Download - Self Tuning Regulators
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uA/B
R/S
y ucR/T
=yS uT uRc
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2n-1 .
cAR BS A+ =1
11
0 1
( )( )
n nn
n nn
A q q a q aB q b q b q b
= + + += + + +
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1 20 1 1
1 20 1 1
( )
( )
n nn
n nn
R q r q rq rS q s q s q s
= + + += + + +
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cA
Diophantine EquationOr
Bezout IdentityOr
Aryabhatta Equation
-
1: A(q) B(q) .
2 : :
1 1
1
1 1
1 11 0 1
0
1 1
0
0 00 00 0
10 1 0
0 000 0 1 0 0
n n
n n n n
n
n n
n n
a ba a b b
ba ba ba bE a b b
b
a bb
=
" "" "
# ## ##
# ## #
# #
2nx2n MatrixIs non-singularIf and only ifA, B are coprime
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3:
1 2 1 1 2 1
2 2 2 2 2 2
0 0 1
1 1
2 2
0 0 0 0
n n n n
n n n n
n n
n n
r rr r
r rE E
s ss s
s s
= =
# # # #
# # # #
2 1 2 20 1 2 1
n nc nA q q = + + +"
-
:
c
c
BT BRy u vAR BS AR BSAT BSu u v
AR BS AR BS
= ++ += + +
B/AcRu Tu Sy= cu
v
u y
STABILITY
TRACKING
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:* 1
* 1 * 10 0 0
* 1 11
* 1 10 1
( ) ( )
( ) ( ) ( )[ ( ) ( )] ,
( ) 1( )
n
nn
mm
A q q Aq
A q y t B q u t d v t d d n mWhereA q aq aqB q b bq b q
=
= + =
= + + += + + +
""
-
(Model Following)9 :
9 - .
Perfect Model-Following
( ) ( )m m m c
m
c m
A y t B u t
BT BT BAR BS A A
=
= =+
BTcA
-
''
'
' '
'
Monic Stable Polynomial with well damped roots
Unstable or poorly damped roots
where,
Let,
And,
Diaphontine Equation
m m
c o m
o m c
o m
B B B
BB
B B B
A A A BR R B
AR B S A A AT A B
+
+
+
=
==
=
= =
+ = = =
-
(-) ( - ) 9:
: 9
ged gedged gedR SR T
S0R0
0
0
, BQ R RAQ S S
+ = =
-
Q
Minimum Degree Solution
Max(degS)=n-1
0
deg 2deg 1deg deg deg deg
c
m m
A AA B A B d =
CausalityConditions
-
PPDM
B ,A : 9: 9: 9
,
'
dna ,
ged gedged ged1 ged ged ged
o m m
m
m
o
m m
A B A
A AB BB A A
B B B
+
== =
=
-
: 1 9 :2 9
: 3 9
= +B B B
A S AA S B RAS R m oged ged ,' '< = +
'
'm o
c
R
S uT uR
B RB A T
y
=+=
=
-
'=B A A Am o c
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B A,
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SLR
A B Am mo , ,T
T S R, ,
=yS uT uRcuy
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RLSUnknown, deg , deg( ) ( ) ( ) ( ), , A n B mA q y t B q u t A B = ==
1 1 Unknown Parameters( ) ( 1) ( ) ( 1) ( )n my t a y t a y t n bu t m n b u t m= + + + + " "
( )( ) 1Ty t t = RLS
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)0d-m,n(xam = 1-m+n= 1-n+m
+ + + + =d m n m n N) , (xam 10
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RTS
: 9
: 1 9
: 2 9
: 3 9
.
A B Ao m m dna , ,
SLRT R S PPDM, ,
=yS uT uRc
-
(Direct Self-Tuning Regulators)
(STR )
0, , ,om mA B A d
?
RLS , ,R S T
cRu Tu Sy= yucu
-
: :
:
:
=t u q B t y q A) ( ) ( ) ( ) (
'
'
'
'
) ( ) ( ) (
) ( ) ( = ) ( ) ( = )) ( () ) ( ( ) ( =
m o
m o
A A S B RAt yS B t y RA t y A A
t yS B t uB Rt yS B t u B B Rt yS Bt tyS t uR B u BR
+
= ++ =
++
+ = +
=t u B t y Ac m m m) ( ) (
-
0
0
0
0 1 0
deg deg deg 1, and constant=( ) ( ( ) ( ))
= ( ) ( )A ll zeors cancelled: (1) is a good choice. Let
=[ ]( ) [ ( ) ( ) ( )
o
o m
dm m
l l
A A B B bA A y t b R u t S y t
R u t S y tB q A
r r r s st u t u t l t y
= = = +
+=
= " "
" " y
* 1 * 10
( )]
( ) ( ) ( ) ( ) ( )To m
t l
t A q A q y t t d
= =
-
:* *
0 0
* 1 * 1
* 1 * 1
0 0
0 1 0
1( ) ( ( ) ( )) ( ) ( )
,1( ) ( )
( ) ( )1( ) ( )
( ) ( )and, d deg deg , deg deg deg( )
=[ ]( ) [ ( ) ( ) (
f fo m
fmo
fmo
o m
l l
f f f
y t Ru t Sy t R u t d S y t dA A
where
u t u tA q A q
y t y tA q A q
A B R S A A d l
r r r s st u t u t l
= + = +
=
== = = =
= " "" y
0
) ( )]
( ) ( )
f
T
t y t l
y t t d
=
"
-
RTS
: 9
: 1 9
: 2 9
.
