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Dampening parameter distributional shifts under robust control and gain scheduling

Mohammad Ramadan, Mihai Anitescu

Abstract

Many traditional robust control approaches assume linearity of the system and independence between the system state-input and the parameters of its approximant low-order model. This assumption implies that robust control design introduces no distributional shifts in the parameters of this low-order model. This is generally not true when the underlying actual system is nonlinear, which admits typically state-input coupling with the parameters of the approximating model. Therefore, a robust controller has to be robust under the 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 and parameter spaces. In computational terms, these objectives are formulated as convex semi-definite programs that standard software packages can efficiently solve. We evaluate the proposed approaches on a simple yet telling gain-scheduling problem, which can be equivalently posed as a robust control problem.

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 low-order model. This assumption implies that robust control design introduces no distributional shifts in the parameters of this low-order model. This is generally not true when the underlying actual system is nonlinear, which admits typically state-input coupling with the parameters of the approximating model. Therefore, a robust controller has to be robust under the 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 and parameter spaces. In computational terms, these objectives are formulated as convex semi-definite programs that standard software packages can efficiently solve. We evaluate the proposed approaches on a simple yet telling gain-scheduling problem, which can be equivalently posed as a robust control problem.

Paper Structure

This paper contains 8 sections, 2 theorems, 34 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Background
  4. Dampened parameter distributional shifts
  5. Data-conforming in the state distribution
  6. Data conforming in the state-input distribution
  7. Numerical simulations
  8. Conclusion

Key Result

corollary 1

In the solution $(\Sigma_\star, K_\star)$ of eq:state-input-data-conforming Robust LQR problem, $\Sigma_\star$ is an upper bound of all the $\Sigma_i$s defined with $K=K_\star$ in eq:Controllability Lyapunov.An analogous result holds for the $(\Sigma_\star, K_\star)$ resulting from eq:state-data-con

Figures (1)

  • Figure 1: Blue circles are the parameters $(A_i(1,2),B_i(2,1))$ over the grid points, given by \ref{['eq:samples2vertex example']}, and used in the control design process. Red crosses, yellow diamonds, and pink triangles correspond to the parameters (of the same indices) of the local approximation matrices over the trajectories achieved after the application of $K_{LQR}$, $K_{robust}$, and $K_{DC}$, respectively.

Theorems & Definitions (6)

  • remark 1
  • corollary 1
  • proof
  • corollary 2
  • proof
  • remark 2