By Petros A. Ioannou, Petar V. Kokotovic

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**Extra resources for Adaptive Systems with Reduced Models**

**Example text**

22) is not exponentially stable. , Re(H(z)} > 0, Vz : Izl =i. The robustness of this identifier wlth respect to parasltlcs Is established as follows. 24) is bounded for all k and where ep(k) = [y(k)-yp(k),y(k-l)-yp(k-l) ..... y(k-n+l)-yp(k-n+l) ] T dpp(k) = [al-al(k),a2-a2(kl ..... an-~n(k),bl-bl(kl ..... bn-bn(k)]T. 21) in the absence of the parasitic input Yu' sup[u(1)-u(i+lII; - i h supI[hi1[ • = i a i is the maximum eigenvalue of Af. 11). (k)]y (k-i) ill i p i=l i l p n + Given ( 3 . 2 5 ) , r [b~-b~(k)]u(k-i) + Yu(k).

Then [xT,u] is ps and therefore the components of MT(t)C are linearly independent. In the pre~enae o f pu~usltla8 this nth order richness of u and hence the linear dependence of MT(t)C can he lost. To prevent this we assume that the input is dominantly rich. 123). 2,7, DISCUSSION AND EXAMPLE Adaptive schemes designed for plants without parasitics, but applied to plants with parasitics are considered robust if the error due to parasitics is 0(B). , as p * 0 . Among the factors cos,non to all the error bounds derived in this chapter ml ~Y~22 ~g' factor ~V determines the robustness property.

0 i 10 , . , . I , 20 Time (~) , , . 4. 5). 34 ' ' ' ' I ' ' ' ' I ' ' ' ' I ' , ~ , ,~o . . 5. 05. L n ~ • • n I I I , , , I . I I I I I n u , 01 > ¢;I n l l l l l l n n l l l l l l l u l l (q) O~ OtP 0~, . . O~ . . O~ . . m . . L O~ i . . I O! 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.