ADAPTIVE ROBUST CONTROL OF UNCERTAIN SYSTEMS WITH TIME‐VARYING STATE DELAY

In this paper, a new adaptive robust control scheme is developed for a class of uncertain dynamical systems with time‐varying state delay, unknown parameters and disturbances. By incorporating adaptive techniques into the robust control method, we propose a continuous adaptive robust controller which guarantees the uniform boundedness of the system and at the same time, the regulating error enters an arbitrarily designated zone in a finite time. The proposed controller is independent of the time‐delay, hence it is applicable to a class of dynamical systems with uncertain time delays. The paper includes simulation studies demonstrating the performance of the proposed control scheme.     

LMI APPROACH TO ROBUST FILTERING FOR DISCRETE TIME‐DELAY SYSTEMS WITH NONLINEAR DISTURBANCES

This paper investigates the problem of robust filtering for a class of uncertain nonlinear discrete‐time systems with multiple state delays. It is assumed that the parameter uncertainties appearing in all the system matrices reside in a polytope, and that the nonlinearities entering into both the state and measurement equations satisfy global Lipschitz conditions. Attention is focused on the design of robust full‐order and reduced‐order filters guaranteeing a prescribed noise attenuation level in an H∞ or l₂‐l∞ sense with respect to all energy‐bounded noise disturbances for all admissible uncertainties and time delays. Both delay‐dependent and independent approaches are developed by using linear matrix inequality (LMI) techniques, which are applicable to systems either with or without a priori information on the size of delays.     

ROBUST ADAPTIVE CONTROL FOR STRICT‐FEEDBACK NONLINEAR SYSTEMS

In this paper, we propose a robust adaptive tracking control based on the backstepping strategy for strict‐feedback nonlinear systems with nonparametric uncertain nonlinearities. It is shown that one can design a stable adaptive control system provided that the uncertain nonlinearities can be decomposed by unknown bounded nonlinear functions and known nonlinear functions. The proposed method can deal with uncertain nonlinearities that appear at the control input term too. It is also shown that suitable choice of design parameters guarantees the convergence of tracking error to any desired bound.     

OPTIMAL REJECTION WITH ZERO STEADY‐STATE ERROR OF SINUSOIDAL DISTURBANCES FOR TIME‐DELAY SYSTEMS

This paper studies the problem of optimal rejection with zero steady‐state error of sinusoidal disturbances for linear systems with time‐delay. Based on the internal model principle, a disturbance compensator is constructed to counterbalance the external sinusoidal disturbances, so that the original system can be transformed into an augmented system without disturbances. Then, with the introduction of a sensitivity parameter and expanding power series around it, the optimal disturbance rejection problem can be simplified to the problem of solving an infinite sum of a linear optimal control series without time‐delay or disturbance. The optimal control law for disturbance rejection with zero steady‐state error consists of accurate linear state feedback terms and a time‐delay compensating term, which is an infinite sum of an adjoint vector series. In the presented approach, iteration is required only for the time‐delay compensation series. By intercepting a finite sum of the compensation series, we obtain an approximate physically realizable optimal control law that avoids complex calculation. A numerical simulation shows that the algorithm is effective and easy to implement.     

ADAPTIVE DYNAMIC MATRIX CONTROL FOR NETWORK‐BASED CONTROL SYSTEMS WITH RANDOM DELAYS

Random transfer delays in network‐based control systems (NCSs) degrade the control performance and can even destabilize the control system. To address this problem, the adaptive dynamic matrix control (DMC) algorithm is proposed. The control algorithm is derived by applying the philosophy behind DMC to a discrete time‐delay model. A method to estimate the network‐induced delays is also presented to facilitate implementation of the control algorithm. Finally, an NCS platform based on the TrueTime simulator is constructed. With it, the adaptive DMC algorithm is compared with the conventional DMC algorithm under different network conditions. Simulation results show that the proposed adaptive DMC algorithm can respond to various network conditions adaptively and achieve better control performance for NCSs with random transfer delays.     

ROBUST H∞ CONTROL FOR UNCERTAIN STOCHASTIC SYSTEMS WITH MARKOVIAN SWITCHING AND TIME‐VARYING DELAYS

This paper is concerned with the problem of robust H_∞ control for uncertain stochastic systems with Markovian jump parameters and time‐varying state delays. A linear matrix inequality approach is developed and state feedback controllers are designed, which guarantee mean square asymptotic stability of the closed‐loop system and a prescribed H_∞ performance level for all modes and admissible uncertainties. A numerical example is provided to demonstrate the application of the proposed method.     

