MIMO systems

Interaction of simple loops

Y1(s)=p11(s)U1(s)+P12(s)U2(s)Y2(s)=p21(s)U1(s)+p22(s)U2(s)

C1

C2

Ysp1

Ysp2

Y1

Y2

2U

1U

Transfer function of a TITO system

Y1(s)=p11(s)U1(s)+p12(s)U2(s)Y2(s)=p21(s)U1(s)+p22(s)U2(s)pij(s) is transfer functions from input j to output i

P(s)=transfer function of the system= the transfer matrix of the system

Effect of interactionC1 controls Y1 by U1, C2 controls Y2 by U2

•

Simulations for steps in setpoints for loop 1C1 = PI control and C2 = P control

2

211 1

1)0(

k

kg cl

The gain decreases as k2 increasesThe gain becomes negative for k2>1

K2=0 second loop disconnectedK2=0.8 the system is unstableK2= 1.6 the system becomes suggish

Could it be better to use the other input/output combination ?

Bristols Relative Gain Array

• Investigate how the static process gain of one loop is infuenced by the gains in other loops

• Second loop in perfect control => Y1(s)=p11(s)U1(s)+p12(s)U2(s)0 =p21(s)U1(s)+p22(s)U2(s)Eliminating gives

Bristols interaction index ʎ for TITO systems

• ʎ =the ratio of the static gains of loop 1 when the second loop is open and when the second loop is closed

• ʎ= • • Interaction for static or low frequency signals• =0 => ʎ=1 no interaction

RGA – Bristols relative gain array

• Comparethe static gain for one output when all other loops are open with the gains when all other outputs are zero.

• P(0) : the static gain for the system• : the transposed of the inverse of P(0)• .* : component-wise multiplication• : the ratio between the open loop and closed loop gain

from input to output • R: symetric, alkle rows ans columns sum to 1.

• First step in the analysez is to calculate the RGA matrix Λ

nnnnn

n

n

n

y

y

y

uuu

21

222212

112111

21

ij Relative gains

jkforconstuj

iij

k

u

y

__

ikforconstyi

j

jkforconstuj

iij

kk

y

u

u

y

____

.

is (i,j) element in the static transfer matrix Ps(s)

thereforeT

ss sPsP ))(()( 1

In Λ the sum of all elements in a row and the sum of all elements i a column = 1

RGA for TITO system

• ʎ is the interaction index• ʎ =1 : no interaction, second loop has no

impact on first loop• 0<ʎ<1 : closed loop has higher gain than open

loop• ʎ >1 : closed loop has lower gain than open

loop• ʎ <0 :gain of first loop changes sign when

second loop is closed.

Pairing : Decide how inputs and outputs should be connected in control loops using

RGA• ʎ =0 no interaction• ʎ = 1 no interaction but loops should be interchanged• ʎ<0.5 loops should be interchanged• 0<ʎ<1 closed loop gains <open loop gains• Corresponding relative gains should be positive and

close to 1• Pairing of signals with negative relative gains should be

avoided.• If gains are outside the interval 0.67<ʎ<1.5 decoupling

can improve the control significantly.

Example

•

• ʎ = -1 => should be paired with . For closed loopp: =>

• Static : • increase for decreasing , is positive for

>0.

Impact of switching loops

U1 controls Y1

U2 controls Y1

Decoupling – design of controllers that reduce the effects of interaction between loops

F11

F12

F21

F22

C11

C12

C21

C22

System

y1

y2

u1

u2

TITO system with interaction from u1 to y1 and y2from u2 to y1 and y2

The dynamic coupling factor Q is

Decoupling eliminates the effect of the interaction from u1 to y2And from u2 to y1

Decoupling - structure

F11

F12

F21

F22

C11

C12

C21

C22

System

Y1

Y2

u1

u2

u1a

u2a

Decoupling if:

Decoupling factor

• The coupling factor is given by

• and are chosen to be 1 then and can be calculated

Decoupled system

F11

F22

1-Q

1-Q

C11

C22

Y1

Y2

u1a

u2a

The controllers can be designed using ordinary SISO rules.Good results requires god models !!