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 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.     

Control for Time‐Varying Delay Systems by Integrating Semi‐Discretization and Hysteresis‐Based Switching

In this paper, we develop an innovative control method for linear systems with time‐varying delay by integrating the semi‐discretization method and the hysteresis‐based switching algorithm. The semi‐discretization method is adopted to design an optimal controller for each fixed time‐delay and form a candidate controller family. The switching algorithm acts as the principal law for switching among various controllers according to the instantaneous value of the time‐delay. A theoretical proof is presented regarding the stability of the switching time‐delay system. It is shown that the most significant factors that affect the system stability are the size of the candidate controller family, the value of the switching coefficient, and the changing rate of the time‐delay. Two case studies are presented to show the effectiveness of the proposed method.     

Passive Control for Stochastic Interval Systems with Interval Time‐Varying Delay

This paper is concerned with the problem of delay‐dependent passive analysis and control for stochastic interval systems with interval time‐varying delay. The system matrices are assumed to be uncertain within given intervals, and the time delay is a time‐varying continuous function belonging to a given range. By the transformation of the interval uncertainty into the norm‐bounded uncertainty, partitioning the delay into two segments of equal length, and constructing an appropriate Lyapunov–Krasovskii functional in each segment of the delay interval, delay‐dependent stochastic passive control criteria are proposed without ignoring any useful terms by considering the information of the lower bound and upper bound for the time delay. The main contribution of this paper is that a tighter upper bound of the stochastic differential of Lyapunov–Krasovskii functional is obtained via a newly‐proposed bounding condition. Based on the criteria obtained, a delay‐dependent passive controller is presented. The results are formulated in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness of the 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.     

Robust Exponential Stability and Stabilization of a Class of Nonlinear Stochastic Time‐Delay Systems

This paper presents new exponential stability and delayed‐state‐feedback stabilization criteria for a class of nonlinear uncertain stochastic time‐delay systems. By choosing the delay fraction number as two, applying the Jensen inequality to every sub‐interval of the time delay interval and avoiding using any free weighting matrix, the method proposed can reduce the computational complexity and conservativeness of results. Based on Lyapunov stability theory, exponential stability and delayed‐state‐feedback stabilization conditions of nonlinear uncertain stochastic systems with the state delay are obtained. In the sequence, the delayed‐state‐feedback stabilization problem for a nonlinear uncertain stochastic time‐delay system is investigated and some sufficient conditions are given in the form of nonlinear inequalities. In order to solve the nonlinear problem, a cone complementarity linearization algorithm is offered. Mathematical and/or numerical comparisons between the proposed method and existing ones are demonstrated, which show the effectiveness and less conservativeness of the proposed method.     

Prediction‐based Adaptive Robust Tracking Control of an Uncertain First‐Order Time‐Delay System

This paper considers the tracking problem of a delayed uncertain first‐order system which is simultaneously subject to (possibly large) known input delay, unknown but bounded time‐varying disturbance, and unknown plant parameter. The proposed predictor adaptive robust controller (PARC) involves prediction‐based projection type adaptation laws with model compensation and prediction‐based continuous robust feedback such that the closed loop system has global exponential convergence with an ultimate bound proportional to delay, disturbance bound, and switching gain. Further, if there are only delay and parameter uncertainties after some finite time, then semi‐global asymptotic tracking is guaranteed. The proposed design is shown to have significant closed loop performance improvement over the baseline controller.     

The Framework for Linear Periodic Time‐Delay Systems Based on Semi‐Discretization: Stability Analysis and Control

Stability analysis and control for linear periodic time‐delay systems are investigated in this paper. In this framework, a semi‐discretization method is used to develop a mapping of the system response in a finite‐dimensional state space. With the mapping, the stability region and stability boundary can be identified by comparing the maximum absolute value of its eigenvalues to 1. More importantly, an efficient stability criterion is presented for periodic neutral systems. In addition, minimization of the maximum absolute value of the mapping's eigenvalues leads to optimal control gains. The tracking control problem is also discussed. Two numerical examples are given to illustrate the effectiveness of the proposed method.     

On the Design of Observer for Nonlinear Time‐Delay Systems

This paper suggests a new method to design observers in a class of nonlinear time‐delay systems with delays in system states. The method is based on an extension of the well‐known state‐dependent Riccati equation (SDRE) technique. The conditions for locally asymptotic stability of the proposed observer are investigated. Some numerical simulations are provided to show the design procedure and the flexibility of the proposed observer.     

H∞ Filtering for Spatially Interconnected Time‐Delay Systems with Interconnected Chains in Finite Frequency Domains

This paper deals with the problem of delay‐dependent H_∞ filtering for spatially interconnected time‐delay systems (SITSs) with interconnected chains in finite frequency domains. First, a multidimensional (N‐D) hybrid time‐delay Roesser model and a delay‐dependent finite frequency bounded ream lemma (BRL) for SITSs with interconnected chains are proposed. Then, using the obtained delay‐dependent finite frequency BRL, a finite frequency H_∞ filter design method can be derived by solving a set of linear matrix inequalities (LMIs). Finally, a practical example is provided to clearly demonstrate the effectiveness of the proposed method.     

Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

This paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.     

