Finite‐Time Stabilization of Coupled Systems on Networks with Time‐Varying Delays via Periodically Intermittent Control

In this paper, finite‐time stabilization of coupled systems on networks with time‐varying delays (CSNTDs) via periodically intermittent control is studied. Both delayed subsystems and delayed couplings are considered; the self‐delays of different subsystems in delayed couplings are not identical. A periodically intermittent controller is designed to stabilize CSNTDs within finite time, and the stabilization duration is closely related to the topological structures of networks. Furthermore, two sufficient criteria are developed to ensure CSNTDs under periodically intermittent control can be stabilized within finite time by using an approach that combines the Lyapunov method with Kirchhoff's Matrix Tree Theorem. Then finite‐time stabilization of coupled oscillators with time‐varying delays is given as a practical application and sufficient criteria is obtained. Finally, a numerical simulation is proposed to support our results and show the effectiveness of the controller.     

Robust Finite‐Time Control for Linear Time‐Varying Delay Systems With Bounded Control

This paper develops a novel finite‐time control design for linear systems subject to time‐varying delay and bounded control. Based on the Lyapunov‐like functional method and using a result on bounding estimation of integral inequality, we provide some sufficient conditions for designing state feedback controllers that guarantee the robust finite‐time stabilization with guaranteed cost control. The conditions are obtained in terms of linear matrix inequalities (LMIs), which can be determined by utilizing the MATLAB LMI Control Toolbox. A numerical example is given to show the effectiveness of the proposed method.     

Robust Exponential Stability Analysis of Uncertain Discrete Time‐Varying Linear Systems

This paper studies the robust exponential stability of uncertain discrete linear time‐varying (UDLTV) systems. The key tool is the recently proposed generating functions. It can be found that a class of improved generating functions (IGFS) can fully characterize the robust exponential stability of UDLTV systems, and the maximum exponential decay rate of system trajectories can be computed by the radius of convergence of the IGFS. Moreover, the application of convex optimization technique and dynamic programming method provides an effective algorithm for the computation of the IGFS. Finally, the numerical example illustrates the efficacy and advantage of the proposed approach.     

Robust Performance Controller Synthesis Based on Discrete‐Time Noncausal Linear Periodically Time‐Varying Scaling

This paper is concerned with the technique called discrete‐time noncausal linear periodically time‐varying (LPTV) scaling for robust stability analysis and synthesis. It is defined through the lifting treatment of discrete‐time systems, and naturally leads to a sort of noncausal operation of signals. In the robust stability analysis of linear time‐invariant (LTI) systems, it has been shown that even static noncausal LPTV scaling induces some frequency‐dependent scaling when it is interpreted in the context of lifting‐free treatment. This paper first discusses in detail different aspects of the effectiveness of noncausal LPTV scaling, with the aim of showing its effectiveness in controller synthesis. More precisely, we study the robust performance controller synthesis problem, where we allow the controllers to be LPTV. As in the LTI robust performance controller synthesis problem, we tackle our problem with an iterative method without guaranteed convergence to a globally optimal controller. Despite such a design procedure, the closed‐loop H_∞ performance is expected to improve as the period of the controller is increased, and we discuss how the frequency‐domain properties of noncausal LPTV scaling could contribute to such improvement. We demonstrate with a numerical example that an effective LPTV controller can be designed for a class of uncertainties for which the well‐known μ‐synthesis fails to derive even a robust stabilization controller.     

Stability of Linear Systems With Time‐Varying Delay: a Generalized Discretized Lyapunov Functional Approach

This paper investigates the stability of linear uncertain systems with time‐varying delay. Stability criteria are derived based on a generalized discretized Lyapunov functional approach. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. The stability conditions are written in the form of linear matrix inequalities. Numerical examples indicate significant improvements over the existing results.     

Delay‐Dependent Stability Criterion for Discrete‐Time Systems with Time‐Varying Delays

The stability analysis problem is considered for linear discrete‐time systems with time‐varying delays. A novel summation inequality is proposed, which takes the double summation information of the system state into consideration. The inequality relaxes the recently proposed discrete Wirtinger inequality and its improved version. Based on construction of a suitable Lyapunov‐Krasovskii functional and the novel summation inequality, an improved delay‐dependent stability criterion for asymptotic stability of the systems is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.     

Delay‐Dependent Exponential Stability for Discrete‐Time Singular Switched Systems with Time‐Varying Delay

In this paper, the exponential stability problem is investigated for a class of discrete‐time singular switched systems with time‐varying delay. By using a new Lyapunov functional and average dwell time scheme, a delay‐dependent sufficient condition is established in terms of linear matrix inequalities for the considered system to be regular, causal, and exponentially stable. Different from the existing results, in the considered systems the corresponding singular matrices do not need to have the same rank. A numerical example is given to demonstrate the effectiveness of the proposed result.     

