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(kkg 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
yyy
uuu
21
222212
112111
21
ij Relative gains
jkforconstuj
iij
kuy
__
ikforconstyi
j
jkforconstuj
iij
kkyu
uy
____
.
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 !!