Scalable Computation of $\mathcal{H}_\infty$ Energy Functions for Polynomial Control-Affine Systems

TL;DR

This paper introduces a scalable numerical method using Kronecker products to compute nonlinear energy functions for high-dimensional polynomial control systems, enabling solutions for systems with over 1000 states.

Contribution

It develops a scalable algorithm based on Al'brekht's power-series method and Kronecker products, with explicit algebraic structures and open-source software for high-dimensional polynomial control systems.

Findings

Successfully computed energy functions for systems with over 1000 states

Demonstrated efficiency and scalability of the proposed algorithms

Provided open-source implementation for practical use

Abstract

We present a scalable approach to computing nonlinear balancing energy functions for control-affine systems with polynomial nonlinearities. Al'brekht's power-series method is used to solve the Hamilton-Jacobi-Bellman equations for polynomial approximations to the energy functions. The contribution of this article lies in the numerical implementation of the method based on the Kronecker product, enabling scalability to over 1000 state dimensions. The tensor structure and symmetries arising from the Kronecker product representation are key to the development of efficient and scalable algorithms. We derive the explicit algebraic structure for the equations, present rigorous theory for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms. The method is illustrated on two simple academic models, followed by a high-dimensional semidiscretized PDE model of dimension as large as $n=1080$.