A novel approach to stability analysis for switched positive linear systems

This paper is concerned with exploring less conservative stability conditions for a class of switched positive linear systems. A switched matrix-parameterized copositive Lyapunov function (SMPCLF) is first introduced, where “matrix-parameterized” implies that the parameters of the constructed Lyapunov function are distributed in a matrix, which is different from the traditional vector-parameterized copositive Lyapunov function. Based on the proposed SMPCLF, a new stability criterion is derived for the underlying systems under arbitrary switching. Furthermore, by performing higher order relaxations in the SMPCLF and its time difference by positive states, the conservativeness can be further reduced. A numerical example is given to demonstrate the effectiveness and advantages of the obtained theoretical results.     

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.     

Stabilization of discrete-time switched singular systems with state,output and switching delays

This paper is concerned with state feedback stabilization of discrete-time switched singular systems with time-varying delays existing simultaneously in the state, the output and the switching signal of the switched controller. On the basis of equivalent dynamics decomposition and Lyapunov–Krasovskii method, exponential estimates for the response of slow states of the closed-loop subsystems running in asynchronous and synchronous periods are first given. Exponential estimates for the response of fast states are also provided by establishing an analytic equation to solve the fast states and using some algebraic techniques. Then, by employing the obtained exponential estimates and the piecewise Lyapunov function approach with average dwell time (ADT) switching, sufficient conditions for the existence of a class of stabilizing switching signals and state feedback gains are derived, which explicitly depend on upper bounds on the delays and a lower bound on the ADT. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained theoretical 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.     

Positive consensus for multi-agent systems with average dwell time switching

This paper considers the positive consensus for a class of multi-agent systems (MASs) with average dwell time (ADT) switching. First, sufficient and necessary conditions are derived for preserving the positivity of the closed-loop MASs. Second, the performance analysis of the consensus error system is accomplished by using the multiple Lyapunov functions (MLFs) approach, and an ADT switching technique designs the corresponding controlled switching signal. Then, both leaderless and leader-following positive consensus are achieved. Furthermore, to reduce the computational complexity, a novel leader-following positive consensus criterion is derived in the form of linear programming (LP). Finally, simulation examples are given to illustrate the effectiveness of the proposed method.     

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.     

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.     

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 adaptive neural control for state-constrained switched nonlinear systems based on nonlinear shifting function technique

This paper addresses the event-triggered tracking control design for state-constrained switched nonstrict feedback nonlinear systems. With the help of a time-varying nonlinear shifting function (TVNSF) introduced into the switched nonlinear system, the proposed solution is seen as a unified tool regardless of whether the constraint conditions are state constraints, output constraint, or even no constraint. Also, by allowing the triggering error to vary with the switching signal in time, the negative effects of the mismatch between the individual controller and the subsystem on system performance are trumped. Moreover, by using constructed individual Lyapunov function that depends on the lower bound of the control gain function of individual subsystem, a novel switching signal satisfying the average dwell time (ADT) is provided to ensure the boundedness of all variables in the closed-loop system. Finally, the proposed theory is carried over into a mass-spring-damper system to verify its effectiveness.     

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.     

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.     

Observer design for a class of nonlinear fractional-order systems with unknown input

A full order fractional-order observer is designed for a class of Lipschitz continuous-time nonlinear fractional-order systems with unknown input. Sufficient conditions of existence for the designed observer and stability of state estimation error system are developed by reconstructing state and using general quadratic Lyapunov function. By applying fractional-order extension of Lyapunov direct method, the stability of the fractional-order state estimation error system is analyzed. Due to the conditions involving a nonlinear matrix inequality, a new sufficient condition with linear matrix inequality (LMI) is reformulated, which makes the full order fractional-order observer implemented easily by using Matlab LMI toolbox. Examples are taken to show the effectiveness of the proposed approach by numerical simulations.     

