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MIMO systems

MIMO systems. Interaction of simple loops. Y 1 (s)=p 11 (s)U 1 (s)+P 12 (s)U 2 (s) Y 2 (s)=p 21 (s)U 1 (s)+p 22 (s)U 2 (s). C1. Y sp1. Y 1. Y sp2. C2. Y 2. Transfer function of a TITO system. Y 1 (s)=p 11 (s)U 1 (s)+ p 12 (s)U 2 (s) Y 2 (s)=p 21 (s)U 1 (s)+p 22 (s)U 2 (s)

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MIMO systems

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  1. MIMO systems

  2. 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 Ysp1 Y1 Ysp2 C2 Y2

  3. 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

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

  5. Simulations for steps in setpoints for loop 1C1 = PI control and C2 = P control K2=0 second loop disconnected K2=0.8 the system is unstable K2= 1.6 the system becomes suggish The gain decreases as k2 increases The gain becomes negative for k2>1 Could it be better to use the other input/output combination ?

  6. Bristols Relative Gain Array • Investigate how the staticprocessgain of one loop is infuenced by the gains in other loops • Second loop in perfectcontrol => Y1(s)=p11(s)U1(s)+p12(s)U2(s) 0 =p21(s)U1(s)+p22(s)U2(s) Eliminating gives

  7. Bristols interactionindex ʎ for TITO systems • ʎ =the ratio of the staticgains of loop 1 when the second loop is open and when the second loop is closed • ʎ= • Interaction for static or lowfrequency signals • =0 => ʎ=1 nointeraction

  8. RGA – Bristols relative gain array • Comparethe staticgain for one output when all other loops are open with the gainswhen all other outputs arezero. • P(0) : the staticgain 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, alklerowsans columns sum to 1.

  9. First step in the analysez is to calculate the RGA matrix Λ Relative gains is (i,j) element in the static transfer matrix Ps(s) therefore In Λ the sum of all elements in a row and the sum of all elements i a column = 1

  10. 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 highergainthan open loop • ʎ >1 : closed loop has lowergainthan open loop • ʎ <0 :gain of first loop changes sign whensecond loop is closed.

  11. 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 gainsshouldbe positive and close to 1 • Pairing of signals with negative relative gains should be avoided. • If gainsareoutside the interval 0.67<ʎ<1.5 decouplingcanimprove the controlsignificantly.

  12. Example • ʎ = -1 => should be paired with . For closed loopp: => • Static : • increase for decreasing , is positive for >0.

  13. Impact of switching loops U1 controls Y1 U2 controls Y1

  14. Decoupling – design of controllers thatreduce the effects of interactionbetween loops System TITO system with interaction from u1 to y1 and y2 from u2 to y1 and y2 The dynamiccoupling factor Q is Decoupling eliminates the effect of the interaction from u1 to y2 And from u2 to y1 u1 F11 y1 C11 F12 C12 F21 C21 u2 F22 C22 y2

  15. Decoupling - structure System u1 u1a F11 Y1 C11 F12 C12 F21 C21 u2 u2a F22 C22 Y2 Decoupling if:

  16. Decoupling factor • The coupling factor is given by • and are chosen to be 1 then and can be calculated

  17. Decoupled system Y1 u1a F11 1-Q C11 u2a F22 1-Q C22 Y2 The controllers can be designed using ordinary SISO rules. Good results requires god models !!

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