MODERN CONTROL SYSTEMS
DR.MOHAMMED ABDULRAZZAQ
mathematical models of systems
2.1 Introduction
2.1.1 Why？
1) Easy to discuss the full possible types of the control
systems —only in terms of the system’s “mathematical
characteristics”.
2) The basis of analyzing or designing the control systems.
2.1.2 What is ？
Mathematical models of systems — the mathematical relationships between the system’s variables.
2.1.3 How get？
1) theoretical approaches
2) experimental approaches
3) discrimination learning
mathematical models of systems
2.1.4 types
1) Differential equations
2) Transfer function
3) Block diagram、signal flow graph
4) State variables
2.2 The input-output description of the physical systems —
differential equations
The input-output description—description of the mathematical
relationship between the output variable and the input variable
of physical systems.
2.2.1 Examples
mathematical models of systems
Example 2.1 : A passive circuit
R
ur
define: input → ur
we have：
L
i
C uc
output → uc。
du
di
 uc  ur i  C c
dt
dt

d 2 uc
du
LC 2  RC c  uc  ur
dt
dt
Ri  L
d 2 uc
duc
L
make : RC  T 1
 T 2  T1T2 2  T1
 uc  ur
R
dt
dt
mathematical models of systems
Example 2.2 : A mechanism
Define: input → F ，output → y. We have:
F
k
m
y
f
dy
d2y
F  ky  f
m 2
dt
d t

d2y
dy
m 2  f
 ky  F
dt
dt
If we make :
f
 T1,
k
we have : T1T2
d2y
dt 2
 T1
m
 T2
f
dy
1
y F
dt
k
Compare with example 2.1: uc→y, ur→F---analogous systems
mathematical models of systems
Example 2.3 : An operational amplifier (Op-amp) circuit
R2
C
i2
ur
R1 i 1
R1
R4
+
Input →ur
R3
output →uc
1
( i3  i2 )dt  R4 ( i3  i2 )......(1)

C
u
i2   i1   r ...........................................( 2)
R1
uc  R3 i3 
i3
uc
i3 
1
( uc  R2 i2 ).....................................( 3)
R3
duc
dur

R2  R3  R2  R3
 uc   R (
 R4 )C
 ur 
(2)→(3); (2)→(1); (3)→(1)： R4C
R

R
1  2 3
dt
dt

make : R4C  T ;
R2  R3
R2  R3
 k; (
 R4 )C  
R1
R2  R3
duc
dur
we have : T
 uc   k (
 ur )
dt
dt
mathematical models of systems
Example 2.4 : A DC motor
La
Ra
ua
( J1, f 1)
w1
ia
M
( J2, f 2)
w2
( J3, f 3)
w3
Input → ua， output → ω1
dia
La
 Ra ia  Ea  ua ....(1)
dt
M  C m ia .........................( 2)
Mf
i1
i2
(4)→(2)→(1) and (3)→(1):
La J 
La f
Ra J 
R f
1  (

) 1  ( a  1)1
CeC m CeC m
CeC m
Ea  C e1 .........................( 3) C e C m

L
Ra
1
d1
a
M
M
MM J
 f 1 .....(4)  ua 
Ce
CeC m
CeC m
dt
mathematical models of systems
J  J1 
J2
here : f  f1 
f2
M
i12
i12
Mf
i1i2


J3
i12 i22
f3
i12 i22
......equivalent moment of inertia
......equivalent friction coefficient
..........................equivalent torque
(can be derived from : 1  i1 2  i1i2 3 )
La
............electric - magnetic time - constant
Make: Te 
Ra
Ra J
Tm 
.......mechanical - electric time - constant
CeC m
Tf 
Ra f
....... friction - electric time - constant
CeC m
mathematical models of systems
the differential equation description of the DC motor is:


TeTm 1  (TeT f  Tm ) 1  (T f  1)1

1
1

ua  (TeTm M  Tm M )
Ce
J
Assume the motor idle: Mf = 0, and neglect the friction: f =
0, we have:
TeTm
d 2
dt 2
d
1
 Tm
 
ua
dt
Ce
Compare with example 2.1 and example 2.2:
uc  y   ;
ur  F  ua ----Analogous systems
mathematical models of systems
Example 2.5 :
A DC-Motor control system
R2
+
ur R1
-
R3
DC
mot or
R3
ua
-
M
w l oad
uk
R1
t r i gger
Uf
-
r ect i f i er
M
t echomet er
+
Input → ur, Output →ω; neglect the friction:
R2
uk 
( ur  u f )  k1 ( ur  u f )........................................(1)
R1
u f   .....................(2)
TeTm
d 2
dt 2
ua  k 2 uk ......................(3)

d
1
1
 Tm
 
ua  (TeTm M  Tm M )......(4)
dt
Ce
J
mathematical models of systems
（2）→（1）→（3）→（4），we have：
d 2

