On the inversion of matrix Routh array

A new matrix which is the product of a chain of factored matrices is constructed in this paper to find the matrix Routh array and the matrix continued fraction inversion for multivariate systems if the partial matrix coefficients are known. A linear transformation matrix ia established from tho matrix Routh array developed to transform the second Cauer matrix form in state-space coordinates to tho phase variable matrix form.     

平面系统稳定性的间接方法*

利用Routh 判据得到的间接检验矩阵特征值的方法讨论平面系统的稳定性问题,对于线性系统,该方法给出平面区间矩阵稳定性的充要条件.对于非线性系统,Jacobian 猜想的已知结果[5],该方法给出平面非线性系统稳定吸引域估计的一种算法.最后,该估计被利用于高维系统二维滑动模的设计.     

平面系统稳定性的间接方法

利用Ｒｏｕｔｈ判据得到的间接检验矩阵特征值的方法平面系统的稳定性问题。对于线性系统,该系统法给出平面区间矩阵稳定性的充要条件。对于非线性系统,基于Ｊａｃｂｉａｎ猜想的已知结果。     

Robust stability and stabilization for singular systems with statedelay and parameter uncertainty

Considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method     

Pascal-Routh polynomials: A first exploration in feedback system design

Following a brief review of the Routh array and a few of its salient properties, the inverse array generating rule is used to produce a family of polynomials--referred to as Pascal-Routh polynomials--for which the first column of the Routh array is the corresponding row from the Pascal triangle of binomial coefficients. A preliminary exploration, illustrated by a design example, is made of the performance of feedback systems with Pascal-Routh characteristic polynomials.     

A new tabular form for determining root distribution of a complexpolynomial with respect to the imaginary axis

The original Routh table dealing with real polynomials is further investigated for complex polynomials. A tabular form for determining root distribution of a complex polynomial with respect to the imaginary axis is developed, and modified procedures for directly treating singularities in the array are proposed. Also, procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. Conditional stability and instability can be distinguished from each other by the criteria developed here     

ON THE STABILITY AND STABILIZATION OF TIME‐VARYING NONLINEAR CONTROL SYSTEMS

Many stability and stabilization problems of nonlinear time‐varying systems lead to asymptotic behavior of (–K + L)‐type systems, which consist of a K‐function and an L‐function. The stability of these systems is of fundamental importance for a series of stabilization problems of time‐varying nonlinear control systems. Even though the asymptotical stability of such systems has been used widely and (in most cases) implicitly, we do not find a rigorous proof, in the literature, and the existing proof for a particular case is questionable. Under quite general conditions, we prove that the solution of these systems tends to 0 as t →. Some generalizations are also obtained. As an immediate consequence, a general theorem is obtained for the stabilization of time‐varying systems. Using the new framework, we examine several stability and stabilization problems. First of all, for cascade systems, two sets of sufficient conditions are obtained for uniformly asymptotical stability and globally asymptotical stability, respectively. Then we consider the stability of ISS and IISS systems. A new concept, namely, strong IISS, is proposed. Several stability properties for autonomous systems are extended to time‐varying systems. Finally, we consider stabilization via detection. A rigorous proof is given for a smooth state feedback time‐varying system with weak detectability to be stabilizable by means of an observer.     

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.     

Comments on the computation of interval Routh approximants

In Bandyopadhyay et al. (1994, 1997), the Routh approximation method was extended to derive reduced-order interval models for linear interval systems. In this paper, the authors show that: 1) interval Routh approximants to a high-order interval transfer function depend on the implementation of interval Routh expansion and inversion algorithms; 2) interval Routh expansion algorithms cannot guarantee the success in generating a full interval Routh array; 3) some interval Routh approximants may not be robustly stable even if the original interval system is robustly stable; and 4) an interval Routh approximant is in general not useful for robust controller design because its dynamic uncertainties (in terms of robust frequency responses) do not cover those of the original interval system     

Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

During the past several years, there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems. This paper aims to briefly survey recent results in this field. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After a brief review of the stability analysis under restricted switching and the multiple Lyapunov function theory, the switching stabilization problem is studied, and a variety of switching stabilization methods found in the literature are outlined. Then the switching stabilizability problem is investigated, that is under what condition it is possible to stabilize a switched system by properly designing switching control laws. Note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature. A necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here.      

Computation of the abscissa of stability by repeated use of the Routh test

The numerical computation of the abscissa of stability of a polynomial, defined as the largest of the real parts of the zeros of the polynomial, is a problem that occurs repeatedly in computer-aided design of dynamical and control systems. The approach developed in this paper, by making repeated use of the Routh test, avoids the computation of zeros and provides a useful method.     

An observer-based approach to controlling time-delay chaotic systems via Takagi-Sugeno fuzzy model

This study presents a fuzzy model-based control criterion for time-delay chaotic systems with uncertainty. The Takagi-Sugeno (T-S) fuzzy model is adopted for representing a chaotic system possessing a time delay. A unified approach for designing a fuzzy observer, which integrates parallel distributed compensation with an adaptive updating method, is first discussed. An observer-based fuzzy control scheme is then developed to deal with stabilization and tracking problems. Based on Lyapunov stability analysis, sufficient conditions that ensure robust control performance of time-delay chaotic systems are derived. These conditions are represented in terms of linear matrix inequalities and adaptive updating laws. Finally, the proposed approach is validated through numerical synchronization, stabilization, and model tracking control simulations.     

