,”IEEE Trans.

    Automat. Contr., vol. 35, no. 7, pp. 777–788, 1990.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • 0 2 4 6 8 10 12 14 16 18 200

    0.2

    0.4

    0.6

    0.8

    1

    0 2 4 6 8 10 12 14 16 18 20−0.3

    −0.2

    −0.1

    0

    0.1

    0.2

    time

    time

    mea

    nva

    lue

    ofx 2

    inpu

    t

    T4g=0

    T4g=0

    T4g=1

    T4g=1

    Fig. 2. Mean values of national incomex2(k) and control effortu(k) obtained by the Monte Carlo simulation of Example 4.

    [2] ——, “Jump linear quadratic Gaussian control: steady-state and testable conditions,” Control Theory and Advanced

    Technology, vol. 6, pp. 289–319, 1990.

    [3] X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, “Stochastic stabilityproperties of jump linear systems,”IEEE Trans.

    Automat. Contr., vol. 37, no. 1, pp. 38–53, 1992.

    [4] O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,”

    J. of Math. Anal. Appl., vol. 179, pp. 154–178, 1993.

    [5] ——, “Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems,”IEEE Trans. Automat.

    Contr., vol. 40, no. 12, pp. 2076–2088, 1995.

    [6] E. F. Costa, J. B. R. do Val, and M. D. Fragoso, “A new approachto detectability of discrete-time infinite Markov jump

    linear systems,”SIAM J. Control Optim., vol. 43, no. 6, pp. 2132–2156, 2005.

    [7] O. L. V. Costa, M. D. Fragoso, and R. P. Marques,Discrete-Time Markovian Jump Linear Systems. New York: Springer-

    Verlag, 2005.

    [8] R. A. Maronna, D. R. Martin, and V. J. Yohai,Robust Statistics: Theory and Methods. John Wiley & Sons, 2006.

    [9] Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust finite-horizonfiltering for stochastic systems with missing

    measurements,”IEEE Signal Processing Lett., vol. 12, no. 6, pp. 437–440, 2005.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • [10] K. D. Kuhn and S. M. Madanat, “A new approach to detectability of discrete-time infinite Markov jump linear systems,”

    Computer-Aided Civil and Infrastructure Engineering, vol. 21, no. 3, pp. 171–178, 2006, postprint available free at:

    http://repositories.cdlib.org/postprints/1260.

    [11] A. El Bouhtouri and K. El Hadri, “Robust stabilization of jump linearsystems with multiplicative noise,”IMA J. Math.

    Contr. Information, vol. 20, no. 1, pp. 1–19, 2003.

    [12] A. Nilim and L. El Ghaoui, “Robust control of Markov decision processes with uncertain transition matrices,”Operations

    Research, vol. 53, no. 5, pp. 780–798, September-October 2005.

    [13] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “A convex programming approach toH2-control of discrete-time

    Markovian jump linear systems,”Int. J. Contr., vol. 66, pp. 557–579, 1997.

    [14] O. L. V. Costa, E. O. Assumpção Filho, E. K. Boukas, and R. P. Marques, “Constrained quadratic state feedback control

    of discrete-time Markovian jump linear systems,”Automatica, vol. 35, no. 4, pp. 617–626, April 1999.

    [15] C. E. de Souza, “Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,”IEEE Trans.

    Automat. Contr., vol. 51, no. 5, pp. 836–841, May 2006.

    [16] L. El Ghaoui and M. Ait Rami, “Robust state-feedback stabilizationof jump linear systems via LMIs,”Int. J. Robust

    Nonlinear Contr., vol. 6, no. 9–10, pp. 1015–1022, November–December 1996.

    [17] P. A. Parrilo, “Exploiting structure in sum of squares programs,”in Proc. 42nd IEEE Conf. Decision Contr., Maui, HI,

    USA, December 2003, pp. 4664–4669.

    [18] J. B. Lasserre, “Global optimization with polynomials and the problemof moments,”SIAM J. Control Optim., vol. 11,

    no. 3, pp. 796–817, February 2001.

    [19] C. W. Scherer and C. W. J. Hol, “Matrix sum-of-squares relaxations for robust semi-definite programs,”Mathematical

    Programming Series B, vol. 107, no. 1–2, pp. 189–211, June 2006.

