Exponential stability of nonlinear differential repetitive processes with applications to iterative learning control

Abstract

This paper studies exponential stability properties of a class of two-dimensional (2D) systems called differential repetitive processes (DRPs). Since a distinguishing feature of DRPs is that the problem domain is bounded in the "time" direction, the notion of stability to be evaluated does not require the nonlinear system defining a DRP to be stable in the typical sense. In particular, we study a notion of exponential stability along the discrete iteration dimension of the 2D dynamics, which requires the boundary data for the differential pass dynamics to converge to zero as the iterations evolve. Our main contribution is to show, under standard regularity assumptions, that exponential stability of a DRP is equivalent to that of its linearized dynamics. In turn, exponential stability of this linearization can be readily verified by a spectral radius condition. The application of this result to Picard iterations and iterative learning control (ILC) is discussed. Theoretical findings are supported by a numerical simulation of an ILC algorithm.

Iterative machine learning for precision trajectory tracking with series elastic actuators

“…For the timevarying case, as in Section II-B, the iteration law designed using the fixed model is anticipated to converge provided the time-variations are sufficiently slow. The convergence conditions for such variations could be developed using Picard-type arguments, e.g., as studied in [29] for the nonlinear case.…”

Rohrs' Example Revisited: On the Robustness of Adaptive Iterative Learning Control

“…As a matter of fact, most ILC systems that achieve perfect tracking are essentially iterative integrators and are prone to instability in the presence of unmodeled dynamics. For example, stability of proportional‐derivative type ILC schemes depend on a specific relative degree assumption . However, in the nonadaptive case, there are easy remedies for this such as the use of the low‐pass Q ‐filter .…”

A Technical Overview of Recent Progresses on Stochastic Iterative Learning Control

“…At the end of this section, we note that the repetitive process has been deeply investigated in the past decades and fruitful results have been obtained, which has shown its effectiveness in design and analysis of corresponding ILC algorithms [44][45][46]. Novel results are expected along this direction by connecting repetitive processes with ILC formulations.…”