# It's the Resolution, Stupid! Part 1

## Do You Wonder What all the Decimals are for, or Where They Went? Do You Know why You Should Care?

07/21/2005

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By Cecil L. Smith, PE

The traditional use of current loops as inputs to control systems is being replaced by measurement devices and/or input systems that provide the measured value in engineering units. In the era of digital technology, this is indeed appropriate. Unfortunately, the resolution of the digital values is often less than what became the norm for current loop inputs; e.g., a resolution of approximately 1 part in 4000.

Normally such a resolution is far beyond what is required on displays to the process operator, management reports, etc. The value chosen for the resolution is usually based on two factors:

1. How many digits after the decimal point do operators really need to see?
2. What is the accuracy of the measurement device itself?

This article examines this issue from the perspective of regulatory control; more specifically, the impact of the resolution of the measured value on the performance of the proportional-integral-derivative (PID) controller. Control calculations often need a greater resolution than is required for data presented to humans. Furthermore, the performance of regulatory control actually depends on the repeatability of the measurement device, not its accuracy.

#### What's the Impact?

The impact of poor resolution on the three modes is as follows:

Proportional mode: Causes the output to change abruptly from one value to another.

Integral or reset mode: Not significantly affected by resolution.

Derivative mode: Causes pulses or 'bumps' to appear in the controller output.

#### Pressure Control Example

This example in Figure 1 is a simple pressure control application in a reactor. One of the reactants is supplied via a gas. This material dissolves in the liquid within the reactor, from which it reacts with one or more components of the feeds. The objective of the pressure control loop is to maintain a constant pressure within the reactor by supplying gas to replenish the gas being consumed by the reaction.

As part of a general instrumentation and controls upgrade, an electronic analog controller was replaced by a controller implemented in a digital system. The measured variable for the pressure is still provided via a current loop. However, the input module within the digital system converted the current loop input into a digital value in engineering units. The result is a value in engineering units but expressed as an integer variable with the decimal point at an understood location.

The measurement range for the reactor pressure was 0 to 10 psig. Expressing the pressure to 0.1 psig was deemed satisfactory for operator displays. The input module was configured to report the pressure to 0.1 psig, which meant that an integer value of 74 is understood to be 7.4 psig.

There are definitely merits in performing the engineering units conversion in the input module. One is that the hand-held monitor used by the instrument technicians would display the value in engineering units. The value displayed on the hand-held monitor is exactly the same as the value of the graphic displays. In most cases, the instrument technicians are able to diagnose I/O problems without any knowledge of the digital control system.

All seemed quite satisfactory until the pressure loop was commissioned. The digital system exhibited a characteristic that was never observed with the conventional controller. Specifically, the output to the pressure control valve did not change in a smooth manner. Instead, the output to the valve would change abruptly from one value to another.

This is one consequence of poor resolution. With the input module configured in this manner, an input value of 0 means 0.0 psig, and an input value of 100 means 10.0 psig. This input has a resolution of 1 part in 100. The accuracy of modern pressure transmitters is far superior to this, and their repeatability is even better.

Why does this lead to abrupt changes in the controller output? Let's examine the proportional mode calculations. The proportional mode equation is actually a proportional-plus-bias equation that can be expressed as follows:

M = KcE + MR

where M  = Controller output, %
Kc= Controller gain, %/%
E = Control error, %
= PV – SP for a direct acting controller
= SP – PV for a reverse acting controller
Mc= Controller output bias, %
PV = Process variable or measured variable, %
SP = Set point or target, %

This is the position form of the proportional mode equation. The impact of resolution is more evident when the equation is expressed in the incremental or velocity form:

∆M = Kc∆E

where  ∆M = Change in controller output, %
∆E = Change in control error, %

The product chemists specified the pressure for reacting conditions; changes were not allowed. Therefore, changes in the control error can only be the result of changes in the PV.

With a resolution of 1 part in 100, the smallest change provided by the input module was 1% of the span. The smallest change in the proportional mode output is the controller gain Kc times the smallest change in the control error. With the controller gain set at 5%/%, the smallest change in the output of the proportional mode equation is (1%) x (5%/%) = 5%. Such changes appear directly in the controller output.

There are two options to achieve smaller changes in the output of the proportional mode equation:

1. Reduce the controller gain Kc The controller gain is a tuning parameter whose value should reflect the characteristicsof the process. For a responsive pressure loop, a value of 5%/% for the controller gain is not unreasonable. Reducing the controller gain also sacrifices loop performance, with the unfortunate result being that the performance of the digital system is inferior to that of the conventional controller.
2. Improve the resolution of the measured value. In this application, the input value could easily be changed to provide the pressure to 0.01 psig. Now the range of the input value is 0 (or 0.00 psig) to 1000 (or 10.00 psig), which provides a resolution of 1 part in 1000. The smallest change in the proportional mode output is now 0.5%, which is closer to what most valves are actually capable of implementing.
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