On the Lyapunov theorem for singular systems

In this paper, we revisit the Lyapunov theory for singular systems. There are basically two well-known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of the work by Lewis. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clarified. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of Takaba et al.     

Duality of 2-D singular systems of Roesser models

In this paper, the properties and concepts of dual systems of the two-dimensional singular Roesser models (2-D SRM) are studied. Two different concepts of the dual systems are proposed for the 2-D SRM. One is derived from the duality defined for two-dimensional singular general models (2-D SGM)-called the S-dual systems; the other one is defined based on 2-D SRM in a traditional sense-called the T-dual systems. It is shown that if a 2-D SRM is jump-mode free or jump-mode reachable, then it can be equivalently transformed into a canonical form of a 2-D SRM, for which the T-duality and the S-duality are equivalent. This will be of some perspective applications in the robust control of 2-D SRM.     

Duality of 2-D singular systems of Roesser models

In this paper, the properties and concepts of dual systems of the two-dimensional singular Roesser models （2-D SRM） are studied. Two different concepts of the dual systems are proposed for the 2-D SRM. One is derived from the duality defined for two-dimensional singular general models （2-D SGM）-called the S-dual systems; the other one is defined based on 2-D SRM in a traditional sense-called the T-dual systems. It is shown that if a 2-D SRM is jump-mode free or jump-mode reachable, then it can be equivalently transformed into a canonical form of a 2-D SRM, for which the T-duality and the S-duality are equivalent. This will be of some perspective applications in the robust control of 2-D SRM.     

A Lyapunov approach to analysis of discrete singular systems

In this paper, a new type generalized Lyapunov equation for discrete singular systems is proposed. Then it is applied to study problems such as pole clustering, controllability and observability for discrete singular systems. First, some necessary and sufficient conditions for pole clustering are derived via the solution of this new type Lyapunov equation. Further, the relationship between the solution of the Lyapunov equation and structure properties of discrete singular systems will be investigated based on these results. Finally, a type of generalized Riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the finite poles of the closed-loop systems are clustered into a specified disk.     

Generalized functional observer

A functional observer and state feedback are proposed for singular systems in the polynomial fraction form that requires no impulsive mode elimination. The order of the compensator is determined by the newly defined generalized observability index that is associated with the McMillan degree of the system. A new generalized Lyapunov equation is also proposed through a realization scheme that can be applied to both ordinary and singular systems. The solution to the equation provides an algebraic approach to the observer of singular systems in the generalized state-space form     

基于观测器的2-D奇异系统Roesser模型的H∞滤波

This paper discusses the problem of the H∞ filtering for discrete time 2-D singular Roesser models (2-D SRM). The purpose is to design an observer-based 2-D singular filter such that the error system is acceptable, jump modes free and stable, and satisfies a pre-specified H∞ performance level. By general Riccati inequality and bilinear matrix inequalities (BMI), a sufficient condition for the solvability of the observer-based H∞ filtering problem for 2-D SRM is given. A numerical example is provided to demonstrate the applicability of the proposed approach. Key words 2-D singular systems, jump modes, general Riccati inequality, bilinear matrix i     

离散广义大系统的Lyapunov稳定性分析

广义大系统的稳定性是广义大系统理论的基本问题之一,对其稳定性的研究要比状态空间大系统复杂得多,因为广义大系统不仅需要考虑稳定性,而且还要考虑正则性和因果性(离散广义系统)及脉冲自由(连续广义系统).本文在所有孤立子系统都是正则的且具有因果关系的条件下,利用Lyapunov方程,应用Lyapunov函数方法,研究了广义离散线性大系统和广义离散非线性大系统的稳定性和不稳定性问题,给出了离散广义大系统稳定性和不稳定性判定定理,得到了离散广义大系统的关联稳定参数域和不稳定域.     

广义区间系统的稳定性与二次能稳分析

对于一类广义区间系统,利用Lyapunov方法,研究了该系统的稳定性.通过广义Lyapunov不等式的建立,得到了此系统结构稳定的充分必要条件是广义Lyapunov不等式有解.在此基础上,建立了广义Riccati不等式,由此不等式得到了此系统二次能稳的充要条件是广义Riccati不等式有解.这些结果将对进一步研究此系统有着重要的基础作用.最后,举例说明了主要结果.     

Analysis of singular systems using orthogonal functions

The use of orthogonal functions to analyze singular systems is investigated. It is shown that the differential-algebraic system equation may be converted to an algebraic generalized Lyapunov equation that can be solved for the coefficients ofx(t)in terms of the orthogonal basis functions. This generalized Lyapunov equation may be considered as a "discrete" equation on the slow subspace of the system, and as a "continuous" equation on its fast subspace. Necessary and sufficient conditions for the existence of a unique solution are given in terms of the relative spectrum of the system. A generalized Bartels/Stewart algorithm based on theQZalgorithm is presented for its efficient solution. Relations are drawn with the invariant subspaces of the system.     

不确定广义大系统有限时间鲁棒分散控制

研究不确定连续广义大系统的有限时间鲁棒分散控制问题,设计系统的有限时间鲁棒分散状态反馈控制器.首先应用广义Lyapunov 函数法,给出不确定广义大系统有限时间鲁棒稳定的充分条件;其次,给出不确定广义大系统应用分散状态反馈控制器鲁棒镇定的充分条件和有限时间鲁棒分散控制器的设计方法;最后,通过仿真例子验证所提出方法的有效性.     

