A note on feedback stabilizability of switching systems under arbitrary switching

The problem of compensating by state feedback the modes of a linear switching system in such a way that it becomes asymptotically stable under arbitrary switching is considered. Two different sufficient conditions for the existence of a family of state feedbacks that, compensating each mode, make the switching system asymptotically stable under arbitrary switching are given. Both conditions are derived from the analysis of the system behavior in terms of eigenvalues and eigenspaces of its modes. In particular, the first one exploits the known fact that stability under arbitrary switching is guaranteed if all modes of the system, beside being Hurwitz or Schur stable, have dynamic matrices which are normal, or equivalently which have a set of orthogonal eigenspaces. The second conditions, exploits the fact that stability under arbitrary switching is guaranteed if all modes of the system, beside being Hurwitz or Schur stable, have dynamic matrices which have the same set of eigenspaces. In both cases, a characterization in terms of equations that involve the coefficients of the dynamic matrices and the input matrices of the modes are given. The case of two-dimensional modes is investigated in details and examples are provided.