Control of an Industrial Rolling Process Using The Theory of Switched Repetitive Processes Jacek Bochniak, Krzysztof Galkowski, Eric Rogers, Joerg Velten Faculty of Electrical Engineering, Computer Science and Telecommunications, University of Zielona Gora, Poland {jbochnia,kgalkows}@uz.zgora.pl School of Electronics and Computer Science, University of Southampton, United Kingdom etar@ecs.soton.ac.uk Faculty of Electrical, Information and Media Engineering, University of Wuppertal, Germany {galkowsk,velten}@uni-wuppertal.de Keywords: Multiprocess dynamics, Switching values, Two-dimensional systems, Stability, Stabilization 1. INTRODUCTION The unique characteristic of a repetitive pro- cess (also termed a multipass process in the early literature) can be illustrated by consider- ing machining operations where the material or workpiece involved is processed by a series of sweeps, or passes, of the processing tool. As- suming the pass length α< +∞ to be con- stant, the output vector, or pass profile, y k (p), p =0, 1,..., (α - 1),(p being the independent spatial or temporal variable), generated on pass k acts as a forcing function on, and hence con- tributes to, the dynamics of the new pass profile y k+1 (p), p =0, 1,..., (α - 1), k =0, 1, .... This, in turn, leads to the unique control prob- lem in that the output sequence of pass profiles generated can contain oscillations that increase in amplitude in the pass-to-pass direction, i.e. in the collection of pass profile vectors {y k } k . Industrial examples include long-wall coal- cutting and metal rolling, see the original pa- pers cited in, for example, (3) for further de- tails. A number of so-called algorithmic exam- ples also exist where adopting a repetitive process setting for analysis has clear advantages over al- ternative approaches to systems related analysis. These include iterative learning control schemes, e.g. (2) and iterative solution algorithms for dy- namic nonlinear optimal control problems based on the maximum principle. In the case of itera- tive learning control for the linear dynamics case, the stability theory for differential (and discrete) linear repetitive processes is one method which can be used to undertake a stability/convergence analysis of a powerful class of such algorithms and thereby produce vital design information concerning the trade-offs required between con- vergence and transient performance. In many practical applications, e.g. metal rolling, or processing operations using multiple operation robot arms, a number of passes may be completed under one regime and then the dynamics change to allow further processing to take place. One way of modeling such a case is by switching the dynamics from one state-space model to an alternative (or alternatives) and this paper continues the development of tools for the analysis of such models. Note also that there is one other form of switching which can occur in repetitive process dynamics, i.e. along the pass. Here, however, we restrict attention to the pass-to-pass case since it has more obvious and immediate practical appli- cations. In this paper we consider a metal rolling process where the workpiece involved is passed through two successive rolling operations which are to be controlled to produce a desired final product. We show that this can be modeled as a discrete linear repetitive process which switched dynamics in the pass-to-pass direction. Then we develop new results on stability and control law design and give an illustrative numerical exam- ple. 2. PROCESS MODELING In its simplest form a multi-roll roll system con- sists of two separate pairs of rolls which are con-