d A B Ao m m dna , , ,0
* *0 0
+ = SLR d t y S d t u R t yf f dna ) ( ) ( ) (
= =A A T y S u T u Rm o c))1( ( , * * * * *
-
. :1 . :2 . :3
. . :4
r0 0R
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NMP Systems9 :
* *0 0
( ) ( ( ) ( )), deg deg deg( ) deg,
,1( ) ( ( ) ( )) ( ) ( )
o m o m
f fo m
A A y t B Ru t Sy t R S AA BLetS B S R B R
y t Ru t Sy t Ru t d S y t dA A
= + = =
= = = + = +
-
RTS
: 9
: 1 9
. . : 2 9
: 3 9
.
d A B Ao m m dna , , ,0
* *0 0
+ = SLR d t y S d t u R t yf f dna ) ( ) ( ) (
= =B A T y S u T u Rm o c) ( , ' * * * * *
SRSR
-
. :1 . :2 : :3
* *
'
*0 0 0
) ( ) (
,
) ) ( )( ) ( ) ( ) ( ) ( ( ) (
mc m
m
m
cc f fo
fm
t u t yB BA
y y e teLt uT t yt y S d t uR S t uR t eBA
t u T dA
d
= =
= + + =
-
RTS
: 9
: 1 9
. : 2 9.
: 3 9
.
A Ao m ,
= SLR t uB t yA dna ) ( ) (
) , ( ,)1()1(m
o o o ct A t T yS uT uRAB
= = =
SRSR
-
1: 2 . 2: :
* * *0 00 0 0
,
( ) ( ( ) ( ) ( ) ( ( ) ( )) ( ))f f co
f
m
cm
Let e y yB B Ru t d S y t d t u te t Rut Sy t t u tA
dA
= + = + =
-
9 9 9 9
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elpicnirP ledoM lanretnI
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=e v Ad - -
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BA
1
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vy u
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9!
: 9.
) (
) (
cd
cd
e u yRB TBSB RA A SB RAe u uSB TA
SB RA A SB RA
+ ++ =+ + =
Ad
'=A R Rd
-
0 0 0
0 0 0
0
0
0 0 0
0
0
0
L e t , ,
I f ,
T h e n ,
( ) ( )
c
c
c
c
c
c
R S AA R B S A
R R BS S A
A R B S AA R B B S A A
X YX Y
X X XY Y XYA YR A B B S B A A
A R B S A
X
X
+ =
= +=
+ = + + =
+ + =
+ =
R, S
-
:
'0+ =B Y R RX Ad
Y R,'
-
:
'
0 00
' 00 0
00 0
00
L e t , D i s t u r b a n c e = S t e p , T h e n1
( 1)A d d e d C -L P o le ,
( )
( 1) ( )L e t , 1
(1 ) 0(1 )
(1)
d
c c
A qR q R
X A q x Aq R q x R y Bqx R y B
xyB
= =
= + = + +
= + + =
+ =
:
00 0
0
00 0
0
(1 )( )(1)
(1 )( )(1)
x RR q x R BB
x RS q x S AB
+= + + += + +
-
. 9. : 9
:
+ =t vB t uB t yA) ( ) ( ) (
)1(B.
