The H2 control problem with internal stability

A solution of the H₂ control problem is presented for linear systems described by rational transfer matrices. These rational matrices are not required to be proper and no assumption is made on the location of their poles and zeros. The control system is considered in the standard configuration, which includes the synthesis model of the plant and the controller. The H₂ control problem consists of internally stabilizing the control system while minimizing the H₂ norm of its transfer function. The notion of internal stability is based on input-output stability and means that the subsystems defined by any pair of input and output terminals within the control system all are input-output stable. The solution proceeds in three steps. Firstly, the set of all controllers that stabilize internally the control system is parametrized. Then the subset of the stabilizing controllers that achieve a finite value of the H₂ norm of the system transfer function is described, also in parametric form. Finally, the optimal controller is obtained by selecting the parameter that minimizes the norm. The existence of these three sets of controllers is established in terms of the given data. The mathematical tool applied are doubly coprime, proper stable factorizations of rational matrices. Based on this description of the plant, two synthesis algorithms are derived: the primal and the dual one. The construction of the optimal controller requires two specific operations with proper stable rational matrices: inner-outer factorization and proper stable projection. The solution obtained is general in the sense that no assumptions on the plant are made other than those securing the outer factors to be square.