«

»

Nov
27

Getting Rid of Dead Time – Beam Me Up Scotty

How can we get rid of dead time in our loops so we can be rich and famous by Friday? PID controllers with dead time compensation are reported to eliminate dead time in terms of a controller seeing the effect of changes in its controller output. For set point changes where all the controller needs to be concerned with is how its output responds to a new set point, the results are impressive for an exact knowledge of the process dead time. However, for unmeasured load disturbances at the process input, the only way to eliminate dead time other than an improvement in the plant or control system design is to accelerate the control system to the speed of light. So unless you have Scotty and Warp Drive on the Starship Enterprise, you are stuck with the dead time from the process equipment, piping, control valves, instrumentation, and digital devices. A dead time compensator can offer some improvement in load rejection by facilitating more aggressive tuning of the PID but with a considerable risk of oscillations from an inaccurate dead time.

If you don’t have time for the details or just want to cut to the chase, here are the recommendations

(1) First improve the PID controller tuning before even considering dead time compensation. Setting Lambda equal to the maximum dead time (Lambda factor equal to the maximum dead time to time constant ratio) is effective for load disturbances at the process input if there are no extenuating circumstances.

(2) Add feedforward control whenever it is possible to measure or infer load disturbances at the process input.

(3) If there is economic justification for further improvement and the dead time can be updated within 25% accuracy for varying operating conditions, trial test and closely monitor a PID with delayed external reset for low dead time to time constant ratios.

(4) For loops with high dead time to time constant ratios, multiple manipulated variables, interactions, or constraints, consider model predictive control.

The ultimate performance achievable in terms of load disturbance rejection depends upon the dead time. In the “Theory” section of Chapter 2 of Advanced Control Unleashed equations are developed that show the minimum peak error is proportional to the dead time and the minimum integrated error is proportional to the dead time squared for unmeasured load upsets. How close the actual performance of a control loop comes to this ultimate performance depends upon PID structure, tuning, and enhancements. This blog focuses on the effect of variations in dead time on the performance and robustness of dead time compensation as an enhancement and Lambda as a tuning rule for disturbance rejection. The two predominant methods of dead time compensation studied here are the Smith Predictor PID and the PID with a delayed external reset.

The Smith Predictor was extensively documented in the 1970s. It provides a new controlled variable that is the response of the process variable to its controller output without dead time. It requires entry of three parameters commonly known as process gain, dead time, and time constant. The Smith Predictor uses these parameters to create models of the process from the controller output. In its most documented form, the Smith predictor subtracts a model of the process with dead time from a model of the process without dead time and adds the net result to the measured process variable to create a new controlled variable. If the model is perfect, the new controlled variable has zero dead time in terms of the controller seeing the effect of its own controller output. Since the maximum allowable controller gain is inversely proportional to dead time, the controller gain can theoretically be increased without limit for a perfect model provided you ignore extenuating circumstances, such as loop interaction, measurement noise, and final element dead band and resolution. One of the practical issues with the Smith Predictor is that the new controlled variable of the PID is no longer the actual process variable. The original process variable must be restored for the operator interface to the PID. Also, performance monitoring or trending must look at the original process variable rather than the new controlled variable used by the PID. Terry Blevins proposed in the 1979 ISA paper “Modifying the Smith Predictor for an Application Software Package” a multiplicative and additive correction of the process variable to deal with changes in the slope (gain) and intercept (bias), respectively in the process model.

The PID with a delayed external reset was informally presented in the 1980s and published in the early 1990s. It simply consists of putting a dead time (DT) block in the external reset. This method only requires that a single parameter commonly known as process dead time be entered as the dead time in the DT block. Terry Blevins documented in the early 1990s how the Smith Predictor for a particular Lambda tuning reduces to this PID with a delayed external reset.

