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Optimal adaptive control and consistent parameter estimates for ARMAX model with quadratic cost. (English) Zbl 0632.93045
The paper deals with an adaptive control problem characterized by a multidimensional ARMAX model and a loss function given by the long run average of quadratic costs. For this problem a suitable adaptive control law is recursively computed on the basis of estimates of the unknown parameters given by the stochastic gradient algorithm.
Under various assumptions, concerning mainly stability properties, it can be shown that the given adaptive control provides a complete solution to the problem in the sense that consistency of parameter estimates and minimality of the quadratic loss function are simultaneously achieved. The adaptive control is obtained using the randomly varying truncation technique and the attenuaing excitation technique as given by H.-F. Chen and L. Guo [Int. J. Control 43, 869-881 (1986; Zbl 0582.93066)]. The optimality and consistency properties of the adaptive control law hold true also when the parameter estimates are obtained by the extended least squares algorithm.
Reviewer: G.Di Masi

93C40 Adaptive control/observation systems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
93E12 Identification in stochastic control theory
93C35 Multivariable systems, multidimensional control systems
93E10 Estimation and detection in stochastic control theory
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