Multi-Objective Output Feedback Control of a Class of Stochastic Hybrid Systems with State Dependent Noise

HAL Id: hal-00578062

https://hal.archives-ouvertes.fr/hal-00578062

Submitted on 18 Mar 2011

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Multi-Objective Output Feedback Control of a Class of Stochastic Hybrid Systems with State Dependent Noise

Samir Aberkane, Jean-Christophe Ponsart, Dominique Sauter

To cite this version:

Samir Aberkane, Jean-Christophe Ponsart, Dominique Sauter. Multi-Objective Output Feedback Control of a Class of Stochastic Hybrid Systems with State Dependent Noise. Mathematical Problems in Engineering, Hindawi Publishing Corporation, 2007, 2007, pp.31561. �10.1155/2007/31561�. �hal- 00578062�

Multi-Objective Output Feedback Control of a Class of Stochastic Hybrid Systems with State Dependent Noise

S. Aberkane J.C. Ponsart and D. Sauter March 15, 2007

This paper deals with dynamic output feedback control of continuous time Active Fault Tolerant Control Systems with Markovian Parameters (AFTCSMP) and state-dependent noise. The main contribution is to formulate conditions for multi-performance design, related to this class of stochas- tic hybrid systems, that take into account the problematic resulting from the fact that the controller only depends on the FDI (Fault Detection and Isolation) process. The specifications and objectives under consideration include stochastic stability, H₂ and H_∞ (or more generally, Stochastic Integral Quadratic Constraints) performances. Results are formulated as matrix inequalities. The theoretical results are illustrated using a classical example from literature.

2000Mathematics Subject Classification: 93E03, 93E15, 93E20, 34D20.

1 Introduction

Control engineers are faced with increasingly complex systems where dependability considerations are sometimes more important than performance. Sensor, actuator or process (plant) failures may dras- tically change the system behavior, resulting in performance degradation or even instability. Thus, fault tolerance is essential for modern, highly complex control systems. Fault Tolerant Control Sys- tems (FTCS) are needed in order to preserve or maintain the performance objectives, or if that turns out to be impossible, to assign new (achievable) objectives so as to avoid catastrophic failures. FTCS have been a subject of great practical importance, which has attracted a lot of interest for the last three decades. A bibliographical review on reconfigurable fault tolerant control systems can be found in [39].

Active fault tolerant control systems are feedback control systems that reconfigure the control law in real time based on the response from an automatic fault detection and identification (FDI) scheme.

The dynamic behaviour of active fault tolerant control systems (AFTCS) is governed by stochastic differential equations (because the failures and failure detection occur randomly) and can be viewed as a general hybrid system [34]. A major class of hybrid systems is jump linear systems (JLS). In JLS, a single jump process is used to describe the random variations affecting the system parameters. This process is represented by a finite state Markov chain and is called the plant regime mode. The theory of stability, optimal control andH₂/H_∞ control, as well as important applications of such systems, can be found in several papers in the current literature, for instance in [5, 6, 7, 8, 9, 12, 13, 14, 22, 23].

To deal with AFTCS, another class of hybrid systems was defined, denoted as active fault tolerant control systems with Markovian parameters (AFTCSMP). In this class of hybrid systems, two random processes are defined: the first random process represents system components failures and the second random process represents the FDI process used to reconfigure the control law. This model was proposed by Srichander and Walker [34]. Necessary and sufficient conditions for stochastic stability of AFTCSMP were developed for a single component failure (actuator failures). In [26], the authors proposed a dynamical model that takes into account multiple failures occurring at different locations in the system, such as in control actuators and plant components. The authors derived necessary and sufficient conditions for the stochastic stability in the mean square sense. The problem of stochastic

