Automated Linear Parameter-Varying Modeling of Nonlinear Systems: A Global Embedding Approach

TL;DR

The paper addresses converting nonlinear dynamical systems into a global LPV representation without approximation, enabling LPV control design. It introduces an automated embedding based on the Second Fundamental Theorem of Calculus to factor nonlinearities into integrals of Jacobians, yielding LPV matrices that depend on a scheduling map $p(t)=\eta(x,u)$ and guaranteeing $\mathcal{B} \subseteq \mathcal{B}_{LPV}$. The implementation in the LPVcore toolbox (via nlss and nlss2lpvss) supports analytical and numerical Jacobian integration, produces self-scheduled simulations, and provides a pathway to LPV controller synthesis. Demonstrations on an unbalanced disk and a 3DOF control moment gyroscope show exact embeddings (RMSE near machine precision) and successful controller designs, with publicly available software for replication.

Abstract

In this paper, an automated Linear Parameter-Varying (LPV) model conversion approach is proposed for nonlinear dynamical systems. The proposed method achieves global embedding of the original nonlinear behavior of the system by leveraging the second fundamental theorem of calculus to factorize matrix function expressions without any approximation. The implementation of the proposed method in the LPVcore toolbox for Matlab is discussed, and its performance is showcased on a comprehensive example of automated LPV model conversion of an unbalanced disk system, which is then used to design an LPV controller that is deployed on the original nonlinear system. In addition, the conversion capabilities are further demonstrated by obtaining an LPV embedding of a three-degree-of-freedom control moment gyroscope. All software implementations are available at www.lpvcore.net.

Figures (8)

• Figure 1: Schematic representation of a self-scheduled simulation with an LPV-SS model and a scheduling map.
• Figure 2: Picture of the unbalanced disk system.
• Figure 3: Simulation of the nonlinear unbalanced disk $\Pi_\mathrm{NL}$ ( ), self-scheduled simulation of the LPV models $\Pi_\mathrm{LPV}^\mathrm{ana}$ ( ) and $\Pi_\mathrm{LPV}^\mathrm{num}$ ( ).
• Figure 4: Generalized plant structure for the controller synthesis of the unbalanced disk.
• Figure 5: Schematic representation of the self-scheduled closed-loop simulation.