This paper considers iterative learning control law design for both trial-to-trial error convergence and along the trial performance. It is shown how a class of control laws can be designed using the theory of linear repetitive processes for this problem where the computations are in terms of linear matrix inequalities (LMIs). It is also shown how this setting extends to allow the design of robust control laws in the presence of uncertainty in the dynamics produced along the trials. Results from the experimental application of these laws on a gantry robot performing a pick and place operation are also given.
Iterative learning control (ILC) is a technique for controlling systems operating in a repetitive (or pass-to-pass) mode with the requirement that a reference trajectory yref(t)yref(t) defined over a finite interval 0≤t≤α0≤t≤α, where αα denotes the trial length, is followed to a high precision. Examples of such systems include robotic manipulators that are required to repeat a given task, chemical batch processes or, more generally, the class of tracking systems.
Since the original work Arimoto, Kawamura, and Miyazaki (1984), the general area of ILC has been the subject of intense research effort. Initial sources for the literature here are the survey papers Bristow, Tharayil, and Alleyne (2006) and Ahn, Chen, and Moore (2007). In ILC, a major objective is to achieve convergence of the trial-to-trial error. It is, however, possible that enforcing fast convergence could lead to unsatisfactory performance along the trial, and here this problem is addressed by first showing that ILC schemes can be designed for a class of discrete linear systems by extending techniques developed for linear repetitive processes (Rogers, Galkowski, & Owens, 2007). This allows us to use the strong concept of stability along the pass (or trial) for these processes, in an ILC setting, as a possible means of dealing with poor/unacceptable transients in the dynamics produced along the trials. The results developed give control law design algorithms that can be implemented via LMIs, and results from their experimental implementation on a gantry robot executing a pick and place operation are also given. Finally, it is shown how the analysis can be extended to robust control where the uncertainty is associated with along the trial dynamics, and again supporting experimental results are given.
Throughout this paper M≻0M≻0 (respectively, ≺0≺0) denotes a real symmetric positive (respectively, negative) definite matrix. Also the identity and null matrices of the required dimensions are denoted by I and 0, respectively.
The stability theory for discrete linear repetitive processes has been used to design ILC laws for both trial-to-trial error convergence and performance along the trial. The resulting computations are in the form of LMIs. Results from the experimental application of these laws to a gantry robot are also reported and show that control over both performance aspects, i.e. trial-to-trial and along the trial, respectively, is possible. A similar conclusion also follows from the results in this paper that extends the algorithms to the case when there is uncertainty associated with the process dynamics.
The results in this paper deal with uncertainty in the trial dynamics. It is important to stress, however, that the approach here is only one way of addressing the problem and alternatives exist such as the lifting approach, see, for example, van de Wijdeven and Bosgra (2008) or formulating the problem in the frequency domain. For eventual application to industrial systems, it is therefore necessary to provide ways of reducing the conservativeness inherent in all of these approaches. In this context, the results here provide a comparative basis, in both simulation and experiments, for any results obtained.
