ON THE ROBUST STABILITY, STABILIZATION ... - Semantic Scholar

SIAM J. CONTROL OPTIM. Vol. 49, No. 3, pp. 1171–1196

c 2011 Society for Industrial and Applied Mathematics

ON THE ROBUST STABILITY, STABILIZATION, AND STABILITY RADII OF CONTINUOUS-TIME INFINITE MARKOV JUMP LINEAR SYSTEMS∗ MARCOS G. TODOROV† AND MARCELO D. FRAGOSO† Abstract. This paper addresses the robust stochastic stability and stabilization of continuoustime Markov jump linear systems (MJLS), with the Markov jump parameters taking values in a countably infinite set. It is assumed that the state and input matrices are subjected to normbounded uncertainty with a prespecified structure, which encompasses the block-diagonal setting. We introduce new robust analysis and synthesis characterizations such that, unlike previous approaches in the MJLS literature, the scaling parameters are treated as decision variables in linear matrix inequalities. As a by-product, new contributions to the theory of stability radii of MJLS are provided. When restricted to the finite case, we further introduce new adjoint linear matrix inequality (LMI) characterizations for each of the robust analysis and synthesis problems, as well as for stability radii. Besides the interest in its own right, the adjoint approach allows us to verify that, in the general MJLS case, there is a gap between the complex stability radius and what can be assessed with scaled versions of the small-gain theorem. This suggests a fundamental limitation of the robustness against linear perturbations that the H∞ control of MJLS may provide. Some numerical examples, which include the robust control of two interconnected oscillators, illustrate the main results. Key words. continuous-time linear systems, Markov jump parameter, robust control, scaling, stability radii, H∞ control AMS subject classifications. 93C05, 93D09, 93D15, 60J27, 60375 DOI. 10.1137/090774410

1. Introduction. In this paper we present new contributions to the robust stability and control of the so-called Markov jump linear systems (MJLS). This class of systems has been the subject of intensive research in recent years and is particularly powerful in capturing a kind of phenomenon that is observed in an enormous range of processes, in nature and industry, by modeling the occurrence of abrupt structural changes. Although the MJLS class seems, prima facie, just an extension of linear systems, it differs from them in many instances. This is due, in particular, to some emergence properties of this class of systems, that certainly can be included in the class of complex systems (roughly, a system composed of interconnected parts that as a whole exhibit one or more properties not obvious from the properties of the individual parts). In the specialized literature, applications can be found spanning solar-thermal stations [54], aeronautics [52], robotics [51], communication systems [48], economics [2, 14, 53], mobile networks [1], and many other fields. We refer to [3, 12] for a detailed discussion and list of references. There is by now in the MJLS literature a fairly large number of different approaches to the problems of robust analysis and control in the face of uncertainty. Previous work includes (but is not limited to) [4, 8, 11, 15, 16, 19, 20, 24, 41, 43, 50, 59, 60, 63, 66] in the continuous-time setting. In [4], different classes of structured ∗ Received by the editors October 20, 2009; accepted for publication (in revised form) March 4, 2011; published electronically May 19, 2011. A preliminary version of this paper appeared in the Proceedings of the 48th Annual IEEE Conference on Decision and Control [59]. This work was partially supported by the Brazilian National Research Council – CNPq, under grant 302501/2010-0, and by FAPERJ under grants E-26/170.008/2008 and E-26/100.437/2009. http://www.siam.org/journals/sicon/49-3/77441.html † National Laboratory for Scientific Computing–LNCC/CNPq, Av. Get´ ulio Vargas 333, Petr´ opolis, Rio de Janeiro, CEP 25651-070, Brazil ([email protected], [email protected]).

1171

1172

MARCOS G. TODOROV AND MARCELO D. FRAGOSO

uncertainty were treated, under the assumption of incomplete knowledge of the transition rates. More recently, [66] investigated the robust stability and stabilization of MJLS with partly unknown transition rates. References [16, 17, 19, 20] addressed the robust stability and stabilization of linear systems with Markov jump parameters driven by Wiener processes. We mention also the norm-bounded uncertainty settings of [10, 13, 64] and the case in which uncertainty stems only from nonexact knowledge of the switching rates (see, e.g., [24, 41]). As for the robust H2 and H∞ control problems, a sample of previous work can be found in, e.g., [8, 11, 15, 43, 50, 63]. A particular problem which has received considerable attention from the robust control community is that of characterizing the stability radius of a given system. Roughly speaking, a stability radius corresponds to the norm of the smallest perturbation (within a prescribed class) that makes a given system unstable. Stability radii can thus be regarded as quantitative measures of how robust a system is, in the face of parametric uncertainty, and therefore as playing a central role in the robustness analysis of dynamic systems. Research on this subject was initiated by Hinrichsen and Pritchard in the late 1980s (see, e.g., [34] or [38, section 5.3] for a historical account). Among the different scenarios studied in the last two decades we mention, without any intention of being exhaustive, the linear time-invariant [32, 33] and time-varying [31, 35] cases, the infinite dimensional setting of [47], and the composite systems frameworks of [39, 42]; see also [22, 23, 36, 37, 44, 45] regarding stochastic systems. As for linear systems with Markov jump parameters, the earlier efforts in this direction seem to have been made in [46, 16, 19, 20], always in the finite case. More recently, [55, 57] featured the introduction of stability radii for MJLS in the countably infinite case. Our general concern in this paper is with the robust stability and stabilization of the continuous-time uncertain system (1.1) x(t) ˙ = A(θt ) + DA (θt ) x(t) + B(θt ) + DB (θt ) u(t), in which θ = {θt , t ≥ 0} is a right-continuous Markov process taking values in a discrete state space. We shall be particularly interested here in advancing the following issues, which are inherent to the robust analysis scenario: (i) the class of uncertain parameters, (ii) the scaling technique, and (iii) the stability radius analysis. In addition, a new framework for robust stability is introduced via a certain adjoint Lyapunov operator. Besides, we focus on the general case in which θ takes values in the set S = {1, 2, . . .}, to which we refer throughout the paper as the infinite case (see, e.g., [26, 28, 29] for further discussion on this setting). Whenever it is relevant, the results obtained here are also specialized to the finite case, in which S = {1, . . . , N }. Regarding the class of uncertainty, we shall be considering (1.2)

DA (i) =

M m=1

m Eim Δm i Fi ,

DB (i) =

M

m m Gm i ∇i Hi ,

θt = i,

m=1

with M < ∞. The uncertain parameters, denoted Δ and ∇, are assumed to be norm-bounded and to possess a prespecified structure for each individual mode of operation. This can be regarded as an extension of the setting commonly encountered in linear robust control theory (see, e.g., [18, 38, 67]) towards the MJLS scenario. For instance, by letting M = 1 we recover the usual setting in which Δ and ∇ are single disturbances. More generally, (1.2) allows for the description of blockdiagonal uncertainty, which is well known to naturally come up in the context of

1173

ROBUST STABILITY AND STABILITY RADII OF MJLS

composite systems (we once again refer to [18, 38, 67] and [39, 42]). In the finite case, we further assume that the transition rates of the Markov process are subject to polytopic uncertainty. This provides a quite general framework for the robust analysis and control of MJLS, which can be regarded as a small step towards a theory of robust control for systems subjected to uncertain coupling and failures. In section 6.1 an example regarding the robust control of two coupled linear oscillators illustrates a situation in which such a class of models arises naturally. When reduced to the finite case, S = {1, . . . , N }, the aforementioned setting still stands as a generalization of the norm-bounded uncertainty scenario found in the current MJLS literature. Also in this case, the paper introduces a new framework for studying robust stability and control of linear systems with Markov jump parameters in terms of the adjoint Lyapunov operator (denoted here as L, which corresponds to the infinitesimal generator of the autocorrelation of the Markovianized state process, x(t)1{θt =i} , i ∈ S). In other words, we explore the fact that, in the finite case, the operators T = (T1 , . . . , TN ) and L = (L1 , . . . , LN ), with λij Pj , Li (Q) = Ai Qi + Qi A∗i + λji Qj (1.3) Ti (P ) = A∗i Pi + Pi Ai + j∈S

j∈S

N for each i ∈ S, are adjoint with respect to the inner product U, V = i=1 tr(Ui Vi∗ ). (For more details, see [29].) Rather surprisingly, the robustness margins obtained via these two approaches are not the same (this is verified with numerical examples in sections 6.1 and 6.2). Furthermore, it is not clear how to extend the adjoint approach to the countably infinite scenario, which further distinguishes it from the finite case. Under the additional assumption of polytopic uncertainty on the transition rates of the Markov process, two alternative methods for the analysis and design of robust controllers are presented in terms of uncertainty-dependent linear matrix inequality methods, which are apparently new in the MJLS literature. Regarding the scaling technique issue, a fundamental contribution of this paper is to make new inroads on this subject in the MJLS setting. In order to properly treat the highly structured class of uncertainty considered here, we carry out a deeper exploration of scaling techniques than what is currently available in the MJLS literature (we refer to, e.g., [4, 19, 20, 21, 64]), aiming towards the more general scenario found in linear robust control theory (as can be found, e.g., in the textbooks [5, 18, 38, 67]). The main tool in this setting (Lemma 3.3) allows us to characterize controllers whose design incorporates the structure of the uncertainty without resorting to a small-gain approach (which, in contrast, was the one carried out in [13, 16, 17, 19, 20, 21, 23, 55, 60]). A highly relevant, and apparently new, consequence of this fact is that, in general, there is a gap between the complex stability radius of MJLS with respect to linear perturbations, and the robust stability margin that can be obtained with scaled versions of the small-gain theorem. Roughly speaking, this interesting property indicates a fundamental limitation on the degree of robustness against linear perturbations that the H∞ control of MJLS may offer. As proven in [33], the same does not hold in the linear time-invariant scenario. This topic is addressed in section 6.2. One scenario in which the adequate treatment of scaling techniques is of utmost importance is revealed in the study of stability radius, which has long been recognized as a challenging research topic in systems theory. In the context of linear stochastic systems, for instance, it was shown in [23, 36, 45, 22] that optimally scaled problems provide the exact assessment of the complex stability radius with respect to nonlinear

