1. MIMO systems

2. 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 C2 Y sp1 Y sp2 Y1Y1 Y2Y2

3. Transfer function of a TITO system

4. Effect of interaction C 1 controls Y 1 by U 1, C 2 controls Y 2 by U 2

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

6. Bristols Relative Gain Array

7. Bristols interaction index ʎ for TITO systems

8. RGA – Bristols relative gain array

9. First step in the analysez is to calculate the RGA matrix Λ Relative gains is (i,j) element in the static transfer matrix P s (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

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 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.

12. Example

13. Impact of switching loops U 1 controls Y 1 U 2 controls Y 1

14. 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

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

16. Decoupling factor

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