Control Systems with Scilab
Aditya Sengupta
Indian Institute of Technology Bombay
apsengupta@iitb.ac.in
December 1, 2010, Mumbai
A simple first order system
// D e f i n i n g a f i r s t o r d e r s y s t e m :
s = %s
// The q u i c k e r a l t e r n a t i v e t o u s i n g s =
poly (0 , ’ s ’ )
K = 1, T = 1
// G a i n and t i m e c o n s t a n t
SimpleSys = s y s l i n ( ’ c ’ , K /(1+ T * s ) )
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CACSD with Scilab
Simulating the system- Step test
t =0:0.01:15;
y1 = c s i m ( ’ s t e p ’ , t , SimpleSys ) ; // s t e p r e s p o n s e
p l o t ( t , y1 )
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Simulating the system- Sine test
u2=s i n ( t ) ;
y2 = c s i m ( u2 , t , SimpleSys ) ;
p l o t ( t , [ u2 ; y2 ] ’ )
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// s i n e r e s p o n s e
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Simulating the system- Sine test
u3=s i n ( 5 * t ) ;
y3 = c s i m ( u3 , t , SimpleSys ) ;
frequency
p l o t ( t , [ u3 ; y3 ] ’ )
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// s i n e r e s p o n s e a t d i f f e r e n t
CACSD with Scilab
Root Locus
e v a n s ( SimpleSys )
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Bode Plot
fMin = 0 . 0 1 , fMax=10
bode ( SimpleSys , fMin , fMax )
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Second Order Systems- Overdamped
p=sˆ2+9 * s+9
O v e rd a mp e d S y s t e m= s y s l i n ( ’ c ’ , 9/ p )
roots (p)
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Second Order Systems- Overdamped
p=sˆ2+9 * s+9
O v e rd a mp e d S y s t e m= s y s l i n ( ’ c ’ , 9/ p )
roots (p)
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Second Order Systems- Overdamped
y4 = c s i m ( ’ s t e p ’ , t , Ov e rd a mp ed S ys t em ) ;
p l o t ( t , y4 )
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Second Order Systems- Overdamped
bode ( OverdampedSystem , fMin , fMax )
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Second Order Systems- Underdamped
q=sˆ2+2 * s+9
U n d er d am p e d S y s t e m = s y s l i n ( ’ c ’ , 9/ q )
roots (q)
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Second Order Systems- Underdamped
y5 = c s i m ( ’ s t e p ’ , t , U nd e r d am p e d Sy s t e m ) ;
p l o t ( t , y5 )
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Second Order Systems- Undamped
r = sˆ2+9
Unda mpedS ys te m = s y s l i n ( ’ c ’ , 9/ r )
roots (r)
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Second Order Systems- Undamped
y6 = c s i m ( ’ s t e p ’ , t , Un damp edSys tem ) ;
p l o t ( t , y6 )
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Second Order Systems- Critically damped
m = sˆ2+6 * s+9
C r i t i c a l l y D a m p e d S y s t e m = s y s l i n ( ’ c ’ , 9/ m )
roots (m)
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Second Order Systems- Critically damped
y7 = c s i m ( ’ s t e p ’ , t , C r i t i c a l l y D a m p e d S y s t e m ) ;
p l o t ( t , y7 )
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Discrete time systems
z = %z
a = 0.1
DTSystem = s y s l i n ( ’ d ’ , a * z / ( z − (1−a ) ) )
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Discrete time systems
u = ones (1 , 50) ;
y = f l t s ( u , DTSystem ) ;
p l o t ( y ) // C l o s e t h i s when done
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Discrete time systems
bode ( DTSystem , 0 . 0 0 1 , 1 )
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Discrete time systems
e v a n s ( DTSystem )
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State space- representation
A
B
C
D
=
=
=
=
[ 0 , 1 ; −1, −0. 5]
[ 0 ; 1]
[1 , 0]
[0]
x0 = [ 1 ; 0 ] // The i n i t i a l
state
SSsys = s y s l i n ( ’ c ’ , A , B , C , D , x0 )
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State space- simulation
t = [0: 0.1: 50];
u = 0.5* ones (1 , l e n g t h ( t ) ) ;
[ y , x ] = c s i m ( u , t , SSsys ) ;
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State space- simulation
scf ( 1 ) ; p l o t ( t , y )
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State space- simulation
scf ( 2 ) ; p l o t ( t , x )
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State space
e v a n s ( SSsys ) // zoom i n
// C o n v e r s i o n from s t a t e s p a c e t o t r a n s f e r f u n c t i o n :
s s 2 t f ( SSsys )
r o o t s ( denom ( a n s ) )
spec (A)
Try this: obtain the step response of the converted transfer
function. Then compare this with the step response of the state
space representation (remember to set the initial state (x0) and
step size (u) correctly.
