Geometric criteria for controllability and near-controllability of driftless discrete-time bilinear systems

Geometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete-time bilinear systems. In this paper, controllability and near-controllability of driftless discrete-time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near-controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near-controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi-input discrete-time bilinear systems and to linear time-invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.     

On almost invariant subspaces for implicit linear discrete-time systems

The concept of almost invariant subspace for an implicit linear discrete-time system is introduced and studied in detail. It is shown also that for regular homogeneous implicit systems the so-called deflating subspaces can be identified with almost invariant subspaces.     

Further decomposition for singular systems and its properties on geometric subspace

This paper focuses on the relationship between the geometric subspaces and the structural decomposition of continuous-time singular systems. The original structural decomposition is not capable of revealing explicitly the invariant geometric subspaces for singular systems. As such, a further decomposition is necessary and is thus investigated in this paper. Under a new decomposition proposed, the supremal output-nulling (A,E, ImB)-invariant subspace of singular systems can be clearly expressed in an explicit form, and some of its applications are also addressed.     

A complement about almost controllability subspaces

We show that, for any almost controllability subspace, there exist a maximal input chain and a family of state feedbacks which lead to the well known ‘feedback-chain’ expression for this subspace.     

Determination of generic dimensions of controllable subspaces and its application

The controllable subspace and its dimension of a structured linear system vary as a function of the free parameters. However, the dimension is stable in the sense that it takes, for almost any system parameters, some maximal constant which is the generic rank of the controllability matrix. In this paper, this maximal constant is called the generic dimension of the controllable subspace. Two simple methods for determining generic dimensions of controllable subspaces are derived. As an application, the results are applied to the determination of system types of linear multivariable unity feedback systems.     

A class of marked invariant subspaces with an application to algebraic Riccati equations

Invariant subspaces of a matrix A are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of A. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.     

Controllability of matrix eigenvalue algorithms: the inverse power method

In this paper we initiate a program to study the controllability properties of matrix eigenvalue algorithms arising in numerical linear algebra. Our focus is on a well-known eigenvalue method, the inverse power iteration defined on projective space. A complete characterization of the reachable sets and their closures is given via cyclic invariant subspaces. Moreover, a necessary and sufficient condition for almost controllability of the inverse power method is derived.     

A homeomorphism between observable pairs and conditioned invariant subspaces

A bijective correspondence between similarity classes of observable systems (C,A) and n-codimensional conditioned invariant subspaces of a pair (C,A) is constructed that leads to a homeomorphism of the spaces. This is applied to the parametrization of inner functions of fixed McMillan degree. Proofs using state space methods as well as using polynomial models are given.     

Controllability of discrete‐time multiagent systems with switching topology

The current theoretical investigation on the controllability of switched multiagent systems mainly focuses on fixed connected topology or union graph without nonaccessible nodes. However, for discrete‐time multiagent systems with switching topology, it is still unknown whether the existing results are valid or not under the condition of arbitrary topology. Based on graph distance partitions and Wonham's geometric approach, we provide the lower and upper bounds for the dimension of controllable subspaces of discrete‐time multiagent systems. Unlike the existing results of controllability with switching topology, the proposed results have the advantage of being applicable to multiagent systems with arbitrary graphic topologies, union graph (strongly connected or not), and coupling weights. We also provide 2 algorithms for computing the lower and upper bounds for the dimension of controllable subspaces, respectively. Furthermore, as a remarkable application, we present how the proposed lower bound can be utilized for achieving the targeted controllability if the dimension of the controllable subspace of the switched system satisfies certain conditions.     

(A, B)-invariant and stabilizability subspaces, a frequency domain description

In a paper of E. Emre and the author a polynomial characterization for (A, B)-invariant subspaces is given. The characterization is used to give a frequency domain criterion for the solvability of the disturbance decoupling problem. In this paper a more elementary and simpler treatment is given. Furthermore, stabilizability subspaces are introduced, are given a frequency domain characterization and are used to solve design problems.     

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.     

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.     

Controllability of multi‐agent systems under directed topology

In this paper, the controllability of multi‐agent systems is studied under leader‐follower framework, where the interconnection between agents is directed. The concept of leader‐follower connectedness is first introduced for directed graph to check controllability, and some graph‐theoretic conditions are derived for constructively designed topologies. Then, distance partition and almost equitable partition are employed to quantitatively study the controllable subspaces. Moreover, the relationship between the state invariance of multi‐agent systems and the existence of almost equitable partition is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Controllability and reachability criteria for switched linear systems

This paper investigates the controllability and reachability of switched linear control systems. It is proven that both the controllable and reachable sets are subspaces of the total space. Complete geometric characterization for both sets is presented. The switching control design problem is also addressed.     

Solving the infinite-dimensional discrete-time algebraic riccati equation using the extended symplectic pencil

In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic pencil (ESP). We present a general form for all linear bounded solutions of the ARS in terms of the deflating subspaces of the ESP. This relation is analogous to the results of the Hamiltonian approach for the continuous-time ARE and to the symplectic pencil approach for the finite-dimensional discrete-time ARE. In particular, we show that there is a one-to-one relation between deflating subspaces with a special structure and the solutions of the ARS. Using the relation between the solutions of the ARS and the deflating subspaces of the ESP, we give characterizations of self-adjoint, nonnegative, and stabilizing solutions. In addition we give criteria for the discrete-time, infinite-dimensional ARE to have a maximal self-adjoint solution. Furthermore, we consider under which conditions a solution of the ARS satisfies the ARE as well.     

Invariant subspaces for LPV systems and their applications

The aim of this paper is to extend the notion of invariant subspaces known in the geometric control theory of the linear time invariant systems to the linear parameter-varying (LPV) systems by introducing the concept of parameter-varying invariant subspaces. For LPV systems affine in their parameters, algorithms are given to compute many parameter varying subspaces relevant in the solution of state feedback and observer design problems.     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstract second-order infinite-dimensional dynamical systems is considered. It is proved using the frequency-domain method, that approximate controllability of second-order system can be verified by the approximate controllability conditions for the corresponding simplified first-order system. General results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends earlier results on approximate controllability of second-order abstract dynamical systems.     

双线性系统可控性综述

双线性系统是一类特殊的非线性系统,广泛存在于现实世界中,如工程、经济、生物、生态等领域,被认为是最接近于线性系统的非线性系统．对双线性系统的研究已历经了近半个世纪. 作为系统最基本的属性,双线性系统可控性的研究一直以来是热点和难点.本文分别对连续双线性系统可控性和离散双线性系统可控性进行讨论, 综述了双线性系统可控性的研究.特别地,报告了近来对离散双线性系统可控性研究的新成果.最后,例举了一些可控的双线性系统例子.