# 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 C2 Y sp1 Y sp2 Y1Y1 Y2Y2.

## Presentation on theme: "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 C2 Y sp1 Y sp2 Y1Y1 Y2Y2."— Presentation transcript:

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 C2 Y sp1 Y sp2 Y1Y1 Y2Y2

Transfer function of a TITO system

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

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 ?

Bristols Relative Gain Array

Bristols interaction index ʎ for TITO systems

RGA – Bristols relative gain array

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

RGA for TITO system

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 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/11/3273826/slides/slide_11.jpg", "name": "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

Example

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

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

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

Decoupling factor

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 !!

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