New stability conditions for discrete-time switched positive linear time-varying systems

In this note, we will devote to investigate the stability of discrete-time switched positive linear time-varying systems (PLTVSs). Firstly, a new asymptotic stability criterion of discrete-time PLTVSs is obtained by using time-varying copositive Lyapunov functions (TVCLFs) and this criterion is then extended to the switched case based on the multiple TVCLFs. Furthermore, the sufficient conditions are derived for stability of discrete-time switched PLTVSs with stable subsystems by means of function-dependent average dwell time and function-dependent minimum dwell time. In addition, the stability sufficient conditions are drawn for the switched PLTVSs which contain unstable subsystems. It is worth noting that the difference of TVCLFs and multiple TVCLFs are both relaxed to indefinite in our work. The theoretical results obtained are verified by two numerical examples.     

Model-based event-triggered control of switched discrete-time systems

In this paper, we aim to study model-based event-triggered control for a class of uncertain switched discrete-time systems composed of stabilizable and unstabilizable subsystems. A nominal model is introduced at the controller side to form a dynamic controller so that it can provide a kind of approximate estimate of the system state for system input even the overall switched discrete-time system is running in open-loop during any two consecutive event-triggered instants. By using multi-Lyapunov function method and the average dwell time switching strategy, stability conditions given in linear matrix inequality form for the closed-loop switched discrete-time system are derived. The design of control gains is also given. Finally, a numerical example and a physical example are provided to verify the effectiveness and usefulness 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.     

H∞ finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance

Finite-time stability concerns the boundness of system during a fixed finite-time interval. For switched systems, finite-time stability property can be affected significantly by switching behavior; however, it was neglected by most previous research. In this paper, the problems of finite-time stability analysis and stabilization for switched nonlinear discrete-time systems are addressed. First, sufficient conditions are given to ensure a class of switched nonlinear discrete-time system subjected to norm bounded disturbance finite-time bounded under arbitrary switching, and then the results are extended to H_∞ finite-time boundness of switched nonlinear discrete-time systems. Finally based on the results on finite-time boundness, the state feedback controller is designed to H_∞ finite-time stabilize a switched nonlinear discrete-time system. A numerical design example is given to illustrate the proposed results within this paper.     

Stabilization of discrete-time switched linear systems with time-varying delays via nearly-periodic impulsive control

In this work, impulsive stabilization problems of discrete-time switched linear systems with time-varying delays are studied. The sequence of impulsive instants is nearly-periodic, i.e., it is close to a periodic impulse and the distance between them is an uncertain bounded term. A time-varying Lyapunov function is introduced to characterize the information of delays, switching signals and impulses, and a stability criterion LMI-based is obtained without any restrictions on the stability of the subsystems. Several design schemes of reduced-order/full-order impulsive controllers with or without time-varying delays are developed. Finally, three numerical examples are provided to illustrate the effectiveness of the established results.     

Stability of discrete-time systems with time-varying delay based on switching technique

In this paper, the stability problem of discrete-time systems with time-varying delay is considered. Some new stability criteria are derived by using a switching technique. Compared with the Lyapunov–Krasovskii functional (LKF) approach, the method used in this paper has two features. First, a switched model, which is equivalent to the original system and contains more delay information, is introduced. It means that the criteria obtained by using the LKF method can be regarded as stability criteria for the switched system under arbitrary switching. Second, when the switching signal is known, the stability problem for the switched model under constrained switching is considered and piecewise LKFs are adopted to obtain stability criteria. Since constrained switching is less conservative than arbitrary switching if the switching signal is known, one can know that the obtained results in this paper are less conservative than some existing ones. Two examples are given to illustrate the effectiveness of the obtained results.     

Input-to-state stability of switched stochastic delayed systems with Lévy noise

This paper focuses on input-to-state stability of a class of switched stochastic delayed systems which are drived by Lévy noise. By multiple Lyapunov function and average dwell time approach, the sufficient conditions of the ψ^λ(t)-weighted input-to-state stability can be obtained if all the subsystems are input-to-state stable. Then utilizing comparison principle and the method of constant variation, the sufficient criteria of the e^λt-weighted input-to-state stability of the switched stochastic delayed systems containing both input-to-state stable subsystems and non-input-to-state stable subsystems can also be derived. Finally, an example is given to illustrate the effectiveness of the proposed results.     

On fault detection of discrete-time switched systems via designing time-varying residual generators

This paper investigates the problem of fault detection (FD) for discrete-time switched systems. Under a dwell time constraint, a switching rule that depends on the measured output is constructed for the system. Time-varying residual generators are designed such that the switched system is asymptotically stable and also with the detection performance under this switching rule. The advantages of the proposed technique are threefold: 1) It has the advantages of both slow switching and fast switching. 2) It can extend the classic design of time-invariant residual generator. 3) It can guarantee the switched system still has the desired fault detection performance even if all subsystems are without it. This feature reduces the performance requirements for each subsystem. A numerical example illustrates the effectiveness of the proposed method.     

Asynchronous switching for switched nonlinear input delay systems with unstable subsystems

We study the input-to-state stability (ISS) of switched nonlinear input delay systems under asynchronous switching. Our results apply to cases where some subsystems of the switched systems are not necessarily stable under the influence of input delay. By making a compromise among the matched-stable period, the matched-unstable period, and the unmatched period and allowing the increase of the Lyapunov–Krasovskii functional (LKF) on all the switching times, the extended stability criteria for switched delay systems in generally nonlinear setting are derived first. Then, we focus on switched nonlinear input delay systems where the presence of the input delay leads to the instability of some subsystems of it. By explicitly constructing input-to-state stable LKF, the sufficient conditions for ISS of switched nonlinear input delay systems under asynchronous switching are presented. Finally, two examples are given to illustrate the effectiveness of the proposed theory.     

