Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.     

On solution concepts and well-posedness of linear relay systems

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two different solution concepts, Filippov solutions and forward solutions. We derive necessary and sufficient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. To be precise, uniqueness of Filippov (and also forward) solutions follows in this case if the first non-zero Markov parameter is positive. By means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two. The influence of the Zeno phenomenon (an infinite number of relay switching times in a finite length time interval) on well-posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems, the consequences of the presence of the Zeno behaviour is typical for many other classes of non-smooth and hybrid systems.     

State representations of linear systems with output constraints

We derive state space representations for linear systems that are described by input/state/output equations and that are subjected to a number of constant linear constraints on the outputs. In the case of a general linear system, the state representation of the constrained system is shown to be essentially nonunique. For linear Hamiltonian systems satisfying a nondegeneracy condition, there is a natural and unique choice of the representation which preserves the Hamiltonian structure. In the linear systems setting we give an algebraic proof that a system withn degrees of freedom underk constraints becomes a system withn−k degree of freedom. Similar results are obtained for linear gradient systems.     

关于一类混合动态系统能控性与镇定的研究:一个例子

线性切换系统是一类重要的混合动态系统模型. 线性切换系统的镇定是一个基本研究问题. 针对一个线性切换系统的具体例子, 分析能控性和可镇定性的关系. 指出能控的线性切换系统是可以通过定常状态反馈镇定的.     

Linear cone-invariant control systems and their equivalence

In this paper, we study invariant control systems that generalise positive systems. A characterisation of linear control systems invariant on polyhedral cones (corner regions) in the state-space, called cone-invariant linear control systems, is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone. The problem of equivalence between control systems invariant on corner regions is introduced. For cone-invariant linear control systems, we study invariance-preserving state-equivalence and invariance-preserving feedback-equivalence and present characterisations of both notions of equivalence.     

Stabilization of nonlinear sandwich systems via state feedback—Discrete‐time systems

A recent paper (IEEE Trans. Aut. Contr. 2010; 55 (9):2156–2160) considered stabilization of a class of continuous‐time nonlinear sandwich systems via state feedback. This paper is a discrete‐time counterpart of it. The class of nonlinear sandwich systems consists of saturation elements sandwiched between linear systems. We focus first on single‐layer sandwich systems, which consist of a single saturation sandwiched between two linear systems. For such systems, we present necessary and sufficient conditions for semi‐global and global stabilization problems by state feedback, and develop design methodologies to achieve the prescribed stabilization. We extend the results to single‐layer sandwich systems subject to additional actuator saturation. Finally, we discuss further extension to general multi‐layer sandwich systems with an arbitrary number of saturations sandwiched between linear systems, both with and without actuator saturation. The design methodologies can be viewed as extensions of classical low‐gain design methodologies developed during the 1990s in the context of stabilizing linear systems subject to actuator saturation. Copyright © 2010 John Wiley & Sons, Ltd.     

Popov–Belevitch–Hautus type controllability tests for linear complementarity systems

It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. This paper deals with the controllability problem of a class of piecewise linear systems, known as linear complementarity systems. By exploiting the underlying structure and employing the results on the controllability of the so-called conewise linear systems, we present a set of inequality-type conditions as necessary and sufficient conditions for controllability of linear complementarity systems. The presented conditions are of Popov–Belevitch–Hautus type in nature.     

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

Many control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.     

A reduction paradigm for output regulation

The goal of this paper is to provide a reduction paradigm for the design of output regulators which can be of interest for nonlinear as well as linear uncertain systems. The main motivation of the work is to provide a systematic design tool to deal with non‐minimum‐phase uncertain systems for which conventional high‐gain stabilization methods are not effective. The contribution of the work is two‐fold. First, this work extends a previous reduction paradigm for output regulation of nonlinear systems. Furthermore, in the case of the uncertain controlled dynamics being linear, we show how the proposed framework leads to a number of systematic design tools of interest for non‐minimum‐phase linear systems affected by severe uncertainties. A numerical control example of a linearized model of an inverted pendulum on a cart is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Asymptotic stability of digitally redesigned control systems

This paper analyses an asymptotic stability of a digitally redesigned control system when the states of the analogue and the digital control systems are approximately matched at every sampling point. The digital redesign is a simple method of converting a given analogue controller to an equivalent digital controller in the sense of state-matching. The concerned state-matching technique is to minimise the norm distance between the discretised closed-loop system matrix of linear analogue control system and that of linear digital control system. It is shown that (i) there exists an upper bound of the norm distance to guarantee the asymptotic stability of the digitally redesigned control system and (ii) the trajectories of the linear analogue and the linear digital control systems coincide at every sampling point if the norm distance is zero. Also, a robustness result is provided in the case that nonlinear perturbations occur in the analogue and the digital control systems. Moreover, design conditions for the developed stability analysis are proposed in terms of linear matrix inequalities.