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(srotalugeR gninuT fleS)
5831
)srotalugeR gninuT-fleS(
) RTS(
9 : 29 : 9)SLR( : 9 : 9 : 9
. :elpicnirP ecnelaviuqE ytniatreC9
.
: :
uA/B
R/S
y ucR/T
=yS uT uRc
:
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
= + + += + + +
""
1 20 1 1
1 20 1 1
( )
( )
n nn
n nn
R q r q rq rS q s q s q s
= + + += + + +
""
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
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
:* 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
: . 1-n
. 1-n2
.
)srotalugeR gninuT-fleS tceridnI(
) RTS(
B A,
?
SLR
A B Am mo , ,T
T S R, ,
=yS uT uRcuy
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
.
: :
.
)0d-m,n(xam = 1-m+n= 1-n+m
+ + + + =d m n m n N) , (xam 10
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
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
elpicnirP ledoM lanretnI
: -
: lortnoC ssecorP
:
=e v Ad - -
-
BA
1
Ad
e
vy u
:
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
tfiL
ffo ekaT
RETSOOB :
: 9
(: ) 9)elissiM lloR oN(
: 9
( : ) 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