Multivariable adaptive parameter and state estimators with convergence analysis

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Abstract
The convergence properties of a very general class of adaptive recursive algorithms for the identification of discrete-time linear signal models are studied for the stochastic case using martingale convergence theorems. The class of algorithms specializes to a number of known output error algorithms (also called model reference adaptive schemes) and equation error schemes including extended (and standard) least squares schemes. They also specialize to novel adaptive Kalman filters, adaptive predictors and adaptive regulator algorithms. An algorithm is derived for identification of uniquely parameterized multivariable linear systems. A passivity condition (positive real condition in the time invariant model case) emerges as the crucial condition ensuring convergence in the noise-free cases. The passivity condition and persistently exciting conditions on the noise and state estimates are then shown to guarantee almost sure convergence results for the more general adaptive schemes. Of significance is that, apart from the stability assumptions inherent in the passivity condition, there are no stability assumptions required as in an alternative theory using convergence of ordinary differential equations.

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