INC 341 – Feedback Control Systems Transfer Functions of Transfer
Functions of
Physical Systems
Physical Systems
S Wongsa
[email protected]
Transfer Functions of Physical Systems
Summary from previous class
 Definition & examples of control systems
D fi iti &
l
f
t l t
 Transfer functions & impedances of mechanical systems
T(s)
1
Js  Ds  K
2
(s)
Transfer Functions of Physical Systems
Today’s goal
 Transfer functions of electrical systems
 Transfer functions of electro‐mechanical (DC motor) systems
 Nonlinearities & linearisation
Transfer Functions of Physical Systems
Models of electrical elements
Z
G
I(s)
Z(s)
G(s)=1/Z(s)
Pictures from ME451: Control Systems, J. Choi, Spring 2010.
V(s)
Transfer Functions of Physical Systems
Kirchhoff’s Voltage & Current Laws
 KCL : A sum of the currents at a node is zero.
i2
i1
i1 ‐ i2 ‐ i3= 0
i1 = i2 + i3
or
i3
 KVL : A sum of the voltages around a loop is zero.
g
p
v1
vs ‐ v1 ‐ v2= 0
vs
or
v2
Transfer Functions of Physical Systems
Example: simple RLC network
Find the transfer function Vc(s)/V(s)
t
L
di (t )
1
 Ri (t )   i ( )d  v(t )
dt
C0
L
LsI ( s )  RI ( s ) 
I (s) 
1
I ( s)  V ( s)
Cs
V (s)s / L
1
R
(s 2  s 
)
L
LC
Vc ( s) 
I ( s)
Cs
vs = v1 + v2
Transfer Functions of Physical Systems
Example: two‐loop RLC network
KVL:
R1 I1 ( s )  ( I1 ( s )  I 2 ( s )) Ls  V ( s )
( I 2 ( s )  I1 ( s )) Ls  I 2 ( s )( R2 
1
)0
Cs
( R1  Ls ) I1 ( s )  LsI 2 ( s )  V ( s )
 LsI
L I1 ( s )  ( R2  Ls
L 
1
) I 2 (s)  0
Cs
Transfer Functions of Physical Systems
Example: two‐loop RLC network
• Node VL
V ( s )  VL ( s ) VL ( s ) VL ( s )  VC ( s )


R1
Ls
R2
1


 G2 VL ( s )  G2VC ( s )  V ( s )G1
 G1 
Ls


• Node V
Node VC
(VL ( s )  VC ( s ))
 VC ( s )Cs
R2
 G2VL ( s )  (G2  Cs )VC ( s )  0
See Examples 2.12, 2.13 & Try Skill‐assessment Exercise 2.6
Transfer Functions of Physical Systems
Operational Amplifier
Characteristics of Op‐Amp
1 Differential input v2‐vv1
1. Differential input, v
2. High input impedance
3. Low output impedance
4. High constant gain amplifier ‐ 100,000 or more for integrated circuit op‐
amps. Mathematically, we often assume that A=.
M th
ti ll
ft
th t A
vo (t )  A(v2 (t )  v1 (t ))
Transfer Functions of Physical Systems
Op‐Amp
 Inverting Op‐Amp
 I a ( s)  0
Vi ( s )
V (s)
 0
Z1
Z2
I1 ( s )   I 2 ( s )
Vo ( s )
Z (s)
 2
Vi ( s )
Z1 ( s )
 Non inverting Op‐Amp


Z1 ( s )
Vo ( s )
V1 ( s )  
 Z1 ( s )  Z 2 ( s ) 
Vo ( s )  A(Vi ( s )  V1 ( s ))
Vo ( s )

Vi ( s) 1  A
A
Z1 ( s )
Z1 ( s )  Z 2 ( s )
A
Vo ( s ) Z1 ( s )  Z 2 ( s )
Z (s)

 1 2
Vi ( s )
Z1 ( s )
Z1 ( s )
Transfer Functions of Physical Systems
Example
Vo ( s )

