Exponential stability of switched systems with impulsive effect

The exponential stability of a class of switched systems containing stable and unstable subsystems with impulsive effect is analyzed by using the matrix measure concept and the average dwell-time approach. It is shown that if appropriately a large amount of the average dwell-time and the ratio of the total activation time of the subsystems with negative matrix measure to the total activation time of the subsystems with nonnegative matrix measure is chosen, the exponential stability of a desired degree is guaranteed. Using the proposed switching scheme, we studied the robust exponential stability for a class of switched systems with impulsive effect and structure perturbations. Simulations validate the main results.     

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.     

Robust stability analysis of switched uncertain systems with multiple time-varying delays and actuator failures

In this paper,the robust stability issue of switched uncertain multidelay systems resulting from actuator failures is considered.Based on the average dwell time approach,a set of suitable switching signals is designed by using the total activation time ratio between the stable subsystem and the unstable one.It is first proven that the resulting closed-loop system is robustly exponentially stable for some allowable upper bound of delays if the nominal system with zero delay is exponentially stable under these switching laws.Particularly,the maximal upper bound of delays can be obtained from the linear matrix inequalities.At last,the effectiveness of the proposed method is demonstrated by a simulation example.     

Synchronization of complex switched delay dynamical networks with simultaneously diagonalizable coupling matrices

This paper studies local exponential synchronization of complex delayed networks with switching topology via switched system stability theory. First, by a common unitary matrix, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when all subnetworks are synchronizable, a delay-dependent sufficient condition is given in terms of linear matrix inequalities （LMIs） which guarantees the solvability of the synchronization problem under an average dwell time scheme. We extend this result to the case that not all subnetworks are synchronizable. It is shown that in addition to average dwell time, if the ratio of the total activation time of synchronizable and non-synchronizable subnetworks satisfy an extra condition, then the problem is also solvable. Two numerical examples of delayed dynamical networks with switching topology are given, which demonstrate the effectiveness of obtained results.     

Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable （While some switching models are probably unstabilizable）, an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

Design of two-layer switching rule for stabilization of switched linear systems with mismatched switching

A two-layer switching architecture and a two-layer switching rule for stabilization of switched linear control systems are proposed, under which the mismatched switching between switched systems and their candidate hybrid controllers can be allowed. In the low layer, a state-dependent switching rule with a dwell time constraint to exponentially stabilize switched linear systems is given; in the high layer, supervisory conditions on the mismatched switching frequency and the mismatched switching ratio are presented, under which the closed-loop switched system is still exponentially stable in case of the candidate controller switches delay with respect to the subsystems. Different from the traditional switching rule, the two-layer switching architecture and switching rule have robustness, which in some extend permit mismatched switching between switched subsystems and their candidate controllers.     

Analysis of robust stability for switched systems with multiple time-delays

The sufficient conditions of delay-dependent exponential stability for switched systems and robust exponential stability for uncertain switched systems with two time-delays are presented by using average dwell time method and free-weighting matrix method.The interaction between different time-delays is considered.The sufficient conditions do not need that every subsystem is stable.The designed methods of the switching law are also given.The sufficient conditions are given in the form of linear matrix inequalities that can be solved easily.The result is proven to be valid by the simulation at last.     

Input-to-state stability of switched nonlinear systems

The input-to-state stability （ISS） problem is studied for switched systems with infinite subsystems. By using multiple Lyapunov function method, a sufficient ISS condition is given based on a quantitative relation of the control and the values of the Lyapunov functions of the subsystems before and after the switching instants. In terms of the average dwell-time of the switching laws, some sufficient ISS conditions are obtained for switched nonlinear systems and switched linear systems, respectively.     

Robust H-infinity integral sliding mode control for a class of uncertain switched nonlinear systems

This paper develops a new method to deal with the robust H-infinity control problem for a class of uncertain switched nonlinear systems by using integral sliding mode control. A robust H-infinity integral sliding surface is constructed such that the sliding mode is robust stable with a prescribed disturbance attenuation level γ for a class of switching signals with average dwell time. Furthermore, variable structure controllers are designed to maintain the state of switched system on the sliding surface from the initial time. A numerical example is given to illustrate the effectiveness of the proposed method.     

Stability of Delayed Switched Systems With State-Dependent Switching

This paper investigates the stability of switched systems with time-varying delay and all unstable subsystems. According to the stable convex combination, we design a state-dependent switching rule. By employing Wirtinger integral inequality and Leibniz-Newton formula, the stability results of nonlinear delayed switched systems whose nonlinear terms satisfy Lipschitz condition under the designed state-dependent switching rule are established for different assumptions on time delay. Moreover,some new stability results for linear delayed switched systems are also presented. The effectiveness of the proposed results is validated by three typical numerical examples.     

Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach

We study the stability properties of switched systems consisting of both Hurwitz stable and unstable linear time-invariant subsystems using an average dwell time approach. We propose a class of switching laws so that the entire switched system is exponentially stable with a desired stability margin. In the switching laws, the average dwell time is required to be sufficiently large, and the total activation time ratio between Hurwitz stable subsystems and unstable subsystems is required to be no less than a specified constant. We also apply the result to perturbed switched systems where nonlinear vanishing or non-vanishing norm-bounded perturbations exist in the subsystems, and we show quantitatively that, when norms of the perturbations are small, the solutions of the switched systems converge to the origin exponentially under the same switching laws.     