stability of AFTCSMP in the presence of noise, parameter uncertainties, detection errors, detection delays and actuator saturation limits has also been investigated in [26, 29, 30, 31]. Another issue related to the synthesis of fault tolerant control laws was also addressed by [27, 32, 33]. In [27], the authors designed an optimal control law for AFTCSMP using the matrix minimum principle to minimize an equivalent deterministic cost function. The problem ofH_∞ and robust H_∞ control (in the presence of norm bounded parameter uncertainties) was treated in [32, 33] for both continuous and discrete time AFTCSMP. The authors defined a single failure process to characterize random failures affecting the system, and they showed that the state feedback control problem can be solved in terms of the solutions of a set of coupled Riccati inequalities. The dynamic/static output feedback counterpart was treated by [1, 2, 3, 4] in a convex programming framework. Indeed, the authors pro- vide an LMI characterization of dynamical/static output feedback compensators that stochastically stabilize (robustly stabilize) the AFTCMP and ensuresH_∞(robustH_∞) constraints. In addition, it is important to mention that the design problem in the framework of AFTCSMP remains an open and challenging problematic. This is due, particulary, to the fact that the controller only depends on the FDI process i.e. the number of controllers to be designed is less than the total number of the closed loop systems modes by combining both failure an FDI processes. The design problem involves searching feasible solutions of a problem where there are more constraints than variables to be solved. Generally speaking, there lacks tractable design methods for this stochastic FTC problem.

Indeed, in [1, 2, 31, 32, 33], the authors make the assumption that the controller must access both failures and FDI processes. However, this assumption is too restrictive to be applicable in practical FTC systems. In this note, and inspired by the work of [15] on mode-independent H_∞ filtering for Markovian jumping linear systems, the assumption on the availability of failure processes, for the synthesis purposes, is stressed. The results are based on a version of the well known Finsler’s lemma and a special parametrization of the Lyapunov matrices. This note is concerned with dynamic output feedback control of continuous time AFTCSMP with state-dependent noise. The main contribution is to formulate conditions for multi-performance design, related to this class of stochastic hybrid sys- tems, that take into account the problematic resulting from the fact that the controller only depends on the FDI process. The specifications and objectives under consideration include stochastic stability, H₂ andH_∞ (or more generally,Stochastic Integral Quadratic Constraints) performances. Results are formulated as matrix inequalities. A coordinate descent-type algorithm is provided and its running is illustrated on a VTOL helicopter example.

This paper is organized as follows: Section 2 describes the dynamical model of the system with appro- priately defined random processes. A brief summary of basic stochastic terms, results and definitions are given in Section 3. Section 4 considers theH₂/H_∞ control problem. In Section 5, a coordinate descent-type algorithm is provided and its running is illustrated on a classical example from literature.

Finally, a conclusion is given in Section 6.

Notations. The notations in this paper are quite standard. R^m×nis the set ofm-by-nreal matrices.

A^T is the transpose of the matrix A. The notation X ≥Y (X > Y, respectively), where X and Y are symmetric matrices, means that X−Y is positive semi-definite (positive definite, respectively);

I and 0 are identity and zero matrices of appropriate dimensions, respectively; E{·} denotes the expectation operator with respect to some probability measureP; L²[0,∞) stands for the space of square-integrable vector functions over the interval[0,∞);k · k refers to either the Euclidean vector norm or the matrix norm, which is the operator norm induced by the standard vector norm; k · k₂ stands for the norm inL²[0,∞); while k · k_E2 denotes the norm inL²((Ω,F, P),[0,∞));(Ω,F, P) is a probability space. In block matrices, ? indicates symmetric terms:

· A B B^T C

¸

=

· A ? B^T C

¸

=

· A B

? C

¸ .

2 Dynamical Model of AFTCSMP with State Dependent Noise

To describe the class of linear systems with Markovian jumping parameters that we deal with in this paper, let us fix a complete probability space (Ω,F, P). This class of systems owns a hybrid state vector. The first component vector is continuous and represents the system states, and the second one is discrete and represents the failure processes affecting the system. The dynamical model of the AFTCSMP with Wiener Process, defined in the fundamental probability space(Ω,F, P), is described by the following differential equations:

ϕ:







dx(t) =A(ξ(t))x(t)dt+B(η(t))u(y(t), ψ(t), t)dt+E(ξ(t), η(t))w(t)dt+P_v

l=1Wl(ξ(t), η(t))x(t)d$l(t) y(t) =C2x(t) +D2(ξ(t), η(t))w(t)

z(t) =C1x(t) +D1(η(t))u(y(t), ψ(t), t)

(1)

where x(t) ∈ Rⁿ is the system state, u(y(t), ψ(t), t) ∈ R^r is the system input, y(t) ∈ R^q is the system measured output, z(t) ∈ R^p is the controlled output, w(t) ∈ R^m is the system external disturbance, ξ(t), η(t) and ψ(t) represent the plant component failure process, the actuator failure process and the FDI process, respectively. ξ(t),η(t) and ψ(t) are separable and mesurable Markov processes with finite state spacesZ ={1,2, ..., z},S ={1,2, ..., s}and R={1,2, ..., σ}, respectively.