ROBUST ADAPTIVE CONTROL WITH MULTIPLE ESTIMATION MODELS FOR STABILIZATION OF A CLASS OF NON‐INVERSELY STABLE TIME‐VARYING PLANTS

This paper presents an indirect adaptive control scheme for nominally stabilizable non‐necessarily inversely stable continuous‐time systems with unmodelled dynamics. The control objective is the adaptive stabilization of the closed‐loop system with the achievement of a bounded tracking‐error between the system output and a reference signal given by a stable filter. The adaptive control scheme includes several estimation algorithms and a supervisor which selects the appropriate estimator at every certain time and keeping it operating for at least a minimum period of residence time. This selection is based on a performance criterion related to a measure of the estimation errors obtained with each estimator. In this way, the performance of the output signal is improved with regard to the performance achieved with a unique estimation algorithm. All the estimators are either of the least‐squares type or gradient type. However, any well‐posed estimation algorithm is potentially valid for application. These estimators include relative dead‐zones for robustness purposes and parameter ‘a posteriori’ modifications to ensure the controllability of the estimated models of the plant, which is crucial for proving the stabilizability of the plant via adaptive pole‐placement designs.     

OPTIMAL TRACKING CONTROL FOR DISCRETE TIME‐DELAY SYSTEMS WITH PERSISTENT DISTURBANCES

Optimal tracking control (OTC) for discrete time‐delay systems affected by persistent disturbances with quadratic performance indexes is considered. Optimal tracking controller is designed based on a sensitivity approximation approach. By introducing a sensitivity parameter, we transform the original OTC problem into a series of difference equations without time‐advance on time‐delay terms. The obtained OTC law consists of analytic feedback and feedforward terms, and a compensation term, which is the sum of the infinite series of adjoint vectors. The compensation term can be obtained with an iterated formula for the adjoint vectors. A simulation example shows that the approximation approach is effective in tracking the reference input and robust with respect to exogenous persistent disturbances.     

DECENTRALIZED STABILIZATION OF H FUZZY CONTROL FOR NONLINEAR MULTIPLE TIME‐DELAY INTERCONNECTED SYSTEMS

A robustness design of fuzzy control is proposed in this paper to overcome the effect of modeling errors between nonlinear multiple time‐delay systems and fuzzy models. In terms of Lyapunov's direct method, a stability criterion is derived to guarantee the UUB (uniformly ultimately bounded) stability of nonlinear multiple time‐delay interconnected systems with disturbances. Based on this criterion and the decentralized control scheme, a set of fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time‐delay interconnected systems and the Hcontrol performance is achieved in the mean time.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

LMI OPTIMIZATION APPROACH FOR DELAY‐DEPENDENT H∞ CONTROL OF TIME‐VARYING DELAY SYSTEMS

In this paper, the robust delay‐dependent H_∞ control for a class of uncertain systems with time‐varying delay is considered. An improved state feedback H_∞ control is proposed to minimize the H_∞‐norm bound via the LMI optimization approach. Based on the proposed result, delay‐dependent criteria are obtained without using the model transformation technique or bounded inequalities on cross product terms. The linear matrix inequality (LMI) optimization approach is used to design the robust H_∞ state feedback control. Some numerical examples are given to illustrate the effectiveness of the approach.     

SENSITIVITY APPROACH TO OPTIMAL CONTROL FOR AFFINE NONLINEAR DISCRETE‐TIME SYSTEMS

This paper deals with the optimal control problem for a class of affine nonlinear discrete‐time systems. By introducing a sensitivity parameter and expanding the system variables into a Maclaurin series around it, we transform the original optimal control problem for affine nonlinear discrete‐time systems into the optimal control problem for a sequence of linear discrete‐time systems. The optimal control law consists of an accurate linear term and a nonlinear compensating term, which is an infinite sequence of adjoint vectors. In the present approach, iteration is required only for the nonlinear compensation series. By intercepting a finite sum of the series, we obtain a suboptimal control law that reduces the complexity of the calculations. A numerical simulation shows that the algorithm can be easily implemented and has a fast convergence rate.     