时变时滞不确定系统的鲁棒无记忆控制

针对控制时滞系统存在时变时滞及参数摄动的情况,通过构造适当形式的Lyapunov函数,求得无记忆反馈控制律.给出单滞后系统稳定化控制,并讨论具有指定衰减度的控制器设计,进而将结论推广到多重滞后系统.     

Adaptive Observer‐Based Global Sliding Mode Control for Uncertain Discrete‐Time Nonlinear Systems with Time‐Delays and Input Nonlinearity

A new discrete‐time adaptive global sliding mode control (SMC) scheme combined with a state observer is proposed for the robust stabilization of uncertain nonlinear systems with mismatched time delays and input nonlinearity. A state observer is developed to estimate the unmeasured system states. By using Lyapunov stability theorem and linear matrix inequality (LMI), the condition for the existence of quasi‐sliding mode is derived and the stability of the overall closed‐loop system is guaranteed. Finally, simulation results are presented to demonstrate the validity of the proposed scheme.     

H∞ Control of Linear Discrete‐Time Systems With Norm‐Bounded Nonlinear Uncertainties

In this paper, robust H_∞ control of a class of discrete‐time uncertain systems in state‐space form with linear nominal parts and norm‐bounded nonlinear uncertainties in both state and output equations is discussed. Such systems have a unique characterisic; that is, the two norm‐bounded nonlinear uncertainties have the equivalent representation by means of time‐varying and norm‐bounded linear uncertainties. To overcome the conservativenss of [5], the two nonlinear uncertainty sets are considered to be different. Then, by converting such systems into related discrete‐time linear systems with time‐varying and norm‐bounded linear uncertainties, we obtain that a sufficient condition for robust H_∞ control of such systems is equivalent to the solvability of the same problem of the related linear uncertain systems, which is solvable by means of a linear algebraic Riccati inequality.     

Adaptive Robust Output Tracking Control of Uncertain Nonlinear Cascade Systems with Disturbance and Multiple Unknown Time‐Varying Delays

In this paper, an adaptive robust controller is designed for a class of uncertain nonlinear cascade systems with multiple time‐varying delays under external disturbance. It is assumed that multiple time‐varying delays are not exactly known and, therefore, the delayed terms must not appear in the adaptation and control laws. Accordingly, by using a Lyapunov‐Krasovskii function, delays are deleted from the adaptation and control laws. A controller based on an adaptive backstepping approach is designed to assure the global asymptotic tracking of the desired output and boundedness of the other states. The proposed controller is proved to be robust against unknown time‐varying delays and external disturbances applying to the system. Simulation results are provided to show the effectiveness of the proposed approach.     

Robust Finite‐Time Stability and Stabilization of Linear Uncertain Time‐Delay Systems

Robust finite‐time stability and stabilization problems for a class of linear uncertain time‐delay systems are studied. The concept of finite‐time stability is extended to linear uncertain time‐delay systems. Based on the Lyapunov method and properties of matrix inequalities, a sufficient condition that ensures finite‐time stability of linear uncertain time‐delay systems is given. By virtue of the results on finite‐time stability, a memoryless state feedback controller that guarantees that the closed‐loop system is finite time stable, is proposed. The controller design problem is solved by using the linear matrix inequalities and the cone complementarity linearization iterative algorithm. Numerical examples verify the efficiency of the proposed methods.     

Robust Stability of Time‐Varying Delay Systems: The Quadratic Separation Approach

In this article, we are interested in analysing the stability of systems that incorporate time‐varying delays in their dynamic. The Lyapunov‐Krasovskii approach is definitely the most popular method to address this issue and many results have proposed new functionals and enhanced techniques for deriving less conservative stability conditions. In the present work, we propose an original approach: the quadratic separation. To this end, the delay operator properties are exploited to provide delay range stability conditions. In particular, L₂‐norm of delay‐dependent operators are computed so as to reduce the conservatism of the approach. Moreover, the main result is able to assess the stability of non‐small delay systems, i.e, it can detect a stability interval for systems that are unstable without any delay. Several examples illustrate the benefit of our methodology.     

Synchronization of Uncertain Complex Networks with Time‐Varying Node Delay and Multiple Time‐Varying Coupling Delays

This paper investigates the synchronization problem of a class of complex dynamical networks via an adaptive control method. It differs from existing works in considering intrinsic delay and multiple different time‐varying coupling delays, and uncertain couplings. A simple approach is used to linearize the uncertainties with the norm‐bounded condition. Simple but suitable adaptive controllers are designed to drive all nodes of the complex network locally and globally synchronize to a desired state. In addition, several synchronization protocols are deduced in detail by virtue of Lyapunov stability theory and a Cauchy matrix inequality. Finally, a simulation example is presented, in which the dynamics of each node are time‐varying delayed Chua chaotic systems, to demonstrate the effectiveness of the proposed adaptive method.     