$(t) = [$₁(t). . . $_v(t)]^T is a v-dimensional standard Wiener process on a given probability space (Ω,F, P), that is assumed to be independent of the Markov processes. The matricesA(ξ(t)),B(η(t)), E(ξ(t), η(t)), D₂(ξ(t), η(t)), D₁(η(t)) and W_l(ξ(t), η(t)) are properly dimensioned matrices which depend on random parameters.

In AFTCS, we consider that the control law is only a function of the mesurable FDI process ψ(t).

Therefore, we introduce a full order dynamic output feedback compensator (ϕ_d) of the form:

ϕ_d: (

dv(t) =A_c(ψ(t))v(t)dt+B_c(ψ(t))y(t)dt

u(t) =C_c(ψ(t))v(t) (2)

Applying the controllerϕ_dto the AFTCSMP ϕ, we obtain the following closed loop system:

ϕcl:







dχ(t) = Λ(ξ(t), η(t), ψ(t))χ(t)dt+ ¯E(ξ(t), η(t), ψ(t))w(t)dt+P_v

l=1Wl(ξ(t), η(t))χ(t)d$l(t)

¯

y(t) = ¯C2(ψ(t))χ(t) + ¯D2(ξ(t), η(t))w(t) z(t) = ¯C1(η(t), ψ(t))χ(t)

(3)

where:

χ(t) = [x(t)^T, v(t)^T]^T; ¯y(t) = [y(t)^T, u(t)^T]^T; Λ(ξ(t), η(t), ψ(t)) =

· A(ξ(t)) B(η(t))Cc(ψ(t)) Bc(ψ(t))C2 Ac(ψ(t))

¸

; E(ξ(t), η(t), ψ(t)) =¯

· E(ξ(t), η(t)) Bc(ψ(t))D2(ξ(t), η(t))

¸

; ¯C2(ψ(t)) =

· C2 0 0 Cc(ψ(t))

¸

; D¯2(ξ(t), η(t)) =

· D2(ξ(t), η(t)) 0

¸

; ¯C1(η(t), ψ(t)) =£

C1 D1(η(t))Cc(ψ(t)) ¤

;Wl(ξ(t), η(t)) =

· Wl(ξ(t), η(t)) 0

0 0

¸ .

Our goal is to compute dynamical output feedback controllers ϕ_d that meet various specifications on the closed loop behavior. The specifications and objectives under consideration include stochas- tic stability,H₂ performance andH_∞ performance (or more generally,Stochastic Integral Quadratic Constraints (SIQC)).

The FDI and the Failure Processes

ξ(t), η(t) and ψ(t) being homogeneous Markov processes with finite state spaces, we can define the transition probability of the plant components failure process as [31, 34]:





p_ij(∆t) =π_ij∆t+o(∆t) (i6=j) p_ii(∆t) = 1−P

i6=j

π_ij∆t+o(∆t) (i=j)

The transition probability of the actuator failure process is given by:





p_kl(∆t) =ν_kl∆t+o(∆t) (k6=l) p_kk(∆t) = 1−P

k6=l

ν_kl∆t+o(∆t) (k=l)

whereπ_ij is the plant components failure rate, and ν_kl is the actuator failure rate.

Given thatξ =kand η=l, the conditional transition probability of the FDI processψ(t) is:





p^kl_iv(∆t) =λ^kl_iv∆t+o(∆t) (i6=v) p^kl_ii(∆t) = 1−P

i6=v

λ^kl_iv∆t+o(∆t) (i=v)

Here, λ^kl_iv represents the transition rate from i to v for the Markov process ψ(t) conditioned on ξ = k ∈ Z and η = l ∈ S. Depending on the values of i, v ∈ R, k ∈ Z and l ∈ S, various interpretations, such as rate of false detection and isolation, rate of correct detection and isolation, false alarm recovery rate, etc, can be given toλ^kl_iv [31, 34].