1174


perturbations, whereas only conservative robustness margins can be attained through unscaled methods. More generally, the advent of scaling allows for the introduction of less conservative analysis and synthesis methods (see, e.g., [19, 20, 21, 37, 40]). However, due to nonconvexity, direct optimization of the scaling parameters may turn out to be a difficult task. Bearing this in mind, in this paper we aim at introducing optimization procedures which depend affinely on the scalings, so that convexity is not lost. As a consequence, here the calculation of estimates for the stability radius is to be achieved by means of standard bisection on linear matrix inequality problems (as in [59, Algorithm 1]), which is rather amenable to computer solution. Regarding stability radius, our goal is to present new contributions on robust analysis of MJLS with respect to linear full-block ˙ = perturbations of the form x(t) ˙ = A(θt ) + E(θt )Δ(θt )F (θt ) x(t), in light of the aforementioned A(θt )x(t) x(t) results introduced here. In the infinite case, this comprises a deeper study of the analysis problem posed in [55, 57], including the exact characterization of both the complex and real stability radii of scalar MJLS, plus the introduction of a spectral approach to the problem. An illustrative example is also presented in section 5.2. In the finite case, we further introduce a new bound for the estimation of stability radius (Theorem 5.8), together with a rather complete characterization of the robust stochastic stability of scalar MJLS. It should be emphasized here that, different from previous work in the literature (such as, e.g., [19, 20, 21, 36]), our results are tailored in such a way so as to avoid the issues of scaling optimization. As shown in [57], direct optimization of the scaling parameters is a difficult nonconvex problem. Besides, in the current MJLS scenario there seem to be no guarantees of existence or uniqueness of the optimal scaling (such as those presented in, e.g., [36]). Instead, the result presented here is based on linear matrix inequalities, in which the scaling parameters represent nothing but additional decision variables. This paper is organized as follows. In section 2 we provide the bare essentials of notation. Section 3 introduces the class of systems to be studied here, followed by some preliminaries. In section 4 we address robust stability analysis and stabilization problems, with special attention to the finite case of polytopic transition rates in section 4.1. Section 5 is devoted to stability radius theory, in both the countably infinite and finite MJLS cases. Some numerical examples can be found in section 6 and brief concluding remarks in section 7. 2. Notation. Let · denote the euclidean norm in the complex n-space Cn . We refer to M(Cm , Cn ) as the Banach space of all n × m complex matrices, equipped with the standard induced matrix norm, ·. The real part of z ∈ C is denoted Re (z). The complex conjugate, transpose, and conjugate transpose of a complex matrix L ¯ L , and L∗ , respectively. We also define Her(L) := L + L∗ . The are indicated as L, Kronecker product of two given complex matrices L and M is denoted L ⊗ M , and the Kronecker sum is denoted L ⊕ M := L ⊗ I + I ⊗ M (see, e.g., [6] for details). We define S := {1, 2, . . .} (unless otherwise stated) and introduce the infinite dimensional Banach space Hm,n sup of all infinite arrays of the form H = (H1 , H2 , . . .) with Hi ∈ M(Cm , Cn ) for every i ∈ S, such that Hsup := supi∈S Hi < ∞. We ˜ n+ further write Hnsup in place of Hn,n sup and define Hsup as the set composed by all matrices n ∗ H = (H1 , H2 , . . .) ∈ Hsup 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). For short, 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 holding for negative Li 0. In the case S = {1, . . . , N }, we denote by Hm,n the space of all H = (H1 , . . . , HN ) such that


1175

˜ n+ stands Hi ∈ M(Cm , Cn ) for every i = 1, . . . , N . Analogously, Hn ≡ Hn,n , and H n ∗ for the set of all H ∈ H such that Hi = Hi > 0 for every i = 1, . . . , N . We denote by diag(H1 , H2 , . . .) the matrix with block-diagonal entries H1 , H2 , . . ., the analogous holding for diag(H1 , . . . , HN ). Concerning the random objects, we fix a complete probability space (Ω, F, P) carrying a right-continuous filtration Ft ⊂ F on t ≥ 0. Also, we denote by E(·) the mathematical expectation and define Ln2 as the space of all second order random variables (Ω, F) → Cn . Throughout λ(A) will stand the paper for λ(A) = sup R (τ ), τ ∈ λ(A) , and the spectrum ofA, with maximal real part R e e λmax (W ) := Re λ(W ) for W = W ∗ ∈ M(Cn , Cn ). In addition to (1.3), throughout the paper we shall refer to the matrix (2.1)

A = Λ ⊗ In2 + diag(A¯i ⊕ Ai )

as well as to the definitions

(2.2) Ei = Ei1 · · · EiM ,

Gi = G1i · · · GM (2.3) , i

Fi∗ = (Fi1 )∗ · · ·

Hi∗ = (Hi1 )∗ · · ·

(FiM )∗ , (HiM )∗ .

3. Basic setting. Consider in (Ω, F, Ft , P) a homogeneous Markov process θ = {θt , t ≥ 0}, with right-continuous sample paths and state space S ⊆ N, such that λij dt + o(dt), i = j, (3.1) P(θt+dt = j|θt = i) = 1 + λii dt + o(dt), i = j, where 0 ≤ λij for i = j, and 0 ≥ λii = − j∈S\{i} λij for all i ∈ S. Unless otherwise stated, we shall be dealing with the countably infinite case S = {1, 2, . . .}, under the additional hypothesis that there is υ < ∞ such that |λii | < υ for all i ∈ S. The initial condition θ0 : Ω → S is assumed to be a random variable with distribution ν = (ν1 , ν2 , . . .), or, in the finite case, ν = (ν1 , . . . , νN ). In this paper we study the uncertain and controlled Markov jump linear system x(t) ˙ = A(θt ) + DA (θt ) x(t) + B(θt ) + DB (θt ) u(t), t ≥ 0, (3.2) x(0) = x0 ∈ Ln2 , P(θ0 = i) = νi , i ∈ S, nu ,n are given state where A = (A1 , A2 , . . .) ∈ Hnsup and B = (B1 , B2 , . . .) ∈ Hsup and input operators, subjected to static disturbances of the form indicated in (1.2), m 1, . . . , M . We shall write that such that Δm i < 1 and ∇i < m1 for all m = m m m em n m gi i Ei ∈ M(C , C ), Fi ∈ M(Cn , Cfi ), Gm , Cn ), and Him ∈ M(Cn , Chi ) i ∈ M(C m m m 4 for a given bounded sequence {(em i , fi , gi , hi ), i ∈ S} in N . We also assume m that supi∈S Eim < ∞ for all m, the same holding for Fim , Gm i , and Hi , so that supi∈S DA (i) and supi∈S DB (i) are finite in (1.2). Our interest lies in statefeedback controllers of the form

(3.3)

u(t) = K(θt )x(t),

sup Ki < ∞, i∈S

for which the Markov property of the augmented state (x, θ) follows, for instance, from [9, Chap. 2] or [65, Chap. 2] (see also [30]). In order to introduce the stability tools which underpin the analysis in this paper, we consider also the nominal system x(t) ˙ = A(θt )x(t), t ≥ 0, (3.4) x(0) = x0 ∈ Ln2 , P(θ0 = i) = νi , i ∈ S,

1176


which, in the infinite case, will always be assumed stochastically stable (SS) in the sense that

∞

(3.5) E x(t)2 dt < ∞ for any x0 ∈ Ln2 and initial distribution ν. 0

Conversely, in the finite case system (3.4) will always be assumed mean square stable (MSS) in the sense that

