Automatic nonlinear MPC approximation with closed-loop guarantees

TL;DR

The paper tackles the challenge of applying nonlinear MPC in real time by converting MPC approximation into a function-approximation problem with uniform error guarantees. It introduces ALKIA-X, a non-iterative, kernel-based method that uses adaptive, localized kernel interpolation and RKHS-norm extrapolation to automatically construct an explicit approximant with guaranteed accuracy. The framework yields fast online evaluation and preserves closed-loop stability and constraint satisfaction when the MPC is designed robust to disturbances. The authors demonstrate the approach on a continuous stirred tank reactor and a cold atmospheric plasma device, achieving substantial offline reductions and online speedups while maintaining safety-critical guarantees. Overall, Alkia-X offers a practical, automatic pathway to deploy nonlinear MPC in high-rate control tasks with deterministic performance bounds.

Abstract

Safety guarantees are vital in many control applications, such as robotics. Model predictive control (MPC) provides a constructive framework for controlling safety-critical systems, but is limited by its computational complexity. We address this problem by presenting a novel algorithm that automatically computes an explicit approximation to nonlinear MPC schemes while retaining closed-loop guarantees. Specifically, the problem can be reduced to a function approximation problem, which we then tackle by proposing ALKIA-X, the Adaptive and Localized Kernel Interpolation Algorithm with eXtrapolated reproducing kernel Hilbert space norm. ALKIA-X is a non-iterative algorithm that ensures numerically well-conditioned computations, a fast-to-evaluate approximating function, and the guaranteed satisfaction of any desired bound on the approximation error. Hence, ALKIA-X automatically computes an explicit function that approximates the MPC, yielding a controller suitable for safety-critical systems and high sampling rates. We apply ALKIA-X to approximate two nonlinear MPC schemes, demonstrating reduced computational demand and applicability to realistic problems.

Figures (7)

• Figure 1: A high level illustration of the proposed framework for approximate MPC with closed-loop guarantees. Alkia-x automatically computes an explicit function that approximates a robust MPC scheme up to any specified accuracy ${\epsilon}$, which yields a fast-to-evaluate approximate MPC with closed-loop guarantees. Notably, the proposed process is non-iterative due to the uniform error bounds inherently guaranteed by Alkia-x.
• Figure 2: Power function $P_X$ on a cube with samples $X$ on its vertices, which are illustrated by the gray asterisks. The maximum of the power function is at the center and is highlighted by the magenta point. This plot is generated with the Matérn kernel with $\nu=3/2$ to compute the power function on the cube $[0,1]^2$.
• Figure 3: Structure of the resulting uniform local cubes and the localized approximating functions on a sub-domain $\mathcal{X}_a$ with length scale $\ell_a$ and grid size $\Delta x_a$. Illustrated are the local cubes $\mathcal{X}_c$, $c \in \{c_1,c_2,c_3,c_4\} \subseteq \mathcal{C}_{a,p_a}$.
• Figure 4: Illustration of the RKHS norm extrapolation. The magenta asterisks show the RKHS norm of the approximating function $\|\widetilde{h}_{X_{a, p}}\|_ {\widetilde{k}_{a}}$\ref{['eq:RKHS_h_a']}. The solid black line is the extrapolating function $\gamma_{a}$\ref{['eq:extrapolation']}, while the dashed blue line depicts its limit value $\overline{\Gamma}_a$\ref{['eq:gamma_bar']}.
• Figure 5: Ground truth MPC feedback law $f$ over the domain $\mathcal{X}=[-0.2,0.2]^2$.

Theorems & Definitions (5)

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