On the reduction of Linear Parameter-Varying State-Space models

TL;DR

The paper addresses the practical reduction of LPV state-space models produced by nonlinear-to-LPV embedding, where parameter-dependent matrices and scheduling complexity drive computational burden. It benchmarks a broad set of state-order and scheduling-dimension reduction methods, including LTIBR, LPVBR, MM, PVOP, LFRBR, PCA, TPCA, KPCA, SDRBR, AE, and DNN, on affine MSD benchmarks to derive actionable guidelines. Key findings show that Moment Matching provides extrapolation capabilities across all benchmarks and can handle non-minimal or unstable models, while SDR balred delivers robust extrapolation with strong accuracy; other methods excel under favorable model properties. The results yield practical guidelines for selecting reduction approaches and point toward future work on integrating state-order and scheduling-dimension reductions for large-scale LPV models.

Abstract

This paper presents an overview and comparative study of the state of the art in State-Order Reduction (SOR) and Scheduling Dimension Reduction (SDR) for Linear Parameter-Varying (LPV) State-Space (SS) models, comparing and benchmarking their capabilities, limitations and performance. The use case chosen for these studies is an interconnected network of nonlinear coupled mass spring damper systems with three different configurations, where some spring coefficients are described by arbitrary user-defined static nonlinear functions. For SOR, the following methods are compared: Linear Time-Invariant (LTI), LPV and LFR-based balanced reductions, moment matching and parameter-varying oblique projection. For SDR, the following methods are compared: Principal Component Analysis (PCA), trajectory PCA, Kernel PCA and LTI balanced truncation, autoencoders and deep neural network. The comparison reveals the most suitable reduction methods for the different benchmark configurations, from which we provide use case SOR and SDR guidelines that can be used to choose the best reduction method for a given LPV-SS model.

Figures (8)

• Figure 1: Visualization of the training and validation data with $u_{\mathrm{train}}(t)$ ( ), $u_{\mathrm{in}}(t)$ ( ) and $u_{\mathrm{out}}(t)$ ( ).
• Figure 2: Visualization of the training grid $\mathcal{P}_{\mathrm{train}}$ ($\mathcolor[rgb]{0.2021, 0.4788, 0.9911}{\times}$) and the validation grids $\mathcal{P}_{\mathrm{in}}$($\mathcolor[rgb]{0.7858, 0.7573, 0.1598}{\times}$) and $\mathcal{P}_{\mathrm{out}}$ ($\mathcolor[rgb]{0.0704, 0.7457, 0.7258}{\times}$).
• Figure 3: Mean of the local $\mathcal{H}_2$ and $\mathcal{H}_\infty$ errors on $\mathcal{P}_{\mathrm{in}}$ with LTI balred ( ), LPV balred ( ), Moment Matching ( ) and Oblique Projections ( ).
• Figure 4: Mean of the local $\mathcal{H}_2$ and $\mathcal{H}_\infty$ errors on $\mathcal{P}_{\mathrm{out}}$ with LTI balred ( ), LPV balred ( ), Moment Matching ( ) and Oblique Projections ( ), where the unstable local model dynamics are excluded from the mean computation and indicated with $\circ$.
• Figure 5: Self-scheduled simulation of the FO ( ) and RO models obtained with LTI balred ( ), LPV balred ( ), Moment Matching ( ) and Oblique Projections ( ) for $r_x = 5$, $u_{\mathrm{out}}(t)$ and with zero initial conditions.