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Kernel-based multi-step predictors for data-driven analysis and control of nonlinear systems through the velocity form

Chris Verhoek, Roland Tóth

TL;DR

This work addresses data-driven analysis and control of nonlinear systems by introducing kernel-based predictors for the velocity form, which describes time-difference dynamics and yields equilibrium-independent stability properties for the primal system. It develops a structured RKHS-based predictor that respects the velocity form's quasi-linear, time-varying structure, and derives both explicit and implicit multi-step predictors, along with ridge-based estimation and LPV embedding. The main contributions include a structured kernelized predictor that preserves velocity-form dependencies, a detailed derivation of the predictor's algebraic form, and a demonstration on a SISO NL example showing improved predictive performance over unstructured kernel methods. The framework enables global stability and performance guarantees in data-driven settings and offers a scalable path for applying kernel methods to nonlinear systems analysis and control through the velocity form and LPV embedding.

Abstract

We propose kernel-based approaches for the construction of a single-step and multi-step predictor of the velocity form of nonlinear (NL) systems, which describes the time-difference dynamics of the corresponding NL system and admits a highly structured representation. The predictors in turn allow to formulate completely data-driven representations of the velocity form. The kernel-based formulation that we derive, inherently respects the structured quasi-linear and specific time-dependent relationship of the velocity form. This results in an efficient multi-step predictor for the velocity form and hence for nonlinear systems. Moreover, by using the velocity form, our methods open the door for data-driven behavioral analysis and control of nonlinear systems with global stability and performance guarantees.

Kernel-based multi-step predictors for data-driven analysis and control of nonlinear systems through the velocity form

TL;DR

This work addresses data-driven analysis and control of nonlinear systems by introducing kernel-based predictors for the velocity form, which describes time-difference dynamics and yields equilibrium-independent stability properties for the primal system. It develops a structured RKHS-based predictor that respects the velocity form's quasi-linear, time-varying structure, and derives both explicit and implicit multi-step predictors, along with ridge-based estimation and LPV embedding. The main contributions include a structured kernelized predictor that preserves velocity-form dependencies, a detailed derivation of the predictor's algebraic form, and a demonstration on a SISO NL example showing improved predictive performance over unstructured kernel methods. The framework enables global stability and performance guarantees in data-driven settings and offers a scalable path for applying kernel methods to nonlinear systems analysis and control through the velocity form and LPV embedding.

Abstract

We propose kernel-based approaches for the construction of a single-step and multi-step predictor of the velocity form of nonlinear (NL) systems, which describes the time-difference dynamics of the corresponding NL system and admits a highly structured representation. The predictors in turn allow to formulate completely data-driven representations of the velocity form. The kernel-based formulation that we derive, inherently respects the structured quasi-linear and specific time-dependent relationship of the velocity form. This results in an efficient multi-step predictor for the velocity form and hence for nonlinear systems. Moreover, by using the velocity form, our methods open the door for data-driven behavioral analysis and control of nonlinear systems with global stability and performance guarantees.
Paper Structure (23 sections, 62 equations, 3 figures)

This paper contains 23 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Velocity forms of nonlinear input-output representations
  3. General form
  4. Structured forms
  5. Computation of the matrix functions $\mathfrak{a}_i(\cdot), \mathfrak{b}_j(\cdot)$
  6. Basis function representation of the IO forms
  7. Regressor form of the multi-step predictor
  8. Exploring the structure of the multi-step predictor
  9. Deriving the model-based regressor form
  10. Kernelized data-driven representations
  11. Preliminaries and definitions
  12. Unstructured kernelized multi-step predictor for \ref{['eq:nlvelioform']}
  13. Structured kernelized multi-step predictor for \ref{['eq:nlvelioform']}
  14. Ridge regression
  15. Structure of $\Psi^\top(W_{L_{[:,i]}}) \Psi(W_{L_{[:,j]}})$
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Visualization of the proposed approach and its utilization in data-driven analysis and controller synthesis for nonlinear systems.
  • Figure 2: Possible realization for a controller that is designed for the velocity form.
  • Figure 3: Simulation results for the example system.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Example 1
  • Definition 1
  • Definition 2