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Kernelized offset-free data-driven predictive control for nonlinear systems

Thomas Oliver de Jong, Mircea Lazar

TL;DR

Kernel methods are employed to parameterize the nonlinear terms of a velocity model, preserving its structure and efficiently learning unknown parameters through a least squares approach, resulting in a offset-free data-driven predictive control scheme formulated as a nonlinear program, but solvable via sequential quadratic programming.

Abstract

This paper presents a kernelized offset-free data-driven predictive control scheme for nonlinear systems. Traditional model-based and data-driven predictive controllers often struggle with inaccurate predictors or persistent disturbances, especially in the case of nonlinear dynamics, leading to tracking offsets and stability issues. To overcome these limitations, we employ kernel methods to parameterize the nonlinear terms of a velocity model, preserving its structure and efficiently learning unknown parameters through a least squares approach. This results in a offset-free data-driven predictive control scheme formulated as a nonlinear program, but solvable via sequential quadratic programming. We provide a framework for analyzing recursive feasibility and stability of the developed method and we demonstrate its effectiveness through simulations on a nonlinear benchmark example.

Kernelized offset-free data-driven predictive control for nonlinear systems

TL;DR

Kernel methods are employed to parameterize the nonlinear terms of a velocity model, preserving its structure and efficiently learning unknown parameters through a least squares approach, resulting in a offset-free data-driven predictive control scheme formulated as a nonlinear program, but solvable via sequential quadratic programming.

Abstract

This paper presents a kernelized offset-free data-driven predictive control scheme for nonlinear systems. Traditional model-based and data-driven predictive controllers often struggle with inaccurate predictors or persistent disturbances, especially in the case of nonlinear dynamics, leading to tracking offsets and stability issues. To overcome these limitations, we employ kernel methods to parameterize the nonlinear terms of a velocity model, preserving its structure and efficiently learning unknown parameters through a least squares approach. This results in a offset-free data-driven predictive control scheme formulated as a nonlinear program, but solvable via sequential quadratic programming. We provide a framework for analyzing recursive feasibility and stability of the developed method and we demonstrate its effectiveness through simulations on a nonlinear benchmark example.

Paper Structure

This paper contains 7 sections, 3 theorems, 27 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries & Problem Statement
  3. Problem Statement
  4. Kernelized Velocity State-space Models
  5. Kernelized Offset-Free DPC
  6. Illustrative Example
  7. Conclusions

Key Result

Lemma III.3

Suppose that the collected set of noiseless state-input-output data $\{x_i,u_i,y_i\}_{i\in[0,s]}$ and the chosen kernel functions are such that the matrices $$ and $$ have full rank. Then the matrices yield the $A_\alpha$, $B_\alpha$ and $C_\alpha$ that minimize $\|\Hat{\mathbf{Z}}^+ - \mathbf{Z}^+\|_2^2$.

Figures (3)

  • Figure 1: $N=20$ step $y$ prediction of kernel model for test data plotted together with the true system trajectory and the error $e:= y-\Hat{y}$.
  • Figure 2: Terminal set $\mathbb{Z}_T$ for $r = \operatorname{col}(0.5,0,0)$ obtained using the kernelized velocity model (blue) and analytic velocity model (red).
  • Figure 3: vKDPC (---) vs. vNMPC (---) trajectories and disturbance signal.

Theorems & Definitions (12)

  • Definition II.1
  • Definition II.2
  • Remark III.1
  • Remark III.2
  • Lemma III.3
  • proof
  • Theorem IV.3
  • proof
  • Theorem IV.4
  • proof
  • ...and 2 more