Tham khảo tài liệu 'adaptive control system part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Adaptive Control Systems 131 Then yields V x f x yT u c C x 6 Ỡ1 x Ag x ớ 0 - ớ rr 1 ờ Thus by choosing the following parameter update law and adaptive controller 0 r c C x ẩi x Ag x ryr u - c C x 01 x A x 0 with Ộ a first-sector third-sector function implies V C x f x - yTộ y Since V is proper in its argument x 0 it follows from that all the solutions C t x r 0 r of the closed loop system and are well-defined and uniformly bounded on 0 to . Furthermore a direct application of LaSalle s invariance principle ensures that all the trajectories C t x r 0 r converge to the largest invariant set E contained in the manifold x 0 x 0 0 . Therefore x r tends to zero as t goes to TO. In other words the cascade system is globally adaptively stabilized by and . Finally Proposition ends the proof of Theorem Remark It is of interest to note that if rank g 0 gj 0 Ag 0 I dim Ớ then lim 0 t 0 0 Indeed on the set E we have g 0 gj 0 Ag O 0 0 r 0 which in turn implies that Ô i ỹ. So E 0 0 0 . The following corollary is an immediate consequence of Theorem where the -system in is void. Corollary Consider a linearly parametrized nonlinear system in the form X f x A x 0 g x u Ag x ớ 6-44 y h x If f g h is strictly C -passive with a positive definite and proper storage function and if holds then system is strongly adaptively C -feedback passive with a proper storage function V. Recursive adaptive passivation In this section we show that the adaptive passivation property can be 132 Adaptive nonlinear control passivation and small gain techniques propagated via adding a feedback passive system with linearly appearing parameters. This is indeed the design ingredient which was used in 19 . More precisely consider a multi-input multi-output nonlinear system of the form with linear parametrization t 10 e 1O G10 e AMW z q z y ý 20 x 2 x ớ Ơ2o x AG2 x 0 m with X fT ZT yT T f G R z G .
