### Lecture 3

```LINEAR CONTROL
SYSTEMS
Ali Karimpour
Associate Professor
lecture 3
Lecture 3
Different representations of control
systems
Topics to be covered include:

High Order Differential Equation. (HODE model)

State Space model. (SS model)

Transfer Function. (TF model)

State Diagram. (SD model)
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lecture 3
High order differential equation (HODE).
‫معادالت ديفرانسيل مرتبه باال‬
dny
d n 1 y
dy
d mu
d m1u
du
 an 1 n 1  ...... a1  a0  m  bm1 m1  ..... b1
 b0
n
dt
dt
dt
dt
dt
dt
y is the output
u is the input
HODE model
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lecture 3
Example 1: A high order differential equation (HODE).
‫ يک معادله ديفرانسيل مرتبه باال‬:1 ‫مثال‬
‫خروجی‬
‫ورودی‬
HODE model
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Example 2: Another high order differential equation.
‫ مثالی ديگر از معادله ديفرانسيل مرتبه باال‬:2‫مثال‬
‫خروجی‬
‫ورودی‬
eb
HODE model
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State Space Models (SS)
‫معادالت فضای حالت‬
For continuous time systems
SS model
For linear time invariant continuous time systems
SS model
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State Space Models
‫معادالت فضای حالت‬
General form of LTI systems in state space form
LTI ‫فرم کلی فضای حالتی سيستمهای‬
 x1  a11
 x  a
 2   21
 .  .
  
. .
 x n  an1
  
a1n   x1  b11 b12 . . b1 p  u1 

 



b
b
.
.
b
. . a2 n   x2   21 22
2 p  u 2 
. .
.  .    .
. . . .  . 
 
  
. .
.  .   .
. . . .  . 
. . ann   xn  bn1 bn 2 . . bnp  u p 


a12 . .
a22
.
.
an 2
SS model
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Example 3: A linear time invariant continuous time systems
)LTI( ‫ يک سيستم خطی غير متغير با زمان‬:3‫مثال‬
output
SS model
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Example 3: Continue
‫ ادامه‬:3‫مثال‬
The equations can be rearranged as follows: ‫ساده سازی‬
di(t ) 1
 v(t )
dt
L
 1
dv(t )
1
1 
1
v(t ) 
  i(t )  

v f (t )
dt
C
R1C
 R1C R2C 
c(t )  v(t )
We have a linear state space model with ‫مدل خطی فضای حالتی‬
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A demonstration robot containing several
servo motors
‫ در ربات‬dc ‫مثالی از موتور‬
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Example 4: DC motor
DC‫ موتور‬:4 ‫مثال‬
J
- be the inertia of the shaft
e(t)
- the electrical torque
ia(t)
k1 ; k2
R
- the armature current
- constants
- the armature resistance
Let x1 (t )   (t ) And
0
d  x1 (t )  


dt  x2 (t )  0

HODE model
x2 (t )  (t )
1 
0 
x
(
t
)


1
  k1 va (t )
k1 k 2  

x2 (t )  


JR 
 JR 
SS model
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lecture 3
Different representations
‫نمايشهای مختلف‬
HODE model
SS model
TF model
SD model
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Linear high order differential equation
‫معادالت ديفرانسيل مرتبه باالی خطی‬
The study of differential equations of the type described
above is a rich and interesting subject. Of all the methods
available for studying linear differential equations, one
particularly useful tool is provided by Laplace Transforms.
‫مطالعه معادالت ديفرانسيل مرتبه باالی خطی فوق موضوع بسيار جالبی‬
.‫بوده و يک روش خاص حل آن استفاده از تبديل الپالس است‬
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Table : Laplace transform table
‫جدول تبديل الپالس‬
u(t) - u(t-τ)
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Table : Laplace transform properties.
‫خواص تبديل الپالس‬
y(t-τ) u(t-τ)
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Transfer Function model ‫مدل تابع انتقال‬
dny
d n 1 y
dy
d mu
d m 1u
du

a

......

a

a
y


b

.....

b
 b0u
n 1
1
0
m 1
1
n
n 1
m
m 1
dt
dt
dt
dt
dt
dt

Taking laplace transform
zero initial condition
s n y ( s )  an 1s n 1 y ( s )  ...... a1sy ( s )  a0 y ( s )  s mu ( s )  bm 1s m 1u ( s )  ..... b1su ( s )  b0u ( s )
A( s) y ( s)  B( s)u ( s)

G( s) 
y ( s ) B( s )

u ( s) A( s)
TF model
Or
Input-output model
HODE model
TF model
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lecture 3
Different representations
‫نمايشهای مختلف‬
HODE model
SS model
TF model
SD model
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But how can we change SS model to TF model?
‫اما چگونه معادالت فضای حالت را به تابع انتقال تبديل کنيم‬
Use Laplace transform
Then
Let initial condition zero
SS model
TF model
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Different representations
‫نمايشهای مختلف‬
HODE model
SS model
TF model
SD model
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TF model properties
‫خواص مدل تابع انتقال‬
1- It is available just for linear systems.
2- It is derived by zero initial condition.
3- It just shows the relation between input
and output so it may lose some information.
4- It can be used to show delay systems but
SS can not.
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Example 5: A thermal system
‫ يک سيستم حرارتی‬:5‫مثال‬
lecture 3
Input signal
1
k
Output signal
Input signal is an step function so:
t (Time)
t 0
1
1
u (s) 
u (t )  
s
t0
k
y(s)
0
?
 g ( s) 
t
u ( s) s  1

k
k


t  0 y( s)  
y(t )  k  ke
s s 1/
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
0
t

0

Dr. Ali Karimpour Feb 2013
Example 6: A system with pure time delay
‫ سيستمی با تاخير ثابت‬:6 ‫مثال‬
lecture 3
h(s)  e sTd
g (s) 
k
s  1
ke sTd
h( s ) g ( s ) 
s  1
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State diagram
‫دياگرام حالت‬
x1 
dx 2
dt

x1 ( s )  sx2 ( s )  x2 (0)
x2 (s)  s 1 x1 (s)  x2 (0)
x2(0)
s -1
x1(s)
s -1
SD model
x2(s)
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State diagram to state space
‫دياگرام حالت به معادالت حالت‬
x1  x2
x 2  4 x1  5 x2  r
c  x1
SD model
SS model
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Different representations
‫نمايشهای مختلف‬
HODE model
SS model
TF model
SD model
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Different representations
‫نمايشهای مختلف‬
HODE
SS
x1  x2
x 2  4 x1  5 x2  r
c  5c  4c  r
c  x1
SD
TF
G(s) 
1
s 2  5s  4
Mason’s rule
Discussed in this lecture
Will be discussed in next lecture
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Exercises
2-1 Find the SS model and TF function model for example 1.
2-2 Find the SS model and TF function model for example 2.
2-3 Find the TF function model for example 3.
2-4 Find the TF function model for example 4.
a) Suppose angular position as output
b) Suppose angular velocity as output
2-5 In example 5 find the output
a) the input is e-2t
b) the input is unit step
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Exercises (Continue)
2-6 In the different representation slide show the validity of
red directions
2-7 Change the equations derived in example 2 to one order 3
HODE.
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