2-1 (a),(b) (pp.50)
Problem: Prove that the systems shown in Fig.
(a) and Fig. (b) are similar.(that is, the format of
differential equation is similar).
Where electric pressure u1 and displacement
x1 are inputs; voltage u2 and displacement x2
are outputs, k1,k2 and k3 are elastic coefficient
of the spring, f is damping coefficient of the
friction
Fig. (a)
Fig. (b)
2-3 (pp. 51)
In pipeline ,the flux q through the valve is
proportional to the square root of the
pressure difference p, that is, q=K p .
suppose that the system changes slightly
around initial value of flux q0.
Problem: Linearize the flux equation.
2-5 (pp. 51)
Suppose that
the
system’s
output
is
t

c(t )  1  e T under the step input r(t)=1(t)
and zero initial condition.
Problems:
1. Determine the system’s transfer function
and the output response c(t) when
r(t)=t, r (t )   (t )
2. Sketch system response curve.
2-6 (pp. 51)
Suppose that the system’s transfer function
is C (s)  2 2
, and the initial
R( s) s  3s  2
condition is c(0)  1,
.
c(0)  0
Problem: Determine the system’s unit
step response c(t) .
2-13 (a), (d) (pp. 53)
Problem: Determine the close loop transfer
function of the system shown in the
following figures, using Mason Formula.
Fig. (a)
Fig. (b)
3-2 (pp. 83)
Suppose that the thermometer can be
characterized by transfer function.
C ( s)
1

R( s) Ts  1
Now measure the temperature of water in
the container by thermometer. It needs one
minute to show 98% of the actual
temperature of water.
Problem: Determine the time constant of
thermometer.
3-4 (pp. 83)
Suppose that the system’s unity step
60t
10t
response is h(t )  1  0.2e  1.2e
Problem:
(1) Solve the system’s close-loop transfer
function.
(2)Determine damp ratio  and un-damped
frequency wn .
3-5 (pp. 83)
Suppose that the system’s unity step
1.2t
h
(
t
)

10[1

1.25
e
sin(1.6t  53.1)]
response is
Problem: Determine the system’s
overshoot  % , peak time t p
and setting time ts
 3-8 (pp. 83)
Suppose that unity step response of a
second –order system is shown as follows.
Problem: If the system is a unity feedback,
try to determine the system’s open loop
transfer function .
3-11 (pp. 84)
Problem: Determine the stability of the
systems described by the following
characteristic equations,using Routh
stability criterion.
(1)
(2)
(3)
s 3  8s 2  24s  100  0
s 3  8s 2  24s  200  0
3s 4  10s 3  5s 2  s  2  0
3-16 (pp. 16)
Suppose that the open loop transfer function
of the unity feedback system is described as
follows.
Problem: Determine the system’s steady2
t
e
state error ss when r(t)=1(t), t, respectively
100
G
(
s
)

(1)
(0.1s  1)(0.5s  1)
(2)
150( s  4)
G(s) 
s ( s  10)( s  5)
(3)
8(0.5s  1)
G( s)  2
s (0.1s  1)
3-19 (a) (pp. 85)
Problem: Determine the system’ steadystate error ess which is shown as follows.
4-2 (pp.108)
The system’s open-loop transfer function is
K.
G( s) H ( s) 
( s  1)( s  2)( s  4)
Problem: Prove that the point s1=-1+j3 is in
the root locus of this system, and determine
the corresponding K.
4-4 (pp.109)
A open-loop transfer function of unity
feedback system is described as
K
G( s) 
s(1  0.02s)(1  0.01s)
Problems :
(1) Draw root locus of the system
(2) Determine the value K when the system
is critically stable.
(3) Determine the value K when the system
is critically damped.
4-7 (pp. 109)
Consider a systems shown as follows:
Where
K (0.25s  1)
G( s) 
s(1  0.5s)
Problems:
1. Determine the range of K when the system has
no overshoot, using locus method.
2. Analysis the effect of K on system’s
dynamic performance.
4-10 (pp. 110)
The open-loop transfer functions of unity
feedback system are described as:
1/ 4( s  a)
(1) G ( s)  2
(a  (0, )
s ( s  1)
2.6
(2) G( s) 
(T  (0, )
s(1  0.1s)(1  Ts)
Problem: Draw root locus with varying
parameters being a and T respectively.
5-2 (1) (pp.166)
A unity feedback system is shown as follows.
Problem: Determine the system’s steadystate output C ss when input signal is
r (t )  2cos(2t  45 )
G(s)  
5
s (5S  1)
2
5-7（3）(pp.167)
Problem: Draw logarithm amplitude frequency
asymptotic characteristics and logarithm phasefrequency characteristic of the following transfer
function。
5
G(s)   2
s (5S  1)
5-8 (pp. 167)
The logarithm amplitude frequency
asymptotic characteristics of a minimum
phase angle system is shown as follows.
Problem: Determine the system’s open
loop transfer function。
5-8（a）
5-8（b）
5-8（c）
5-8（d）
5-10 (pp. 168)
The system’s open loop amplitude-phase
curve is shown as follows，where P is the
number of poles in right semi-plane of
G（s）H（s）.
Problem: Determine the stability of the closeloop system。
5-10（a）
5-10（b）
5-10（c）
5-12（1）,（2）(pp.168-169)
The open loop transfer function of the unity
feedback system is shown below：
100
1.G ( s ) 
s (0.2 s  1)
50
2. G ( s) 
(0.2 s  1)( s  2)( s  0.5)
Problem: Determine the system’s stability
using logarithm frequency stability criterion,
the phase angle margin and amplitude
margin of the steady system。