Input-Output Finite-Time Stability of Discrete-Time Impulsive Switched Linear Systems with State Delays

This paper is concerned with the problem of input-output finite-time stability (IO-FTS) for discrete impulsive switched systems with state delays. Sufficient conditions are presented for the existence of IO-FTS for such systems under the cases of certain switching, arbitrary switching, and uncertain switching. All the obtained results are formulated in a set of linear matrix inequalities (LMIs). Two numerical examples are given to illustrate the effectiveness of the proposed results.     

Exponential Stability of Switched Systems with Unstable Subsystems: A Mode-Dependent Average Dwell Time Approach

This article studies the exponential stability of switched systems with unstable subsystems. By using the multiple Lyapunov function (MLF) method combined with mode-dependent average dwell time (MDADT) techniques, less conservative exponential stability conditions are derived in terms of a set of solvable linear matrix inequalities (LMIs). Compared to the existing results, unstable subsystems are considered based on MDADT in this paper. It is revealed that switched systems can be exponentially stable under slow switching schemes and also in the presence of fast switching of unstable subsystems. A numerical example and its simulations are also given to verify the effectiveness of the proposed method.     

Control Synthesis of Discrete-Time Switched Linear Systems with Input Saturation Based on Minimum Dwell Time Approach

This paper investigates the control synthesis problem for discrete-time switched linear systems with input saturation. Based on minimal dwell time approach, state feedback controller and dynamic output feedback controller are respectively designed in terms of LMIs. The corresponding optimization problems are formulated to obtain a bigger attractive region. Compared with previous results, our proposed results can recover part of attractive region. Finally, numerical example is presented to illustrate the effectiveness and value of our obtained results.     

Robust Stabilization of Discrete-Time Positive Switched Systems with Uncertainties and Average Dwell Time Switching

This paper studies robust problems of a class of discrete-time positive switched systems with uncertainties. The uncertainties refer to interval and polytopic uncertainties. By means of the multiple linear copositive Lyapunov functions approach, the robust stability of autonomous systems with average dwell time is solved. Then, the control synthesis of non-autonomous systems with average dwell time is discussed. State-feedback and output-feedback controllers are designed to guarantee the robust stabilization of the considered systems, respectively. All present conditions are solvable in terms of linear programming. Finally, several simulation examples are given to illustrate the validity of the design.     

Stability Analysis of Uncertain Discrete-Time Piecewise Linear Systems with Time Delays

This paper considers the stability analysis of uncertain discrete-time piecewise linear systems with time delays based on piecewise Lyapunov-Krasovskii functionals. It is shown that the stability can be established for the control systems if there is a piecewise Lyapunov-Krasovskii functional, and moreover, the functional can be obtained by solving a set of linear matrix inequalities （LMIs） that are numerically feasible. A numerical example is given to demonstrate the efficiency and advantage of the proposed method.     

Stability Analysis and Stabilization of Discrete-Time 2D Switched Systems

This paper is concerned with stability analysis and stabilization problems for two-dimensional (2D) discrete switched systems represented by a model of Roesser type. First, sufficient conditions for the exponential stability of the 2D discrete switched system are derived via the average dwell time approach. Then, based on this result, a state feedback controller is designed to achieve the exponential stability of the corresponding closed-loop system. All the results are presented in linear matrix inequalities (LMIs) form. A numerical example is given to illustrate the effectiveness of the proposed method.     

Finite-Time Stability of Time-Delay Switched Systems with Delayed Impulse Effects

This paper investigates the finite-time stability problem for a class of time-delay switched systems with nonlinear perturbation and delayed impulse effects. Based on the Lyapunov function method and the technique of inequalities, stability criteria are established to guarantee that the state trajectory of the system does not exceed a certain threshold over a pre-specified finite time interval. Compared with existing results for related problems, the obtained results can be applied to a larger class of hybrid delayed systems, including those in which all of the subsystems are stable and unstable. Two examples are given to demonstrate the validity of the main results.     

Finite-Time Stability and Stabilization of Fractional Order Positive Switched Systems

This paper is concerned with finite-time stability analysis and control synthesis of fractional order positive switched systems. By using the linear copositive Lyapunov function integrated with average dwell time switching technique, the finite-time stability of fractional order positive switched systems is first addressed. Then, the finite-time boundedness of fractional order positive switched systems with exogenous input is discussed. Finally, the stabilization of the considered systems is proposed, where a control strategy based on linear programming is designed. Several implemental algorithms are provided to reduce the conservativeness of results. Two numerical examples are given to show the effectiveness of the findings of theory.     

Stability Analysis of Switched Positive Systems: A Switched Linear Copositive Lyapunov Function Method

This brief addresses the stability problem of discrete-time switched positive systems. The main contribution lies in two aspects. First, a novel method [the switched linear copositive Lyapunov function (SLCLF)] is proposed to reduce the conservatism of the stability criterion obtained by means of the common linear copositive Lyapunov function. Second, some necessary and sufficient conditions are established for the existence of the switched linear copositive Lyapunov function, assuming that a switched linear positive system is given. A numerical example shows the advantage of the obtained results.      

Switched Discrete-Time Delay Systems: Delay-Dependent Analysis and Synthesis

A delay-dependent analysis and synthesis approach is established for a class of linear discrete-time switched delay systems with convex bounded parameter uncertainties in all system matrices. New results are established for both constant and time-varying delays using switched Lyapunov–Krasovskii functionals. A delay-dependent analysis of the uncertain switched delay system is developed to guarantee that it is asymptotically stable with an ℒ₂ gain smaller than a prescribed constant level. Delay-dependent switched control feedback is then designed, based on state and output measurements, to render the corresponding switched closed-loop system delay-dependent asymptotically stable with a prescribed ℒ₂ gain measure. The developed results are cast as linear matrix inequalities (LMIs) and tested on representative examples.     

