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Towards safe and tractable Gaussian process-based MPC: Efficient sampling within a sequential quadratic programming framework

Manish Prajapat, Amon Lahr, Johannes Köhler, Andreas Krause, Melanie N. Zeilinger

TL;DR

This work proposes a robust GP-MPC formulation that guarantees constraint satisfaction with high probability and proposes a sampling-based GP-MPC approach that iteratively generates consistent dynamics samples from the GP within a sequential quadratic programming framework.

Abstract

Learning uncertain dynamics models using Gaussian process~(GP) regression has been demonstrated to enable high-performance and safety-aware control strategies for challenging real-world applications. Yet, for computational tractability, most approaches for Gaussian process-based model predictive control (GP-MPC) are based on approximations of the reachable set that are either overly conservative or impede the controller's safety guarantees. To address these challenges, we propose a robust GP-MPC formulation that guarantees constraint satisfaction with high probability. For its tractable implementation, we propose a sampling-based GP-MPC approach that iteratively generates consistent dynamics samples from the GP within a sequential quadratic programming framework. We highlight the improved reachable set approximation compared to existing methods, as well as real-time feasible computation times, using two numerical examples.

Towards safe and tractable Gaussian process-based MPC: Efficient sampling within a sequential quadratic programming framework

TL;DR

This work proposes a robust GP-MPC formulation that guarantees constraint satisfaction with high probability and proposes a sampling-based GP-MPC approach that iteratively generates consistent dynamics samples from the GP within a sequential quadratic programming framework.

Abstract

Learning uncertain dynamics models using Gaussian process~(GP) regression has been demonstrated to enable high-performance and safety-aware control strategies for challenging real-world applications. Yet, for computational tractability, most approaches for Gaussian process-based model predictive control (GP-MPC) are based on approximations of the reachable set that are either overly conservative or impede the controller's safety guarantees. To address these challenges, we propose a robust GP-MPC formulation that guarantees constraint satisfaction with high probability. For its tractable implementation, we propose a sampling-based GP-MPC approach that iteratively generates consistent dynamics samples from the GP within a sequential quadratic programming framework. We highlight the improved reachable set approximation compared to existing methods, as well as real-time feasible computation times, using two numerical examples.
Paper Structure (16 sections, 5 theorems, 20 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 16 sections, 5 theorems, 20 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Robust GP-MPC problem
  4. Efficient sampling-based GP-MPC
  5. Sequential Quadratic Programming
  6. Gaussian processes with derivatives
  7. Sequential GP sampling
  8. Discussion of Sampling-based GP-MPC
  9. Convergence of the SQP algorithm
  10. Simultaneous GP sampling and optimization
  11. Reachable set approximation
  12. Efficient receding-horizon implementation
  13. Simulation results
  14. Comparison of uncertainty propagation using a pendulum
  15. Safe closed-loop control with car dynamics
  16. ...and 1 more sections

Key Result

Lemma 1

(chowdhury_kernelized_2017) Let assump:q_RKHS hold and let $\sqrt{\beta_{}} = B_g + 4 \lambda \sqrt{\gamma_D + 1+ \log(1/p)}$, where $\gamma_D$ is the maximum information gain as defined in chowdhury_kernelized_2017. Then, for all $z \in \mathcal{Z}$, with probability at least $1-p$, it holds that

Figures (3)

  • Figure 1: Conditioning on the past SQP iterations to construct a sample from GP. In each SQP iteration $j$ in \ref{['alg:sampling_gpmpc_sqp']} (or forward-sampling iteration in \ref{['prop:SQPsampling']}), consistent function and derivative values of $g^n$ are obtained by sampling from $\mathcal{GP}_{[\underline{g_{}}, \overline{g_{}}]}(0, k_{\mathrm{d}}; \mathcal{D}^n_{0:j-1})$. Initial training points are denoted by black stars; sampled values at iteration $j \in \{1,2\}$, by $\{ \text{circles, crosses} \}$, respectively; sampling locations, by gray vertical bars.
  • Figure 2: Comparison of uncertainty propagation for a given input sequence $\bm{u}$ by different GP-MPC methods in the pendulum example. The blue-shaded area represents the true uncertainty propagation for $2\times 10^5$ dynamics sampled from $\mathcal{GP}_{[\underline{g_{}}, \overline{g_{}}]}(0, k; \bar{\mathcal{D}})$, the solid black line represents the true system, and the dashed line shows the angular velocity constraint. Cautious GP-MPC significantly under-approximates (orange, \ref{['fig:cautius']}), Robust tube-based GP-MPC significantly over-approximates (red, \ref{['fig:safeMPC']}), whereas Robust sampling-based (proposed) GP-MPC provides a close approximation (green, \ref{['fig:samplingMPC']}) to the true uncertainty set, even with only 20 dynamics samples.
  • Figure 3: Demonstration of safe closed-loop control using the proposed sampling-based GP-MPC with car dynamics. The ego car (brown) successfully overtakes other vehicles (black) from left to right. The solid blue line represents the resulting closed-loop trajectory. The dashed cyan line is a reference along the y-axis, while multiple thin lines show trajectory prediction with various trajectory samples used in sampling-based GP-MPC (\ref{['alg:sampling_gpmpc_closedloop']}).

Theorems & Definitions (11)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • Proposition 5
  • ...and 1 more