(3.6) lim E x(t)2 = 0 for any x0 ∈ Ln2 and initial distribution ν, t→∞

a setup which is motivated by the following remark. Remark 3.1. It is a well-known fact that the SS and MSS properties are equivalent for system (3.4), in the finite case (see, e.g., [25]). In the infinite MJLS case, however, it was proved in [28] that stochastic and mean square stability are no longer equivalent. In order to preserve the usual terminology in the literature, in this paper we shall address the robust SS of infinite MJLS, together with the robust MSS of finite MJLS. Finally, some results from [27, 29] regarding the mean square and stochastic stability of (3.4) are summarized in what follows.1 Lemma 3.2. The following assertions are equivalent: (i) System (3.4) is stochastically stable as in (3.5). (ii) Re {λ(A)} < 0. ˜ n+ such that Ti (P ) 0 for all i ∈ S. (iii) There is P = (P1 , P2 , . . .) ∈ H sup Moreover, when S = {1, . . . , N }, the above reduces to one of the following equivalent statements: (iv) System (3.4) is mean square stable as in (3.6). ˜ n+ such that Ti (P ) < 0 for all i = 1, . . . , N . (v) There is P = (P1 , . . . , PN ) ∈ H ˜ n+ such that Li (Q) < 0 for all i = 1, . . . , N . (vi) There is Q = (Q1 , . . . , QN ) ∈ H 3.1. Structured uncertainty. Let us write that, for each i ∈ S and m ∈ m {1, . . . , M }, the uncertain parameters Δm i and ∇i in (1.2) are constrained to adm m missible sets Δi and ∇i . A key assumption in this paper is that, for each i ∈ S m and m ∈ {1, . . . , M }, Δm i and ∇i fall within one of the following three classes (the m m dummy pi stands for either ei or gim ): pm m ∈ M C i (3.7) Dscalar m = δI ; |δ| < 1 p ×p pm i i i of complex perturbations which are a scalar multiple of an appropriately sized identity matrix; pm i m ; D < 1 (3.8) Ddiag pm ×pm = D = diag(δ1 , . . . , δpi ) ∈ M C i

i

of diagonal perturbations; or the full-block class (qim stands for either fim or hm i ) m m D ∈ M Cqi , Cpi ; D < 1 (3.9) Dfull m = pm i ×qi of matrices in which all entries are uncertain. The scaling technique employed in this paper is based on the standard approach of introducing positive definite decision variables Sim and Zim which commute with the uncertainty, i.e., such that (3.10)

m m m Δm i Si = Si Δi

and

m m m ∇m i Zi = Zi ∇i

1 The connection between Lemma 3.2 and the decay of the candidate Lyapunov function V (x(t), θt ) = E x(t), P (θt )x(t) has been extensively discussed in [27].


1177

m are always satisfied in suitable sets Sm i and Zi . From now on, by the scaling classes which correspond to (3.7), (3.8), and (3.9), we shall, respectively, refer to pm i Rscalar (3.11) ; R > 0 , = R ∈ M C m m pi ×pi pm i m Rdiag (3.12) ; R>0 , pm ×pm = R = diag(s1 , . . . , spi ) ∈ M C i

i

Rfull = (0, ∞)

(3.13)

m scalar m in such a way that, e.g., Δm = Rscalar = Ddiag m implies Si m , ∇i i = Dem em gim ×gim i ×ei i ×ei

diag m full m implies Zm = Rfull. Throughout this m implies Si i = Rgim ×gim , and Δi = Dem i ×fi paper the role of such scaling classes will be related to the following preliminary result. m Lemma 3.3. Given i ∈ S and m ∈ {1, . . . , M }, suppose Δm i ∈ Δi . Then, for any given compatible E and F, we have

(3.14)

m ∗ ∗ m −1 Her(EΔm F i F) ≤ ESi E + F (Si )

and (3.15)

∗ m m −1 ∗ E Her(EΔm i F) ≤ F Si F + E(Si )

m m satisfied for any Sim ∈ Sm i . Furthermore, the analogous holds for ∇i ∈ ∇i if m m Sim ∈ Sm i is replaced by Zi ∈ Zi . m Proof. Bearing in mind (3.10), together with the fact that Δm i ∈ Δi implies m m Δm i < 1, notice that any Si ∈ Si satisfies m m ∗ m −1 ∗ m −1 ∗ Her (EΔm Si EΔm + Her (EΔm i F) ≤ EΔi − F (Si ) i − F (Si ) i F) ∗ ∗ ∗ m −1 m ∗ ∗ ∗ m −1 F = ESim Δm F = EDim Sim (Δm i ) E + F (Si ) i (Δi ) E + F (Si )

≤ ESim E∗ + F∗ (Sim )

−1

F,

which corresponds to (3.14); the proof of (3.15) follows analogously, bearing in mind ∗ m ∗ ∗ m ∗ m the fact that Her(EΔm i F) = Her(F (Δi ) E ) with (Δi ) = Δi < 1. Finally, m m m m m the same proof with Δi ∈ Δi replaced by ∇i ∈ ∇i and Zi ∈ Zm i in lieu of yields the validity of the last statement. Sim ∈ Sm i m m m Once the classes Δm i and ∇i , comprising, respectively, all Δ i and ∇i1 , are specM ified from among (3.7), (3.8), and (3.9), we define the sets Δ i = Δi = (Δi , . . . , Δi ); m m m Δm and ∇i = ∇i = (∇1i , . . . , ∇M , as well as the sets of all i ∈ Δi i ); ∇i ∈ ∇i admissible uncertainties affecting system (3.2), as (3.16) ∇ = (∇1 , ∇2 , . . .); ∇i ∈ ∇i , Δ = (Δ1 , Δ2 , . . .); Δi ∈ Δi , or, in the finite case, (3.17) ΔN = (Δ1 , . . . , ΔN ); Δi ∈ Δi ,

∇N = (∇1 , . . . , ∇N ); ∇i ∈ ∇i . Finally, we also define the sets of all admissible scalings as S = Si = (Si1 , . . . , SiM ); i m m 1 M m m Si ∈ Si and Zi = Zi = (Zi , . . . , Zi ); Zi ∈ Zi , along with (3.18) S = (S1 , S2 , . . .); Si ∈ Si , Z = (Z1 , Z2 , . . .); Zi ∈ Zi , or SN and ZN in the finite case, with (S1 , S2 , . . .) substituted by (S1 , . . . , SN ), and (Z1 , Z2 , . . .) accordingly replaced by (Z1 , . . . , ZN ).

1178


The choice of a particular disturbance class allows one to describe more precisely which perturbations are admissible for a given mode of operation. A concrete example which illustrates the ideas just introduced will be treated in section 6.1. Before proceeding further, it is necessary to introduce the following notation: f 1 M 1, . . . , S (3.19) ϕei (S) = diag Si1 Ie1i , . . . , SiM IeM , ϕ , (S) = diag S I I M f f i i i i i i (3.20)

ϕgi (Z) = diag Zi1 Igi1 , . . . , ZiM IgiM ,

ϕhi (Z) = diag Zi1 Ih1i , . . . , ZiM IhM , i

which may be simplified by noticing that full Sm i = R full Zm i = R

m

m

gim

hm i

Sim ∈ M(Cei ) = M(Cfi )

=⇒

Zim ∈ M(C

=⇒

) = M(C

)

=⇒

Sim Iem = Sim Ifim = Sim , i

=⇒

Zim Igim = Zim Ihm = Zim . i

4. Robust stability and stabilization. In the first part of this section we study, in the countably infinite setting, the stability of the system M x(t) ˙ = A(θt ) + m=1 E m (θt )Δm (θt )F m (θt ) x(t), t ≥ 0, (4.1) x(0) = x0 ∈ Ln2 , P(θ0 = i) = νi , i ∈ S, with respect to the following notion of robustness. Definition 4.1. System (4.1) is said to be robustly stochastically stable (robustly SS) in Δ if, for every Δ = (Δ1 , Δ2 , . . .) ∈ Δ as in (3.16), (3.5) is satisfied. The following result provides us with a sufficient condition for the robust SS of system (4.1), in terms of the feasibility of a set of infinitely coupled LMIs. Theorem 4.2. System (4.1) is robustly SS for any Δ = (Δ1 , Δ2 , . . .) ∈ Δ ˜ n+ and S = (S1 , S2 , . . .) ∈ S such that the whenever there are P = (P1 , P2 , . . .) ∈ H sup LMI problem Ti (P ) + Fi∗ ϕfi (S)Fi Pi Ei (4.2) 0, i ∈ S, E∗i Pi −ϕei (S) is satisfied. Proof. Following along the same lines as [58, Theorem 2.2], it is not difficult to show that the congruence transformation

I ψi · := I Pi Ei ϕei (S)−1 · ϕei (S)−1 E∗i Pi satisfies supi∈S ψi (I) < ∞ with ψi (I) ≥ I for all i ∈ S, so that, from Lemma 3.3 in conjuction with (1.2), the (Schur’s complement–like) application of ψi (·) to the left-hand side of (4.2) yields 0 Ti (P ) + Fi∗ ϕfi (S)Fi + Pi Ei ϕei (S)−1 E∗i Pi = Ti (P ) +

