Robust stability of positive switched systems with dwell time

This paper studies robust stability of positive switched systems (PSSs) with polytopic uncertainties in both discrete-time and continuous-time contexts. By using multiple linear copositive Lyapunov functions, a sufficient condition for stability of PSSs with dwell time is addressed. Being different from time-invariant multiple linear copositive Lyapunov functions, the Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. Then, robust stability of PSSs with polytopic uncertainties is solved. All conditions are solvable via linear programming. Finally, illustrative examples are given to demonstrate the validity of the proposed results.     

Robust H ∞ control of stochastic linear switched systems with dwell time

The theory of H _∞ control of switched systems is extended to stochastic systems with state‐multiplicative noise. Sufficient conditions are obtained for the mean square stability of these systems where dwell time constraint is imposed on the switching. Both nominal and uncertain polytopic systems are considered. A Lyapunov function, in a quadratic form, is assigned to each subsystem that is nonincreasing at the switching instants. During the dwell time, this function varies piecewise linearly in time following the last switch, and it becomes time invariant afterwards. Asymptotic stochastic stability of the set of subsystems is thus ensured by requiring the expected value of the infinitesimal generator of this function to be negative between switchings, resulting in conditions for stability in the form of LMIs. These conditions are extended to the case where the subsystems encounter polytopic‐type parameter uncertainties. The method proposed is applied to the problem of finding an upper bound on the stochastic L₂‐gain of the system. A solution to the robust state‐feedback control problem is then derived, which is based on a modification of the L₂‐gain bound result. Two examples are given that demonstrate the applicability of the proposed theory. Copyright © 2013 John Wiley & Sons, Ltd.     

Equivalence of several stability conditions for switched linear systems with dwell time

In this paper, several equivalent stability conditions for switched linear systems with dwell time are presented. Both continuous‐time and discrete‐time cases are considered. For the continuous‐time case, the conditions that are convex in system matrices are presented in terms of infinite‐dimensional linear matrix inequalities (LMIs), which are not numerically testable. Then, by adopting the sum of square (SOS) and piecewise linear approach, computable conditions are formulated in terms of SOS program and LMIs. Compared to the literature, less conservative results can be obtained through solving these conditions for the same polynomial degree or discretized order. For the discrete‐time case, the stability conditions, which are convex in system matrices, are numerically testable. The convexity comes at the price of increment of computational complexity. Furthermore, by adopting the convexification approach, sufficient stability conditions of switched linear systems with polytopic uncertainties are derived, both for continuous‐time and discrete‐time cases. At last, several examples are given to demonstrate the correctness and advantages of our results.     

Stability of linear discrete switched systems with delays based on average dwell time method

This paper deals with the problem of exponential stability for a class of linear discrete switched systems with constant delays.The switched systems consist of stable and unstable subsystems.Based on the average dwell time method, some switching signals will be found to guarantee exponential stability of these systems.The explicit state decay estimation is also given in the form of the solutions of linear matrix inequalities(LMIs).An example relating to networked control systems(NCSs) illustrates the effect...     

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.     

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.     

Asynchronously switched control of switched linear systems with average dwell time

This paper concerns the asynchronously switched control problem for a class of switched linear systems with average dwell time (ADT) in both continuous-time and discrete-time contexts. The so-called asynchronous switching means that the switchings between the candidate controllers and system modes are asynchronous. By further allowing the Lyapunov-like function to increase during the running time of active subsystems, the extended stability results for switched systems with ADT in nonlinear setting are first derived. Then, the asynchronously switched stabilizing control problem for linear cases is solved. Given the increase scale and the decrease scale of the Lyapunov-like function and the maximal delay of asynchronous switching, the minimal ADT for admissible switching signals and the corresponding controller gains are obtained. A numerical example is given to show the validity and potential of the developed results.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

A dwell time approach to the stability of switched linear systems based on the distance between eigenvector sets

In this article, a sufficient condition on the minimum dwell time that guarantees the stability of switched linear systems is given. The proposed method interprets the stability of switched linear systems through the distance between the eigenvector sets of subsystem matrices. Thus, an explicit relation in view of stability is obtained between the family of the involved subsytems and the set of admissible switching signals.     

Finite‐time stability of switched positive linear systems

This brief paper addresses the finite‐time stability problem of switched positive linear systems. First, the concept of finite‐time stability is extended to positive linear systems and switched positive linear systems. Then, by using the state transition matrix of the system and copositive Lyapunov function, we present a necessary and sufficient condition and a sufficient condition for finite‐time stability of positive linear systems. Furthermore, two sufficient conditions for finite‐time stability of switched positive linear systems are given by using the common copositive Lyapunov function and multiple copositive Lyapunov functions, a class of switching signals with average dwell time is designed to stabilize the system, and a computational method for vector functions used to construct the Lyapunov function of systems is proposed. Finally, a concrete application is provided to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Average dwell time based stability analysis for nonautonomous continuous‐time switched systems

Inspired by the idea of multiple Lyapunov functions and the average dwell time, we address the stability analysis of nonautonomous continuous‐time switched systems. First, we investigate nonautonomous continuous‐time switched nonlinear systems and successively propose sufficient conditions for their (uniform) stability, global (uniform) asymptotic stability, and global (uniform) exponential stability, in which an indefinite scalar function is utilized to release the nonincreasing requirements of the classical multiple Lyapunov functions. Afterwards, by using multiple Lyapunov functions of quadratic form, we obtain the corresponding sufficient conditions for (uniform) stability, global (uniform) asymptotic stability, and global exponential stability of nonautonomous switched linear systems. Finally, we consider the computation issue of our current results for a special class of nonautonomous switched systems (ie, rational nonautonomous switched systems), associated with two illustrative examples.     