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CACSD with Scilab
State space
e v a n s ( SSsys ) // zoom i n
// C o n v e r s i o n from s t a t e s p a c e t o t r a n s f e r f u n c t i o n :
s s 2 t f ( SSsys )
r o o t s ( denom ( a n s ) )
spec (A)
Try this: obtain the step response of the converted transfer
function. Then compare this with the step response of the state
space representation (remember to set the initial state (x0) and
step size (u) correctly.
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CACSD with Scilab
Discretizing continuous time systems
Simp leSys Di sc r = s s 2 t f ( d s c r ( SimpleSys , 0 . 1 ) )
// S i n c e d s c r ( ) r e t u r n s a s t a t e s p a c e model
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CACSD with Scilab
Discretizing continuous time systems
t = [0: 0.1: 10];
u = ones ( t ) ;
y = f l t s ( u , Si mp le SysDi scr ) ;
plot (t , y)
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Multiple subsystems
SubSys1 = s y s l i n ( ’ c ’ , 1/ s )
SubSys2 = 2 // System w i t h g a i n 2
Series
= SubSys1 * SubSys2
Parallel = SubSys1+SubSys2
Feedback = SubSys1 / . SubSys2 // Note s l a s h −dot , n o t dot−s l a s h
// A l s o t r y t h e a b o v e s t e p u s i n g 2 i n s t e a d o f S u b s y s 2
Hint: put a space after the dot and then try.
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Multiple subsystems
SubSys1 = s y s l i n ( ’ c ’ , 1/ s )
SubSys2 = 2 // System w i t h g a i n 2
Series
= SubSys1 * SubSys2
Parallel = SubSys1+SubSys2
Feedback = SubSys1 / . SubSys2 // Note s l a s h −dot , n o t dot−s l a s h
// A l s o t r y t h e a b o v e s t e p u s i n g 2 i n s t e a d o f S u b s y s 2
Hint: put a space after the dot and then try.
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Nyquist Plot
G = s y s l i n ( ’ c ’ , ( s −2) / ( s+1) ) ;
H = 1
n y q u i s t ( G*H )
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An Example
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Modelling the system
mẍ(t) + b ẋ(t) + kx(t) = F (t)
Taking the Laplace transform:
ms 2 X (x) + bsX (s) + kX (s) = F (s)
X (s)
1
=
F (s)
ms 2 + bs + k
We will use a scaling factor of k and represent the system as:
G (s) =
ms 2
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k
+ bs + k
CACSD with Scilab
Modelling the system in Scilab
m
b
k
s
=
=
=
=
1 // Mass :
10 // Damper :
20 // S p r i n g :
%s
kg
Ns/m
N/m
// System :
System = k / ( m * s ˆ2 + b * s + k )
// We u s e k a s o u r s c a l i n g f a c t o r .
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CACSD with Scilab
A second order model
Comparing the system:
ms 2
k
+ bs + k
with the standard representation of a second order model:
ωn2
s 2 + 2ζωn s + ωn2
We have
r
ωn =
and
ζ=
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k
m
b/m
2ωn
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Understanding the system
wn = s q r t ( k / m )
zeta = ( b / m ) / ( 2 * wn )
We note that this is an overdamped system since ζ > 1
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// S t e p r e s p o n s e ( open l o o p t r a n s f e r f u n c t i o n ) :
t = 0:0.01:3;
y = c s i m ( ’ s t e p ’ , t , System ) ;
plot (t , y)
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// S t e p r e s p o n s e o f t h e s y s t e m w i t h u n i t y f e e d b a c k :
t = 0:0.01:3;
yFeedback = c s i m ( ’ s t e p ’ , t , System / . 1 ) ;
p l o t ( t , yFeedback )
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// Comparing t h e open l o o p t r a n s f e r f u n c t i o n and c l o s e d l o o p
transfer function :
p l o t ( t , [ y ; yFeedback ] )
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Steady State error
From the final value theorem:
lim f (t) = lim sF (s)
t→∞
s→0
Rs = 1/ s
Gs = System
// The s t e a d y s t a t e v a l u e o f t h e s y s t e m i s :
css = h o r n e r ( s * System * Rs , 0 )
After adding the feedback loop:
css = h o r n e r ( s * ( System / . 1 ) * Rs , 0 )
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Steady State error
From the final value theorem:
lim f (t) = lim sF (s)
t→∞
s→0
Rs = 1/ s
Gs = System
// The s t e a d y s t a t e v a l u e o f t h e s y s t e m i s :
css = h o r n e r ( s * System * Rs , 0 )
After adding the feedback loop:
css = h o r n e r ( s * ( System / . 1 ) * Rs , 0 )
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Lets take a look at the root locus
e v a n s ( System )
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Finding the gain at a point on the root locus
Say we have the root locus in front of us.
