Stability, convergence, and performance of an adaptive control algorithm applied to a randomly varying system

The stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system are investigated. The authors demonstrate that: loss functions on the input-output process converge to their expectation with respect to an invariant probability at a geometric rate, and hence, a form of stochastic exponential asymptotic stability is established; and when the parameter variation and measurement noise are small, it is shown that the performance is nearly optimal, and if an excitation signal is added in the control law, near consistency of the parameter estimates is obtained. Further results include central limit theorems and the law of large numbers of the input-output and parameter processes.<>