Ling Hou

Finite-time and practical stability of a class of stochastic dynamical systems

In practice, one is not only interested in qualitative characterizations provided by Lyapunov and Lagrange stability, but also in quantitative information concerning system behavior, including estimates of trajectory bounds over finite and infinite time intervals. This type of information has been ascertained in a systematic manner using the notions of finite-time stability and practical stability. In the present paper we generalize some of the existing finite-time stability and practical stability results for deterministic dynamical systems determined by ordinary differential equations to dynamical systems determined by an important class of stochastic differential equations. We consider two types of stability concepts: finite-time and practical stability in the mean and in the mean square. We demonstrate the applicability of our results by means of several examples.

Stability analysis of pulse-width-modulated feedback systems

In this paper the author presents new Lyapunov and Lagrange stability results for pulse-width-modulated (PWM) feedback systems with type 2 modulation. The linear plant considered herein is assumed to be Hurwitz stable. An optimization procedure is incorporated that improves the stability bounds significantly. The applicability of the present results is demonstrated by means of two specific examples.

Moment stability of discontinuous stochastic dynamical systems

In this note, we establish new Lyapunov and Lagrange stability results in the pth mean for a class of discontinuous stochastic dynamical systems. We apply these results in the qualitative analysis of a class of digital feedback control systems that are subjected to multiplicative and additive disturbances in the plants. The present results constitute natural extensions of our earlier results for discontinuous deterministic dynamical systems.

On the Continuity of the Lyapunov Functions in the Converse Stability Theorems for Discontinuous Dynamical Systems

In our previous paper (Ye et al., 1998) we established, among other results, a set of sufficient conditions for the uniform asymptotic stability of invariant sets for discontinuous dynamical systems (DDS) defined on metric space, and under some additional minor assumptions, we also established a set of necessary conditions (a converse theorem). This converse theorem involves Lyapunov functions which need not necessarily be continuous. In the present paper, we show that under some additional very mild assumptions, the Lyapunov functions for the converse theorem need actually be continuous. This improvement in the regularity properties of the Lyapunov functions shows that the stability results in our previous paper (Ye et al., 1998) (under the additional mild assumptions) are rather robust

Moment stability of pulse-width-modulated feedback systems subjected to random disturbances

We study the stability properties of pulse-width-modulated (PWM) feedback systems with stable plants, subjected to multiplicative and additive random disturbances (modeled by the derivative of a Wiener process). We show that when the parameters of the pulse-width modulator are within a computable range and the random disturbances are sufficiently small, then the PWM feedback system is globally asymptotically stable in the pth mean. We also show that in the presence of additive disturbances, such PWM feedback systems are bounded in the pth mean for arbitrarily large disturbances.

Stability results for finite-dimensional discrete-time dynamical systems involving non-monotonic Lyapunov functions

In and in a more recent paper we established results for the uniform stability and the uniform asymptotic stability in the large involving non-monotonic Lyapunov functions for continuous-time dynamical systems. In the present paper we continue this work by addressing finite-dimensional discrete-time dynamical systems. Similarly as in and, we prove that in general, the results presented herein are less conservative than the corresponding standard Lyapunov stability results (henceforth called classical Lyapunov stability results) for finite-dimensional discrete-time dynamical systems. We present two specific examples to demonstrate the applicability of our results.

Stability analysis of pulse-width-modulated feedback systems with type 2 modulation

In this paper the author presents new Lyapunov and Lagrange stability results for pulse-width-modulated (PWM) feedback systems with type 2 modulation. The linear plant considered herein is assumed to be Hurwitz stable. An optimization procedure is incorporated that improves the stability bounds significantly. The applicability of the present results is demonstrated by means of two specific examples.

Finite-Dimensional Dynamical Systems

We present the principal stability and boundedness results for continuous dynamical systems, discrete-time dynamical systems, and discontinuous dynamical systems involving monotonic and non-monotonic Lyapunov functions. We apply the results of Chapter 3 to arrive at these results. When considering various stability types, our focus is on invariant sets that are equilibria. Our results constitute sufficient conditions (the Principal Stability and Boundedness Results) and necessary conditions (Converse Theorems). We demonstrate the applicability of all results by means of numerous examples.

Fundamental Theory: The Principal Stability and Boundedness Results on Metric Spaces

We present the Principal Lyapunov and Lagrange Stability Results, including Converse Theorems for continuous dynamical systems, discrete-time dynamical systems and discontinuous dynamical systems defined on metric spaces. All results presented involve the existence of either monotonic Lyapunov functions or non-monotonic Lyapunov functions. We show that the results involving monotonic Lyapunov functions reduce to corresponding results involving non-monotonic Lyapunov functions. Furthermore, in most cases, the results involving monotonic Lyapunov functions are in general more conservative than the corresponding results involving non-monotonic Lyapunov functions.We present stability results (sufficient conditions) for uniform stability, local and global uniform asymptotic stability, local and global exponential stability, and instability of invariant sets. We also present Converse Theorems (necessary conditions) for most of the enumerated stability types. Furthermore, we present Lagrange stability results (sufficient conditions) for the uniform boundedness and the uniform ultimate boundedness of motions of dynamical systems, as well as corresponding Converse Theorems (necessary conditions).The results of this chapter constitute the fundamental theory for the entire book because most of the general results that we develop in the subsequent chapters concerning finite-dimensional systems and infinite-dimensional systems can be deduced as consequences of the results of the present chapter.

Fundamental Theory: Specialized Stability and Boundedness Results on Metric Spaces

We present several important specialized stability and boundedness results for dynamical systems defined on metric spaces. It turns out that a number of the results that we will develop in the subsequent chapters concerning finite-dimensional and infinite-dimensional systems can be deduced as consequences of corresponding results of the present chapter.For autonomous and periodic dynamical systems we show that when an invariant set is stable (asymptotically stable) then it is also uniformly stable (uniformly asymptotically stable). For autonomous dynamical systems we also present necessary and sufficient conditions for the stability and the asymptotic stability of invariant sets.For dynamical systems determined by continuous-time and discrete-time semigroups defined on metric spaces we establish stability and boundedness results which comprise the LaSalle-Krasovskii invariance theory.We present for both continuous-time and discrete-time dynamical systems a comparison theory for the various Lyapunov and Lagrange stability types. This comparison theory enables us to deduce the qualitative properties of a complex dynamical system (the object of inquiry) from the qualitative properties of a simpler dynamical system (the comparison system).For general continuous-time dynamical systems we establish a Lyapunov-type result which ensures the uniqueness of motions of a dynamical system.