Quadratic Maximization of Reachable Values of Stable Discrete-Time Affine Systems

TL;DR

This paper presents a method to exactly solve a quadratic maximization problem over the reachable set of stable discrete-time affine systems by efficiently selecting a finite subset of quadratic programs, ensuring optimality.

Contribution

It introduces an integer-valued function to minimize overapproximations, enabling finite-time exact solutions for a class of quadratic maximization problems in control systems.

Findings

Method successfully solves the problem for 2D non-diagonalizable systems.

Experimental results on random instances validate the approach.

The approach guarantees finding the global maximum within the constructed finite set.

Abstract

In this paper, we solve a maximization problem where the objective function is quadratic and the constraints set is the reachable values set of a stable discrete-time affine system. This problem is equivalent to solve an infinite number of linearly constrained quadratic maximization programs. To solve exactly and in finite time this problem, we have to safely extract a finite number of them. Safely means that we must guarantee that the optimal solution can be found within this extracted family. This family has to be the smallest possible. Therefore, we construct an integer-valued function defined on the solutions of the discrete Lyapunov equation. Those integers represent overapproximations of the number of quadratic programs to solve to obtain an optimal solution of our specific maximization problem. The integer-valued function is minimized in order to get the smallest possible overapproximation. The method proposed in the paper is first illustrated on a class of nondiagonalizable systems of dimension two and finally experimented on randomly generated instances of the problem.