For other applications, a constraint controller could also be based on electrical current, pressure, level, density, etc. Here, instead of switching from a controller to a ramp limiter, the control is switched from one controller to another, which is still a linear device. A PID controller is defined as being a linear piece of equipment unless the output is saturated at either 0% or 100%, or any other external device would limit the controller output. On the other hand, a ramp limiter is a non-linear function block. Mathematically, a linear function is defined as K*y = K*x.
When the ramp limiter takes over the PID and prevents it from moving overly fast, then the previous equation becomes K*y K*x+bias and is not satisfied all the way through 0% to 100% any more. As a matter of fact, whenever a rule-based function-- such as “if X then conclude Y; else conclude Z”-- is being used in a control scheme, a non-linearity is inserted. When a non-linearity is present, the behaviour of the closed-loop becomes harder to predict and is definitely different throughout the full range of the controller’s 0% to 100% output. In fact, this type of rule-based mechanism is most likely suitable when the process is so non-linear that an operator can beat the linear controller (PID) by manipulating the output in manual mode. Alternative types of controllers such as Fuzzy logic and Expert System could then be used to achieve better performance.
Making the Transition from One Operating Point to Another Smoother
As for the third solution, which aims to make the transition from one operating zone to another production rate smoother, it is somewhat not as smooth as people would expect it to be. As we saw in Figure 1, the ramp creates an overshoot for a non-self-regulating process and likely produces the same on self-regulating process, depending on tuning’s aggressiveness, when looking at the process value (PV). But now, if we look at the output’s behavior (See Figure 2 below), the pattern exhibits sharp breaks (OUT2, OUT3 and OUT5) and all these three involve a ramp device.
|FIGURE 2: SMOOTHING THE CURVE|
|Transitioning from one operating zone to another isn’t as smooth as it could be. The output’s behavior pattern exhibits sharp breaks (OUT2, OUT3 and OUT5) and all these three involve a ramp device.|
WHAT IS SO awful about having a sharp break? Referring to the fundamental of frequency domain according to Fourier theorem, “Any signal can be synthesized by the sum of sinusoids of different frequencies that are multiple integer (harmonics) of the periodic signal to be reproduced.” Moreover, the more sinusoids of high frequency are added, the sharper the synthesized signal will get. For instance, to generate a square wave of 2Hz, all odd multiple integer frequencies (also called harmonics) of the square wave [ (1X2Hz)+(3X2Hz)+(5X2Hz)+(7X2Hz)+(nX2Hz)…] will be added up together and create the 2Hz square wave. Moreover, the higher =the frequencies, the sharper the corner of those square waves get. Therefore, based on Fourier Theory, during a setpoint ramping those two sharp breaks will generate lots of high frequency harmonics (many integer multiple frequencies), which in return will cause the controller to have a higher risk of oscillation. This is why those abrupt changes are so violent.
Why is the ramp effect worse on a non-self-regulating process type than on a self-regulating one? The non-self-regulating is, in other words, an integrating process. The down side of an integrating process type is that it will create more Lag (i.e., an important phase shift). As a matter of fact, an integrating function causes a phase shift of 90 deg. Therefore, in theory, when using two integrators back to back, a 180 deg phase shift takes place since a perfect oscillator is being produced. In practice, a third integrator function may be required to obtain a perfect oscillator. Since a ramp can be mathematically reproduced by the integration of a step, that ramp could represent that third component needed for the oscillation to occur.
When referring to the first three statements, we began by saying that none of them are well served by the use of a ramp function. Moreover, a ramp will insert non-linearity, which is really not suitable in a regulatory control scheme using PID. As stated previously, a PID controller is by definition a linear piece of equipment; as long as this PID controller will be working in a linear environment, its behavior will be predictable. Finally, a ramp will postpone the setpoint response and make it overshoot. So why would we use those ramp functions to turn something linear to non-linear, when there are many other means to achieve the same goal?
Implementation of a Lead/Lag Filter Within an Hour
Depending on the DCS or PLC, the implementation may vary from one minute to one hour. Some DCSes have this feature as a built-in function in the PID controller. In that case, the only task that is left to do is to specify the Lead value. The Lag will inherit the same value as the integral time of the PID controller. In other cases, most DCSes and PLCs will have a Lead/Lag function available and minimum programming will be required to insert the Lead/Lag block in the control scheme. Finally, if nothing exists, we can create programmatically the whole Lead/Lag tool within an hour.
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