Weighted least squares and continuous time adaptive LQG control

The adaptive linear quadratic Gaussian control problem where the linear transformation of the state, A, and the linear transformation of the control, B, are unknown is solved assuming only that (A,B) is controllable and (A,Q₁^1/2) is observable where Q ₁ determines the quadratic form for the state in the integrand of the cost functional. A weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimates is uniformly controllable and observable. A diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent. This family of estimates also identifies (A,B) for deterministic systems. A lagged certainty equivalence control using this family of estimates is shown to be self-optimizing for an ergodic, quadratic cost functional