Novel LMI conditions for observer-based stabilization of Lipschitzian nonlinear systems and uncertain linear systems in discrete-time

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Abstract

In this paper, it is shown that the observer-based control of uncertain discrete-time linear systems is conditioned by the solvability of three linear matrix inequalities that must hold simultaneously. The developed theory is then extended to Lipschitz discrete-time nonlinear systems. We show that the observer-based control problem, which is originally a non-convex issue, can be decomposed into two separate convex problems formulated as a set of numerically tractable linear matrix inequalities conditions. The new proposed linear matrix inequalities are neither iterative nor subject to any equality constraint. Illustrative examples are given to indicate the novelty and effectiveness of the proposed design.

Introduction

Usually, the design of feedback systems is achieved under the assumption that the system states are available for feedback. However, this unrealistic assumption is not always verified, and hence, the construction of the unmeasured states through the knowledge of the system inputs and outputs still an unavoidable task to solve any desired control problem. As a matter of fact, state estimation is not quite limited to stabilization exercises, but it is also a crucial task that permits to detect the system faults, evaluate the performances of industrial processes, or identify unknown parameters of inherently complex dynamical systems. The name of observer is referred to as a dynamical system that uses the information of the system inputs and outputs to reconstruct the unmeasured states of the system under consideration. For deterministic and stochastic linear systems, the theory of observers is well developed thanks to the pioneer works of Kalman [1] and Luenberger [2]. However, for uncertain linear systems, there is no generic procedure to solve the observation issue, which motivates the research in this area for the past decades, see for example [3]. When parts of the system dynamic is not completely known and the state vector is not entirely available for feedback, the available results are limited to some cases including matched uncertainties [4] norm-bounded uncertainties [5], [6] and uncertainties of dyadic types [7], [8].

The dynamic output feedback for discrete-time uncertain systems has been the subject of numerous papers, see for example [9], [10]. Reduced order observer-based compensators for continuous-time systems was discussed in [11]. Conceptually, the observer-based control of uncertain linear systems is recognized to be a non-convex issue since the computation of the observer and the controller gains is usually conditioned by the solution of some matrix inequalities which are not numerically tractable [12]. Available techniques that have been devoted to observer-based stabilization of uncertain linear systems can be classified into three categories: Lyapunov-based techniques as in [13], iterative linear matrix inequalities (ILMIs) procedures as proposed in [14], and convex optimization-based algorithms with equality constraints as recently discussed in [15]. Roughly speaking, the Lyapunov-based design leads in general to complicated analysis and necessitates many computational steps to solve the entire problem. Even ILMIs algorithms give a straightforward method to solve the observer-based problem, the computational algorithms are at least two steps procedures that permit to find, in convex optimization setting, the observer and the controller gains. Therefore, ILMIs cannot be classified as convex inequalities because they cannot be solved simultaneously. Linear matrix inequality algorithms subject to equality constraints as used in [15] permit to reverse the observer-based issue to a convex one but, in the meantime, they may increase the conservatism of the conditions under the presence of significant uncertainties. In this paper, new sufficient LMIs conditions, that guarantee the stability of discrete-time uncertain linear systems under the action of observer-based feedbacks, are proposed. By introducing new scalar variables, the original non-convex problem is decomposed into two separate convex issues: observer design and controller design. It will be shown that the determination of the observer and the controller gains is conditioned by the solvability of three linear matrix inequalities that must hold simultaneously. In comparison with existing results, the proposed LMIs are neither subject to any equality constraint nor iterative. Furthermore, the proposed design is novel in the sense that the observer design issue is decoupled from the controller design problem by introduction of new free parameters that link the two separate problems. Subsequently, We show that the results can be extended to discrete-time nonlinear systems whose nonlinearities are globally Lipschitz. The novelty of the proposed LMI-based procedure is tested through numerical examples.

Throughout this paper, the notation A>0 (respectively A<0) means that the matrix A is positive definite (respectively negative definite). We denote by A′ the matrix transpose of A. We note by I and 0 the identity matrix, and the null matrix of appropriate dimensions, respectively. IR stands for the set of real numbers, and “★” is used to notify a matrix element that is induced by transposition.

Section snippets

System description and preliminaries

Consider the uncertain linear systemxk+1=(A+ΔA(k))xk+(B+ΔB(k))uk,yk=(C+ΔC(k))xk+(D+ΔD(k))uk,where xkRn is the state vector, ukRm is the control input, and ykRp is the system output. The nominal matrices ARn×n, BRn×m, CRp×n, and DRp×m are constant known matrices and ΔA(k)Rn×n, ΔB(k)Rn×m, ΔC(k)Rp×n, and ΔD(k)Rp×m are partially known uncertainties defined as follows:ΔA(k)=MAFA(xk,k)NA;FA(xk,k)FA(xk,k)I,ΔB(k)=MBFB(xk,k)NB;FB(xk,k)FB(xk,k)I,ΔC(k)=MCFC(xk,k)NC;FC(xk,k)FC(xk,k)I,ΔD(k)=

Extension to discrete-time nonlinear systems

Observer design for systems with globally Lipschitz nonlinearities has been considered in many research papers, see for example [17], [18], [19]. In this section we show that the complexity of the stabilization problem by the use of high-gain observers is equivalent to our initial problem discussed in the previous sections. It is important to stress that many of the developed algorithms require that the Lipschitz constants of nonlinearities should be sufficiently small to increase the chance of

First illustrative example

To illustrate the powerfulness of the proposed LMIs, let us consider the discrete-time system with the following state matrices:A=10.10.4110.5-0.301,B=0.10.3-0.40.50.60.4,C=111111,MA=0000.10.3100.20,NA=0000.200.400.10,MC=000.3000.8,NC=000000000.2.By the use of the Matlab LMI package, a solution of LMIs of Theorem 1 isP1=1.8720-0.7033-0.5523-0.70334.9703-0.5303-0.5523-0.53031.5312,P1=6.1099-0.20910.1894-0.20910.61350.17350.18940.17352.0335,Y1=1.16521.1241-1.28261-1.2499-1.1599-0.7464,Y2=-2.3531-

Application to stabilization of one-link flexible joint robot

The single-link flexible joint robot is described by the following dynamics [17]:θ˙m=ωm,ω˙m=kJm(θ-θm)-BJm+KτJmu,θ˙m=ω,ω˙=kJ(θ-θm)-mghJsin(θ),where Jm represents the inertia of the actuator (d.c. motor), and J stands for the inertia of the link. θm and θ are the angles of rotations of the motor and the link, respectively. θ˙m and θ˙ are their angular velocities. k,Kτ,m,g, and h are positive constants, see Table 1. For the parameters given in Table 1, we can write system (47) as follows:

Conclusion

New sufficient linear matrix inequality conditions are proposed to solve the dynamic output feedback for discrete-time systems subject to both Lipschitz nonlinearities or norm-bounded uncertainties. The proposed design is novel, in the sense that, the numerically tractable conditions are neither iterative nor subject to equality constraints. Furthermore, the proposed design is straightforward and covers general systems with uncertainties in all the state matrices. The proposed design is

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