Solver‐free optimal control for linear dynamical switched system by means of geometric algebra

Abstract

An algorithm for finding a control of a linear switched system by means of Geometric Algebra is designed. More precisely, we develop a switching path searching algorithm for a two‐dimensional linear dynamical switched system with a non‐singular matrix whose integral curves are formed by two sets of centralized ellipses. It is natural to represent them as elements of Geometric Algebra for Conics and construct the switching path by calculating switching points, i.e., intersections and contact points. For this, we use symbolic algebra operations or, more precisely, the wedge and inner products that are realizable 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 vector normalization, 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 the optimality of the switching path with respect to the number of switches. Two examples are provided to demonstrate the search for the intersections of the conics and, consequently, a description is presented of the construction of a switching path in both cases.

Survey of New Applications of Geometric Algebra

“…In 116 the authors exploit GA for conics ability to represent transformed objects such as rotated ellipses to propose a novel algorithm for finding the optimal control of a switched dynamical systems with purely imaginary eigenvalues. Such a geometric approach guarantees the optimality of the switching path and eliminates the need for any solver, thus yielding results with minimal numerical errors.…”