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.     

A novel approach to L1 filter design for asynchronously switched positive linear systems with dwell time

In this paper, the L₁ filtering problem is studied for continuous‐time switched positive linear systems (SPLSs) with a small delay existing in the switching of the filter and the subsystem. Unlike the existing literature concerned with asynchronous problems of SPLSs, the synchronous and asynchronous filters will be designed separately, which implies less conservative results. By introducing a class of clock‐dependent Lyapunov function (CDLF), which jumps down when the modes of the filter or the subsystem change and may increase or decrease during the asynchronous interval, clock‐dependent sufficient conditions characterizing a nonweighted L₁‐gain performance of the filter error systems are established. Then, based on the L₁ analysis results, a pair of error‐bounding filters are designed to estimate the outputs of SPLSs. The filter gains can be obtained by solving a set of linear programming. Finally, two numerical examples are presented to show the effectiveness and advantages of the results.     

Delay‐dependent robust stability analysis for switched time‐delay systems with polytopic uncertainties

In this paper, sufficient conditions are provided for the stability of switched retarded and neutral time‐delay systems with polytopic‐type uncertainties. It is assumed that the delay in the system dynamics is time‐varying and bounded. Parameter‐dependent Lyapunov functionals are employed to obtain criteria for the exponential stability of the system in the form of linear matrix inequality (LMI). Free‐weighting matrices are then provided to express the relationship between the system variables and the terms in the Leibniz–Newton formula. Numerical examples are presented to show the effectiveness of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

Output regulation of switched linear multi-agent systems: an agent-dependent average dwell time method

The output regulation problem of switched linear multi-agent systems with stabilisable and unstabilisable subsystems is investigated in this paper. A sufficient condition for the solvability of the problem is given. Owing to the characteristics of switched multi-agent systems, even if each agent has its own dwell time, the multi-agent systems, if viewed as an overall switched system, may not have a dwell time. To overcome this difficulty, we present a new approach, called an agent-dependent average dwell time method. Due to the limited information exchange between agents, a distributed dynamic observer network for agents is provided. Further, a distributed dynamic controller based on observer is designed. Finally, simulation results show the effectiveness of the proposed solutions.     

Simultaneous reduced-order control and fault detection for discrete-time switched systems with relaxed persistent dwell time regularity

This paper focuses on the problem of simultaneous control and fault detection (FD) for discrete-time switched systems. The main goal is to develop a control/detection unit (CDU) associated with a switching law to control the system and detect faults simultaneously. When the switched systems are with partial measurable states, we directly use these states to construct partial control and FD signals. Next, the reduced-order CDU is designed to generate the other control and FD signals. Compared with existing results based on full-order CDU, the proposed results lead to less conservatism and reduce design complexity. The switching law is constructed in the frame of persistent dwell time (PDT) switching. A novel switching number constraint condition is introduced, which further relaxes the restrictions on the dwell time of switching processes of PDT. The less restriction on dwell time degrades the performance requirement of each subsystem and upgrades the degree of freedom for switching law design. Based on the proposed results, a class of nonweighted performance indexes is introduced to characterize the fault sensitivity and robustness. Finally, the effectiveness of the proposed method is illustrated by an example.     

Robust control and fault detection for continuous‐time switched systems subject to a dwell time constraint

This paper is concerned with the simultaneous robust control and fault detection problem for continuous‐time switched systems subject to a dwell time constraint. To meet the control and detection objectives under the constraint, the controller/detectors matching different time intervals are first constructed in an output feedback framework. A state‐dependent switching law that obeys the dwell time constraint is then designed such that the closed‐loop switched system is asymptotically stable and also with the robust and detection performance. Further, the proposed switching law is dependent only on the partial measurable states of the closed‐loop system, which is applicable when the states of system mode are fully unavailable. Thus, our result extends the existing ones in state‐dependent switching and state‐feedback frameworks. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

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 switched systems with limiting average dwell time

In this paper, the stability problems of a class of switched systems with limiting average dwell time (ADT) are concerned. The common ADT is improved to a form of limit, and the limiting ADT even can be infinite. Different from previous results, in order to take full advantage of stabilizing switchings, switching‐dependent switched parameters are first used to describe the relationship of two consecutive activated switchings. Then, stability criteria of switched systems with limiting ADT are established, which are less conservative comparing with the existing results. Additionally, some stability criteria of switched systems including continuous‐time and discrete‐time cases are derived. Finally, the validity and effectiveness of our results are elucidated by numerical examples.