By Petros A. Ioannou, Petar V. Kokotovic
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Additional info for Adaptive Systems with Reduced Models
3. SERIES PARALLEL ADAPTIVE IDENTIFIER Consider now the series-parallel ,. 59) where B (k-l) ffi [y(k-l) ,y(k-2) ..... y(k-n) ,u(k-l) ..... 60) and ~ ( k ) ffi [$1(k),~2(k) . . . $n(k),~l(k) . . . 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).
Control and Opt~ization, Vol. 15, No. i, pp. 5-24, January 1977.  B . D . O . Anderson, "Exponential Stability of Linear Equations Arising in Adaptive Identification," IEEE Trans. on Automatic Control, Vol. AC-22, pp. 83-88, February 1977.  J . C . Yuan and W. M. Wonham, "Probing Signals for Model Reference Identification," IEEE T ~ s . on Automatic Control, pp. 530-538, August 1977.  P . V . Kokotovlc and A. H. Haddad, "Controllability and Time-Optlmal Control of Systems with Slow and Fast Modes," IEEE Trans.
6, pp. 552-560, November 1974. 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).