Multi-input square iterative learning control with input rate limits and bounds

Abstract

We present a simple modification of the iterative learning control algorithm of Arimoto et al. (1984) for the case where the inputs are bounded and time-rate-limited. The Jacobian error condition for monotonicity of input-error, rather than output-error, norms, is specified, the latter being insufficient to assure convergence, as proved herein. To the best of our knowledge, these facts have not been previously pointed out in the iterative learning control literature. We present a new proof that the modified controller produces monotonically decreasing input error norms, with a norm that covers the entire time interval of a learning trial.

Iterative learning control with saturation constraints

“…Accounting for the fact that each iteration of in the implementation requires an experiment, we use the Newton method primarily because of its fast convergence (in fact Newton method converges quadratically despite the linear convergence rate of computationally lighter first order methods). Furthermore, one can exploit the structure of the problem in (13) to reduce the computational cost of the Newton method. Note that each step of the Newton method requires to solve the linear set of equalities…”

“…However, the cost function in (13) cannot be evaluated analytically because it contains w which is unknown (but constant) and cannot be measured. This prohibits using off-the-shelf optimization solvers for the solution of (13). On the other hand, when w appears in the expressions that need to be evaluated in the iterations of the optimization scheme specifically in the form Gu − w, then it may be possible to evaluate the expression because, in fact, Gu − w = e and e can be measured at each iteration (i.e., when the process is ran with the current candidate for the optimizing value of u in the problem in (13).…”

“…This prohibits using off-the-shelf optimization solvers for the solution of (13). On the other hand, when w appears in the expressions that need to be evaluated in the iterations of the optimization scheme specifically in the form Gu − w, then it may be possible to evaluate the expression because, in fact, Gu − w = e and e can be measured at each iteration (i.e., when the process is ran with the current candidate for the optimizing value of u in the problem in (13). For example, although the gradient of the cost function in the problem in (13) contains w it can still be evaluated with the knowledge of e since it is equal to G T Gu−G T w = G T (Gu−w) = −G T e. We have made extensive use of this observation and its variants in the implementations of optimization schemes for the problem in (13) discussed next.…”

“…On the other and, this analogy has not been fully exploited especially for systems with constraints (such as saturation constraints). Driessen et al proposed a learning scheme for multi-input multi-output square systems with bounded input constraints [13]. Xu et al [14] used a composite energy function based ILC algorithm without the global Lipschitz condition required by [13].…”

“…Driessen et al proposed a learning scheme for multi-input multi-output square systems with bounded input constraints [13]. Xu et al [14] used a composite energy function based ILC algorithm without the global Lipschitz condition required by [13]. However, both these approaches require that the optimum of the unconstrained ILC problem is also optimal for the corresponding constrained problem.…”