Reliable guaranteed cost control for parameterized interconnected systems with LMI characterization

RELIABLE GUARANTEED COST CONTROL FOR PARAMETERIZED INTERCONNECTED SYSTEMS WITH LMI CHARACTERIZATION Gisela Pujol ∗,1Jos´e Rodellar ∗∗,1 Josep M. Rossell ∗∗,1 ∗Escola Universitaria d’Enginyeria T`ecnica Industrial de Barcelona, Universitat Polit`ecnica de Catalunya, Spain ∗∗ Departament de Matem`atica Aplicada III, Universitat Polit`ecnica de Catalunya, Spain Abstract: This paper presents the design of a reliable decentralized state feedback control for a class of uncertain interconnected polytopic continous systems. A model of failures in actuators is adopted which considers outages or partial degradation in independent actuators. The control is developed using the concept of guaranteed cost control and a new LMI characterization using polytopic Lyapunov functions. Copyright c 2005 IFAC Keywords: Reliability, Interconnected Systems, LMI 1. INTRODUCTION The decentralized control of interconnected systems in the presence of uncertainties has attracted considerable attention in the last years. With the aim of stabilizing the overall system while ensuring a satisfactory performance, the guaranteed cost control approach (GCC) has been recently considered (see (Gong et al., 1996), (Mukaidini et al., 2002), (Xie et al., 1999) and (Yang et al., 2000)). The guaranteed cost control is concerned with the design of a state feedback controller such that the closed-loop system is stable and an upper bound of a quadratic cost function is minimized. Another important issue when dealing with interconnected systems is the design of fault-tolerant control systems. Reliable control is concerned with the design of a closed-loop system to maintain key properties, in spite of sensor or actuator outage or 1This work was funded by CICYT-Spain under project DPI2002-04018-C02-01. The partial support of the Government of Catalonia’s grant ACI053-2002 is also appreciated. partial degradation. Two main approaches have been proposed in the literature. One uses multiple controllers in a redundant control scheme ((Siljak, 1980), (Yang et al., 1998)). The other approach seeks a reliable control design without redundancy by ensuring stability and some performance bounds for specified class of admissible failures of particular control components (Veillette, 1995). Within this approach, the class of failures have been usually modelled as outages. This model considers that the control set can be partitioned into two subsets: one subset includes the actuators whose failures are admissible in the control design; and the complementary subset with the actuators that are assumed to keep a normal operation (see (Liao et al., 2002)). A more general failure scheme is considered in (Yang et al., 2000), where the authors present a model of actuator failures which takes into account the outage case and also the possibility of partial failures. In (Yang et al., 2000) a centralized reliable control is designed for a class of uncertain nonlinear systems without subsystem interconnec- tions. In the present paper, we adopt this failure model extended within a reliable decentralized control scheme for uncertain interconnected linear systems. Linear matrix inequalities (LMI) techniques have been used in the context of guaranteed cost control problems and reliable control design ((Yang et al., 2001), (Liao et al., 2002)). The currently known LMI characterizations are potentially conservative in the sense that