Nonfragile Observer for Discrete-Time Switched Nonlinear Systems with Time Delay

This paper investigates the problem of the nonfragile observer design for discrete-time switched nonlinear systems with time delay. Based on the average dwell-time approach and linear matrix inequality (LMI) techniques, an exponential stability criterion for the discrete-time switched delay system with Lipschitz nonlinearity is derived. Based on several technical lemmas, the discrete-time observer design can be transferred to the problem of solving a set of LMIs. Furthermore, in cases when the gain of the state observer varies, a kind of nonfragile observer is proposed, and the solution to the observer gain is also obtained by solving a set of LMIs. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

Stability Analysis of Switched Linear Systems with Uncertainty and Delays

Combining the common quadratic Lyapunov functional approach and free-weighting matrix approach, this paper is devoted to the stability analysis of continuous-time switched linear systems (CTSLS) with uncertainty and time-delays. A particular class of matrix inequalities, the so-called Lyapunov–Metzler inequalities are introduced for the CTSLS to investigate the stability and performance in the presence of the uncertainty and delays. We provide sufficient conditions in terms of the Linear Matrix Inequality criterions to guarantee delay-dependent asymptotically stability under the uncertainty of the CTSLS. The combination of switching rule and switching output feedback controllers which will be designed to stabilize the CTSLS and satisfy a prespecified \({\mathcal{L}}_{2}\) gain performance. A example used jitterbug tools is provided to illustrate the effectiveness of the proposed method.     

Stabilization of Discrete-Time Positive Switched Systems

This paper addresses the stabilization for a class of discrete-time positive switched systems. Based on the average dwell time switching strategy, the stabilization of the underlying systems in the autonomous form is studied. Further, the control synthesis of non-autonomous systems is discussed. The state-feedback and output-feedback controllers are constructed, respectively. All the present conditions are solvable in terms of linear programming. Finally, four simulation examples illustrate the validity of the design.     

Sliding Mode Control of Uncertain Stochastic Hybrid Delay Systems with Average Dwell Time

This paper investigates the problem of sliding mode control (SMC) for uncertain switched stochastic system with time-varying delay. The system under consideration is concerned with the stochastic dynamics and deterministic switching laws. An integral sliding surface is constructed and the stable sliding mode is derived. A sufficient condition for mean-square exponential stability of the sliding mode is developed under a class of switching laws based on the average dwell time method. Variable structure controllers are designed to guarantee the existence of the sliding mode from the initial time. An illustrative example is used to demonstrate the effectiveness of the proposed scheme.     

Stability Analysis and Synthesis of Discrete Impulsive Switched Systems with Time-Varying Delays and Parameter Uncertainty

This paper studies the problem of stability and stabilization for discrete impulsive switched systems with time-varying delays and norm-bounded parameter uncertainty. By using the Lyapunov–Krasovskii functional technique and the method of linear matrix inequalities (LMIs), some delay-dependent criteria on asymptotic stability are established. A stabilization condition using feedback control is formulated to stabilize the closed-loop system. Some numerical examples are given to illustrate the main results.     

Exponential Stability of Nonlinear Impulsive and Switched Time-Delay Systems with Delayed Impulse Effects

The exponential stability problem is considered for a class of nonlinear impulsive and switched time-delay systems with delayed impulse effects by using the method of multiple Lyapunov–Krasovskii functionals. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for stabilizing delayed impulses and destabilizing delayed impulses. It is shown that even if all the subsystems governing the continuous dynamics without impulse input delays are not exponential stable, if impulsive and switching signal satisfy a dwell-time upper bound condition, stabilizing delayed impulses can stabilize the systems in the exponential stability sense. Moreover, it is also shown that if the magnitude of the delayed impulses is sufficiently small, the exponential stability properties can be derived irrespective of the size of the impulse input delays under some conditions. The opposite situation is also developed. The efficiency of the proposed results is illustrated by two numerical examples.     

\mathcal{H}_{\infty} Filtering for Short-Time Switched Discrete-Time Linear Systems

In this paper, the $\mathcal{H}_{\infty}$ filtering problem for a class of short-time switched discrete-time linear systems is investigated. For such systems, switching always occurs in some short interval. Since the error state may attain large unacceptable values in short-time switching intervals, besides the asymptotic stability of error dynamics, the boundedness of error state is also significant for short-time switched systems. Thus the designed filter is composed of two parts: asymptotic filter, based upon the existing results, ensures the asymptotic stability of the system during normal, relatively long interval, and finite-time filter ensures system to be finite-time bounded during the short interval of switching, which is the main concern in this paper. By introducing the concept of finite-time boundedness, the proposed filter is formulated as a set of sub-filters ensuring the error dynamics $\mathcal{H}_{\infty}$ finite-time bounded in the short switching interval. Finally, a numerical example is provided to illustrate the effectiveness of this approach.     

Quadratic Stabilization of Uncertain Discrete-Time Switched Systems via Output Feedback

Quadratic stabilization of discrete-time switched systems with norm-bounded time-varying uncertainties is studied. A robust switching rule is proposed to stabilize switched systems by using a designed switched static or dynamic output feedback controller. All the switching rules adopted are constructively designed and state dependent, and they do not rely on any uncertainties.     

Robust Stability and $$H_{\infty }$$ Control of Discrete-Time Uncertain Impulsive Systems with Time-Varying Delay

This paper considers the robust stability and \(H_{\infty }\) control problems for a class of discrete-time uncertain impulsive systems with time-varying delay. Sufficient conditions for the robust stability, stabilization and \(H_\infty \) control of the considered systems are developed. Some numerical examples are presented to show the effectiveness of the theoretical results.