Exponential stability of switched positive systems with unstable modes and distributed delays

This paper studies the exponential stability of switched positive system consisting of unstable subsystems with distributed time-varying delay. Unlike the existing results concerning delays, switching behaviors dominating the system can be either stabilizing or destabilizing. The distributed delay is supposed to be slowly varying and upper-bounded. To tackle the difficulties brought by both the switching behaviors with mixed effects and the distributed delay, a multiple discretized Lyapunov–Krasovskii functional is employed to derive sufficient conditions for the exponential stability of the system. Specifically, by adjusting the ratio of the stabilizing switching behaviors, the state divergence caused by unstable subsystems and destabilizing switching behaviors can be compensated. Simulation examples demonstrate the effectiveness of the 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.     

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.     

On delay-independent stabilization analysis for a class of switched time-delay systems with the state-driven switching strategy

In this paper, in view of the state-driven switching method and Lyapunov stability theorem, the sufficient stability conditions with delay-independence have been derived for switched time-delay system under a simple switching rule. The particular method can be applied to cases, whose all individual subsystems are unstable. Finally, one example is performed to illustrate the feasibility and effectiveness of the proposed techniques.     

Control of multi-scroll Chen system

This paper investigates the chaos control problem of a new multi-scroll chaotic system. Two nonlinear control methods are studied, namely high-order and predictive types of control. The proposed methodologies offer the possibility of stabilizing unstable periodic orbits and unstable equilibrium points from the state equations. For this purpose, we apply the high-order control method for stabilizing a desired unstable periodic orbit, while the predictive control method is applied for stabilization problem of unstable equilibrium points. In particular, these approaches are effective and easy to be implemented since we only need to apply small perturbations to the system dynamics. The multi-scroll Chen system is used as representative example to show the working principle of these methods. Numerical simulation results indicate the potential engineering applications of the proposed control methods for various multi-scroll chaos-based practical applications.     

Energy analysis of a class of state-dependent switched systems with all unstable subsystems

In this paper, we investigate the stability and periodicity of a class of state-dependent switched systems with all unstable subsystems by means of energy analysis. We firstly transform the unstable subsystems reversibly into the form of second order mechanical systems, and then construct energy functions by calculating the sum of kinetic and potential energies of each subsystem. After that, two switching lines, derived from the lines with the largest and smallest energy drops, make the stable phase trajectory approach to the equilibrium point at the fastest speed. In addition, we explore possible dynamic behaviors of the switched system under a pair of switching line including asymptotic stability, instability and periodicity. Furthermore, based on the bisection method and nested intervals theorem, we design a state-dependent switching law, which makes the switched system periodic initiated from a stable switching law. Finally, numerical simulation examples are provided to illustrate the effectiveness and less conservativeness of the proposed method with practical significance.     

Robust stability of switched delayed logical networks with all unstable modes

Logical network is a special kind of positive systems with finite state space and logical operations. This paper analyzes the robust stability of switched delayed logical networks (SDLNs) with all unstable modes. The main mathematical tool is the semi-tensor product of matrices, which is used to convert the dynamics of SDLNs with all unstable modes into an equivalent algebraic state space representation. Following the algebraic state space representation framework, the concept of robustly stable parts is proposed, based on which, a new criterion is established for the robust stability of SDLNs with all unstable modes. Finally, two examples are given for the verification of main results.     

On delay-dependent stabilization analysis for the switched time-delay systems with the state-driven switching strategy

For the switched time-delay systems, the delay-dependent stability criteria will be derived under a state-driven switching law. A linear state transformation was introduced to transfer the switched time-delay system. On delay dependent stabilization analysis, we apply the Lyapunov-Krasovskii functionals to analyze the stabilization of the switched time-delay systems. This method can be applied to cases when all individual switched systems are unstable. Finally, one example is exploited to illustrate the proposed schemes.     

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.     

Controller design for discrete-time hybrid linear parameter-varying systems with semi-Markov mode switching

The paper is concerned with the stability and stabilization problems for a family of hybrid linear parameter-varying systems with stochastic mode switching. The switching phenomenon is modeled by a semi-Markov stochastic process which is more generalized than a Markov stochastic process. With the construction of a Lyapunov function that depends on both the parameter variation and operating mode, numerical testable stability and stabilization criteria are established in the sense of σ-error mean square stability with the aid of some mathematical techniques that can eliminate the terms containing products of matrices. To test the effectiveness of the designed stabilizing controller, we apply the developed theoretical results to a numerical example.     

Stability criteria of a class of nonlinear impulsive switching systems with time-varying delays

This paper investigates stability problems of a class of nonlinear impulsive switching systems with time-varying delays. Based on the common Lyapunov function method and Razumikhin technique, several stability criteria are established for nonlinear impulsive switching systems with time-varying delays. Our results show that switching systems can be stabilized by impulsive switching signals even if the system matrices are all unstable. In the absence of impulses, some of our results reduce to similar stability criteria for nonimpulsive switching systems in some recent research articles. Several examples with simulations are given to illustrate the efficiency of our results.     

Fixed-time stability of dynamical systems with impulsive effects

The present article is concerned with the fixed-time stability(FxTS) analysis of the nonlinear dynamical systems with impulsive effects. The novel criteria have been derived to achieve stability of the non-autonomous dynamical system in fixed-time under the effects of stabilizing and destabilizing impulses. The fixed time stability analysis due to the presence of destabilizing impulses in dynamical system, that leads to behavior of perturbing the systems’ stability, have not been addressed much in the existing literature. Therefore, two theorems are constructed here, for stabilizing and destabilizing impulses separately, to estimate the fixed-time convergence precisely by using the concept of Lyapunov functional and average impulsive interval. The theoretical derivation shows that the estimated fixed-time in this study is less conservative and more accurate as compared to the existing FxTS theorems. Further, the theoretical results are applied to the impulsive control of general neural network systems. Finally, two numerical examples are given to validate the effectiveness of the theoretical results.     

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.     

Robust H∞ filtering for switched stochastic systems under asynchronous switching

This paper investigates the problem of robust H_∞ filtering for switched stochastic systems under asynchronous switching. The so-called asynchronous switching means that the switching between the filters and system modes is asynchronous. The aim is to design a filter ensuring robust exponential mean square stability and a prescribed H_∞ performance level for the filtering error systems. Based on the average dwell time approach and piecewise Lyapunov functional technique, sufficient conditions for the existence of the robust H_∞ filter are derived, and the proposed filter can be obtained by solving a set of LMIs(linear matrix inequalities). Finally, a numerical example is given to show the effectiveness of the proposed approach.     

Stabilizing two-dimensional stochastic systems through sliding mode control

This paper presents a stabilizing control for two-dimensional stochastic differential equations. The stability concept in this study is the stability in probability. To ensure such a stability, the control is designed based on the sliding mode technique, and applied to account stochastic systems. This finding has a practical implication—the proposed control can be used to stabilize a real-time automotive electronic throttle valve. The proposed approach is verified by data collected from experiments.     

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.     

Stability analysis of stochastic switching singular systems with jumps

This paper deals with the stability problem of a class of stochastic switching singular systems with jumps, where all the continuous subsystems can be unstable. By introducing multiple time-varying discretized Lyapunov functions and designing novel switching signals via mode-dependent dwell times, sufficient conditions ensuring mean square and almost sure stabilities of the considered systems are obtained. On the basis of these findings, some existing results in the literature are further generalized. We finally provide a numerical example to verify the applicability of the proposed theoretical results.     

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.     

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.