Scalable Computation of $\mathcal{H}_\infty$ Energy Functions for Polynomial Drift Nonlinear Systems

TL;DR

This paper introduces a scalable tensor-based method for computing energy functions in polynomial nonlinear systems, enabling efficient solutions to complex PDEs in high-dimensional models.

Contribution

It develops a tensor-structured approach using Kronecker products to solve Hamilton-Jacobi-Bellman equations for energy functions in polynomial nonlinear systems.

Findings

Successfully computed degree 3 energy functions in 1023 dimensions

Achieved degree 4 energy functions in 127 dimensions

Demonstrated scalability and efficiency on a reaction-diffusion model

Abstract

This paper presents a scalable tensor-based approach to computing controllability and observability-type energy functions for nonlinear dynamical systems with polynomial drift and linear input and output maps. Using Kronecker product polynomial expansions, we convert the Hamilton-Jacobi-Bellman partial differential equations for the energy functions into a series of algebraic equations for the coefficients of the energy functions. We derive the specific tensor structure that arises from the Kronecker product representation and analyze the computational complexity to efficiently solve these equations. The convergence and scalability of the proposed energy function computation approach is demonstrated on a nonlinear reaction-diffusion model with cubic drift nonlinearity, for which we compute degree 3 energy function approximations in $n=1023$ dimensions and degree 4 energy function approximations in $n=127$ dimensions.