By D. H. Jacobson, D. H. Martin, M. Pachter, T. Geveci
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2 . 4 ) e x i s t s because e A t + 0 as t + ~. 4) is positive d e f i n i t e . _t = 0 ~ I I ceAtx 12dr >~ 0 0 with equality only i f ceAtx = 0 for a l l t > O , which holds Only for x = 0 i f (A,C) is completely observable. 4), is positive definite. 2). 1), we have , A't V(x(t)) = Xoe , At (A P+PA)e xo : - ICeAtxol2 Of degree n-nc independent of the choice of K, and the factor det i %1 - A~+BIK~ of degree nc . Since 0 kll 0 [krl 0 k12. . kln kr2" k l l . . 0 k21 .... k2n 0 0 i ! 01 krl .... 10) that the matrix AI-BIKI has the same form as At, with the elements ani j of the full rows being modified by the amount -~ij" Thus by adjusting the choice of KI these full rows of AI-BIKI can be arbitrarily set. In particular we may choose K~ so that A~-B~K~ becomes the nc x nc companion matrix corresponding to any real monic polynomial of degree nc. The following theorem is due to Luenberger, and its proof is a clever extension of that sketched above for single-input systems. THEOREM 2 . 1 . 2 state exists dlme~ion Given any c o m p l e t e l y c o n t r o l l a b l e n a~d i n p u t dimension r ~ n w i t h r a n k B = r , t h e r e a state tra~ fo~ation ( 2 . 1 . 2 ) and a n o n - s i n g u l a r t~ans f o r m a t i o n T i n c o n t r o l s p a c e R r , s u c h t h a t w i t h n~w s t a t e variable~ = Sx, •the. 10). SBT- I The i n t c g ~ nl,n 2 .