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This chapter first introduces the unique features and control problems for repetitive processes by reference to two physical examples - long-wall coal cutting and metal rolling. Two so-called algorithmic examples are considered next, i.e. problem areas where adopting a repetitive process approach to modelling and analysis has clear advantages over alternatives. All these examples are shown to be special cases of the general abstract model in a Banach space setting on which the stability theory for linear repetitive processes is based. Finally, the links at the modelling/structural level with well known 2D discrete and standard, termed 1D in this setting, linear systems are detailed.

This chapter presents the stability theory for the Banach space based abstract model of Definition 1.2.1, together with the results of applying it to the various forms of differential and discrete linear repetitive processes introduced in the previous chapter. Also the question of testing the resulting stability conditions is addressed by a number of routes. These lead in most cases to computationally feasible tests and, in one case of applications interest, computable bounds on expected performance.

In this chapter we investigate the role of Lyapunov equations in the stability related analysis of discrete linear repetitive processes. There are two types of Lyapunov equation which, in common with the 2D systems literature, are termed 1D and 2D respectively. The key difference here is that in the latter case the matrices involved have constant entries whereas those in the former have entries which are functions of a complex variable. Moreover, the 1D equation gives necessary and sufficient conditions for stability along the pass whereas the 2D equation gives, in general, sufficient but not necessary conditions. The role of these equations in the extraction of computable measures of the performance of stable processes is also investigated.

In this chapter we investigate the role of Lyapunov equations in the stability related analysis of differential linear repetitive processes. As for the discrete processes of the previous chapter, there are two types of Lyapunov equation, again termed 1D and 2D respectively. Also it is shown that the Lyapunov equation approach can be extended to deal with one case of dynamic boundary conditions.

In this chapter we study the stability robustness of discrete unit memory linear repetitive processes of the form defined by Example 1.2.10 and their differential counterparts of Example 1.2.3. In the former case, a range of techniques are considered which draw on results already known in 2D systems theory. For the differential case, it is only an LMI approach which can be applied. In subsequent chapters, it will be shown that (of those currently available) only the LMI approach can be extended to permit control law design, including the case when there is uncertainty associated with the process model.

In this chapter, we first investigate controllability and observability of discrete and differential linear repetitive processes. As with the 2D linear systems case, it will be shown that more than one physically relevant controllability property can be defined in the discrete case and also that clear differences exist with 2D discrete linear systems. Observability is only addressed for differential processes since for the discrete case this property is almost identical to that for 2D discrete linear systems described by the Roesser or Fornasini-Marchesini state-space models. The next topic covered in this chapter is the extension of the 1D theory of system equivalence to linear repetitive processes using the 2D transfer-function matrix descriptions of Chap. 1. Finally, a behavioral approach is used to define poles and zeros for these processes which, in contrast to those in the 2D/nD systems literature, have interpretations in terms of the existence of exponential trajectories in the dynamics, i.e. a natural generalization of the well known 1D linear systems case.

In this chapter a substantial volume of results on the control of differential and discrete linear repetitive processes are developed and illustrated, where applicable, by numerical examples. The control laws (or controllers) used make use of both current and previous pass (or passes in the non-unit memory case) and the resulting design algorithms fall, in the main, into two general classes. The first of these execute the design using Linear Matrix Inequalities and the second by minimizing a suitably formulated cost function.

In this chapter the theme is again control law (or controller) design but here we allow uncertainty in the model structure. The uncertainty allowed is described by particular model structures, some of which have already been introduced in Chap. 5.

In this chapter, the application of repetitive process theory to iterative learning control is considered. The first part uses the stability theory of Chap. 2 to establish a fundamental link between convergence of a powerful class of such schemes and the resulting error dynamics. Following this, the norm optimal approach is developed and an experimental verification on a chain conveyor system described. Finally, some highly promising results on robust control are obtained and experimentally verified on a gantry robot system.

In this chapter we first review progress in terms of the results given in this monograph. Then we proceed to discuss areas for further work. These include further development/extension of the results given here and new applications areas.