Fractional variable order discrete-time systems,their solutions and properties

In this paper, fractional variable order discrete state-space systems based on different definitions of fractional variable order difference are investigated.The general solution of these systems is derived. Moreover, necessary and sufficient conditions for reachability and observability are given and proven. The sufficient conditions for controllability are proposed too.     

New results in stability of a class of hereditary linear systems

Some basic connections between bounded-input, bounded-output stability and exponential asymptotic stability are presented for a class of linear time-varying hereditary systems. For that class of systems, controllability and observability are closely related to that of an equivalent linear system. This eases the analytical treatment of such properties which are then used to derive the stability results. The requirement of boundedness of inputs and outputs in the BIBO definition is modified by requesting that their ‘energy content’ over a fixed-length interval should be bounded independently of the position of the interval. For the class of hereditary linear systems studied, it is found that EAS and BIBO stability are equivalent under uniform complete controllability and observability.     

Stability of the Kalman filter for continuous time output error systems

The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of continuous time output error systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability alone is sufficient to ensure the asymptotic stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered systems themselves. The exponential or polynomial convergence of the Kalman filter is then further analyzed for particular cases of stable or unstable output error systems.     

Composite system controllability and observability

A differential operator approach to modeling the dynamical behavior of linear, time-invariant, multivariable systems is employed in order to develop some new techniques for determining the controllability and observability properties of series, parallel, and feedback composite systems. The tests outlined involve polynomial matrix manipulations and rely on some recent results which deal with the controllability and observability of systems in differential operator form.     

非线性控制系统的特性（Ⅱ）

本文采用与线性系统理论比较的方法来阐述非线性控制系统理论。文章通过例子分析了非线性控制系统在能控性、能观性、抗干扰性、解耦和稳定性等方面的特性，从而进一步说明非线性控制系统理论在提法、论述和结论上的特点。文章的第一部分介绍了非线性控制系统的能控性和能观性，这一部分揭示非线性控制系统在干扰解耦和无交互作用控制方面的特性。     

Computation of controllability and observability Gramians in modeling of discrete‐time noncommensurate fractional‐order systems

This paper presents a new algorithm for computation of controllability and observability Gramians for an expanded state space form of integer‐order approximator to linear time‐invariant discrete‐time noncommensurate fractional‐order systems. The introduced methodology can significantly reduce the time complexity of the Gramians' calculation, being the main computational burden in modeling of discrete‐time fractional‐order systems by means of a high integer‐order expanded state space approximator and the balanced truncation reduction method. Simulation experiments illustrate an efficiency of the introduced methodology, in particular for low‐dimension fractional‐order systems and high implementation lengths.     

Observability conditions of switched linear singular systems

The observability problem of switched linear singular(SLS) systems is studied in this paper. Based on the observability definition, the unobservable subspaces of given switching laws are investigated under the condition that all subsystems are regular. A necessary condition and a sufficient condition for observability of SLS systems are given. It is shown that the observability and controllability are dual for some special SLS systems with circulatory switching laws. The method developed here is applicable to the observability analysis of normal switched linear systems.     

用多元有理函数矩阵研究系统的结构性质

本文着重介绍了各种结构化矩阵，经比较，说明多元有理函数矩阵（ＲＦＭ）能起初地描述线性物理系统，通过对多元有理函数域Ｆ（ｚ）上的能控制能观性的讨论，说明ＲＦＭ是研究与系统结构相关性质的有力工具，也说明了研究Ｆ（ｚ）上能控制观性的必要性。     

Observability conditions of switched linear singular systems

The observability problem of switched linear singular(SLS) systems is studied in this paper. Based on the observability definition, the unobservable subspaces of given switching laws are investigated under the condition that all subsystems are regular. A necessary condition and a sufficient condition for observability of SLS systems are given. It is shown that the observability and controllability are dual for some special SLS systems with circulatory switching laws. The method developed here is applicable to the observability analysis of normal switched linear systems.     

Necessary and sufficient conditions for controllability and observability of switched impulsive control systems

Many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This note first introduces the impulsive behavior into switched linear systems and studies the controllability and observability of such systems. Necessary and sufficient criteria for reachability and controllability are established. It is proved that reachability is equivalent to controllability for such systems. Then, necessary and sufficient criteria for observability and determinability of such systems are established by duality. It is also proved that observability is equivalent to determinability. Our criteria are of geometric type, they can be transformed into algebraic form conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.     

