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Control Engineering Practice Problem Set 3: EEL 205, Essays (university) of Control Systems

Indian Institute of Technology Control Systems

COntrol System tutorials academic year 2017

Typology: Essays (university)

2018/2019

Uploaded on 09/02/2019

harneet-sudan 🇮🇳

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EEL 205 Control Engineering

Practice Problem Set 3

1. A hydraulic system has elements analogous to the

resistor, inductor, and capacitor of an electrical

circuit. These are respectively known as hydraulic

resistor (R, attributable to friction and turbulence

internal to the fluid), hydraulic inertor (I, attributable

to mass of fluid in flow - which is never zero !), and

hydraulic capacitor (C, attributable to storage and

compression). Consider the pressure sensor shown in

Fig. P1, which is used to obtain a measurement Ps of a source pressure PP, both relative to a low sump pressure

P0. What is the transfer function between the actual and sensed pressure values ? Let the fluid flow in the pipe

be given by Q, as shown.

1

IC $s2+RC $s+1

2. The fluid from a constant source of flow QS is

modulated for pressure using a valve, the flow Qv

through which is a nonlinear function of the pressure

differential across it, given by

Qv=KA $P0.5

where A is the valve cross section area controlled by a servomechanism. Obtain the nonlinear equation that

determines pressure PS in presence of effective resistance R and capacitance C. Linearise the equation about a

set point given by pressure PS0 and valve opening A0.

PS(s)

A(s)=−K$[PS0−P0]0.5

Cs +0.5KA0$[PS0−P0]−0.5

3. An electromagnetic speed sensor, mounted on the rotary part

of a system, is in the shape of a conducting disk, across

which a magnetic field of uniform flux density B is applied

(Fig. P3). With the rotary system shaft driven at torque T

Nm and angular speed

ω

rad/s, obtain an expression for the

emf

ε

(t) between brushes located at the disk periphery

(radius rD) and the shaft surface (radius rs). The electrical

signal serves as a measure of the speed signal

ν

(t).

[(t)=(B/2)(rD

2−rs

2)$*(t)]

4. Fig. P4 shows the cross

section for one type of

electromechanical

actuator that is used in

large steam flow

systems. It consists of a

coil of 500 turns around

a soft iron plunger of

radius 2cm, and an air

gap annulus of 0.5mm

around it. The plunger

has mass M, and is

supported by a spring of

EEL 205/PS3

SUMP

P

0

P

s

RI

C

P1

Q

SUMP

P

0

Q

S

P

S

R

C

P2

1R A( )

ε

B

r

s

r

D

ω, Τ

P3

STEAM FLOW

CORE

PLUNGER

COIL

(500 turns)

(mass M)

(constant K)

SPRING

SUPPORT

P4

2 cm

0.5mm air gap

5cm x

Discover Essays (university) of Control Systems Indian Institute of Technology

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EEL 205 Control Engineering

Practice Problem Set 3

1. A hydraulic system has elements analogous to the

resistor, inductor, and capacitor of an electrical circuit. These are respectively known as hydraulic resistor ( R , attributable to friction and turbulence internal to the fluid), hydraulic inertor ( I , attributable to mass of fluid in flow - which is never zero !), and hydraulic capacitor ( C , attributable to storage and compression). Consider the pressure sensor shown in Fig. P1 , which is used to obtain a measurement Ps of a source pressure PP , both relative to a low sump pressure P 0. What is the transfer function between the actual and sensed pressure values? Let the fluid flow in the pipe be given by Q , as shown.

1 IC $ s^2 + RC $ s + 1

2. The fluid from a constant source of flow Q S is

modulated for pressure using a valve, the flow Q (^) v through which is a nonlinear function of the pressure differential across it, given by

Q v = KA $ P 0.

where A is the valve cross section area controlled by a servomechanism. Obtain the nonlinear equation that determines pressure PS in presence of effective resistance R and capacitance C. Linearise the equation about a set point given by pressure PS 0 and valve opening A 0. P (^) S ( s ) A ( s ) =^

− K $ [ P S 0 − P 0 ]^ 0.

Cs + 0.5 KA 0 $ [ P (^) S 0 − P 0 ]−0.

3. An electromagnetic speed sensor, mounted on the rotary part

of a system, is in the shape of a conducting disk, across which a magnetic field of uniform flux density B is applied (Fig. P3 ). With the rotary system shaft driven at torque T

Nm and angular speed ω rad/s, obtain an expression for the

emf ε( t ) between brushes located at the disk periphery

(radius r D ) and the shaft surface (radius r s ). The electrical

signal serves as a measure of the speed signal ν( t ).

