Asymptotic stability of controlling uncertain dynamical systems

Asymptotic stabilization of general uncertain dynamical systems is investigated. A new class of continuous feedback controls is proposed to guarantee asymptotic stability for any uncertain systems whose nominal system is uniformly asymptotically stable. The analysis is based on a new theoretical result on asymptotical stability. The required information about uncertain dynamics in the system is merely that the uncertainties are bounded in euclidean norm by a known function of the system state.     

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     

Componentwise asymptotic stability of linear constant dynamical systems

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vectorgamma(t)(CWASγ) of systemS: dot{x} = Ax + Bu, t geq 0, wheregamma(t) > 0(componentwise inequality) andgamma(t) rightarrow 0ast rightarrow + infty.Sis CWASγ if for eacht_{0} geq 0and for each|x(t_{0})| leq gamma (t_{0}) (|x (t_{0})|with the components|x_{i}(t_{0})|the free response ofSsatisfies|x(t)| leq gamma (t)for eacht geq t_{0}. Forgamma(t){underline { underline delta} } alphae^{-beta t}, t geq 0, withalpha > 0andbeta > 0(scalar), the CWEAS (E= exponential) may be defined.Sis CWAS γ (CWEAS) if and only ifdot{gamma}(t) geq bar{A}gamma(t), t geq 0 (bar{A}alpha < 0); A {underline { underline delta} } (a_{ij})andbar{A}has the elements aijand|a_{ij}|, i neq j. These results may be used in order to evaluate in a more detailed manner the dynamical behavior ofSas well as to stabilizeScomponentwise by a suitable linear state feedback.     

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.     

Attractors and partial stability of nonlinear dynamical systems

This paper represents a study of the key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations between the local properties and the unification of the methodologies of their analysis. The use of recent results of stability theory and techniques of geometric control enables one to generalize Lyapunov-like sufficient conditions and propose simplified solutions for the problems of set attractivity and partial stability.     

Robust stability criteria for dynamical systems including delayedperturbations

Considers the problem of robust stability of uncertain time-delay dynamical systems. A new robust stability criteria for linear dynamical systems subject to delayed time-varying and nonlinear perturbations is derived. The results obtained in this note are less conservative than the ones reported so far in the literature. Some analytical methods are employed to investigate the bound on the perturbations so that the systems are stable. A numerical example is given to demonstrate the utilization of the authors' results     

Uniform asymptotic stability of hybrid dynamical systems with delay

We formulate a model for hybrid dynamical systems with delay, which covers a large class of delay systems. Under several mild assumptions, we establish sufficient conditions for uniform asymptotic stability of hybrid dynamical systems with delay via a Lyapunov-Razumikhin technique. To demonstrate the developed theory, we conduct stability analyses for delay sampled-data feedback control systems including a nonlinear continuous-time plant and a linear discrete-time controller.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

Probability of stability of a class of linear dynamical systems

In this paper, simple formulas for the probability of stability of a class of linear dynamical systems arc presented. The theoretical results thus obtained are compared with certain empirical results obtained by Ashby (1952).     

On the stability of convolution feedback systems with dynamical feedback

This paper considers distributed n-inputn-output convolution feedback systems characterized by $\text{y = G}{}_1\text{1e}$, $\text{z = G}{}_2\text{1y}$ and e = u ? z, where the forward path transfer function $\text{G}\text{?}{}_1$ and the feedback path transfer function $\text{G}\text{?}{}_2$ both contain a real single unstable pole at different locations. Theorem 1 gives necessary and sufficient conditions for both input-error and input-output stability. In addition to usual conditions that guarantee input-error stability a new condition is found which results in the fact that input-error stability will guarantee input-output stability. These conditions require to investigate only the open-loop characteristics. A basic device is the consideration of the residues of different transfer functions at the open-loop unstable poles. An example is given.     

Optimal control and robust stability of uncertain impulsive dynamical systems

This paper studies the problem of optimal feedback control and robust stability for uncertain impulsive dynamical systems. By using algebraic inequalities, Riccati and Hamilton‐Jacobi inequalities, the conditions are derived under which not only the uncertain impulsive dynamical system has robust asymptotic stability but also the value of the optimal hybrid performance functional can be estimated. An example is also given to illustrate our results. © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

The disturbance decoupling problem with stability for switching dynamical systems

The disturbance decoupling problem with stability is dealt with by means of the geometric approach for switching systems. The existence of feedbacks which decouple the disturbance and, at the same time, assure stability is difficult to characterize, since the action of the feedback couples with that of the switching law. Under suitable conditions, it is shown that the above requirement can be dealt with in separate ways and this allows us to state a checkable necessary condition and, on that basis, also a sufficient condition for solvability of the problem.     

Exponential stability of a class of nonlinear dynamical systems with uncertainties

Exponential stability for a class of nonlinear dynamical systems with uncertainties is investigated. Based on the stability of the nominal systems, a new approach to synthesizing a class of continuous state feedback controllers for uncertain nonlinear dynamical systems is proposed. By such a class of feedback controllers, we can guarantee exponential stability of uncertain nonlinear dynamical systems. Our approach can give a clear insight into system analysis. Finally, an illustrative example is given to demonstrate the utilization of the approach developed in this paper.     

Input-to-state stability for a class of hybrid dynamical systems via averaging

Input-to-state stability (ISS) properties for a class of time-varying hybrid dynamical systems via averaging method are considered. Two definitions of averages, strong average and weak average, are used to approximate the time-varying hybrid systems with time-invariant hybrid systems. Closeness of solutions between the time-varying system and solutions of its weak or strong average on compact time domains is given under the assumption of forward completeness for the average system. We also show that ISS of the strong average implies semi-global practical (SGP)-ISS of the actual system. In a similar fashion, ISS of the weak average implies semi-global practical derivative ISS (SGP-DISS) of the actual system. Through a power converter example, we show that the main results can be used in a framework for a systematic design of hybrid feedbacks for pulse-width modulated control systems.     

A study of the boundaries of stability regions in two-parameter dynamical systems

We consider dynamical systems defined by autonomous and periodic differential equations that depend on two scalar parameters. We study the problems of constructing boundaries of stability regions for equilibrium points in the plane of parameters. We identify conditions under which a point on the boundary of a stability region has one or more smooth boundary curves coming through it. We show schemes to find the basic scenarios of bifurcations when parameters transition over the boundaries of stability regions. We distinguish types of boundaries (dangerous or safe). The main formulas have been obtained in the terms of original equations and do not require to pass to normal forms and using theorems on a central manifold.     

Robust stability of a class of uncertain nonlinear dynamical systems with time-varying delay

We consider the problem of robust stability of a class of uncertain nonlinear dynamical systems with time-varying delay. Based on the assumption that the nominal system (i.e. the system in the absence of uncertainty and delay) is stable, we derive some sufficient conditions on robust stability of uncertain nonlinear dynamical systems with time-varying delay. Some analytical methods and the Bellman-Gronwall inequality are used to investigate such sufficient conditions. The notable features of the results obtained in this paper are their simplicity in testing the stability of uncertain dynamical systems with delay and their clarity in giving an insight into system analysis. Our results are also applicable to perturbed time-delay dynamical systems without exact knowledge of the delay. In addition, a numerical example is given to demonstrate the validity of our results.     

A criterion for structural stability of quadruples of matrices defining generalized dynamical systems

A characterization in terms of the rank of a matrix is proved for structurally stable quadruples related to generalized dynamical systems

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under a local equivalence relation by realizing a geometric approach.