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.     

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.     

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, the linear-quadratic problem of linear time-varying systems with delays are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples illustrate that the algorithms are valid.     

Contraction-based model predictive control for stochastic nonlinear discrete-time systems with time-varying delays via multi-dimensional Taylor network

For stochastic nonlinear systems with time-varying delays, the existing robust control approaches are unnecessarily conservative in most practical scenarios. Within this context, a mathematically rigorous and computationally tractable tube-based model predictive control scheme is proposed in the framework of contraction theory. A contraction metric is systematically constructed via convex optimization by forming a differential LyapunovKrasovskii function on tangent space. It guarantees the perturbed actual solution trajectories to be contained within a robust positive invariant tube centered along the reference trajectories and results in an explicit exponential bound on the deviation. The application scenarios of the control contraction metric controller are extended from constant delay systems into time-varying delay systems thereby. Compared with the existing robust mechanism for time-delay systems based on min-max optimization formulation with a linear feedback controller, the proposed scheme greatly reduces the design conservativeness and yields a larger region of attraction. A sparse multi-dimensional Taylor network (MTN) is designed to parameterize the family of the geodesic. Compared to conventional NNs and MTN surrogates, sparse MTN features a more concise topology that enhances its computational efficiency conspicuously. Results of the numerical simulations verify the effectiveness of the proposed method.     

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.     

Reachable set estimation for discrete-time bilinear systems with time-varying delays

The problem of reachable set estimation is studied for discrete-time bilinear system in this paper. Time-varying delays and bounded input disturbances are both considered in bilinear system. The aim is to find reachable set that converges from all the states of system with initial conditions. By constructing Lyapunov–Krasovskii functional, sufficient delay-dependent less conservative stable conditions of reachable set estimation are obtained for bilinear delay system via the reciprocally convex combination and delay partition approaches. The derived theorem can guarantee that all the states of system with initial conditions from some domain are bounded in an ellipsoid and all the states from other domain are converged exponentially within a ball. One simulation example is presented to illustrate the correctness of the derived result in this paper.     

Exponential synchronization for coupled complex networks with time-varying delays and stochastic perturbations via impulsive control

This paper is concerned with the problem of exponential synchronization of coupled complex networks with time-varying delays and stochastic perturbations (CCNTDSP). Different from previous works, both the internal time-varying delay and the coupling time-varying delay are taken into account in the network model. Meanwhile, an impulsive controller is designed to realize exponential synchronization in mean square of CCNTDSP. Combining the Lyapunov method with Kirchhoff’s Matrix Tree Theorem, some sufficient criteria are obtained to guarantee exponential synchronization in mean square of CCNTDSP. Furthermore, we apply the theoretical results to study exponential synchronization of stochastic coupled oscillators with the internal time-varying delay and the coupling time-varying delay. And a synchronization criterion is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness and feasibility of our theoretical results and the superiority of impulsive control.     

Exponential stability of switched linear impulsive time-varying system and its application

This paper aims at the problem of exponential stability for switched linear impulsive time-varying system. By constructing two different switched discretized Lyapunov functions, some new sufficient conditions ensuring the global exponential stability of switched linear impulsive time-varying system are provided, which can be employed to the case when all subsystems are unstable. Furthermore, we apply theoretical results to the consensus of multi-agent system with switching topologies. Finally, numerical examples demonstrate the effectiveness of given results.     

Stabilization of switched linear singular systems with state reset

This paper investigates the stability and stabilization of switched linear singular systems with state reset at switching instants. Based on the dynamics decomposition of singular subsystems, a sufficient stability condition for the system with the given state reset is obtained. Then, the stabilization problem by state reset is investigated and an algorithm for computing the reset matrices is presented. The obtained results extend some previous works on both singular switched systems and reset control for normal switched systems. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.     

Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality

In this paper, the problem of stability analysis for linear systems with time-varying delays is considered. By the consideration of new augmented Lyapunov functionals, improved delay-dependent stability criteria for asymptotic stability of the system are proposed for two cases of conditions on time-varying delays with the framework of linear matrix inequalities (LMIs), which can be solved easily by various efficient convex optimization algorithms. The enhancement of the feasible region of the proposed criteria is shown via three numerical examples by the comparison of maximum delay bounds.     

