Optimal actuator placement for minimizing the worst-case control energy

TL;DR

This paper addresses the challenge of optimally placing actuators in linear systems to minimize the worst-case control energy, providing a complete solution for systems with positive definite matrices.

Contribution

It offers a complete solution for the optimal actuator placement problem specifically for systems with positive definite matrices, highlighting the complexity of the general case.

Findings

Optimal actuator placement minimizes worst-case control energy.

Complete solution provided for positive definite system matrices.

Highlights the difficulty of the general problem.

Abstract

We consider the actuator placement problem for linear systems. Specifically, we aim to identify an actuator which requires the least amount of control energy to drive the system from an arbitrary initial condition to the origin in the worst case. Said otherwise, we investigate the minimax problem of minimizing the control energy over the worst possible initial conditions. Recall that the least amount of control energy needed to drive a linear controllable system from any initial condition on the unit sphere to the origin is upper-bounded by the inverse of the smallest eigenvalue of the associated controllability Gramian, and moreover, the upper-bound is sharp. The minimax problem can be thus viewed as the optimization problem of minimizing the upper-bound via the placement of an actuator. In spite of its simple and natural formulation, this problem is difficult to solve. In fact, properties such as the stability of the system matrix, which are not related to controllability, now play important roles. We focus in this paper on the special case where the system matrix is positive definite. Under this assumption, we are able to provide a complete solution to the optimal actuator placement problem and highlight the difficulty in solving the general problem.