Multiple switching-time-dependent discretized Lyapunov functions/functionals methods for stability analysis of switched time-delay stochastic systems

This paper presents novel approaches for stability analysis of switched linear time-delay stochastic systems under dwell time constraint. Instead of using comparison principle, piecewise switching-time-dependent discretized Lyapunov functions/functionals are introduced to analyze the stability of switched stochastic systems with constant or time-varying delays. These Lyapunov functions/functionals are decreasing during the dwell time and non-increasing at switching instants, which lead to two mode-dependent dwell-time-based delay-independent stability criteria for the switched systems without restricting the stability of the subsystems. Comparison and numerical examples are provided to show the efficiency of the proposed 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.     

Global stabilization of switched nonlinear systems in non-triangular form and its application

This paper investigates the problem of global stabilization of switched nonlinear systems in non-triangular form whose subsystems are not assumed to be asymptotically stabilizable. The use of multiple Lyapunov functions (MLFs) method permits removal of a common restriction in which the nonlinear structures in the non-switched nonlinear systems are restricted to a triangular structure when applying backstepping. Using the MLFs method and the adding a power integrator technique, we design state-feedback controllers for individual subsystems and construct a switching law to guarantee asymptotic stability of the closed-loop switched system. As an application of the proposed design method, the global stabilization problem of a continuously stirred tank reactor (CSTR) system and two inverted pendulums which cannot be handled by the existing methods is investigated.     

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.     

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.     

Dissipative observers for discrete-time nonlinear systems

This paper addresses the problem of designing a state observer for a class of nonlinear discrete-time systems using the dissipativity theory. We show that the dissipative observation methodology, originally proposed by one of the authors for continuous-time nonlinear systems, can be extended to the discrete-time case. For constructing a convergent observer, the methodology is applied to the nonlinear estimation error dynamics, which is decomposed into a discrete-time Linear Time-Invariant (LTI) subsystem in the forward loop, connected to a time-varying static nonlinearity in the feedback loop. In order to assure asymptotic stability of the closed-loop, complementary dissipativity conditions are imposed on each of the subsystems: (i) the static nonlinearity is required to be dissipative with respect to a quadratic supply rate, and (ii) the observer gains are designed such that the LTI system is dissipative with respect to a complementary supply rate. As in the continuous time framework, the proposed method includes as special cases, unifies and generalizes some observer design methods proposed previously in the literature. A great advantage of the Dissipative Observer Design Method proposed here is that it leads to Matrix Inequalities for the design of the observer gains, and these can be usually converted into Linear Matrix Inequalities (LMI’s). The results are illustrated using Chua’s Chaotic system.     

Impulsive observer design for a class of switched nonlinear systems with unknown inputs

This paper investigates hybrid observer design of a class of unknown input switched nonlinear systems. The distinguishing feature of the proposed method is that the stability of all subsystems of the error switched systems is not necessarily required. First, an output derivative-based method and time-varying coordinate transformation are considered to eliminate the unknown input. Then in order to maintain a satisfactory estimation performance, an impulsive full-order and switched reduced-order observer are developed with a pair of upper and lower dwell time bounds and constructing time-varying Lyapunov functions combined with convex combination technique. In addition, the time-varying Lyapunov functions method is also used to analyze the stability of a class of error switched nonlinear systems with stable subsystems. Finally, two examples are presented to demonstrate the effectiveness of the proposed method.     

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.     

Global finite-time stabilization for switched stochastic nonlinear systems via output feedback

This paper is concerned with the problem of global finite-time stabilization via output feedback for a class of switched stochastic nonlinear systems whose powers are dependent of the switching signal. The drift and diffusion terms satisfy the lower-triangular homogeneous growth condition. Based on adding a power integrator technique and the homogeneous domination idea, output-feedback controllers of all subsystems are constructed to achieve finite-time stability in probability of the closed-loop system. Distinct from the existing results on switched stochastic nonlinear systems, the delicate change of coordinates are introduced for dominating nonlinearities. Moreover, by incorporating a multiplicative design parameter into the coordinate transformations, the obtained control method can be extended to switched stochastic nonlinear systems with nonlinearities satisfying the upper-triangular homogeneous growth condition. The validity of the proposed control methods is demonstrated through two examples.     

Non-weighted H∞ performance for 2-D FMLSS switched system with maximum and minimum dwell time

This paper investigates the H_∞ performance of two-dimensional (2-D) switched system represented by Fornasini–Marchesini local state-space (FMLSS) model with maximum and minimum dwell time approach. By using the multiple Lyapunov function approach, and designing a set of switching signals subject to maximum and minimum dwell time characteristic, respectively, for all stable subsystems or both stable and unstable subsystems exist, we give the sufficient condition on exponential stability of the given switched system, and propose the sufficient condition which can guarantee that the given switched system is exponentially stable and has a specified H_∞ disturbance attenuation level γ. All the results obtained are on normal noise attenuation index of strictly non-weighted form, which are better than the existing results on weak one of weighted form from the physical point of view. Finally, numerical examples are presented to display the effectiveness of the proposed results.