PIECEWISE LYAPUNOV FUNCTIONS FOR SWITCHED SYSTEMS WITH AVERAGE DWELL TIME

The stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems are investigated by using piecewise Lyapunov functions incorporated with an average dwell time approach. It is shown that if the average dwell time is chosen sufficiently large and the total activation time ratio between Hurwitz stable and unstable subsystems is not smaller than a specified constant, then exponential stability of a desired degree is guaranteed. The above result is also extended to the case where nonlinear norm‐bounded perturbations exist.     

Stability of switched neural networks with time-varying delays

This paper investigates the globally exponential stability of switched neural networks with time-varying delays. By virtue of mode-dependent average dwell time, some significant criteria of exponential stability are obtained for delayed switched neural networks with only stable subsystems or with both stable subsystems and unstable subsystems. The proposed theoretical results could be utilized not only to verify the globally exponential stability of switched neural networks, but also to design appropriate switching signal to guarantee the globally exponential stabilization. They can explicitly reflect the effect of mode-dependent average dwell time on the stability of switched neural networks. In contrast to the previous results, the proposed criteria are more straightforward and effective in the real-world application. Three numerical examples are introduced to illustrate the effectiveness of the proposed results.

     

Stability analysis of switched systems under dynamical dwell time control approach

This article investigates the stability of a class of switched systems using dynamical dwell time approach. First, the condition for stability of switched systems whose subsystems are stable are presented with dynamical dwell time approach, which is shown to be less conservative in switching law design than dwell time approach. Then the proposed approach is extended to the switched systems with both stable and unstable subsystems. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.     

Input/output-to-state stability of switched nonlinear systems with an improved average dwell time approach

This paper investigates the input/output-to-state stable (IOSS) property of the switched systems under average dwell time (ADT) switching signals in two cases: 1) all of the subsystems are IOSS, 2) parts of the subsystems are IOSS, and proposes a number of new results on stability analysis. First, we present a new IOSS result for the switched nonlinear systems whose subsystems are IOSS with an improved ADT method. Second, extending the improved ADT method to unforced nominal switched nonlinear systems in which parts of subsystems are stable, we establish a new stability analysis result. IOSS property of switched nonlinear systems in which parts of subsystems are IOSS, we show that if the average dwell time is large enough and if the fraction of time where one of the non-IOSS system is active is not too big, then IOSS property of the switched system can be established. It should be pointed that the main results obtained in this paper have some advantages over the exiting ones. Finally, two illustrative examples with simulation verify the correctness and validity of our results.     

Unified mode‐dependent average dwell time stability criteria for discrete‐time switched systems

In this article, a unified mode‐dependent average dwell time (MDADT) stability result is investigated, which could be applied to switched systems with an arbitrary combination of stable and unstable subsystems. Combined with MDADT analysis method, we classified subsystems into two categories: switching stable subsystems and switching unstable subsystems. State divergence caused by switching unstable subsystems could be compensated by activating switching stable subsystems for a sufficiently long time. Based on the above considerations, a new globally exponentially stability condition was proposed for discrete‐time switched linear systems. Under the premise of not resolving the LMIs, the MDADT boundary of the new stability condition is allowed to be readjusted according to the actual switching signal. Furthermore, the new stability result is a generalization of the previous one, which is more suitable for the case of more unstable subsystems. Some simulation results are given to show the advantages of the theoretic results obtained.     

Stability of neutral stochastic switched time delay systems: An average dwell time approach

This paper considers a class of stochastic systems referred to as stochastic switched systems of neutral type with time‐varying delay, which combines switched systems with neutral stochastic systems. The systems consist of subsystems of two forms: (i) only stable subsystems and (ii) both stable subsystems and unstable subsystems. By establishing an integral inequality, the exponential stability in pth(p≥1)‐moment for such systems with only stable subsystems is first considered. Then, by using an average dwell time approach, the exponential stability in pth(p≥1)‐moment for the second form is addressed. An important finding of this study is that when the average dwell time is chosen to be sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable subsystems, the exponential stability in pth(p≥1)‐moment for such systems can be guaranteed. Two major advantages of these new results are that the differentiability or continuity of the delay function is not required compared with the existing results in the literature, and the proposed approaches can be used to consider the case when the neutral item and the stochastic perturbation are simultaneously presented. An example is provided to verify the effectiveness and potential of the theoretic results obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust exponential stability of uncertain discrete time impulsive switching systems with state delay

A new class of hybrid impulsive and switching models are introduced and their robust exponential stability and control synthesis are addressed. The proposed switched system is composed of stable subsystems and unstable subsystems, which not only involves state delay and norm-bounded time-varying parameter uncertainties, but also contains the impulsive switching effects between the subsystems. Based on the extension of the system dimension and the concept of average dwell time, a kind of practically useful switching rule is presented which guarantees the desired robust exponential stability. A switched state feedback controller is also given.     

Graph‐Based Dwell Time Computation Methods for Discrete‐Time Switched Linear Systems

Analytical computation methods are proposed for evaluating the minimum dwell time and the average dwell time guaranteeing the asymptotic stability of a discrete‐time switched linear system whose switchings are assumed to respect a given directed graph. The minimum and average dwell time can be found using the graph that governs the switchings, and the associated weights. This approach, which is used in a previous work for continuous‐time systems having non‐defective subsystems, has been adapted to discrete‐time switched systems and generalized to allow defective subsystems. Moreover, we present a novel method to improve the dwell time estimation in the case of bimodal switched systems. In this method, scaling algorithms to minimize the condition number have been used to give better minimum dwell time and average dwell time estimates.