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

Repetitive and 2-D systems theory approach for modeling in gas networks

“…The most accurate linear model can be realized in some neighborhood of the known basic regime ) , ( ), , ( t x p t x Q of the considered process. It can be shown [4] that the linearized model in the neighborhood of the known (preassigned/basic) regime ) , ( p Q has the following form…”

“…In this paper the mathematical model and corresponding optimization problem of gas network units are introduced on the basis of the so-called repetitive processes [2] and the 2-D system theory setting [3], which is a starting point for the further investigation of complex networks and which provide a fairly well-established mathematical framework. Some aspects of control theory for multidimensional systems are investigated in [4,5] and application of it to gas networks have been considered in [6]. In this paper we develop substantial new results on optimal control of differential linear repetitive processes with constraints which we motivate from the introduced linearization gas pipeline model in the presence of constraints The analysis is based on generalizing the well known maximum and ε -maximum principles.…”

Delay-dependent stability and stabilisation of continuous 2D delayed systems with saturating control

“…Example 2: Some dynamical processes in gas absorption, water stream heating, and air drying can be described by the Darboux equation with time delays (Dymkou, Rogers, Dymkov, & Galkowski, 2008): 0 , a 1 , a 2 , a 3 , and b are real coefficients, τ 2 is a time delay, and f(x, t) is the input function. Defining…”

Repetitive models in gas transportation networks

“…For a representative survey of these see, for example, [5]. The main elements of the theory of constructive optimization for the repetitive processes were developed in [6], [7]. Some aspects of control theory for multidimensional systems are investigated in [8], [9] and application of it to gas networks have been considered in [10].…”