Solver-free optimal control for Linear Dynamical Switched System by means of Geometric Algebra

TL;DR

This paper introduces a solver-free algorithm using Geometric Algebra to determine optimal switching paths in 2D linear switched systems, avoiding numerical solvers and ensuring minimal switches.

Contribution

It develops a novel, symbolic algebra-based method for control of linear switched systems using Geometric Algebra, eliminating the need for numerical solvers.

Findings

Successfully finds switching points using symbolic algebra operations.

Constructs optimal switching paths with minimal switches.

Demonstrates the method on two example systems.

Abstract

We design an algorithm for control of a linear switched system by means of Geometric Algebra. More precisely, we develop a switching path searching algorithm for a two-dimensional linear dynamical switched system with non-singular matrix whose integral curves are formed by two sets of centralised ellipses. Then it is natural to represent them as elements of Geometric Algebra for Conics (GAC) and construct the switching path by calculating the switching points, i.e. intersections and contact points. For this, we use symbolic algebra operations, more precisely the wedge and inner products, that are realisable by sums of products in the coordinate form. Therefore, no numerical solver to the system of equations is needed. Indeed, the only operation that may bring in an inaccuracy is a vector normalisation, i.e. square root calculation. The resulting switching path is formed by pieces of ellipses that are chosen respectively from the two sets of integral curves. The switching points are either intersections in the first or final step of our algorithm, or contact points. This choice guarantees optimality of the switching path with respect to the number of switches. On two examples we demonstrate the search for conics' intersections and, consequently, we describe a construction of a switching path in both cases.