Dampening parameter distributional shifts under robust control and gain scheduling

Abstract

Many traditional robust control approaches assume linearity of the system and independence between the system state-input and the parameters of its approximant (possibly lower-order) model. This assumption implies that the application of robust control design to the underlying system introduces no distributional shifts in the parameters of its approximant model. This is generally not true when the underlying system is nonlinear, which may require different approximant models with different parameter distributions when operated at different regions of the state-input space. Therefore, a robust controller has to be robust under the approximant model with parameter distribution that will be experienced in the future data, after applying this control, not the parameter distribution seen in the learning data or assumed in the design. In this paper, we seek a solution to this problem by restricting the newly designed closed-loop system to be consistent with the learning data and slowing down any distributional shifts in the state-input space of the underlying system, and therefore, in the parameter space of its approximant model. In computational terms, the objective of dampening the shifts in the parameter distribution is formulated as a convex semi-definite program that can be solved efficiently by standard software packages. We evaluate the proposed approach on a simple yet telling gain-scheduling problem, which can be equivalently posed as a robust control problem.