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Adaptive Robust Control for Uncertain Systems with Ellipsoid-Set Learning

Xuehui Ma, Shiliang Zhang, Zhiyong Sun, Xiaohui Zhang, Sabita Maharjan

TL;DR

This work develops an ellipsoid-set based framework for adaptive robust control of uncertain linear systems where process/observation noises and states are bounded within ellipsoids and system parameters are uncertain. By discretizing the parameter space into a finite candidate set and maintaining Bayesian a-posteriori weights, the authors decouple learning from control, solving per-candidate SDP-based robust MPCs and aggregating them into an adaptive robust control law. The approach yields reduced conservativeness through state-set learning and achieves parameter identification with convergence guarantees, demonstrated on adaptive robust LQR and LQT tasks. The results indicate improved performance over fixed-set robust controllers and competitive proximity to an ideal oracle controller, with attention to computational complexity and potential extensions for large-scale systems.

Abstract

Despite the celebrated success of stochastic control approaches for uncertain systems, such approaches are limited in the ability to handle non-Gaussian uncertainties. This work presents an adaptive robust control for linear uncertain systems, whose process noise, observation noise, and system states are depicted by ellipsoid sets rather than Gaussian distributions. We design an ellipsoid-set learning method to estimate the boundaries of state sets, and incorporate the learned sets into the control law derivation to reduce conservativeness in robust control. Further, we consider the parametric uncertainties in state-space matrices. Particularly, we assign finite candidates for the uncertain parameters, and construct a bank of candidate-conditional robust control problems for each candidate. We derive the final control law by aggregating the candidate-conditional control laws. In this way, we separate the control scheme into parallel robust controls, decoupling the learning and control, which otherwise renders the control unattainable. We demonstrate the effectiveness of the proposed control in numerical simulations in the cases of linear quadratic regulation and tracking control.

Adaptive Robust Control for Uncertain Systems with Ellipsoid-Set Learning

TL;DR

This work develops an ellipsoid-set based framework for adaptive robust control of uncertain linear systems where process/observation noises and states are bounded within ellipsoids and system parameters are uncertain. By discretizing the parameter space into a finite candidate set and maintaining Bayesian a-posteriori weights, the authors decouple learning from control, solving per-candidate SDP-based robust MPCs and aggregating them into an adaptive robust control law. The approach yields reduced conservativeness through state-set learning and achieves parameter identification with convergence guarantees, demonstrated on adaptive robust LQR and LQT tasks. The results indicate improved performance over fixed-set robust controllers and competitive proximity to an ideal oracle controller, with attention to computational complexity and potential extensions for large-scale systems.

Abstract

Despite the celebrated success of stochastic control approaches for uncertain systems, such approaches are limited in the ability to handle non-Gaussian uncertainties. This work presents an adaptive robust control for linear uncertain systems, whose process noise, observation noise, and system states are depicted by ellipsoid sets rather than Gaussian distributions. We design an ellipsoid-set learning method to estimate the boundaries of state sets, and incorporate the learned sets into the control law derivation to reduce conservativeness in robust control. Further, we consider the parametric uncertainties in state-space matrices. Particularly, we assign finite candidates for the uncertain parameters, and construct a bank of candidate-conditional robust control problems for each candidate. We derive the final control law by aggregating the candidate-conditional control laws. In this way, we separate the control scheme into parallel robust controls, decoupling the learning and control, which otherwise renders the control unattainable. We demonstrate the effectiveness of the proposed control in numerical simulations in the cases of linear quadratic regulation and tracking control.
Paper Structure (17 sections, 6 theorems, 135 equations, 9 figures, 4 tables)

This paper contains 17 sections, 6 theorems, 135 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Controller Design
  4. The Ellipsoid-Set Learning for State Estimation
  5. Robust N$_{p}$-Steps Predictive Control
  6. Convergence of the Weights for Parameter Vectors
  7. Adaptive Robust Control Law
  8. Extension to Linear Quadratic Tracking Control
  9. Simulations and Results
  10. Example 1: adaptive robust linear quadratic regulation control
  11. Example 2: adaptive robust linear quadratic tracking control
  12. Conclusion
  13. Appendix
  14. The ellipsoid set for observation
  15. Proof for Lemma 1 and derivation for parameter p
  16. ...and 2 more sections

Key Result

Lemma 1

$\forall p(k,\bm{\theta}_i)\in(0,\infty)$, we have $\mathcal{X}(k|k-1,\bm{\theta}_i) \supset \mathcal{X}'(k|k-1,\bm{\theta}_i) \oplus \mathcal{W}(k-1)$, where the ellipsoid center and shape matrix of $\mathcal{X}(k|k-1,\bm{\theta}_i)=E(\hat{\bm{x}}(k|k-1,\bm{\theta}_i), \bm{P}(k|k-1,\bm{\theta}_i))$ and The optimal scalar parameter $p(k,\bm{\theta}_i)\in(0,\infty)$ that minimize the size of ellip

Figures (9)

  • Figure 1: Geometrical illustration of the learning progress of the ellipsoid set for the system state at the k-th instant.
  • Figure 2: The convergence of a-posteriori probabilities for candidate system parameters.
  • Figure 3: The learning progress of ellipsoid set for system states by our adaptive robust control for the system with unknown parameters.
  • Figure 4: The learning progress of ellipsoid set for system states by the ideal optimal robust control for the system with known parameters.
  • Figure 5: Control performances under robust control, our adaptive robust control, and the ideal optimal robust control. The green, red, and blue areas show ranges of the system states during 100 rounds of simulations. The triangle/circle/star-marked lines represent the result from a one-time simulation.
  • ...and 4 more figures

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 5
  • Lemma 3
  • ...and 9 more