Limiting Forms of Optimum Stochastic Linear Regulators

Of interest are the limiting forms of the optimum stochastic regulator which minimize the steady-state expectation, Es{x′Qx+u′Ru}, for the linear process, x˙=Ax+Bu+Gv, given noisy observations y = Hx+w (with v and w being independent white noise processes) as the control weighting matrix, R and/or the spectral density matrix W of the observation noise w tend to zero. It is found that as R vanishes, the optimum regulator can be synthesized by a system using at most n-k integrators, where n is the order of the system and k is the rank of B. Similarly, when W vanishes, the regulator can sometimes be realized with at most n-r integrators, where r is the rank of H. The structure of the regulator is given for each of these cases.