Input Matrix Optimization for Desired Reachable Set Warping of Linear Systems

TL;DR

This paper develops a method to optimize the input matrix of linear systems to effectively warp their reachable sets along specific directions, balancing theoretical rigor and heuristic approaches.

Contribution

It introduces a finite optimization framework for input matrix selection to control reachable set warping, validated on aerospace and oscillator models.

Findings

Finite linear optimization problems can be used to optimize input matrices under certain assumptions.

Heuristic methods perform well when assumptions are relaxed.

Validated on a fighter jet model and a damped oscillator.

Abstract

Shaping the reachable set of a dynamical system is a fundamental challenge in control design, with direct implications for both performance and safety. This paper considers the problem of selecting the optimal input matrix for a linear system that maximizes warping of the reachable set along a direction of interest. The main result establishes that under certain assumptions on the dynamics, the problem reduces to a finite number of linear optimization problems. When these assumptions are relaxed, we show heuristically that the same approach yields good results. The results are validated on two systems: a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues. The paper concludes with a discussion of future directions for reachable set warping research.