Arbitrary Switching Rule Research Articles

Necessary and Sufficient Condition on Controllability and Reachability of Fractional-Order Linear Switched System

Fractional-order linear switched system (FLSS) is an important system of hybrid systems. In this paper, by using the analytical solution of FLSS, the necessary and sufficient condition on controllability and reachability of FLSS is given, respectively. The condition shows that if every subsystem is controllable (reachable), then the whole system is also controllable (reachable) for arbitrary switching rules. And if the whole system is controllable (reachable) for arbitrary switching rules, then every subsystem is controllable (reachable).

Finite-Time Stability Analysis for a Class of Continuous Switched Descriptor Systems

Finite-time stability has more practical application values than the classical Lyapunov asymptotic stability over a fixed finite-time interval. The problems of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems are considered in this paper. Based on the average dwell time approach and the multiple Lyapunov functions technique, the concepts of finite-time stability and boundedness are extended to continuous switched descriptor systems. In addition, sufficient conditions for the existence of state feedback controllers in terms of linear matrix inequalities (LMIs) are obtained with arbitrary switching rules, which guarantee that the switched descriptor system is finite-time stable and finite-time bounded, respectively. Finally, two numerical examples are presented to illustrate the reasonableness and effectiveness of the proposed results.

Robust H∞ reliable control for uncertain switched systems with circular disk pole constraints

This paper addresses the issue of optimal robust H∞ reliable control with circular disk pole constraints for switched systems with actuator faults and arbitrary switching rules. The design method of the state feedback controller is proposed, which guarantees that the robust H∞ performance is minimum and the closed poles are located in a specified circular disk. The corresponding parameters of the controller are obtained by using the linear matrix inequalities (LMIs) optimization. Finally, a simulation example is provided to validate the effectiveness of the proposed approach.

Synchronization of Switched Interval Networks and Applications to Chaotic Neural Networks

This paper investigates synchronization problem of switched delay networks with interval parameters uncertainty, based on the theories of the switched systems and drive-response technique, a mathematical model of the switched interval drive-response error system is established. Without constructing Lyapunov-Krasovskii functions, introducing matrix measure method for the first time to switched time-varying delay networks, combining Halanay inequality technique, synchronization criteria are derived for switched interval networks under the arbitrary switching rule, which are easy to verify in practice. Moreover, as an application, the proposed scheme is then applied to chaotic neural networks. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Stabilization of impulsive switched linear systems with saturated control input

This paper considers the stabilization problem of impulsive switched linear systems with saturated control input, in which the impulses are viewed as control or disturbances. Under arbitrary switching rule and dwell time switching rule, the local stabilization conditions are obtained in terms of LMIs, respectively. The corresponding optimization problems are formulated to obtain a bigger attractive region. Our results show that impulsive control can recover part of attractive region, or allow a wide range of switching signals. If the impulses occur as disturbances, we have to increase the dwell time of switching signals to offset the effects of disturbances. Numerical examples are given to show the effectiveness of the results proposed.

${\cal H}_2$ and ${\cal H}_\infty$ Performance Optimization of Singularly Perturbed Switched Systems

This paper addresses the ${\cal H}_2$ and ${\cal H}_\infty$ performance problems for singularly perturbed switched linear systems in continuous time. We first propose a new and unified approach for stability analysis and norms computations for singularly perturbed linear time invariant systems. We generalize this approach to the case of singularly perturbed switched linear systems to obtain numerically well conditioned conditions for the design of a state-switching control. Such a switching strategy guarantees that the closed-loop switched system is globally asymptotically stable for sufficiently small values of the perturbation parameter and provides an upper bound to an ${\cal H}_2$ and ${\cal H}_\infty$ performance index. We show that this approach can also be used for singularly perturbed switched linear systems with arbitrary switching rules and we provide a comparison with existing methods.

Stability of Discrete-Time Switched Homogeneous Systems on Cones and Conewise Homogeneous Inclusions

This paper presents a stability analysis of switched homogeneous systems on cones under arbitrary and optimal switching rules with extensions to conewise homogeneous or linear inclusions. Several interrelated approaches, such as the joint spectral radius approach and the generating function approach, are exploited to derive necessary and sufficient stability conditions and to develop suitable algorithms for stability tests. Specifically, the generalized joint spectral radius and the generalized joint lower spectral radius are introduced to characterize the radii of domains of strong and weak attraction. Furthermore, strong and weak generating functions and their radii of convergence are employed to derive stability conditions; their analytic properties, numerical approximations, and convergence analysis are established. Extensions to conewise homogeneous or linear inclusions are made to address state-dependent switching dynamics. Relations between different stability notions in the strong or weak sense are studied; Lyapunov techniques are used for stability analysis of the conewise linear inclusions.

