A behavioral approach for LPV data-driven representations

TL;DR

The paper tackles direct data-driven analysis and control for discrete-time LPV systems, focusing on the LPV-SA subclass with shifted-affine scheduling by embedding the kernel representation into a data-driven LPV-IO/SS framework within the behavioral setting. It derives a finite-horizon data-driven representation and a necessary-and-sufficient LPV-GPE condition to certify data richness, and it provides a formal solution to the LPV data-driven simulation problem. The main contributions include (i) a computable data-driven LPV-SA representation from finite data, (ii) a LPV-GPE condition that certifies full horizon representation from data, and (iii) a trajectory-based data-driven simulation algorithm applicable to LPV and LPV-embedded nonlinear systems, demonstrated on a mass-spring-damper example and a nonlinear embedding with predictive control. The results enable direct data-driven analysis and control for LPV systems and lay groundwork for extending to broader scheduling dependencies and noisy data, with practical impact for nonlinear-to-LPV modeling and data-centric control design.

Abstract

In this paper, we present a data-driven representation for linear parameter-varying (LPV) systems, which can be used for direct data-driven analysis and control of such systems. Specifically, we use the behavioral approach to develop a data-driven representation of the finite-horizon behavior of LPV systems for which there exists a kernel representation with shifted-affine scheduling dependence. Moreover, we provide a necessary and sufficient rank-based test on the available data that concludes whether the data fully represents the finite-horizon LPV behavior. Using the proposed data-driven representation, we also solve the data-driven simulation problem for LPV systems. Through multiple examples, we demonstrate that the results in this paper allow us to formulate a novel set of direct data-driven analysis and control methods for LPV systems, which are also applicable for LPV embeddings of nonlinear systems.

Figures (8)

• Figure 1: Schematics of a mass-spring-damper system with a spring that is varying with a measurable signal $p$. The input of the system is a force $u$, while the output is the measured position $y$.
• Figure 2: Illustration of the (a) LPV system versus (b) its LTI embedding and (c) their corresponding behaviors. The behavior of the LTI embedding over-approximates the behavior of the LPV system.
• Figure 3: Input, scheduling, and output sequences of $\mathcal{D}_{N_\mathrm{d}}$ in Example \ref{['exmp:counter']}.
• Figure 4: Schematic representation of the simulation problem: With a length-$T_\mathrm{i}$ initial trajectory $(w_\mathrm{i},p_\mathrm{i})$ (depicted by the red data points), determine the response $(w_\mathrm{r},p_\mathrm{r})$ (depicted by the yellow data points) using only a given data set $\mathcal{D}_{N_\mathrm{d}} = (\breve{w}_{[1,{N_\mathrm{d}}]}, \breve{p}_{[1,{N_\mathrm{d}}]})$ (depicted by the blue data points).
• Figure 5: Data-dictionary measured from the LPV MSD system with ${N_\mathrm{d}}=161$.

Theorems & Definitions (18)

• Example 1
• Lemma 1: Dimension of $\left.\mathfrak{B}_{p}\right|_{[1,L]}$
• proof
• Theorem 1: LPV-SA Fundamental Lemma
• proof
• Example 2
• Lemma 2: Initial state characterization
• proof
• Example 3
• Example 4