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Adaptive Economic Model Predictive Control for linear systems with performance guarantees

Maximilian Degner, Raffaele Soloperto, Melanie N. Zeilinger, John Lygeros, Johannes Köhler

Abstract

We present a model predictive control (MPC) formulation to directly optimize economic criteria for linear constrained systems subject to disturbances and uncertain model parameters. The proposed formulation combines a certainty equivalent economic MPC with a simple least-squares parameter adaptation. For the resulting adaptive economic MPC scheme, we derive strong asymptotic and transient performance guarantees. We provide a numerical example involving building temperature control and demonstrate performance benefits of online parameter adaptation.

Adaptive Economic Model Predictive Control for linear systems with performance guarantees

Abstract

We present a model predictive control (MPC) formulation to directly optimize economic criteria for linear constrained systems subject to disturbances and uncertain model parameters. The proposed formulation combines a certainty equivalent economic MPC with a simple least-squares parameter adaptation. For the resulting adaptive economic MPC scheme, we derive strong asymptotic and transient performance guarantees. We provide a numerical example involving building temperature control and demonstrate performance benefits of online parameter adaptation.
Paper Structure (12 sections, 7 theorems, 43 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 7 theorems, 43 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Proposed Approach
  4. Parameter adaptation
  5. Terminal cost design
  6. Adaptive Economic MPC scheme
  7. Theoretical Analysis
  8. Asymptotic performance
  9. Transient performance bound
  10. Discussion
  11. Numerical example
  12. Conclusion

Key Result

Proposition 1

Suppose that $x_k\in \mathbb{Z}$ and $u_k\in\mathbb{U}$ for all $k\in\mathbb{N}$. Then, for all $T\in\mathbb{N}$, it holds that with the one-step parametric prediction error Moreover, the difference between two successive parameter estimates satisfies

Figures (3)

  • Figure 1: Evolution of the parameter estimate (black solid). Bounds of the parameter set are the dashed black lines, and the dotted blue line is the true parameter value.
  • Figure 2: Evolution of the state $[x]_1$ (top) and the input $u$ (bottom) for the initial phase (left) and after parameter convergence (right). The constraint sets $\mathbb{X}, \mathbb{U}$ are shown as black, dashed lines.
  • Figure 3: Accumulated cost of AE-MPC (blue) and E-MPC (orange).

Theorems & Definitions (13)

  • Remark 1: Open-loop stability and soft-constraints
  • Proposition 1: LMS bounds
  • proof
  • Proposition 2: Terminal cost
  • proof
  • Theorem 3: Asymptotic average performance
  • proof
  • Corollary 4: Asymptotic convergence
  • Theorem 5: Transient performance bound
  • proof
  • ...and 3 more