Delay‐dependent robust stability and stabilization for uncertain discrete singular systems with delays

The robust stability and robust stabilization for time‐delay discrete singular systems with parameter uncertainties is discussed. A delay‐dependent linear matrix inequality (LMI) condition for the time‐delay discrete systems to be nonsingular and stable is given. Based on this condition and the restricted system equivalent transformation, the delay‐dependent LMI condition is proposed for the time‐delay discrete singular systems to be admissible. With this condition, the problems of robust stability and robust stabilization are solved, and the delay‐dependent LMI conditions are obtained. Numerical examples illustrate the effectiveness of the method given in the paper. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

The problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust stability and H∞ control for uncertain discrete Markovian jump singular systems with mode‐dependent time‐delay

The robust stochastic stability, stabilization and H_∞ control for mode‐dependent time‐delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a standard linear system, and delay‐dependent linear matrix inequalities (LMIs) conditions for the mode‐dependent time‐delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with γ‐disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust H_∞ control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Delay‐range‐dependent robust stability and stabilization for uncertain systems with time‐varying delay

This paper concerns delay‐range‐dependent robust stability and stabilization for time‐delay system with linear fractional form uncertainty. The time delay is assumed to be a time‐varying continuous function belonging to a given range. On the basis of a novel Lyapunov–Krasovskii functional, which includes the information of the range, delay‐range‐dependent stability criteria are established in terms of linear matrix inequality. It is shown that the new criteria can provide less conservative results than some existing ones. Moreover, the stability criteria are also used to design the stabilizing state‐feedback controllers. Numerical examples are given to demonstrate the applicability of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Delay‐dependent robust H∞ controller synthesis for discrete singular delay systems

In this paper, the problems of delay‐dependent robust stability analysis, robust stabilization and robust H_∞ control are investigated for uncertain discrete‐time singular systems with state delay. First, by making use of the delay partitioning technique, a new delay‐dependent criterion is given to ensure the nominal system to be regular, causal and stable. This new criterion is further extended to singular systems with both delay and parameter uncertainties. Then, without the assumption that the considered systems being regular and causal, robust controllers are designed for discrete‐time singular time‐delay systems such that the closed‐loop systems have the characteristics of regularity, causality and asymptotic stability. Moreover, the problem of robust H_∞ control is solved following a similar line. The obtained results are dependent not only on the delay, but also on the partitioning size and the conservatism is non‐increasing with reducing partitioning size. These results are shown, via extensive numerical examples, to be much less conservative than the existing results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Delay‐dependant robust stability and stabilization conditions for a class of Lur'e singular time‐delay systems

The problem of the delay‐dependant robust stability and stabilization is dealt with for a class of Lur'e singular system with state time‐delays in this paper. The new concept of generalized robust quadratic stability and generalized robust quadratic stabilization is presented. Furthermore, the sufficient condition of the robust stability and stabilization is also proposed based on linear matrix inequalities. The synthesis problem addressed is to design a memoryless state feedback control law such that the resulting closed‐loop system is regular, impulse free and asymptotically stable for all admissible uncertainties. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust stabilization for uncertain discrete singular systems with state delay

The robust stabilization problem for uncertain discrete singular time‐delay systems is addressed in this paper. In terms of strict linear matrix inequality and a finite sum inequality, a delay‐dependent criterion for the nominal systems to be admissible is obtained. Based on the criterion, a state feedback controller, which guarantees that, for all admissible uncertainties, the resulting closed‐loop system is regular, causal and stable, is constructed. An explicit expression for the desired controller is also given. The obtained results include both delay‐independent and delay‐dependent cases. Some numerical examples are introduced to show the effectiveness of the given results. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust stabilization and H∞ control of uncertain stochastic time‐delay systems with nonlinear perturbation

This paper investigates the problem of delay‐dependent robust stochastic stabilization and H_∞ control for uncertain stochastic nonlinear systems with time‐varying delay. System uncertainties are assumed to be norm bounded. Firstly, by using novel method to deal with the integral terms, robustly stochastic stabilization results are obtained for stochastic uncertain systems with nonlinear perturbation, and an appropriate memoryless state feedback controller can be chosen. Compared with previous results, the new technique can sufficiently utilize more negative items information. Then, robust H_∞ control for uncertain stochastic system with time‐varying delay and nonlinear perturbation is considered, and the controller is designed, which will guarantee that closed‐loop system is robustly stochastically stable with disturbance attenuation level. Finally, two numerical examples are listed to illustrate that our results are effective and less conservative than other reports in previous literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite‐time robust simultaneous stabilization of a set of nonlinear time‐delay systems

This paper investigates the finite‐time robust simultaneous stabilization problem of a set of nonlinear time‐delay systems with general forms and proposes some new simultaneous stabilization results. First, by developing an equivalent form and applying augmented technique, this paper obtains an augmented equivalent form of the original systems. Secondly, based on the equivalent form, we study finite‐time simultaneous stabilization problem and present some new stabilization results by constructing some suitable Lyapunov functionals. Thirdly, using the simultaneous stabilization results obtained, this paper investigates the finite‐time robust simultaneous stabilization problem for the set systems and proposes a delay‐dependent robust simultaneous stabilization result. Finally, the study of an illustrative example shows that the results obtained by this paper work well in the finite‐time robust simultaneous stabilization the set systems. It is shown that, by using the method in this paper, the developed conditions do not contain delay terms, which can avoid solving nonlinear mixed matrix inequalities and reduce effectively computational burden in studying nonlinear time‐delay systems.     

