It's the Resolution, Stupid! Part 2

The consequence of this is a large value for the controller gain. With a measured variable span of 20°C, the value of the controller gain of 5%/% would be a reasonable value for a temperature controller. But with a measured variable span of 100°C, the value of the controller gain would be 25%/%. Such large values for the controller gain are occasionally encountered in control loops, and in most cases, this is the result of a measured variable span that is wider than required for control.

In any loop with a large value for the controller gain, careful attention must be paid to the resolution. Consider the following for a temperature control application:

• Measured variable span of 0°C to 100°C
• Input module provides the temperature in engineering units, but as an integer value with the temperature expressed to 0.1°C
• Controller gain is 25%/%

The smallest change in the measured input is 0.1°C, which is 0.1% of the input span of the measured variable. With a controller gain of 25%/%, a change of 0.1°C in the temperature would cause the controller output to change by 2.5%. In reactor temperature control applications, this is likely to attract some attention. Figure 1 illustrates the performance of a PI controller for a disturbance to a temperature loop. The resolution on the temperature measurement is 0.1°F, which is 0.1% of the measurement span of 100°F. With a controller gain of 2.6%/%, a temperature change of 0.1°F (also 0.1% of span) causes the controller output to change by 2.6%. Such abrupt changes are clearly visible in the controller output, but otherwise the resolution has little impact on the performance of the loop.

However, the impact of the resolution is much greater on the performance of the PID controller illustrated in Figure 2. The addition of derivative has reduced the maximum departure from set point (150°F) from 2.3°F with PI to 1.2°F with PID, which in applications such as reactor temperature control is very appealing. The tuning coefficients are reasonable:

1. The gain for PID is 50% higher than the gain for PI.
2. The value of the derivative time is less than 1.0 minute and about one-tenth of the reset time.

However, there are numerous distinct "bumps" in the controller output. It is also evident from Figure 2 that the bumps are associated with 0.1°F changes in the temperature. The controller has a derivative mode smoothing factor of 0.1 (which is a derivative gain limit of 10).

The following changes will reduce the"bumps," but at the expense of performance:

1. Reduce the controller gain. This will increase the maximum departure from the set point.
2. Reduce the derivative time. The controller gain will also have to be reduced. These reduce the benefits of PID control.
3. Increase the derivative mode smoothing factor. Not all controllers permit this and, even in those that do, this will also reduce the benefits of PID control.
4. Add smoothing to the process variable input for the temperature. Addition of smoothing to temperature measurements where the probe is inserted into a thermowell is always suspicious (the thermowell normally provides far more smoothing than is required).
5. Improve the resolution of the input from the temperature measurement device.

The latter is the only viable approach.

Traditional installations relied on current loop inputs. If an A/D converter with a resolution of 1 part in 4000 is applied to a current loop input with a span of 100°F, the resolution in engineering units is 0.025°F. The performance of the PID controller is illustrated in Figure 3. The bumps are still present, but they are much smaller.

Resolution with A/D Converters

The customary industrial practice with A/D converters gives resolutions such as the following:

1. An input of 4 mA gives a raw value or "count" of 0; an input of 20 mA gives a raw value of 4096. The resolution is 1 part in 4096.
2. An input of 4 mA gives a raw value of 0; an input of 20 mA gives a raw value of 4000. The resolution is 1 part in 4000. One advantage of this approach is that some over-range is provided. That is, inputs up to almost 20.4 mA can be sensed. If the A/D is bipolar, an over-range on the 4 mA end can also be provided.
3. An input of 4 mA gives a raw value of 800; an input of 20 mA gives a raw value of 4000. The resolution is 1 part in 3200. There is some sacrifice in resolution, but some over-range can be provided at both ends, even with a unipolar A/D.

PID Control With Resolution of 0.025ºF

Figure 3: The Higher Resolution Improves the Control. The Bumps Are Still Present, But They Are Much Smaller.

There are a number of other factors that go into the decision as to which approach to use. But for purposes here, let's consider the resolution to be 1 part in 4000 for an input range of 4 to 20 ma.

A number of inexpensive modules are available that convert the input from a thermocouple or RTD to a 4 to 20 mA signal. Most of these products provide linearization and, for thermocouples, reference junction compensation. The measurement range on these modules often reflects the characteristics of the sensor. Consequently, for a type J thermocouple, the lower range value is often -200°F and the upper range value is 1320°F. The resolution of this input is

This is not as good as the input module that provides the input value as the temperature in °F to 0.1°F. To obtain better resolution, a narrower span is required for the module that produces the milliamp signal.

Another input application where the resolution must be examined carefully is for inputs from weight transmitters or load cells.

Modern load cells have very high resolutions. For example, a weight of up to 6,000 kg can be indicated to 0.1 kg. This measurement has a resolution of 1 part in 60,000!