非线性控制系统的特性（Ⅰ）

本文采用与线性系统理论相比较的方法，阐述非线性控制系统理论。文中通过一些例子来分析非线性控制系统在能控性、能观性、抗干扰性、解耦和稳定性等方面的特性，从而进一步说明非线性控制系统理论在提法、论述和结论上的特点。第一部分介绍非线性系统的能控性和能观性。     

Lagrange stabilisation for uncertain phase-controlled systems

In this article, we are concerned with the Lagrange stabilisation problem for phase-controlled systems with parameter uncertainties. By using the Kalman–Yakubovich–Popov lemma, the frequency-domain conditions for Lagrange stability of nominal phase-controlled systems are converted into linear matrix inequality (LMI) conditions. The allowable parameter uncertainty bounds for controllability and observability, as well as LMI conditions for Lagrange stability of uncertain phase-controlled systems, are presented. A controller design strategy based on the LMI method is proposed such that the uncertain phase-controlled systems are Lagrange stabilised. A numerical example is provided to demonstrate the effectiveness of the proposed results.     

Controllability and observability for time‐varying switched impulsive controlled systems

This paper is concerned with the controllability and observability for linear time‐varying switched impulsive systems. First, some new results about the variation of parameters for time‐varying switched impulsive systems are derived. Then less conservative sufficient conditions and necessary conditions for state controllability and observability of such systems are established. And for such system without impulsive control input, sufficient and necessary conditions for controllability and observability are derived. Furthermore, corresponding criteria applied to time‐varying impulsive systems are also discussed and examples are presented to show the effectiveness of the proposed results. Copyright © 2009 John Wiley & Sons, Ltd.     

System theoretic properties of a class of spatially invariant systems

In this paper we develop new readily testable criteria for system theoretic properties such as stability, controllability, observability, stabilizability and detectability for a class of spatially invariant systems. Our approach uses the well-established theory developed to solve infinite-dimensional systems. The theoretical results are illustrated by several examples.     

Disturbance decoupling by measurement feedback with stability for infinite-dimensional systems

The concepts of controllability and stabilizability subspaces are extended to infinite-dimensional linear systems. Under certain assumptions one obtains a nice generalization of the finite-dimensional theory, and, by dualizing, similar results are obtained for the concepts of complementary observability and complementary detectability subspaces. These concepts are used to solve the infinite-dimensional version of the disturbance-decoupling problem, the disturbance-decoupled estimation problem and the disturbance-decoupling problem with measurement feedback, all requiring various stability requirements. The solution is in terms of geometric concepts such as (A, B)- and (C,A)- invariant subspaces and controllability and complementary observability subspaces.     

Linear time-variable systems: Balancing and model reduction

A "uniformly balanced" realization for linear time-variable systems is defined. This representation is characterized by the fact that its controllability, and observability Gramians are equal and diagonal. Existence and uniqueness of the uniformly balanced realization is studied. Such a framework has many remarkable properties and leads to a novel method for approximating time-variable systems, where the subsystems of the balanced realization can be taken as a reduced model. The reduced model is examined from the point of view of stability, controllability, and observability.     

The h‐Difference Approach to Controllability and Observability of Fractional Linear Systems with Caputo–Type Operator

In the paper a class of h‐difference linear control systems with n fractional orders is studied. The Caputo‐type h‐difference operator is used. The formula of the solution to the stated initial value problem of the given system is proved. The problems of controllability and observability in a finite number of steps of h‐difference linear control systems with n fractional orders are studied.     

New necessary and sufficient conditions for local controllability and local observability of 2-D separable denominator systems

New necessary and sufficient conditions for local controllability and observability of 2-D separable denominator systems (SDS's) are presented on the basis of the reduced-dimensional decomposition of 2-D SDS's. It is proved that local controllability of a 2-D SDS is equivalent to controllability of two 1-D systems which are a special decomposition pair of the 2-D SDS. Furthermore, local observability of a 2-D SDS is equivalent to a full rank condition of the coefficient matrices of the 2-D SDS in addition to observability of two 1-D systems which are another special decomposition pair of the 2-D SDS. Thus, these new conditions which use only 1-D controllability matrices, 1-D observability matrices, and coefficient matrices of the 2-D SDS are much simpler than any previous conditions which use complex 2-D controllability and observability matrices.     

非线性控制系统的特性（Ⅲ）

本文采用线性系统理论比较的方法来阐述非线性控制系统理论。文章通过一些例子来分析非线性控制系统的能控性、能观性、抗干扰性、解耦和稳定性等方面的特性，从而进一步说明非线性控制系统理论在提法、论述和结论上的特点。