Tm
d
1
1
TeTm 2  Tm
 (1  k1k 2 C )  k1k 2
ur 
(Te M  M )
e
dt
Ce
J
dt
2.2.2 steps to obtain the input-output description (differential
equation) of control systems
1) Identify the output and input variables of the control systems.
2) Write the differential equations of each system’s component
in terms of the physical laws of the components.
* necessary assumption and neglect.
* proper approximation.
3) dispel the intermediate(across) variables to get the inputoutput description which only contains the output and input
variables.
mathematical models of systems
4) Formalize the input-output equation to be the “standard” form:
Input variable —— on the right of the input-output equation .
Output variable —— on the left of the input-output equation.
Writing the polynomial—according to the falling-power order.
2.2.3 General form of the input-output equation of the linear
control systems
——A nth-order differential equation:
Suppose:
input → r ，output → y
y ( n )  a1 y ( n1)  a2 y ( n2)      an1 y (1)  an y
 b0 r ( m )  b1r ( m 1)  b2 r ( m 2)      bm 1r (1)  bm r .........n  m
mathematical models of systems
2.3 Linearization of the nonlinear components
2.3.1 what is nonlinearity？
The output of system is not linearly vary with the linear variation
of the system’s (or component’s) input → nonlinear systems (or
components).
2.3.2 How do the linearization？
Suppose: y = f(r)
The Taylor series expansion about the operating point r0 is:
f ( r )  f ( r0 )  f
(1)
f ( 2 ) ( r0 )
f ( 3) ( r0 )
2
( r0 )( r  r0 ) 
( r  r0 ) 
( r  r0 )3    
2!
3!
 f ( r0 )  f (1) ( r0 )( r  r0 )
make : y  f ( r )  f ( r0 ) and : r  r  r0
we have : y  f ' ( r0 )r ............linearization equation
mathematical models of systems
Examples:
Example 2.6 : Elasticity equation
F ( x )  kx
suppose : k  12.65;   1.1; operating point x0  0.25
F ' ( x )  kx 1  F ' ( x0 )  12.65  1.1 0.250.1  12.11
we have :
that is :
F ( x )  F ( x0 )  12.11( x  x0 )
ΔF  12.11x ..............linearization equation
Example 2.7 : Fluxograph equation
Q( p)  k p
Q —— Flux;
p —— pressure difference
mathematical models of systems
because : Q' ( p ) 
thus : Q 
k
2 p
k
p...........linearization equation
2 p0
2.4 Transfer function
Another form of the input-output(external) description of control
systems, different from the differential equations.
2.4.1 definition
Transfer function: The ratio of the Laplace transform of the
output variable to the Laplace transform of the input variable with
all initial condition assumed to be zero and for the linear systems,
that is:
mathematical models of systems
C ( s)
G( s) 
R( s )
C(s) —— Laplace transform of the output variable
R(s) —— Laplace transform of the input variable
G(s) —— transfer function
Notes:
* Only for the linear and stationary(constant parameter) systems.
* Zero initial conditions.
* Dependent on the configuration and coefficients of the systems,
independent on the input and output variables.
2.4.2 How to obtain the transfer function of a system
1) If the impulse response g(t) is known
mathematical models of systems
We have:
Because:
Then:
Example 2.8 :
G ( s )  Lg ( t )
C ( s)
G( s) 
, if r ( t )   ( t )  R( s )  1
R( s )
G ( s )  C ( s )  Lg( t )
g( t )  5  3e
2t
5
3
2( s  5)
 G( s)  

s s  2 s( s  2)
2) If the output response c(t) and the input r(t) are known
We have:
Lc( t )
G( s) 
Lr ( t )
mathematical models of systems
Example 2.9:
c( t )  1  e  3t
Then:
1
........Unit step function
s
1
1
3
 C ( s)  

s s  3 s( s  3)
.........Unit step response
r ( t )  1( t )  R(s) 
C ( s ) 3 s( s  3)
3
G( s) 


R( s )
1s
s3
3) If the input-output differential equation is known
•Assume: zero initial conditions;
•Make: Laplace transform of the differential equation;
•Deduce: G(s)=C(s)/R(s).
mathematical models of systems
Example 2.10:



2 c( t ) 3 c( t ) 4c( t )  5 r ( t ) 6r ( t )

2 s 2C ( s )  3 sC ( s )  4C ( s )  5 sR( s )  6 R( s )

C(s)
5s  6
G(s) 
 2
R(s) 2 s  3 s  4
4) For a circuit
* Transform a circuit into a operator circuit.
* Deduce the C(s)/R(s) in terms of the circuits theory.
Chapter 2 mathematical models of systems
Example 2.11: For a electric circuit:
R1
R2
C1
ur
R1
C2
uc
ur ( s)
1
1
1
// ( R2 
)
sC1
sC 2
sC 2
U c ( s) 
 U r ( s) 
1
1
1
R1 
// ( R2 
)
R2 
sC1
sC 2
sC 2

