Iterative Learning Control for Deterministic Systems

Recently, increased attention has been given in the media and trade journals to the relationship between a country’s performance in the industrial sector and its success in the international marketplace. It has been generally observed that future excellence in manufacturing and industrial automation is essential to improving quality and competitiveness in the industrial sector. An important factor in providing the technological base necessary to achieve this excellence is the development of advanced industrial control systems. As more sophisticated control strategies are developed for practical systems (such as robotic manipulators, motion control systems, and process control systems), it will be possible for the manufacturing and industrial control industries to reduce production costs and increase their competitiveness.

In this chapter we give an overview of the field of iterative learning control. We begin with an introduction to the concept of learning control. Then we summarize the past work that has been reported in the literature. We conclude the chapter with a general formulation and statement of the learning control problem.

Consider again the iterative learning control configuration shown in Figure 3.1. We will now restrict our discussion to the case where the plant is a causal, linear time-invariant (LTI) dynamical system S. We suppose S is represented by the operator T _s , so that y = T _s u. We also restrict the learning controller L to be causal and LTI. Although such a plant may not be encountered in practice, we will consider this class as a starting point to gain insight into the problem. As noted in the previous chapter, other assumptions include: (i) the desired response y _d (t) is defined on the interval (t₀, t _f ), where t _f may be infinity; and (ii) the initial conditions are reset at the beginning of each trial (although this fact is never used explicitly in any of our proofs, it is used implicitly because we usually suppose that all the initial conditions are zero). To avoid cluttered notation, the time dependence of the various signals will often be suppressed. Additionally, time may be continuous or discrete in our formulation. A final comment is that we are implicitly assuming that the plant T _s is known.

In the previous chapter we were interested in a qualitative assessment of the nature of a linear time-invariant (LTI) learning scheme. Because we wanted to get some idea of the best possible performance, we assumed the system dynamics were known. In this chapter we partially relax this assumption. We now consider the learning control problem when the plant has a known structure, but unknown parameters. Our approach to this problem will be based on parameter estimation techniques. Previously we found that, if the plant is known, then iteration is not necessary. However, if the plant is not known, then it seems reasonable to expect that learning would be useful in improving the system response. Below we describe a learning control scheme for this situation. We begin with a description of the scheme. Then we present our main result, followed by some comments on the implications of the result.

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.

As we have noted, the real usefulness of iterative learning control is for problems in which we wish to control the transient response behavior of nonlinear or time-varying systems. In this case it makes sense to consider learning controllers that also have a nonlinear or time-varying structure, such as the learning control scheme demonstrated in the previous chapter. A non-trivial question, however, is what type of nonlinear system should be considered. The class of all nonlinear systems is very large and it is not clear what structure would work best for learning control applications. One answer to this question is to consider the class of nonlinear systems called artificial neural networks. Artificial neural networks, with their nonlinear structure and their ability to learn internal representations and recall associations, are good candidates for nonlinear learning control.

In this research monograph we have presented a number of results related to the analysis and design of iterative learning control systems, motivated by the problem of transient response control in deterministic systems. The monograph began with a description of the concept of learning control and its motivation, a summary of the status of the problem, and a general problem formulation.