Distributed Observer-based Guaranteed Cost Control Design For Large Scale Interconnected Systems GHAZI BEL HAJ FREJ
ASSEM THABET
Research Center for Automatic Control, CRAN UMR7039 Modeling Analysis and Control of Systems, MACS LR16ES22 University of Lorraine, FRANCE University of Gabes, TUNISIA Modeling Analysis and Control of Systems, MACS, LR16ES22 Email:
[email protected] University of Gabes, TUNISIA Email:
[email protected]
MOHAMED BOUTAYEB Research Center for Automatic Control, CRAN UMR7039 University of Lorraine, FRANCE Email:
[email protected]
Abstract—This paper focuses on the problem of observerbased distributed guaranteed cost control for large scale nonlinear interconnected systems with a chosen quadratic cost function. The aim is to ensures closed-loop stability and guaranteed cost for all planned parameter changes. For that, differential mean value theorem is used to introduce a general condition on the nonlinear time-varying interconnections functions. The obtained design procedures are formulated in the form of Linear matrix inequalities (LMIs) by using the Lyapunovs direct method stability analysis. Effectiveness of the proposed scheme is verified through simulation results on a power system with three interconnected machines. Keywords—Large Scale System, Nonlinear interconnected systems, Observer-based Control, Guaranteed cost function, Lyapunov stability.
I.
I NTRODUCTION
Analysis and control of dynamical nonlinear interconnected systems has been an active area of research during the past years. This type of structure is widely used in literature, see [1] [2] [3] [4], and the references therein. Examples include power systems, chemical process, social structures such as internet, where a large number of systems communicate for a general objective. Other examples include transportation, communication networks, groups of robots and others [5]. Meanwhile, stability analysis and control synthesis for practical engineering systems has achieved many results in theory research based on Lypunov method, and it still a challenging subject. The control scheme, in many applications needs the knowledge of the system state at each instant [6] [7]. However, in some cases, the entirely information of the plant is frequently not accessible. For that, the observer-based control is usually applied in such situation for monitoring and controlling a large scale system [8]. Also, in control engineering domain, it is always advantageous to design a control system which is not only stable, but also guarantees an adequate degree of performances [9]. With the increasing penetration of renewable energy and communications networks as new research areas, dispatch
MOHAMED AOUN Modeling Analysis and Control of Systems, MACS LR16ES22 University of Gabes, TUNISIA Email:
[email protected] models are becoming so large-scale that conventional centralized approach would face critical challenges and would be often impractical for computational reasons [10]. The complexity of large-scale mathematical models combining with the huge size of this class of systems make the task more difficult [11]. Despite the progress of software tools, the number of operations to be executed in the real-time implementation still a challenge. To handle these problems, the design of distributed estimation and control schemes is a solution for coping with these limitations. Indeed, various distributed state estimation and control design has recently attracted significant attention [12], [13]. Some benefits results are already published, and for a nice overview, the reader can refer to this list of papers: [14], [15], [6], [7]. Many works related on decentralized observer-based control assume that the nonlinear function of interconnection is uncertain and satisfies some conditions. The designed observer is totally decentralized and the control protocol uses only local information of the subsystem. In the case of distributed control, the control protocol design uses relative information from agent’s neighbors to finish the designed task. For that, the nonlinear interconnection of the studied class of system is assumed to be known and satisfies some conditions. The designed observer is no longer decentralized but affected by external information which are extracted from the interconnection with other subsystems. In [17] and [18], authors developed a decentralized scheme to control the interconnected system where control units of subsystems are not aware of each other. In this work, we assume that the control scheme makes use of information exchange between the controllers allocated with regarded control structure. The goal of this paper is to design a distributed guaranteed cost controller with distributed observer for the studied class of systems. Based on the differential mean value theorem (DMVT), the dynamics of the estimation error can be written using the nonlinear function of interconnection terms as a class of Linear parameter varying (LPV) systems [16]. Non restrictive sufficient conditions on nonlinear interconnection
function are provided. Stability of the estimation error is analyzed using the Lyapunov method. The necessary and sufficient conditions for the existence of distributed observerbased guaranteed cost controllers are synthesized basing on the linear matrix inequality (LMI) approach. The observer and control gains guaranteeing the global convergence of the considered scheme are computed by LMIs. The remainder of this paper is organized as follows. A model of linear subsystems interconnected with nonlinear interconnections and some limitations of existing work are introduced in Section II. Next, in section III, the main result for designing a distributed observer-based guaranteed cost controller is presented. Finally, a numerical examples is used to demonstrate the effectiveness of the proposed scheme. II.
SYSTEM MODEL AND PROBLEM FORMULATION
A linear plant S with nonlinear interconnection composed of N of subsystems Si , i = 1, .., N , can be described as follows Si : x˙ i = Ai xi + Bi ui + gi (t, x), yi = Ci xi
(1)
Vectors xi ∈ Rni , ui ∈ Rmi and yi ∈ Rpi describe respectively the subsystem state, input and output. Ai , Bi and Ci are constant matrices of adequate dimensions. gi (t, x) is the non-linear interconnection term for the ith subsystem. The global system S is characterized by the following representation x˙ = Ax + Bu + g(t, x) S: (2) y = Cx Vectors xT = (xT1 , ..., xTN ), u = (uT1 , ..., uTN ) and y = T ) represent the global state, input and output re(y1T , ..., yN PN PN P spectively ( N i=1 ni = n , i=1 mi = m , i=1 pi = p). Matrices A = diag{A1 , ..., AN }; B = diag{B1 , ..., BN }; C = diag{C1 , ..., CN } are constant matrices of adequate dimensions. g(t, x) = (g1 (t, x)T , ...., gN (t, x)T )T is the global nonlinear interconnection function. In litterature, researches treating the field of observer design of interconnected models, are based on the fact that the nonlinear function gi (t, x) is uncertain and satisfies one of these expressions: -
gi (t, x)T gi (t, x) ≤ α2i xT ΓTi Γi x known as the Quadratic inequalities ([3], [4], [19]). kgi (t, x)k ≤ αi kxk known as the function norm ([20]).
