State and input observability for structured linear systems: A graph-theoretic approach

This paper deals with the state and input observability analysis for structured linear systems with unknown inputs. The proposed method is based on a graph-theoretic approach and assumes only the knowledge of the system's structure. Using a particular decomposition of the systems into two subsystems, we express, in simple graphic terms, necessary and sufficient conditions for the generic state and input observability. These conditions are easy to check because they are based on comparison of integers and on finding particular subgraphs in a digraph. Therefore, our approach is suited to study large-scale systems.     

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.     

广义区间动力系统的能控性

讨论了广义区间动力系统正则性、I-能控与C-能控问题.由于上述问题等价于判别某 个区间矩阵为列满秩,首先得到判别区间矩阵为列满秩的充分条件与充分必要条件,进一步得 到了判别广义区间系统为正则的充分必要条件、I-能控的充分条件与C-能控的充分必要条件. 通过数值实例说明所得到的结果相对于已有结果更具有一般性及有效性.由对偶原理,得到相 应的能观性的判据.     

How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

Approximate observability of infinite dimensional bilinear systems using a Volterra series expansion

In this paper, the approximate observability of a class of infinite dimensional bilinear systems is investigated. The observability conditions are discussed based on a Volterra series representation of this class of systems. A testable observability criterion is derived for the case where the infinitesimal generator is self-adjoint or Riesz-spectral operator. The theoretical results are illustrated by examples.     

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.     

基于矩阵方法的有界Petri网系统的能观性分析

Petri网和有限自动机是离散事件动态系统的两类主要研究内容.而Petri网系统的能观性分析与判别是基于Petri网的实际系统设计、优化、监测及控制的重要基础.以往关于Petri网能观测性的研究缺乏定量化的充要判别条件.本文利用代数矩阵方法研究了带有输出的有界Petri网系统的能观性问题.首先,基于矩阵的半张量积,将带有输出的有界Petri网系统的动态行为以线性方程组的形式建立了数学模型.然后,针对初始标识和当前标识,介绍了两种能观性定义.最后,基于矩阵运算建立了关于有界Petri网系统能观性的几个充分必要条件,并给出严格证明.数值算例验证了理论结果.本文提出的方法实现了有界Petri网系统能观性的矩阵运算,易于计算机实现.     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Observability Conditions by Polynomial Ideals

In this paper, we give global and local observability conditions of polynomial systems. Each observability is defined in terms of distinguishability of a pair of initial states. The set of pairs of initial states are represented as an affine variety. By using this affine variety, a global observability condition is derived. Then, a local observability condition is presented based on the minimal decomposition of the affine variety and dimension theory.     

线性广义系统的区间观测器设计

本文研究了连续时间线性广义系统的区间观测器设计问题.首先根据正系统的稳定性判据提出了一种基于线性矩阵不等式的广义区间观测器直接设计法,然后通过引入更多的设计自由度进一步放宽了区间观测器的设计条件,扩大了设计方法的适用范围.所提出的设计方法无需坐标变换,是一种直接设计方法.最后,通过两个仿真算例验证了所提出方法的有效性.