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Robust Adaptive MPC Using Uncertainty Compensation

Ran Tao, Pan Zhao, Ilya Kolmanovsky, Naira Hovakimyan

TL;DR

The proposed control framework leverages an adaptive controller to compensate for the matched uncertainties and to provide guaranteed uniform bounds on the error between the states and control inputs of the actual system and those of a nominal i.e., uncertainty-free system.

Abstract

This paper presents an uncertainty compensation-based robust adaptive model predictive control (MPC) framework for linear systems with both matched and unmatched nonlinear uncertainties subject to both state and input constraints. In particular, the proposed control framework leverages an L1 adaptive controller (L1AC) to compensate for the matched uncertainties and to provide guaranteed uniform bounds on the error between the states and control inputs of the actual system and those of a nominal i.e., uncertainty-free, system. The performance bounds provided by the L1AC are then used to tighten the state and control constraints of the actual system, and a model predictive controller is designed for the nominal system with the tightened constraints. The proposed control framework, which we denote as uncertainty compensation-based MPC (UC-MPC), guarantees constraint satisfaction and achieves improved performance compared with existing methods. Simulation results on a flight control example demonstrate the benefits of the proposed framework.

Robust Adaptive MPC Using Uncertainty Compensation

TL;DR

The proposed control framework leverages an adaptive controller to compensate for the matched uncertainties and to provide guaranteed uniform bounds on the error between the states and control inputs of the actual system and those of a nominal i.e., uncertainty-free system.

Abstract

This paper presents an uncertainty compensation-based robust adaptive model predictive control (MPC) framework for linear systems with both matched and unmatched nonlinear uncertainties subject to both state and input constraints. In particular, the proposed control framework leverages an L1 adaptive controller (L1AC) to compensate for the matched uncertainties and to provide guaranteed uniform bounds on the error between the states and control inputs of the actual system and those of a nominal i.e., uncertainty-free, system. The performance bounds provided by the L1AC are then used to tighten the state and control constraints of the actual system, and a model predictive controller is designed for the nominal system with the tightened constraints. The proposed control framework, which we denote as uncertainty compensation-based MPC (UC-MPC), guarantees constraint satisfaction and achieves improved performance compared with existing methods. Simulation results on a flight control example demonstrate the benefits of the proposed framework.
Paper Structure (14 sections, 7 theorems, 77 equations, 8 figures, 1 algorithm)

This paper contains 14 sections, 7 theorems, 77 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Overview of the UC-MPC Framework
  4. ${\mathcal{L}_1}$AC With Uniform Performance Bounds
  5. UC-MPC: Robust MPC via Uncertainty Compensation
  6. Simulation Case Study
  7. UC-MPC Design
  8. Simulation Results
  9. Conclusion
  10. Proofs
  11. Proof of \ref{['them:x-xref-bnd']}
  12. Proof of \ref{['lem:ref-id-bnd']}
  13. Proof of \ref{['lem:refine_bnd_xi_w_Txi']}
  14. Proof of \ref{['them:xi-uai-bnd']}

Key Result

Lemma 1

Given the uncertain system eq:dynamics-uncertain subject to assump:lipschitz-bnd-fi and the reference system eq:ref-system subject to the conditions eq:l1-stability-condition-Lfeq:l1-stability-condition with a constant $\gamma_1>0$, with the ${\mathcal{L}_1}$AC defined via eq:state-predictoreq:adapt where $\rho$, $\rho_{u_\textup{a}}$, and $\gamma_2$ are defined in eq:rho-defn, eq:rho-u-defn, eq:g

Figures (8)

  • Figure 1: Diagram of the proposed UC-MPC framework
  • Figure 2: Tracking performance under MPC, TMPC and UC-MPC (ours).
  • Figure 3: Zoomed-in view of tracking performance on $\theta$ under MPC, TMPC and UC-MPC.
  • Figure 4: Actual and estimated uncertainties under UC-MPC. For $i=1,2$, the symbols $f_j$ and $\{B^\dagger\hat{\sigma}\}_i$ denotes the $i$th element of $f$ and $B^\dagger\hat{\sigma}$, respectively.
  • Figure 5: Trajectories of constrained states (top), and control inputs (middle and bottom) under MPC, TMPC, and UC-MPC. Green dash-dotted lines illustrate the constraints specified in \ref{['eq:cts-F16']}.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • ...and 7 more