Γi are bounding matrices and αi are interconnection bounds to be maximized. In this work, we assume that the nonlinear interconnected function is known and is differentiable with respect of x and satisfy an assumption that will be presented in the next section. The idea is to show that the high level general optimal stabilization problem for the class of interconnected systems can be formulated as an LMIs problem using DMVT. III.
D ISTRIBUTED STATE OBSERVER - BASED FEEDBACK CONTROL DESIGN
A. Preliminaries and problem formulation Before proposing the design scheme, some mathematical steps are presented. First, assume that the Jacobian matrix of
g satisfies the coming condition [16]: ∂gij (t, x) ≤ ̺ijk ∂xk ∂g (t,x) ij i where ηjk = minZ∈Rn (Z) and ̺ijk ∂xk ∂g (t,x) ij (Z) . maxZ∈Rn ∂xk i ≤ ηjk
(3) =
The proposed distributed observer of the overall system (2), composed from N local observers, is given by : x ˆ˙ = Aˆ x + Bu + L(y − yˆ) + g(t, x ˆ) (4) yˆ = C x ˆ Note that x ˆ is the state observer of the global model. L is assumed to be the observation gain matrix. The observation error between the real state and the observed one of the overall system is ε=x−x ˆ (5) Therefore, from (4) and (2), the dynamic of the global state estimation error is expressed by: ε˙ = (A − LC)ε + ∆g
(6)
where ∆g = g(t, x) − g(t, x ˆ). Notations. In the rest of the paper, the following notations will be used: •
•
the set Co(x, y) = {λx + (1 − λ)y, 0 ≤ λ ≤ 1} is the convex hull of x, y; T s−components z }| { 0, ..., 0, 1 , 0, ..., 0 ∈ Rs , s ≥ 1, is |{z} es (j) = j th
the vector of the canonical basis of Rs .
N is deAssumption 1. The bounded convex domain Dq,n fined as follows: N = Dq,n
1 1 1 N {v = (v11 , ..., v1n , ..., vqn , ..., vqn ) i i i : ηjk ≤ vjk ≤ ̺jk , i = 1, ..., N ; j = 1, ..., q; k = 1, ..., n, }
(7)
N is defined as: The set of vertices of Dq,n N = {α = (α111 , ..., α11n , ..., α1qn , ...αN ΥDq,n qn ) : i i i αij ∈ {ηij , ̺ij }}
(8)
Assumption 2. (Mean Value Theorem [16]). Consider Ψ : Rn → Rq . Let x, y ∈ Rn . We assume that Ψ is differentiable on Co(x, y). Then, there are constant vectors z1 , ..., zq ∈ Co(x, y), zi 6= x, zi 6= y for i = 1, ..., q such that q,n X ∂Ψ i Ψ(x) − Ψ(y) = eq (i)eTn (j) (zi ) (x − y). (9) ∂x j i,j=1
By applying Assumption 1 on the function g [21] [16], the existence of zi ∈ Co(x, xˆ), for all i = 1...n, such that: q,n X ∂gi eq (i)eTn (j) (zi ) ε (10) ∆g = g(t, x) − g(t, xˆ) = ∂x j i,j=1
(11)
with J is a quadratic cost performance. Q = QT > 0 and R = RT > 0 are a known weighting matrices. The main result related to the convergence analysis of the estimation error and enuring the guaranteed cost criterion is summarized in the following theorem..
with ΞTz = [ΞTz1 , ..., ΞTzn ]. From (10) and (11), the global estimation error dynamics (6) pass into ε˙ = (A − LC)ε (12)
Theorem 1. The global system is stable in the sense of Lyapunov and the cost performance (19) is guaranteed if there exist matrices P0 = P0T , W and Z of appropriate dimensions such that the following LMI is feasible:
where A = A + Ξz . At this point, the task of designing an observer for the studied class of nonlinear systems (2) is transformed into a resolution of stability problem for an LPV system(12).
Diag(F (α1 ), ..., F (α2 )) < 0, f or i = 1, ..., 2N qn N αi ∈ VDq,n where Y11 Y12 i F (α ) = 0 for the augmented system (16) and the cost function (22) if the closed loop system is quadratically stable [22]. The closed loop value of the cost function (22) satisfies the bound J < J˜ for all admissible non-linearities. The Lyapunov function V (˜ x) related to this problem is given by: ˜ (23) V (˜ x) = x ˜T P x Pc 0 P = is the Lyapunov matrix. 0 P0 T Pc = Pc = diag{Pci } and P0 = P0T = diag{P0i } are Lyapunov symmetric definite positive matrices. In the rest of the paper, the objective is to propose conditions for which d ˜x < 0 V (˜ x) + x ˜T Q˜ (24) dt Noting that g˜(t, x ˜) = g˜, then from (23) and according to (24), we have: ˜ T P x˜ + x ˜ +x ˜x + h) ˜x + h) ˜x < 0 (A˜ ˜T P (A˜ ˜T Q˜
(25)
The equation (25) can be rewritten as: ˜T P x ˜x + h ˜ x + x˜T Q˜ ˜+x ˜T P ˜ h
