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Robust Adaptive MPC Under Nonlinear Time-Varying Uncertainties: An Uncertainty Compensation Approach

Ran Tao, Pan Zhao, Ilya Kolmanovsky, Naira Hovakimyan

TL;DR

This paper introduces an uncertainty compensation-based robust adaptive model predictive control framework for linear systems with nonlinear time-varying uncertainties that ensures constraint satisfaction while delivering enhanced performance compared to existing methods.

Abstract

This paper introduces an uncertainty compensation-based robust adaptive model predictive control (MPC) framework for linear systems with nonlinear time-varying uncertainties. The framework integrates an L1 adaptive controller to compensate for the matched uncertainty and a robust feedback controller, designed using linear matrix inequalities, to mitigate the effect of unmatched uncertainty on target output channels. Uniform bounds on the errors between the system's states and control inputs and those of a nominal (i.e., uncertainty-free) system are derived. These error bounds are then used to tighten the actual system's state and input constraints, enabling the design of an MPC for the nominal system under these tightened constraints. Referred to as uncertainty compensation-based MPC (UC-MPC), this approach ensures constraint satisfaction while delivering enhanced performance compared to existing methods. Simulation results for a flight control example and a spacecraft landing on an asteroid demonstrate the effectiveness of the proposed framework.

Robust Adaptive MPC Under Nonlinear Time-Varying Uncertainties: An Uncertainty Compensation Approach

TL;DR

This paper introduces an uncertainty compensation-based robust adaptive model predictive control framework for linear systems with nonlinear time-varying uncertainties that ensures constraint satisfaction while delivering enhanced performance compared to existing methods.

Abstract

This paper introduces an uncertainty compensation-based robust adaptive model predictive control (MPC) framework for linear systems with nonlinear time-varying uncertainties. The framework integrates an L1 adaptive controller to compensate for the matched uncertainty and a robust feedback controller, designed using linear matrix inequalities, to mitigate the effect of unmatched uncertainty on target output channels. Uniform bounds on the errors between the system's states and control inputs and those of a nominal (i.e., uncertainty-free) system are derived. These error bounds are then used to tighten the actual system's state and input constraints, enabling the design of an MPC for the nominal system under these tightened constraints. Referred to as uncertainty compensation-based MPC (UC-MPC), this approach ensures constraint satisfaction while delivering enhanced performance compared to existing methods. Simulation results for a flight control example and a spacecraft landing on an asteroid demonstrate the effectiveness of the proposed framework.
Paper Structure (31 sections, 10 theorems, 93 equations, 14 figures, 2 tables, 1 algorithm)

This paper contains 31 sections, 10 theorems, 93 equations, 14 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Overview of the UC-MPC Framework
  4. Equivalent Nominal System and Connection with Tube MPC
  5. Unmatched Uncertainty Attenuation
  6. ${\mathcal{L}_1}$AC with Uniform Performance Bounds
  7. Individual Performance Bounds
  8. Scaling for Improved Bounds
  9. UC-MPC: Robust Adaptive MPC via Uncertainty Compensation
  10. Nominal MPC Formulation
  11. Stability and Recursive Feasibility of Nominal MPC
  12. Non-emptiness of the Tightened Constraint Sets
  13. Formal Guarantees of UC-MPC
  14. Computational Complexity Analysis
  15. Simulation Case Studies
  16. ...and 16 more sections

Key Result

Lemma 1

(scherer1997multiobjective) The system in eq:kx_closed is internally stable and has a PPG bound of $\beta > 0$, if there exist $\lambda > 0$, $\mu > 0$, and a symmetric matrix $P$, s.t. the following two inequalities hold:

Figures (14)

  • Figure 1: Diagram of the proposed UC-MPC framework. UC-MPC leverages an ${\mathcal{L}_1}$AC to estimate and compensate for the matched uncertainty and a robust feedback controller to mitigate the effect of the unmatched uncertainty.
  • 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 matched uncertainties $f(t,x(t))$ under UC-MPC. $f(t,x(t))$ has two elements, corresponding to 1 and 2 in figure.
  • Figure 5: Trajectories of constrained states (top), and control inputs (middle and bottom) under MPC, TMPC, and UC-MPC. Gray dash-dotted lines illustrate the constraints specified in \ref{['eq:cts-F16']}.
  • ...and 9 more figures

Theorems & Definitions (27)

  • Remark 1: Matched and unmatched uncertainties
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 4: Design of $K_x$
  • Remark 5
  • Lemma 3
  • Lemma 4
  • ...and 17 more