Robust Adaptive Fuzzy Control of Nonlinear Systems with Unknown and Time‐Varying Saturation

In this paper, a robust adaptive fuzzy control approach is proposed for a class of nonlinear systems in strict‐feedback form with the unknown time‐varying saturation input. To deal with the time‐varying saturation problem, a novel controller separation approach is proposed in the literature to separate the desired control signal from the practical constrained control input. Furthermore, an optimized adaptation method is applied to the dynamic surface control design to reduce the number of adaptive parameters. By utilizing the Lyapunov synthesis, the fuzzy logic system technique and the Nussbaum function technique, an adaptive fuzzy control algorithm is constructed to guarantee that all the signals in the closed‐loop control system remain semiglobally uniformly ultimately bounded, and the tracking error is driven to an adjustable neighborhood of the origin. Finally, some numerical examples are provided to validate the effectiveness of the proposed control scheme in the literature.     

Quasi‐Time‐Dependent Controller for Discrete‐Time Switched Linear Systems With Mode‐Dependent Average Dwell‐Time

This paper is concerned with the stability and stabilization problem of a class of discrete‐time switched systems with mode‐dependent average dwell time (MDADT). A novel Lyapunov function, which is both mode‐dependent (MD) and quasi‐time‐dependent (QTD), is established. The new established Lyapunov function is allowed to increase at some certain time instants. A QTD controller is designed such that the system is globally uniformly asymptotically stable (GUAS) and has a guaranteed performance index. The new QTD robust controller designed in this paper is less conservative than the mode independent one which is frequently considered in literatures. Finally, a numerical example and a practical example are provided to illustrate the effectiveness of the developed results.      

Observer‐Based Reliable Control for Discrete‐Time Switched Linear Systems with Faulty Actuators

This paper deals with the issue of reliable control for discrete‐time switched linear systems with faulty actuators by utilizing a multiple Lyapunov functions method and estimate state‐dependent switching technique. A solvability condition for the reliable control problem is given in terms of matrix inequality with an extra matrix variable. This condition allows the reliable control problem for each individual subsystem to be unsolvable. For each subsystem of such a switched system, we design an observer and an observer‐based controller. A switching rule depending on the observer state is designed which, together with the controllers, can guarantee the stability of the closed‐loop switched system for all admissible actuator failures. The observers, controllers, and switching law are explicitly computed by solving linear matrix inequalities (LMIs). The proposed design method is illustrated by two numerical examples.     

Robust L2−L∞ Filtering for Stochastic Systems with Discrete and Distributed Time‐Varying Delays

This paper is devoted to the problem of robust L₂–L_∞ filtering for a class of stochastic systems with both discrete and distributed time‐varying delays. The objective is to design a full‐order filter such that the resulting filtering error system is stochastically asymptotically stable with a prescribed L₂–L_∞ performance satisfied. Delay‐dependent sufficient condition for the existence of the filter is obtained in terms of linear matrix inequalities (LMIs). And the filter design method is proposed, while the explicit expression for the desired filter is also given. Numerical examples are included to illustrate the benefit and the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Discrete‐time Adaptive ILC for Non‐parametric Uncertain Nonlinear Systems with Iteration‐Varying Trajectory and Random Initial Condition

This paper presents a new discrete‐time adaptive iterative learning control approach (AILC) for a class of time‐varying nonlinear systems with nonparametric uncertainties and non‐repeatable external disturbances by incorporating a novel iterative estimate scheme. A major distinct feature of the presented approach is that uncertainties can be completely compensated for, using only I/O data. Another distinct feature is that the pointwise convergence is achieved over a finite time interval without requiring the matching condition on initial states and reference trajectory. Rigorous mathematical analysis is developed, and simulation results illustrate the effectiveness of the proposed approach.     

Robust Stability Analysis for Systems with Time‐Varying Delays: A Delay Function Approach

This paper is concerned with the robust stability of time‐varying delay systems with structured uncertainties. Stability conditions are provided through a Lyapunov‐Krasovskii functional (LKF) method. The proposed method introduces a linear function of the time‐varying delay to construct the LKF. With this function, two‐dimensional partition is conducted on the integral domain in the derivative of LKF. Quadratic convex combination then is employed to present stability criteria in the form of linear matrix inequalities (LMIs). The method not only exploits the information of delay at different time instants, but also enables the handling of its derivative to reduce conservatism. Numerical examples are given to show the effectiveness of our method.