The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.     

Stability Analysis of Distributed-Order Hilfer–Prabhakar Systems Based on Inertia Theory

The notion of a distributed-order Hilfer–Prabhakar derivative is introduced, which reduces in special cases to the existing notions of fractional or distributed-order derivatives. The stability of two classes of distributed-order Hilfer–Prabhakar differential equations, which are generalizations of all distributed or fractional differential equations considered previously, is analyzed. Sufficient conditions for the asymptotic stability of these systems are obtained by using properties of generalized Mittag-Leffler functions, the final-value theorem, and the Laplace transform. Stability conditions for such systems are introduced by using a new definition of the inertia of a matrix with respect to the distributed-order Hilfer–Prabhakar derivative.     

Novel criterion for stability of discrete-time systems in a state-space realization utilizing saturation nonlinearities

A new criterion for the global asymptotic stability of discrete-time systems in a state-space realization utilizing saturation nonlinearities is presented. The form of the criterion is unconventional in that it offers the flexibility of using even an unsymmetric Lyapunov matrix P. A recently reported criterion is shown to be a special case of the present criterion. An example shows the effectiveness of the new criterion.     

Generalization of the Jurdjevic-Quinn theorem and stabilization of bilinear control systems with periodic coefficients: I

The Jurdjevic-Quinn theorem on the global asymptotic stabilization of the origin is generalized to nonlinear time-varying affine control systems with periodic coefficients. The proof is based on the Krasovskii theorem on the global asymptotic stability for periodic systems and the introduced notion of “commutator” for two vector fields one of which is time-varying. The obtained sufficient conditions for stabilization are applied to bilinear control systems with periodic coefficients. We construct a control periodic in t in the form of a quadratic form in x that asymptotically stabilizes the zero solution of a bilinear periodic system with a time-invariant drift.     

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for L ₁-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in L ₁. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.     

Global asymptotic stability analysis of discrete-time Cohen–Grossberg neural networks based on interval systems

The global asymptotic stability of discrete-time Cohen–Grossberg neural networks (CGNNs) with or without time delays is studied in this paper. The CGNNs are transformed into discrete-time interval systems, and several sufficient conditions for asymptotic stability for these interval systems are derived by constructing some suitable Lyapunov functionals. The conditions obtained are given in the form of linear matrix inequalities that can be checked numerically and very efficiently by using the MATLAB LMI Control Toolbox. Finally, some illustrative numerical examples are provided to demonstrate the effectiveness of the results obtained.     

On the stability of time-varying differential equations *

This paper deals with stability of time-varying differential equations. New asymptotic stability conditions for time-varying continuous systems with more general assumptions are given. The Gronwall's inequality and its discrete-time versions play a crucial role in our investigation.     

On stability for switched linear positive systems

This paper addresses the stability properties of switched linear positive systems in continuous-time as well as in discrete-time. In the discrete-time case, some sufficient and necessary conditions for asymptotic stability are derived for pairs of second order systems. Similar conditions are also established for a finite number of second order systems. Furthermore, for higher order systems, some results on stability are provided in a similar manner. In particular, in this case, a common linear Lyapunov function guaranteeing the stability of the switched positive systems can be easily located by means of geometry properties. In the continuous-time case, a finite number of second order systems are considered. Some equivalent conditions for stability of such systems are developed.     

Stochastic stability for Markovian jump linear systems associated with a finite number of jump times

This paper deals with a stochastic stability concept for discrete-time Markovian jump linear systems. The random jump parameter is associated to changes between the system operation modes due to failures or repairs, which can be well described by an underlying finite-state Markov chain. In the model studied, a fixed number of failures or repairs is allowed, after which, the system is brought to a halt for maintenance or for replacement. The usual concepts of stochastic stability are related to pure infinite horizon problems, and are not appropriate in this scenario. A new stability concept is introduced, named stochastic τ-stability that is tailored to the present setting. Necessary and sufficient conditions to ensure the stochastic τ-stability are provided, and the almost sure stability concept associated with this class of processes is also addressed. The paper also develops equivalences among second order concepts that parallels the results for infinite horizon problems.     

