The H2 Control Problem for Descriptor Systems

A solution of the H2 control problem is presented for linear descriptor systems. The solution proceeds in two steps. Firstly, the set of all controllers that stabilize the control system is parametrized. The mathematical tool applied are doubly coprime, proper stable factorizations of rational matrices. The factors are expressed in terms of stabilizing descriptor feedback and output injection gains, which represent degrees of freedom that can be used in the subsequent optimization. In a coordinate system in which dynamic and non-dynamic modes are separated, the corresponding gains are used to regularize the problem and minimize the norm. Finally, a projection result is applied to obtain an optimal controller. Two synthesis procedures are derived, a primal and the dual one. The symmetries thus obtained provide further insight as well as further computational advantages.