M

(Fim )∗ Sim Fim + Pi Eim (Sim )−1 (Eim )∗ Pi

m=1

≥ Ti (P ) +

M m=1

m Her (Pi Eim Δm + λij Pj i Fi ) = Her Pi Ai + DA (i) j∈S


1179

m for all Δm i ∈ Δi , m = 1, . . . , M , yielding from Lemma 3.2 the desired result. Let us now consider the robust stabilization of system (3.2), with DA and DB as in (1.2), by means of a static state-feedback control law (x, θ) → u such as (3.3). In this case, the closed-loop system is of the form

ˆ t ) + DA (θt ) + DB (θt )K(θt ) x(t) x(t) ˙ = A(θ

(4.3)

ˆ t ) ≡ A(θt ) + B(θt )K(θt ). Hence, due to Lemma 3.2, we get that a given with A(θ u controller K = (K1 , K2 , . . .) ∈ Hn,n sup guarantees the robust SS of system (4.3) for given Δ = (Δ1 , Δ2 , . . .) ∈ Δ and ∇ = (∇1 , ∇2 , . . .) ∈ ∇ whenever there is P = ˜ n+ such that T˘i (P ) := A˘∗ Pi + Pi A˘i + (P1 , P2 , . . .) ∈ H sup i j∈S λij Pj 0 on i ∈ S, M m m m m m m ˆ ˘ where Ai := Ai + m=1 Ei Δi Fi + Gi ∇i Hi Ki . A sufficient condition for the existence of a robust controller for system (3.2) is stated in the following theorem. ˜ n+ , W = (W1 , W2 , . . .) ∈ Theorem 4.3. Suppose there are X = (X1 , X2 , . . .) ∈ H sup u , S = (S , S , . . .) ∈ S, and Z = (Z , Z , . . .) ∈ Z such that Hn,n 1 2 1 2 sup ⎡ (4.4)

Ti ⎢ Fi X i ⎢ ⎣ Hi Wi λi ⊗ Xi

⎤ Xi Fi∗ Wi∗ Hi∗ (λi ⊗ Xi )∗ ⎥ −ϕfi (S) 0 0 ⎥ 0, h ⎦ 0 −ϕi (Z) 0 0 0 −Di (X)

i ∈ S,

is satisfied, where Ti = Her(Ai Xi + Bi Wi ) + λii Xi + Ei ϕei (S)E∗i + Gi ϕgi (Z)G∗i , 1/2 1/2 1/2 (λi ⊗ Xi )∗ = λi1 Xi · · · λi(i−1) Xi λi(i+1) Xi · · · , Di (X) = diag(X1 , . . . , Xi−1 , Xi+1 , . . .). Then, the static state-feedback control law (3.3), with K(θt ) := W (θt )X(θt )−1 , ensures the robust SS of (3.2) for any Δ = (Δ1 , Δ2 , . . .) ∈ Δ and ∇ = (∇1 , ∇2 , . . .) ∈ ∇. Proof. After an application of the congruence transformation2 ⎡ (4.5)

⎤∗

I

⎡

I

⎤

⎢ ϕf (S)−1 Fi Xi ⎥ ⎢ ϕf (S)−1 Fi Xi ⎥ i i ⎢ ⎢ ⎥ ⎥ ⎣ ϕh (Z)−1 Hi Wi ⎦ · ⎣ ϕh (Z)−1 Hi Wi ⎦ i i Di (X)−1 (λi ⊗ Xi ) Di (X)−1 (λi ⊗ Xi )

to (4.4), we immediately have that (4.6) Her(Ai Xi + Bi Wi ) +

λij Xi Xj−1 Xi + Ei ϕei (S)E∗i

j∈S

+

Xi Fi∗ ϕfi (S)−1 Fi Xi

+ Gi ϕgi (Z)G∗i + Wi∗ Hi∗ ϕhi (Z)−1 Hi Wi 0,

2 It is not difficult to verify that the transformation is uniformly positive (see [58]), just as in the proof of Theorem 4.2. Hence, application of (4.5) corresponds to the uniform Schur’s complement operation devised in [58, Theorem 2.2].

1180


so that, according to Lemma 3.3 and (1.2), 0 Her(Ai Xi + Bi Wi ) +

λij Xi Xj−1 Xi +

+

Eim Sim (Eim )∗

m=1

j∈S M

M

m m ∗ ∗ m ∗ m −1 m Xi (Fim )∗ (Sim )−1 Fim Xi + Gm Hi Wi i Zi (Gi ) + Wi (Hi ) (Zi )

m=1

≥ Her

Ai + DA (i) Xi + Bi + DB (i) Wi + λij Xi Xj−1 Xi j∈S

must hold for all Δ = (Δ1 , Δ2 , . . .) ∈ Δ and ∇ = (∇1 , ∇2 , . . .) ∈ ∇. Letting Pi := Xi−1 and Ki := Wi Pi , this implies that Pi−1 {A˘∗i Pi + Pi A˘i + j∈S λij Pj }Pi−1 0, from which the robust SS of system (4.3) is verified, yielding the desired result. 4.1. The finite case. In the last part of this section some additional results regarding the case S = {1, . . . , N } will be stated. In order to preserve the usual terminology in the MJLS literature, in the finite case we also replace robust stochastic stability, as introduced in Definition 4.1, by the following equivalent notion of robust stability. (It should be kept in mind that, as pointed out in Remark 3.1, in the infinite case these two notions are not equivalent.) Definition 4.4. System (4.1) is said to be robustly mean square stable (robustly MSS) in ΔN if S = {1, . . . , N }, and, for every Δ = (Δ1 , . . . , ΔN ) ∈ ΔN as in (3.17), we have (3.6) satisfied. We shall also make use of the additional hypothesis that the transition rates matrix Λ = [λij ] belongs to the polytope (4.7)

Π=

Λξ =

L =1

ξ( )Λ( );

ξ( ) ≥ 0,

L

ξ( ) = 1 ,

=1

in which {Λ( ) = [λij ( )], = 1, . . . , L} is a set of known transition rate matrices. In this case, an obvious consequence of Theorem 4.2 goes as follows. Corollary 4.5. System (4.1) is robustly MSS for any Δ = (Δ1 , . . . , ΔN ) ∈ ˜ n+ and S( ) = ΔN and Λ = [λij ] in Π whenever there are P = (P1 , . . . , PN ) ∈ H N (S1 ( ), . . . , SN ( )) ∈ S such that, for each = 1, . . . , L and i = 1, . . . , N , the coupled LMI problems N Her(Pi Ai ) + j=1 λij ( )Pj + Fi∗ ϕfi S( ) Fi Pi Ei (4.8) 0 and pi > 0, it follows that ρ < ρˆx implies ρ < ρˆ, so that ρˆ ≥ ρˆx . Moreover, noticing that the parametrization si → ρ−1 |bi |pi can be chosen near ρ ≈ ρˆ, we have ρˆ = ρˆx . Next, define the perturbation ⎧ ¯ ⎨ρ bi , b = 0, i ρ (5.14) δi := |bi | ⎩ 0, otherwise. ˜ 1+ such that Then, whenever 0 ≤ ρ < ρˆ, there has to be p = (p1 , p2 , . . .) ∈ H sup (5.15) 0 Ti (p) + 2ρ|bi |pi = (ai + bi δiρ )pi + pi (ai + bi δiρ ) + λij pj j∈S

1185


from the very definition of ρˆ. When ρ = ρˆ, on the other hand, (5.15) ceases to be feasible (from the definition), so that δ ρˆ = (δ1ρˆ, δ2ρˆ, . . .) is a destabilizing perturbation with δ ρˆsup = ρˆ. Therefore, invoking Theorem 5.2, the validity of the first equality in (5.10) follows from rC (a, b) ≥ ρˆx = ρˆ. ˜ 1+ satisfying Suppose now ρ < ρˆR , so that there must be p = (p1 , p2 , . . .) ∈ H sup λij pj 0 Ti (p) + 2ρ|Re (bi )|pi ≥ Ti (p) + 2ρRe (bi )pi ≥ Her pi (ai + bi δi ) + j∈S

as long as δi ∈ R with δi < ρ for all i ∈ S. Therefore, stochastic stability is guaranteed for whatever real disturbance δ ∈ H1sup such that δsup < ρˆR , yielding rR (a, b) ≥ ρˆR . Finally, letting ⎧ ⎪ Re (bi ) > 0, ⎨ρˆR , (5.16) ηî := −ρˆR , Re (bi ) < 0, ⎪ ⎩ 0, otherwise, ˜ 1+ we get, invoking the definition of ρˆR , that there cannot exist p = (p1 , p2 , . . .) ∈ H sup such that (5.17) (ai + bi ηî )pi + pi (ai + bi ηî ) + λij pj = Ti (p) + 2ρˆR |Re (bi )|pi 0 j∈S

is feasible. In other words, ηˆ = (ˆ η1 , ηˆ2 , . . .) is a real destabilizing perturbation with ˆ η sup = ρˆR , which yields rR (a, b) = ρˆR . Remark 5.6. Due to the fact that Ti (p) + 2ρ|Re (bi )|pi ≤ Ti (p) + 2ρ|bi |pi , it is immediate that ρ ≤ ρˆ ⇒ ρ ≤ ρˆR , so that ρˆ ≤ ρˆR in Theorem 5.5. In the unstructured case, where bi = 1 for all i ∈ S, system (5.9) reduces to (5.18) x(t) ˙ = a(θt ) + δ(θt ) x(t), t ≥ 0, about which we can prove that the real and complex stability radii are equal in (5.10). Furthermore, in this case the radii can also be exactly characterized with the spectrum of a matrix (instead of the feasibility of LMIs). The results are summarized in the following theorem. Theorem 5.7. The stability radii of system (5.18) are given by (5.19)

