Lyapunov equation for infinite-dimensional discrete bilinear systems

Mean-square stability for discrete systems requires that uniform convergence is preserved between input and state correlation sequences. Such a convergence preserving property holds for an infinite-dimensional bilinear system if and only if the associate Lyapunov equation has a unique strictly positive solution.     

Strong stability does not imply similarity to a contraction

We shall exhibit a strongly asymptotically stable discrete linear system whose model operator is not similar to a contraction.     

Controllability of discrete time inhomogeneous bilinear systems

In this paper we derive a set of conditions which are both necessary and sufficient for complete controllability of a class of inhomogeneous discrete time bilinear systems.     

Stabilizing controllers for discrete bilinear systems

In this paper, we study stabilizing controllers for time varying bilinear systems. Here, the feedback function, f in our paper is for larger classes than those given in the current literature. We establish existence theorems for stabilizing bilinear systems by output feedback from a large class. The main theorems basically state that under broad conditions, the zero state of the systems can be made asymptotically stable by output feedback. Following these stabilizing theorems, a new design procedure for the feedback control law is presented. Application of the new theorems is illustrated by a simulation example.     

Lagrange stability and boundedness of discrete event systems

Recently it has been shown that the conventional notions of stability in the sense of Lyapunov and asymptotic stability can be used to characterize the stability properties of a class of logical discrete event systems (DES). Moreover, it has been shown that stability analysis via the choice of appropriate Lyapunov functions can be used for DES and can be applied to several DES applications including manufacturing systems and computer networks (Passino et al. 1994, Burgess and Passino 1994). In this paper we extend the conventional notions and analysis of uniform boundedness, uniform ultimate boundedness, practical stability, finite time stability, and Lagrange stability so that they apply to the class of logical DES that can be defined on a metric space. Within this stability-theoretic framework we show that the standard Petri net-theoretic notions of boundedness are special cases of Lagrange stability and uniform boundedness. In addition we show that the Petri ent-theoretic approach to boundedness analysis is actually a Lyapunov approach in that the net-theoretic analysis actually produces an appropriate Lyapunov function. Moreover, via the Lyapunov approach we provide a sufficient condition for the uniform ultimate boundedness of General Petri nets. To illustrate the Petri net results, we study the boundedness properties of a rate synchronization network for manufacturing systems. In addition, we provide a detailed analysis of the Lagrange stability of a single-machine manufacturing system that uses a priority-based part servicing policy.     

A stability criterion for a class of discrete nonlinear systems with random parameters

In this paper, discrete nonlinear models with random parameters are studied. A new frame to classify and analyze discrete stochastic nonlinear systems has been developed from deterministic nonlinear systems to stochastic nonlinear systems. This frame is broad and includes a large class of stochastic nonlinear systems. A stability criterion developed for this frame is a non-Lyapunov method of stability analysis and is easily applied. In addition, this derived sufficient condition of stability is obtained without the assumption of stationarity for the random noise as frequently assumed in the literature.     

Generic uniform observability analysis for bilinear systems

In this paper, we study the property of generic uniform observability for structured bilinear systems. More precisely, to check whether or not a structured bilinear system is generically uniformly observable, we provide a group of necessary conditions and a second group of sufficient ones. These conditions are expressed in quite simple graphic-terms. On one hand, all the given conditions are easy to check because they deal with finding paths in a digraph. On the other hand, the proposed method is based on a graph-theoretic approach and assumes only a knowledge of the system’s structure. This makes the suggested approach particularly well suited to study large scale systems or systems with unknown parameters, as may be the case during a conception stage.     

Remarks on -input bounded-state stability of linear controllable infinite-dimensional systems

It is shown that every eP-input bounded-state stable linear (infinite-dimensional) system x_k+1=A_kx_k+B_ku_k is uniformly power equistable, if it is uniformly equicontrollable.     

On the stability of bilinear free systems

Problems of stability of stationary and nonstationary free bilinear discrete systems over a field GF(p) are investigated. Criteria of asymptotic stability of a system status are found, and some properties of the above-mentioned notions are studied.     

Observability analysis for structured bilinear systems: A graph-theoretic approach

This paper is devoted to the generic observability analysis for structured bilinear systems using a graph-theoretic approach. On the basis of a digraph representation, we express in graphic terms the necessary and sufficient conditions for the generic observability of structured bilinear systems. These conditions have an intuitive interpretation and are easy to check by hand for small systems and by means of well-known combinatorial techniques for large-scale systems.     

Uniform stability of discrete impulsive systems

This article studies discrete impulsive systems. The uniform stability (uniformly asymptotic stability and quasi-exponential stability) criteria are established for linear discrete impulsive system and a class of nonlinear discrete impulsive systems, respectively. These stability criteria are expressed in terms of fairly simple algebraic conditions so that they are easy to be tested. Moreover, by using the stability criteria, a discrete impulsive synchronization scheme is designed to synchronize two discrete chaotic systems. Finally, some examples are given to illustrate results obtained by us.     

Razumikhin-type stability theorems for discrete delay systems

In this paper, by employing the Razumikhin technique and Lyapunov functions, Razumikhin-type theorems that guarantee the uniform stability, uniformly asymptotic stability and uniformly exponential stability for the general discrete delay systems are established, respectively. Moreover, Razumikhin-type uniformly exponential stability theorem gives the estimation of the convergence speed. As theoretic application, the Razumikhin-type uniformly exponential stability result is further studied and used to show some well-known stability results for some kinds of discrete delay systems. Finally, examples are also worked through to illustrate our results.     

On the regional control problem of the bilinear wave equation

The present research deals with regional optimal control problem of the bilinear wave equation evolving on a spatial domain $\mathrm{\Omega }\subset {\mathrm{ℝ}}^n,n\ge 1$. Such an equation is excited by bounded controls that act on the velocity term. It addresses the tracking of a desired state all over the time interval $[0,T]$ only on a subregion $\omega$ of $\mathrm{\Omega }$ with minimum energy. Then, we prove that an optimal control exists and is characterized as a solution to an optimality system. Algorithm for the computation of such a control is given and successfully illustrated through simulations.     

Weak and strong stabilisation of bilinear systems in a Banach space

In this paper, we study the feedback stabilisation of homogeneous bilinear systems on a Banach state-space. Sufficient conditions for weak and strong stabilisation are given. A decay rate of the stabilised state is explicitly estimated. Applications to heat and transport equations are provided.     

Robust stability and stabilisation of 2D discrete state-delayed systems

In this paper, we first present sufficient stability and robust stability conditions for discrete linear state-delayed 2D systems in terms of linear matrix inequalities. All results are obtained with Fornasini–Marchesini delay model but appropriate transformation to the corresponding Roesser form is provided as well. Generalisation to the multiple state-delayed case is also given. Then the stabilisation and robust stabilisation using static state feedback are studied. Stabilising feedback gain matrices are constructed based on the solutions of certain linear matrix inequalities. An numerical example is used to illustrate the effectiveness of the approach.     

广义双线性系统的二阶终端滑模控制

利用Lyapunov稳定性理论研究了一类广义双线性系统的镇定问题.通过构造特殊的二阶终端滑模超曲面,设计相应的变结构控制器,使闭环系统在有限时间内实现滑动模运动,系统的状态在平衡点渐近稳定.该设计方法能有效削弱系统的高频抖振.仿真结果验证了设计方法的可行性.     

Simple stability criteria for nonlinear time-varying discrete systems

We present two sufficient conditions for asymptotic stability of nonlinear time-varying discrete systems. Our main results generalize Mori's result for linear discrete time-invariant systems to nonlinear time-varying systems.