they use a common Lyapunov function regardless of the parameter values. In order to reduce conservatism in the case of constant parameters, the notion of parameter-dependent function was introduced (Barmish, 1989). The Lyapunov conditions leads to nonconvex optimization problem which does not seem tractable in general. In (Apkarian et al., 2000) and (Tuan et al., 2003), this weakness is overcome using auxiliary variables, as well as replacing the functions of singular Lyapunov for multiple functions to obtain more robust tools and to reduce the conservatism in the control problems, and provides additional flexibility in a wide range of problems. In (Tuan et al., 2003), a way to ”convexifying” the general affine Lyapunov problem has been proposed, leading to a parametric linear matrix inequality. This paper has two objectives. One is related to the decentralized reliable guaranteed cost control problem for interconnected continous systems. To our knowledge, there are not available results dealing with reliable control for uncertain interconnected systems with guaranteed cost. We present a control design that allows to consider reliability and guaranteed cost. This reliable control shows that the admission of control failures imposes some restriction in the control weighting matrices in the performance criterion. Thus the designer can take some trade-off between control performance and admitted reliability. The second objective is to give a new LMI characterization and its relation with polytopic systems and polytopic Lyapunov functions. In the centralized control design of uncertain systems with guaranteed cost and reability, matrix inequalities are obtained which lead to LMI’s through a linealization process. When dealing with decentralized control of interconnected systems this process needs to be extended. In this paper, we obtain a general result that can be applied to different classes of problems, such as guaranteed cost, H2or H∞ control design. We apply this result in the design of decentralized reliable control of uncertain interconnected systems with guaranteed performance. The notation throughout the paper is fairly standard. In symmetric block matrices or long matrix expressions, we use ∗as an ellipsis for terms that are induced by symmetry, e.g. S+ (∗)∗ M Q ≡S+STMT M Q . We use bold to denote dependence on the parameter α, e.g. A=A(α); Idenotes the identity matrix. 2. PROBLEM STATEMENT Consider a class of large-scale interconnected system composed of Npolytopic subsystems described by the following state equations:    ˙xi=Ai(α)xi+Bi(α)ui+X j6=i Gijgij(t, xj) xi(0) = xi0, i = 1,...,N . (1) We define: •xi∈Rnistate of the ith subsystem •ui∈Rsicontrol of the ith subsystem •Ai(α)∈Rni×niparametric state matrix •Bi(α)∈Rni×siparametric control matrix •gij(t, xj)∈Rliunknown interconnection vector function •Gij ∈Rni×liconstant interconnection matrix It is assumed that the unknown vectors gij(t, xj) are continuous and sufficiently smooth in xjand piecewise continuous in t. Uncertain model The parameter uncertainties considered here are assumed to be of the following form: Ai(α) = L X k=1 αkAik, Bi(α) = L X k=1 αkBik.