Finite-Horizon Learning Control

In the two preceding chapters we have presented a general development of the learning control problem for linear time-invariant (LTI) systems. The results presented can be applied to both finite- and infinite-horizon problems. However, as we have noted, any practical implementation of an iterative learning controller will be a finite-time problem. That is, the duration of each trial will be fixed to some time less than infinity. This is inherent in the nature of repetitive operations. In addition, our formulation and results are equally valid for continuous-time and discrete-time systems. However, to implement a learning control scheme we will have to use a microprocessor-based controller. For these reasons, it is reasonable to restrict our attention to discrete-time plants that are operated repetitively on a finite time horizon. In this chapter we consider the learning control problem for such systems. Our results are given for LTI, single-input, single-output plants, but can be generalized to multiple-input, multiple-output systems. We first give a learning control scheme with memory, using an l_∞, criterion. Then we show that this scheme can actually be modified to give a single-trial convergence rate. In the third section we present a learning control scheme for a discrete-time system with multirate sampling. These techniques are illustrated with examples. The final section of the chapter discusses extensions of these results to linear time-varying systems.