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).

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