    [20] M. Kojima, “Sums of squares relaxations of polynomial semidefiniteprograms,” Department of Mathematical and

    Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, Research Report B-397, November 2003.

    [21] M. Kojima and M. Muramatsu, “An extension of sums of squares relaxations to polynomial optimization problems over

    symmetric cones,”Mathematical Programming Series B, vol. 110, no. 2, pp. 315–336, July 2007.

    [22] D. Henrion and J. B. Lasserre, “Convergent relaxations of polynomial matrix inequalities and static output feedback,”

    IEEE Trans. Automat. Contr., vol. 51, no. 2, pp. 192–202, February 2006.

    [23] M. Choi, T. Y. Lam, and B. Reznick, “Sum of squares of real polynomials,” in Proceedings of Symposia in Pure

    Mathematics, vol. 58.2, Providence, RI, 1995, pp. 103–126.

    [24] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter-dependent Lyapunov functions for robust stability

    of polytopic systems: an LMI approach,”IEEE Trans. Automat. Contr., vol. 50, no. 3, pp. 365–370, March 2005.

    [25] ——, “Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions,”

    Automatica, vol. 43, no. 2, pp. 309–316, February 2007.

    [26] C. W. Scherer, “Higher-order relaxations for robust LMI problems with verifications for exactness,” inProc. 42nd IEEE

    Conf. Decision Contr., Maui, HI, USA, December 2003, pp. 4652–4657.

    [27] ——, “Relaxations for robust linear matrix inequality problems with verifications for exactness,”SIAM J. Matrix Anal.

    Appl., vol. 27, no. 2, pp. 365–395, 2005.

    [28] R. C. L. F. Oliveira and P. L. D. Peres, “LMI conditions for the existence of polynomially parameter-dependent Lyapunov

    functions assuring robust stability,” inProc. 44th IEEE Conf. Decision Contr. — Eur. Control Conf. ECC 2005, Seville,

    Spain, December 2005, pp. 1660–1665.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.

  • [29] J. B. R. do Val, J. C. Geromel, and A. P. Gonçalves, “TheH2-control for jump linear systems: cluster observations of the

    Markov state,”Automatica, vol. 38, no. 2, pp. 343–349, February 2002.

    [30] S. Boyd and L. Vandenberghe,Convex Optimization. Cambridge, UK: Cambridge University Press, 2004.

    [31] P.-A. Bliman, “An existence result for polynomial solutions of parameter-dependent LMIs,”Syst. Contr. Lett., vol. 51, no.

    3-4, pp. 165–169, March 2004.

    [32] P.-A. Bliman, R. C. L. F. Oliveira, V. F. Montagner, and P. L. D.Peres, “Existence of homogeneous polynomial solutions

    for parameter-dependent linear matrix inequalities with parameters in the simplex,” in Proc. 45th IEEE Conf. Decision

    Contr., San Diego, CA, December 2006, pp. 1486–1491.

    [33] M. C. de Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,”Syst. Contr. Lett.,

    vol. 37, no. 4, pp. 261–265, July 1999.

    [34] J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimizationover symmetric cones,”Optimization Methods

    and Software, vol. 11–12, pp. 625–653, 1999, URL: http://sedumi.mcmaster.ca/.

    [35] J. Löfberg, “YALMIP : A toolbox for modeling and optimization in MATLAB,” inProceedings of the IEEE

    CCA/ISIC/CACSD Multiconference, Taipei, Taiwan, 2004, URL: http://control.ee.ethz.ch/˜joloef/yalmip.php.

    [36] W. P. Blair and D. D. Sworder, “Feedback control of a class of linear discrete systems with jump parameters and quadratic

    cost criteria,”Int. J. Contr., vol. 21, no. 5, pp. 833–841, 1975.

    http://dx.doi.org/10.1080/00207170802136178 OLIVEIRA, R. C. L. F. ; Vargas, A. N. ; do Val, J.B.R. ; PERES, P. L. D. . Robust stability, H2 analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix. International Journal of Control, v. 82, p. 470-481, 2009.