120
2
12
12
(2 1) 3
(3 2) 0
ss X X F
XsX
°
®
°
¯
Solve for
1
X
:
2. A dynamic system with input
f
and output
x
is described by
1
32()xx x ft
(a) Find the state-space form.
(b) Find the transfer function directly from the state-space form.
(c) Decide if the system is stable by examining the state matrix in Part (a).
Solution
(a) There are two state variables:
1
xx
,
2
xx
. The state-variable equations are
12
212
63 3
xx
xxxf
®
¯
3. A system’s transfer function is defined as
2
() 2 1
() 31
Ys s
Us ss
(a) Find the input-output equation.
(b) Find the state-space form directly from the input-output equation.
(c) Find the transfer function directly from the state-space form.
Solution
(a)
32yyy uu
(b) State-space form is found as
4. A system is described by its governing equations
11 21
221
3( ) ( )
3( ) 0
xx x x ft
xxx
®
¯
where
f
is the input, while
1
x
and
1
x
are the outputs.
(a) Obtain the state-space form.
(b) Find the transfer matrix directly from the state-space form.
(c) Find the transfer matrix using the governing equations, and compare with Part (b).
Solution
(a) There are four state variables: 11
xx ,
22
xx
,31
xx ,
42
xx
. The state-variable equations are
13
24
312
412
43
3 3
xx
xx
xxxf
xxx
°
°
®
°
°
¯
122
5. A system’s input-output equation is given as
1
4
32yy yuu
(a) Find the state-space form.
(b) Find the transfer function directly from the state-space form.
Solution
(a) State-space form is found as
1
1
24
01 0
, , , ,
31
x
uuu
x
ªº
½ ªº
®¾ «»
«»
¬¼
¯¿ ¬¼
xAxB x A B
123
6. Repeat Problem 5 for the input-output equation
21
32
yyy uu
.
Solution
(a) State-space form is found as
1
33 3
222 2
01 0
, , , ,
x
uuu
x
ªºªº
½
®¾ «»«»
¯¿ ¬¼¬¼
xAxB x A B
7. A system’s transfer function is given as
2
32
() ( 1)
() 231
Ys s s
Us sss
(a) Find the input-output equation.
(b) Find the state-space form directly from Part (a).
(c) Find the transfer function from Part (b).
Solution
(a)
23yyyyuu
(b) State-space form is found as
(c) Using the state-space form,
1
() ( )
Gs s D
CI A B
8. Find the input-output equation of a SISO system whose state-space form is given as
u
yDu
®
¯
xAxB
Cx
with
1
2
13
3
2
0
01
, , , 1 , 0
1uu D
ªº
ªº ªº
«»
«» ¬¼
¬¼
¬¼
AB C
Solution
We will first find the transfer function, as
9. Repeat Problem 8 for
>@
01 0
, , , 1 1 , 1
23 2 uu D
ªºªº
«»«»
¬¼¬¼
AB C
Solution
We will first find the transfer function, as
10. Find all possible input-output equations for a system with state-space form
u
u
®
¯
xAxB
yCxD
where
01 0 20 0
, , , ,
14 1 01 0
uu
ª º ªº ª º ªº
« » «» « » «»
¬ ¼ ¬¼ ¬ ¼ ¬¼
AB CD
Solution
We will first find the transfer matrix, as
11. Find the transfer function for the system in Figure 4.36 using
(a) Block diagram reduction,
(b) Mason’s rule.
Figure 4.36 Problem 11.
Solution
(a) The inner loop is replaced with
1
1
323
1
s
ss
. With this, the larger loop is replaced with
In Problems 12–15, block diagram representation of a system is provided. Find the transfer function using Mason’s
rule.
12. Figure 4.37
Figure 4.37 Problem 12.
Solution
All paths are coupled: one forward path, two feedback loops. By Mason’s rule, we have
sss
13. Figure 4.38
Figure 4.38 Problem 13.
Solution
There are two forward paths and four loops, as shown in Figure Review4No13:
Forward path Gain Loop Gain
12456
1123
FGGG
242
1
G
1236
24
FG
5675
3
GH
14. Figure 4.39
Figure 4.39 Problem 14.
Solution
There are three forward paths and two loops:
Forward path gain Loop gain
14
FG
1
G
In the general case of Mason’s rule, we have
13 13
1DGGHGGH
ªº
15. Figure 4.40
Figure 4.40 Problem 15.
Solution
There are two forward paths and two loops:
Forward path gain Loop gain
14
FG
1
kG
128
In the general case of Mason’s rule, we have
In Problems 16–19, the input-output equation or the state-space form of a system model is provided. Construct the
appropriate block diagram, and directly use it to find the transfer matrix.
16. State-space form is
u
yDu
®
¯
xAxB
Cx
with
>@
1
2
11
322
010 0
, 0 0 1 , 0 , 1 2 0 , 0 ,
11
x
xDuu
x
ªº
½ ªº
«»
°° «»
®¾ «»
«»
°° «»
«»
¬¼
¯¿ ¬¼
xA BC
Solution
The block diagram is shown in Figure Review4No16.
Figure Review4No16
17. Input-output equation is
23xx xuu
.
Solution
Choosing
1
xx
,
2
xx
, we first find the state-variable equations and the output equation, as
>@
12
1
221
2
21
3
xx
xxxu
yx x
°
®
°
¯
18. Input-output equation is
23xx xuu
.
Solution
Choosing
1
xx
,
2
xx
, we first find the state-variable equations and the output equation, as
>@
12
1
221
2
111
21
222
3
xx
xxxu
yxxu
°
®
°
¯
130
19. State-space form is
u
u
®
¯
xAxB
yCxD
with
1
2
01 0 10 0
, , , , ,
13 1 02 0
xuu
x
½ ª º ªº ª º ªº
®¾ « » «» « » «»
¬ ¼ ¬¼ ¬ ¼ ¬¼
¯¿
xA BC D
Solution
The block diagram is shown in Figure Review4No19.
Figure Review4No19
20. A system is atable if the poles of the overall transfer function lie in the left half-plane; these poles are the same
as the eigenvalues of the state matrix. Consider the block diagram representation of a system shown in Figure
4.41, where
0k!
is a parameter. Determine the range of values of
k
for which the system is stable.
Figure 4.41 Problem 20.
Solution
The overall transfer function is formed by Mason’s rule, as
2
3
2
() 3( 2)
Ys ss
21. Repeat Problem 20 for the block diagram in Figure 4.42, where
0k!
is a parameter.
Figure 4.42 Problem 21.
Solution
The overall transfer function is formed by Mason’s rule, as
22. Derive the linearized model for a nonlinear system described by
1
111 2
2
212
(0) 2
1cos , (0) 2
1
x
xxxx t
x
xxx
°
®
°
¯
Solution
Step 1: The operating point
12
,xx
satisfies
Add
11 2
11 1
12
01
2 0
01
xx x xx x
xx
°
®
°
¯
23. Repeat Problem 22 for
12 1
32
21 2
2sin (0) 1
, (0) 0
31
xx t x
x
xx x
°
®
°
¯
133
Solution
Step 1: The operating point
12
,xx
satisfies
2
1
3
12
0 1
031
xx
xx
°
®
°
¯