×

Adaptive control with the stochastic approximation algorithm: Geometry and convergence. (English) Zbl 0591.93063

The paper deals with linear systems that can be described by the equation \[ y(t+1)=\sum^{p}_{i=0}[a_ iy(t-i)+b_ iu(t-i)+c_ iw(t- i)+w(t+1) \] where y, u and w are the output, input and noise, respectively. It is assumed that \((a_ 0,...,a_ p,b_ 0,...,b_ p,c_ 0,...,c_ p)\) are unknown parameters and the goal is to minimize the variance of the output process.
A well-known adaptive control algorithm for this problem is introduced. However, the algorithm uses a recursive stochastic approximation scheme to obtain estimates of the unknown parameters. Namely, the behaviour of these estimates is studied. It is proved that if the system does not have a reduced-order minimum variance controller then the estimates converge with positive probability to some value different from the true parameter. This can happen even with probability one. Further, when the system processes reduced-order minimum variance controllers, then convergence to a minimum variance controller in a Cesaro sense is shown.
Reviewer: V.Kankova

MSC:

93E20 Optimal stochastic control
62L20 Stochastic approximation
93C40 Adaptive control/observation systems
62F12 Asymptotic properties of parametric estimators
62J10 Analysis of variance and covariance (ANOVA)
93E10 Estimation and detection in stochastic control theory
93E25 Computational methods in stochastic control (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI