Learning control algorithms for tracking "slowly" varying trajectories

To date, most of the available results in learning control have been utilized in applications where a robot is required to execute the same motion over and over again, with a certain periodicity. This is due to the requirement that all learning algorithms assume that a desired output is given a priori over the time duration t in [0,T]. For applications where the desired outputs are assumed to change "slowly", we present a D-type, PD-type, and PID-type learning algorithms. At each iteration we assume that the system outputs and desired trajectories are contaminated with measurement noise, the system state contains disturbances, and errors are present during reinitialization. These algorithms are shown to be robust and convergent under certain conditions. In theory, the uniform convergence of learning algorithms is achieved as the number of iterations tends to infinity. However, in practice we desire to stop the process after a minimum number of iterations such that the trajectory errors are less than a desired tolerance bound. We present a methodology which is devoted to alleviate the difficulty of determining a priori the controller parameters such that the speed of convergence is improved. In particular, for systems with the property that the product matrix of the input and output coupling matrices, CB, is not full rank. Numerical examples are given to illustrate the results.