Solutions of the Singular Stochastic Regulator Problem

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Abstract
The limiting form of the optimum stochastic regulator is determined which minimizes 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) when either the control weighting matrix, R, or the spectral density matrix, W, of the observation noise, w, is singular. It is shown that as R tends from a positive-definite matrix to a non-negative definite matrix, the optimum regulator can be synthesized by a system using at most n − ks integrators, where n is the order of the system and ks equals the rank of B minus the rank of BR. Similarly, when W tends from a positive-definite matrix to a non-negative definite matrix, the optimum regulator can sometimes be synthesized by a system using at most n − rs integrators, where rs equals the rank of H minus the rank of WH. The structure of the regulator is given for each of these cases.

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