Suboptimal Output Consensus for Time‐Delayed Singular Multi‐Agent Systems

The current paper investigates guaranteed‐cost output consensus analysis and design problems for high‐order linear time‐invariant singular multi‐agent systems with constant time delays, which can realize suboptimal output consensus control. Firstly, a new output consensus protocol with a suboptimal index and a single delay is proposed to realize the tradeoff design between output consensus regulation performances and control energy consumptions. Then, sufficient conditions for guaranteed‐cost output consensus and consensualization are derived in terms of linear matrix inequalities by a combined tool from the Lyapunov‐Krasovskii approach and the free‐weighting matrix technique, respectively, and the output consensus function is determined on the basis of the First Equivalent Form. Finally, a numerical example is performed to demonstrate the effectiveness and conservativeness of theoretical 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 Stability and Static Output Feedback Control of 2‐D Discrete Systems with Interval Time‐varying Delays

This paper is concerned with the problems of delay‐dependent stability and static output feedback (SOF) control of two‐dimensional (2‐D) discrete systems with interval time‐varying delays, which are described by the Fornasini‐Marchesini (FM) second model. The upper and lower bounds of delays are considered. Applying a new method of estimating the upper bound on the difference of Lyapunov function that does not ignore any terms, a new delay‐dependent stability criteria based on linear matrix inequalities (LMIs) is derived. Then, given the lower bounds of time‐varying delays, the maximum upper bounds in the above LMIs are obtained through computing a convex optimization problem. Based on the stability criteria, the SOF control problem is formulated in terms of a bilinear matrix inequality (BMI). With the use of the slack variable technique, a sufficient LMI condition is proposed for the BMI. Moreover, the SOF gain can be solved by LMIs. Numerical examples show the effectiveness and advantages of our results.     

Adaptive Finite‐Time Robust Control of Nonlinear Delay Hamiltonian Systems Via Lyapunov‐Krasovskii Method

This paper investigates the adaptive finite‐time robust control problem of a class of nonlinear time‐delay Hamiltonian systems via the Lyapunov‐Krasovskii (L‐K) method, and proposes some delay‐dependent results on the issue. Different from existing works, this paper first presents a time‐varying finite‐time stability (FTS) criterion via an L‐K functional approach, and obtains two FTS conditions by constructing specific L‐K functionals. Then, the adaptive finite‐time robust control problem is investigated for nonlinear time‐delay port‐controlled Hamiltonian (PCH) systems, and a control design procedure is presented. Finally, the effectiveness of the results is demonstrated by an illustrative example.     

Chattering Free Sliding Mode Control for Uncertain Discrete Time‐Delay Singular Systems

The problem of chattering free sliding mode control for a class of uncertain discrete singular systems with state delay is investigated in this paper. As a component of the solution, a new least squares support vector machine (LS‐SVM) reaching law is proposed. In terms of linear matrix inequalities, a delay‐dependent condition for sliding mode dynamics to be regular, causal, and asymptotically stable is established, and the chattering problem that appears in traditional variable structure systems is eliminated. Numerical examples are provided to demonstrate the applicability of the proposed methods.     

Delay‐distribution‐dependent robust stability of uncertain systems with time‐varying delay

By employing the information of the probability distribution of the time delay, this paper investigates the problem of robust stability for uncertain systems with time‐varying delay satisfying some probabilistic properties. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a priori. In terms of the probability distribution of the delay, a new type of system model with stochastic parameter matrices is proposed. Based on the new system model, sufficient conditions for the exponential mean square stability of the original system are derived by using the Lyapunov functional method and the linear matrix inequality (LMI) technique. The derived criteria, which are expressed in terms of a set of LMIs, are delay‐distribution‐dependent, that is, the solvability of the criteria depends on not only the variation range of the delay but also the probability distribution of it. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Stabilization of Fractional‐Order Linear Systems with State and Input Delay

This paper describes a variable structure control for fractional‐order systems with delay in both the input and state variables. The proposed method includes a fractional‐order state predictor to eliminate the input delay. The resulting state‐delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional‐order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.     

Finite‐time stability and stabilization of semi‐Markovian jump systems with time delay

Semi‐Markovian jump systems are more general than Markovian jump systems in modeling practical systems. On the other hand, the finite‐time stochastic stability is also more effective than stochastic stability in practical systems. This paper focuses on the finite‐time stochastic stability, exponential stochastic stability, and stabilization of semi‐Markovian jump systems with time‐varying delay. First, a new stability condition is presented to guarantee the finite‐time stochastic stability of the system by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. Second, the stability criterion is further proved to guarantee the exponential stochastic stability of the system. Moreover, a controller design method is also presented according to the stability criterion. Finally, an example is provided to illustrate that the proposed stability condition is less conservative than other existing results. Additionally, we use the proposed method to design a controller for a load frequency control system to illustrate the effectiveness of the method in a practical system of the proposed method.     

Adaptive Neural Dynamic Surface Control for Stochastic Nonlinear Time‐Delay Systems with Input and Output Constraints

This paper presents an adaptive neural tracking control approach for uncertain stochastic nonlinear time‐delay systems with input and output constraints. Firstly, the dynamic surface control (DSC) technique is incorporated into adaptive neural control framework to overcome the problem of ‘explosion of complexity’ in the control design. By employing a continuous differentiable asymmetric saturation model, the input constraint problem is solved. Secondly, the appropriate Lyapunov‐Krasovskii functional and the property of hyperbolic tangent functions are used to deal with the unknown time‐delay terms, RBF neural network is utilized to identify the unknown systems functions, and barrier Lyapunov functions (BLFs) are designed to avoid the violation of the output constraint. Finally, based on adaptive backstepping technique, an adaptive neural control method is proposed, and it decreases the number of learning parameters. Using Lyapunov stability theory, it is proved that the designed controller can ensure that all the signals in the closed‐loop system are 4‐Moment (or 2 Moment) semi‐globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a small neighborhood of the origin. Two simulation examples are provided to further illustrate the effectiveness of the proposed approach.