Stability Of Discrete-time Nonlinear Systems Research Articles

Output feedback stabilization of nonlinear discrete-time systems with time-delay

This article considers both the static output feedback stabilization issue and output-feedback guaranteed cost controller design of a class of discrete-time nonlinear systems with time-delay. First, by static output feedback controller, the new sufficient conditions for static output feedback stabilization of a class of discrete-time nonlinear systems with time-delay are presented. Then, we establish the new delay-independent sufficient conditions for existence of the guaranteed cost control by static output-feedback controller in terms of matrix inequalities. Finally, two examples are given to show the effectiveness of our proposed approaches.2000 MSC 93D15; 93C55; 34K20; 34D23.

Exponential stabilization of nonlinear discrete-time systems

In this paper, we give sufficient conditions for the exponential stabilizability of a class of perturbed non-autonomous difference equations with slowly varying coefficients. Under appropriate growth conditions on the perturbations, we establish explicit results concerning the feedback exponential stabilizability.

Homogeneous Polynomially Nonquadratic Stabilization of Discrete-Time Takagi–Sugeno Systems via Nonparallel Distributed Compensation Law

This paper considers stability of discrete-time nonlinear systems in Takagi–Sugeno (T–S) form. This problem has been studied for more than 20 years with many sufficient conditions, and the asymptotically necessary and sufficient (ANS) conditions with respect to the common-quadratic Lyapunov, function, having being obtained. This paper considers general forms of homogeneous polynomially nonquadratic Lyapunov (HPNQL) function and homogeneous polynomially parameterized nonparallel distributed compensation (HPP-non-PDC) law. By generalization of the procedure based on Pólya’s theorem and techniques used for parameter-dependent linear matrix inequality (PD-LMI) which have been studied previously in different contexts, ANS stability conditions with respect to the general HPNQL function are obtained.

On Stabilization of Discrete-time Nonlinear Systems under Information Constraints

This paper deals with stabilization and optimal control of interconnected discrete-time nonlinear systems under given information constraints on the controller structure. As a prominent example, the considered information constraints on the controller structure include decentralized and distributed control over a given communication network. Firstly, we briefly recall the recently introduced notion of structured control Lyapunov functions (CLFs), which provides a suitable tool for stabilizing controller synthesis under information constraints. Then, we show that the controller synthesis problem using structured CLFs can be formulated as a convex optimization problem. Furthermore, we present solutions for solving this problem efficiently under several different types of information constraints.

Input–output-to-state stability for discrete-time systems

This paper presents Lyapunov characterizations of uniform output-to-state stability and uniform input–output-to-state stability (IOSS) (with respect to disturbances) for discrete-time (DT) nonlinear systems. We show the equivalence of the following three properties for DT systems: uniform IOSS, existence of a smooth Lyapunov function for uniform IOSS, and existence of a (state-)norm estimator. This equivalence result is a DT counterpart of Krichman et al. [2001. Input–output-to-state stability. SIAM Journal on Control and Optimization 39, 1874–1928. Theorem 2.4] and a generalization of Jiang and Wang [2002. A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58. Theorem 1.1] and Jiand and Wang [2001. Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869. Theorem 1, 1 ⇔ 4 ].

Global stabilization of non-linear discrete-time systems by linear feedback

Global stabilization of non-linear discrete-time systems by linear feedback

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].

State feedback stabilization of nonlinear discrete-time systems with time-varying time delay

This paper deals with the class of nonlinear discrete-time systems with varying time delay. The problems of stability and stabilizability for this class of systems are considered. Given an upper bound and a lower bound on the time-varying delay, sufficient conditions for checking the stability of this class of systems are developed. A control design algorithm is also provided for determining the state feedback controller gain. All the results developed in this paper are in the LMI formalism which makes it easier to obtain them using existing tools. A numerical example is provided to show the effectiveness of the results established.

A necessary condition for local asymptotic stability of discrete-time nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in the control systems literature. In this paper, we investigate the problem of asymptotic stability for discrete-time nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Sundarapandian [1] for discrete-time nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Lin and Byrnes [2] for discrete-time nonlinear autonomous systems. We illustrate our result with several examples.

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.

A new stability criterion for discrete-time neural networks: Nonlinear spectral radius

Abstract In this paper, the exponential stability of nonlinear discrete-time systems is studied. A novel notion of nonlinear spectral radius is defined. Under the assumption of Lipschitz continuity for the activation function, the developed approach is applied to stability analysis of discrete-time neural networks. A series of sufficient conditions for global exponential stability of the neural networks are established and an estimate of the exponential decay rate is also derived for each case.

Robust stability of discrete-time nonlinear system with time-delay

The robustly asymptotical stability problem for discrete-time nonlinear systems with time-delay was investigated. Positive definite matrix are constructed through Lyapunov functional. With the identity transform, property of matrix inverse and S-procedure, a new sufficient condition independent of the size of time-delay for robust stability of discrete-time nonlinear systems with time-delay is established. With Schur complement, another equivalent sufficient condition for robust stability of discrete-time nonlinear systems with time-delay is given. Finally, a sufficient condition dependent on the size of time-delay for robust stability of discrete-time nonlinear systems with time-delay is obtained. A unified approach is used to cast the robust stability problem into a convex optimization involving linear matrix inequalities.

