By Cornelius T. Leondes
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Extra resources for Advances in Theory and Applications : System Identification and Adaptive Control, Pt 3 of 3
JOHNSON 44 VI. GENERALIZATION OF THE NEW APPROACH: CONSIDERATION OF NONZERO [δΒ], [ δ η , [ÖC] The adaptive controller (47) and (50) was derived under the assumptions 7 [δΒ] = 0, [δί ] = 0, [5C] = 0 in (9). However, as stated earlier, the controller (47), (50) will, in fact, tolerate a moderate range of [OF], [δΒ]. In this section we will show how the same DAC techniques used to derive (47) and (50) can be generalized to accommodate the cases of nonzero [δΒ], [OF], In the case of [δΒ] this will require a mild nonlinearity (multiplier) in the controller.
Nevertheless, in the time-invariant case of (47) and (50), it is interesting to calculate and examine the controller "transfer-function" structure implied by (47) and (50), to see if any novel structural properties or interpretations emerge. For this purpose, the results are easier to see if one examines various particular cases of (47) and (50). The one case examined below yields structural results which are representative of the general case. In the particular case of the universal adaptive controller (77) and (78), as used for all 22 different plants associated with Figs.
At this point we will proceed as in (27) and invoke the assumption/approximation — = 0 and ^ ^ = 0, almost every where. (97) dt dt As a consequence of (97), the successive time derivatives of (96) along solutions of (8) are (i) Zi = <9 ,Dz> ( i ) 2 z>((p ,D z) 1 ^ (98) (i) * ζ = <φ , D z). ( ί) It follows from (98) and the Cayley-Hamilton theorem that if φ * 0 the associated Zi obeys (at least) the ρ-th-order linear, constant coefficient differential equation (P)
Advances in Theory and Applications : System Identification and Adaptive Control, Pt 3 of 3 by Cornelius T. Leondes