An Improved Method for Designing Optimal Linear Compensators

In linear dynamical systems with white Gaussian plant and observation noise and quadratic cost criteria the well-known separation theorem of stochastic control holds. The result is that the determination of the system matrix for the compensator depends on the solution of two Riccati equations, one arising from a deterministic regulator problem and the other from a filtering problem. It is shown in this paper that if the given system is single-input, single-output and time-invariant, one can achieve substantial savings in computation time. Assuming controllability and observability and using the standard controllable representation of the system matrix, we show that at each iteration step of the solution of the algebraic matrix Riccati equation by Newton's method the number of variables to be solved for reduces from the customary n(n+1)/2 to n. Moreover, the number of operations to determine these n variables is on the order of n
³
/16 as opposed to n
³
/3 in ordinary matrix inversion. The observability matrix is used as a similarity transformation so that the Riccati equation for the filtering problem is placed in the same format and the above procedure may be used again. The results obtained are easily extended to the case in which the system is either single-input or single-output. The system matrix of a fifteenth-order compensator was determined using 0.92 seconds of computer time on the IBM System/360, Model 67.