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Adaptive Control Design and Analysis Part 8

Tham khảo tài liệu 'adaptive control design and analysis part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 262 Chapter 5 Continuous-Time Model Reference Adaptive Control adaptive control case it results in 0 t G B2 ko ft2 A p Ơ or di or A for some k0 0 for closed-loop stability there is an additional condition that V G 0 V for some V T or d or A 0 which depends on the unknown Go s and cannot be large. On the other hand the parameter projection algorithm 3.182 can be used to ensure the system properties as that from a switching cr-modification when the parameter bounds are available. Feedback Unmodeled Dynamics The results in Section 5.5 can be generalized to the following plant with more general unmodeled dynamics and disturbances y t G0 s u t i G0 s Am s Aou s u t - i G s Amy .s Aa s y t Go s Fd s di t dz t 5.340 where z G0 s Amỉ s Aaj s i t represents feedback unmodeled dynamics and Fd s di t represents input disturbances 370 . This plant model captures some dynamics which cannot be modeled in the form of 5.177 . Robustness to limWoo d t 0 A special class of disturbances are those d í G L such that lim oo d t 0 but d t ị La for any a e l oo e.g. d t ĩaạ 2ý t 0 . For such disturbances the adaptive control schemes in Sections 5.3.2 5.3.3 5.3.4 5.3.5 and 5.4 may not ensure closed-loop signal boundedness. For robustness with respect to such disturbances we should use the modified adaptive schemes in Section 5.5. Consider the plant 5.157 that is F s y - dy t kpZ s u du t 5.341 where d t G L dy t G L and lim oo d t 0 limt oo dy t 0. With the controller 5.34 the closed-loop error equation is 5.159 in which the equivalent output disturbance is í í í1 km i w w R A r 5-342 Using the adaptive laws with a switching ơ-modification see Section 5.5 the controller 5.34 ensures closed-loop signal boundedness see e.g. Theorem 5.7 Discussion 263 5.8 for d t G L . Consider the gradient algorithm 5.201 - 5.202 . The time derivative of v 0 p in 5.95 along 5.201 - 5.202 satisfies V - 2t71 í ẽr í ớ í - 2 T2 i p t p i 5.343 where ơi t p ỡT í ớ t 0 and Ơ2 f p t p t 0. The question is whether .