-
: =e v Ad
+ =t v t u B A t yA Ad d)) ( ) ( ( ) (
+ =t eB t uB t yAf f) ( ) ( ) (. 9
. . . -
-
- :
-
STR
0' ' '
1 1
'1 0
( ) ( ) ( )( ( ) ( )), deg deg( ) ( ) (1) ( ), deg
Process:Desired Responce:
Design Equation:
Integral Action:
,
(
1)
m m c m
o m
A q y t B q u t v t d A BA q y t A u t d A d
AR BS B A A B b BR R B R B q R B
A R b S
+ +
+ + +
= + = =
+ = == = =
+ ='1 0' '1 0
* 1 * 1 '* 1 1 * 10 0
( ) ( ) ( )
= ( ) ( )
...... ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( )]
o m
o m
m
A AA A y t AR y t b Sy t
BR u t b R v tA q A q y t d b R q q u t S q y t
= + +
+ = +
-
* * *0
Note That, (1) (1) (1) (1) (1)o m o mb S A A A A= =
* 1 1 '* 10
'* 1 *
Let, ( ) (1) (1) (1 ) ( )
(1) (1) ( )
o m
o m
b S q A A q S qA A S q
= + = +
* 1 * 10 0
'* 1 * 1 '* 1 *0* 1 * 1 * 1 *
( ) ( ) ( ) (1) (1) ( )
[ ( ) ( ) ( ) ( ) ( )]
( ) ( ) ( ) ( ) ( )
m mA q A q y t d A A y tb R q q u t S q y tR q q u t S q y t
+ = + = +
-
* 1 * 10* 1 * 10
(1) (1)( ) ( ) ( ) ( ) ( ) ( ) (*)( ) ( )
mf f
m
A Ay t d y t R q u t S q y tA q A q
+ = +
1
* 1 * 10
1 ( )( ) ( )m
q u tA q A q
1
* 1 * 10
1 ( )( ) ( )m
q y tA q A q
-
:* 1 * 1 * 1 * * 1 * 1
0 0 ( ) ( ) ( ) ( ) ( ) (1) (1) ( ) ( ) ( ) ( )m mR q q u t S q y t A A y t A q A q u t + + =
Integrator Windup
* 1 * 1 * * 1 * 1 * 10( )[ ( ) (1) ( )] (1) (1) ( ) ( ) ( ) [ ( ) ( ) ( )] ( )
( ) sat ( ) o m c m oA q u t A u t A A y t S q y t R q q A q u tu t u t
= = (**)
-
RTS
. ) *( : 1 9 1 : 2 9
. ) * *(
-
seidutS esaC
9 9 9 9 9
-
TANK LEVEL CONTROL SYSTEM
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
60
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 100 200 300 400 500 600 700 800 900 10000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
0 100 200 300 400 500 600 700 800 900 1000-40
-30
-20
-10
0
10
20
30
40
time(seconds)
STR
0 100 200 300 400 500 600 700 800 900 10000.8
0.9
1
1.1
Identification model parameterst
e
t
a
1
0 100 200 300 400 500 600 700 800 900 10000
0.05
0.1
time(seconds)
t
e
t
a
2
-
TANK LEVEL CONTROL SYSTEM
10
20
30
40
50
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 50 100 150 200 250 300 350 400 450 5000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
10 250
0.8
0.9
1
1.1Identification model parameters landa=0.99 Ts=1sec
t
e
t
a
1
0 50 100 150 200 250 300 350 400 450 5000
0.05
0.1
time(seconds)
t
e
t
a
2
0 50 100 150 200 250 300 350 400 450 500-4
-3
-2
-1
0
1
2
3
4
5
6Controller parameters Am=[1 -0.92]
time(seconds)
STR
-
TANK LEVEL CONTROL SYSTEM
20
30
40
50
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 50 100 150 200 250 300 350 400 450 5000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
10 250
0.9
1
1.1
1.2
1.3Identification model parameters landa=0.99 Ts=1sec
t
e
t
a
1
0 50 100 150 200 250 300 350 400 450 5000
0.01
0.02
0.03
time(seconds)
t
e
t
a
2
0 50 100 150 200 250 300 350 400 450 500-4
-3
-2
-1
0
1
2
3
4
5
6Controller parameters Am=[1 -0.92]
time(seconds)
STR
-
TANK LEVEL CONTROL SYSTEM
20
30
40
50
60
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 100 200 300 400 500 600 700 800 900 10000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
0.8
0.85
0.9
0.95
1
1.05Identification model parameters landa=0.9999 Ts=1sec
t
e
t
a
1
0 100 200 300 400 500 600 700 800 900 10000
0.02
0.04
0.06
0.08
0.1
time(seconds)
t
e
t
a
2
0
2
4
Controller parameters S,T & R
0
0.05
0.1
0 100 200 300 400 500 600 700 800 900 1000
-1
0
1
time(seconds)
R1R0
T
S1S0
-
TANK LEVEL CONTROL SYSTEM
10 350
20
30
40
50
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 100 200 300 400 500 600 7000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
-
TANK LEVEL CONTROL SYSTEM
350 10
20
30
40
50
60
T
a
n
k
L
e
v
e
l
(
c
m
)
Setpoint & Plant Output
0 100 200 300 400 500 600 7000
50
100Control Signal
time (seconds)
i
n
l
e
t
v
a
l
v
e
p
o
s
i
t
i
o
n
Plant outputsetpoint
-
METSYS LORTNOC ERUSSERP
opiiii
-
METSYS LORTNOC ERUSSERP
opiiii
-
METSYS LORTNOC ERUSSERP
-
METSYS LORTNOC ERUSSERP
-
METSYS LORTNOC WOLF
-
METSYS LORTNOC WOLF
-
METSYS LORTNOC ERUTAREPMET
-
TEMPERATURE CONTROL SYSTEM
-
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ffo ekaT
RETSOOB :
-
: 9
(: ) 9)elissiM lloR oN(
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-
( : ) 9
) (K
p s s+ =
-
9
-
: 9.
-
- -
-
- -
-
9
: 3 2
2
) (2.141 44.932.302 103.4s Gs
s s+ + =+
-
9:
-
9
-
9
-
. 6 : RTS 9
-
RTS: 9 9 9 9 9 9