Class Of Discrete-time Singular Systems Research Articles

A State Space Decomposition Filtering Method for a Class of Discrete-Time Singular Systems

This paper develops a state space decomposition method for a class of discrete-time singular systems. The singular matrix in this system is allowed to vary not only in the value but also in the dimension. An orthogonal factorization is used to rewrite the original space as a direct sum of its two subspaces. Correspondingly, the singular system is replaced by two reduced-order systems, whose states are then calculated respectively. Based on this construction, the prediction of part of the state, optimal filtering result, an optimal one step smoothing result is proposed. Lastly, a numerical example is presented to demonstrate the effectiveness of the new method.

Iterative learning control for a class of discrete-time singular systems

This paper is concerned with the iterative learning control problem for a class of discrete-time singular systems. According to the characteristics of the systems, a closed-loop PD-type learning algorithm is proposed and the convergence condition of the algorithm is established. It is shown that the algorithm can guarantee the system output converges to the desired trajectory on the whole time interval. Moreover, the presented algorithm is also suitable for discrete-time singular systems with state delay. Finally, the validity of the presented algorithm is verified by two numerical examples.

StochasticH∞Control for Discrete-Time Singular Systems with State and Disturbance Dependent Noise

This paper is concerned with the stochasticH∞state feedback control problem for a class of discrete-time singular systems with state and disturbance dependent noise. Two stochastic bounded real lemmas (SBRLs) are proposed via strict linear matrix inequalities (LMIs). Based on the obtained SBRLs, a state feedbackH∞controller is presented, which not only guarantees the resulting closed-loop system to be mean square admissible but also satisfies a prescribedH∞performance level. A numerical example is finally given to illustrate the effectiveness of the proposed theoretical results.

Iterative learning control of a class of discrete-time singular system

Singular systems are also called descriptor systems. Compared with normal systems, singular systems have become one of effective tools which can describe and characterize varieties of real systems, because they can better describe physical properties of the systems. Up to now, the analyses and syntheses of singular systems have been widespread applied to linear matrix inequality (LMI) method. The method requires known systems to have accurate mathematical model information, however, real systems have difficulties in obtaining their accurate mathematical models, owing to the fact that real systems are frequently subjected to all kinds of interferences, uncertainties, and nonlinear factors. Specially, in the case of singular systems, obtained LMI conditions often have a constraint equation or LMI is semi-definite, which makes it more difficult to solve LMI. Therefore, in order to avoid the above two problems occurring in settling state tracking issue for singular systems, in the meantime, for convenience of computating control algorithm and information storage, in this paper we propose a discrete-time iterative learning control algorithm for a class of discrete-time singular system with repetitive running characteristics in finite time interval. The specific process is divided into two steps. First, the class of discrete-time singular system is decomposed into normal discrete-time state equation and algebraic equation form by nonsingular transformation. Accordingly, the singular system state is also decomposed into two parts. Among them, the dimension of the first part state is equal to singular matrix rank and another is equal to system dimension minus singular matrix rank. In addition, the control law of last iterative learning is modified by using two tracking errors at two different times: one error is real-time tracking error generated from the comparison between the first part state and its desired state and another is tracking error at a previous time generated from comparison between the second part state and its desired state. And thus a new control law of next iterative learning is obtained, such that, as for any given real singular system, its state may completely track the desired state as long as selected learning gain can satisfy the convergence condition of the algorithm. Further, the convergence of the control algorithm is theoretically proved by compression mapping method, and thus its sufficient convergence condition is given in the sense of -norm. The results indicate that the proposed iterative learning control algorithm can make system state realize the perfect tracking of desired state as iteration number gradually increases in finite time interval, and the convergence of the algorithm only depends on system parameters and learning gain rather than initial value of control variable. The simulation example finally verifies the effectiveness of the proposed algorithm.

Input‐to‐state stability of discrete‐time singular systems based on quasi‐min–max model predictive control

This study is concerned with robust quasi-min–max model predictive control (MPC) for a class of discrete-time singular systems with persistent disturbance and input constrains. To deal with the persistent disturbance, the authors introduce the notion of input-to-state stability (ISS) of discrete-time singular system for the first time. The optimal control can be obtained by solving a quasi-min–max optimal problem of a finite horizon cost function. On the basis of the proposed dual-mode MPC approach, it can be proved that the closed-loop discrete-time singular system is ISS, regular and causal. Finally, a numerical simulation shows the feasibility and effectiveness of the proposed method.

Non-fragile robust H∞ control for uncertain discrete-time singular systems with time-varying delays

In this paper, the problem of delay-dependent non-fragile robust H∞ control for a class of discrete-time singular systems with state-delay and parameter uncertainties is investigated. Based on singular value decomposition approach, a delay-dependent sufficient condition for the H∞ control problem for a class of discrete-time singular systems is proposed by constructing generalized Lyapunov–Krasovskii function and a new difference inequality. A memoryless state feedback controller under controller gain perturbations is designed, which guarantees that, for all admissible uncertainties, the resultant closed-loop system is regular, causal, and stable with an H∞ norm bound constraint. Numerical examples in the last will show that our results have the better performance in conservativeness than some results reported in the literature.

Delay-dependent stability analysis for discrete-time singular Markovian jump systems with time-varying delay

This article considers the problem of delay-dependent stability analysis for a class of discrete-time singular systems with Markovian jump and time-varying delay. The transition probabilities in Markov chain are assumed to be partially unknown. In terms of linear matrix inequality approach, the delay-dependent criteria are proposed to ensure the underlying system to be regular, causal and stochastically stable. Several numerical examples are given to demonstrate the effectiveness and less conservatism of the obtained results.

Further Results on H ∞ Filtering for a Class of Discrete-Time Singular Systems with Interval Time-Varying Delay

In this paper, the problem of delay-dependent robust H∞ filtering for discrete-time singular systems with time-varying delay and polytopic uncertainties is investigated. Without introducing the free-weighting matrices to deal with the cross terms, a delay-dependent H∞ performance analysis result is established for error singular system by considering the relationship between the time-varying delay and its bounds at first. Based on this result, a linear robust H∞ filter is designed such that the filtering error singular system is regular, causal and asymptotically stable with a H∞ norm bound. Numerical examples are given to show the advantages of the proposed results.

Further results onH ∞ control for discrete-time uncertain singular systems with interval time-varying delays in state and input

SUMMARY This paper investigates the state feedback robust H ∞ control problem of a class of discrete-time singular systems with norm-bounded uncertainties and interval time-varying delays in state and input. A new bounded real lemma for discrete-time singular systems with a pair of time-varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robust H ∞ control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.

Exponential Estimates and Stabilization of Discrete‐Time Singular Time‐Delay Systems Subject to Actuator Saturation

This paper is concerned with exponential estimates and stabilization of a class of discrete‐time singular systems with time‐varying state delays and saturating actuators. By constructing a decay‐rate‐dependent Lyapunov‐Krasovskii function and utilizing the slow‐fast decomposition technique, an exponential admissibility condition, which not only guarantees the regularity, causality, and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities (LMIs). Under the proposed condition, the exponential stabilization problem of discrete‐time singular time‐delay systems subject actuator saturation is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed‐loop system is guaranteed. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

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.