The Insensitivity of the Iterative Learning Control Inverse Problem to Initial Run When Stabilized by a New Stable Inverse

In routine feedback control, the input command is the desired output. The actual output is a convolution integral of the forcing function—which essentially is never equal to the command. If it were equal, control system designers would be solving an inverse problem. For systems that execute the same task repeatedly, Iterative Learning Control (ILC) addresses the inverse problem by adjusting the command the next time the task is executed, based on the error observed during the current run, aiming to converge to that command that produces the desired output. For a majority of real world systems, this asks to solve an ill-conditioned inverse problem whose exact solution is an unstable control action. A simple robot example performing a one-second maneuver asks to invert a Toeplitz matrix of Markov parameters that is guaranteed full rank, but has a condition number of $$10^{52}$$ 10 52 . The authors and co-workers have developed a stable inverse theory to address this difficulty for discrete-time systems. By not asking for zero error for the first or first few time steps (the number determined by the number of discrete-time zeros outside the unit circle), the inverse problem has stable solutions for the control action. Incorporating this understanding into ILC, the stable control action obtained in the limit as the iterations tend to infinity, is a function of the tracking error produced by the command in the initial run. By picking an initial input that goes to zero approaching the final time step, the influence becomes particularly small. And by simply commanding zero in the first run, the resulting converged control minimizes the Euclidean norm of the underdetermined control history. Three main classes of ILC laws are examined, and it is shown that all ILC laws converge to the identical control history. The converged result is not a function of the ILC law. All of these conclusions apply to ILC that aims to track a given finite time trajectory, and also apply to ILC that in addition aims to cancel the effect of a disturbance that repeats each run.