Transformation-Free Fixed-Structure Model Reduction for LPV Systems

TL;DR

This work addresses LPV model reduction by transforming the problem into a fixed-structure controller synthesis task using a generalized plant, enabling gradient-based optimization to obtain a reduced LPV model with a direct error bound. By restricting the reduced model to a prescribed structure (e.g., block-diagonal modal forms) and solving a bi-linear matrix inequality via gradient methods, the approach provides both accuracy control and structural interpretability. The method is demonstrated on a mass-spring-damper chain, showing that unstructured reductions closely match the full-order model in open-loop, while structured reductions incur higher approximation errors but offer favorable modal interpretation for controller design. In closed-loop, controllers designed from reduced models perform well at low frequencies, with minor degradation when structure is imposed, illustrating practical trade-offs between structure and performance in LPV control design.

Abstract

In this paper, we propose a model reduction technique for linear parameter varying (LPV) systems based on available tools for fixed-structure controller synthesis. We start by transforming a model reduction problem into an equivalent controller synthesis problem by defining an appropriate generalized plant. The controller synthesis problem is then solved by using gradient-based tools available in the literature. Owing to the flexibility of the gradient-based synthesis tools, we are able to impose a desired structure on the obtained reduced model. Additionally, we obtain a bound on the approximation error as a direct output of the optimization problem. The proposed methods are applied on a benchmark mechanical system of interconnected masses, springs and dampers. To evaluate the effect of the proposed model-reduction approach on controller design, LPV controllers designed using the reduced models (with and without an imposed structure) are compared in closed-loop with the original model.

Figures (8)

• Figure 1: Equivalent block diagrams representing the model approximation problem (left) and a fixed structure controller synthesis problem (right)
• Figure 2: Mass-spring-damper system containing $N$ blocks with horizontal displacement $x_i$ for $i=1,...,N$ and equal mass ($m = 1\,\text{kg}$), dampers ($d = 0.75\,\text{Ns/m}$) and externally scheduled springs ($k_i = k_0 + \rho_i k_\rho$ with $k_0 = 0.5\,\text{N/m}$, $k_\rho = 0.3\,\text{N/m}$ and $\rho_i \in \mathcal{P} = \left[ -1, 1 \right]$ for $i = 1,...,N$). System input is the external force $F_\text{u}$. System output is the displacement $x_N$.
• Figure 3: Frequency response of the original model $G$ (shown with solid blue curves), reduced model $G_{\textnormal{red}}$) (shown with dashed red curves) and the reduced model with a block diagonal structure in state-matrix $G_{\textnormal{red-modal}}$ (shown with dash-dotted green curves). The different curves correspond to the different values of the scheduling parameter $\rho$ from the grid $\{-1,-0.5,0,0.5,1\}$.
• Figure 4: Frequency response of the error system $G-G_{\textnormal{red}}$ (shown with dashed red curves) and the error system $G-G_{\textnormal{red-modal}}$ (shown with dash-dotted green curves). The different curves correspond to the different values of the scheduling parameter $\rho$ from the grid $\{-1,-0.5,0,0.5,1\}$.
• Figure 5: Step response of the original model $G$(shown with solid blue curves), reduced model $G_{\textnormal{red}}$ (shown with dashed red curves) and the reduced model $G_{\textnormal{red-modal}}$ (shown with dash-dotted green curves). The different curves correspond to the different values of the scheduling parameter $\rho$ from the grid $\{-1,-0.5,0,0.5,1\}$.