Real Solutions to Control, Approximation, and Factorization Problems

During the past decades much of finite-dimensional systems theory has been generalized to infinite dimensions. However, there is one important flaw in this theory: it only guarantees complex solutions, even when the data is real. We show that the standard solutions of many classical problems with real data are also real. We call a (possibly matrixor operator-valued) holomorphic function G real (real-symmetric) if G(z̄) = G(z) for every z. We show that if such a function can be presented as G = NM−1, where N,M ∈ H∞, then we have G = NRM R , where NR,MR ∈ H ∞ are real and weakly right coprime. Consequently, if a real function G has a stabilizing compensator (i.e., a function K such that [ I −K −G I ]−1 ∈ H∞), then G has a real doubly coprime factorization and a Youla parameterization of all real stabilizing controllers. If a system of the form ẋ = Ax + Bu, y = Cx + Du or of the form xn+1 = Axn + Bun, yn = Cxn + Dun has real (possibly unbounded, constant) coefficients A, B, C and D, then the system is stabilizable iff it is stabilizable by a real state-feedback operator. This holds for both exponential stabilization and output stabilization. A real stabilizing state-feedback operator is then given by the standard LQR feedback operator, hence the standard (complex) formulae can be used to find this real solution. Analogous results are established for other optimization, factorization, approximation and representation problems too.