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We design an algorithm for control of a linear switched system by means ofGeometric Algebra. More precisely, we develop a switching path searchingalgorithm for a two-dimensional linear dynamical switched system withnon-singular matrix whose integral curves are formed by two sets of centralisedellipses. Then it is natural to represent them as elements of Geometric Algebrafor Conics (GAC) and construct the switching path by calculating the switchingpoints, i.e. intersections and contact points. For this, we use symbolicalgebra operations, more precisely the wedge and inner products, that arerealisable by sums of products in the coordinate form. Therefore, no numericalsolver to the system of equations is needed. Indeed, the only operation thatmay bring in an inaccuracy is a vector normalisation, i.e. square rootcalculation. The resulting switching path is formed by pieces of ellipses thatare chosen respectively from the two sets of integral curves. The switchingpoints are either intersections in the first or final step of our algorithm, orcontact points. This choice guarantees optimality of the switching path withrespect to the number of switches. On two examples we demonstrate the searchfor conics' intersections and, consequently, we describe a construction of aswitching path in both cases.

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