Download Structural analysis and design of multivariable control by Yih T. Tsay, Leang-San Shieh, Stephen Barnett PDF

Show description

3) in Eq. 2) and Dr(k) in Eq. 3) is the following: Theorem 3 . 1 Let A(X) be a c o l u m n - r e d u c e d X - m a t r i x r e p r e s e n t e d by Eq. ( 3 . 2 ) , l e t D (k) be d e f i n e d i n Eq. ( 3 . 3 ) . 4) 43 where Ur(k) X-matrix, Ur(k) = [CT:I~r(X)+DDr(X)] -I. is a unimodular and T C St(X) are defined in Eqs. 14c), respectively. Proof: From Eqs. 2)~ we obtain A-l(k) - C(kln-A)-IB+D - lqr(k)D:l(X)+D - Nr (k)D:l(k) where Nr(k) ffi cz:l~r(k)¢cmxm[x]| Nr(k) - Nr(X)÷DDr(A) | T c is a transformation matrix which transforms (A,B) into the canonical controller pair.

1 6 ) . 17) Proof; From Eq. 13d)~ we have [D(rk) (~)] T B T - k [ ~ ( r k - l ) ( x ) ] T + [ ¢ ( r k ) ( x ) ] T ( X I n - A c ) T . -l~ , and Pci(-1) ~ Onxl" Therefore2 k=0 J 1 ~TI [ D(k)(~)]TB~pci(:rj - k) = k~ 1 ( k - l ) ! CJ)cx i ),T a Pci(-l) ~r " tVr i J PciCj-k-1) 0mxl Thus, we conclude t h a t B l P c i j , 0 ~ J ~ t i - I , is a l e f t Jordan chain of Dr(k). xj = 0 ; reversed upper t r i a n g u l a r T o e p l i t z m a t r i x [ I I ] with f i r s t [Tp] i i m m column [ a l i 2 , .

__ o:j . ^& ^ ^ ; n=n-KI-~ m . o l o . 1e) Dc Proof: Lemma 3 . 2 can be e s t a b l l s h e d by d i r e c t The q u a d r u p l e ( A c , B c , C c , ; c ) is referred oontroller) form r e a l i z a t i o n o f A-ICk). ~ B ~ verification. T:IBc, C ~ CcTc trans£ormatlen matrix, and D " so t h a t the to as a c o n t r o l l e r ,n g e n e r a l , A • (not a canonical we c ~ . 2) A-I(A) . C(kln-A)-IB+D Let Dr(k) be the right characteristic k-matrix of (A,B) in Eq. 2). 3) in Eq. 2) and Dr(k) in Eq.