Symmetry Reduction

Abstract

In this chapter, we exploit the symmetries in the interconnection structure to reduce the number of decision variables in the stability and performance tests of earlier chapters. We describe symmetries with the graph-theoretic notion of an automorphism, a permutation of the nodes that leaves the graph unchanged. The equivalence classes of nodes that can be reached from one another by an automorphism are called orbits. The main idea in this chapter is to assign one decision variable per orbit rather than one per node. This reduction leads to major computational savings for interconnections that are rich in symmetries, since they possess few orbits. We further propose a class of transformations that enrich the symmetries in the interconnection. As a special case, we revisit cyclic interconnections and derive a stability test that was presented without proof in Chap. 2.