Common joint linear copositive Lyapunov functions for positive switched systems

Unlike many papers focusing on common linear copositive Lyapunov functions (CLCLFs) for positive switched systems, this paper studies the existence of a class of common weak linear copositive Lyapunov functions called common joint linear copositive Lyapunov functions (CJCLFs). Necessary and sufficient conditions for the existence of CJCLFs are established. The conditions are easily verifiable via the straightforward computation, and can be applied to the asymptotic stability problem of the positive switched linear system when each switching mode is only Lyapunov stable.     

On linear copositive Lyapunov functions for switched positive systems

This paper addresses the existence of a common linear copositive Lyapunov function for discrete-time switched positive systems. Our results reveal that the existence of such a function is equivalent to the Schur stability of a kind of special matrices, these matrices consist of column vector of system matrices in an appropriate manner. A simple example is provided to illustrate the implication of our results.     

Finite-time stability of positive switched time-delay systems based on linear time-varying copositive Lyapunov functional

In this paper, we deal with the finite-time stability of positive switched linear time-delay systems. By constructing a class of linear time-varying copositive Lyapunov functionals, we present new explicit criteria in terms of solvable linear inequalities for the finite-time stability of positive switched linear time-delay systems under arbitrary switching and average dwell-time switching. As an important application, we apply the method to finite-time stability of linear time-varying systems with time delay.     

Finite-time stabilization of continuous-time switched positive delayed systems

This paper deals with the problems of finite-time stability and stabilization for continuous-time switched positive linear time-delay systems under mode-dependent average dwell time switching signals. First, finite-time stability conditions are established by constructing an multiple piecewise copositive Lyapunov–Krasovskii functional. Then, finite-time stabilization is achieved by designing a state-feedback controller in the form of linear programming. This numerical construction approach proposed for controller cancels the restriction of the multiple piecewise copositive Lyapunov–Krasovskii functional on controllers, which can decrease the conservatism. Finally, two numerical examples are given to show the advantages of our methods.     

Estimator design of discrete-time switched positive linear systems with average dwell time

This paper is concerned with the problem of state estimation for a class of discrete-time switched positive linear systems (SPLS) with average dwell time (ADT) switching. By utilizing the multiple linear copositive Lyapunov function (MLCLF) approach, the ADT switching is introduced to tackle the state estimation of the underlying system. Some sufficient conditions of the existence of the estimator are derived in terms of a set of linear matrix inequalities for the underlying systems with ADT switching. The results for the SPLS under arbitrary switching can be easily obtained by reducing MLCLF to the common linear copositive Lyapunov function used for the system in the literature. Finally, a numerical example is given to show the validity of the obtained theoretical results.     

New stability conditions for discrete-time switched positive linear time-varying systems

In this note, we will devote to investigate the stability of discrete-time switched positive linear time-varying systems (PLTVSs). Firstly, a new asymptotic stability criterion of discrete-time PLTVSs is obtained by using time-varying copositive Lyapunov functions (TVCLFs) and this criterion is then extended to the switched case based on the multiple TVCLFs. Furthermore, the sufficient conditions are derived for stability of discrete-time switched PLTVSs with stable subsystems by means of function-dependent average dwell time and function-dependent minimum dwell time. In addition, the stability sufficient conditions are drawn for the switched PLTVSs which contain unstable subsystems. It is worth noting that the difference of TVCLFs and multiple TVCLFs are both relaxed to indefinite in our work. The theoretical results obtained are verified by two numerical examples.     

Output regulation for a class of positive switched systems

In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme. The feature of the developed approach is that each subsystem is not required to has a solution to the problem. Moreover, two types of controllers and switching laws are devised. The first one depends on the state together with the external input and the other depends only the error. The conditions ensuring the solvability of the problem for positive switched systems are presented in the form of linear matrix equations plus linear inequalities under some mild constraints. Two examples are finally given to show the performance of the proposed control strategy.     