A NEW DESIGN APPROACH TO DELAY‐DEPENDENT ROBUST H∞ CONTROL FOR UNCERTAIN TIME‐DELAY SYSTEMS

A new design approach to delay‐dependent robust stabilization and robust H∞ control for a class of uncertain time‐delay systems is provided in this paper. The sufficient conditions for delay‐dependent robust stabilization and robust H∞ control are derived based on a new state transformation and given in terms of linear matrix inequalities (LMI). Numerical examples are presented to show that the proposed results can be less conservative and can be used to deal with not only small but also large delay systems.     

Delay‐dependent robust stabilization for uncertain singular systems with multiple time‐varying state delays

In this note, the problem of delay‐dependent robust stabilization for singular systems with multiple time‐varying state delays has been investigated, and the problem is solved via state feedback controller in terms of a linear matrix inequality technique. Numerical examples are given to show the validity of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust Exponential Stability of Discrete‐Time Delay Impulsive Systems with Parametric Uncertainties

This paper investigates robust exponential stability for discrete‐time delay impulsive systems with parametric uncertainties. The parametric uncertainties in the systems are assumed to be time varying and norm bounded. Using Lyapunov functionals, some robust exponential stability criteria are given. It is shown that the time interval between the nearest two impulses should be small enough, i.e., impulses must act frequently, when the impulses are employed to stabilize the original impulse‐free system that is not robustly stable. Conversely, when the original system without impulses is robustly stable, the time interval between the nearest two impulses should be large enough to let the system with impulsive perturbations retain its stability property. It should be noted that this is the first time that impulsive robust exponential stabilization results are given via Lyapunov functionals for discrete‐time uncertain delay impulsive systems. Some examples, including an example which cannot be studied by the existing results, are also presented to illustrate the effectiveness of the obtained results.     

AN LMI APPROACH TO ROBUST STABILITY OF UNCERTAIN DISCRETE SINGULAR TIME‐DELAY SYSTEMS

The problem of robust stability analysis for uncertain discrete singular time‐delay systems is investigated in this paper. By decomposing the nominal system into slow and fast subsystems, a linear matrix inequality (LMI) condition is proposed for a discrete singular time‐delay system to be regular, causal and stable. Based on this, an LMI criterion is obtained for robust stability of an uncertain discrete singular time‐delay system. Two numerical examples are provided to demonstrate the feasibility of the proposed approach.     

Finite‐time adaptive robust control of nonlinear time‐delay uncertain systems with disturbance

This paper investigates finite‐time adaptive robust control problem for a general class of nonlinear time‐delay systems with uncertain and external disturbance via the Lyapunov‐Krasovskii (L‐K) method and presents some delay‐independent and delay‐dependent results on the issue. First, by applying the orthogonal decomposition method, this paper presents an equivalent form. Based on which, we study the finite‐time adaptive robust control problem for the systems by constructing a specific L‐K functional and designing a suitable controller. Different from existing works, this paper studies finite‐time adaptive robust control problem for general nonlinear delay system and presents a delay‐dependent sufficient condition on the problem. Finally, an illustrative example is given to show the effectiveness of the result in this paper.     

Robust exponential stability of uncertain stochastic systems with probabilistic time‐varying delays

This paper is concerned with the problem of delay‐distribution–dependent robust exponential stability for uncertain stochastic systems with probabilistic time‐varying delays. Firstly, inspired by a class of networked systems with quantization and packet losses, we study the stabilization problem for a class of network‐based uncertain stochastic systems with probabilistic time‐varying delays. Secondly, an equivalent model of the resulting closed‐loop network‐based uncertain stochastic system is constructed. Different from the previous works, the proposed equivalent system model enables the controller design of the network‐based uncertain stochastic systems to enjoy the advantage of probability distribution characteristic of packet losses. Thirdly, by applying the Lyapunov‐Krasovskii functional approach and the stochastic stability theory, delay‐distribution–dependent robust exponential mean‐square stability criteria are derived, and the sufficient conditions for the design of the delay‐distribution–dependent controller are then proposed to guarantee the stability of the resulting system. Finally, a case study is given to show the effectiveness of the results derived. Moreover, the allowable upper bound of consecutive packet losses will be larger in the case that the probability distribution characteristic of packet losses is taken into consideration.     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

Robust delay‐dependent ℋ︁∞ control of time‐delay systems with state and input delays

This paper studies the design problem of robust delay‐dependent ??_∞ controller for a class of time‐delay control systems with time‐varying state and input delays, which are assumed to be noncoincident. The system is subject to norm‐bounded uncertainties and ??₂ disturbances. Based on the selection of an augmented form of Lyapunov–Krasovskii (L‐K) functional, first a Bounded Real Lemma (BRL) is obtained in terms of linear matrix inequalities (LMIs) such that the nominal, unforced time‐delay system is guaranteed to be globally asymptotically stable with minimum allowable disturbance attenuation level. Extending BRL, sufficient delay‐dependent criteria are developed for a stabilizing ??_∞ controller synthesis involving a matrix inequality for which a nonlinear optimization algorithm with LMIs is proposed to get feasible solution to the problem. Moreover, for the case of existence of norm‐bounded uncertainties, both the BRL and ??_∞ stabilization criteria are easily extended by employing a well‐known bounding technique. A plenty of numerical examples are given to illustrate the application of the proposed methodology of this note. The achieved numerical results on the maximum allowable delay bound and minimum allowable disturbance attenuation level are exhibited to be less conservative in comparison to those of existing methods in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

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.