Strong pro-fibrations and ANR objects in pro-categories

The notions of pro-fibration and approximate pro-fibration for morphisms in the pro-category pro-Top of topological spaces were introduced by S. Mardeši? and T.B. Rushing. In this paper we introduce the notion of strong pro-fibration, which is a pro-fibration with some additional property, and the notion of ANR object in pro-Top, which is approximately an ANR-system, and we consider the full subcategory ANR of pro-Top whose objects are ANR objects. We prove that the category ANR satisfies most of the axioms for fibration category in the sense of H.J. Baues if fibrations are strong pro-fibrations and weak equivalences are morphisms inducing isomorphisms in the pro-homotopy category pro-H(Top) of topological spaces. We give various applications. First of all, we prove that every shape morphism is represented by a strong pro-fibration. Secondly, the fibre of a strong pro-fibration is well defined in the category ANR, and we obtain an isomorphism between the pro-homotopy groups of the base and total systems of a strong pro-fibration, and hence obtain the pro-homotopy sequence of a strong pro-fibration. Finally, we also show that there is a homotopy decomposition in the category ANR.     

基于有限区域量化的系统稳定性分析与设计

针对基于输出反馈和具有有限区域信号量化的离散时间系统,进行了系统稳定性分析与量化参数设计的研究.首先,分别对状态观测误差系统和对象系统在有限区域对数量化作用下的系统渐近稳定性进行了分析,得到了相应的稳定性条件,接着针对对数型量化器,给出了两个系统稳定性之间的内在关联,同时得到了各有限区域量化器的量化区间上界值.在此基础上,得到了保证各子系统稳定的系统通信速率比值.最后,给出了在时变量化作用下基于状态观测的控制策略和仿真例子.     

The method of parametric Lyapunov functions for Markov dynamic systems

The notion of parametric Lyapunov function is introduced for Markov dynamic systems. The existence of a function of this kind is shown to be a necessary and sufficient condition for the strong stochastic stability of an equilibrium. In terms of parametric Lyapunov functions, a sufficient criterion is proved for asymptotic strong stochastic stability in the case of Feller Markov chains. Some examples are given showing the efficiency of the method proposed.     

Stability in part of the variables of “partial” equilibria of systems with aftereffect

The problem of stability in part of the variables of a “partial” equilibrium (this means that a given part of the phase vector coordinates is zero) is considered for nonlinear nonstationary systems of functional differential equations with aftereffect. The notions of stability in part of the variables, which admit more general (compared with the known ones) assumptions about the values of the supremum-norm of the components of the initial vector function corresponding to the variables that do not determine the given equilibrium, are introduced. The stability and asymptotic stability conditions of the the type mentioned above are obtained in the context of the method of Lyapunov-Krasovskii functionals; this conditions allow generalization of several well-known results.     

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

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.     

渐近紧随机动力系统及其极限集的性质

通过对随机动力系统极限行为的研究,推广了传统动力系统的相关定义和理论.由于在无界区域上,Sobolevr紧嵌入的缺乏,Crauel,Debussche及Flancloli等人在研究有界区域上的随机演化系统时所引入的渐近紧的概念不再适用.通过对Ladyzlmnskaya,Rosa等人在对确定性系统的研究中所提出的渐近紧概念的推广,引入了随机动力系统渐近紧的概念,并以相应的示例及严格的理论推导证明了此概念的合理性和必要性.最后,作为这一新的概念的应用,证明了渐近紧随机动力系统极限集的性质.     

Polynomial matrix-fraction representations for linear time-varying systems

This paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time-varying discrete-time systems. It is shown that the time-varying analog of the polynomial model always has a free state-space. Necessary and sufficient conditions for the existence of Bezout factorizations are obtained in terms system theoretic properties of state-space realizatios.     

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.     

Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions

This paper deals with the stability analysis of a class of uncertain switched systems on non-uniform time domains. The considered class consists of dynamical systems which commute between an uncertain continuous-time subsystem and an uncertain discrete-time subsystem during a certain period of time. The theory of dynamic equations on time scale is used to study the stability of these systems on non-uniform time domains formed by a union of disjoint intervals with variable length and variable gap. Using the concept of common Lyapunov function, sufficient conditions are derived to guarantee the asymptotic stability of this class of systems on time scale with bounded graininess function. The proposed scheme is used to study the leader–follower consensus problem under intermittent information transmissions.     

Stability of equilibria of discrete-time systems in terms of invariant sets

We suggest a new approach to the verification of the stability (or asymptotic stability) of the equilibria of time-invariant discrete-time systems based on stability and asymptotic stability criteria stated in terms of invariant sets. A set-theoretic method for the verification of the conditions in these criteria is presented.     

Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities

This paper considers the problem of global asymptotic stability of a class of nonlinear uncertain discrete-time state-delayed systems. The class of systems under investigation involves multiple state delays, norm-bounded parameter uncertainties, and generalized overflow nonlinearities which cover usual types of overflow arithmetic used in practice. A new criterion for the global asymptotic stability of such systems is presented. A numerical example is given to illustrate the applicability of the criterion presented.