1 ˆ , rC (a) = rR (a) = ρˆ = − Re λ(A) 2

in which Aˆ = Λ + 2 diag{Re (ai )}, and ρˆ stands for the supremum of all ρ > 0 such that the LMI problem (5.20)

Ti (p) + 2ρpi 0,

i ∈ S,

˜ 1+ . is feasible on p = (p1 , p2 , . . .) ∈ H sup Proof. In view of (5.10) in Theorem 5.5, it remains only to prove the last equality in (5.19). Due to Lemma 3.2, we know that the perturbed MJLS (5.21) x˙ ρ (t) = a(θt ) + ρ xρ (t), t ≥ 0,

1186


˜ 1+ such that is SS if and only if there is p = (p1 , p2 , . . .) ∈ H sup (5.22)

Tiρ (p) ≡ (ai + ρ)pi + pi (ai + ρ) +

λij pj = Ti (p) + 2ρpi 0,

i ∈ S,

j∈S

or, equivalently, if and only if Aˆρ := Λ + diag (ai + ρ) ⊕ (ai + ρ) is Hurwitz, (5.23)

ˆ + 2ρ < 0, Re λ(Aˆρ ) = Re λ(Aˆ + 2ρI) = Re λ(A)

from which the result follows easily, i.e., ρˆ = sup ρ > 0; (5.22) is feasible = sup ρ > 0; Re λ(Aˆρ ) < 0 1 ˆ ˆ . = − Re λ(A) = sup ρ > 0; ρ < − 12 Re λ(A) 2 Proof of Theorem 5.4. Since the first statement was already proven in [61, Theorem 1], it remains only to prove the first inequality in (5.7). According to Theorem 5.7, we have 1 ˜ = sup ρ > 0; T˜i (p) + 2ρpi 0 is feasible , − Re λ(A) 2 where T˜i (p) := 2˜ ai pi + j∈S λij pj = λmax (Ai + A∗i )pi + j∈S λij pj . Thus, bearing in mind Theorem 5.2, together with uniform Schur complements (see [58, Theorem ˜ 1+ 2.2]), it suffices to prove that the existence of p = (p1 , p2 , . . .) ∈ H sup such that n+ 2 ˜ ˜ Ti (p) + 2ρpi 0 guarantees there is P ∈ Hsup such that Ti (P ) + ρ si I + Pi2 /si 0 holds true. In fact, letting Pi = pi I and si = pi /ρ for all i ∈ S yields

(5.24)

Ti (P ) + ρ2 si I +

1 2 Pi = (Ai + A∗i )pi + λij pj I + 2ρpi I si j∈S ! " ∗ ≤ λmax (Ai + Ai )pi + λij pj + 2ρpi I j∈S

= T˜i (p) + 2ρpi I 0,

so that (5.24) implies (5.7). We close our study of the countably infinite case with the following example. 5.2. Example: Stability radii of an infinite MJLS. Consider the pure-birth Markov process with unit transition rates λi(i+1) = −λii = 1, and λij = 0 otherwise, √ on S = {1, 2, . . .}. Then, according to Theorem 5.7, the scalar system3 (here j = −1) $ # 1 jθt (5.25) x(t) ˙ = −1 + e + δ(θt ) x(t), t ≥ 0, 10 is robustly SS for any δ = (δ1 , δ2 , . . .) satisfying supi∈S |δi | < ρ whenever there is p = (p1 , p2 , . . .) such that 2Re (ai ) + 2ρ − 1 pi + pi+1 < −ε, with pi > ε for all i ∈ S, 3 Notice that, since 1 rad is not a rational multiple of 2π, the nominal system is intrinsically of infinite dimension in the sense that exp(jθt ) cannot repeat itself as time goes by.


1187

and such that supi∈S pi < ∞. That is, there must exist a bounded sequence p1 , p2 , . . . satisfying the (normalized by ε) recursion # $ cos i (5.26) 1 < pi+1 < (1 − 2 {Re (ai ) + ρ}) pi − 1 = 3 − 2ρ − pi − 1 5 for all i ∈ S. Therefore, it is not difficult to see that the comparison system # $ cos i (5.27) ri+1 = 3 − 2ρ − ri − 1, i = 1, 2, . . . , r1 > 1, 5 limi ri = ∞, in order must satisfy ri > 1 for all i ∈ S with% for pi+1 to always exist ∞ cos i such that (5.26) is satisfied. That is, 3 − 2ρ − = ∞ must hold, which is 5 i=1 ∞ i = ∞. Finally, this holds true if and only if equivalent to i=1 log 3 − 2ρ − cos 5 # # $ $ cos i cos i log 3 − 2ρ − log 3 − 2ρ + ∞= + 5 5 i i # $ 2 cos i log (3 − 2ρ)2 − = 25 i is satisfied. A sufficient condition for this to hold is that, for all i ∈ S, √ onehas 2 i 1 2 ≥ (3 − 2ρ) − > 1, which is true whenever 0 ≤ ρ < 3 − 26/5 /2. (3 − 2ρ)2 − cos 25 25 On the other hand, it is not difficult to check that (5.27) cannot escape to infinity n i oscillates between finite bounds as n increases, if ρ = 1. In fact, since i=1 cos 5 ∞ cos i 2 while diverges to infinity, we have from [7, Chap. VI, Sec. 40] that %∞ i=1cos i 5 1 − = 0. Summarizing, we have proven that i=1 5 (5.28)

0.9901 < rC (a) = rR (a) ≤ 1.

Finally, let us consider the structured stability radii of the system # $ 1 (5.29) x(t) ˙ = −1 + ejθt + b(θt )δ(θt ) x(t), t ≥ 0, 10 with bi ≡ sign(cos i). Then, due to the fact that |bi | = |Re bi | = 1 for all i ∈ S, we have that both (5.11) and (5.12) become equivalent to (5.20). Hence, in this example all stability radii lie within (0.9901, 1] in (5.28). Moreover, as shown in theorems 5.5 and 5.7, there are perturbations (ˆ ρ, ρˆ, . . .), and (ˆ ρ sign cos 1, ρˆ sign cos 2, . . .), with 0.9901 < ρˆ ≤ 1, such that the uncertain systems (5.25) and (5.29) cease to be SS, respectively. 5.3. Stability radius (finite case). Let us consider the problem posed in section 5, regarding now finite MJLS. In order to once again preserve the usual terminology in the finite case, we replace (5.2) in Definition 5.1 by (5.30) rK (A, E, F ) = inf Δsup ; Δ ∈ Hf,e , system (5.1) is not MSS . To begin with, we present the following consequence of Theorem 5.2 and of a particular version of Corollary 4.5. Theorem 5.8. In the finite case, the stability radii of system (3.4) with respect to perturbations such as Ai Ai + Ei Δi Fi satisfy (5.31)

max{ρˆx , σ ˆx } ≤ rC (A, E, F ) ≤ rR (A, E, F )

1188


in which, in this case, ρˆx is the supremum of all ρ > 0 such that the LMIs Ti (P ) + ρ2 si Fi∗ Fi Pi Ei (5.32) < 0, i = 1, . . . , N, Ei∗ Pi −si Ie ˜ n+ and s = (s1 , . . . , sN ) ∈ H ˜ 1+ , and σ ˆx stands are feasible on P = (P1 , . . . , PN ) ∈ H for the supremum of all σ > 0 such that the LMIs Li (Q) + σ 2 si Ei Ei∗ Qi Fi∗ (5.33) < 0, i = 1, . . . , N, Fi Qi −si If ˜ n+ and s = (s1 , . . . , sN ) ∈ H ˜ 1+ . are feasible for some Q = (Q1 , . . . , QN ) ∈ H Proof. Bearing in mind Theorem 5.2, it remains only to prove that system (5.1) ˆ . This follows from the fact that, due to Schur is robustly MSS whenever Δsup < σ ∗ complements and Lemma 3.3, (5.33) yields 0 > Li (Q) + σ 2 si Ei Ei∗ + s−1 i Qi Fi Fi Qi ≥ N Her (Ai + Ei Δi Fi )Qi + j=1 λji Qj . Remark 5.9. In contrast to what was mentioned in Remark 5.3, the connection between (5.33) and H∞ control does not seem to be trivial. In other words, it is not clear how the margins obtained with (5.33) should yield the disturbances-tostate gain of system (4.1) sufficiently small, in order that stochastic stability not be compromised. More importantly, we do not know whether (5.33) could be possibly regarded as a guarantee of the robustness of system (3.4) with respect to disturbances such as Ai x Ai x + Ei Δi (Fi x), with nonlinear Δi : Cf → Ce . 5.3.1. Stability radius of scalar MJLS (finite case). In the finite case, the main feature surrounding the scenario treated in section 5.1 is that (5.32) and (5.33) become equivalent (which is not true in general, as it will be explicitly checked in section 6.2). The following finite-dimensional counterpart of Theorem 5.5 is the precise statement of this fact. Theorem 5.10. In the finite case, the stability radii of system (5.9) are given by (5.34)

rC (a, b) = ρˆ = σ ˆ,

rR (a, b) = ρˆR = σ ˆR ,

in which ρˆ and σ ˆ are the maximal ρ > 0 and σ > 0 such that the LMI problems Tiρ (p) < 0 and Lσi (q) < 0, i = 1, . . . , N , are, respectively, satisfied for some p = ˜ 1+ , where (p1 , . . . , pN ) and q = (q1 , . . . , qN ) in H (5.35)

Tiρ (p) := Ti (p) + 2ρ|bi |pi ,

Lσi (q) := Li (q) + 2σ|bi |qi ,

the analogous holding for ρˆR , σ ˆR , and the LMI problems (5.36)

Ti (p) + 2ρ|Re (bi )|pi < 0,

Li (q) + 2σ|Re (bi )|qi < 0,

i = 1, . . . , N.

