Robust Stability and Stabilization of Switched Systems with Dwell Time

Sufficient conditions are derived for the robust stability of discrete-time, switched, linear systems with dwell time in the presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each of the subsystems. This function is allowed to be time-varying and piecewise linear during the dwell time and it becomes time invariant afterwards. Asymptotic stability conditions are obtained in terms of linear matrix inequalities for the nominal set of subsystems. These conditions are then extended to the case where the subsystems encounter polytopic type parameter uncertainties. The developed method is applied to l 2-gain analysis where a bounded real lemma is derived, and to H ∞ control and estimation, both for the nominal and the uncertain cases.

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

This work outlines a practical process, based on dwell times, for ensuring the stability of non-linear switched systems with a different stable equilibrium for each mode. Relevant Lyapunov theorems are discussed, followed by a derivation of an algorithm that determines dwell times for each system mode for guaranteeing stability. An analysis of algorithm parameters and potentially troublesome systems is then included. Finally, two examples are drawn from power electronics and electric machinery and processed by the proposed method. Application of this approach will both grant confidence in switched system stability and warn of possible unstable system behavior when appropriate.

Robust stability of switched positive linear systems with interval uncertainties via multiple time-varying linear copositive Lyapunov functions

Path-complete methods utilize a set of positive definite functions and a specially constructed graph in order to evaluate, among others, stability of switching systems. This tool is shown to be general, e.g., path-complete criteria are universal for linear switching systems and quadratic templates. In this work, we extend the approach to polyhedral Lyapunov functions, and introduce a simple parameterization that can be sufficient for stability analysis. Moreover, we indicate ways of obtaining less conservative stability criteria by partial graph extensions, all evaluated by solving Linear Programs (LPs).

The use of avionics networks for flight control system has reduced aircraft weight, system installation, integration and maintenance costs as well as introduced time delays and structured uncertainties. This paper studies the robust stability and stabilisation problem of uncertain networked flight control system with random time delays. Some new sufficient criteria with simple mathematical forms for robust stability and stabilisation of networked flight control system are derived by a free weighting matrix approach. A memoryless state feedback robust controller and the maximum bound of time delay are obtained based on the solution of linear matrix inequalities (LMIs) which are established by the Lyapunov-Krasovskii stability theory. Numerical examples are given to illustrate the effectiveness of design procedure.

This article investigates the robust exponential stabilization of a class of switched positive systems with uncertainties by co‐designing controllers for subsystems and dwell time switching strategy. The uncertainties refer to interval uncertainties. One of the distinguishing features is that none of the forced individual subsystems is assumed to be stabilized. The other feature is that the stabilization for the unforced switched system can not be solvable by designing dwell time switching signal. First, for switched positive systems without uncertainties, a type of multiple time‐varying co‐positive Lyapunov functions is used, and the computable sufficient conditions on the state feedback controller for each subsystem are derived in the framework of dwell time strategy. Moreover, the exponential stabilization problem is solved by confining the lower and upper bounds of the dwell time, restricting the upper bound of derivative of considered Lyapunov function of the active subsystem, and decreasing the Lyapunov function values at successive switching instants of the overall switched system. Then, the results are extended to the robust exponential stabilization case for switched positive systems with uncertainties, and sufficient conditions are given in the same framework of the dwell time for that of the forced switched systems by further designing state feedback controllers. Finally, illustration examples are given to illustrate the effectiveness of the proposed methods.

This paper addresses the stability, as well as an upper bound on the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain of linear switched state-multiplicative stochastic systems. The switching law is assumed to obey a dwell time constraint. The results are achieved by assigning each subsystem its own Lyapunov function. The Lyapunov function is time varying during the dwell time and becomes time invariant afterwards. The Lyapunov function is required to be non-increasing at the switching instances. If the dwell time approaches zero-the standard quadratic conditions are recovered. Stability and performance are guaranteed in the mean-square sense by requiring the expected value of the Lyapunov function to decrease, both for nominal systems and systems that entail polytopic-type parameter uncertainties. The theory is then extended to state-feedback controller design. The conditions are all given in terms of linear matrix inequalities.

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.

Switched linear systems with dwell time constraints constitute a very popular class of switched systems. In this study, impact of uncertain parameters on the stability of Parametrised switched linear (SL) systems with dwell time is investigated. Sufficient conditions are first proposed to analyse the global asymptotic stability of SL systems with dwell time. The conditions are then generalised to the case with parametric uncertainties to determine upper bounds on parametric uncertainties such that robust asymptotic stability is assured under dwell time constraints. Finally, an off-line computational algorithm is proposed to enlarge the estimated stability intervals. Effectiveness and simplicity of the computations are shown by a numerical example.

The authors consider the class of nonnegative discrete-time systems and analyze the matrix theoretical results involving nonnegative maintainability with respect to the problem of robust stabilization of interval discrete systems. Using state feedback, a controller which may include an observer such that the closed-loop interval system is robustly stabilized and at the same time becomes nonnegative is designed. The procedure is to characterize the set of robust controllers by means of matrix inequalities, and to choose a suitable controller by solving an optimization algorithm. >

In this paper, the command tracking problem for uncertain Euler-Lagrange (EL) systems with multiple partial loss of effectiveness (PLOE) actuator faults is considered. Compared to the existing fault-tolerant controllers for Euler-Lagrange systems, the proposed adaptive controller accounts for parametric uncertainties in both the system parameters and multiple time-varying actuator fault parameters. The closed-loop fault-tolerant system is treated as a switched dynamical system, and a switched system stability is established using multiple Lyapunov functions. It is shown that all signals are bounded in each sub-interval and at the switching instances, and asymptotic tracking can be obtained for a finite number of fault occurrences. Finally, a simulation example on a robotic manipulator is presented to show the effectiveness of the proposed method.

The standard approach to the problem of controlling linear systems with large parameter uncertainty is to seek a controller that stabilizes the system and achieves a required performance over the whole polytope of uncertainty. In the case where the latter polytope is large, the design may become very conservative. We present an alternative approach where the uncertainty polytope is divided into overlapping smaller regions and where each of these regions is assigned to a separate subsystem. Assuming that there is online information on which of the regions the parameters of the system move to, a recently developed method for H∞ design of switched system with dwell time is applied. A Lyapunov Function (LF) in a quadratic form, which is non-increasing at the switching instants, is assigned to each subsystem. This function is used to determine the stability and to find a bound on the L2 -gain of the switched system. The obtained results are used to solve the corresponding robust H∞ state-feedback and static output-feedback control problems.

This paper is concerned with the problem of robust stabilization for a class of uncertain two-dimensional (2D) discrete switched systems with state delays under asynchronous switching. The asynchronous switching here means that the switching instants of the controller experience delays with respect to those of the system. The parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee the exponential stability. The dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach