multivariable systems relative gain array (rga) distillation control course december 4-8, 2004

Multivariable systems

Relative Gain Array (RGA)

Distillation Control Course

December 4-8, 2004

April 4-8, 2004 KFUPM-Distillation Control Course

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2. MIMO (multivariable case)Direct Generalization:

“Increasing L from 1.0 to 1.1 changes yD from 0.95 to 0.97, and xB from 0.02 to 0.03”

“Increasing V from 1.5 to 1.6 changes yD from 0.95 to 0.94, and xB from 0.02 to 0.01”

Steady-State Gain Matrix

B

2221

1211

B

D

Δx2outputonΔL1inputofEffect

0.10.1

0.10.2

1.51.6

0.020.01

1.01.1

0.020.031.51.6

0.950.94

1.01.1

0.950.97

0gg

0gg0G

ΔV

ΔL0G

Δx

ΔY

40s1

0.1

40s1

0.150s1

0.1

50s1

20.

G

:dynamicsincludealsoCan

s

B

D

x

y

(Time constant 50 min for yD)

(time constant 40 min for xB)


3

Effect of Feedback Control

C(s)ys u

G(s)d

y

G(s): process (distillation column)

C(s): controller (multivariable or single-loop PI’s)

y: output , ys: setpoint for output u: input d: effect of disturbance on output


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( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )u

u

= × +

=- - s

Process : y s G s s d s (1)

Controller : s C s y s y s (2)

( )( ) ( ) No disturbance supression

=

= ¬

No Control, u s 0 :

y s d s

Negative feedback

With feedback control Eliminate u(s) in (1) and (2), ys=0

sdC(s)sGIsy 1

S(S)=“sensitivity function” = supression disturbances


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( ) ( ) ( )

( )

s s s

s

y d

Ideal:-Want 0

S

S

= ×

=In practice: Cannot do anything with fast changes, that is, S(jw)= I at high frequency

S is often used as performance measure

1 11 12 1

2 21 2

y S (s) S (s) d (s)

y S (s) . . d (s)

21 jwS

One Direction

Log- scale

1

0.1

Small resonance peak (small peak = large GM and PM)

Bandwidth (Feedback no help)

Slow disturbance d1: 90% effect on y2 removed

w (log)w B

τ


6

“Summing up of Channels”

“maximum singular value” (worst direction)

0.1

1 w (log)wB

“minimum singular value” (best direction)

(s)

(S)


7

Analysis of Multivariable processes (Distillation

Columns)What is different with MIMO processes to SISO:

The concept of “directions” (components in u and y have different magnitude”

Interaction between loops when single-loop control is used

INTERACTIONS

Process Model

G

y1

y2

g12

g21

g11

g22u2

u1

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 11 1 12 2

2 21 1 22 2

" "

( )

( )

Open loop

y s g s u s g s u s

y s g s u s g s u s

-

= +

= +


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Consider Effect of u1 on y1

1) “Open-loop” (C2 = 0): y1 = g11(s)·u1

2) Closed-loop” (close loop 2, C2≠0)

Change caused by “interactions”

( ) uæ ö× ÷ç ÷= -ç ÷ç ÷ç + ×è ø

12 21 21 11 1

22 2

g g Cy g s

1 g C


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Limiting Case C2→∞ (perfect control of y2)

( )( )

CL

×= = = =

- -

def1 1 RGAOL 11

1112 21 12 211 1

1122 11 22

y /μ g 1RelativeGain λ

g g g gy /μ g 1g g g

122

2112111 μ

g

ggsgy

( )( )

( )( )

( )( )

( )( )

u

é ùê úê úé ù ê úê ú= = =ê úê ú ê úë û ê úê úë û

1 1 1 2OL OL

1 1 1 211 12 CL CL

21 22 2 1 2 2OL OL

2 1 2 2CL CL

y /u y /u

y /u y /uλ λRGA Λ

λ λ y y /u

y u y /u

How much has “gain” from u1 to y1 changed by closing loop 2 with perfect control

The relative Gain Array (RGA) is the matrix formed by considering all the relative gains

21

12RGA

2

0.10.2

0.10.11

1λ,

0.10.1

0.10.2G

0.5

11

Example from before


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Property of RGA:

Columns and rows always sum to 1

RGA independent of scaling (units) for u and y.

Note: RGA as a function of frequency is the most important for control!


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Use of RGA:

(1) Interactions

From derivation: Interactions are small if relative gains are close to 1

Choose pairings corresponding to RGA elements close to 1

Traditional: Consider Steady-state

Better: Consider frequency corresponding to closed-loop time constant

But: Avoid pairing on negative steady-state relative gain – otherwise you get instability if one of the loops

become inactive (e.g. because of saturation)


12

Example:

0.2 0.1G o

0.1 0.1

1 1 2

2 1 2

y = 0.2 u -0.1 u

y = 0.1u -0.1u

× ×

u

u

u

u

1 1

2 2

1 2

2 1

2 -1RGA =

-1 2

Only acceptablepairings :

y

y

Not recommended :

y

y

é ùê úê úë û

«

«

«

«

With integral action :

Negative RGA individual

loop unstable + overall system unstable

when loops saturate


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(2) Sensitivity measure

But RGA is not only an interaction measure:

Large RGA-elements signifies a process that is very sensitive to small changes (errors) and therefore fundamentally difficult to control

example

1 1 91 90G= RGA

0.9 0.91 90 91

Large (BAD!)

21

12 12

0.9

1Relativechange - makesmatrixsingular!

1ˆThen g g 1 0.91

90

l=

æ ö÷ç= + =÷ç ÷çè ø

11.1%

90


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Singular Matrix: Cannot take inverse, that is, decoupler hopeless.

Control difficult


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multivariable systems relative gain array (rga) distillation control course december 4-8, 2004

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