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Published bySally Shutts Modified over 2 years ago

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MIMO systems

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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

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Transfer function of a TITO system

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

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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 ?

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Bristols Relative Gain Array

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Bristols interaction index ʎ for TITO systems

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RGA – Bristols relative gain array

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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

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RGA for TITO system

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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

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Example

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

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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

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

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Decoupling factor

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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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What does C store? >>A = [1 2 3] >>B = [1 1] >>[C,D]=meshgrid(A,B) c)1 1 1 1 1 1 a) 1 2 3 1 1 1 d)1 2 3 1 1 1 b) 1 2 3 1 2 3.

What does C store? >>A = [1 2 3] >>B = [1 1] >>[C,D]=meshgrid(A,B) c)1 1 1 1 1 1 a) 1 2 3 1 1 1 d)1 2 3 1 1 1 b) 1 2 3 1 2 3.

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