A necessary condition for local asymptotic stability of discrete-time nonlinear systems with parameters

In this paper, we derive a necessary condition for local asymptotic stability of equilibria of discrete-time nonlinear systems with parameters. Our necessary condition is analogous to the necessary condition obtained by Byrnes and Sundarapandian [1] for local asymptotic stability of equilibria of continuous-time nonlinear systems with parameters. As a corollary of our general result, we deduce the necessary condition obtained by Lin and Byrnes [2] for local asymptotic stability of equilibria of discrete-time nonlinear systems. We also illustrate our result with some examples.

Input-to-state stability for discrete-time nonlinear systems

In this work we study the input-to-state stability ( iss) property for discrete-time nonlinear systems. It is shown that most iss results for continuous-time nonlinear systems in the current literature can be extended to the discrete-time case. Several equivalent characterizations of iss are introduced and two iss small-gain theorems are proved for nonlinear and interconnected discrete-time systems. ISS stabilizability is discussed and comparisons with the continuous-time case are made. As in the continuous time framework, where the notion iss found wide applications, we expect that this notion will provide a useful tool in areas related to stability and stabilization for nonlinear discrete time systems as well.

Robust stability and stabilization of discrete-time non-linear systems: The LMI approach

The purpose of this paper is to convert the problem of robust stability of a discrete-time system under non-linear perturbation to a constrained convex optimization problem involving linear matrix inequalities (LMI). The nominal system is linear and time-invariant, while the perturbation is an uncertain non-linear time-varying function which satisfies a quadratic constraint. We show how the proposed LMI framework can be used to select a quadratic Lyapunov function which allows for the least restrictive non-linear constraints. When the nominal system is unstable the framework can be used to design a linear state feedback which stabilizes the system with the same maximal results regarding the class of non-linear perturbations. Of particular interest in this context is our ability to use the LMI formulation for stabilization of interconnected systems composed of linear subsystems with uncertain non-linear and time-varying coupling. By assuming stabilizability of the subsystems we can produce local control laws under decentralized information structure constraints dictated by the subsystems. Again, the stabilizing feedback laws produce a closed-loop system that is maximally robust with respect to the size of the uncertain interconnection terms.

Suboptimal control and stability of nonlinear discrete-time systems

In this paper suboptimality bounds and stability of nonlinear discrete-time infinite-horizon controllers are investigated. The nominal plant is supposed to be controlled by means of a feedback control law which is optimal with respect to some given criterion. The use of the Taylor–Maclaurin formula appears to be the key to establish the suboptimality properties of the decentralized control and at the same time to derive upper and lower bounds on the suboptimality.

Input-to-state stability for discrete-time nonlinear systems

In this paper the input-to-state stability (ISS) property is studied for discrete-time nonlinear systems. We show that many ISS results for continuons-time nonlinear systems in earlier papers (Sontag, 1989; Sontag, 1990; Sontag and Wang, 1996; Jiang et al ., 1994; Coron et al ., 1995) can be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties. Utilizing the notion of ISS, we present a small gain theorem for nonlinear discrete time systems. ISS stabilizability is discussed and connections with the continuous-time case are made. As in the continuous time case, where the notion ISS found wide applications, we expect that this notion will provide a useful tool in areas related to stability for nonlinear discrete time systems as well.

Feedback stabilization of nonlinear discrete-time systems

It is the merit of Aeyels (Systems Control Lett. 5 (1985) 289-294) to have shown a way in which center manifold theory can be used in a constructive manner to find a smooth feedback control for stabilizing an equilibrium of a continuous-time system described by a nonlinear ordinary differential equation . In this paper we are going to extend Aeyels' approach to nonlinear discrete-time systems described by equations of the type where we assume that f is sufficiently smooth and satifies f(0)=0. In critical cases. i.e. in situations where the linearization of the system in the neighborhood of the equilibrium includes non-controllable modes, we derive conditions under which there exists a stabilizing quadratic feedback control.

Discrete-time nonlinear H∞ control with measurement feedback

This paper deals with the problem of H ∞ control via measurement feedback with internal stability for discrete-time nonlinear systems. Using the theory of discrete-time dissipative systems and differential game, we first present a parameterization of a family of static state feedback H ∞ nonlinear controllers. Then, we provide sufficient conditions for the existence of a dynamic output compensator which solves the problem of disturbance attenuation with internal stability for discrete-time nonlinear systems. The theory developed in this paper incorporates the well-known results for discrete-time linear systems, and generalizes some recent advances in H ∞ control of continuous-time nonlinear systems to their discrete-time counterparts.

Zero-state observability and stability of discrete-time nonlinear systems

This paper studies discrete-time invariant autonomous systems ∑:xk + 1 = tf(xk)yk = h(xk)We investigate the problem of when asymptotic stability of ∑ can be characterized by means of zero-state observability of ∑ and square summability of the output function h(xk). The relationship among observability, square summable series and stability of discrete-time autonomous systems is exposed. The main result developed in this paper unifies a series of stability theorems recently proposed in the literature, and generalizes Morse's Lemma [Morse, A. S. (1990). Towards a unified theory of parameter adaptive control tunability. IEEE Trans. Autom. Contr., 35, 1002–1012.] in two directions: from continuous-time linear systems to their discrete counterparts and from linear systems to nonlinear systems.