# Adaptive Systems with Reduced Models by Petros A. Ioannou, Petar V. Kokotovic

By Petros A. Ioannou, Petar V. Kokotovic

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94) i] and A=diag(Xi). 97) S W S 8 ffi AW S where y(0) = 0, Zs(0) = O, Ws(0) = 0. 99) where ~ ~ [ ( ~ . ( t ) - a . (t)-b_),(b(t)-b s l 1 ) T ] , and v = [y,~T]T, s s P ffi[U,Ws] . 92) and el, ¢, T are the output and parameter errors, respectively. 13) with E = 0 and, similarly as before, obtain the following result. 105) =gf m2 fl where gf -- IIA;lll IIAflBf II I[hll IIRII T~2 . 2,6, PARAMETERIZED ADAPTIVE OBSERVER The parameterlzed adaptive observer [ii],[12] is based on a state repre- sentatlon of the plant in which the unknown parameters are contained in an algebraic equation and an error criterion is defined.

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). +l>j +r,>,l Lb j u (k) where P is a small positive parameter, (3. I) The states x(k) and z(k) are n and m vectors and u(k), y(k) are the scalar input and output, respectively. The permutation and/or scaling of states necessary to put two-time-scale continuous systems into specific forms is discussed in [2],[3].

12') that the unknown parameters as,b s are functions of B. 12). 16) where p~(k) = [al(k) ... ~n(k) bl(k) ... bn(k)], 8 (k-l) = [yp(k-l) ... yp(k-n) P u(k-l) ... 18) 43 and yp(k), y~(k)_ are the a posteriori and a priori output of the estimation model, respectively, at the instant k. 5, 1~i starting with an arbitrary positive definite matrix Fp(ko). 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.