Abstmct-We derive a broad range of theoretical results concerning the performance and limit.tioas of a class of rnJoe adaptive fikxs. Applicrtions of theae filters have been proposed m many d i f f m t enginedng contexts which have in common the Idlowing idealized identificrtion problem: A system has a vector input x ( t ) and a scrln output z(t) = h'x(t), where h is an unknown timeinvariant coefficient vector. From a knowledge of x(t) and z(t) it is reqM,to estimate h. The filta considered

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... 8djusts an &ate general nonlinear, function, and K is the loop gai n. The effectiveness of the filta is detemim$ by the conveqence p m p t i e a of the m i d i g m e n t vector, r = hk. With weak nondegenency requirements on x(r) we prove the exponential convergence to zero of the norm IIr(t)II. For ail values of K, we give uppet and lower bounds on the convergence rate which n e tight in that both bounds have s i m h qualitative dependence on K. The dependence of these bounds on K is unexpected and imporhnt since it reveils bade limitations of the filters which rte not predicted by the conventional approximate method of analysis, the "method of averaging." By analyzing the effects of added forcing term u(t) in the control equation we obtain uppes bounds to the effects on the convergence process of various important departures from the idealized model as when noise is present as an d d i t i o d component of z(t), the coefficient vector h is time-vrrying, and the integrators in a hardware implementation have f d t e menory.