Sufficient conditions for simultaneous stabilization of three linear systems within the framework of nest algebras

In this paper, we study the problem of the simultaneous stabilization of three time-varying discrete-time linear systems within the framework of nest algebras. From the perspective of strong transitivity, we establish sufficient condition for the existence of time-varying simultaneously stabilizing controllers based on the coprime factorization. In particular, we also derive a sufficient condition for simultaneous stabilizability of three systems, where the systems are pairwise simultaneously stabilizable. Additionally, the sufficient conditions given in this paper lead to a constructive controller design to stabilize simultaneously three systems. These results hold as well in the time-invariant case. Two examples are included for illustration.