Vi ( s)
 ( R2 
Z1 ( s )  R1 
1
)
sC2
R1

 ( R2C2 s  1)
R1C2 s
1
R (1 / C2 s )
, Z 2 ( s)  2
C1s
R2  (1 / C2 s )
Vo ( s ) C2C1 R2 R1s 2  (C2 R2  C1 R2  C1 R1 ) s  1

Vi ( s)
C2C1 R2 R1s 2  (C2 R2  C1 R1 ) s  1
See Example 2.14
Transfer Functions of Physical Systems
DC motor
Lorentz law:
Faraday’s law:
Current‐carrying conductor in magnetic field will induce electromagnetic
force acting on the conductor.
g
Electromotive force (EMF) is generated in a conductor moving in a magnetic field.
o g a ag e c e d
Tm  2 Fr  N  2(iBlr ) N  (2 BNlr )i
vb  2VBlN  2 ( m r ) BlN  ( 2 BNlr ) m
Tm  K t i
Multiple windings N
From Dorf R. C. & Bishop R. H., Modern Control Systems, 9th ed. Addison‐Wesley, 2000. & 2.004 Dynamics & Control II, MIT OCW, Fall 2007.
vb  K bm
Transfer Functions of Physical Systems
DC motor with mechanical load
Inductance & resistance of windings
Equations of motion
 Electrical:
Ea ( s )  I a ( s ) Ra  La sI a ( s )  Vb ( s )
 I a ( s ) Ra  La sI a ( s )  K b s m ( s )
Load
 Mechanical:
I a ( s) 
( J m s 2  Dm s ) m ( s )  Tm
( J m s 2  Dm s ) m ( s )
Kt
 K t I a (s)
Viscous fiction in motor bearings
g
Ea ( s )  I a ( s ) Ra  La sI a ( s)  K b s m ( s)


( J s 2  Dm s )
  ( Ra  La s ) m
 K b s   m ( s )
Kt


Transfer Functions of Physical Systems
Realistic electrical properties
Inductance & resistance of windings


( J s 2  Dm s )
 K b s    m ( s )
Ea ( s )   ( Ra  La s ) m
Kt


La  Ra ,i.e. La  0
Load
Viscous fiction in motor bearings
g
 m (s)
Ea ( s )

K t /( Ra J m )
 ( J s  Dm )

Ea ( s )  s Ra m
 K b    m ( s )
Kt


R J
D R

 s a m s   m a  K b     m ( s )
 Kt

 Kt

K K 
1 
 Dm  b t  
s s 
Jm 
Ra  
 
 m ( s)
K t /( Ra J m )

K K 
1 
 Dm  t b 
s s 
Ra 
 Jm 
Transfer Functions of Physical Systems
How to find electrical constants of a motor
Inductance & resistance of windings
 Given the equation
I a ( s ) Ra  La sI a ( s )  K bm ( s )  Ea ( s )
La  0
Load
Ia(s)=Tm/Kt
Ra
Tm ( s )  K bm ( s )  Ea ( s )
Kt
Viscous fiction in motor bearings
g
Ra
Tm (t )  K bm (t )  ea (t )
Kt
 At steady‐state, Ra
Tm  K bm  ea
Kt
Transfer Functions of Physical Systems
How to find electrical constants of a motor
Under a constant dc voltage ea, the torque and speed of a motor can be measured using a dynamometer, resulting in torque‐speed curves.
Ra
Tm  K bm  ea
Kt
m= 0
K t Tstall

Ra
ea
ea1  ea2
Tm= 0
ea increases
Kb 
ea
noload
Transfer Functions of Physical Systems
Example: Exercise 2.23, pp. 79‐80.
 m (s)
Ea ( s )

K t /( Ra J m )

K K 
1 
 Dm  t b 
s s 
J
Ra 
m 

K t Tstall

Ra
ea
Kb 
ea
no load
Try Skill‐assessment Exercise 2.11
Transfer Functions of Physical Systems
Nonlinearity vs Linearity
 A linear system processes two properties: superposition & homogeneity.
homogeneity
superposition
x1(t)
y1(t)
x2(t)
y2(t)
x1(t)+x2(t)
y1(t)+y2(t)
Transfer Functions of Physical Systems
Some physical nonlinearities

Nonlinear resistors, e.g.
‐ Varistors / VDR (Voltage Dependent Resistor) used to protect circuits against excessive transient voltages. ‐ Thermistors: special solid temperature‐sensing element where its resistance proportional to its p p
temperature.

