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Adaptive Tube MPC: Beyond a Common Quadratically Stabilizing Feedback Gain

Anchita Dey, Shubhendu Bhasin

Abstract

This paper proposes an adaptive tube framework for model predictive control (MPC) of discrete-time linear time-invariant systems subject to parametric uncertainty and additive disturbances. In contrast to conventional tube-based MPC schemes that employ fixed tube geometry and constraint tightening designed for worst-case uncertainty, the proposed approach incorporates online parameter learning to progressively refine the parametric uncertainty set and update the parameter estimates. These updates are used to adapt the components of the MPC optimization problem, including the prediction model, feedback gain, terminal set, and tube cross-sections. As the uncertainty set contracts, the required amount of constraint tightening reduces and the tube shrinks accordingly, yielding less conservative control actions. Recursive feasibility, robust constraint satisfaction, and closed-loop stability are formally established. Furthermore, the framework does not require the existence of a common quadratically stabilizing linear feedback gain for the entire parametric uncertainty set, thereby relaxing a standard assumption in existing tube-based MPC formulations. Numerical examples illustrate the effectiveness of the proposed approach.

Adaptive Tube MPC: Beyond a Common Quadratically Stabilizing Feedback Gain

Abstract

This paper proposes an adaptive tube framework for model predictive control (MPC) of discrete-time linear time-invariant systems subject to parametric uncertainty and additive disturbances. In contrast to conventional tube-based MPC schemes that employ fixed tube geometry and constraint tightening designed for worst-case uncertainty, the proposed approach incorporates online parameter learning to progressively refine the parametric uncertainty set and update the parameter estimates. These updates are used to adapt the components of the MPC optimization problem, including the prediction model, feedback gain, terminal set, and tube cross-sections. As the uncertainty set contracts, the required amount of constraint tightening reduces and the tube shrinks accordingly, yielding less conservative control actions. Recursive feasibility, robust constraint satisfaction, and closed-loop stability are formally established. Furthermore, the framework does not require the existence of a common quadratically stabilizing linear feedback gain for the entire parametric uncertainty set, thereby relaxing a standard assumption in existing tube-based MPC formulations. Numerical examples illustrate the effectiveness of the proposed approach.
Paper Structure (12 sections, 34 equations, 8 figures, 1 algorithm)

This paper contains 12 sections, 34 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. COCP components without adaptation
  4. Disturbance set characterization
  5. Terminal set construction
  6. Homothetic tube parameterization
  7. Adaptive tube MPC framework
  8. Parameter learning and uncertainty set refinement
  9. Reformulated COCP with adaptation
  10. Recursive Feasibility and Stability Analysis
  11. Numerical Example
  12. Conclusion

Figures (8)

  • Figure 1: State and control input.
  • Figure 2: Frobenius norm of the parameter estimation error.
  • Figure 3: Elements of the terminal cost weight matrix, and the stabilizing linear feedback gain.
  • Figure 4: Barycentric representation of the parametric uncertainty sets containing the convex combination of the true parameter (shown with a blue dot).
  • Figure 5: (a), (b) The polytopes used to define the cross-sectional shape of the homothetic tubes, (c) The terminal sets.
  • ...and 3 more figures