For notational simplicity, we will denote A(ξ(t)) = A_i when ξ(t) = i ∈ Z, B(η(t)) = B_j and D₁(η(t)) = D_1j when η(t) = j ∈ S, E(ξ(t), η(t)) = E_ij, D₂(ξ(t), η(t)) = D_2ij and W_l(ξ(t), η(t)) = W_lij when ξ(t) = i∈ Z, η(t) = j ∈ S and A_c(ψ(t)) = A_ck, B_c(ψ(t)) = B_ck, C_c(ψ(t)) = C_ck when ψ(t) = k ∈ R. We also denote x(t) = x_t, y(t) = y_t, z(t) = z_t, w(t) = w_t, ξ(t) = ξ_t, η(t) = η_t, ψ(t) =ψ_t and the initial conditionsx(t₀) =x₀,ξ(t₀) =ξ₀,η(t₀) =η₀ and ψ(t₀) =ψ₀.

3 Definitions and Basic Results

In this section, we will first give a basic definition related to stochastic stability notion and then we will summarize some results about exponential stability in the mean square sense of the AFTCSMP.

3.1 Stochastic Stability

For system (1), whenu_t≡0for all t≥0, we have the following definition.

Definition 1: System (1) is said to be internally exponentially stable in the mean square sense (IESS), if there exist positive constantsα andβ such that the solution of

dx_t=A(ξ_t)x_tdt+ Xv

l=1

W_l(ξ_t, η_t)x_td$_lt

satisfies the following inequality E©

kx_tk²ª

≤βkx₀k²exp [−α(t−t₀)] (4) for arbitrary initial conditions(x₀, ξ₀, η₀, ψ₀).

The following theorem gives a sufficient condition for internal exponential stability in the mean square sense for the closed loop system (3).

Theorem 1: The closed loop system (3) is IESS for t ≥ t₀ if there exists a Lyapunov function ϑ(χ_t, ξ_t, η_t, ψ_t, t) such that

K₁kχ_tk²≤ϑ(χ_t, ξ_t, η_t, ψ_t, t)≤K₂kχ_tk² (5) and

Lϑ(χ_t, ξ_t, η_t, ψ_t, t)≤ −K₃kχ_tk² (6)

for some positive constants K₁, K₂ and K₃, where L is the weak infinitesimal operator of the joint Markov process {χ_t, ξ_t, η_t, ψ_t}.

Remark 1: The proof of the Markovian property of the joint process {χ_t, ξ_t, η_t, ψ_t} is given for example in [25, 31, 34].

A necessary condition for internal exponential stability in the mean square sense for the closed loop systemϕ_cl is given by theorem 2.

Theorem 2: If the system (3) is IESS, then for any given quadratic positive definite function W(χ_t, ξ_t, η_t, ψ_t, t) in the variables χ_t which is bounded and continuous ∀t ≥ t₀, ∀ξ_t ∈ Z, ∀η_t ∈ S and ∀ψ_t ∈ R, there exists a quadratic positive definite function ϑ(χ_t, ξ_t, η_t, ψ_t, t) in χ_t that satisfies the conditions in theorem 1 and is such that Lϑ(χ_t, ξ_t, η_t, ψ_t, t) =−W(χ_t, ξ_t, η_t, ψ_t, t).

Remark 2: The proofs of these theorems follow the same arguments as in [31, 34] for their proposed stochastic Lyapunov functions, so they are not shown in this paper to avoid repetition.

The following proposition gives a necessary and sufficient condition for internal exponential stability in the mean square sense for the system (3).

Proposition 1: A necessary and sufficient condition for IESS of the system (3) is that there exists symmetric positive-definite matricesP_ijk,i∈Z,j∈S and k∈R such that:

Λ˜^T_ijkP_ijk+P_ijkΛ˜_ijk+ Xv

l=1

W^T_lijP_ijkW_lij +X

h∈Z h6=i

π_ihP_hjk+X

l∈S l6=j

ν_jlP_ilk+X

v∈R v6=k

λ^ij_kvP_ijv <0 (7)

∀i∈Z,j∈S andk∈R, where

Λ˜_ijk= Λ_ijk−0.5I





 X

h∈Z h6=i

π_ih+X

l∈S l6=j

ν_jl+X

v∈R v6=k

λ^ij_kv





 (8)

¤ Proof: The proof of this proposition is easily deduced from theorems 1 and 2. ¥ Proposition 2: If the system (3) is IESS, for every w = {w_t;t ≥ 0} ∈ L₂[0,∞), we have that χ = {χ_t;t ≥ 0} ∈ L₂((Ω,F, P),[0,∞)), i.e., E©R_∞

0 χ^T_tχ_tdtª

< ∞, for any initial conditions

(χ₀, ξ₀, η₀, ψ₀). ¤

Proof: The proof of this proposition follows the same lines as for the proof of proposition 4 in [3].¥

We conclude this section by recalling a version of the well known Finsler’s lemma that will be used in the derivation of the main results of this paper.