ˆ=σ ˆx as defined in Theorem 5.8. Moreover, ρˆ = ρˆx and σ Proof. For any u = (u1 , . . . , uN ) and v = (v1 , . . . , vN ) in H1 , let u, v := N ¯i vi . Then, bearing in mind Theorem 5.5, the first two equalities in (5.34) i=1 u follow from the easily verifiable fact that T ρ = (T1ρ, . . . , TNρ ) and Lρ = (Lρ1 , . . . , LρN ) are adjoint operators in the Hilbert space H1 , ·, · . That is, T ρ (p), q = p, Lρ (q)

for all p, q ∈ H1 ,

so that the LMI problems Tiρ (p) < 0 and Lρi (q) < 0, i = 1, . . . , N , are equivalent, yielding ρˆ = σ ˆ . The proof of the last two equalities in (5.34) follows, mutatis mutandis, from Theorem 5.5 together with duality of the corresponding operators in (5.36).


1189

As for the unstructured setting, the next result follows as an easy consequence of Theorem 5.7. Corollary 5.11. In the finite case, the stability radii of system (5.18) satisfy 1 ˆ rC (a) = rR (a) = ρˆ = σ , ˆ = − Re λ(A) 2

(5.37)

in which Aˆ = Λ + 2 diag{Re (ai )}, with ρˆ and σ ˆ standing for the maximal ρ and σ such that the LMI problems Ti (p) + 2ρpi < 0,

(5.38)

Li (q) + 2σqi < 0,

i = 1, . . . , N,

˜ 1+ . Moreover, are, respectively, feasible on p = (p1 , . . . , pN ) and q = (q1 , . . . , qN ) in H ˆ=σ ˆx are as defined in Theorem 5.8. ρˆ = ρˆx and σ Proof. This follows immediately from Theorem 5.7, together with the fact that ρˆ = σ ˆ in Theorem 5.10. Finally, in light of the above results, we remark that the statement of Theorem 5.4 holds equally in the finite case, with SS appropriately replaced by MSS. The proof is entirely analogous to that presented at the end of section 5.1. 6. Numerical examples. In this section the proposed theory is illustrated with some numerical examples. For simplicity, we shall restrict ourselves to the finite case. 6.1. Composite systems. In this example we study the interaction of the damped harmonic oscillators (6.1)

y¨ + 3y˙ + y = 0,

z¨ + 2z˙ + z = 0

in the form (6.2)

y¨ + 3y˙ + y = −φ(αz + β z), ˙

z¨ + 2z˙ + z = −φ(γy + δ y) ˙

in which α, β, γ, and δ are uncertain parameters satisfying |α|2 +|β|2 < 1, |γ|2 +|δ|2 < 1 (so that, for instance, α and β may depend on each other, as do γ and δ). The uncertainty is normalized by the constant φ, which models the gain of the (uncertain) connection between the oscillators, in a worst-case scenario. We assume the coupling to be subject to failures, according to a homogeneous Markov process θ = {θt , t ≥ 0} taking values in a two-state set, S = {1, 2}. Mode 1 is chosen to represent the nominal behavior (6.1), while θt = 2 leads the system to the coupled dynamics in (6.2); for simplicity, all transition let us assume for now that

1 . Letting x ≡ y, y, ˙ z, z ˙ , notice that the rates are equal to one, so that Λ = −1 1 −1 coupled system can be easily rewritten in the uncertain MJLS form (4.1), with M = 2 and 0 1 0 0 −3 0 0 Ai ≡ −1 (6.3) Δ22 = [ γ δ ] , , Δ12 = [ α β ] , 0 0 0 1 0 −1 −2

0

(6.4)

E21

=

0 −φ 0 0

,

F21

=

[ 00 00 10 01 ] ,

E22

=

0 0 0 −φ

,

F22 = [ 10 01 00 00 ] ,

along with (6.5)

E11

=

E12

0 =

0 0 0

,

F11 = F12 = [ 0 0 0 0 ] ,

1190


so that Δ11 and Δ21 do not affect the system. It is interesting to point out that, even though the oscillators (6.1) are (individually) asymptotically stable, the intermittent forces in the right-hand side of (6.2) can drive the overall system to an unstable behavior. In fact, if α = 0.9727, β = 0.2276, γ = 0.9713, δ = 0.2335, a coupling gain such as φ = 1.6064 is easily verified (with the aid of Lemma 3.2) to destabilize system (4.1) in the mean square sense. Therefore, in this example interconnection may be regarded as an undesirable phenomenon which compromises the system behavior. Our first goal here will be to provide a bound on the maximal coupling gain for which the overall system is always stable in the mean square sense. That is, we want to obtain a bound on the quantity (6.6)

φmax = sup { > 0, such that (6.1)–(6.2) is MSS whenever 0 ≤ φ < } ,

which can be regarded as a measure of robustness for system (4.1), (6.3), (6.4), with respect to the interconnection gain. full Bearing in mind the definitions in section 3.1, we have Δm 2 ∈ D1×2 for each 1 2 full full m = 1, 2, so that S2 = S2 = R . Besides, (6.5) allows us to choose Δm 1 = D1×1 , m full so that S2 = R for m = 1, 2. By means of a bisectional procedure, in the same vein as that proposed in [59, Algorithm 1], we found that the smallest φ such that the LMI problem (4.8) ceases to be feasible is approximately 1.5653. On the other hand, by running the analogous procedure for (4.9), feasibility was lost only with φ = 1.5888. Hence, the MSS of system (4.1), (6.3), (6.4) is guaranteed whenever 0 ≤ φ < 1.5888 ≤ φmax < 1.6064 in (6.6). Suppose now that we wish to increase the robustness of system (4.1) by the action of control. For simplicity, let us assume that the controller, which supposedly becomes inactive in the event of interconnection (θt = 2), has access to the entire state of the system. Let us further assume that the input gain of the first mode of operation is uncertain and given by 0 0 0 0 1 01 03 1 01 00 1 0 1 0 1 00 01 10 + = (3 I2 ) [ 10 01 ] B1 + DB (1) = 10 00 + ] + [ 0 2 01 00 4 03 02 4 00 01 4 01 00 01 01 with |1 |2 + |2 |2 < 1 and |3 |2 < 1, so that 0 0 0 0 1 01 00 1 00 01 1 2 1 0 0 0 B1 = 0 0 , B2 = 0 0 , G1 = , G1 = , 4 00 01 4 01 00 0 1 00

m with H1m = 10 01 , Gm 2 = [ 0 0 0 0 ] , and H2 = [ 0 0 ] for all m = 1, 2. The per2 scalar turbation classes may thus be accordingly chosen as ∇11 = Ddiag 2×2 and ∇1 = D2×2 , diag 1 2 scalar for which the corresponding scaling classes are Z1 = R2×2 and Z1 = R2×2 . As for the second mode of operation, we might as well choose ∇12 = ∇22 = Dfull 1×1 , with Z12 = Z22 = Rfull . In this case, the maximal robustness margins of the closed-loop system (i.e., the closed-loop counterparts of (6.6)) obtained from bisection on (4.10) and (4.11) are 2.0454 and 2.2153, respectively. Finally, let us assume that the transition rates are of the form

−1 1

−1.5 1.5

−1.5 1.5

1 (6.7) Λ = ξ(1) −1 1 −1 + ξ(2) 1.5 −1.5 + ξ(3) 1 −1 + ξ(4) 1.5 −1.5 , in which ξ( ), = 1, 2, 3, 4, are uncertain parameters, as in (4.7). In this case, with the aid of Proposition 4.9, we found that the closed-loop system is robustly MSS for any 0 ≤ φ < 1.75 (which is smaller than the maximal robustness margin obtained via


1191

−36.6085 0 0 bisection) if the controller gains K1 = −63.9904 0 0 −31.7974 −27.8944 , K2 = [ 00 00 00 00 ] are implemented in the feedback loop. Hence, the proposed control law is decentralized and given by

