self tuning regulators

(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


10

20

30

40

50

T

a

n

k

L

e

v

e

l

(

c

m

)


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


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


20

30

40

50

T

a

n

k

L

e

v

e

l

(

c

m

)


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


10 250

0.9

1

1.1

1.2


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


20

30

40

50

60

T

a

n

k

L

e

v

e

l

(

c

m

)


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


0.8

0.85

0.9

0.95

1


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


10 350

20

30

40

50

T

a

n

k

L

e

v

e

l

(

c

m

)


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



350 10

20

30

40

50

60

T

a

n

k

L

e

v

e

l

(

c

m

)


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


METSYS LORTNOC ERUSSERP

opiiii

METSYS LORTNOC ERUSSERP

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

: 3 2

2

) (2.141 44.932.302 103.4s Gs

s s+ + =+

. 6 : RTS 9

RTS: 9 9 9 9 9 9

self tuning regulators

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