Convergence properties of adaptive observers

Convergence characteristics of the adaptive observer, to be acceptable, must be insensitive to initial conditions. In this paper the problem of optimization with respect to the set of adaptive gains is considered. It is found that sensitivity of an optimal solution is highly dependent upon the definition of the cost function. The adaptive observer is shown, to be characterized as a linear autonomous system with (periodic) time-variable parameters. A cost function is defined in terms of the elements of a transition matrix so as to yield a suboptimal solution insensitive to initial conditions. Other factors affecting the convergence properties are discussed. Particular attention is given to the case of periodic signals for which the eigenvalues of a constant matrix can be used as design criteria.