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Problem 7.05PP
Give the state description matrices in modal carionical form for the transfer functions of Problem. 
Make sure that all entries in the state matrices are real valued by keeping any pairs of complex 
conjugate poles together, and realize them as a separate subblock in control canonical form. 
Give the state description matrices in control-canonical form for the following transfer functions; 
Problem
"■> « ' ) = wmTTT-
"F+3+2"
"F+3+2"
(d) C(i) =
Step-by-step solution
step 1 of 10
(a)
Consider the numerator part of the gain.......(1)
Consider the denominator part of the gain.
o ( i)= i" + o ,i" ' '+ +lG(5)
G W =
»’ + 3 i + 2 
8s + I
.(11)
( i + l ) ( j + 2)
Apply partial fraction in equation (11).
*»+l A B
( i + l ) ( * + 2 ) ° ( i+ l ) '^ ( j+ 2 ) * '
Find A from equation (12).
, 8s + l
(s + l)(» + 2 ) 
8s+ I
' " i
-8+ 1
-1 + 2
A = -7 ...... (13)
Find B from equation (12).
„ 8 s+ l
+0.288 
D = -0 .07 (41)
Substitute equations (33), (35), (37). (39) and (41) in equation (32).
(5 + 10)(5^+5 + 25) -1,3465 + 3.472 1.421 -0.075 + 0.288 
5 '(5 + 2 )(5 ’ + 5+36) s‘ (4 + 2) 5*+5+36
step 9 of 10
Draw the block diagram from equation (42).
Figure 3
step 10 of 10 A
Write the state equation in control canonical form from figure 3.
"4l 0 0 0 0 0 4j 1
42 1 0 0 0 0 4i 0
= 0 0 -2 0 0 4i + 1
4. 0 0 0 -1 -36 4. 1
.45. 0 0 0 1 0 .45. 0
0 0 0 0 0 r
1 0 0 0 0 0
JC = 0 0 -2 0 0 x+ 1 tt
0 0 0 -1 -36 1
0 0 0 1 0 0
(43)
Write the output equation in control canonical form from figure 3.
y = [-1.346 3.472 1.421 -0.07 0.288] + 0
y = [-1.346 3.472 1.421 -0.07 0.288]x+0 (44)
Write the state description matrices in control canonical form from equations (16), (17), (43) and 
(44).
0 0 0 0 0 ■
1 0 0 0 0
A .= 0 0 -2 0 0
0 0 0 -1 -36
0 0 0 1 0
0
B ,= 1
11
p
c .= -1.346 3.472 1.421
2>c = 0]
Hence, the value of state description matrices in control canonical form is