A transfer function solution of the L.Q.G. control problem

A general transfer function approach to the solution of the continuous stationary Kalman-Bucy filter and the time invariant linear deterministic and stochastic optimal control problems is presented. A generalised spectral factor is defined by which the equivalence between the time domain and the frequency domain approaches to these problems is established. An explicit expression for this generalised spectral factor is given. In the case where the number of system inputs and outputs is much smaller than the order of the system, the computation involved is much faster and requires less storage than is required to solve the Riccati equation of the time domain approach. The theory is also extended to the case of coloured signals and linear dynamical operators in the performance index and a simple solution to the singular optimal control problem is derived.