There is a chance we can all get on the same page as to what really is going on in a process’s dynamic response. There is a lot of confusion that can be resolved if we have a fundamental understanding of the source of process dynamics and how to represent it. Such an understanding can improve process, mechanical and automation system design and better diagnostics and maintenance.
We start with the need to understand the change in the controlled variable for a step change in manipulated variable when the controller is not in automatic or cascade or supervisory mode. It is critical to realize that the PID algorithm works in percent of the controlled and manipulated variable in 99% of the industrial process controllers and the PID gain is without dimensions. There are a few PID algorithms in a few programmable logic controllers (PLCs) and some academic publications that work in engineering units of the controlled and manipulated variables with direr consequences. For example, changing flow measurement units for a PID algorithm working in engineer units while keeping the same flow span from a flow rate per second to flow rate per hour requires changing the PID gain settings by a factor of 3600.
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Open loop gain
The gain needed to determine the PID tuning is best termed the open loop gain. It is the product of the manipulated variable, process and measurement gain that can be estimated as the change in controlled variable percent divided by the change in manipulated variable percent. For a control valve or variable frequency drive, the manipulated variable gain is the slope of the installed flow characteristic plotted versus percent controller output. For a cascade loop, it is the secondary loop setpoint span divided by 100%. The process gain is the change in process variable for a change in the manipulated variable. The measurement gain is 100% divided by the measurement span for a linear measurement. For self-regulating processes, open loop gain is without dimensions since it is the ratio of the final change in controlled variable in percent divided by the step change in manipulated variable in percent. For integrating processes, open loop gain units are one per second since it is the ratio of the rate of change of the controlled variable in percent per second divided by the step change in manipulated variable in percent.
It is common for the open loop gain to be referred to as the process gain. I prefer the term open loop gain, so we are aware of the contributing factors of the controlled and manipulated variable spans and for valves and VFDs, the nonlinearity of the installed flow characteristic.
A tricky part to getting the process gain contribution to the open loop gain right is realizing how the engineering units (eu) change depending upon whether the process has a self-regulating response or an integrating response where the process variable settles out at a final value or ramps, respectively for a step change in the manipulated variable. For a self-regulating process, the process gain for manipulated flows is the controlled variable eu per second divided by manipulated variable gain. The subsequential multiplication of the process gain by the manipulated flow gain in eu per second per percent output cancels out the seconds giving a result in process variable eu divided by percent controller output. The final multiplication by the measurement gain that is 100% divided by measurement span in process eu gives an open loop gain in controlled variable percent divided by manipulated variable percent giving a dimensionless open loop gain.
For an integrating process for instance by manipulating flow to control level, there is no seconds in the process gain and thus no cancellation of seconds from manipulated variable gain resulting units one per second for the open loop gain. Note that open loop gain of near-integrating processes has the same units as those for true integrating processes since this open gain is obtained by dividing the process gain by the process primary time constant. This conversion to a near-integrating process tuning rules saves a lot of time in a test for tuning since you just need to see the max excursion rate well before the inflection point in the open loop response. More importantly, it facilitates the use of aggressive integrating process tuning rules including derivative action to help correct load disturbances that are changes in a process input. Note that much of the control literature mistakenly shows disturbances as being on the process output rather than on a process input. This leads to false conclusion about the value of a slower primary process time constant and tuning rules misguidance. Process output disturbances are rare and are usually associated with a process measurement problem.
Time constants
Often, literature leads one to believe there is one process time constant. In fact, there are many process time constants due to mixing, mass transfer, and heat transfer, and many instrumentation time constants due to sensor lags, damping settings, signal filters, and valve response time. Hopefully, the primary time constant is in the process and is large so it slows load disturbances and enables a large PID gain especially when response is computed as near-integrating. The ultimate limit to the peak and integrated error is inversely proportional to the primary process time constant. While a large process time constant does slow down a setpoint response, setpoint feedforward and a setpoint lead-lag can significantly help. Also, set point changes in primary loops are occasional while load disturbances are frequent.
Furthermore, load disturbance feedforward gain and dynamic compensation is challenging and many load disturbances notably in stream composition are not measured eliminating the opportunity for a feedforward.
The conversion factor for the equivalent dead time from time constants smaller than the largest time constant increases as the ratio of the smaller to largest time constant decreases. While there are figures that show the conversion factor, I generally advocate simply including all constants smaller than 20% of the primary times constant as equivalent dead time, since we usually missing sources of deadtime.
Deadtime
There are many sources of deadtime. Dynamic simulations typically include transportation delays but often are missing mixing delays (e.g., turnover times) and frequently are neglecting measurement update rates, I/O scan rates, algorithm execution rates, and valve response delays and the equivalent dead times from measurement and final control element lags and of course, filters. An interesting complication is the valve response dead time that does not show up for a step change larger than valve resolution limit and lost motion. There is also a slower response for small step changes for some positioners with poor sensitivity. You can see why the deadtime is usually significantly underestimated. The consequences are severe, since the controller can’t see to do anything during the total-loop deadtime. The ultimate limit to the peak error and integrated error are proportional to the dead time and the deadtime squared. Also, the ultimate period and hence the best reset and rate time is proportional to the total loop deadtime.
Deeper understanding
I encourage you to invest a few hours and take a deep dive into the sources of process dynamics that would show there was some advantage to taking a Calculus course. It is going to bring up memories of how to set up and solve simple differential equations. Fortunately, only short ordinary differential equations are used where the factor multiplying the process variable derivative is the primary time constant (left side of equation), the factor multiplying the manipulate variable input is the process gain and the subtraction of process variable (right side of equation) is the negative feedback for a self-regulating process. If there is no subtraction of the process variable, the process response is integrating (e.g., level, gas pressure, and batch loops). If the process variable is added instead of subtracted, the process response is runaway due to positive feedback (e.g., temperature loops in highly exothermic reactors). Please invest the time in reading Appendix F – First Principle Process Relationships from my book Tuning and Control Loop Performance Fourth Edition with some improvements to the engineer units. It can be career-altering.
The following excerpt from Appendix F shows the dynamic response from ordinary differential equations where Y is the process output (controlled variable) and X is the process input (manipulated variable).
We can identify process dynamics in the material or energy balance ordinary differential equations.
If the sign of the unity coefficient of the process output on the right side is negative, the process has negative feedback. As the process output changes, the negative feedback slows down and eventually halts the excursion of the process output at its new steady state when it balances out the effect of the process input and the disturbances.
τp dY / dt = Kp X-Y
The integration of this equation provides the time response of a change in process output (ΔY) for a step change in the process input (ΔX). The step occurs at t=0.
ΔY = Kp * (1-e-t/τp) * ΔX
If the process output does not appear on the right side, there is no process feedback. As the process output changes, there is no feedback to slow it down or speed up, so it continues to ramp. There is no steady state. The ramping will stop only when X is zero, which occurs when the manipulated variable is equal to the load.
dY / dt = Kip * X
ΔY = Kip t ΔX
If the sign of the unity coefficient of process output on the right side is positive, the process has positive feedback. As the process output changes, the positive feedback speeds up the excursion unless disturbances counteract the effect of the process input and output.
τp1 dY / dt = Kp * X+Y
ΔY = Kp * (et/τp1-1) * ΔX