(2) The parameter vector αis in the simplex Π defined by Π = {α∈RL, L X k=1 αk= 1, αk≥0, k = 1, . . . , L}. (3) That is, Ai(α)∈co{Ai1,...,AiL}and Bi(α)∈ co{Bi1, . . . , BiL}are convex combinations of the matrix Akand Bk, respectively. Interconnection assumptions Assumption 1. There exist known constant matrices Wij such that, for all xj∈Rnj, ||gij(t, xj)|| ≤ ||Wij xj|| (4) for all i,jand for all t≥0, where || || denotes the Euclidean norm. Assumption 2. For all i,Wi:= N P j=1,j6=i WT ji Wji >0. The Assumption 1 allows some linear structure to the interconnection vector gij in terms of the state vector xj. The Assumption 2 ensure that almost two subsystems must be interconnected, because at least one Wij must be nonzero, for each i. Cost function Consider the following cost function associated with system (1): J(x, u) = N X i=1 Z∞ 0xT iQixi+uT iRiuidt (5) where Qi∈Rni×niand Ri∈Rsi×siare given constant symmetric positive definite matrices, for all i. Failure model Let uF idenote the vectors with the signals from the siactuators which control the ith subsystem. Here we consider the following failure model: uF i= Λiui+φi(ui), i = 1, . . . , N (6) where Λi=diag(λi1,...,λisi)∈Rsi×siis a diagonal positive definite matrix. The uncertain function φi(ui) = (φi1(ui1),...,φisi(ui si)) satisfies, for each i, φ2 ij(uij)≤γ2 ij u2 ij , j = 1,...,si(7) where γij ≥0. If (7) holds, then ||φi(ui)||2≤ ||Γiui||2, i = 1,...,N (8) where Γi=diag(γi1,...,γisi)∈Rsi×siis a diagonal positive semidefinite matrix. The value of λij represents the percentage of failure in the jth actuator of the controller of the ith subsystem. Each actuator can fail independently. If λij = 1 and γij = 0, it corresponds to the normal case for the jth actuator of the ith subsystem (uF ij =uij). When this is true for all j, we have Λi=Isiand Γi= 0 and it corresponds to the normal case in the ith canal (uF i=ui). When λij =γij, (6) and (7) cover the outage case (uF ij = 0) because φij =−λij uij verify (7). The case φi(ui) = −Λiuicorresponds to the outage of the whole controller of the ith system. Other cases correspond to partial failures or partial degradations of the actuators. Control objective (Reliable guaranteed cost control) The objective of this paper is to design a set of decentralized feedback control laws ui(t) = Kixi(t) (i= 1,...,N) and obtain a Lyapunov function defined by Xi, for the interconnected systems (1) with uncertainties model (2)-(3) and Assumptions 1 and 2, in such a way that, in the presence of the failures described by (6) and (7), the following property is satisfied: N X i=1 d dtxT iXixi+xT iQixi+ (uF i)TRiuF i<0. (9) This inequality leads to a bound for the cost function (5) in the form J(x, uF)≤¯ J, where ¯ J is some specified constant, and to ensure that the closed-loop system              ˙xi=Ai(α)xi+Bi(α)uF i+ N X j=1,j6=i Gij gij uF i= Λiui+φi(ui) ui=Kixi (10) is asymptotically stable. Definition 3. The set of above feedback control laws uiis said to be a reliable guaranteed cost control. 3. MAIN RESULTS 3.1 Instrumental tools Lemma 4. (Projection lemma). For a symmetric matrix Ψ and matrices P,Qwith appropriate dimensions, there exists a matrix Xsuch that: Ψ + PTXTQ+QTXP < 0⇔   NT PΨNP<0 NT QΨNQ<0 with NPand NQany matrices whose columns form bases of Pand Qrespectively. Lemma 5. (Schur Complement). Consider a symmetric matrix M=P1P2 PT 2P3. Then, M < 0⇔(P1−P2P−1 3PT 2<0 P3<0invertible. 3.2 LMI characterization Here we give a general theorem that allows a LMI characterization. The subindex iis omitted because the results can be applied to interconnected systems but also to a single system. Theorem 6. The following statements, involving symmetric positive definite matrix variables X, Yand general matrix variables V,N,K, are equivalent. (i) There exist X,Ksuch that (A+BK)TX+ (∗)∗ ∗ GTXQ11 ∗ C Q21 Q22 !