By Cornelius T. Leondes
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Additional info for Advances in Theory and Applications : System Identification and Adaptive Control, Pt 3 of 3
0 0 0 0 ... 0 c N( A n r l n 2 - a nA " - . . - a 2I ) b u 0 It is readily verified  that the consolidated (n + l)-order system (128) is completely observable, for any "constant" u(t), if and only if c N( A nl n 2 - a n A " - ... - a 2 I)bu Φ 0. (129) Satisfaction of (129) thus requires two simultaneous conditions: u(t)#0 (130a) 56 C. D. JOHNSON and 1 N 2 C n C A " - - c t N A - - ... , the plant (124) does not have an open-loop "zero" at the origin. The canonical system (128) is a very special type of bilinear dynamical system for which a novel full-order observer design has recently been developed .
Substituting (84c) into (84a) yields the counterpart of (13) as χ = A N(t)x + B N ( t ) u p + B N ( t ) u b + [öB(t)]u (85) + B N ( t ) (u d + u f ) + [F N(t) + ÔF(t)]H(t)z. It follows from (85) that u b and (u d + u f ) should ideally satisfy the cancellation conditions (86a) BN(t)u b^-[ÖB(t)]u (86b) B N ( t ) (u d + u f ) == - [ F N ( t ) + öF(t)]H(t)z. An approximate method for satisfying (86) is to model the right sides of (86) by polynomial splines, as in (26), and use the technique of (7), (8), (14).
W e call such a parameter domain the "adaptability domain" for the closed-loop system. The adaptability domain for the plant (72) and adaptive controller (77) and (78), with λ ο ί = - 5 , i = 1, 2, 6, is shown in Fig. 6. The effectiveness of a (model-reference) adaptive controller, with respect to δΑ perturbations in linear plants, is measured by: 1) the extent of the adaptability domain, and 2) the degree to which actual response mimics ideal-model response within the adaptability domain; see Fig.
Advances in Theory and Applications : System Identification and Adaptive Control, Pt 3 of 3 by Cornelius T. Leondes