Nonlinear springs, f k  k1 x  k3 x 3
Sources: http://en.wikipedia.org/wiki/Varistor
http://automationwiki.com/index.php?title=Thermistors
A. Carrella et al., Using nonlinear springs to reduce the whirling of a rotating shaft, Mechanical Systems and Signal Processing, Volume 23, Issue 7, October 2009, Pages 2228–223.5
Transfer Functions of Physical Systems
Case study : Antenna Control
Transfer Functions of Physical Systems
Saturation
Unity‐gain amplifier
Ea

 The obtained velocity is limited. Transfer Functions of Physical Systems
Deadzone

 Deadzone  lower amplitude
Transfer Functions of Physical Systems
Backlash
The gears finally connect

Transfer Functions of Physical Systems
Linearisation by small signal analysis
We linearise a nonlinear system for small‐signal inputs (0) about the steady‐state solution, known as equilibrium.
Using the Taylor series and neglecting higher‐order terms, we get
 x ‐ deviation variable
f ( x)  f ( x0 ) 
df
dx
( x  x0 )
x  x0
Linearisation of f(x)
Linearisation of f(x)
Transfer Functions of Physical Systems
Example
Linearisation of a simple pendulum
The motion equation:
d 2 MgL
J 2 
sin   T
dt
2
Transfer Functions of Physical Systems
Example
Linearisation of a simple pendulum
J
d 2 MgL

sin   T
dt 2
2
 Equilibrium Equilibrium  0  0  T0  0. Define
D fi      0 , T  T  T0
 Linearisation
MgL
M
L
MgL
M
L
M L
MgL
MgL
M
L

sin  
sin  0 
cos  0 (   0 ) 
2
2
2
2
 Linearised differential equation
NB: we express each variable in deviation from the steady state operating point
steady‐state operating point.
J
d 2 MgL

  T
dt 2
2
 Linearised
Li
i d transfer function
t
f f ti
     0 , T  T  T0
See Example 2.28 & Try Skill‐assessment Exercise 2.13
( Js 2 
MgL
) ( s )  T ( s )
2
 ( s )
1/ J

T ( s ) s 2  MgL
2J
Transfer Functions of Physical Systems
Readings
 Jirka Routbal et al., Linearization: Students Forget the Operating Point, IEEE Transactions on Education, Vol. 53, No. 3, August 2010, pp. 413‐418.  Creating model of engineering systems: http://www.freestudy.co.uk/control/t1.pdf
 Introduction: Simulink Modeling: http://control.me.cmu.edu/ctms/index.php?example=Introduction&section
=SimulinkModeling
 Using Simulink to Model Continuous Dynamical Systems
http://www.mathworks.com/academia/student_center/tutorials/sltutorial_launch
http://www
mathworks com/academia/student center/tutorials/sltutorial launch
pad.html#
Transfer Functions of Physical Systems
Summary
 Transfer function of electrical systems
f f
i
f l
i l
KCL
KVL
v1
i2
i1
i3
vs
vo (t )  A(v2 (t )  v1 (t ))
v2
 Transfer function of DC motor with initial load, bearings and negligible inductance
Inductance & resistance of windings
 m (s)
Ea ( s )
Load
Vi
Viscous fiction in motor bearings
fi i i
b i
 Linearisation
f ( x)  f ( x0 ) 
df
dx
( x  x0 )
x  x0

K t /( Ra J m )

K K 
1 
 Dm  t b 
s s 
J
Ra 
m 

Transfer Functions of Physical Systems
Next class
 First‐order systems response
First order systems response
 Second‐order systems response
‐ types of 2nd‐order systems & their behaviours
4 1 4 8 of Ch4
4.1‐4.8 of Ch4.
‐ time domain specifications
 Effects of additional poles & zeros
 Block diagrams (5.1‐5.3 of Ch5.)
Please read these topics before coming to the next class!