Simultaneous Stabilization of Second Order Linear Switched Systems Based on Superstability and D-Decomposition Technique

The considered problem is to simultaneously stabilize a family of second order linear systems by static linear state feedback when applied to switched systems. The proposed synthesis approach is based on a known design method where a static regulator is found as a solution to the linear programming problem. This regulator makes all matrices from the family forming switched systems superstable in the closed loop state, which in turn guarantees exponential stability of the switched system. This approach is generalized for the case where not all matrices in the family can simultaneously be made superstable: for non-superstabilizable matrices one determines using D-decomposition linear bounds on the set of stabilizing regulators, which are used in the linear programming problem. The designed switched system properties are briefly studied. An example of a design problem solution using the proposed approach is presented.