Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds

doi:10.1016/j.jfranklin.2010.07.003

Journal of the Franklin Institute

Volume 347, Issue 8, October 2010, Pages 1511-1525

https://doi.org/10.1016/j.jfranklin.2010.07.003 Get rights and content

Abstract

The robust absolute stability problem for norm uncertain and structured uncertain discrete Lur’e systems is considered in this paper by using Lyapunov function method. A sufficient condition of absolute stability for discrete Lur’e systems is established in terms of linear matrix inequalities (LMIs) or the equivalent frequency-domain condition. We compare the result with the Popov-like criterion (Tsypkin criterion) and extended strictly positive real (ESPR) lemma. Furthermore, sufficient conditions on absolute stability for discrete Lur’e systems with norm and structured uncertainties are also presented based on linear matrix inequalities. Estimates of the maximum bounds of all admissible perturbations are given by generalized eigenvalue problems. Finally, several numerical examples are worked out to illustrate the efficiency of the main results.

Introduction

As is well known, a Lur’e system is a feedback system, in which the forward path is a linear time invariant system and the feedback path is an uncertain nonlinearity with bounded sector constrains (see [19]). Since the absolute stability of Lur’e system has been introduced and studied by Lur’e and Postnikov in 1940s, this study is still one of the important problems in control theory. Due to the explicit physical significance and important role in applications of Lur’e systems, there are many researchers to commit themselves to study the absolute stability of such systems up to now [8], [9], [10], [11], [12], [14], [16], [19]. If the plant in the forward path is a discrete system, then the Lur’e system is discrete. Since Popov derived a frequency-domain condition for the absolute stability of continuous-time systems, consideration effort has been devoted to establish similar criteria for discrete-time systems [10], [12], [16], [18]. For discrete-time Lur’e systems, a stabilization condition on the existence of the state feedback controller was established in terms of LMIs by using Finsler's lemma [12]. Moreover, since the uncertainties are ubiquitous in many real systems, there are more and more interest in the robust absolute stability problems for uncertain Lur’e systems. Recently, there are many papers to devote to this problem and present many elegant results, see [3], [4], [9], [11], [13] and the references therein for more details. However, many of the approaches given in these references are based on frequency-domain and therefore have generally computational costly. Very recently, in the light of state space (i.e., time-domain), we gave some conditions on guaranteeing simultaneously the robust absolute stability and disturbance rejection of uncertain Lur’e systems with persistent bounded disturbances (see [5], [6], [7]). In networked control systems, if the controlled plant is a continuous-time Lur’e system, it will form a discrete-time Lur’e system after the output passing the network and being sampled. So, there are some explicit physical significance and important roles in applications of discrete-time Lur’e systems. Now, a consideration problem is how to establish conditions (frequency-domain or time-domain) on absolute stability for discrete-time Lur’e systems. Especially, the robust absolute stability problem for uncertain discrete Lur’e systems is more and more of interest.

Motivated by Kalman–Yakubovich–Popov Lemma in [17] and Popov-like criterion (Tsypkin criterion) in [10], combining frequency-domain with time-domain methods, we will consider in this paper the absolute stability of discrete Lur’e systems, and based-on this, we will study the robust absolute stability problem of discrete Lur’e systems with norm and structured uncertainties and an estimate of the maximum bound of all admissible perturbations.

In this paper, we present a new sufficient condition in terms of LMIs for absolute stability of discrete Lur’e systems, this condition do not require observable condition in Tsypkin criterion. By ESPR lemma, this condition can be equivalent to a frequency-domain condition. By using the obtained result, we establish an estimate of the maximum bound of all admissible perturbations guaranteeing robust absolute stability of uncertain discrete Lur’e systems in terms of a generalized eigenvalue problem, which can be solved easily by Matlab tool (see [1] for more details). Finally, some numerical examples are worked out to illustrate the efficiency of the main results.

Section snippets

Preliminaries

We use the following notations. R is the set of all real numbers, Rⁿ is the set of all n-tuples of real numbers. N is the set of all natural numbers. Denote by A^T, A⁻¹ and $He A = \frac{1}{2} (A + A^{T})$ the (conjugate) transpose, the inverse and the Hermitian part of a matrix A (if it is invertible), respectively. I_n denotes the unit matrix with n dimensions. Without confusing, I denotes the unit matrix of appropriate dimensions. We denote a state-space realization of a transfer function G(z)=C(zI−A)⁻¹B+D by $G (z)$

The case of nominal systems

We will give an LMI condition on absolute stability w.r.t. $ϕ \in F [0, μ]$ for the discrete Lur’e system as shown in Fig. 1, where the plant is certain G(z), (A,B) is controllable and A is Schur stability (i.e. all eigenvalues are in unit disk).

Theorem 1

For the system as shown in Fig. 1, if there exist positive definite matrix P and positive constant scalars n_i, i=1,…,m (or $N ≕ diag [n_{1}, n_{2}, \dots, n_{m}]$ be positive-definite) satisfying the following condition: $[\begin{matrix} A^{T} PA - P & A^{T} PB - C^{T} & A^{T} C^{T} (I + N) - C^{T} N \\ B^{T} PA - C & B^{T} PB - 2 μ^{- 1} & B^{T} C^{T} (I + N) \\ (I + N) CA - NC & (I + \end{matrix}]$

Examples

Example 1

We consider a simple example to illustrate efficiency of the obtained results. In this section we consider firstly SISO Lur’e system (3) with output uncertainties, where $A = [\begin{matrix} - 0.8 & 0.3 & 0 \\ - 0.3 & - 0.9 & 0 \\ 0 & 0 & - 0.6 \end{matrix}], B = [\begin{matrix} 0.1 \\ 0 \\ 0 \end{matrix}], C = [0.1 0 0], M_{C} = [0.1 0 0.2]$ $μ = 1.2, N_{C} = I$ Thus, solving the generalized eigenvalue problem in Theorem 2, we can obtain the minimum $δ_{\inf} = 21.2993$ of $δ$ satisfying GEP in Theorem 2. Here, we obtain $P = [\begin{matrix} 16.6299 & 0.9701 & 0 \\ 0.9701 & 17.9188 & 0 \\ 0 & 0 & 10.2769 \end{matrix}]$ $α_{1} = 0.6992, α_{2} = 0.7889$ and n = 2.3764. Thus, from Theorem 2, this result

Conclusion

This paper has been concerned with robust absolute stability of norm and structured uncertain discrete-time Lur’e systems. By using Lyapunov method and Popov-like (Tsypkin) criterion, we gave based-on LMI conditions of robust absolute stability of uncertain discrete Lur’e systems. Furthermore, we presented an estimate of bounds of the maximum admissible perturbations guaranteeing robust absolute stability of uncertain discrete Lur’e systems, which is given in terms of a generalized eigenvalue

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^☆: This work is supported by the National Natural Science Foundation of China under Grants (No. 60504018 and 60874012).

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Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds☆

Abstract

Introduction

Section snippets

Preliminaries

The case of nominal systems

Examples

Conclusion

Appl. Math. Comput.

Nonlinear Anal.

Syst. Control Lett.

Linear Matrix Inequalities in System and Control Theory

Robust solutions to least squares problems with uncertain data

SIAM J. Matrix Anal. Appl.

Absolute stability criteria for multiple slope-restricted monotonic nonilinearities

IEEE Trans. Automat. Control

Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis

IEEE Trans. Automat. Control

Analysis of disturbances rejection for Lur’e system

Appl. Math. Mech.