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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 . suppose that the system changes slightly around initial value of flux q0.
Problem: Linearize the flux equation.
p
2-5 (pp. 51)
Suppose that the system’s output is
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,
2. Sketch system response curve.
( ) 1t
Tc t e
( ) ( )r t t
2-6 (pp. 51)
Suppose that the system’s transfer function is , and the initial
condition is
Problem: Determine the system’s unit
step response c(t) .
2
( ) 2
( ) 3 2
C s
R s s s
.
(0) 1, (0) 0c c
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.
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.
( ) 1
( ) 1
C s
R s Ts
3-4 (pp. 83)
Suppose that the system’s unity step response is
Problem:
(1) Solve the system’s close-loop transfer
function.
(2)Determine damp ratio and un-damped frequency .
60 10( ) 1 0.2 1.2t th t e e
nw
3-5 (pp. 83)
Suppose that the system’s unity step response is
Problem: Determine the system’s overshoot , peak time
and setting time
1.2( ) 10[1 1.25 sin(1.6 53.1 )]th t e t
% pt
st
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)
3 28 24 100 0s s s
3 28 24 200 0s s s 4 3 23 10 5 2 0s s s s
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 steady-state error when r(t)=1(t), t, respectively
(1)
(2)
(3)
sse2t
100( )
(0.1 1)(0.5 1)G s
s s
150( 4)( )
( 10)( 5)
sG s
s s s
2
8(0.5 1)( )
(0.1 1)
sG s
s s
3-19 (a) (pp. 85)
Problem: Determine the system’ steady-state error which is shown as follows.sse
4-2 (pp.108)
The system’s open-loop transfer function is
Problem: Prove that the point s1=-1+j3 is in the root locus of this system, and determine the corresponding K.
.
( ) ( )( 1)( 2)( 4)
KG s H s
s s s
4-4 (pp.109)
A open-loop transfer function of unity feedback system is described as
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.
( )(1 0.02 )(1 0.01 )
KG s
s s s
4-7 (pp. 109)
Consider a systems shown as follows:
(0.25 1)( )
(1 0.5 )
K sG s
s s
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.
Where
4-10 (pp. 110)
The open-loop transfer functions of unity feedback system are described as:
Problem: Draw root locus with varying parameters being a and T respectively.
2
1/ 4( )(1) ( ) ( (0, )
( 1)
2.6(2) ( ) ( (0, )
(1 0.1 )(1 )
s aG s a
s s
G s Ts s Ts
5-2 (1) (pp.166)
A unity feedback system is shown as follows.
Problem: Determine the system’s steady-state output when input signal is ssC
( ) 2cos(2 45 )r t t
5-7( 3) (pp.167)
Problem: Draw logarithm amplitude frequency asymptotic characteristics and logarithm phase-frequency characteristic of the following transfer function。
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
2
5( )
(5 1)G s
s S
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 close-loop 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:
Problem: Determine the system’s stability using logarithm frequency stability criterion, the phase angle margin and amplitude margin of the steady system。
1001. ( )
(0.2 1)
502. ( )
(0.2 1)( 2)( 0.5)
G ss s
G ss s s