Tensor-based reduction of linear parameter-varying state-space models

TL;DR

This paper introduces a tensor-based joint reduction method for LPV state-space models, effectively reducing both state order and scheduling dimension simultaneously, which improves computational efficiency for control applications.

Contribution

It presents the first systematic tensor-based approach for joint reduction of state and scheduling dimensions in LPV models, extending Petrov-Galerkin projection with tensor decomposition techniques.

Findings

Effective reduction demonstrated on mass-spring-damper systems

Tensor methods outperform traditional separate reduction approaches

Scalability shown on higher-order models

Abstract

The Linear Parameter-Varying (LPV) framework is a powerful tool for controlling nonlinear and complex systems, but the conversion of nonlinear models into LPV forms often results in high-dimensional and overly conservative LPV models. To be able to apply control strategies, there is often a need for model reduction in order to reduce computational needs. This paper presents the first systematic approach for the joint reduction of state order and scheduling signal dimension of LPV state space models. The existing methods typically address these reductions separately. By formulating a tensorial form of LPV models with an affine dependency on the scheduling variables, we leverage tensor decomposition to find the dominant components of state and scheduling subspaces. We extend the common Petrov-Galerkin projection approach to LPV framework by adding a scheduling projection. This extension enables the joint reduction. To find suitable subspaces for the extended Petrov-Galerkin projection, we have developed two different methods: tensor-based LPV moment matching, and an approach through Proper Orthogonal Decomposition. Advantages of the proposed methods are demonstrated on two different series-interconnected mass-spring-damper systems with nonlinear springs: one primarily used for comparison with other methods and a more elaborate higher-order model designed to assess scalability.