Modeling and Control

For a constant flow and set of process operating conditions, is the observed total loop deadtime relatively constant? We know from last week’s blog, the deadtime also depends upon the sensor time constant and hence it’s fouling. Less recognized is that it depends upon whether a step change is made in the controller output versus its setpoint.

The closed loop deadtime (e.g. deadtime in automatic mode) is generally greater than the open loop deadtime (e.g. deadtime in manual mode).

The deadtime from control valve stick-slip and backlash is the valve resolution and deadband, respectively divided by the rate of change of the controller output. For small step changes (particularly for pneumatic positioners), the response time also gets incredibly slow. For a large step change in controller output, the dead time from stick-slip and backlash is zero and the response time is minimal (except for large actuators). Next week, we will discuss some other ramifications of step size.

For a step change in controller setpoint, there is a kick from proportional action (for a PID structure with proportional action on error) and a ramp from reset action. If the kick is not enough to get the valve to move then the loop has to wait on reset action and the chosen closed loop time constant. Thus the deadtime identified for a setpoint change depends upon the controller tuning. Equations 2-47 through 2-50 in the book Advanced Control Unleashed show the development of an equation to estimate the increase in the deadtime from a control valve based on the open loop deadtime. While, these equations are for deadband, they can be used for stick-slip if you consider that half of a deadband is roughly equal to a resolution limit, which is often the case for the best throttle valves (e.g. sliding stem valves with diaphragm actuators). Note the presence of a detuning factor Kx that is approximately the inverse of the Lambda factor (the ratio of closed loop to open loop time constant).

For adaptive controllers or on-demand tuning software that rely upon setpoint changes, very sluggish initial tuning or an unnecessarily large closed loop time constant specified will lead to a larger identified deadtime and overly conservative settings that tends to keep the loop deadtime larger and hence the controller detuned.

Dead time is bad news because the controller has no effect on the process during this time interval. The minimum peak error for a disturbance is basically how far the process is driven away from set point during the total loop deadtime by the process upset. The minimum peak error from a load upset can be estimated as the average rate of change of the process variable multiplied by the dead time. The minimum integrated error is proportional to the deadtime squared. These relationships for peak and integrated error are developed in Equations 2-38 through 2-44 of Advanced Control Unleashed. If this is not enough to get you to rush out and buy a copy, I am offering for a limited time a $0.25 rebate (generous considering the royalties are donated to a university). Just send me your receipt in a self-addressed and with enough postage to get to my secret island hideaway.

What steps can be taken to make the real loop deadtime step forward? Last week we found that a step made in the controller setpoint rather than its output for a controller gain less than one increases the deadtime because of the time it takes for the controller output to work through the valve deadband and resolution. Slower tuning makes the deadtime larger. Subsequent increases in Lambda factors for additional robustness can get the user into a downward spiral in terms of loop performance (slower tuning -> larger deadtime -> slower tuning -> larger deadtime).

Additionally, small steps in the signal to control valves, particularly those with pneumatic positioners, have a dramatic effect on valve response time and hence loop deadtime. The following tests show that the response time of positioner can increase from 1 second to 100 seconds when the step size is decreased from 10% to 0.2%. While you may not be making such a small change in controller output, consider that a 1% change in setpoint to a controller with a 0.2 gain translates to a 0.2% step in the signal to the valve.

Effect of Step Size on Positioner Response

For slow loops like tank level and temperature, the time it takes for a change in the process variable to work through the resolution limit or noise band of the measurement creates another increase in deadtime. For a 1980s vintage DCS with 12 bit A/D (one sign bit) wide range thermocouple cards, the resolution limit of about 0.25 degrees adds significant deadtime besides loop A/D noise. The additional deadtime can be estimated as the measurement resolution divided by the rate of change of the process variable. For a temperature loop changing 0.05 degrees per minute from a step change in controller output, a resolution limit of 0.25 degrees can add 5 minutes of loop deadtime.

I remember trying to use an auto tuner on level loops on large tanks and waiting what seemed like forever for the measurement to get out of the noiseband. I quickly realized that I needed to take larger steps to drive the level faster before the auto tuner or my brain timed out.

Where tight control is needed for slow level and temperature loops, the controller is normally tuned with a controller gain much larger than one. This is a tip that the step changes in the controller output should be large so you are not waiting till the cows come home to see the process variable stir. For more Texas talk, see my Control Talk Column “Puzzler Roundup” in the July issue of Control magazine.