The results presented here show that for a perfect model and the same controller tuning the PID with a delayed external reset performed better for processes with a small dead time to time constant ratio (time constant dominant), whereas the Smith Predictor performed better for processes with a large dead time to time constant ratio (dead time dominant). The Smith Predictor did not do as well for small dead time to time constant ratios because the control error seen in the controlled variable by the PID is much smaller than the actual control error in the process variable. In both cases, the improvement was not as impressive as the improvement gained from setting Lambda equal to the dead time rather than the time constant. Surprisingly the improvement in load disturbance rejection from dead time compensation was greater for processes with small dead time to time constant ratios. This goes against the conventional wisdom that the best opportunity for dead time compensation is for dead time dominant loops. The results can be explained in terms of the ultimate limit for performance of dead time dominant loops being lower. The reduction in the peak excursion from more aggressive tuning settings is negligible for dead time dominant processes because the peak error is essentially the open loop error.

Another startling result was how quickly a Smith Predictor erupted into rapidly growing oscillations in the controller output when the model dead time was more than twice the actual process dead time. The fast full scale oscillations in the controller output resembled on-off control. While it is relatively well known that dead time compensators are sensitive to model mismatch, the effect was expected to be gradual and thought to be more in terms of a model dead time being too small. The concern for rapid deterioration for a model dead time being too large was raised in Good Tuning – a Pocket Guide and was documented for model predictive control in Models Unleashed. While a PID with delayed external reset is also adversely affected by a dead time mismatch in both directions, this PID develops a small amplitude high frequency dither rather than a full scale oscillation in controller output for an excessively high model dead time. The consequence is less severe and may be adequately handled by the addition of a small dither filter inserted in the PID controller output, but this was not tested.

PID controller tuning for self-regulating processes without extenuating circumstances can develop oscillations when the identified (model) process dead time is too small. PID controllers with dead time compensation and model predictive controllers can develop oscillations when the identified (model) dead time is too large as well as too small.

In order to get the performance benefit from dead time compensation, the PID must be tuned more aggressively. In other words, a PID with dead time compensation will perform the same as a PID without dead time compensation if they are tuned the same. While the improvement in integrated absolute error (IAE) for load upsets from more aggressive tuning (higher controller gain and lower reset time) can be accurately estimated for a regular PID, the equation does not work well for a dead time compensator. Furthermore, a dead time compensator soon reaches a point of diminishing returns. For example, the improvement in load rejection of a Smith Predictor from a controller gain that is quadrupled may not be noticeable whereas for a regular PID, it normally results in a four fold reduction in IAE. It is important to remember there is a tradeoff between performance and robustness for any feedback controller in that as you make controller tuning more aggressive to improve load rejection you make the controller more sensitive to changes in the process gain, dead time, or time constant.

A nonlinear gain from the installed characteristic of a control valve has been widely discussed. However, the nonlinearity of the process gain of the temperature or composition response is the inverse and consequently the combined effect is less than documented when these loops directly manipulate a control valve. The variability of dead time is often larger than the variability of the process gain or time constant because the dead time is inversely proportional to a rate (e.g. flow rate or pumping rate or rate of change of a signal) and has many different sources (e.g. valve deadband or resolution, piping transportation delay, mixing delay, process lags in series, sensor lags, signal filters, and discrete communication or scan intervals). Thus, it is problematic to compute the dead time accurately enough to get the benefit of a dead time compensator.

In all of the following test results AC1 is always an uncompensated PID with Lambda equal to the process time constant (lag), which is equivalent to a Lambda factor of one.

The first set of test results illustrates the effect of different tuning. Here AC2 is an uncompensated PID with Lambda equal to the process dead time (delay), which is equivalent to a Lambda factor set equal to the dead time to time constant ratio.

Tuning Rule Test 1

The second set of test results shows how well a Smith Predictor can do. Here AC2 is a Smith Predictor PID with the gain doubled and the reset time halved after Lambda has again been set equal to the process dead time. In other words, this AC2 has twice the proportional and integral action of the uncompensated AC2 in the first set of test results.

Smith Predictor Test 2

The third set of test results shows how well a PID with a delayed external reset can do. Here AC2 is a PID with delayed external reset with the gain doubled and the reset time halved after Lambda has again been set equal to the process dead time. In other words, this AC2 has twice the proportional and integral action of the uncompensated AC2 in the first set of test results.

Delay Comp Test 3

For discussion of the test results and configuration, request from me a copy of the Advanced Application Note 003 titled “Compensation of Dead Time in PID Controllers.”