Some results on simultaneous stabilization with multiple domains of stability

In this paper, the problem of simultaneous stabilization in the case where each plant has a different domain of stability associated with it is studied. Specifically, it is supposed that one is given plants P₀, P₁,…, P_r together with domains of stability D₀,…, D_r, and the objective is to find a common controller C such that the poles of the closed-loop transfer matrix H(P_i, C) are all inside the corresponding domain of stability D_i, for i = 0,…, r. This formulation of the simultaneous stabilization problem generalizes earlier problem formulations wherein it was assumed that all the domains of stability were the same, and is believed to be a more realistic formulation of the problem of preserving system stability in the event of sensor or actuator failure. In the paper, necessary and sufficient conditions for the existence of a common controller are derived, in the case where one of the domains (say D₀) is a subset of the rest. The case of two plants P₀, P₁ with D₀ a subset of D₁ is studied in great detail, and the following result is established: the two plants can be simultaneously stabilized in the above generalized sense if and only if they can be simultaneously stabilized with respect to the larger of the two regions. More precisely there exists a controller C such that the poles of H(P_i, C) lie inside D_i for i = 0, 1 if and only if there exists a controller such that the poles of bothH(P₀, C) and of H(P₁, C) lie inside the larger region D₁.