Optimal adaptive control and consistent parameter estimates for ARMAX model with quadratic cost

We consider the multi-dimensional ARMAX model A(z)yn =B(z)un + C(z)wn with loss function J(u) = limsup 1/n ??i=1 n (yi ??Q1yi+ui ??Q2ui) n???? where the coefficients in the matrix polynomials A(z), B(z) and C(z) are unknown. Conditions used here are:1) stability of A(z) and full rank of Ap :2) strictly positive realness of C(z) - ??I and 3) controllability and observability of a matrix triple consisting of coefficients in A(z), B(z) and Q1. On the basis of the estimates given by the stochastic gradient algorithm for unknown parameters an adaptive control is recursively defined. It is proved that the parameter estimates are strongly consistent and the quadratic loss function reaches its minimum. This paper also includes some general theorems on parameter estimation, which the results about adaptive control are essentially based on.