Download Lecture-7: Modeling of Real World Systems

Feedback Control Systems (FCS)
Lecture-7
Mathematical Modeling of Real World Systems
Dr. Imtiaz Hussain
email: [email protected]
URL :http://imtiazhussainkalwar.weebly.com/
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Modelling of Mechanical Systems
• Automatic cruise control
• The purpose of the cruise control system is to maintain a constant vehicle
speed despite external disturbances, such as changes in wind or road grade.
• This is accomplished by measuring the vehicle speed, comparing it to the
desired speed, and automatically adjusting the throttle.
• The resistive forces, bv, due to rolling resistance and wind drag act in the
direction opposite to the vehicle's motion.
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Modelling of Mechanical Systems
u  mv  bv
• The transfer function of the systems would be
V (s)
1

U ( s ) ms  b
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Electromechanical Systems
• Electromechanics combines electrical and mechanical
processes.
• Devices which carry out electrical operations by using
moving parts are known as electromechanical.
–
–
–
–
Relays
Solenoids
Electric Motors
Switches and e.t.c
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Potentiometer
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Potentiometer
• R1 and R2 vary linearly with θ between the
two extremes:
R1 

 max
Rtot
 max  
R2 
Rtot
 max
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Potentiometer
• Potentiometer can be used to sense
angular position, consider the circuit
of figure-1.
Figure-1
• Using the voltage divider principle we
can write:
eout
R1
R1

ein 
ein
R1  R2
Rtot
eout 

 max
R1 
ein

 max
Rtot
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D.C Drives
• Speed control can be achieved using
DC drives in a number of ways.
• Variable Voltage can be applied to the
armature terminals of the DC motor .
• Another method is to vary the flux per
pole of the motor.
• The first method involve adjusting the
motor’s armature while the latter
method involves adjusting the motor
field. These methods are referred to as
“armature control” and “field control.”
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Example-2: Armature Controlled D.C Motor
Ra
Input: voltage u
Output: Angular velocity 
La
B
u
ia
eb
T
J

Electrical Subsystem (loop method):
dia
u  Ra ia  La
 eb ,
dt
Mechanical Subsystem
Tmotor  Jω  Bω
where eb  back-emf voltage
Example-2: Armature Controlled D.C Motor
Ra
La
Power Transformation:
Torque-Current:
Voltage-Speed:
B
u
Tmotor  K t ia
ia
eb
T
J

eb  K b ω
where Kt: torque constant, Kb: velocity constant For an ideal motor
Kt  Kb
Combing previous equations results in the following mathematical
model:
 dia
 Ra ia  K b ω  u
 La
dt

 Jω
   B-K t ia  0
Example-2: Armature Controlled D.C Motor
Taking Laplace transform of the system’s differential equations with
zero initial conditions gives:
La s  Ra I a(s)  K b Ω(s)  U(s)

Js  B Ω(s)-K t I a(s)  0
Eliminating Ia yields the input-output transfer function
Kt
Ω(s)

U(s) La Js 2  JRa  BLa s  BRa  K t K b
Example-2: Armature Controlled D.C Motor
Reduced Order Model
Assuming small inductance, La 0

K t Ra 
Ω(s)

U(s) Js  B  K t K b Ra 
Example-3: Armature Controlled D.C Motor
If output of the D.C motor is angular position θ then we know
Ra
d

dt
or
La
B
( s )  s ( s )
u
ia
eb
J
T
θ
Which yields following transfer function

K t Ra 

U(s) sJs  B  K t K b
(s)
Ra 
Example-3: Field Controlled D.C Motor
Ra
Rf
if
ef
Lf
Tm
J
B ω
Applying KVL at field circuit
ef  if Rf  Lf
Mechanical Subsystem
Tm  Jω  Bω
di f
dt
La
ea
Example-3: Field Controlled D.C Motor
Power Transformation:
Torque-Current: Tm  K f i f
where Kf: torque constant
Combing previous equations and taking Laplace transform (considering
initial conditions to zero) results in the following mathematical model:

 E f ( s )  R f I f ( s )  sL f I f ( s )


 Js( s )  B( s )  K f I f ( s )
Example-3: Field Controlled D.C Motor
Eliminating If(S) yields
Kf
Ω(s)

