Constrained Optimal Control Theory for Differential Linear Repetitive Processes

Abstract

Abstract. Differential repetitive processes are a distinct class of continuous-discrete twodimensional linear systems of both systems theoretic and applications interest. These processes complete a series of sweeps termed passes through a set of dynamics defined over a finite duration known as the pass length, and once the end is reached the process is reset to its starting position before the next pass begins. Moreover the output or pass profile produced on each pass explicitly contributes to the dynamics of the next one. Applications areas include iterative learning control and iterative solution algorithms, for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modeling of numerous industrial processes such as metal rolling, long-wall cutting, etc. In this paper we develop substantial new results on optimal control of these processes in the presence of constraints where the cost function and constraints are motivated by practical application of iterative learning control to robotic manipulators and other electromechanical systems. The analysis is based on generalizing the well-known maximum and -maximum principles to them.Key words. two-dimensional systems, optimal control, constraints AMS subject classifications. 93C05, 93C15 DOI. 10.1137/060668298 1. Introduction. Repetitive processes are a distinct class of two-dimensional (2D) systems of both systems theoretic and applications interest. The unique characteristic of such a process is a series of sweeps, termed passes, through a set of dynamics defined over a fixed finite duration known as the pass length. On each pass an output, termed the pass profile, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This, in turn, leads to the unique control problem in that the output sequence of pass profiles generated can contain oscillations which increase in amplitude in the pass-to-pass direction.To introduce a formal definition, let α < +∞ denote the pass length (assumed constant). Then in a repetitive process the pass profile y k (t), 0 ≤ t ≤ α, generated on pass k acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile y k+1 (t), 0 ≤ t ≤ α, k ≥ 0.Physical examples of repetitive processes include long-wall coal cutting and metalrolling operations (see, for example, the references cited in [17]). Also in recent years applications have arisen where adopting a repetitive process setting for analysis has distinct advantages over alternatives. Examples of these so-called algorithmic applications include classes of iterative learning control (ILC) schemes (see, for example, [13]) and iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle [15]. In the case of iterative learning control for the linear

Optimisation problem for some class of hybrid differential-difference systems with delay

“…In particular, our interest is the optimal control problem where the state variables at the final interval α α , + [11]). Also, it is worth mentioning here that the approach of the supporting control functions setting described in [12] can be used for the design of the numerical algorithms with good conditioning properties. These problems are subject of ongoing work and will be reported in due course.…”