Tham khảo tài liệu 'adaptive control 2011 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 68 Adaptive Control I-------- Virtual controlled system Fig. 1. Virtual controlled system with a virtual filter 1 t z1 t i t -bzufi-1 t zi t vZ c j zi-j t 6 j 1 where c i 0i -ar-i 1 i r-1 r-1 c r -a0 pj-1c j j 1 1 Pr-2 i-1 di Pr-i-1 Pr-i j-1c j j 1 the system 1 can be transformed into the following virtual system which has uf given from a virtual input filter as the control input Michino et al. 2004 see 1 t az t cT ny t bzuf1 t c 1 w t ny t An t cn 1 t Cdn w t 7 y t 1 t dTw t where ny T nT T 2 3 L r T and cT 1 0 L 0 cn cT 0 L 0 1 T. cd1 and Cdn are a vector and a matrix with appropriate dimensions respectively. Further An is given by the form of An Auf 0 0 Qn Adaptive output regulation of unknown linear systems with unknown exosystems 69 Since A. and Q_ are stable matrices A is a stable matrix. uf n n Virtual error system Now consider a stable filter of the form zcf t Acf Zcf t AA t uf t 0TZcf t uf1 t 8 where cc - 0 L 0 1 T and Aq 0 Im-1xm-1 Pc c0 - a0 - Pc0 L am-1 - Pcm-1 . B Pr. - B_ are chosen such that A. is stable. c0 c1 cm-1 cf Let s consider transforming the system 7 into a one with uf given in 8 as the input. Define new variables X1 and X2 as follows X1 - 1m a m iP l 1 1 L a1 1 a0 1 X2 - 1P am-1nym-1 L a1ny a0 . Since it follows from the Cayley-Hamilton theorem that Am am-1Am 1 - a1Am a0I - 0 10 we have from 2 and 7 that X1 t azX1 t cTx2 t bzuf t 11 X2 t - AnX2 t cqX1 t where uf- u am-1ufm-1 L a1u f1 a0uf1 12 Further we have from 10 that e m am-1e m-1 - a1e a0e - X1. 13 70 Adaptive Control Therefore defining E e e L e m 1 the following error system is obtained E t AEE t X1 t X 1 t azX1 t cTx t bzuf t X 2 t AnX 2 t cqX1 t e t 1 0 L 0 E t . 14 Obviously this error system with the input uf and the output e has a relative degree of m 1 and a stable zero dynamics because An is stable . Furthermore there exists an appropriate variable transformation such that the error system 14 can be represented by the following form Isidori 1995 0 ze t Aze ze
