Stability analysis and observer design for a class of nonlinear systems with multiple time-delays
© Dong and Yang; licensee Springer 2013
Received: 5 December 2012
Accepted: 7 May 2013
Published: 23 May 2013
In this paper, we propose a simple and useful approach to design an observer for multiple time-delays nonlinear systems in a triangular form. By constructing a new Lyapunov-Krasovskii functional and using the differential mean-value theorem, the sufficient conditions for the existence of such an observer are derived, which guarantee that the estimation error converges asymptotically towards zero. The observer gain is independent of the time-delay. A numerical example is provided to illustrate the result.
Time-delay, as well as nonlinearities, is often encountered in various systems which render the control design more difficult . During the past decades, a lot of significant advances have been proposed in stability analysis and feedback control for time-delay systems, e.g., [1–7] and reference therein. Among these schemes, the system states are assumed to be precisely known for the control design, which is not true in some practical cases as some commercial control systems are not equipped with enough sensors. This inspires the issue of observer design for control systems, which is an active research topic in the control community.
Different types of observers have been proposed, e.g., Luenberger observer , adaptive observer , high-gain observer . The observer design problem for time-delay systems has been widely investigated in the recent years. For time-delay systems, most of the state observation methods developed in the literature concern the linear case; we refer the reader to some recent advances and their extensions [11–13]. However, the problem of state estimation of time-delay systems in the nonlinear case has been rarely studied. For an overview of recent works, see, e.g., [14–16]. In , a new approach to the nonlinear observer design problem in the presence of delayed output measurements was presented. The proposed nonlinear observer possesses a state-dependent gain which is computed from the solution of a system of first-order singular partial differential equations. In , the authors established a new method for the observer design problem for a class of Lipschitz time-delay systems. The obtained synthesis conditions are expressed in terms of linear matrix inequalities (LMIs) easily tractable and are less restrictive than those obtained in . In , the problem of observer design for a class of multi-output nonlinear system was considered. A new state observer design methodology for linear time-varying multi-output systems was presented. Furthermore, the same methodology was extended to a class of multi-output nonlinear systems and some sufficient conditions for the existence of the proposed observer were obtained, which guaranteed that the error of state estimation converged asymptotically to zero. For further results on observation of time-delay systems, we refer the reader to [20–23] and the references therein.
In this paper, we investigate observer design for nonlinear systems written in a triangular form. Our main task is to design the observer for a class of nonlinear systems with multiple time-delays. The observer is convergent, whatever the size of the delay. The design method of observer for the class of nonlinear systems with multiple time-delays is proposed, and the gain matrix is obtained. The observer gain is independent of the time-delay. The sufficient conditions are presented, which guarantee that the estimation error converges asymptotically towards zero.
This paper is arranged as follows. In Section 2, the system description and some lemmas are given. In Section 3, we present the observer synthesis method for a class of nonlinear systems with multiple time-delays. In Section 4, we propose an illustrative example in order to show the validity of our method. Finally, some conclusions are given in Section 5.
The notation used in this paper is fairly standard. Throughout this paper, R stands for the set of real numbers. The notation (<0) means that the matrix A is symmetric and positive definite (negative definite). stands for the matrix transpose of matrix A. denotes the Euclidean norm for a vector or a matrix. denotes the infinity norm for a matrix.
2 System description
To complete the system description, the following assumptions are considered.
Assumption 1 For all , , the entries of are bounded.
Assumption 2 For all , , the entries of , , are bounded.
We set and .
The following lemmas are necessary for the proof of the main statement.
Proof Let , where . Since is symmetric and positive-definite for all , one gets that is invertible. It is easy to verify that is stabilizable and is observable. According to ref. , we obtain that the matrix is the unique solution of ARE (3) which is always symmetric and positive-definite for .
Lemma 3 
where , .
Lemma 4 (Lyapunov-Krasovskii stability theorem )
then the trivial solution of (9) is uniformly stable. If for , then it is uniformly asymptotically stable. In addition, if , then it is globally uniformly asymptotically stable.
3 Observer design
In the sequel, we introduce our main contribution which consists of a new feasibility condition for the observer synthesis problem of a class of nonlinear time-delays systems. The convergence analysis is performed by the use of a Lyapunov-Krasovskii functional.
the observer error that results from (1) and (10) converges asymptotically towards zero.
From (13), we have . According to Lemma 4, we deduce that the observer error converges asymptotically towards zero. This ends the proof of Theorem 1. □
where A, C and are given by (2), and is a lower-triangular matrix.
the observer error that results from (16) and (17) converges asymptotically towards zero.
From (19), we have . This ends the proof of Theorem 2. □
When , , (16) becomes (22). So, (22) is the special case of (16).
where A and C are defined as in (2). and , , are real and lower-triangular matrices and is an input-injection vector of dimension n.
the estimation error that results from (23) and (26) converges asymptotically towards zero.
Proof The matrices A and can be seen as the matrix Jacobian. Therefore, the proof becomes straightforward as it was developed before. □
Remark 4 Those results obtained can be extended to multiple time-delays nonlinear systems in upper-triangular form.
Remark 5 In , the sufficient conditions which guarantee that the estimation error converges asymptotically towards zero are given in terms of a linear matrix inequality. Comparing with , our results are less conservative and more convenient to use.
4 Numerical example
It is easy to obtain that , . Let , . It is easy to verify that (19) holds.
According to Theorem 2, the estimation error converges asymptotically towards zero.
The main purpose of this paper is to offer a systematic algorithm for designing an observer for a class of nonlinear systems with multiple time-delays. By using an improved Lyapunov-Krasovskii functional and the differential mean-value theorem, we present the sufficient conditions for the existence of the observer, which guarantee that the estimation error converges asymptotically towards zero. The new design plays an important role in obtaining a nonrestrictive synthesis condition and rendering our approach application to a broader class of systems, namely the class of nonlinear time-delay systems in a lower-triangular form. The proposed design is valid whatever the size of the delay. Finally, the efficiency of the proposed method is shown by a numerical example.
This work was supported by the Natural Science Foundation of Tianjin under Grant 11JCYBJC06800.
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