Global Robust Stability of Switched Interval Neural Networks with Discrete and Distributed Time‐Varying Delays of Neural Type

By combing the theories of the switched systems and the interval neural networks, the mathematics model of the switched interval neural networks with discrete and distributed time‐varying delays of neural type is presented. A set of the interval parameter uncertainty neural networks with discrete and distributed time‐varying delays of neural type are used as the individual subsystem, and an arbitrary switching rule is assumed to coordinate the switching between these networks. By applying the augmented Lyapunov‐Krasovskii functional approach and linear matrix inequality (LMI) techniques, a delay‐dependent criterion is achieved to ensure to such switched interval neural networks to be globally asymptotically robustly stable in terms of LMIs. The unknown gain matrix is determined by solving this delay‐dependent LMIs. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

New Delay Dependent Stability Analysis for Switched Neutral Systems with Mode-Dependent Delays under Arbitrary Switching Laws

In this paper, the stability analysis of switched uncertain neutral systems with mode-dependent delays under arbitrary switching rules is presented. Based on common Lyapunov functional, and combined with the analysis of matrix inequalities, the delay dependent stability conditions are obtained in the form of linear matrix inequalities(LMIs)which can be easily solved by LMI toolbox in Matlab. Finally, a numerical example illustrate that the proposed criteria are effective.

MODEL REDUCTION FOR DISCRETE-TIME SWITCHED LINEAR TIME-DELAY SYSTEMS VIA THE H∞ ROBUST STABILITY

This paper describes the problem of model order-reduction for a class of hybrid switched linear time-delay systems composed of linear discrete-time delayed invariant subsystems with a switching rule. The paper investigates a novel delay term-dependent approach to model reduction. It consists of evaluating the error approximation performance between the original system and the order-reduced system; the stability problem is solved under arbitrary switching rule. The model order-reduction conditions are given in the form of linear matrix inequalities with a term-independent delay. A numerical example shows the effectiveness of the proposed approach.

Stability and Stabilization of Two Time Scale Switched Systems in Discrete Time

In this technical note, stability and stabilization of two time scale switched linear systems in the singular perturbation form are addressed in discrete time. We show that, under an arbitrary switching rule, stability of the slow and fast switched subsystems is not sufficient to assess stability of the original two time scale switched system, even if the singular perturbation parameter tends to zero. Therefore, we propose LMI based conditions that guarantee the asymptotic stability of the two time scale switched system using switched quadratic Lyapunov functions. These conditions express the fact that the coupling between fast and slow subsystems has to be taken into account in addition to stability properties of the two subsystems, when the switching rule is arbitrary. The presented conditions are extended to state feedback control design. A numerical example illustrates the features of the proposed approach.

A Separation Principle for Linear Switching Systems and Parametrization of All Stabilizing Controllers

In this paper, we investigate the problem of designing a switching compensator for a plant switching amongst a (finite) family of given configurations ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ). We assume that switching is uncontrolled, namely governed by some arbitrary switching rule, and that the controller has the information of the current configuration <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> . As a first result, we provide necessary and sufficient conditions for the existence of a switching compensator such that the closed-loop plant is stable under arbitrary switching. These conditions are based on a separation principle, precisely, the switching stabilizing control can be achieved by separately designing an observer and an estimated state (dynamic) compensator. These conditions are associated with (non-quadratic) Lyapunov functions. In the quadratic framework, similar conditions can be given in terms of LMIs which provide a switching controller which has the same order of the plant. As a second result, we furnish a characterization of all the stabilizing switching compensators for such switching plants. We show that, if the necessary and sufficient conditions are satisfied then, given any arbitrary family of compensators K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (s), each one stabilizing the corresponding LTI plant ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) for fixed i, there exist suitable realizations for each of these compensators which assure stability under arbitrary switching.