Adaptive event-triggering distributed filter of positive Markovian jump systems based on disturbance observer

This paper presents an adaptive event-triggered filter of positive Markovian jump systems based on disturbance observer. A new adaptive event-triggering mechanism is constructed for the systems. A positive disturbance observer is designed for the systems to estimate the disturbance. A distributed output model of each subsystem of positive Markovian jump systems is introduced. Then, an adaptive event-triggering distributed filter is designed by employing stochastic copositive Lyapunov functions. All presented conditions are solvable in terms of linear programming. Under the designed disturbance observer and the distributed filter, the corresponding error system is stochastically stable. The filter design approach is also developed for discrete-time positive Markovian jump systems. The contribution of the paper lies in that: (i) A new adaptive event-triggering mechanism is established for positive systems, (ii) A positive disturbance observer is designed for the disturbance of positive Markovian jump systems, and (iii) The designed distributed filter can guarantee the stochastic stability of the error while existing filters in literature only achieve the stochastic gain stability of the error. Finally, two examples are given to illustrate the effectiveness of the proposed design.     

H∞ tracking control for a class of asynchronous switched nonlinear systems with uncertain input delay

This paper studies the problems of stability and H∞ model reference tracking performance for a class of asynchronous switched nonlinear systems with uncertain input delay. First, it is assumed switched controller and corresponding piecewise Lyapunov function are unknown but the derivative of piecewise Lyapunov function has a condition; this condition implies that the nominal system (system without input delay and disturbance) is exponentially stable by any switched controller which satisfies this condition. With this assumption, a proper Lyapunov–Krasovskii functional is constructed. By employing this new functional and average dwell time technique, the delay-dependent input-to-state stability criteria are derived under a certain delay bound; in addition, a mechanism which finds the upper bound of input delay is proposed. Finally, a kind of state feedback control law which fulfils condition of aforesaid piecewise Lyapunov function is introduced to guarantee the input-to-state stability and H∞ model reference tracking performance. Simulation examples are presented to demonstrate the efficacy of results.     

New approaches to positive observer design for discrete-time positive linear systems

This paper aims at providing new design approaches for positive observers of discrete-time positive linear systems based on a construction method of linear copositive Lyapunov function for positive systems. First, an efficient positive observer design approach is proposed by using linear programming such that the observer error system is exponentially stable. Furthermore, an interval observer design is proposed for uncertain positive systems. Then, the results are extended to positive time delay systems. In contrast with the previous design approaches, the new design method provides a general observer design with lower computational burden. Finally, three comparison examples are given to show the merit of the new design approach.     

Practical exponential stability of switched homogeneous positive nonlinear systems with stable and unstable modes

In this article, we primarily investigate practical exponential stability of switched homogeneous positive nonlinear systems (SHPNSs) that have partial unstable modes and perturbation. First of all, the max-separable Lyapunov function (MSLF) technique and a special pre-setting switching sequence are used to define a number of significant stability conditions that ensure the state trajectories of the system converge to a confined region in continuous-time and discrete-time domains. In addition, the key results include several earlier findings as special situations and can be directly applied to general switched systems, not necessarily positive. Finally, two important instances are provided to further illustrate the validity of the theoretical findings.     

Stability analysis for nonlinear switched singular systems via T-S fuzzy modeling

The stability for discrete nonlinear switched singular systems with unstable subsystems is investigated. First, by constructing an appropriate multiple discontinuous Lyapunov function, and utilizing the characteristics of mode-dependent average dwell time switching signals, new stability results for nonlinear switched singular system are established. Then, we adopt the T-S fuzzy modeling method to approximate the nonlinear switched singular systems and get general stability conditions in forms of linear matrix inequalities. Compared to the current results, our technique is more flexible and we also get tighter dwell time boundaries. Furthermore, a numerical example demonstrates the effectiveness of the proposed method.     

Hybrid control strategy for positive switched delay systems with unstable modes

In this paper, the problem of hybrid control strategy (HCS) for time-varying delay positive switched linear systems (PSLS) with unstable modes is studied. Firstly, the HCS, which includes minimum switching strategy and discretized state feedback controller, is applied to PSLS with time-varying delay for the first time. Secondly, by using the discretized multiple linear copositive Lyapunov-Krasovskii functional, a sufficient condition of globally uniformly asymptotically stable (GUAS) under the HCS is given. Finally, the HCS is extended to discrete-time positive switched time delay systems, and a delay independent stabilization condition is obtained in the discrete system. The effectiveness of the HCS is verified by two simulation examples.     