<0 (11) (ii) There exist Y,Vand Nsuch that     −(V+VT)∗ ∗ ∗ ∗ AV+Y+BN−Y∗ ∗ ∗ 0GTQ11 ∗ ∗ CV 0Q21 Q22 ∗ V0 0 0 −Y     <0 (12) −Y∗ ∗ GTQ11 ∗ 0Q21 Q22 !<0 (13) (iii) Consider A=A(α), B=B(α), X=X(α) as in (2). There exist Y=Y(α), Vand N such that     −(V+VT)∗ ∗ ∗ ∗ AkV+Yk+BkN−Yk∗ ∗ ∗ 0GTQ11 ∗ ∗ CV 0Q21 Q22 ∗ V0 0 0 −Yk     <0 (14) −Yk∗ ∗ GTQ11 ∗ 0Q21 Q22 !<0.(15) The matrix Q=Q11 ∗ Q21 Q22 has to be a negative defined matrix. PROOF. Taking into account that PL k=1 αk= 1, the inequality (12) is true when it holds in the vertices of the simplex Π, so (ii) and (iii) are equivalent. To see the equivalence between (i) and (ii), consider a new variable Ni:= KiVi and apply Projection Lemma 4 to (12). Consider ¯ A=A+BK,P= (Id, 0,0,0,0), Q= −Id, ¯ AT,0, CT, Idand ψ=    0∗ ∗ ∗ ∗ Y−Y∗ ∗ ∗ 0GTQ11 ∗ ∗ 0 0 Q21 Q22 ∗ 0 0 0 0 −Y     . The null spaces bases of Pand Qare NP=    0000 I000 0I0 0 0 0 I0 000I     ,NQ=    ¯ AT0CTId I0 0 0 0I0 0 0 0 I0 0 0 0 I     . By NT PψNP<0, we have    −Y∗ ∗ ∗ GTQ11 ∗ ∗ 0Q21 Q22 ∗ 0 0 0 −Y   <0.(16) By Schur complement, (16) is equivalent to (13). From NT QψNQ<0, we have    ¯ AY +Y¯ AT −Y∗ ∗ ∗ GTQ11 ∗ ∗ CYQ21 Q22 ∗ Y0 0 −Y   <0. By Schur complement, ¯ AY +Y¯ AT ∗ ∗ GTQ11 ∗ CYQ21 Q22 !<0.(17) Consider Q=Q11 ∗ Q21 Q22 . Applying Schur complement again, ¯ AY +Y¯ AT −(GYCT)Q−1GT CY<0. Using now that Y=X−1, we have X¯ A+¯ ATX−(XG CT)Q−1GTX C<0, which is equivalent to (11).  3.3 Stability problem In this subsection, we apply Theorem 6 to stability characterization. We introduce an alternative characterization of the fundamental Lyapunov stability theorem for linear interconnected systems. It introduces a new transformation on the Lyapunov variables which helps to reduce the typical degree of conservatism in some problems. The obtained result will be the base of the development of the reliable guaranteed control in the next section. Consider the polytopic Lyapunov function: V(x, α) = N X i=1 xT iXixi,(18) where Xi=PL k=1 αkXik,α∈Π and Xik>0, for all k= 1,...,L and for all i= 1,...,N. This function has to satisfy: (i) V(x, α)>0, (ii) ˙ V(x, α)<0. Let us evaluate condition (ii). From (18), ˙ V(x, α) = N X i=1 ˙xT iXixi+xT iXi˙xi<0. By using the feedback controls ui=Kixiin (1), ˙ V= N X i=1 xT i(Ai+Bi)Ki)TXi+Xi(Ai+BiKi)xi+ + N X i=1 X j6=i gT ij(t, xj)GT ijXixi+xT iXiGij gij(t, xj)<0. Now, using (4), this inequality leads to (Ai+BiKi)TXi+ (∗)∗ ∗ GT iXiQi 11 ∗ CiQi 21 Qi 22 !<0 (19) with Gi= (Gi1,...,GiN ), Ci=Id,Qi 11 = diag(−Id, , . . . , −Id), Qi 21 = 0, Qi 22 =−W−1 i. This step involves tedious manipulations. The details are omitted here, but can be found in (Pujol, 2004). So that, if (19) holds for each i= 1,...,N, then ˙ V < 0 and the system is stable. In order to obtain an efficient LMI condition, we now use Theorem 6 to characterize the stability. Theorem 7. Under assumptions 1 and 2, consider the system (1) with uncertain model (2) and (3), and consider the state-feedback controls ui= Kixi. Assume that there exist symmetric positive definite matrices {Yik}kand matrices Viand Ni such that the LMI system     −(Vi+VT i)∗ ∗ ∗ ∗ AikVi+Yik+BikNi−Yik∗ ∗ ∗ 0GT iQi 11 ∗ ∗ CiVi0Qi 21 Qi 22 ∗ Vi0 0 0 −Yik     <0 (20) −Yik∗ GT iQi 11 <0 is feasible for all k= 1,...,L and i= 1,...,N. Then, the system is stable with Lyapunov function (18) and      Ki=NiV−1 i Xi(α) = L X k=1 αkY−1 ik PROOF. The proof is an immediate application of Theorem 6, for each k. If we use a Lyapunov function parameter independent, we can consider Yiinstead of Yikand the same expression is obtained. 