Iterative learning control with discrete‐time nonlinear nonminimum phase models via stable inversion

Abstract

Output reference tracking can be improved by iteratively learning from past data to inform the design of feedforward control inputs for subsequent tracking attempts. This process is called iterative learning control (ILC). This article develops a method to apply ILC to systems with nonlinear discrete-time dynamical models with unstable inverses (i.e. discrete-time nonlinear non-minimum phase models). This class of systems includes piezoactuators, electric power converters, and manipulators with flexible links, which may be found in nanopositioning stages, rolling mills, and robotic arms, respectively. As these devices may be required to execute fine transient reference tracking tasks repetitively in contexts such as manufacturing, they may benefit from ILC. Specifically, this article facilitates ILC of such systems by presenting a new ILC synthesis framework that allows combination of the principles of Newton's root finding algorithm with stable inversion, a technique for generating stable trajectories from unstable models. The new framework, called Invert-Linearize ILC (ILILC), is validated in simulation on a cart-and-pendulum system with model error, process noise, and measurement noise. Where preexisting Newton-based ILC diverges, ILILC with stable inversion converges, and does so in less than one third the number of trials necessary for the convergence of a gradient-descent-based ILC technique used as a benchmark.

Control of the formation and amplification of pulse bursts in regenerative amplifiers

“…\end{cases}} \end{equation}$$The initial correction is set to zero, that is, $$\Delta {\bf u}^0 = {\bf 0}$$, and thus $${\bf u}^0 = {\bf u}^s$$. To obtain a suitable learning gain L , one can use standard model‐based approaches from the ILC literature [26–29]. Since we want to leverage the available system knowledge and the semi‐analytical structure of (33) makes it hard to provide strict results on its invertibility, a pseudo‐inverse learning law [27] $$$$\begin{equation} {\bf L} = {\left(\beta {\bf I} + {\bf G}_\textrm {S}^T{\bf G}_\textrm {S} \right)}^{-1} {\bf G}_\textrm {S}^T, \end{equation}$$$$with the regularization parameter $$\beta > 0$$ is used.…”

“…The initial correction is set to zero, that is, Δu 0 = 0, and thus u 0 = u s . To obtain a suitable learning gain L, one can use standard model-based approaches from the ILC literature [26][27][28][29].…”

“…together with the system's linearized behaviour in Equation (29) in the vicinity of the operating point (x s , u s ) due to the optimization problem Equation (20), one obtains the closed-loop dynamics…”