ROBUST STABILITY AND STABILIZATION OF A CLASS OF SINGULAR SYSTEMS WITH MULTIPLE TIME‐VARYING DELAYS

This paper deals with the problem of robust stability and robust stabilization for uncertain continuous singular systems with multiple time‐varying delays. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties. The purpose of the robust stabilization problem is to design a feedback control law such that the resulting closed‐loop system is robustly stable. This problem is solved via generalized quadratic stability approach. A strict linear matrix inequality (LMI) design approach is developed. Finally, a numerical example is provided to demonstrate the application of the proposed method.     

ROBUST H∞ CONTROL AND QUADRATIC STABILIZATION OF DISCRETE‐TIME SWITCHED SYSTEMS WITH NORM‐BOUNDED TIME‐VARYING UNCERTAINTIES

We focus on robust H_∞ control analysis and synthesis for discretetime switched systems with norm‐bounded time‐varying uncertainties. Sufficient conditions are derived to guarantee quadratic stability along with a prescribed H_∞‐norm bound. Each of them can be dealt with as a linear matrix inequality (LMI) which can be tested with efficient algorithms. All the switching rules are constructively designed, and do not rely on any uncertainties.     

STABILITY ANALYSIS OF UNCERTAIN DISCRETE‐TIME SYSTEMS WITH TIME‐VARYING STATE DELAY: A PARAMETER‐DEPENDENT LYAPUNOV FUNCTION APPROACH

This paper presents several new robust stability conditions for linear discrete‐time systems with polytopic parameter uncertainties and time‐varying delay in the state. These stability criteria, derived by defining parameter‐dependent Lyapunov functions, are not only dependent on the maximum and minimum delay bounds, but also dependent on uncertain parameters in the sense that different Lyapunov functions are used for the entire uncertainty domain. It is established, theoretically, that these robust stability criteria for the nominal and constant‐delay case encompass some existing result as their special case. The delay‐dependent and parameter‐dependent nature of these results guarantees the proposed robust stability criteria to be potentially less conservative.     

STABILIZATION AND H∞ CONTROL FOR UNCERTAIN STOCHASTIC TIME‐DELAY SYSTEMS VIA NON‐FRAGILE CONTROLLERS

This paper considers the problems of robust non‐fragile stochastic stabilization and H_∞ control for uncertain time‐delay stochastic systems with time‐varying norm‐bounded parameter uncertainties in both the state and input matrices. Attention is focused on the design of memoryless state feedback controllers which are subject to norm‐bounded uncertainties. For both the cases of additive and multiplicative controller uncertainties, delay‐independent sufficient conditions for the solvability of the above problems are obtained. The desired state feedback controller can be constructed by solving a certain linear matrix inequality.     

A NEW LMI CONDITION FOR DELAY‐DEPENDENT ROBUST STABILITY OF STOCHASTIC TIME‐DELAY SYSTEMS

This paper studies robust stability for a class of uncertain nonlinear stochastic time‐delay systems. In terms of a linear matrix inequality, an improved delay‐dependent condition guaranteeing that a stochastic delay system will be exponentially stable in the mean square is proposed. This condition is less conservative than existing ones in the literature and is demonstrated by means of an example.     

DELAY‐DEPENDENT ROBUST CONTROL OF DESCRIPTOR SYSTEMS WITH TIME DELAY

This paper deals with the problem of stability and robust control for both certain and uncertain continuous‐time singular systems with state delay. Systems with norm‐bounded parameter uncertainties are considered. Robust delay‐dependent stability criteria and linear memoryless state feedback controllers based on linear matrix inequality are obtained. By choosing some Lyapunov‐Krasovskii functionals, neither model transformation nor bounding for cross terms is required in the derivation of our delay‐dependent results. Finally, numerical example is provided to illustrate the effectiveness of the proposed method.     

IMPROVED CONDITIONS FOR DELAY‐DEPENDENT ROBUST STABILITY AND STABILIZATION OF UNCERTAIN DISCRETE TIME‐DELAY SYSTEMS

This paper provides improved delay‐dependent conditions for the robust stability and robust stabilization of discrete time‐delay systems with norm‐bounded parameter uncertainties. It is theoretically established that the proposed conditions are less conservative than those discussed in the literature. The new approach proposed in this paper in a derivation of delay‐dependent conditions and involves the use of neither model transformation nor bounding techniques for some cross terms. A numerical example is provided to demonstrate the reduced conservatism of the proposed conditions.