By Petros A. Ioannou, Petar V. Kokotovic
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Extra info for Adaptive Systems with Reduced Models
3?  G. Kreisselmeler, "Adaptive Observers with Exponential Rate of Convergence," IEEE Trans. on Automatlc Control, Vol. AC-19, pp. 117-119, April 1974.  G. Kreisselmeier, "The Generation of Adaptive Law Structures for Globally Convergent Adaptive Observers," IEEE Trans. on Automatia Control, Vol. AC-24, No. 3, pp. 510-512, June 1979. 3. 3,1, CHARACTERIZATION DISCRETE-TIME IDENTIFIERS OF THE MISMATCH Linear discrete systems possessing the two-time scale property analogous to the continuous systems analyzed in Chapter 2, have m fast eigenvalues 0(~) and n slow eigenvalues 0(1).
14) Suppose now that an identifier is designed for the nth-order dominant part to identify the vectors a s = [aI ... an]T, b s = [bI ... bn]T under the assumption that , = 0, that is, there are no parasitics. that ~(k) = 0 and, hence, Yu(k) =0. system with parasitics. This implies This identifier is then applied to the actual Our purpose is to examine the performance of "parallel" and "series parallel" identifiers in the presence of the parasitic input and obtain bounds for their parameter and output errors.
Bn(k)]. 5 but I~i. 64) is the standard least squares algorithm which is not exponentially stable, since F (k) does not retain positive definites hess for this value of I. 65) the composite error Zs(k) ffi [y(k)-Ys(k),Zl-al(k) ..... an-an(k),bl-bl(k) ..... 68) APs(k ) = [aI ... - bn ]T-~s(k). 71) is proven in  (with ~ ~ ) . 49) is applicable. 71) and is not a function of 52 Zs(k o) as in the parallel case. 71) k-k IIZs(k)II -< ClC 2 k-I °IIZs(ko) ll + Z clck-l-illBs(i) lIIlYu(i+l)II. 73), due to the boundedness of u, the stability of A s , and the boundedness of Fs(k), Bs(k ) is bounded.