Design of Iterative Learning Controllers for Linear Discrete Systems with Multiple Time Delays

Abstract

In a number of practical control problems, there is a class of repetitive dynamical systems, such as robotic control systems, neuromuscular stimulation systems, and so on. For such repetitive dynamical systems, the so-called iterative learning control laws have been introduced. Generally speaking, by employing an iterative learning control algorithm, one can gradually improve or perfect the system performance of a specified task, based on the previous performances of the identical tasks.It is well known that the main advantage of the iterative learning control strategy is to require less a priori knowledge about the system dynamics and less computational effort than many other types of control strategies. Therefore, the problem of iterative learning control for repetitive dynamical systems has received considerable attention, and many results have been obtained. In particular, there are some works in which an iterative learning control scheme has been applied to the analysis and design of continuous dynamical systems with time-delay. On the other hand, the iterative learning control problem of discrete dynamical systems is also considered in the control literature, and some methods are presented to deal with such a class of iterative learning control problems. However, few efforts are made to consider the problem of iterative learning control for discrete dynamical systems with time-delay. It seems that for discrete timedelay dynamical systems, the similar results have not been reported yet in the control literature.In this paper, we consider a discrete-time dynamical system with time-delay, described byn is the current value of the state, u(m) ∈ R m is the control (or input) vector, y(m) ∈ R l is the output vector, A, B, C are constant matrices of appropriate dimensions, and the matrix Eµ, µ = 1, 2, . . . , r, represent the delayed state perturbations. Moreover, the time-varying delay hµ(m) is assumed to be any bounded nonnegative integer function which is not required to be known for the system designer. That is, 0 ≤ hµ(m) ≤hµ, wherehµ is a positive integer.The initial condition for the system with time delays is given by (1) is also assumed to be unknown at each iteration.For discrete time-delay system (1), it is supposed that a desired output trajectory yr(m) ∈ R l is given for a finite time interval m ∈ Γ N 0 := {0, 1, 2, . . . , N}. Then, the error between the desired output and the actual output trajectories of the discrete time-delay system can be represented by e(m) = yr(m) − y(m)· · · · · · · · · · · · · · · · · · · · · · · · · · · (3)where m ∈ Γ Here, we use the superscript k to denote the iteration number of learning processes. Therefore,[k] (m) represent the corresponding vectors at the kth iteration. Now, the main objective of this paper is to find the iterative learning control laws for discrete time-delay system (1) with the unknown initial state such that the output error e(m) between the given desired output yr(m) and the actual output y(m) is identical for all m ∈ Γ N 0 , through the iterativ...

Adaptive iterative learning control for nonlinearly parameterized systems with unknown time-varying delays

“…Although so many approaches have been developed in the field of ILC, only a few of them are concerning timedelay systems [26][27][28][29]32], and they are investigated only within the framework of classical ILC. To the best of our knowledge, up to now little work is reported from the viewpoint of AILC to deal with nonlinear time-delay systems, especially in the case where the unknown timevarying parameters and the unknown time-varying delays appear in the nonlinearly parameterized systems.…”