Lemma 1 [15]: Given matrices Ψ_ijk = Ψ^T_ijk ∈ R^n×n and H_ijk ∈ R^m×n, ∀i ∈ Z, j ∈ S and k ∈ R, then

x^T_tΨ_ijkx_t<0, ∀x_t∈Rⁿ: H_ijkx_t= 0, x_t6= 0; (9) if and only if there exist matricesL_ijk ∈R^n×m such that:

Ψ_ijk+L_ijkH_ijk+H_ijk^T L^T_ijk<0, ∀i∈Z, j ∈S, k ∈R. (10) Note that conditions (10) remain sufficient for (9) to hold even when arbitrary constraints are imposed to the scaling matricesL_ijk.

4 The Control Problem

4.1 H_∞ Control

Let us consider the system (3) with

z_t=z_∞t=C_∞1x_t+D_∞1(η_t)u(y_t, ψ_t, t) z_∞t stands for the controlled output related toH_∞ performance.

In this section, we deal with the design of controllers that stochastically stabilize the closed-loop system and guarantee the disturbance rejection, with a certain levelµ >0. Mathematically, we are concerned with the characterization of compensators ϕ_d that stochastically stabilize the system (3) and guarantee the following for allw∈L²[0,∞):

kz_∞k_E2=E

½Z _∞

0

z^T_∞tz_∞tdt

¾_1/2

< µkwk₂ (11)

whereµ >0is a prescribed level of disturbance attenuation to be achieved. To this end, we need the auxiliary result given by the following proposition.

Proposition 3: If there exists symmetric positive-definite matricesP_∞ijk, i∈Z,j ∈S and k∈ R such that

Λ˜^T_ijkP_∞ijk+P_∞ijkΛ˜_ijk+ Xv

l=1

W^T_lijP_∞ijkW_lij+ ¯C_∞1jk^T C¯_∞1jk+µ⁻²P_∞ijkE¯_ijkE¯_ijk^T P_∞ijk

+X

h∈Z h6=i

π_ihP_∞hjk+X

l∈S l6=j

ν_jlP_∞ilk+X

v∈R v6=k

λ^ij_kvP_∞ijv <0 (12)

∀i∈Z,j∈S andk∈R.

then the system (3) is IESS and satisfies kz_∞k_E2=E

½Z _∞

0

z^T_∞tz_∞tdt

¾_1/2

< µkwk₂ (13)

¤ Proof The proof of this proposition follows the same arguments as in [2]. ¥ Remark 3: Given fixed matrices U ≥ 0, V =V^T and Q, the previous characterization extends to more general quadratic constraints onw_t and z_t of the form

J_SIQC =E (Z _tf

t0

µ z_t w_t

¶_T µ

U Q Q^T V

¶ µ z_t w_t

¶ dt

)

<0 (14)

These constraints are known asstochastic quadratic integral constraints [11].

The following proposition gives a sufficient condition to (14) to be hold.

Proposition 4: If there exists symmetric positive-definite matrices P_ijk, i∈ Z, j ∈ S and k ∈ R

such that 

 Θ_ijk P_ijkE_ijk+C^T_1jkQ C^T_1jk∆

? V 0

? ? −Σ



<0 (15)

where







U= ∆Σ⁻¹∆^T

Θ_ijk = ˜Λ^T_ijkP_ijk+P_ijkΛ˜_ijk+P_v

l=1W^T_lijP_ijkW_lij+ P

h∈Z h6=i

π_ihP_hjk+P

l∈S l6=j

ν_jlP_ilk+ P

v∈R v6=k

λ^ij_kvP_ijv

then

J_SIQC <0

¤ Proof: Let us consider the following quadratic Lyapunov function

ϑ(χ_t, ξ_t, η_t, ψ_t) =χ^T_tP(ξ_t, η_t, ψ_t)χ_t (16) then:

