A Stochastic Design Approach for Iterative Learning Observers for the Estimation of Periodically Recurring Trajectories and Disturbances

Many repetitive control problems are characterized by the fact that disturbances have the same effect in each successive execution of the same control task. Such disturbances comprise the lumped representation of unmodeled parts of the open-loop system dynamics, a systematic model-mismatch or, more generally, deterministic yet unknown uncertainty. In such cases, well-known strategies for iterative learning control are based on enhancing the system behavior not only by exploiting data gathered during a single execution of the task but also using information from previous executions. The corresponding dual problem, namely, iterative learning state and disturbance estimation has not yet received the same amount of attention. However, it is obvious that improved estimates for the aforementioned states and disturbances which periodically occur in each execution will be a means to achieve an improved accuracy and, therefore, in future work also to optimize the control accuracy. In this paper, we present a joint design procedure for observer gains in two independent dimensions, a gain for processing information in the temporal domain during a single execution of the task (also named trial) and a gain for learning in the iteration domain (i.e., from trial to trial).

I. Introduction

Two-dimensional (2D) systems with the time domain and the iteration domain as two independent dimensions have been widely used in the last decades to derive iterative learning control (ILC) procedures [1]–[3]. Such approaches can be applied effectively for enhancing the control accuracy of repetitive tasks that are characterized by identical reference trajectories of finite length during each successive execution of a control task, where before the restart of the execution a reset to (nearly) the same initial conditions takes place. Such tasks occur widely in pick and place operations of manufacturing processes as well as during welding executed by robots. They can also be found in other areas such as rehabilitation or the control of wind power plants. ILC has the unique feature that it does not only exploit past data that are classically available in control tasks from the current execution of the task under consideration. In addition, it also exploits information from previous evaluations and is hence able to outperform control implementations that only exploit current trial data. By using information from one or multiple previous trials, control structures can be implemented which classically would even be acausal.