Tham khảo tài liệu 'adaptive control design and analysis part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 342 Chapter 7 Indirect Adaptive Control Under the singularity-free condition from it follows that i t e L and ĩ t 1 V i L2 so that limt- oo t 1 v i 0 - From the adaptive law or we have ỡp t G L and Op t 1 Op t G L2 so that limí_ẠOO ớp Ể 1 Op tỴ 0 . Therefore A t G L and A t 1 A t G L2 so that limt_ oo A t 1 A i 0 . With for A A A2 4 -1 it can be shown that at each fixed t det AI A tỴ A2 -1. It follows that the system w t 1 A t w t is exponentially stable 77 .4 Since Aự t G L it follows that lim oo A t 0 so that the system w t 1 A t w i A t w t is exponentially stable. Given that G L n L2 where mo t ựl ộT t ị t we have Mi 1s s 1 IWi 1 IW w 1 IMí Í Hallow Ila I5 í I mo t as from . Since A t 2 e L2 so that lim oo Aự t 2 0 the system w t 1 j4 i cơ t A t w t yylự t cư t is exponetially stable. Hence the solution w i of system is bounded and so are u t and y t that is all closed-loop signals are bounded. From closed-loop signal boundedness with G L2 it follows that e t G L2 and IimẾ 13ũe í 0 and from and - that e t p z p z_ t Z z z z n u t so that ton P z p z- y t - Z z z z-n u t fim p z p z k y t - Z z z -fe u t 0 for any finite k in particular for k n riq 1 zk zn n - i Ac z the first equality of follows from limt oo e t n fc 0 limi_ oo p z p t p z p t n - A z_ y t 0 and lirnt x Z z z ty - Z z z t n -fc _fcM t 0 for p z p t p z p and Z z z t Z z z . To prove asymptotic output tracking of ym t by y t . we first introduce the following composite polynomial operators 125 of two time-varying polynomial 4 A linear system x t 1 A t x t is exponentially stable if A t is bounded max Aj A t 1 J for some Ỗ 0 all i and any t 6 0 1 2 . and A i 1 A í II is sufficiently small for any t e 0 1 2 . 77 or for any t Í1 and some ti 0 . This is a special case of the discrete-time counterpart of Lemma with xxtj-1 A r 1 A t p vT 7o for any t 0 T 0 some 7o 0 V 0 small and p e 1 oo . Discrete-Time Adaptive Control .
