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.     

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.     

Reachable set estimation for discrete‐time switched system with application to time‐delay system

This paper addresses the problem of reachable set estimation and synthesis for a class of discrete‐time switched linear systems with time delay and bounded peak disturbance. Combined with the feature of mode‐dependent average dwell time switching, a new algorithm is developed to estimate the reachable set of switched system, which is both quasi‐time‐dependent and mode‐dependent. Then, the proposed method is applied to time‐delay system and a sufficient condition is presented to guarantee the asymptotic stability and estimate the bounding ellipsoid. Furthermore, the quasi‐time‐dependent controller is designed to stabilize the system and restrict the closed‐loop system states to an ellipsoidal bound. Examples are presented to illustrate the effectiveness and advantages of the obtained theorems.     

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.     

Discrete‐time repetitive controller for time‐delay systems with disturbance observer

A novel discrete‐time repetitive controller design for time‐delay systems subject to a periodic reference and exogenous periodic disturbances is presented. The main idea behind the proposed approach is to take advantage of the plant delay in the controller design, and not to compensate for the effect of this delay. To facilitate this concept, we introduce an appropriate time‐delay and a compensator in a positive feedback connection with the plant, such that a generator for periodic signals is constructed. Then a proportional controller is used to stabilize the closed‐loop system. The tracking control capability is thus guaranteed according to the internal model principle (IMP). In addition, to attenuate external periodic disturbances, a disturbance observer (DO) is developed to simultaneously achieve reference tracking and disturbance rejection. The possible fractional delay due to the digital discretization is handled by using a fractional delay filter approximation. The proposed controller has a simple structure, in which only a proportional parameter and a low‐pass filter are required to be chosen. The closed‐loop stability conditions and a robustness analysis under model uncertainties are studied. Numerical simulations and practical experiments on a servo motor system are conducted to verify the feasibility and simplicity of the proposed controller. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Semi‐global output feedback control for nonlinear systems with uncertain time‐delay and output function

This paper considers semi‐global output feedback control for more general nonlinear systems with unknown time‐delay and output function whose derivative is unbounded from above. By introducing a new observer and using the backstepping design method and the Razumikhin stability theorem, an output feedback controller is constructed to achieve a semi‐global stability. Copyright © 2016 John Wiley & Sons, Ltd.     

An impulsive‐switched‐system approach to aperiodic sampled‐data systems with time‐delay control

This paper devotes to the stability of aperiodic sampled‐data systems with time‐delay control, where the delays can impose a positive effect on the stability of the systems. The systems are modeled as impulsive switched systems with fixed switching laws. A novel separation theorem is presented to determine the Schur property of a matrix product and then used to obtain a less conservative stability criterion for the impulsive switched systems with fixed switching laws. By the separation theorem and a loop‐functional approach, some new stability and stabilization criteria for aperiodic sampled‐data systems with time‐delay control are provided in terms of linear matrix inequalities. Finally, the stability and stabilization results are tested on some classical numerical examples to illustrate the efficiency of the proposed method.     

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

In this paper, the robust stability of a fractional‐order time‐delay system is analyzed in the frequency domain based on finite spectrum assignment (FSA). The FSA algorithm is essentially an extension of the traditional pole assignment method, which can change the undesirable system characteristic equation into a desirable one. Therefore, the presented analysis scheme can also be used as an alternative time‐delay compensation method. However, it is superior to other time‐delay compensation schemes because it can be applied to open‐loop poorly damped or unstable systems. The FSA algorithm is extended to a fractional‐order version for time‐delay systems at first. Then, the robustness of the proposed algorithm for a fractional‐order delay system is analyzed, and the stability conditions are given. Finally, a simulation example is presented to show the superior robustness and delay compensation performance of the proposed algorithm. Moreover, the robust stability conditions and the time‐delay compensation scheme presented can be applied on both integer‐order and fractional‐order systems.     

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.     

Dynamic anti‐windup compensator for fractional‐order system with time‐delay

This paper describes the results of introducing an additional dynamic element to an anti‐windup compensator from control quality and stability area anslysis viewpoint. The analyzed system consists of a first‐order plant with time delay and a fractional‐order PI controller, to present the discussed approach. The controller is tuned based on Hermite‐Biehler and Pontryagin theorems. In the paper, the stability analysis and tracking performance are presented based on both simulation and experimental results. The experiments have been performed using Inteco Modular Servo System with performance evaluated on the basis of the selected performance criterion, namely the Integral of Absolute Error, to verify the applicability of the proposed method. The results have proven that use of the additional dynamic element provides a wider range of controller parameters to ensure stability of the closed‐loop system and better tracking performance in comparison to the system without anti‐windup compensation or system with a standard anti‐windup compensator. It is actually the first time that this type of analysis for dynamic element compensation in anti‐windup framework has been presented for fractional‐order systems. In addition, all the obtained results are referred to the experimental data.     