Now we wish to calculate some system parameters at some
particular point along the root locus. How do we find the position
of this point?
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Finding the gain at a point on the root locus
The root locus is the plot of the location of the poles of:
KG (s)
1 + KG (s)
as k varies.
That is to say, we need to find the solution of the denominator
going to zero:
1 + KG (s) = 0
or
K=
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−1
G (s)
CACSD with Scilab
Finding the gain at a point on the root locus
We can find the location of a given point on the root locus using
the locate() command.
We then need to multiply the [x; y] coordinates returned by this
command with [ 1, % i ] so that we obtain the position in the
complex plane as x + iy
We then simply evaluate -1/G(s) at the given position using
horner()
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Try this now:
e v a n s ( System )
Kp = −1/ r e a l ( h o r n e r ( System , [ 1 %i ] * l o c a t e ( 1 ) ) )
// C l i c k on a p o i n t on t h e r o o t l o c u s
css = h o r n e r ( s * ( Kp * System / . 1 ) * Rs , 0 )
// T h i s i s t h e s t e a d y s t a t e v a l u e o f t h e c l o s e d l o o p s y s t e m
// w i t h g a i n Kp .
(Choose any point on the root locus right now, we will shortly see
how to decide which point to choose)
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CACSD with Scilab
Achieving specific parameters- a proportional controller
Lets say we wish to achieve the following parameters:
OS = 0 . 3 0
tr = 0 . 0 8
// O v e r s h o o t
// R i s e t i m e
We know from theory, for a second order system,
− ln OS
ζ=p
π2
ωn =
1
tr
p
1 − ζ2
+ (ln OS)2
!
p
2
1
−
ζ
π − tan−1
ζ
In Scilab:
zeta = −l o g ( OS ) / s q r t ( %pi ˆ2 + l o g ( OS ) ˆ 2 )
wn = ( 1 / ( tr * s q r t ( 1 − zeta ˆ 2 ) ) ) * ( %pi − a t a n ( s q r t ( 1 − zeta ˆ 2 ) )
/ zeta )
We use these values of ζ and ωn to decide which point to choose
on the root locus.
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CACSD with Scilab
Achieving specific parameters- a proportional controller
Lets say we wish to achieve the following parameters:
OS = 0 . 3 0
tr = 0 . 0 8
// O v e r s h o o t
// R i s e t i m e
We know from theory, for a second order system,
− ln OS
ζ=p
π2
ωn =
1
tr
p
1 − ζ2
+ (ln OS)2
!
p
2
1
−
ζ
π − tan−1
ζ
In Scilab:
zeta = −l o g ( OS ) / s q r t ( %pi ˆ2 + l o g ( OS ) ˆ 2 )
wn = ( 1 / ( tr * s q r t ( 1 − zeta ˆ 2 ) ) ) * ( %pi − a t a n ( s q r t ( 1 − zeta ˆ 2 ) )
/ zeta )
We use these values of ζ and ωn to decide which point to choose
on the root locus.
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CACSD with Scilab
Achieving specific parameters- a proportional controller
Lets say we wish to achieve the following parameters:
OS = 0 . 3 0
tr = 0 . 0 8
// O v e r s h o o t
// R i s e t i m e
We know from theory, for a second order system,
− ln OS
ζ=p
π2
ωn =
1
tr
p
1 − ζ2
+ (ln OS)2
!
p
2
1
−
ζ
π − tan−1
ζ
In Scilab:
zeta = −l o g ( OS ) / s q r t ( %pi ˆ2 + l o g ( OS ) ˆ 2 )
wn = ( 1 / ( tr * s q r t ( 1 − zeta ˆ 2 ) ) ) * ( %pi − a t a n ( s q r t ( 1 − zeta ˆ 2 ) )
/ zeta )
We use these values of ζ and ωn to decide which point to choose
on the root locus.
