By Petros A. Ioannou, Petar V. Kokotovic
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Additional info for Adaptive Systems with Reduced Models
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.
Cn ]T. 37) or v;Ck+l) " (a+C)ep(k) + 0p(k)App(k) + YuCk+l). 39) or 0T(k)F (k)0 (k)b(a+c) ep(k+l) ffi A- P P P i + b If- I l+0~(k)~(k)0p(k) ep(k) + O~(k)Fp(k)0p(k)] 0T(k)App(k) 1 + o~(u)Fp(k)op(k)J 0~(k)Fp(k)ep(k) ] + b I-l+0~(k)Fp(k)0p(k)jYu(k+l). 32), then App(k+l) = App(k) - Fp(k)Op(k) [(a+C)ep(k) 1+ e~(k)Fp(k)Op(k) + O~(k)App(k) + Yu(k-l-l)]. 42) is not linear since A(k) depends on 8p(k) which includes yp(k), a part of Zp(k). As in  we define Ap(k) to be Ap(k) with 8p(k) replaced by e(k) = [y(k),y(k-l) .
2 and u = S s i n t + S s i n 2 . 5). 33 . . i I0 . . i . . 20 T , m e (r~eC) i , i h i 30 40 (o) ~4 w ,co a i i m 11,~ i m . i I . . ' 2O i " i n ' ~ (sL,m ) ' ' ' ' 30 40 (b) IC O~ . . ~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.