## Recursive identification and adaptive prediction in linear stochastic systems.(English)Zbl 0756.93081

This paper deals with adaptive $$k$$-step forward predictors for ARMAX systems of the usual form: $A(z^{-1})y(t)=z^{-\tau} B(z^{-1}) u(t)+C(z^{-1})e(t).$ Here $$z^{-1}$$ is the unit delay operator, $$A(z^{-1})$$, $$B(z^{-1})$$ and $$C(z^{-1})$$ are polynomials in $$z^{- 1}$$, $$\tau$$ is a pure delay, and $$y(t)$$, $$u(t)$$ and $$e(t)$$ are the system’s output, input and innovation sequences. Both implicit and explicit adaptive predictors are considered. By making use of extended stochastic Lyapunov function and martingale limit theorems, the authors establish some basic (mainly convergence) properties of several adaptive $$k$$-step ARMAX predictors based on extended least-squares (also called pseudo-linear regression), stochastic gradient, and monitored recursive maximum likelihood algorithms. The prime application of the derived results is to adaptive control of ARMAX systems, a topic that is also addressed in the present paper.

### MSC:

 93E12 Identification in stochastic control theory 93C40 Adaptive control/observation systems 60G42 Martingales with discrete parameter
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