Algebraic Riccati Tensor Equations with Applications in Multilinear Control Systems

TL;DR

This paper extends control theory to continuous-time multilinear tensor systems, introducing algebraic Riccati tensor equations, tensor decompositions, and numerical methods, with applications in stability and robustness analysis.

Contribution

It develops the algebraic Riccati tensor equation for continuous-time MLTI systems and proposes tensor-based numerical solutions and stability criteria.

Findings

Unique positive semidefinite solution to ARTE under stabilizability and detectability

Tensor-based Newton method effectively solves ARTE

Tensor versions of bounded real lemma and small gain theorem established

Abstract

In a recent paper by Chen et al. [8], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decomposition. Building on this new research direction, the purpose of our paper is to extend the study to continuous-time MLTI control systems. Specifically, we define Hamiltonian tensors and symplectic tensors, and we establish the Schur-Hamiltonian tensor decomposition and the symplectic tensor singular value decomposition (SVD). Based on these concepts, we propose the algebraic Riccati tensor equation (ARTE) and demonstrate that it has a unique positive semidefinite solution if the system is stabilizable and detectable. To find numerical solutions to the ARTE, we introduce a tensor-based Newton method. Additionally, we establish the tensor versions of the bounded real lemma and the small gain theorem. A first-order robustness analysis of the ARTE is also conducted. Finally, we provide a numerical example to illustrate the proposed theory and algorithms.