What Have I Learned? – Einstein and the Ultimate Limits for Loop Performance

With all of the advanced algorithms and smart instrumentation available today, we can sometime lose sight of what are the real limits to loop performance. While it doesn’t take an Einstein to figure this out, as a former physicist, I found an interesting analogy.

Einstein’s reasoning that nothing can travel faster than the speed of light lead to incredible insights and revolutionary equations. For example if you substitute the speed of light for velocity into the equation for kinetic energy, you now have the famous equation that relates mass and energy (energy is equal to mass multiplied by speed of light squared). You also end up with a unification of space and time and warping by gravitational fields.

The absolute limit to feedback control system performance is the total dead time in the loop, which is the summation of all the final element, process, measurement, I/O, and controller execution time delays. A feedback control system cannot correct for something it hasn’t seen yet and hasn’t been able to change yet in the process (see “Funny you should Ask a Process Control Engineer” in the Funny Thing E-book). http://www.modelingandcontrol.com/FunnyThing/page-123.asp

The fastest closed loop time constant (Lambda) possible is the deadtime. If you substitute deadtime for Lambda into the controller gain equations for Lambda tuning, you end up with the Simplified Internal Model Control and factored Ziegler Nichols equations for the highest controller gain with a relatively smooth response. This unification of equations for controller gain was documented in Appendix C of New Directions in Bioprocess Modeling and Control. This Appendix also provides the derivation that the performance achieved in terms of integrated absolute error (IAE) for an unmeasured load upset is proportional to reset time and inversely proportional to controller gain. BioprocessModelingControlBookAppendixC

The implications of this for sample delays in terms of there being an additional implied dead time for detuned controllers is explored in Advanced Application Note 5.


The hype of some advanced process control (APC) algorithms may lead one to believe this limit can be violated. Many of the early APC algorithms significantly increased the loop deadtime (See “Advanced Control Algorithms- Beware of False Prophecies in the Funny Thing E-book). While model predictive control (MPC) can potentially help dead time dominant systems, the original execution time (e.g. 1 minute) of separate MPC software packages was so large their applicability was restricted to slow processes. With the advent of the MPC embedded in the DCS, the execution time can be as fast as 1 second which means MPC can be applicable to all but the fastest processes (e.g. liquid pressure control and furnace pressure loops).



Deadtime compensators such as the Smith Predictor can make the PID algorithm think there is no deadtime in the loop. You can get fooled as well if the PID faceplate shows the compensated PV that has the deadtime removed from the consequences of its own actions instead of the original PV. Deadtime compensators allow the user to increase the controller gain. If the deadtime compensation is perfect, the increase in controller gain can be huge. However, many sources of deadtime are variable and unknown.



For PID controllers an underestimate of deadtime can lead to instability if one goes for the gusto of ultimate performance and pushes the limit beyond the original unfactored Ziegler Nichols equation for controller gain. For deadtime compensators and model predictive control, you can also get into some oscillations for overestimates of deadtime.


Finally, there is another limit to control loop performance, signal resolution. You can’t control to a degree finer than the resolution of the final element, measurement, or I/O. The resolution limit of digital devices today is nearly negligible (e.g. 16 bit A/D) but some older DCS (e.g. 12 bit A/D) could cause noticeable stair-steps in the temperature response from wide range thermocouple input cards and the standard input card. It is strange to me that the standard input card for many variable speed drives still uses an 8 bit A/D that significantly restricts the resolution of the final element. Today most of the resolution limit seen in control loops is from control valves. The principal cause is stick-slip and is usually lost in the smoothing afforded by process volumes and the compression of data (except for pH and other high gain processes) unless you have on-off or isolation valves posing as control valves or choose to save money by not buying digital positioners.




Not well recognized is that for PID control of integrating processes, valve deadband causes an implied resolution limit in the PV for a reversal of direction that is the deadband multiplied by the integrating process gain and size of the correction needed in the controller output to balance out the disturbance. Real control valves with digital positioners have a deadband that is less than twice the resolution limit.