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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. 6.61. 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 tateSpace 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 statespace 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 nl, 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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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