The interpretation is:
Î»i,j = 1.0 no interaction with other loops
Î»i,j > 1.0 reduced control effectiveness
Î»i,j < 1.0 interaction will extend the loop period and raises gain
The relative gain can be computed by plant test or by using a process model. Î»1,1 is slightly greater than 1.0 for the problem as described. Here is why: The numerator is âˆ†Flow/âˆ†Valve with the remaining 5 valves held constant. The numerator is slightly more than the denominator which is âˆ†Flow/âˆ†Valve with the remaining 5 other controllers regulating flow. Lets say we evaluate the relative gain at a typical operating condition with FV1 = 50% and increasing to 55%.
This movement will increase the flow thus the derivative is a positive number. In the numerator the plenum pressure will drop somewhat reducing the flow to the other 5 ducts. In the denominator the flow will drop less as the other 5 controllers will further reduce the plenum pressure in order to keep their flow at setpoint, thereby reducing the control effectiveness of the first controller. From the perspective of FC1 there is a reduced loop gain when FC2...FC6 controllers are in Automatic.
Of course we made huge assumptions here and we could be way off base. It could be that some other interacting control loop is causing the plenum pressure to oscillate and we really need to be looking at that loop. Or it could be tuning problems, valve problems, or process problems. We also do not know specifically how good the control needs to be. Some loops oscillate their whole life and still are acceptable.
We assumed 6 simple single control loops without any known installation or application problems. The simplicity of 6 single loop controllers is appealing, and it may be quite possible to simply retune them. Adaptive tuning might be helpful for initial tuning if the remaining 5 controllers are in manual mode, but as we can see from the RGA analysis putting the other 5 controllers in AUTO will impact the loop gain and thus the optimal tuning. Tuning features might not be available depending on PID software or availability of third party tuning packages. The controller tuning will be especially important here and care should be taken to insure the tuning is stable at all loads and that an oscillation in one control loop will not excite the other loops into a 6-piece orchestra of oscillations.
The multivariable approach would surely handle the interaction problem, but just how do you make a step test with FV1 while keeping FC2....FC6 in tight control. It would seem that you almost need to fix the control problem before you can do the step tests that will lead to fixing the control problem. This is a common issue when planning the step tests. The way we have assumed the problem, it does not look like we have the hardware or software to jump into the multivariable controller solution. Additionally, what happens if the multivariable controller is out of service? Say one flowmeter or valve is failed, can the controller continue to work with the remaining 5 loops? Or is there a fallback scheme? Our potentially false assumptions could mean that our multivariable control solution is off base. Based on our assumptions a multivariable controller structure with 6 CV's (FT1...FT6), 6 MV's (FV1...FV6) and possibly 1 FF (Plenum Pressure) would seem the most likely structure. Decoupling is inherent in the multivariable solution assuming a good feedforward model is captured in the step testing.
Of course cost, operator training, and support over the life of the plant may need to be factored in. 6 simple loops appears to be the easiest to understand and support if it can be made to work.
In that line of thinking, a few other things that could help are: to characterize those butterfly valves so that the controller output is roughly linear to the flow through the valve (this applies to either multivariable or PID control). If we have a Plenum pressure measurement, we could implement feedforward compensation (in the PID) so that when the plenum pressure falls, the valves all open appropriately, even before the flow measurement responds. This would decouple the system. In some PID algorithms, connecting the feedforward calculation into the "BIAS" will allow for smooth Auto/Manual transitions making the feedforward easier for the operator to manage. Many single loop controllers could accommodate this addition with relatively minor wiring and software configuration changes. There are a number of ways to decouple the PID controllers, implicitly by simple tuning or explicitly by wiring feedforward signals or by decoupling.
If the interaction is via the plenum pressure, why not put in a pressure controller to keep the pressure constant. It may manipulate the blower, but could add hardware and increase energy consumption a bit. If the pressure controller is fast compared to upsets or the 6 flow controllers, it would surely end the interaction problem.
This little "what if" anaysis shows a few of the approaches that might be considered.
Keeping the solution simple, but effective, is the real goal.
e can’t believe our vortex shedding flowmeter! We are having issues with the believability of our current vortex shedding flowmeter used in NH3 service when compared to a mass flowmeter. The vortex shedder has been compensated with square-root P/T compensation. The flow valve is in manual, the line pressure is constant, with dropping temperature only. But the reading from the vortex shedding flowmeter decreased and the mass flow increased. Looking at trend data, the NH3 vaporizer pressure dipped. Because the vortex meter uses the calculation: (volume)=M(mass)/D(density), where D is inversely affected by temperature, we have come to the conclusion that the vortex meter is not compensating for increased M that the vaporizer is pushing into the line. How would I compensate for this in a formula?
Insert and Orifice
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Determine the individual flows by inserting an orifice, one at a time in each channel and measure. Calculate an orifice size for each duct that will provide constant flow at constant air pressure. If the flows are not constant, this simple solution will not work, and flow control and actuated dampers will be called for.