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Problem 7.43PP
X/ = Ax/ + B/u/,
y/ = Cixi, where / = 1.2.
Give a set of state equations for the combined systems in Fig.
Consider the systems shown in Fig., employing series, parallel, and feedback configurations,
(a) Suppose we have controllable-observable realizations for each subsystem;
(b) For each case, determine what condition(s) on the roots of the polynomials Ni and Di is 
necessary for each system to be controllable and observable. Give a brief reason for your 
answer in terms of pole-zero cancellations.
Figure Block diagrams; (a) series; (b) parallel; (c) feedback
.................*1 J~r7r~w5n j,
Figure Block diagrams; (a) series; (b) parallel; (c) feedback
Step-by-step solution
step 1 of 7
(a)
Series connection:
Consider the controllable and observable realizations for any system, 
i , = A ,* ,+ B ,» , ...... (1)
...... (2)
Where, 
i = land2.
Refer figure 7.94 (a) in the textbook and write state equations for series connection from 
equations (1) and (2).
X, = A ,x ,+ B ,h (3)
x , = A ,x ,+ B ,C ,X |...... (4)
Write the state equation of series connection in matrix form from equations (3) and (4).
(5)
Write the output equation of series connection.
y = c ,x , ...... (6)
Write the output equation of series connection in matrix form from equation (6).
- [ o ....
Hence, the state equations of the series connection is ' A, 0 B ,'i r s
B ,C , A .
JC +
0
u
y = [0 c,]
step 2 of 7
Parallel connection:
Refer figure 7.94 (b) in the textbook and write state equations for parallel connection from 
equations (1) and (2).
X, = A ,x ,+ B ,h (8)
X,=A2AC2 + B,U ...... (9)
Write the state equation of parallel connection in matrix fonn from equations (8) and (9).
Ki-i*.' :][;]•&
( 10 )
Write the output equation of parallel connection.
> = C |X ,+ C jX j...... (11)
Write the output equation of parallel connection in matrix form from equation {11).
' ■ [ 4 ' ■ f t ] ......
■ ( 12)
Hence, the state equations of the parallel connection is
A, 0 
0 A,
B,
y = [C, C,]
Step 3 of 7
Feedback connection:
Refer figure 7.94 (c) in the textbook and write state equations for feedback connection from 
equations (1) and (2).
X, ■ A ,x i - A ,C jX j+ B , r ...... (13)
x , = A ,x ,+ B ,C ,X |...... (14)
Write the state equation of feedback connection in matrix form from equations (13) and (14).
.
Write the output equation of feedback connection. 
y = C ,*, ...... (16)
(15)
Step 4 of 7 ^
Write the output equation of feedback connection in matrix form from equation (16).
-̂ = |C, oil - | (17),.[c, .|[^]
Hence, the state equations of the parallel connection is irs ■ A, -A ,C ,'
'“ ' r
B,C, A, 0
y = [C, 0]
0
step 5 of 7
(b)
Refer 7.94(a) in the textbook and write the transfer function of the series connection. 
= (18)
^ f o r a ( s ) and ^ f o r CM) in equations (18).
' A W ' 'A ( * ) 
y(s) Af,MAT,(5) 
U(s) D ,( ,)D ,(s ) (19)
Therefore, from equation (19), there are no cancellations in poles and zeros part of the transfer 
functions in series combination.
For observability the parts [AT^(j)and£); W should be coprime o therw iseQ (j) | is covered 
from the output.
For controllability the parts | ( j)a n d P 2 ( j ) should be ooprime o th e rw is e ( j )~ | is covered 
from the output.
Step 6 of 7
Refer 7.94(b) in the textbook and write the transfer function of the parallel connection.
(20)
Substitute f ' . M
^ ( s )
M lfor