[( t ) = ( B /2)( r D^2 − r s^2 ) $ *( t )]

4. Fig. P4 shows the cross

section for one type of electromechanical actuator that is used in large steam flow systems. It consists of a coil of 500 turns around a soft iron plunger of radius 2cm, and an air gap annulus of 0.5mm around it. The plunger has mass M , and is supported by a spring of

EEL 205/PS

SUMP

P

0

PP

Ps R I

P1 C

Q

SUMP

P

0

Q

S

P

S R

C

P

1 R A ( )

ε

B

rs

rD

P

STEAM FLOW

CORE

PLUNGER

COIL

(500 turns)

(mass M) (constant K)

SPRING

SUPPORT

P

2 cm

0.5mm air gap

5cm x

constant K. The position of the plunger beyond the coil is given by a variable x ( t ), which in the rest position shown in the figure, has a value x 0. If the electromagnetic force on the plunger by a DC current i ( t ) in the coil is given by ½ i^2 ( t ).d L ( x )/d x , and if this tends to draw the plunger inwards towards a minimum reluctance position, obtain the nonlinear system dynamic equation of this actuator. Linearise the equation about x 0 and current i 0 , and obtain the transfer function for small valve opening in response to small current.

6332 i 0 (0.05 + x 0 ) 2

$ Ms^2 + K +

6332 i 02 (0.05 + x 0 ) 3

− 1

5. Fig. P5 shows a system involving forward transfer

functions a and c , together with feedback transfer functions b , d , and e. Obtain the sensitivity transfer functions of the system with respect to each of the feedback blocks. ab (1 − cd ) (1 − ab )(1 − cd ) (^) − eac ,^

cd (1 − ab ) (1 − ab )(1 − cd ) (^) − eac ,^

eac (1 − ab )(1 − cd ) (^) − eac

6. As we all know, automobile designers take pains to make the passenger

comfortable by incorporating appropriate tires, shock absorbers, and leaf springs in a vehicle configuration. However one parameter that they never can freeze on is the weight of the passenger s !! That remains arbitrary, and hence the objective of this analysis.

The model in Fig. P6 represents the dynamics of one wheel of a four-wheeler automobile. M 1 is roughly ¼ the mass of the automobile (with the questionable weight of passengers included !). K 1 and D 1 respectively represent the leaf spring and shock absorber associated with a single wheel. M 2 is the mass of the wheel, and K 2 represents the tire stiffness. Obtain the transfer function of X 1 ( s )/ X ( s ) - which represents the effect of the bumpiness of the terrain on the passenger - and most important of all, its sensitivity to M 1.

S (^) M^ G ( 1 s )^ = − M 1 s^2 ( M 2 s^2 + D 1 s + K 1 + K 2 ) ( M 1 s^2 + D 1 s + K 1 )( M 2 s^2 + D 1 s + K 1 + K 2 ) (^) − ( D 1 s + K 1 ) 2

EEL 205/PS

a c

b d

e

R s ( )^ Y s ( )

P

M 1

M 2

K 1 D^ 1

K 2

x

x 2

x 1

Control Engineering Practice Problem Set 3: EEL 205, Essays (university) of Control Systems

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Download Control Engineering Practice Problem Set 3: EEL 205 and more Essays (university) Control Systems in PDF only on Docsity!

EEL 205 Control Engineering

Practice Problem Set 3

1. A hydraulic system has elements analogous to the

2. The fluid from a constant source of flow Q S is

Q v = KA $ P 0.

− K $ [ P S 0 − P 0 ]^ 0.

3. An electromagnetic speed sensor, mounted on the rotary part

Nm and angular speed ω rad/s, obtain an expression for the

emf ε( t ) between brushes located at the disk periphery

signal serves as a measure of the speed signal ν( t ).

[( t ) = ( B /2)( r D^2 − r s^2 ) $ *( t )]

4. Fig. P4 shows the cross

EEL 205/PS

SUMP

P

PP

P1 C

Q

SUMP

P

Q

P

C

P

B

B

P

STEAM FLOW

STEAM FLOW

CORE

CORE

PLUNGER

COIL

SPRING

SUPPORT

P

5. Fig. P5 shows a system involving forward transfer

6. As we all know, automobile designers take pains to make the passenger

EEL 205/PS

M 1

M 2

K 1 D^ 1

K 2

P

[( t ) = ( B /2)( r D^2 − r s^2 ) $ *( t )]