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

Abstract

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.