Aditya Sengupta, EE, IITB
CACSD with Scilab
Achieving specific parameters- a proportional controller
Lets say we wish to achieve the following parameters:
OS = 0 . 3 0
tr = 0 . 0 8
// O v e r s h o o t
// R i s e t i m e
We know from theory, for a second order system,
− ln OS
ζ=p
π2
ωn =
1
tr
p
1 − ζ2
+ (ln OS)2
!
p
2
1
−
ζ
π − tan−1
ζ
In Scilab:
zeta = −l o g ( OS ) / s q r t ( %pi ˆ2 + l o g ( OS ) ˆ 2 )
wn = ( 1 / ( tr * s q r t ( 1 − zeta ˆ 2 ) ) ) * ( %pi − a t a n ( s q r t ( 1 − zeta ˆ 2 ) )
/ zeta )
We use these values of ζ and ωn to decide which point to choose
on the root locus.
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CACSD with Scilab
Achieving specific parameters- a proportional controller
e v a n s ( System )
s g r i d ( zeta , wn )
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Achieving specific parameters- a proportional controller
Find the value of proportional gain at some point on the root locus:
Kp = −1/ r e a l ( h o r n e r ( System , [ 1 %i ] * l o c a t e ( 1 ) ) )
// C l i c k n e a r t h e i n t e r s e c t i o n
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Achieving specific parameters- a proportional controller
PropSystem = Kp * System / . 1
yProp = c s i m ( ’ s t e p ’ , t , PropSystem ) ;
p l o t ( t , yProp )
p l o t ( t , o n e s ( t ) , ’ r ’ ) , // Compare w i t h s t e p
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A PI controller
Note the steady state error and overshoot.
In order to eliminate the steady state error, we need to add an
integrator- that is to say, we add a pole at origin.
In order to have the root locus pass through the same point as
before (and thus achieve a similar transient performance), we add
a zero near the origin
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A PI controller
PI = ( s+1)/ s
e v a n s ( PI * System )
s g r i d ( zeta , wn ) // These v a l u e s a r e from t h e P r o p o r t i o n a l
controller
Kpi = −1/ r e a l ( h o r n e r ( PI * System , [ 1 %i ] * l o c a t e ( 1 ) ) )
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A PI controller
PISystem = Kpi * PI * System / . 1
yPI = c s i m ( ’ s t e p ’ , t , PISystem ) ;
p l o t ( t , yPI )
p l o t ( t , o n e s ( t ) , ’ r ’ ) , // Compare w i t h s t e p
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A PD controller
We wish to achieve the following parameters:
OS = 0 . 0 5
Ts = 0 . 5
From theory, we know the corresponding values of ζ and ωn are:
s
(log OS)2
ζ=
(log OS)2 + π/2
ωn =
4
ζTs
zeta = s q r t ( ( l o g ( OS ) ) ˆ 2 / ( ( l o g ( OS ) ) ˆ2 + %pi ˆ 2 ) )
wn = 4 / ( zeta * Ts )
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A PD controller
PD = s + 20
e v a n s ( PD * System )
s g r i d ( zeta , wn ) // Zoom f i r s t , t h e n e x e c u t e n e x t l i n e :
Kpd = −1/ r e a l ( h o r n e r ( PD * System , [ 1 %i ] * l o c a t e ( 1 ) ) )
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A PD controller
PDSystem = Kpd * PD * System / . 1
yPD = c s i m ( ’ s t e p ’ , t , PDSystem ) ;
p l o t ( t , yPD )
p l o t ( t , o n e s ( t ) , ’ r ’ ) , // Compare w i t h s t e p
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A PD controller
Note the improved transient performance, but the degraded steady
state error:
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A PID Controller
We design our PID controller by eliminating the steady state error
from the PD controlled system:
PID = ( s + 2 0 ) * ( s+1)/ s
e v a n s ( PID * System )
s g r i d ( zeta , wn ) // These v a l u e s a r e from t h e PD s y s t e m
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A PID Controller
Kpid = −1/ r e a l ( h o r n e r ( PID * System , [ 1 %i ] * l o c a t e ( 1 ) ) )
PIDSystem = Kpid * PID * System / . 1
yPID = c s i m ( ’ s t e p ’ , t , PIDSystem ) ;
p l o t ( t , yPID )
p l o t ( t , o n e s ( t ) , ’ r ’ ) , // Compare w i t h s t e p
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A Comparison
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References
Control Systems Engineering, Norman Nise
Modern Control Engineering, Katsuhiko Ogata
Digital Control of Dynamic Systems, Franklin and Powell
Master Scilab by Finn Haugen.
http://home.hit.no/f̃innh/scilab scicos/scilab/index.htm
Mass, damper, spring image: released into the public domain
by Ilmari Karonen.
http://commons.wikimedia.org/wiki/File:Mass-SpringDamper.png
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