Constrained optimal iterative learning control for mixed-norm cost functions

Iterative learning control (ILC) is a technique for determining feedforward signals for systems that execute a task repeatedly. One approach towards designing ILC algorithms is to pose it as an optimization problem. Traditionally, norm optimal iterative learning control (NOILC) algorithms use ℓ₂-norm-type cost functions. However, many applications require optimizing non-smooth cost functions, e.g., in trajectory tracking where it is desirable to minimize the peak tracking error, i.e., its ℓ_∞-norm. In this paper, we explore the performance of a class of non-smooth cost functions along with constraints which can be recast into the constrained optimal ILC (COILC) framework. For linear systems with constraints (linear in the feedforward input) and certain cost functions (such as ℓ₂, ℓ_∞ norms of tracking error and control effort), this optimization problem can be formulated as a quadratic program (QP) or a linear program (LP). These COILC problems can then be solved with a modified interior-point-type method. In this manuscript, we derive ILC algorithms for linear systems (and linear constraints) with (1) a pure ℓ_∞ norm cost, (2) a mixed ℓ₂ - ℓ_∞ norm cost. We compare the results to the traditional ℓ₂ norm (NOILC) in simulation and experiment to illustrate the effect of the choice of the cost function on the design of the optimized feedforward control effort and hence the optimal error profile.