Optimal state feedback and stabilizing compensators are real when data is real

Infinite-dimensional control theory often provides complex solutions to various control problems. However, in practical applications one would like to have real solutions. We show that the standard solutions are real given real data. We call a (possibly matrix- or operator-valued) holomorphic function G real (or real-symmetric) if G(z) = 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 = N_RM_R^-1, where N_R, M_R ∈ H^∞ are real and weakly right coprime. Thus, if a real function G has a stabilizing compensator (a function K such that [_-G^I_I^-K]^-1 ∈ H^∞), then G has a real doubly coprime factorization and a Youla parameterization of all real stabilizing controllers. If a LTI system of the form ẋ = Ax + Bu, y = Cx + Du or of the form x_n+1 = Ax_n + Bu_n, y_n = Cx_n + Du_n 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 and output stabilization. The stabilizing state-feedback operator is the standard LQR feedback operator, hence the standard (complex) formulae can be used to find this real solution. Analogous results hold for other optimization, factorization and approximation problems too.

I. Introduction

In much of mathematical infinite-dimensional control theory only complex formulae for the solutions are obtained. This is particularly natural when complex function theory or related tools are applied. However, in practical applications one would usually like to obtain solutions that are real numbers, real sequences, real-symmetric functions-or that are matrices (or operators) having such entries. We show how for many output-feedback, state-feedback and other control problems, the standard methods yield real solutions if the original system or transfer function is real (real-symmetric, i.e., . Both state-space and frequency-domain problems are treated, including optimal control, stabilization, factorization and approximation.