G( s) 
1
T1T2 s  (T1  T2  T12 ) s  1
2
 U r ( s)
U c ( s)
1

U r ( s ) T1T2 s 2  (T1  T2  T12 ) s  1
here : T1  R1C 1 ;
T2  R2 C 2 ;
T12  R1C 2
1/ C1s
R2
1/ C2s
uc( s)
Chapter 2 mathematical models of systems
Example 2.12: For a op-amp circuit
R2
ur R1
R1
+
R2
C
uc
ur R1
R1
+
1/ Cs
uc
1
R2 
U c ( s)
R2Cs  1
sC
G( s) 


U r ( s)
R1
R1Cs
1
)................. .PI-Controller
s
k  R2 ;   R2C ...... Integral time consta nt
R1
  k (1 
here :
Chapter 2 mathematical models of systems
5) For a control system
• Write the differential equations of the control system;
• Make Laplace transformation, assume zero initial conditions,
transform the differential equations into the relevant algebraic
equations;
• Deduce: G(s)=C(s)/R(s).
Example 2.13 the DC-Motor control system in Example 2.5
R2
+
ur R1
-
R3
DC
mot or
R3
ua
-
w l oad
uk
R1
t r i gger
Uf
M
-
r ect i f i er
M
t echomet er
+
Chapter 2 mathematical models of systems
In Example 2.5, we have written down the differential equations
as:
R2
uk 
( ur  u f )  k1 ( ur  u f )..................................(1)
R1
u f   ....................(2)
TeTm
d 2
ua  k 2 uk ...................(3)

Tm
d
1
 Tm
 
ua 
(Te M  M )......(4)
dt
Ce
J
dt 2
Make Laplace transformation, we have:
U k ( s )  k1[U r ( s )  U f ( s )]...................................................(1)
U f ( s )  ( s )...............(2)
U a ( s )  k 2U k ( s )..............(3)
TeTm s  Tm
1
(TeTm s  Tm s  1)( s ) 
U a ( s) 
M ( s )......(4)
Ce
J
(2)→(1)→(3)→(4), we have:
2
Chapter 2 mathematical models of systems
TeTm s  Tm
1
1
[TeTm s  Tm s  (1  k1k 2 )]( s )  k1k 2 U r ( s ) 
M ( s)
Ce
Ce
J
k1k 2 1
Ce
( s )
G( s) 

U r ( s ) T T s 2  T s  (1  k k  1 )
e m
m
1 2
Ce
2
La
here : Te 
...........electric  magnetic time - constant
Ra
Ra J
Tm 
......mechanical  electric time - constant
CeC m
Chapter 2 mathematical models of systems
2.5 Transfer function of the typical elements of linear systems
A linear system can be regarded as the composing of several
typical elements, which are:
2.5.1 Proportioning element
Relationship between the input and output variables:
c ( t )  kr ( t )
C ( s)
G( s ) 
k
Transfer function:
R( s )
Block diagram representation and unit step response:
R( s)
C( s)
k
r( t )
Examples:
amplifier, gear train,
tachometer…
C( t )
k
1
t
t
Chapter 2 mathematical models of systems
2.5.2 Integrating element
Relationship between the input and output variables:
t
1
c( t )   r ( t )dt ..........TI : integral time constant
TI 0
C ( s)
1
G( s) 

Transfer function:
R( s ) TI s
Block diagram representation and unit step response:
R( s)
1
TI s
r( t )
1
t
C( s)
1
Examples:
C( t )
TI
Integrating circuit, integrating
motor, integrating wheel…
t
Chapter 2 mathematical models of systems
2.5.3
Differentiating element
Relationship between the input and output variables:
dr ( t )
c( t )  TD
dt
Transfer function:
C ( s)
G( s) 
 TD s
R( s )
Block diagram representation and unit step response:
R( s)
TDs
r( t )
1
C( s)
C( t )
TD
t
Examples:
differentiating amplifier, differential
valve, differential condenser…
t
Chapter 2 mathematical models of systems
2.5.4 Inertial element
Relationship between the input and output variables:
dc( t )
T
 c( t )  kr ( t )
dt
C ( s)
k
Transfer function: G ( s ) 