Constrained state feedback control of piecewise affine systems

This paper deals with constrained state feedback control of piecewise affine systems, an important class of hybrid systems. Our objective is to design stabilizing feedback gains that guarantees, for any arbitrary switching rule, the respect of the control and state constraints. The existence of a solution is first expressed geometrically, in terms of two polyhedral sets inclusion. The geometric conditions is then reformulated algebraically as linear matrix inequalities, which provides an effective way to calculate feedback gains answering the problem.

Robust pole placement controller design in LMI region for uncertain and disturbed switched systems

This paper concerns the state feedback control for continuous-time, disturbed and uncertain linear switched systems with arbitrary switching rules. The main result of this work consists in getting a LMI (Linear Matrix Inequalities) condition guaranteeing a robust pole placement according to some desired specifications. Then, external disturbance attenuation with a fixed rate according to the H ∞ criterion is ensured. This is obtained thanks to the existence of a common quadratic Lyapunov function for all sub-systems. Finally, an academic example illustrates the efficiency of the developed approach.

Stabilization of a class of discrete-time switched systems via observer-based output feedback

In this paper, observer-based static output feedback control problem for discrete-time uncertain switched systems is investigated under an arbitrary switching rule. The main method used in this note is combining switched Lyapunov function (SLF) method with Finsler’s Lemma. Based on linear matrix inequality (LMI) a less conservative stability condition is established and this condition allows extra degree of freedom for stability analysis. Finally, a simulation example is given to illustrate the efficiency of the result.

OBSERVER-BASED STABILIZATION OF DISCRETE-TIME UNCERTAIN SWITCHING SIMILAR COMPOSITE SYSTEMS

Reduced-order design of decentralized observer-based output switching controllers is presented for uncertain discrete-time switching similar composite systems with arbitrary switching rule. First, a reduced-order control design model is constructed with the dimension equal to a subsystem's dimension of the original system. Then, asymptotically stabilizing switching controller is designed. It is proved that when this controller is implemented into each subsystem of the original system then such decentralized controller asymptotically stabilizes the overall closed-loop system under arbitrary switching.

Robust Stability of Switched Cohen–Grossberg Neural Networks With Mixed Time-Varying Delays

By combining Cohen-Grossberg neural networks with an arbitrary switching rule, the mathematical model of a class of switched Cohen-Grossberg neural networks with mixed time-varying delays is established. Moreover, robust stability for such switched Cohen-Grossberg neural networks is analyzed based on a Lyapunov approach and linear matrix inequality (LMI) technique. Simple sufficient conditions are given to guarantee the switched Cohen-Grossberg neural networks to be globally asymptotically stable for all admissible parametric uncertainties. The proposed LMI-based results are computationally efficient as they can be solved numerically using standard commercial software. An example is given to illustrate the usefulness of the results.

State feedback control of switched linear systems: An LMI approach

This paper addresses the problem of state feedback control of continuous-time switched linear systems with arbitrary switching rules. A quadratic Lyapunov function with a common matrix is used to derive a stabilizing switching control strategy that guarantees: (i) the assignment of all the eigenvalues of each linear subsystem inside a chosen circle in the left-hand half of the complex plane; (ii) a minimum disturbance attenuation level for the closed-loop switched system. The proposed design conditions are given in terms of linear matrix inequalities that encompass previous results based on quadratic stability conditions with fixed control gains. Although the quadratic stability based on a fixed Lyapunov matrix has been widely used in robust control design, the use of this condition to provide a convex design method for switching feedback gains has not been fully investigated. Numerical examples show that the switching control strategy can cope with more stringent design specifications than the fixed gain strategy, being useful to improve the performance of this class of systems.

Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty

With the development of intelligent control, switched systems have been widely studied. Here we try to introduce some ideas of the switched systems into the field of neural networks. In this Letter, a class of switched Hopfield neural networks with time-varying delay is investigated. The parametric uncertainty is considered and assumed to be norm bounded. Firstly, the mathematical model of the switched Hopfield neural networks is established in which a set of Hopfield neural networks are used as the individual subsystems and an arbitrary switching rule is assumed; Secondly, robust stability analysis for such switched Hopfield neural networks is addressed based on the Lyapunov–Krasovskii approach. Some criteria are given to guarantee the switched Hopfield neural networks to be globally exponentially stable for all admissible parametric uncertainties. These conditions are expressed in terms of some strict linear matrix inequalities (LMIs). Finally, a numerical example is provided to illustrate our results.