By Petros A. Ioannou, Petar V. Kokotovic
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Additional resources for Adaptive Systems with Reduced Models
That IIAIIU = 0(I). 1) are close to the eigenvalues of All. 1). -I We assume that A I exists and It is obvious then that for ~ sufficiently small the n eigenThey are the n "slow" elgen- The remaining m eigenvalues are of 0(~) magnitude. 1) has this two-time scale property. The following lemm, from  extended in [1,2] gives such a bound. 4) A21 + PAll- ~A22 P - ~PAI2P = 0 -i having the property that P÷-A21AII as ~ ÷ 0 . 4) follows from the implicit function theorem. 5) y(k) ffi ClX(k) + ~C2z(k) so that the parasitics are weakly observable from the plant output.
I , , , I , , , ' o~ b"£ 36 The improvement of the bound by making the convergence rate m 2 larger may require trial and error selection of the adaptive gains. 135) (1 + n k 2 max ~[G]) 2 gives more information about the dependence of the error bound on other quantities. In this case m 2 is the largest when max %[G] - mln %[G]. 135) that the error bound will be higher when the order n of the dominant part of the plant is higher. 2,8, REFERENCES  P. V. Kokotovic, R. E. , and P. Sannuti, "Singular Perturbations and Order Reduction in Control Theory - An Overview," Automatica, Vol.
Narendra, "On the Uniform Asymptotic Stability of Certain Linear Nonautonomous Differential Equations," SIAM J. 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.
Adaptive Systems with Reduced Models by Petros A. Ioannou, Petar V. Kokotovic