˙ 31.7974 z + 27.8944 z˙ if θt = 1, (6.8) u(t) = − 63.9904 y + 36.6085 y, with u(t) = [ 0 0 ] otherwise. One final comment is as follows. Due to Lemma 3.2, the controlled system is robustly MSS for all Δ ∈ ΔN and ∇ ∈ ∇N if and only if the spectrum of the perturbed matrix A& := Λ ⊗ In2 + diag (Ai + Di ) ⊕ (Ai + Di ) is entirely contained in the open left half plane, with Λ ∈ Π as in (6.7), and ' ( M M m m m m Gi ∇i Hi Eim Δm Ki + (6.9) Di := Bi + i Fi . m=1

m=1

In fact, after randomly generating 1029 different samples of Λ ∈ Π, as well as of Δ ∈ ΔN and ∇ ∈ ∇N such that Δsup = ∇sup = 0.99, we found that the MSS of the closed-loop system was not lost. The worst setup in this case, discovered by 1.2653 , α = γ = 1 = −0.99, β = δ = 2 = 0, random search, is that of Λ = −1.2653 1.4472 −1.4472 & and 3 = −0.5140 + 0.8461j, for which Re λ(A) = −0.3679. A plot of the outcomes of the experiment, in Figure 6.1, shows that all of the selected samples’ spectra lie entirely in the open left half plane.

Imaginary axis

20

10

0

−10

−20

−80 −60 −40 −20 0 Real axis

20

40

Fig. 6.1. Union of the uncertain spectra of 1029 different samples of A.

6.2. On the complex stability radius. When reduced to the single-mode case (S = {1}), it is easy to prove that the feasibility of (5.32) is equivalent to that of (5.33) for given ρ = σ. In fact, this corresponds to the LMI problems ∗ AQ + QA∗ + σ 2 EE ∗ QF ∗ A P + P A + ρ2 F ∗ F P E < 0 and 0 and Q > 0, respectively.4 In the general case that S = {1}, however, it is possible to verify that such an equivalence does not hold true in general (although it does hold for the scalar case studied in section 4 The

equivalence is easily proven, with the aid of Schur complements, by letting Q ≡ ρ2 P −1 .

1192


5.3.1, as proved in Theorem 5.10). In fact, let S = {1, 2} with |λij | ≡ 10 for all (i, j) ∈ S × S and consider, for each given ω ∈ [0, 1], the uncertain system (ω) (ω) x(t) ˙ = A(ω) (θt ) + E (ω) (θt )Δ(θt )F (ω) (θt ) x(t), with A(ω) = (A1 , A2 ), E (ω) = (ω)

(ω)

(ω)

(ω)

(E1 , E2 ), and F (ω) = (F1 , F2 ) given by −2 ω−1 1 (ω) (ω) A1 = 2 ω−1 , A2 = , 3 −ω −ω −2 4 ω (ω) (ω) (ω) (ω) , F2 = [ 1−ω E1 ≡ E1 , F1 ≡ F1 , E2 = 1

1].

Then, by running [59, Algorithm 1] for each 0 ≤ ω ≤ 1, we obtained the result plotted ˆx = σ ˆx (ω). Also, in Figure 6.2. As it shows, ρˆx = ρˆx (ω) is in general different from σ neither of the robustness margins dominates the other. This indicates that, in the general nonscalar case, the LMI problems (5.32) and (5.33) are not equivalent. 1.4

1

x

ρ, σ

x

1.2

0.8 0.6 0.4

0

0.2

0.4

ω

0.6

0.8

1

Fig. 6.2. Comparison between ρˆx (solid line) and σ ˆx (dashed line) for different values of ω.

In particular, let ω = 0, so that A(0) ≡ (A1 , A2 ), E (0) ≡ (E1 , E2 ), and F (0) ≡ (F1 , F2 ) are given by −2 −1 0 0 1 A1 = 2 −1 , A2 = , F1 = [ 0 0 ], E2 = , and F2 = [ 1 1 ] . 3 , E1 = 0

−2

0

4

0

1

Then, by employing [59, Algorithm 1], we found that ρˆx and σ ˆx in Theorem 5.8 are, respectively, such that ρˆx < 0.9516 and σ ˆx > 1.3474. This shows that ρˆx can be strictly smaller than rC (A, E, F ) in (5.3). Besides, in this case it is not difficult to verify that the (real) perturbation Δ0 = (Δ01 , Δ02 ) = (0, 1.3827) destabilizes the system. In conclusion, (6.10)

ρˆx < 0.9516 < 1.3474 < σ ˆx ≤ rC (A, E, F ) < 1.3827 = Δ0 sup .

7. Conclusion. In this paper we addressed the robust stability and stabilization problems for a class of Markov jump linear systems (MJLS) subjected to structured uncertainty on the system parameters. As a by-product, special attention was devoted to the open problem of stability radius analysis. The general case in which the Markov process takes values in a countably infinite set has been treated, with special sections regarding the finite case. Among the main features of this work, we highlight the following:


1193

• The consideration of static multiperturbations with additional structure (i.e., not necessarily full-blocks); in particular, this encompasses the block-diagonal case. • An adjoint approach to the robust mean square stability of MJLS, which does not seem to have a trivial connection with H∞ control. Besides being of interest in its own right, it was shown here that this approach can sometimes yield less conservative results (section 6.1). • Consideration of the robust control of MJLS subjected to simultaneous uncertainty in the system data and in the switching rates (section 4.1). The corresponding LMI problems have the advantage of being uncertainty-dependent. • A new lower bound on the stability radii of finite MJLS which is verified to be less conservative in a particular example (section 6.2). • A spectral characterization of stability radii (Theorem 5.4), which is tighter than the one presented in references [57] and [61]. • Presentation of a fairly complete characterization of the complex and real stability radii of scalar MJLS. • An example of computation of stability radii for an infinite MJLS in terms of infinitely coupled linear matrix inequalities (section 5.2). Finally, an interesting conclusion which can be drawn from the proposed contributions to stability radius theory is that, different from the situation studied in, e.g., [23] and [36], the complex stability radius of MJLS with respect to linear perturbations can be strictly larger than the margin provided via the small-gain theorem, as shown in the example of section 6.2. In other words, as exemplified in (6.10), ρˆx can be strictly smaller than rC (A, E, F ) in (5.3) (on the other hand, as also shown in (6.10), the alternative robust stability margin σ ˆx introduced in this paper can sometimes provide tighter estimates for the complex stability radius of MJLS). This further suggests that there might exist a fundamental limitation of H∞ control theory, if one’s ultimate goal relies on its application to the robust control of MJLS (as in, e.g., [49]). REFERENCES [1] F. Abdollahi and K. Khorasani, A decentralized H∞ routing control strategy for mobile networked multi-agents, in Proceedings of the 2009 American Control Conference, St. Louis, MO, 2009, pp. 1555–1560. [2] W. P. Blair and D. D. Sworder, Continuous-time regulation of a class of econometric models, IEEE Trans. Systems Man Cybernet., 5 (1975), pp. 341–346. [3] E. K. Boukas, Stochastic Switching Systems: Analysis and Design, Birkh¨ auser, Boston, 2005. [4] E. K. Boukas, P. Shi, and K. Benjelloun, On stabilization of uncertain linear systems with jump parameters, Internat. J. Control, 72 (1999), pp. 842–850. [5] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Stud. Appl. Math. 15, SIAM, Philadelphia, 1994. [6] J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits and Systems, 25 (1978), pp. 772–781. [7] T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan, London, 1908. [8] Y. Cao and J. Lam, Robust H∞ control of uncertain Markovian jump systems with time-delay, IEEE Trans. Automat. Control, 45 (2000), pp. 77–83. [9] C. G. Cassandras and J. Lygeros, Stochastic Hybrid Systems, Taylor & Francis, Boca Raton, FL, 2007. [10] O. L. V. Costa and E. K. Boukas, Necessary and sufficient conditions for robust stability and stabilizability of continuous-time linear systems with Markovian jumps, J. Optim. Theory Appl., 99 (1998), pp. 359–379. [11] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, Continuous-time state feedback H2 control of Markovian jump linear systems via convex analysis, Automatica J. IFAC, 35 (1999), pp. 259–268.

