Constrained optimal iterative learning control with mixed-norm cost functions

Abstract

Iterative learning Control (ILC) is a widely used design technique for determining feedforward control inputs to systems that perform the same task repeatedly. This feedforward control design problem can be posed as an optimization problem in the Norm Optimal ILC (NO-ILC) framework. Although NO-ILC problems are usually formulated without constraints, they may be extended to enforce constraints by posing an analogous constrained optimization problem, termed Constrained Optimal ILC (CO-ILC). Typical NO-ILC and CO-ILC algorithms use 2-norm type cost functions (i.e., minimizing tracking error and control effort 2-norms), which are smooth and have analytical gradient expressions for design of the update law. However, in many applications, the max (∞) norm of tracking error is critical, which is a nonsmooth function and thus gradient-based ILC methods cannot be directly used. In this manuscript, we design a CO-ILC algorithm to explore the performance of using non-smooth type cost functions and compare them with the case using 2-norm cost function with actuator (or more generally state) box constraints. Specifically, CO-ILC algorithms for linear system with linear constraints are derived using (1) an ∞-norm cost function, (2) a mixed (2−∞)-norm cost function and (3) a sequential (2−∞)-norm cost function; the performance results of these are then compared with the traditional CO-ILC using 2-norm cost function on a precision motion control stage experimentally. We also provide proofs for robust monotone convergence of the proposed CO-ILC algorithms for a class of uncertainty models.