If I had just 12 wishes for practices to improve pH control systems they would be:

1. Review and improve electrode design (glass thickness, shape, and formulation and reference type and electrolyte)

2. Check and improve electrode location for dead time (transport delay) and velocity

3. Review and improve calibration practices

4. Check and improve upstream loop tuning and valve resolution to reduce size and speed of pH disturbances

5. Verify valve resolution by small step tests

6. Improve valve resolution (add digital positioner, reduce seating and packing friction, and verify positioner feedback mechanism to ascertain it actually tracks internal trim position)

7. For great reliability, maintainability, and onstream time, consider middle signal selection of 3 electrodes

8. Check dip tube and injector design for time delay for emptying and refilling upon closure of reagent valve (consider reagent injection into recirculation line)

9. Check for fully filled reagent pipeline downstream of valve

10. For flow disturbances and startup, consider flow feedforward

11. For steep titration curves, consider linear reagent demand control

12. If tank is not mixed well enough for pH, consider adding inline system upstream or in a recirculation line

I originally considered the first item further down on the list because it takes time for a plant trial to confirm improvements for radical changes in electrode construction but then I considered the system is only as good as the measurement and the trial could be started while the other items are pursued. Electrode construction is particularly important for high temperature, high ionic strength, low water content, high pH, and low pH streams.

Is smooth good? We like smooth trend charts but is that what is really going on in the process? Do we want a smooth talking measurement or the straight story?

My first clue dates back to a startup of a world class intermediate plant when smooth temperature recordings were traced back to sand in thermowells from when the pipelines were sandblasted during construction. Then, in a downstream plant a report came in that temperature sensors on extruders were to be now installed in large blocks of metal rather than the melt because the trends were smoother. Many years later in a lab, a biochemist proudly showed how he had smoothed out his temperature recording on the bioreactor by partially retracting the sensor in its thermowell. Finally, I heard horror stories about thermocouples installed in glass lined thermowells on exothermic reactors.

The concern is not restricted to temperature. Rugged (thick glass) and most high temperature electrodes are extremely slow. pH electrodes installed in overflow lines and behind baffles in a vessel have an environment so still that process buildup makes the electrode smooth out changes even when the flow restarts or the agitator speed is increased. Just a 10 millimeter film on a pH electrode can increase its response time from 10 seconds to 100 seconds. Coated electrodes are slow electrodes. Multiple electrodes should expose the foul up but then again the smoothest response I have seen was for 3 electrodes all installed with their protective caps still on.

The easiest way to slow down a measurement is to increase the filter time constant in the DCS. Here the sky is the limit particularly for pressure systems that blow their rupture disks. For some fast gas pressure systems, putting in a faster transmitter will make the trend recordings look worse even though the pressure loop is doing a better job because it is seeing the disturbances better.

For pressure, flow, and inline composition and temperature control, the measurement time constant is probably already the largest time constant in the loop. An increase in this time constant due to coatings or filter times not only makes the trend chart smoother but allows the user to increase the controller gain which furthers the deception. You and the controller are seeing an attenuated version of the real world.

Other time proven ways to make trend charts look smoother to impress friends and relatives is to increase the process variable scale range, decrease the time scale range, and increase the compression, update time, and exception trigger for data reporting.

A smooth loop could be good news or bad news, which leads me to my Top Ten List.

Top Ten Good News Bad News

(10) The good news is that smart instrumentation has been approved. The bad news is it is a dumb installation.
(9) The good news is that the control valves are not oscillating. The bad news is the loops are all in manual.
(8) The good news is the new project manager is a process control engineer. The bad news is you are the project manager.
(7) The good news is that all the process variables are drawing a straight line. The bad news is they are off scale.
(6) The good news is that digital positioners have been added to all of the control valves. The bad news is the position measurement is a “smooth talker”.
(5) The good news is that the loops are no longer oscillating in automatic. The bad news is the plant is shutdown due to the loops being in automatic.
(4) The good news is that your group’s name has “advanced” in it. The bad news is the name is “Advanced Aged Engineers”.
(3) The good news is that you have reached the level to work through “others”. The bad news is there are no “others”.
(2) The good news is that you will have a creative new office. The bad news is it has virtual walls.
(1) The good news is that you have been offered a retirement package. The bad news it is a gift certificate.

We wind up our series on measurement and valve dynamics with the timely question do we want a large time constant anywhere?

A large time constant in the measurement or valve slows down what the controller can see and manipulate, respectively. A large time constant in the process slows down the effect of disturbances at the input to the process. It gives a chance for the controller to catch up. In fact the ultimate integrated absolute error is proportional to the dead time squared divided by the process time constant. Is this the whole answer?

The process time constant must be downstream of the manipulated variable otherwise this process time constant acts to slow down the effect of the controller’s reaction to the upset similar to a slow valve. You can spot a slow valve or large intervening slow process time constant by a fast initial excursion from a disturbance followed by a slow recovery.

We have been talking about open loop time constants (time constant for an output change for a controller in manual). There is also a closed loop time constant (time constant for a set point change to a controller in automatic). We may want a fast closed loop time constant if this loop is a critical loop (e.g. reactor pressure) that doesn’t upset other loops or this loop is a secondary loop (e.g. flow) in a cascade control system. If the action of this loop upsets other loops, then you can reduce the interaction by increasing the closed loop time constant of the loop.