Lϑ(χ_t, ξ_t, η_t, ψ_t) = µ χ_t

w_t

¶_T µ

Θ(ξ_t, η_t, ψ_t) P(ξ_t, η_t, ψ_tE(ξ_t, η_t, ψ_t

? 0

¶ µ χ_t w_t

¶

(17) adding and subtractingEn R_tf

t0 Lϑ(χ_t, ξ_t, η_t, ψ_t)dt o

to J_SIQC, we get

J_SIQC =E (Z _tf

t0

µ χ_t w_t

¶_TÃ

Φ(ξ_t, η_t, ψ_t) +

µ C₁(η_t, ψ_t) 0

0 I

¶_T µ

U Q Q^T V

¶ µ C₁(η_t, ψ_t) 0

0 I

¶! µ χ_t w_t

¶ dt

)

− E

½Z _tf

t0

Lϑ(χ_t, ξ_t, η_t, ψ_t)dt

¾

(18) where

Φ(ξ_t, η_t, ψ_t) =

µ Θ(ξ_t, η_t, ψ_t) P(ξ_t, η_t, ψ_tE(ξ_t, η_t, ψ_t

? 0

¶

From Dynkin’s formula, we have E©

ϑ(χ_tf, ξ_tf, η_tf, ψ_tf)ª

−ϑ(χ_t0, ξ_t0, η_t0, ψ_t0) =E

½Z _tf

t0

Lϑ(ξ_t, η_t, ψ_t)dt

¾

(19) Assuming, without loss of generality, thatϑ(χ_t0, ξ_t0, η_t0, ψ_t0) = 0, it follows from (18) and (19) that if

Φ(ξ_t, η_t, ψ_t) +

µ C₁(η_t, ψ_t) 0

0 I

¶_T µ

U Q Q^T V

¶ µ C₁(η_t, ψ_t) 0

0 I

¶

<0 (20)

then (14) holds.

Finally, by factorizingU≥0as

U= ∆Σ⁻¹∆^T

and by Schur complement property, (20) is equivalent to (15). Hence, the proof is complete. ¥ The H_∞ output feedback fault tolerant control method to be developed in this paper is based on Lemma 1 and with the following parametrization of the Lyapunov matricesP_∞ijk:

P_∞ijk=M^T_kN_∞ijk⁻¹ M_k>0 (21) where N_∞ijk are symmetric positive definite matrices and M_k are nonsingular matrices. This parametrization is inspired by the work of [15] on mode-independent H_∞ filtering for Markovian jumping linear systems.

The next proposition presents the main result of this section, which is derived from proposition 3 with P_∞ijk as in (21) and using Finsler’s lemma together with appropriate parametrization of matrices

M_k and the dynamical controller matrices.

Proposition 5: If there exists matricesR_k,S_k,D_k,X_k,Y_k,Z_kand symmetric matricesN_∞ijk,i∈Z, j∈S andk∈R such that









Ω_k+ Ω^T_k ? ? ? ?

A_ijk−N_∞ijk δ_ijkN_∞ijk ? ? ? 0 E^T_ijk −µ²I ? ?

C_ijk 0 0 −I ?

Ξ_ijk 0 0 0 −Γ_ijk







<0 (22)

where

Ω_k=

· R_k 0 S_k+D_k D_k

¸

(23) A_ijk =

· R_kA_i+R_kB_jZ_k R_kB_jZ_k S_kA_i+Y_kC₂+X_k+S_kB_jZ_k X_k+S_kB_jZ_k

¸

(24) E_ijk =

· R_kE_ij S_kE_ij +Y_kD_2ij

¸

(25) C_ijk=£

C_∞1+D_∞1Z_k D_∞1Z_k ¤

(26) Γ_ijk=diag{k_1ijk,k_2ijk,k_3ijk,k_4ijk} (27)



































































k_1ijk=£

N_∞1jk, . . .N_∞(i−1)jk,N_∞(i+1)jk, . . .N_∞zjk¤ k_2ijk=£

N_∞i1k, . . .N_∞i(j−1)k,N_∞i(j+1)k, . . .N_∞isk¤ k_3ijk=£

N_∞ij1, . . .N_∞ij(k−1),N_∞ij(k+1), . . .N_∞ijr¤ k_4ijk= [N_∞ijk, . . . ,N_∞ijk]

