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Robust adaptive NMPC using ellipsoidal tubes

Johannes Buerger, Mark Cannon

TL;DR

The algorithm employs successive linearization around nominal predicted trajectories and accounts for uncertainties in predicted states due to linearization, model errors, and disturbances using ellipsoidal sets, and shows that the ellipsoidal embedding of model uncertainty scales favourably with system dimensions in numerical simulations.

Abstract

We propose a computationally efficient nonlinear Model Predictive Control (NMPC) algorithm for safe, learning-based control. The system model is represented as an affine combination of basis functions with unknown parameters, and is subject to additive set-bounded disturbances. Our algorithm employs successive linearization around nominal predicted trajectories and accounts for uncertainties in predicted states due to linearization, model errors, and disturbances using ellipsoidal sets. The ellipsoidal tube-based approach ensures that constraints on control inputs and system states are satisfied. Robustness to uncertainty is ensured using bounds on linearization errors and a backtracking line search. We show that the ellipsoidal embedding of model uncertainty scales favourably with system dimensions in numerical simulations. The algorithm incorporates set membership parameter estimation, and provides guarantees of recursive feasibility and input-to-state practical stability.

Robust adaptive NMPC using ellipsoidal tubes

TL;DR

The algorithm employs successive linearization around nominal predicted trajectories and accounts for uncertainties in predicted states due to linearization, model errors, and disturbances using ellipsoidal sets, and shows that the ellipsoidal embedding of model uncertainty scales favourably with system dimensions in numerical simulations.

Abstract

We propose a computationally efficient nonlinear Model Predictive Control (NMPC) algorithm for safe, learning-based control. The system model is represented as an affine combination of basis functions with unknown parameters, and is subject to additive set-bounded disturbances. Our algorithm employs successive linearization around nominal predicted trajectories and accounts for uncertainties in predicted states due to linearization, model errors, and disturbances using ellipsoidal sets. The ellipsoidal tube-based approach ensures that constraints on control inputs and system states are satisfied. Robustness to uncertainty is ensured using bounds on linearization errors and a backtracking line search. We show that the ellipsoidal embedding of model uncertainty scales favourably with system dimensions in numerical simulations. The algorithm incorporates set membership parameter estimation, and provides guarantees of recursive feasibility and input-to-state practical stability.
Paper Structure (10 sections, 9 theorems, 48 equations, 2 figures, 1 table)

This paper contains 10 sections, 9 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Optimal control problem
  3. Linearization error bounds
  4. Ellipsoidal tube MPC
  5. Online MPC optimization
  6. Terminal constraints and terminal cost
  7. Recursive feasibility and stability
  8. Online learning of system parameters
  9. Numerical results
  10. Conclusion

Key Result

Lemma 1

Condition (eq:tube_mem_cond_beta) holds for all $e\in\mathcal{E}(V,\beta_k^2)$ if for all $j\in\mathbb{N}_{\nu_{1}}$, $q\in\mathbb{N}_{\nu_\theta}$, with $\lambda_k$ defined by where $\Psi^{(r)} = (V^{-1} - w^{(r)} w^{(r)\, \top} \sigma^{-2})^{-1}$ and where $\sigma$ is a constant whose design is discussed in Section sec:termset.

Figures (2)

  • Figure 1: Average time required for one iteration (solid lines) and total time to convergence (dashed lines), as a function of $n_x=n_\theta$. Shaded regions indicate the upper and lower bounds on observed computation time per iteration.
  • Figure 2: Suboptimality of the first iteration at $t=0$, as a fraction of the optimal cost for no uncertainty: meam values (solid lines) and 2 standard deviation bounds (shaded regions); and mean cost evaluated for closed loop trajectories over $10$ time steps (dashed lines), as a function of $n_x=n_\theta$.

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 13 more