Optimal actuator design via Brunovsky's normal form

TL;DR

This paper introduces a novel approach using Brunovsky's normal form to optimize actuator design for linear systems, simplifying the problem and enabling effective numerical solutions.

Contribution

It reformulates the actuator design problem without diagonalizability restrictions, providing a clearer framework and insights into the problem's symmetries and solution existence.

Findings

The reformulation simplifies the optimization problem.

Numerical experiments reveal intrinsic symmetries and challenges.

Evolutionary algorithms outperform gradient-based methods in this context.

Abstract

In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky's normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem's intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms - this is alleviated by using an evolutionary algorithm.