Ξ_ijk = [z_1ijk,z_2ijk,z_3ijk,z_4ijk]^T z_1ijk =

h√

π_i1Ω^T_k, . . . ,√π_i(i−1)Ω^T_k,√π_i(i+1)Ω^T_k, . . . ,√ π_izΩ^T_k

i

z_2ijk =

h√ν_j1Ω^T_k, . . . ,√ν_j(j−1)Ω^T_k,√ν_j(j+1)Ω^T_k, . . . ,√ν_jsΩ^T_k i

z_3ijk =

·q

λ^ij_k1Ω^T_k, . . . , q

λ^ij_k(k−1)Ω^T_k, q

λ^ij_k(k+1)Ω^T_k, . . . , q

λ^ij_krΩ^T_k

¸

z_4ijk = h

Φ^T_1ijk, . . . ,Φ^T_vijk i

Φ_lijk=

"

R_kW_lij 0 S_kW_lij 0

#

δ_ijk=−



P

h∈Z h6=i

π_ih+P

l∈S l6=j

ν_jl+ P

v∈R v6=k

λ^ij_kv





(28)

andk_4ijk,z_4ijk containv elements. Then, the system (3) is IESS and satisfies (13).

Moreover, the transfer functions matrices of suitable controllers are given by H_k(s) =Z_k¡

sI−D⁻¹_k X_k¢₋₁

D⁻¹_k Y_k, ∀k∈R. (29)

¤ Proof: It will be shown that if the inequalities (22) hold, then the controllers (29) ensure that conditions (12) of proposition 3 are satisfied with a matrixP_∞ijk>0 as in (21).

First, note that with a matrixP_∞ijk>0as in (21), the inequalities (12) are equivalent to

Π^T_ijkΥ_ijkΠ_ijk <0 (30)

where

Π_ijk=









I 0 0 0

N_∞ijk⁻¹ M_k 0 0 0

0 I 0 0

0 0 I 0

0 0 0 I







 (31)

Υ_ijk=









0 ? 0 C^T_∞1jk R_∞ijk M_kΛ_ijk δ_ijkN_∞ijk M_kE_ijk 0 0

? ? −µ²I 0 0

? ? ? −I 0

? ? ? ? −S_∞ijk









(32)

















































R_∞ijk = [R_∞1ijk,R_∞2ijk,R_∞3ijk,R_∞4ijk] R_∞1ijk=

h√

π_i1M^T_k, . . . ,√π_i(i−1)M^T_k,√π_i(i+1)M^T_k, . . . ,√ π_izM^T_k

i

R_∞2ijk=

h√ν_j1M^T_k, . . . ,√ν_j(j−1)M^T_k,√ν_j(j+1)M^T_k, . . . ,√ν_jsM^T_k i

R_∞3ijk=

·q

λ^ij_k1M^T_k, . . . , q

λ^ij_k(k−1)M^T_k, q

λ^ij_k(k+1)M^T_k, . . . , q

λ^ij_krM^T_k

¸

R_∞4ijk= h

W^T_1ijM^T_k, . . . ,W^T_vijM^T_k i

S_∞ijk =diag[S_∞1ijk,S_∞2ijk,S_∞3ijk,S_∞4ijk] S_∞1ijk=£

N_∞1jk, . . .N_∞(i−1)jk,N_∞(i+1)jk, . . .N_∞zjk¤ S_∞2ijk=£

N_∞i1k, . . .N_∞i(j−1)k,N_∞i(j+1)k, . . .N_∞isk¤ S_∞3ijk=£

N_∞ij1, . . .N_∞ij(k−1),N_∞ij(k+1), . . .N_∞ijr¤ S_∞4ijk= [N_∞ijk, . . . ,N_∞ijk]

(33)

S_∞4ijk contains v elements.