Stabilizing a class of mixed states for stochastic quantum systems via switching control

The global stabilization of a class of mixed states for finite dimensional stochastic quantum systems with degenerate measurement operator is investigated in this paper. We construct a measurement operator and control Hamiltonian that make the target state one of the system equilibria. Based on the proposed Lyapunov function, a control law is designed following Lyapunov’s method to steer system state to the target mixed state from an initial state in the convergence domain, which is obtained through the analyses of invariant set based on LaSella’s invariance principle. When the initial state isn’t in the convergence domain, a constant control is used to steer the system state so that it enters the convergence domain in finite time. The constant control and the control designed by Lyapunov method compose a switching control strategy, which can steer system state to the target mixed state from any arbitrary state in the state space, i.e., the target mixed state is globally stable under the switching control. The convergence of the switching control is proved based on state sample trajectories. Moreover, the numerical experiments on a three dimensional stochastic quantum system are performed to demonstrate the effectiveness of switching control.     

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.     

Stabilization of discrete-time switched singular time-delay systems under asynchronous switching

This paper is concerned with the problem of state feedback stabilization of a class of discrete-time switched singular systems with time-varying state delay under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the subsystems. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence of a class of stabilizing switching laws is first derived to guarantee the closed-loop system to be regular, causal and exponentially stable in the presence of asynchronous switching. The stabilizing switching laws are characterized by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.     

Exponential stabilization of switched linear systems subject to actuator saturation with stabilizable and unstabilizable subsystems

This paper is concerned with the exponential stabilization of switched linear systems subject to actuator saturation with both stabilizable subsystems and unstabilizable subsystems for continuous-time case and discrete-time case, respectively. Sufficient conditions for the exponential stabilization under dwell time switching under the cases of continuous-time and discrete-time are established by using a novel class of multiple time-varying Lyapunov function. The existence conditions for stabilizing controllers are presented in terms of linear matrix inequalities (LMIs) for the continuous-time case and the discrete-time case, respectively. Two optimization problems are proposed for obtaining the maximal attraction region. The problem of exponential stabilization for switched system subject to actuator saturation with asynchronous switching controller is also studied. Several numerical examples are presented to prove the validity of the obtained results.     

Hybrid state observer‐based event‐triggered control for switched linear systems with quantized input

The paper investigates the design of hybrid state observer-based event-triggered controller for switched linear systems subject to quantized input and unknown but bounded additional disturbance and measurement noise. Firstly, by introducing a hybrid state observer and constructing a mode-dependent event-triggered mechanism, we design event-triggered controller for the considered switched linear systems. Then, by modeling the closed-loop system as an augmented asynchronous switched time-delay system, we deal with the asynchronous control problem caused by the switching between two consecutive trigger instants for the switched linear system. Thirdly, based on merging signal technique and multiple Lyapunov functional method, we obtain the sufficient criteria to guarantee the stability of the switched system when the switching signal meets an average dwell time condition, and further establish the hybrid observer-based event-triggered controller gains. Finally, a simulation example illustrates the validity of the results.     

Stochastic finite-time stabilization for discrete-time positive Markov jump time-delay systems

In this paper, the problems of stochastic finite-time stability and stabilization of discrete-time positive Markov jump systems are investigated. To deal with time-varying delays and switching transition probability (STP), stochastic finite-time stability conditions are established under mode-dependent average dwell time (MDADT) switching signal by developing a stochastic copositive Lyapunov-Krasovskii functional approach. Then a dual-mode dependent output feedback controller is designed, thus stochastic finite-time stabilization is achieved based on linear programming technique. Finally, two examples are given to show the effectiveness of our results.     

Finite-time asynchronous filtering for switched linear systems with an event-triggered mechanism

This paper investigates the event-triggered finite-time H_∞ filtering for a class of continuous-time switched linear systems. Considering that the system may switch within an inter-event interval, the asynchronous problem is taken into account for the system and filter modes. By adopting the average dwell time (ADT) technique and multiple Lyapunov functions, new conditions are obtained to guarantee that the filtering error system is finite-time bounded with a prescribed disturbance attenuation performance. Further, the finite-time H_∞ filter together with event-triggered mechanism is co-designed for the switched linear systems. Finally, a numerical example is provided to demonstrate the effectiveness of the method proposed in this paper.