E f (s) Js  B ( L f s  R f )
If angular position θ is output of the motor
Ra
Rf
if
ef
Lf
Tm
La
J
B θ
(s)
E f (s)

Kf
sJs  B ( L f s  R f )
ea
Example-4
An armature controlled D.C motor runs at 5000 rpm when 15v applied at the
armature circuit. Armature resistance of the motor is 0.2 Ω, armature
inductance is negligible, back emf constant is 5.5x10-2 v sec/rad, motor torque
constant is 6x10-5, moment of inertia of motor 10-5, viscous friction coefficient
is negligible, moment of inertia of load is 4.4x10-3, viscous friction coefficient
of load is 4x10-2.
La
Ra
ea 15 v
ia
N1
Bm
eb
T
Jm
BL

JL
N2
L
1. Drive the overall transfer function of the system i.e. ΩL(s)/ Ea(s)
2. Determine the gear ratio such that the rotational speed of the load is
reduced to half and torque is doubled.
System constants
ea = armature voltage
eb = back emf
Ra = armature winding resistance = 0.2 Ω
La = armature winding inductance = negligible
ia = armature winding current
Kb = back emf constant = 5.5x10-2 volt-sec/rad
Kt = motor torque constant = 6x10-5 N-m/ampere
Jm = moment of inertia of the motor = 1x10-5 kg-m2
Bm=viscous-friction coefficients of the motor = negligible
JL = moment of inertia of the load = 4.4x10-3 kgm2
BL = viscous friction coefficient of the load = 4x10-2 N-m/rad/sec
gear ratio = N1/N2
Example-4
Since armature inductance is negligible therefore reduced order transfer
function of the motor is used.
Kt
ΩL(s)

U(s)
J eq Ra  Beq La s  Beq Ra  K t K b

La
Ra
ea 15 v

ia
N1
Bm
eb
T
Jm
BL

JL
N2
2
N 
J eq  J m   1  J L
 N2 
L
2
N 
Beq  Bm   1  BL
 N2 
Example-5
A field controlled D.C motor runs at 10000 rpm when 15v applied at the field
circuit. Filed resistance of the motor is 0.25 Ω, Filed inductance is 0.1 H,
motor torque constant is 1x10-4, moment of inertia of motor 10-5, viscous
friction coefficient is 0.003, moment of inertia of load is 4.4x10-3, viscous
friction coefficient of load is 4x10-2.
Ra
Rf
ef
if
Lf
La
ea
Tm
Bm ωm
N1
Jm
BL
JL
N2
L
1. Drive the overall transfer function of the system i.e. ΩL(s)/ Ef(s)
2. Determine the gear ratio such that the rotational speed of the load is
reduced to 500 rpm.
Example-5
+
+
e
kp
r
_
La
Ra
+
ea
_
N1
+
ia
JM
BM
T
eb
_
BL
θ
JL
N2
-
if = Constant
c
Numerical Values for System constants
r = angular displacement of the reference input shaft
c = angular displacement of the output shaft
θ = angular displacement of the motor shaft
K1 = gain of the potentiometer shaft = 24/π
Kp = amplifier gain = 10
ea = armature voltage
eb = back emf
Ra = armature winding resistance = 0.2 Ω
La = armature winding inductance = negligible
ia = armature winding current
Kb = back emf constant = 5.5x10-2 volt-sec/rad
K = motor torque constant = 6x10-5 N-m/ampere
Jm = moment of inertia of the motor = 1x10-5 kg-m2
Bm=viscous-friction coefficients of the motor = negligible
JL = moment of inertia of the load = 4.4x10-3 kgm2
BL = viscous friction coefficient of the load = 4x10-2 N-m/rad/sec
n= gear ratio = N1/N2 = 1/10
System Equations
e(t)=K1[ r(t) - c(t) ]
or
E(S)=K1 [ R(S) - C(S) ]
(1)
Ea(s)=Kp E(S)
(2)
Transfer function of the armature controlled D.C motor Is given by
θ(S)
Ea(S)
=
Km
S(TmS+1)
System Equations (contd…..)
Where
K
Km =
RaBeq+KKb
And
Tm
RaJeq
=
RaBeq+KKb
Also
Jeq=Jm+(N1/N2)2JL
Beq=Bm+(N1/N2)2BL
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END OF LECTURES-7
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