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Additional resources for Advances in Theory and Applications : System Identification and Adaptive Control, Pt 3 of 3
5, where it can be seen that the adaptive performance obtained is virtually the same as that obtained in Fig. 4 for "constant" parameter perturbations. and Fig. 6. Adaptability domain for plant (72) with adaptive controller (77) and (78) = - 5 , i = 1 , 2 , ... , 6 . A NEW APPROACH TO ADAPTIVE CONTROL 35 Adaptability Domains in Parameter Space. For sufficiently large parameter perturbations Ss^ (constant or time varying), the general adaptive controller (47) and (50) (with a fixed gain Ko) will eventually be "overpowered" and a serious loss of performance (or instability) will then result.
A n ) are the coefficients of the characteristic polynomial of A η 1 det(XI - A) = λ - Οηλ*- - α 2 λ - ax. ,010) ( φ ) . (128b) where On 1 0 (Vi 0 1 0 0 A= ... 0 crfm 0 c^A-aJPa ... 0 c^-c^A-a^bu . (128c) ax 0 0 ... 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.
4. The results of exercising the closed-loop simulation for a variety of disturbances and rather large "constant" parameter perturbations , 6 a 2 are shown in Fig. 4. Introduction of Time-Varying Perturbations. It was stated below (6) [see also remarks below (27)] that in spite of the technical assumptions (27), the final adaptive controller (47) and (50) can, in fact, accommodate a significant degree of nonconstant behavior of [δΑ], To demonstrate this feature the plant parameter perturbations 5ai, 5 a 2 in this example were next allowed to vary with time in various linear and sinusoidal fashions using the same linear, time-invariant adaptive controller (77) and (78) with the same numerical gains k^ used in Fig.