R( s ) Ts  1
Block diagram representation and unit step response:
R( s)
k
Ts  1
r( t )
1
t
C( s)
Examples:
C( t )
inertia wheel, inertial load (such
as temperature system)…
k
T
t
Chapter 2 mathematical models of systems
2.5.5 Oscillating element
Relationship between the input and output variables:
T
2
d
c( t )
2
dt
2
dc( t )
 2T
 c( t )  kr ( t )
dt
0 1
C ( s)
k
 2 2
Transfer function: G ( s ) 
R( s ) T s  2Ts  1
0 1
Block diagram representation and unit step response:
R( s)
r( t )
1
C( s)
T 2 s 2  2Ts  1
C( t )
Examples:
oscillator, oscillating table,
oscillating circuit…
k
1
t
t
Chapter 2 mathematical models of systems
2.5.6 Delay element
Relationship between the input and output variables:
c( t )  kr ( t   )
C ( s)
Transfer function: G ( s ) 
 ke s
R( s )
Block diagram representation and unit step response:
R( s)
ke
C( s)
s
Examples:
gap effect of gear mechanism,
threshold voltage of transistors…
C( t )
r( t )
k
1
t

t
Chapter 2 mathematical models of systems
2.6 block diagram models (dynamic)
Portray the control systems by the block diagram models more
intuitively than the transfer function or differential equation
models
.2.6.1 Block diagram representation of the control systems
Si gnal
( var i abl e)
X( s)
Component
( devi ce)
G( s)
X3( s)
Adder ( compar i son)
E( s) =x1( s) +x3( s) - x2( s)
X1( s)
+
+
E( s)
-
Examples:
X2( s)
Chapter 2 mathematical models of systems
Example 2.14 For the DC motor in Example 2.4
In Example 2.4, we have written down the differential equations
as:
dia
La
 Ra ia  Ea  ua ....(1) M  C m ia .........................( 2)
dt
d
Ea  C e .........................( 3) M  M  J
 f  .....(4)
dt
Make Laplace transformation, we have:
U a ( s )  Ea ( s )
La sI a ( s )  Ra I a ( s )  Ea ( s )  U a ( s )  I a ( s ) 
.............(5)
La s  Ra
M ( s )  C m I a ( s )......................................................................................(6)
Ea ( s )  C e ( s ).......................................................................................(7)
1
M ( s )  M ( s )  J s ( s )  f  ( s )   ( s ) 
[ M ( s )  M ( s )]......(8)
Js  f
Chapter 2 mathematical models of systems
Draw block diagram in terms of the equations (5)～(8):
M (s )
Ua( s)
-
1
La s  Ra
I a( s)
M( s)
Cm
Ea( s)
-
(s )
1
Js  f
Ce
M(s)
Consider the Motor as a whole:
Ua(s)
1
Ce
TeTms2  (Tm  TeTf )s  Tf 1
1
(TeTms  Tm)
J
TeTms2  (Tm  TeTf )s  Tf 1
-
(s)
Chapter 2 mathematical models of systems
Example 2.15
The water level control system in Fig 1.8:
1
k4
k3
Ce
k2e s
k1 TeTm s 2  Tm s  1
T1 s  1 T2 s  1
s
Desi r ed
wat er l evel
I nput hi
e
ampl i f i er
ua
Mot or

Gear i ng
Feedback si gnal hf
Tm
(Te s  1)
J

M ( s)
2
TeTm s  Tm s  1

Act ual
wat er l evel
Q Wat er Out put h
Val ve
cont ai ner
Fl oat

Chapter 2 mathematical models of systems
The block diagram model is:
M (s )
Tm
(Te s  1)
J
TeTm s 2  Tm s  1
Hi ( s)
E( s)
-
K
Ua( s)
1
Ce
TeTm s 2  Tm s  1
k3
k4
k 2 e  s
(s) s (s) T1s 1 Q( s) T2 s  1
-
Hf ( s)

H( s)
Chapter 2 mathematical models of systems
Example 2.16
The DC motor control system in Fig 1.9
1
k p (1 
)
TI s
Desi red
rot at e speed
Ref erence
i nput ur
k De
s
1
Ce
TeTm s 2  Tm s  1
Act uat or
Error
uk
e
Regul at or
ua
a
Tri gger
Rect i f i er
-
DC
mot or
Act ual
rot at e speed
Out put 
Techomet er
Feedback si gnal uf

Ts  1
Tm
(Te s  1)
J

M ( s)
2
TeTm s  Tm s  1
Chapter 2 mathematical models of systems
The block diagram model is:
M (s)
Tm
( T e s  1)
J
TeTm s 2  Tm s  1
Ur ( s)
E( s)
Uf ( s)
1
k p (1  )
TI s
Uk( s)
kD e
s
Ua( s)
1
Ce
TeTm s 2  Tm s  1

Ts1
-
 (s ) 1
s
 (s )
Chapter 2 mathematical models of systems
2.6.2 Block diagram reduction
purpose: reduce a complicated block diagram to a simple one.
2.6.2.1 Basic forms of the block diagrams of control systems
Chapter 2-2.ppt
Chapter 2 mathematical models of systems