Advanced stability criteria for linear systems with time-varying delays

This paper investigates a stability problem for linear systems with time-varying delays. By constructing suitable augmented Lyapunov–Krasovskii functionals, improved stability criteria under various conditions of time-varying delays are derived within the framework of linear matrix inequalities (LMIs). Moreover, to reduce the computational burden caused by the non-convex term including h²(t), how to deal with it is applied by estimating it to the convex term including h(t). Finally, three illustrative examples are given to show the effectiveness of the proposed criteria.     

Improved stability criteria for linear systems with time-varying delays

This paper is concerned with the stability analysis of linear systems with time-varying delays. First, by introducing the quadratic terms of time-varying delays and some integral vectors, a more suitable Lyapunov-Krasovskii functional (LKF) is constructed. Second, two new delay-dependent estimation methods are developed in the stability analysis of linear system with time-varying delays, which include a reciprocally convex matrix inequality and an integral inequality. More information about time-varying delays and more free matrices are introduced into the two estimation approaches, which play a key role for obtaining an accurate upper bound of the integral terms in time derivative of LKFs. Third, based on the novel LKFs and new estimation approaches, some less conservative criteria are derived in the form of linear matrix inequality (LMI). Finally, three numerical examples are applied to verify the advantages and effectiveness of the newly proposed methods.     

Positivity and exponential stability of discrete-time coupled homogeneous systems with time-varying delays

This paper investigates the positivity and stability of discrete-time coupled homogeneous systems with time-varying delays. First, an explicit criterion is given for the positivity of discrete-time coupled homogeneous delay systems. Then, by using the properties of homogeneous functions, a sufficient condition is presented for ensuring the stability of the considered systems. Moreover, the obtained result is applied to study the stability of positive singular systems with time-varying delay. It should be noted that it is the first time that the stability result is given for discrete-time coupled homogeneous positive systems with time-varying delays. Two numerical examples are presented to demonstrate the effectiveness of the derived results.     

State bounding estimation of positive singular discrete-time systems with unbounded time-varying delays

This paper investigates the problem of state bounding for linear positive singular discrete-time systems with unbounded time-varying delay. Our approach is based on the comparison principle and the spectral analysis of matrices. Using the proposed approach, we provide new sufficient conditions for the regularity, causality, positivity and the existence of the smallest ultimate upper bound of such systems. The conditions are given in terms of linear programming problems, which can be solved by LP optimal toolbox. Numerical examples with simulation are given to demonstrate the advantage and validity of the proposed theoretical results. The proposed method is the first trial in the state bounding problem of linear singular discrete-time systems with unbounded time-varying delays.     

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.     

Stabilization of discrete-time switched singular time-delay systems under asynchronous switching

This paper is concerned with the problem of state feedback stabilization of a class of discrete-time switched singular systems with time-varying state delay under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the subsystems. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence of a class of stabilizing switching laws is first derived to guarantee the closed-loop system to be regular, causal and exponentially stable in the presence of asynchronous switching. The stabilizing switching laws are characterized by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.     

An improved robust stabilization method for discrete-time fuzzy systems with time-varying delays

This note focuses on the robust stabilization of discrete-time fuzzy uncertain systems with time-varying delays under a delayed nonparallel distributed compensation scheme. The key idea is twofold: first, the linear matrix inequalities (LMI) proposed here are shown to generalize some previous similar results available in recent literature, and second, the design of control parameters is decoupled from the proposed fuzzy-basis dependent Lyapunov–Krasovskii functional (FBDLKF) by means of Finsler?s lemma. Finally, a numerical example is provided to illustrate the effectiveness of this method.     

Consensus of discrete-time second-order agents with time-varying topology and time-varying delays

This paper studies the consensus problem of multiple agents with discrete-time second-order dynamics. It is assumed that the information obtained by each agent is with time-varying delays and the interaction topology is time-varying, where the associated direct graphs may not have spanning trees. Under the condition that the union graph is strongly connected and balanced, it is shown that there exist controller gains such that consensus can be reached for any bounded time-delays. Moreover, a method is provided to design controller gains. Simulations are performed to validate the theoretical results.     

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.