A geometric theory for derivative feedback

The authors use a singular system setting to provide a geometric theory for dynamical systems under derivative feedback. They define the relevant subspace and provide computational design techniques in terms of a generalized Sylvester or Lyapunov equation for which efficient solution techniques are well-known. The authors provide both geometric and algebraic characterizations of the effects of derivative feedback, drawing connections with previous work in state-variable systems as well as extending that work to singular systems     

一般2-D线性常系数离散状态空间模型渐近稳定性的一类Lyapunov方法

本文给出了一般2-D线性常系数离散状态空间模型(2-D GM)的渐近稳定性的定义以及相应的判据,并借助于Lyapunov方程给出了2-D Roesser Model(2-D RM)渐近稳定条件,推广和简化了文献[1,7]的有关结论。最后通过行列式的一个简单性质,证明了关于2-DGM渐近稳定性的判定可以转化为一类特殊的2-D RM的相应问题,从而得到了判定2-D GM渐近稳定性的Lyapunov方法。     

不确定2-D 奇异系统Roesser 模型 鲁棒能稳的矩阵不等式方法

考虑具有参数不确定性的2-D奇异系统Roesser模型(简称2-D SRM)鲁棒能稳问题．通过静态状态反馈控制律，使得对所有容许的不确定参数，闭环系统容许、稳定、无跳跃模．通过求解矩阵不等式，给出了不确定2-D奇异系统鲁棒能稳问题可解的充分条件及静态状态反馈控制律设计的代数表达式．最后通过算例验证了方法的有效性．     

2-D奇异线性离散系统渐近稳定性的一类Lyapunov 方法

引入2－D奇异一般离散状态空间模型的Lyapunov方程,探讨了该模型的渐近稳定性、特征多项式的根式以及2－D Lyapunov矩阵方程间的关系,给出了系统渐近稳定性的充分条件。     

Adaptive Fault Tolerant Control of Multi-time-scale Singularly Perturbed Systems

This paper studies the fault tolerant control, adaptive approach, for linear time-invariant two-time-scale and three-time-scale singularly perturbed systems in presence of actuator faults and external disturbances. First, the full order system will be controlled using ε-dependent control law. The corresponding Lyapunov equation is ill-conditioned due to the presence of slow and fast phenomena. Secondly, a time-scale decomposition of the Lyapunov equation is carried out using singular perturbation method to avoid the numerical stiffness. A composite control law based on local controllers of the slow and fast subsystems is also used to make the control law ε-independent. The designed fault tolerant control guarantees the robust stability of the global closed-loop singularly perturbed system despite loss of effectiveness of actuators. The stability is proved based on the Lyapunov stability theory in the case where the singular perturbation parameter is sufficiently small. A numerical example is provided to illustrate the proposed method.     

2-D奇异系统Roesser模型的鲁棒H∞控制

考虑具有参数不确定性的2-D奇异系统Roesser模型(简称2-D SRM)鲁棒H∞控制问题．在给出界实引理的另一等价形式的基础上,通过求解矩阵不等式,给出了不确定2-D奇异系统鲁棒H∞控制问题可解的充分条件及静态状态反馈控制律设计的代数表达式．最后通过一个仿真算例验证了方法的有效性．     

线性时滞系统时滞独立稳定的充分条件

利用Lyapunov稳定性理论，通过一个推广的Lyapunov矩阵方程得出时滞独立稳定的充分条件，基于这个充分条件建立了几个判定线性时滞系统稳定性的简单判据，并推导出系统具有任意指定收敛速度指数稳定的充分条件。计算例子说明了所得结果的有效性。     

On robust stability of 2-D discrete systems

Presents a study on robust stability of two-dimensional (2-D) discrete systems in the Fornasini-Marchesini (F-M) state space setting. A measure of stability robustness of a stable F-M model is defined. The relation of this measure to its counterpart in the Roessor state space and related computational issues are addressed. Three lower bounds of the stability-robustness measure defined are derived using a one-dimensional parameterization approach and a 2-D Lyapunov approach. A numerical example is included to illustrate the main results obtained     

Geometric structure and feedback in singular systems

The output-nulling (A, E, R(B))-invariant subspaces are defined for singular systems, rigorously justifying the name and demonstrating that special cases of these geometric objects are the familiar subspace of admissible conditions and the supremal (A, E, R(B ))-invariant subspace. A novel singular-system-structure algorithm is used to compute them by numerically efficient means. Their importance for describing the possible closed-loop geometric structure in terms of the open-loop geometric structure is shown. An approach to spectrum assignment in singular systems that is based on a generalized Lyapunov equation is introduced. The equation is used to compute feedback gains to place poles and assign various closed-loop invariant subspaces while guaranteeing closed-loop regularity     

Exponential stability of output‐based event‐triggered control for switched singular systems

This paper presents the exponential stability of output‐based event‐triggered control for switched singular systems. An event‐triggered mechanism is introduced based on measure output, by employing the Lyapunov functional method and average dwell time approach, some sufficient conditions for exponential stability of the switched singular closed‐loop systems are derived. Furthermore, dynamic output feedback controller parameters are obtained. Lastly, a numerical example is given to illustrate the validity of the proposed solutions.