Define n×n nonsingular matrices U_k and V_k such that U_kV_k = D_k and let the matrix M_k be parameterized as follows

M_k =

· R⁻¹_k R⁻¹_k V_k^T

−U_k⁻¹S_kR⁻¹_k U_k⁻¹V_k^T − U_k⁻¹S_kR⁻¹_k V_k^T

¸₋₁

(34) Moreover, let the transformation matrix

T_k =

· R^T_k S^T_k 0 U_k^T

¸

(35) Note that

M^−T_k T_k=ג_k =

· I 0 V_k V_k

¸

(36) and M_k and T_k as defined above are nonsingular. Indeed, from (22) it follows that Ω_k+ Ω^T_k < 0, which implies that the matrices R_k and D_k are nonsingular. Therefore, in view of the definition of the matricesU_k andV_k, the matricesT_k andM^−T_k T_k are nonsingular and thusM^−T_k are nonsingular matrices as well.

Next, introduce the matrices

F_k=diag©

ג_k,ג_k,I,I,ג_kª

(37)

ג_k ={ג_k, . . . ,ג_k} (38)

whereג_k contains ((z−1) + (s−1) + (r−1) +v) blocks ג_k.

Performing the congruence transformationF^T_k(·)F_k on Υ_ijk, inequality (30) is equivalent to

κ^TF^T_kΥ_ijkF_kκ <0, κ=F⁻¹_k Π_ijkς, ς 6= 0 (39)

considering that ©

κ:κ=F⁻¹_k Π_ijkς, ς6= 0ª

={κ:H_ijkF_kκ= 0, κ6= 0} (40) where

H_ijk =£

M_k −N_∞ijk 0 0 0 ¤

(41) we have that (39) is equivalent to

κ^TF^T_kΥ_ijkF_kκ <0, ∀κ6= 0 :H_ijkF_kκ= 0 (42) By lemma 1, (42) holds iff the following inequalities are feasible for some matricesL_ijk of appropriate dimensions

F^T_kΥ_ijkF_k+L_ijkH_ijkF_k+F^T_kH^T_ijkL^T_ijk <0 (43) Without loss of generality, let the matricesL_ijk be rewritten asL_ijk=F^T_kL_ijk, then (43) is equivalent to

F^T_k ¡

Υ_ijk+L_ijkH_ijk+H^T_ijkL^T_ijk¢

F_k<0 (44) Setting

L_ijk =£

I 0 0 0 0 ¤

(45) inequalities (44) become

F^T_kΥ_ijkF_k <0 (46) where

Υ_ijk =









M_k+M^T_k ? ? ? ? M_kΛ_ijk− N_∞ijk^T δ_ijkN_∞ijk ? ? ? 0 E^T_ijkM^T_k −µ²I ? ?

C_∞1jk 0 0 −I ?

R^T_∞ijk 0 0 0 −S_∞ijk









(47)

Consider the following state-space realization for the controllers (2)

A_ck =V_kD⁻¹_k X_kV_k⁻¹, B_ck=V_kD⁻¹_k Y_k, C_ck =Z_kV_k⁻¹ (48) and let the matrixN_∞ijk be defined as

N_∞ijk=T_k^TM⁻¹_k N_∞ijkM^−T_k T_k (49) By performing straightforward matrix manipulations, it can be easily shown that

T_k^TΛ_ijkM^−T_k T_k =A_ijk, T_k^TE_ijk=E_ijk, C_∞1jkM^−T_k T_k=C_ijk (50) T_k^TM^−T_k T_k= Ω_k, ג^T_kR_∞ijkג_k= Ξ^T_ijk (51) ג^T_kS_∞ijkג_k= Γ_ijk (52) Next, taking into account (37), (38) and (49)-(52), it can be readily verified that (46) is identical to (22). Thus (12) is satisfied with P_∞ijk = M^T_kN_∞ijk⁻¹ M_k. Finally, the controller transfer matrix of

(29) is readily obtained from (48). ¥

From practical point of view, the controller that stochastically stabilizes the AFTCMP and at the same time guarantees the minimum disturbance rejection is of great interest. This controller can be obtained by solving the following optimization problem:

O_∞:





























min

τ >0,N∞ijk=N^T_∞ijk>0,Rk,Sk,Dk,Xk,Yk,Zk

τ s.t:











Ω_k+ Ω^T_k ? ? ? ?

A_ijk−N_∞ijk δ_ijkN_∞ijk ? ? ? 0 E^T_ijk −τI ? ?

C_ijk 0 0 −I ?

Ξ_ijk 0 0 0 −Γ_ijk











<0