3.4 Reliable guaranteed cost control By Definition 3, the system (10) has reliable guaranteed cost control ui=Kixiif there exist symmetric variable matrices Xiand gain matrices Kisuch that the following inequality Ξi∗ ∗ GT iXi−I∗ BT iXi+RiΛiKi0Ri−I!<0,(21) is feasible, where Ξi= (Ai+BiΛiKi)TXi+Xi(Ai+BiΛiKi)+ +Wi+Qi+KT iΓ2 iKi+KT iΛiRiΛiKi. This characterization involves tedious manipulations where (4) and (8) are used. The details are omitted here, but can be found in (Pujol, 2004). By Schur complement, inequality (21) is equivalent to (22), defined in figure 1. In order to obtain an LMI characterization, it is necessary to separate the terms in Xiand Kiin Ξifrom (21) to be linear. After some manipulations, we obtain that this is possible if Ri−Id is a negative definite matrix and invertible. In this way, we obtain a restriction on the cost function. This means that we may loose some freedom in prescribing the control performance to achieve a reliable control. In accordance with this result, we introduce the following assumption before constructing the reliable guaranteed cost control. Assumption 3. The cost control matrix Rimust be invertible and verify Ri−Id < 0. Taking this assumption into account, we obtain that (22) is equivalent to:       (Ai+ˆ BiKi)TXi+ (∗)∗∗∗ ET iXiQi 11 ∗ ∗ Ci0Qi 22 ∗ FiKi0 0 Qi 33       <0 (23) where ˆ Bi=Bi(I+ (Id −Ri)−1Ri)Λi Fi= (Λi,Γi, RiΛi),Ei= (Gi,Bi), Ci= (I, I)T Qi 11 =−I∗ 0Ri−I, Qi 22 =−Q−1 i∗ 0−W−1 i Qi 33 = −R−1 i∗ ∗ 0−I∗ 0 0 Ri−I!. We present now the main result. Theorem 8. Under Assumptions 1, 2 and 3, consider the system (10) with the polytopic structure (2) and (3). Suppose that, for each i= 1, . . . , N, there exist symmetric positive definite matrices {Yik}k=1,...,L and matrices Viand Nithat make the LMI system       −(Vi+VT i)∗ ∗ ∗ ∗ ∗ AikVi+Yik+ˆ BikNi−Yik∗ ∗ ∗ ∗ 0ET ikQi 11 ∗ ∗ ∗ CiVi0 0 Qi 22 ∗ ∗ FT iNi0 0 0 Qi 33 ∗ Vi0 0 0 0 −Yik       <0 (24) −Yik∗ ET ikQi 11 <0 feasible for all k= 1,...,L and all i= 1, . . . , N. Then, the set of state feedback controls ui= Kixiis a reliable guaranteed cost control with Lyapunov function V(x, α) = PN i=1 xT iXi(α)xi, where        Ki=NiV−1 i Xi(α) = L X k=1 αkY−1 ik. (25) Moreover, for any xi(0) = xi0, the cost function satisfies J < N X i=1 xT i0Xi(α)xi0.(26) PROOF. Applying the Theorem 6, we obtain (23) from (24).  The cost bound (26) depends on the initial conditions. In order to eliminate this dependence, the mean value of the cost function is sought over all possible values xi0. This is equivalent to: E(J)< N X i=1 tr(Xi(α)) = N X i=1 L X k=1 tr(Xik)αk, where tr denotes the trace of a matrix. This equation allows to find an optimum value. Consider-                 (Ai+BiΛiKi)TXi+ (∗)∗ ∗ ∗ ∗ ∗ ∗ GT iXi−I∗ ∗ ∗ ∗ ∗ BT iXi+RiΛiKi0Ri−I∗ ∗ ∗ ∗ ΛiK0 0 −R−1 i∗ ∗ ∗ ΓiKi0 0 0 −I∗ ∗ I0 0 0 0 −Q−1 i∗ I0 0 0 0 0 −W−1 i                 <0.(22) Fig. 1. ing ¯ J:= maxkPN i=1 tr(Xik), we have the cost function bounded by E(J)<¯ J. 4. CONCLUSIONS In this work, a solution for a reliable decentralized guaranteed cost control problem for interconnected systems with polytopic uncertainties has been presented. Failures are described by a model which considers possible outage or partial failures in every actuator of each decentralized controller. The control design involves two steps. First, an LMI characterization is presented. Second, a sufficient condition is given for the existence of a decentralized reliable guaranteed cost control set. 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