A linear programming algorithm for invariant polyhedral sets of discrete-time linear systems

In this paper we formulate necessary and sufficient conditions for an arbitrary polyhedral set to be a positively invariant set of a linear discrete-time system. Polyhedral cones and linear subspaces are included in the analysis. A linear programming algorithm is presented that enables practical application of the results stated in this paper.     

Stochastic stability of Positive Markov Jump Linear Systems

This paper investigates on the stability properties of Positive Markov Jump Linear Systems (PMJLS’s), i.e. Markov Jump Linear Systems with nonnegative state variables. Specific features of these systems are highlighted. In particular, a new notion of stability (Exponential Mean stability) is introduced and is shown to be equivalent to the standard notion of 1-moment stability. Moreover, various sufficient conditions for Exponential Almost-Sure stability are worked out, with different levels of conservatism. The implications among the different stability notions are discussed. It is remarkable that, thanks to the positivity assumption, some conditions can be checked by solving Linear Programming feasibility problems.     

On linear co-positive Lyapunov functions for sets of linear positive systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.     

On relationships among passivity,positive realness,and dissipativity in linear systems

The notions of passivity and positive realness are fundamental concepts in classical control theory, but the use of the terms has varied. For LTI systems, these two concepts capture the same essential property of dynamical systems, that is, a system with this property does not generate its own energy but only stores and dissipates energy supplied by the environment. This paper summarizes the connection between these two concepts for continuous and discrete time LTI systems. Beyond that, relationships are provided between classes of strictly passive systems and classes of positive real systems. The more general framework of dissipativity is introduced to connect passivity and positive realness and also to survey other energy-based results. The frameworks of passivity indices and conic systems are discussed to connect to passivity and dissipativity. After surveying relevant existing results, some clarifying results are presented. These involve connections between classes of passive systems and finite-gain _L2$L_2$ stability as well as asymptotic stability. Additional results are given to clarify some of the more subtle conditions between classes of these systems and stability results. This paper surveys existing connections between classes of passive and positive real systems and provides results that clarify more subtle connections between these concepts.     

Positive invariance, monotonicity and comparison of nonlinear systems

In this paper, the interrelation between positive invariance, monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied. First, necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v(x)≤w with respect to nonlinear systems are established. Then, various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established. Finally, necessary and sufficient conditions for a, not necessarily monotone, nonlinear system to be a comparison system are established. No specific algebraic structure is necessary for developing the main results of this work. Numerical examples illustrating the applicability of all these results to the case of discrete-time dynamical systems are also given.     

On the simultaneous diagonal stability of linear discrete-time systems

This paper studies the set of time-invariant linear discrete-time systems in which each system has a diagonal quadratic Lyapunov function. First, it is shown that there is generally no common diagonal quadratic Lyapunov function for such a set of systems even if the set is assumed to be commutative. It is also shown that the commutativity assures the existence of a common diagonal quadratic Lyapunov function inside the set of 2×2 systems or the set of nonnegative systems. Then, two simple topological results are presented concerning the simultaneous diagonal stability on the set of nonnegative systems. The first is a measure of the difference of matrices that assures the simultaneous diagonal stability. The second is a measure of the commutativity of matrices.     

REACHABILITY AND CONTROLLABILITY OF DISCRETE-TIME LINEAR DYNAMICAL SYSTEMS

The concepts of reachability and controllability do not coincide but reflect a fundamental difference between the dynamical characteristics of different classes of multi-variable systems. This difference is discussed in relation to discrete-lime linear dynamical systems and is illustrated by a simple numerical example.     

REACHABILITY AND CONTROLLABILITY OF DISCRETE-TIME LINEAR DYNAMICAL SYSTEMS

The concepts of reachability and controllability do not coincide but reflect a fundamental difference between the dynamical characteristics of different classes of multi-variable systems. This difference is discussed in relation to discrete-time linear dynamical systems and is illustrated by a simple numerical example.     

Regulator problem for linear systems with constraints on control and its increment or rate

This paper discusses the problem of constraints on both control and its rate or increment, for linear systems in state space form, in both the continuous and discrete-time domains. Necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate (for the continuous-time case), such that the system evolves respecting incremental or rate constraints. A pole assignment technique is then used to solve the inverse problem, giving stabilizing state feedback controllers that respect non-symmetrical constraints on both control and its increment or rate. An illustrative example shows the application of the method on the double integrator problem.     

On the existence of a positive realization

The positive realization problem for linear systems is to find conditions, for a given transfer function with nonnegative impulse response, to have a realization such that the resulting system is a positive system. Recently, it has been shown that, under a mild assumption on the long-term behaviour of the impulse response, this problem is related to the maximum modulus poles only. In this paper necessary and sufficient conditions for positive realizability of discrete-time systems are given. They show that also nondominant poles play a role in the most general case. Positive realizability conditions for the continuous-time case are also given.     

Positivity of discrete singular systems and their stability: An LP-based approach

In this paper we present an efficient approach to the analysis of discrete positive singular systems. One of our main objectives is to investigate the problem of characterizing positivity of such systems. Previously, this issue was not completely addressed. We provide easily checkable necessary and sufficient conditions for such problem to be solved. On the other hand, we study the stability of discrete positive singular systems. Note that this is not a trivial problem since the set of admissible initial conditions is not the whole space but it is represented by a special cone. All the conditions we provide are necessary and sufficient and are based on a reliable computational approach via linear programming.     

Necessary conditions for positive realizability of continuous-time linear systems

This paper deals with the positive realization problem. The problem is to find, from a given transfer function, a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Necessary conditions are investigated and a new one is given, together with some related results.     

Complete controllability of continuous-time recurrent neural networks

This paper studies controllability for the class of control systems commonly called (continuous-time) recurrent neural networks. It is shown that, under a generic condition on the input matrix, the system is controllable, for every possible state matrix. The result holds when the activation function is the hyperbolic tangent.     

A reduced-order observer for non-linear discrete-time systems

In this note, a simple and useful reduced-order observer for a large class of non-linear discrete-time systems is presented. Neither detectability of the linear part nor resolution of the Sylvester constraint is required. From the stability analysis we show that the proposed state observer has strong tracking properties under very general conditions and generalizes results in (M. Boutayeb et al., IEEE. Trans. Automat. Control 42(4) (1997) 581–586). High performances, from the design and implementation point of view, will be shown through three numerical examples of physical processes largely used in industrial applications.     

Global stabilization of linear discrete-time systems with bounded feedback

This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less than or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (“single hidden layer neural networks”) of simple saturation functions.     

Solvability of static output-feedback stabilization for LTI positive systems

This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.     

线性离散周期系统研究综述

由于线性离散周期时变系统在理论和实践双重方面的重要意义,因此成为控制领域研究的重要课题. 本文对目前线性离散周期系统的研究成果加以总结.以系统分析和综合为线索,着重介绍了线性离散周期系统的时不变重构、系统的结构属性、稳定性、零点、极点、模型降解及实现、故障诊断等问题所存在的典型方法. 并对线性离散周期系统领域仍存在的问题和未来的发展方向作了进一步的展望.