(654d) Towards Model Predictive Dual Control

We present a novel approach to optimal adaptive control that involves closed-loop system identification with minimal deterioration of output regulation. The control algorithm excites the system in order to improve the parameter estimates, but does not disturb the system when the estimates are judged good enough. Without requiring persistent excitation, the method seeks a balance between excitation for identification and output regulation.

The algorithm is designed such that standard optimization codes can be used in the implementation, avoiding the use of techniques like stochastic dynamic programming, which often lead to complicated control designs. We base our method on model predictive control (MPC) and formulate a controller that can be realized with small changes to an existing MPC framework.

The current implementation estimates parameters online using a recursive least squares algorithm. At each sampling time, an optimal input is computed based on the current estimate, leading to a certainty-equivalence type controller. The computation of the input involves solving a nonlinear programming (NLP) problem, minimizing a finite time-horizon objective function whose terms include future input and output magnitudes, future rates of change in input, as well as a measure of future parameter estimate error variances. The constraints include the model with its estimated parameters, bounds on future inputs and outputs, and a modified least squares algorithm implemented as a set of equality constraints. Adding the estimation equations as constraints informs the NLP of how the input sequence will affect the future parameter estimate quality. This leads to a controller with dual features (after Feldbaum [1]).

We illustrate the method on an example system with unknown input gain. The algorithm intelligently excites the system only when the parameter estimate is uncertain. We also show how a slight increase in system excitation by the controller can greatly enhance parameter identification with minimal effect on the system output.

[1] Feldbaum, A. A. Dual control theory. I-IV. Automation and Remote Control, 1961.