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### Chapter 15

PID Controllers Applied to MIMO Processes

Example of a 2×2 MIMO Process

- Two inputs: Setpoints for flow controller on steam and reflux.
- Two outputs: Composition of products B and D

Configuration Selection (Choosing the u/y Pairings)

- That is, which manipulated variable is to be used to control which controlled variable.
- Choosing an inferior configuration can dramatically reduce control performance.
- For many processes, configuration selection is a difficult and challenging process (e.g., dual composition control for distillation).

Example of Single Loop PID Controllers Applied to 2×2 Process

- L is adjusted by PID controller to maintain composition of D at its setpoint.
- Steam flow is adjusted by PID controller to maintain composition of B at its setpoint.

Coupling Effect of Loop 2 on y Process1

Example of Coupling Process

- L is adjusted to maintain the composition of D which causes changes in the composition of B.
- The bottom loop changes the flow rate of steam to correct for the effect of the reflux changes which causes changes in the composition of D.

The Three Factors that Affect Configuration Selection Process

- Coupling
- Dynamic response
- Sensitivity to Disturbances

Steady-State Coupling Process

Relative Gain Array Process

- When l11 is equal to unity, no coupling is present.
- When l11 is greater than unity, coupling works in the opposite direction as the primary effect.
- When l11 is less than unity, coupling works in the same direction as the primary effect.

Numerator of Processl11

Denominator of Processl11

RGA Example Process

D Processc1 = 1.0Dy2 = K21Dc1 = 0.05

D Processc2 = -Dy2/K22 = -0.05/2 =-0.025

( ProcessDy1)coup = Dc2 K12 = -0.025(0.1) =-.0025

Calculation of RGA Process

RGA Calculation for 2×2 System Process

RGA Analysis Process

- RGA is a good measure of the coupling effect of a configuration if all the input/output relationships have the same general dynamic behavior.
- Otherwise, it can be misleading.

Example Showing Dynamic Factors Process

Dynamic Example Process

- Note that the off-diagonal terms possess dynamics that are 10 times faster than the diagonal terms.
- As a result, adjustments in c1 to correct y1 result in changes in y2 long before y1 can be corrected. Then the other control loop makes adjustments in c2 to correct y2, but y1 changes long before y2. Thus adjustments in c1 cause changes in y1 from the coupling long before the direct effect.

Dynamic RGA Process

Dynamic RGA for Direct (a) and Reverse (b) Pairings Process

- Consider the frequency, w, corresponding to desired closed loop response which indicates b better than a

Overall Dynamic Considerations Process

- Pairings of manipulate and controlled variables should be done so that each controlled variable responds as quickly as possible to changes in its manipulated variable.

Sensitivity to Disturbances Process

- In general, each configuration has a different sensitivity to a disturbance. Note that thick and thin line represent the results for different configurations

Configuration Selection Process

- It is the combined effect of coupling, dynamic response, and sensitivity to disturbances that determines the control performance for a particular control configuration for a MIMO process.

Configuration Selection for a C Process3 Splitter

(L,V) Configuration Applied to the C Process3 Splitter

Reflux Ratio Applied to the Overhead of the C Process3 Splitter

Configuration Selection Example Process

- L, L/D, and V are the least sensitive to feed composition disturbances.
- L and V have the most immediate effect on the product compositions followed by L/D and V/B with D and B yielding the slowest response.

Control Performance Process

Analysis of Configuration Selection Example Process

- Note that (L,V) is the worst configuration in spite of the fact that it is the least susceptible to disturbances and the fastest acting configuration, but it is the most coupled.
- Even though (D,V) had an RGA of 0.06, it had decent control performance.
- (L,B) is best since it has good decoupling and the overhead product is most important.

Tuning Decentralized Controllers Process

- When a particular loop is 3 times or more faster than the rest of the loops, tune it first.
- When tuning two or more loop with similar dynamics, use ATV identification with online tuning

One-Way Decoupler Process

Overview Process

- The combined effect of coupling, sensitivity to disturbances, and dynamic response determine the performance of a configuration
- Implement tuning of fast loops first and use a single tuning factor when several loops are tuned together.
- One-way decoupling can be effective when the most important controlled variable suffers from significant coupling.

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