Constrained biogeography-based optimization for invariant set computation

We discuss the application of biogeography-based optimization (BBO) to invariant set approximation. BBO is a recently developed evolutionary algorithm (EA) that is motivated by biogeography, which is the study and science of the geographical migration of biological species. Invariant sets are sets in the state space of a dynamic system such that if the state begins in the set, then it remains in the set for all time. Invariant sets have applications in many constrained control problems, and their computation amounts to a constrained optimization problem. We therefore frame the invariant set computation problem as a constrained optimization problem, and we use a constrained BBO algorithm to solve it. We study three specific invariant set problems: the approximation of the maximum invariant ellipsoid, the approximation of the maximum invariant semi-ellipsoid, and the approximation of the maximum invariant cylinder, which has application to sliding mode control. We find that BBO outperforms linear matrix inequality (LMI) algorithms for the first and third of these problems. For the second problem, LMI performs better than BBO, but BBO only requires 65% of the computational effort.