Order-dependent and delay-dependent conditions for stability and stabilization of fractional-order time-varying delay systems using small gain theorem

In this paper, the stability and stabilization problems for fractional-order time-varying delay systems are investigated. Firstly, by casting the stability problem as one of robust stability analysis problems and utilizing the small gain theorem, an order-dependent and delay-dependent stability condition for fractional-order time-varying delay systems is developed. Taking advantage of the information of order and delay, the stability condition is less conservative than the existing results. Then, state feedback controllers that stabilize fractional-order time-varying delay systems are developed. To tackle the computational difficulty of the controller design method, a local optimization algorithm is proposed. Finally, numerical examples are provided to illustrate that the proposed criteria are valid and less conservative than the existing ones.     

Interconnected jumping time‐delay systems: Mode‐dependent decentralized stability and stabilization

In this paper, we deal with the problems of mode‐dependent decentralized stability and stabilization with ??_∞ performance for a class of continuous‐time interconnected jumping time‐delay systems. The jumping parameters are governed by a finite state Markov process and the delays are unknown time‐varying and mode‐dependent within interval. The interactions among subsystems satisfy quadratic bounding constraints. To characterize mode‐dependent local stability behavior, we employ an improved Lyapunov–Krasovskii functional at the subsystem level and express the stability conditions in terms of linear matrix inequalities (LMIs). A class of local decentralized state‐feedback controllers is developed to render the closed‐loop interconnected jumping system stochastically stable. Then, we extend the feedback strategy to dynamic observer‐based control and establish the stochastic stabilization via LMIs. It has been established that the developed results encompass several existing results as special cases which are illustrated by simulation of examples. Copyright © 2011 John Wiley & Sons, Ltd.     

An Overview of Finite/Fixed-Time Control and Its Application in Engineering Systems

The finite/fixed-time stabilization and tracking control is currently a hot field in various systems since the faster convergence can be obtained. By contrast to the asymptotic stability, the finite-time stability possesses the better control performance and disturbance rejection property. Different from the finite-time stability, the fixed-time stability has a faster convergence speed and the upper bound of the settling time can be estimated. Moreover, the convergent time does not rely on the initial information. This work aims at presenting an overview of the finite/fixed-time stabilization and tracking control and its applications in engineering systems. Firstly, several fundamental definitions on the finite/fixed-time stability are recalled. Then, the research results on the finite/fixed-time stabilization and tracking control are reviewed in detail and categorized via diverse input signal structures and engineering applications. Finally, some challenging problems needed to be solved are presented.      

具输入饱和因子的广义系统的镇定

广义系统的镇定问题是广义系统理论的基本问题之一.为了研究具输入饱和因子 的广义系统的镇定,首先利用代数方法,借助大系统分解原理,对具输入饱和因子的广义系统 进行综合,给出闭环系统渐近稳定的判别条件.其次,给出了闭环系统E一渐近稳定的概念,利 用广义Lyapunov函数方法,研究具输入饱和因子的广义系统,提出了合适的反馈控制,得到 了具输入饱和因子的广义系统E一渐近稳定的判别定理.     

Robust H-infinity fuzzy control for uncertainMarkovian jump nonlinear singular systems withwiener process

This paper deals with the problems of robust stochastic stabilization and H-infinity control for Markovian jump nonlinear singular systems with Wiener process via a fuzzy-control approach. The Takagi-Sugeno (T-S) fuzzy model is employed to represent a nonlinear singular system. The purpose of the robust stochastic stabilization problem is to design a state feedback fuzzy controller such that the closed-loop fuzzy system is robustly stochastically stable for all admissible uncertainties. In the robust H-infinity control problem, in addition to the stochastic stability requirement, a prescribed performance is required to be achieved. Linear matrix inequality (LMI) sufficient conditions are developed to solve these problems, respectively. The expressions of desired state feedback fuzzy controllers are given. Finally, a numerical simulation is given to illustrate the effectiveness of the proposed method.     

Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints

Stability analysis and control of linear impulsive systems is addressed in a hybrid framework, through the use of continuous-time time-varying discontinuous Lyapunov functions. Necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas. Extensions to the stability of aperiodic systems under minimum, maximum and ranged dwell-times are then derived. By exploiting further the particular structure of the stability conditions, the results are non-conservatively extended to quadratic stability analysis of linear uncertain impulsive systems. These stability criteria are, in turn, losslessly extended to stabilization using a particular, yet broad enough, class of state-feedback controllers, providing then a convex solution to the open problem of robust dwell-time stabilization of impulsive systems using hybrid stability criteria. Relying finally on the representability of sampled-data systems as impulsive systems, the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled-data systems are straightforwardly solved using the same ideas. Several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach.     

一类新的Takagi-Sugeno模糊时滞系统的稳定性准则

本文提出一种新的时滞划分方法—变时滞划分法,以解决连续延时Takagi-Sugeno模糊系统的稳定性和镇定性问题.不同于已有的文献,用可变参数将时变时滞区间[0,d(t)]划分为若干个可变子区间,并得出模糊时滞系统的新的时滞相关稳定性准则.本文提出的新方法能充分利用时滞子区间的内部信息,因此新的时滞相关稳定性准则比以往结果具有更小的保守性.基于Lyapunov稳定性理论,以线性矩阵不等式形式给出T--S模糊系统的新的时滞相关稳定性准则,并将稳定性和镇定性研究结果扩展到具有不确定参数的T--S模糊系统.仿真实例证明了本文方法降低保守性的有效性.     

Quadratic stability and stabilization of bimodal piecewise linear systems

This paper deals with quadratic stability and feedback stabilization problems for continuous bimodal piecewise linear systems. First, we provide necessary and sufficient conditions in terms of linear matrix inequalities for quadratic stability and stabilization of this class of systems. Later, these conditions are investigated from a geometric control point of view and a set of sufficient conditions (in terms of the zero dynamics of one of the two linear subsystems) for feedback stabilization are obtained.