Fast adaptive fault estimation and accommodation for nonlinear time‐varying delay systems

This paper studies the problem of fault estimation and accommodation for a class of nonlinear time‐varying delay systems using adaptive fault diagnosis observer (AFDO). A novel fast adaptive fault estimation algorithm that does not need the derivative of the output vector is proposed to enhance the performance of fault estimation. Meanwhile, a delay‐dependent criteria is obtained based on free weighting matrix method with the purpose of reducing the conservatism of the AFDO design. On the basis of fault estimation, an observer‐based fault‐tolerant controller is designed to guarantee the stability of the closed‐loop system. In terms of matrix inequality, we derive sufficient conditions for the existence of the adaptive observer and fault‐tolerant controller. Simulation results are presented to illustrate the efficiency of the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Reliable control for a class of uncertain singular systems with interval time‐varying delay

This paper is concerned with the reliable controller design problem for a class of singular systems with interval time‐varying delay and norm‐bounded uncertainties. A more practical model of actuator failures than outages is considered. First, by constructing a novel Lyapunov–Krasovskii functional combined with Finsler's Lemma, an improved delay‐range‐dependent stability criterion for the nominal unforced singular time‐delay system is established in terms of linear matrix inequality (LMI). Then, based on this criterion, an LMI condition for the design of a reliable state feedback controller is presented such that, for all parameter uncertainties and actuator failures, the resultant closed‐loop system is regular, impulse‐free and stable. Numerical examples are proposed to illustrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ stabilization for networked semi‐Markovian jump systems with randomly occurring uncertainties via improved dynamic event‐triggered scheme

This paper aims to solve the H_∞ stabilization problem for networked semi‐Markovian jump systems subject to randomly occurring uncertainties by an improved event‐triggered technique. A new measurement error that is defined as the difference value between the latest transmitted data and the mean value of both current data and latest transmitted data is introduced into the event‐triggered condition. Compared with traditional dynamic event‐triggered scheme, more unexpected data could be avoided to be transmitted, which is demonstrated in the simulation through sufficient comparison experiments. Furthermore, by employing a Lyapunov‐Krasovskii functional method and a free‐weighting matrix method, sufficient conditions are derived to guarantee the stabilization of the closed‐loop semi‐Markovian jump time‐delay system with uncertainties and a prescribed performance index. Then, a codesign method for H_∞ controller gains and event‐triggered parameters is presented. Finally, simulations are given to verify the effectiveness of our improved dynamic event‐triggered scheme.     

A computational approach to stability of polytopic families of quasi‐polynomials with single delay

This paper addresses stability of families of linear, time‐invariant systems with single time delay. The families are generated by system parameters uncertain within known intervals, and appearing linearly (affinely) in the systems' characteristic functions, which are quasi‐polynomials. First, easily implementable conditions for delay‐independent stability for such families are given. Later, for families that fail to be stable independently of delay, a computationally tractable method is presented to compute all possible time‐delay intervals for which the families are stable. The method is based on the edge theorem and provides the complete stability picture of a given family. Numerical examples illustrate the application of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Pseudo‐predictor feedback control of discrete‐time linear systems with a single input delay

This paper studies the problems of stabilization of discrete‐time linear systems with a single input delay. By developing the methodology of pseudo‐predictor feedback, which uses the (artificial) closed‐loop system dynamics to predict the future state, memoryless state feedback control laws are constructed to solve the problem. Necessary and sufficient conditions are obtained to guarantee the stability of the closed‐loop system in terms of the stability of a class‐difference equations. It is also shown that the proposed controller achieves semi‐global stabilization of the system if its actuator is subject to either magnitude saturation or energy constraints under the condition that the open‐loop system is only polynomially unstable. Numerical examples have been worked out to illustrate the effectiveness of the proposed approaches. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic Stability and Finite‐Time Stability of Networked Control Systems: Analysis and Synthesis

This paper focuses on the problems of asymptotic stability and finite‐time stability (FTS) analysis, along with the state feedback controller design for networked control systems (NCSs) with consideration of both network‐induced delay and packet dropout. The closed‐loop NCS is modeled as a discrete‐time linear system with a time‐varying, bounded state delay. Sufficient conditions for the asymptotic stability and the FTS of the closed‐loop NCS are provided, respectively. Based on the stability analysis results, a mixed controller design method, which guarantees the asymptotic stability of the closed‐loop NCS in the usual case and the FTS of the closed‐loop NCS in the unusual case (that is, in some particular time intervals, large state delay occurs), is presented. A numerical example is provided to illustrate the effectiveness of the proposed mixed controller design 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.     

SLIDING MODE CONTROL SYNTHESIS OF UNCERTAIN TIME‐DELAY SYSTEMS

Sliding mode control synthesis is developed for a class of uncertain time‐delay systems with nonlinear disturbances and unknown delay values whose unperturbed dynamics is linear. The synthesis is based on a new delay‐dependent stability criterion. The controller constructed proves to be robust against sufficiently small delay variations and external disturbances. An admissible upperbound such that the corresponding closedloop system remains globally asymptotically stable for each delay value less than this up‐perbound is derived. Performance issues of the controller are illustrated in a simulation study.