Signal Processing and Linear Systems-B.P.Lathi copy

The node equation a t the intermediate node is but i3

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Unformatted text preview: erver canonical form [Eq. (13.15a)], (iii) cascade realization [Eq. (13.15b)] a nd (iv) parallel realization [Eq. (13.15c)]. T hese r ealizations a re d epicted in Figs. 13.5a, 13.5b, 13.5c, a nd 13.5d, respectively. As mentioned earlier, t he o utput o f e ach i ntegrator serves as a n atural s tate variable. ' -., ~ ... .u... n § 0 1. Canonical Forms ~. ~ n ~ n Here we s hall realize t he s ystem using t he first (controller) canonical form discussed in Sec. 6.6-1. I f we choose t he s tate v ariables t o b e t he t hree i ntegrator o utputs X l, X 2, a nd X 3, t hen, a ccording t o Fig. 13.5a, g. !' ~ e- O> 0 ". '"0 el E. ~ (13.16a) 11: (l E. N' 0> >t. Also, t he o utput y is given by 0 (13.16b) " " .'..., S- 0 '" " E quations (13.16a) a re t he s tate e quations, a nd Eq. (13.16b) i s t he o utput e quation. In m atrix form we h ave '< 1t 3 Er t Il >< 0> [ .§. " 'f>. " :: ] [: 0 - 19 - 12 X3 : ] [ ::] , -8 v + [ :] / 1 X3 ' -v-' ' A a nd (13.17a) B (13.17b) We c an also realize H {s) b y using t he s econd (observer) canonical form (discussed i n A ppendix 6.1), as shown in Fig. 13.5b. I f we label t he o utput o f t he t hree i ntegrators from left t o r ight as t he s tate v ariables V l, V 2, a nd V 3, t hen, a ccording t o Fig. 13.5b, Vl = - 12v3 + 10/ 19v3 V2 = Vl - V3 = + 2/ (13.18a) V 2 - 8V3 a nd t he o utput y is given by y= Hence (13.18b) V3 [=] [:: ~::] [::] [1:] / + F ig. 1 3.5 C anonical, cascade, a nd p arallel realizations o f t he s ystem in E xample 13.4. V3 0 1 -8 '---v-' A V3 0 ~ (13.19a) i!& Q 13 794 S tate-Space A nalysis a nd Y=~[::l c 13.2 A S ystematic P rocedure for D etermining S tate E quations 795 a nd t he o utput e quation is y = ~ Zl (13.19b) - 2Z2 + ~ Z3 (13.23b) Therefore, the equations in t he m atrix form are V3 Observe closely the relationship between the state-space descriptions of H (s) t hat use t he controller canonical form [Eqs. (13.17)] and those using t he observer canonical form [Eqs. (13.19)]. The A matrices in these two cases are t he t ranspose of one another; also, t he B of one is the transpose of C in the other, a nd vice versa. Hence (13.24a) ( A)T=A. (13.24b) ( Bf=c (13.20) ( C)T=B o Solve Example 13.4 using MATLAB. Caution: T he convention of MATLAB for labeling s tate variables X l, X 2, . .. , X n in a block diagram, such a s shown in Fig. 13.5a, is reversed. W hat we label X l is X n , a nd X2 is T his is no coincidence. This duality relation is generally true. I 2 . Series Realization T he three integrator outputs T he s tate equations are W I, W 2, and W3 in Fig. 13.5c are t he s tate variables. (13.21a) (13.21b) (13.21c) a nd t he o utput equation is y= • C omputer E xample C 13.1 W3 T he elimination of W2 from Eq. (13.21c) by using Eq. (13.21b) converts these equations i nto t he desired state form X n-l, a nd s o on. n um=[2 1 0]; d en=[1 8 1 9 1 2]; [ A,B,C,Dj=tf2ss(num,den) % I n o rder t o f ind t he t ransfer f unction f rom A , B , C , a nd D...
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