Event-triggered controller design for positive T-S fuzzy systems with random time-delay

This paper is concerned with the design of event-triggered controller for positive Takagi-Sugeno (T-S) fuzzy systems with a random time-delay. The random time-delay is described as a Markov process. A controller switched at different event-triggered instant is proposed. By constructing a new event-triggered instant-dependent linear co-positive Lyapunov function, the design criteria of event-triggered controller is derived to ensure the positivity and stability of the closed-loop system. These criteria can be solved by linear programming (LP) technique. A positive lower bound on the inter-execution time is ensured, which means that there is Zeno-free phenomenon. Finally, the simulation has demonstrated the effectiveness and merit of the proposed results.     

Actuator saturation control of continuous-time positive switched T–S fuzzy systems

The saturated control problem is investigated for positive switched Takagi–Sugeno (T–S) fuzzy systems with partially controllable subsystems in this paper. Based on the parallel distribution compensation (PDC) algorithm and the convex hull technology, new fuzzy control schemes are proposed for continuous-time positive switched T–S fuzzy systems (PSTSFSs) with actuator saturation. By the multiple linear co-positive Lyapunov function and the slow-fast combined mode-dependent average dwell time (MDADT) approach, sufficient conditions for the stability of continuous-time closed-loop PSTSFSs are developed, which is an extension of the results in previous literature. Furthermore, the least conservative estimation of the attraction domain of PSTSFSs is transformed into an optimization problem. Finally, three simulation examples are given to illustrate the effectiveness of the proposed saturated control schemes.     

Multiple switching-time-dependent discretized Lyapunov functions/functionals methods for stability analysis of switched time-delay stochastic systems

This paper presents novel approaches for stability analysis of switched linear time-delay stochastic systems under dwell time constraint. Instead of using comparison principle, piecewise switching-time-dependent discretized Lyapunov functions/functionals are introduced to analyze the stability of switched stochastic systems with constant or time-varying delays. These Lyapunov functions/functionals are decreasing during the dwell time and non-increasing at switching instants, which lead to two mode-dependent dwell-time-based delay-independent stability criteria for the switched systems without restricting the stability of the subsystems. Comparison and numerical examples are provided to show the efficiency of the proposed results.     

Stability of mode-dependent linear switched singular systems with stable and unstable subsystems

This paper studies the E-exponential stability of mode-dependent linear switched singular systems with stable and unstable subsystems. First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established. Considering the feature of mode-dependent average dwell time switching, we adopt the switching strategy where fast switching and slowing switching are respectively applied to unstable and stable subsystems. Compared with the existing results, our approach is more flexible and tighter bounds can be obtained. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.     

Input-to-state stability of switched stochastic delayed systems with Lévy noise

This paper focuses on input-to-state stability of a class of switched stochastic delayed systems which are drived by Lévy noise. By multiple Lyapunov function and average dwell time approach, the sufficient conditions of the ψ^λ(t)-weighted input-to-state stability can be obtained if all the subsystems are input-to-state stable. Then utilizing comparison principle and the method of constant variation, the sufficient criteria of the e^λt-weighted input-to-state stability of the switched stochastic delayed systems containing both input-to-state stable subsystems and non-input-to-state stable subsystems can also be derived. Finally, an example is given to illustrate the effectiveness of the proposed results.     

Impulsive observer design for a class of switched nonlinear systems with unknown inputs

This paper investigates hybrid observer design of a class of unknown input switched nonlinear systems. The distinguishing feature of the proposed method is that the stability of all subsystems of the error switched systems is not necessarily required. First, an output derivative-based method and time-varying coordinate transformation are considered to eliminate the unknown input. Then in order to maintain a satisfactory estimation performance, an impulsive full-order and switched reduced-order observer are developed with a pair of upper and lower dwell time bounds and constructing time-varying Lyapunov functions combined with convex combination technique. In addition, the time-varying Lyapunov functions method is also used to analyze the stability of a class of error switched nonlinear systems with stable subsystems. Finally, two examples are presented to demonstrate the effectiveness of the proposed method.