Performance-driven Constrained Optimal Auto-Tuner for MPC

TL;DR

The paper tackles tuning MPC cost function parameters under a hard performance constraint by modeling the unknown performance function with a Gaussian process and enforcing safety through Lipschitz-based optimistic/pessimistic sets. It introduces COAt-MPC, a safe, goal-directed auto-tuner that samples only from the pessimistic set while targeting a goal in the optimistic set, yielding finite-time convergence to the constrained optimum with high probability. Theoretical guarantees show the constraint is satisfied with probability at least 1-δ at all iterations and that the optimum under the constraint is reached within a finite budget, independent of discretization granularity. Empirically, COAt-MPC outperforms constrained and unconstrained baselines in autonomous racing, achieving lower constraint violations and competitive or better cumulative regret, with faster convergence on an RC platform. The work advances safe, data-efficient MPC tuning and suggests scalable extensions to higher-dimensional parameter spaces.

Abstract

A key challenge in tuning Model Predictive Control (MPC) cost function parameters is to ensure that the system performance stays consistently above a certain threshold. To address this challenge, we propose a novel method, COAT-MPC, Constrained Optimal Auto-Tuner for MPC. With every tuning iteration, COAT-MPC gathers performance data and learns by updating its posterior belief. It explores the tuning parameters' domain towards optimistic parameters in a goal-directed fashion, which is key to its sample efficiency. We theoretically analyze COAT-MPC, showing that it satisfies performance constraints with arbitrarily high probability at all times and provably converges to the optimum performance within finite time. Through comprehensive simulations and comparative analyses with a hardware platform, we demonstrate the effectiveness of COAT-MPC in comparison to classical Bayesian Optimization (BO) and other state-of-the-art methods. When applied to autonomous racing, our approach outperforms baselines in terms of constraint violations and cumulative regret over time.

Figures (6)

• Figure 1: COAt-MPC overview. COAt-MPC proposes a set of cost function weights $\boldsymbol{\theta}_n$, which are evaluated on the system. It gets a performance function sample, which is used to update the posterior belief and acquire a new set of cost function weights. The process is repeated until convergence.
• Figure 2: Pessimistic and optimistic operators evaluated at $\theta_0$ (figure from gooseprajapat2024safe). This figure demonstrates how the pessimistic and optimistic sets are computed when only evaluated in a single point $\theta_0$. The operators make use of the GP upper and lower confidence bounds, as well as the $L$-Lipschitz continuity. In this example, $d(\theta, \theta_0)$ is the Euclidean distance function, where $\theta_0$ is fixed. Additionally, $\tau$ is set arbitrarily.
• Figure 3: COAt-MPC illustration. (i) The grey, dashed line represents the true function. (ii) The red, dashed line represents the constraint. (iii) The blue line represents the Gaussian Process mean, and the shaded blue area represents the confidence bounds ($\mu_{n}(\boldsymbol{\theta}) \pm \sqrt{\beta_{n}} \sigma_{n}(\boldsymbol{\theta})$). (iv) The cross markers represent the samples, with yellow denoting the first sample and red the COAt-MPC recommended sample. (v) The green dot denotes the goal at each iteration. The algorithm learns the pessimistic (green bar) and optimistic (orange bar) sets, and explores the parameter space while satisfying the performance constraint. At $n=5$, the goal is outside of the pessimistic set, although it is inside the optimistic set. COAt-MPC expands the pessimistic set by approaching the goal. It reaches the goal at $n=15$, and by further expanding the sets, it discovers the maximum of the function at $n=28$.
• Figure 4: 1:28 scale RC racecar Carron2022ChronosAC and track used in the experiments.
• Figure 5: Lap time and cumulative regret over time with their standard deviation. COAt-MPC converges in only 20-30 iterations and achieves the lowest cumulative regret with the RC platform. Cross markers indicate each method's minimum laptime. The initial laptime (iteration 0) is the same for all methods.

Theorems & Definitions (9)

• Corollary 1: Theorem 2 pmlr-v70-chowdhury17a
• Theorem 1
• Lemma 1
• proof
• Corollary 2: Theorem 1 prajapat2024safe
• proof
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm:convergence']}