1194


[12] O. L. V. Costa, M. D. Fragoso, and R. P. Marques, Discrete-Time Markov Jump Linear Systems, Probab. Appl. (N.Y.), Springer-Verlag London, Ltd., London, New York, 2005. [13] D. P. de Farias, J. C. Geromel, and J. B. R. do Val, A note on the robust control of Markov jump linear uncertain systems, Optimal Control Appl. Methods, 23 (2002), pp. 105–112. [14] J. B. R. do Val and T. Bas¸ar, Receding horizon control of jump linear systems and a macroeconomic policy problem, J. Econom. Dynam. Control, 23 (1999), pp. 1099–1131. [15] J. Dong and G. Yang, Robust H2 control for continuous-time Markov jump linear systems, Automatica J. IFAC, 44 (2008), pp. 1431–1436. [16] V. Dragan and T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative noise, Stoch. Anal. Appl., 20 (2002), pp. 33–92. [17] V. Dragan, T. Morozan, and A. Stoica, Mathematical Methods in Robust Control of Linear Stochastic Systems, Math. Concepts Methods Sci. Eng. 50, Springer, New York, 2006. [18] G. E. Dullerud and F. Paganini, A Course in Robust Control Theory, Texts Appl. Math. 36, Springer-Verlag, New York, 2000. [19] A. El Bouhtouri and K. El Hadri, Robust stabilization of jump linear systems subject to structured uncertainties in the state and input matrices, IMA J. Math. Control Inform., 17 (2000), pp. 281–293. [20] A. El Bouhtouri and K. El Hadri, Robust stabilization of jump linear systems with multiplicative noise, IMA J. Math. Control Inform., 20 (2003), pp. 1–19. [21] A. El Bouhtouri and K. El Hadri, Robust stabilization of discrete-time jump linear systems with multiplicative noise, IMA J. Math. Control Inform., 23 (2006), pp. 447–462. [22] A. El Bouhtouri, D. Hinrichsen, and A. J. Pritchard, Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations, Automatica J. IFAC, 36 (2000), pp. 1033–1040. [23] A. El Bouhtouri and A. J. Pritchard, Stability radii of linear systems with respect to stochastic perturbations, Systems Control Lett., 19 (1992), pp. 29–33. [24] L. El Ghaoui and M. A. Rami, Robust state-feedback stabilization of jump linear systems via LMIs, Internat. J. Robust Nonlinear Control, 6 (1996), pp. 1015–1022. [25] X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37 (1992), pp. 38–53. [26] M. D. Fragoso and J. Baczynski, Optimal control for continuous-time linear quadratic problems with infinite Markov jump parameters, SIAM J. Control Optim., 40 (2001), pp. 270– 297. [27] M. D. Fragoso and J. Baczynski, Lyapunov coupled equations for continuous-time infinite Markov jump linear systems, J. Math. Anal. Appl., 274 (2002), pp. 319–355. [28] M. D. Fragoso and J. Baczynski, Stochastic versus mean square stability in continuous time linear infinite Markov jump parameter systems, Stoch. Anal. Appl., 20 (2002), pp. 347–356. [29] M. D. Fragoso and O. L. V. Costa, A unified approach for stochastic and mean square stability of continuous-time linear systems with Markovian jumping parameters and additive disturbances, SIAM J. Control Optim., 44 (2005), pp. 1165–1191. [30] M. K. Ghosh, A. Arapostathis, and S. I. Marcus, Optimal control of switching diffusions with application to flexible manufacturing systems, SIAM J. Control Optim., 31 (1993), pp. 1183–1204. [31] D. Hinrichsen, A. Ilchmann, and A. J. Pritchard, Robustness of stability for linear timevarying systems, J. Differential Equations, 77 (1989), pp. 254–286. [32] D. Hinrichsen and A. J. Pritchard, Stability radii for linear systems, Systems Control Lett., 7 (1986), pp. 1–10. [33] D. Hinrichsen and A. J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett., 8 (1986), pp. 105–113. [34] D. Hinrichsen and A. J. Pritchard, Real and complex stability radii: A survey, in Control of Uncertain Systems, Progr. Syst. Control Theory 6, D. Hinrichsen and B. M˚ artensson, eds., Springer-Verlag, Basel, 1990, pp. 119–162. [35] D. Hinrichsen and A. J. Pritchard, Robust stability of linear evolution operators in Banach spaces, SIAM J. Control Optim., 32 (1994), pp. 1503–1541. [36] D. Hinrichsen and A. J. Pritchard, Stability radii of systems with stochastic uncertainty and their optimization by output feedback, SIAM J. Control Optim., 34 (1996), pp. 1972–1998. [37] D. Hinrichsen and A. J. Pritchard, Stochastic H ∞ , SIAM J. Control Optim., 36 (1998), pp. 1504–1538. [38] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness, Texts Appl. Math. 48, Springer-Verlag, New York, 2005.


1195

[39] D. Hinrichsen and A. J. Pritchard, Composite systems with uncertain couplings of fixed structure: Scaled Riccati equations and the problem of quadratic stability, SIAM J. Control Optim., 47 (2009), pp. 3037–3075. [40] D. Hinrichsen and N. K. Son, Stability radii of positive discrete-time systems under affine parameter perturbations, Internat. J. Robust Nonlinear Control, 8 (1998), pp. 1169–1188. [41] 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 J. IFAC, 42 (2006), pp. 2159–2168. [42] M. Karow, D. Hinrichsen, and A. J. Pritchard, Interconnected systems with uncertain couplings: Explicit formulae for μ-values, spectral value sets, and stability radii, SIAM J. Control Optim., 45 (2006), pp. 856–884. [43] M. S. Mahmoud and P. Shi, Robust stability, stabilization and H∞ control of time-delay systems with Markovian jump parameters, Internat. J. Robust Nonlinear Control, 13 (2003), pp. 755–784. [44] T. Morozan, Stability radii of some stochastic differential equations, Stoch. Stoch. Rep., 54 (1995), pp. 281–291. [45] T. Morozan, Stability radii of some discrete-time systems with independent random parameters, Stoch. Anal. Appl., 15 (1997), pp. 375–386. [46] T. Morozan, Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations, Stoch. Anal. Appl., 16 (1998), pp. 661–682. [47] A. J. Pritchard and S. Townley, Robustness of linear systems, J. Differential Equations, 77 (1989), pp. 254–286. [48] T. Sathyan and T. Kirubarajan, Markov-jump-system-based secure chaotic communication, IEEE Trans. Circuits Syst. I, 53 (2006), pp. 1597–1609. [49] P. Seiler and R. Sengupta, An H∞ approach to networked control, IEEE Trans. Automat. Control, 50 (2005), pp. 356–364. [50] P. Shi and E. K. Boukas, H∞ control for Markovian jumping linear systems with parametric uncertainty, J. Optim. Theory Appl., 95 (1997), pp. 75–99. [51] A. A. G. Siqueira and M. H. Terra, Nonlinear and Markovian H∞ controls of underactuated manipulators, IEEE Trans. Control Syst. Technol., 12 (2004), pp. 811–826. [52] A. Stoica and I. Yaesh, Jump-Markovian based control of wing deployment for an uncrewed air vehicle, J. Guidance Control Dynam., 25 (2002), pp. 407–411. [53] L. E. O. Svensson and N. Williams, Optimal monetary policy under uncertainty: A Markov jump-linear-quadratic approach, Federal Reserve of St. Louis Review, 90 (2008), pp. 275– 293. [54] D. D. Sworder and R. O. Rogers, An LQ-solution to a control problem associated with a solar thermal central receiver, IEEE Trans. Automat. Control, 28 (1983), pp. 971–978. [55] M. G. Todorov and M. D. Fragoso, Output feedback robust stabilization of continuous-time infinite Markov jump linear systems, in Proceedings of the 46th Annual IEEE Conference on Decision & Control (New Orleans, LA), IEEE Press, Piscataway, NJ, 2007, pp. 3685– 3690. [56] M. G. Todorov and M. D. Fragoso, Infinite Markov jump bounded real lemma, Systems Control Lett., 57 (2008), pp. 64–70. [57] M. G. Todorov and M. D. Fragoso, On the stability radii of continuous-time Markov jump linear systems, in Proceedings of the 2008 American Control Conference, Seattle, WA, 2008, pp. 4621–4626. [58] M. G. Todorov and M. D. Fragoso, Output feedback H∞ control of continuous-time infinite Markovian jump linear systems via LMI methods, SIAM J. Control Optim., 47 (2008), pp. 950–974. [59] M. G. Todorov and M. D. Fragoso, On the robust stability, stabilization, and stability radii of continuous-time Markov jump linear systems, in Proceedings of the 48th Annual IEEE Conference on Decision & Control (Shanghai, China), IEEE Press, Piscataway, NJ, 2009, pp. 3864–3869. [60] M. G. Todorov and M. D. Fragoso, Robust stability and stabilization of continuous-time infinite Markov jump linear systems, in Proceedings of the 2009 European Control Conference, Budapest, Hungary, 2009, pp. 3227–3232. [61] M. G. Todorov and M. D. Fragoso, On the stability radii of continuous-time infinite Markov jump linear systems, Math. Control Signals Systems, 22 (2010), pp. 23–38. [62] C. Van Loan, How near is a stable matrix to an unstable matrix?, Contemp. Math., 47 (1985), pp. 465–478. [63] J. Xiong and J. Lam, Robust H2 control of Markovian jump systems with uncertain switching probabilities, Internat. J. Systems Sci., 40 (2009), pp. 255–265.

1196


[64] J. Xiong, J. Lam, H. Gao, and D. W. C. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica J. IFAC, 41 (2005), pp. 897– 903. [65] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010. [66] L. Zhang and E. K. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica J. IFAC, 45 (2009), pp. 463–468. [67] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1996.