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Steady-State Process Gain

When we speak of a controller’s gain, we refer to the aggressiveness of its proportional control action: the ratio of output change to input change. However, we may go a step further and characterize each component within the feedback loop as having its own gain (a ratio of output change to input change):

Steady-state process gain

The gains intrinsic to the measuring device (transmitter), final control device (e.g. control valve), and the process itself are all important in helping to determine the necessary controller gain to achieve robust control. The greater the combined gain of transmitter, process, and valve, the less gain is needed from the controller. The less combined gain of transmitter, process, and valve, the more gain will be needed from the controller. This should make some intuitive sense: the more “responsive” a process appears to be, the less aggressive the controller needs to be in order to achieve stable control (and vice-versa).

These combined gains may be empirically determined by means of a simple test performed with the controller in manual mode, also known as an “open-loop” test. By placing the controller in manual mode (and thus disabling its automatic correction of process changes) and adjusting the output signal by some fixed amount, the resulting change in process variable may be measured and compared. If the process is self-regulating, a definite ratio of PV change to controller output change may be determined.

For instance, examine this process trend graph showing a manual “step-change” and process variable response:

Steady-state process graph

Here, the output step-change is 10% of scale, while the resulting process variable step-change is about 7.5%. Thus, the “gain” of the process (Note) (together with transmitter and final control element) is approximately 0.75, or 75% (Gain = 7.5%/10% ). Incidentally, it is irrelevant that the PV steps down in response to the controller output stepping up. All this means is the process is reverse-responding, which necessitates direct action on the part of the controller in order to achieve negative feedback. When we calculate gains, we usually ignore directions (mathematical signs) and speak in terms of absolute values.

We commonly refer to this gain as the steady-state gain of the process, because the determination of gain is made after the PV settles to its self-regulating value.

Since from the controller’s perspective the individual gains of transmitter, final control element, and physical process meld into one over-all gain value, the process may be made to appear more or less responsive (more or less steady-state gain) just by altering the gain of the transmitter and/or the gain of the final control element.

Note : The general definition of gain is the ratio of output change over input change (ΔOut/ΔIn ). Here, you may have noticed we calculate process gain by dividing the process variable change (7.5%) by the controller output change (10%). If this seems “inverted” to you because we placed the output change value in the denominator of the fraction instead of the numerator, you need to keep in mind the perspective of our gain measurement. We are not calculating the gain of the controller, but rather the gain of the process. Since the output of the controller is the “input” to the process, it is entirely appropriate to refer to the 10% manual step-change as the change of input when calculating process gain.

Consider, for example, if we were to reduce the span of the transmitter in this process. Suppose this was a flow control process, with the flow transmitter having a calibrated range of 0 to 200 liters per minute (LPM). If a technician were to re-range the transmitter to a new range of 0 to 150 LPM, what effect would this have on the apparent process gain?

To definitively answer this question, we must re-visit the process trend graph for the old calibrated range:

Steady-state process example

We see here that the 7.5% PV step-change equates to a change of 15 LPM given the flow transmitter’s span of 200 LPM. However, if a technician re-ranges the flow transmitter to have just three-quarters that amount of span (150 LPM), the exact same amount of output step-change will appear to have a more dramatic effect on flow, even though the physical response of the process has the same as it was before:

Steady-state process example - 2

From the controller’s perspective – which only “knows” percent (For Analog Devices) of signal range – the process gain appears to have increased from 0.75 to 1, with nothing more than a re-ranging of the transmitter. Since the process is now “more responsive” to controller output signals than it was before, there may be a tendency for the loop to oscillate in automatic mode even if it did not oscillate previously with the old transmitter range. A simple fix for this problem is to decrease the controller’s gain by the same factor that the process gain increased: we need to make the controller’s gain 3 4 what it was before, since the process gain is now 4 3 what it was before.

The exact same effect occurs if the final control element is re-sized or re-ranged. A control valve that is replaced with one having a different Cv value, or a variable-frequency motor drive that is given a different speed range for the same 4-20 mA control signal, are two examples of final control element changes which will result in different overall gains. In either case, a given change in controller output signal percentage results in a different amount of influence on the process thanks to the final control element being more or less influential than it was before. Re-tuning of the controller may be necessary in these cases to preserve robust control.

If and when re-tuning is needed to compensate for a change in loop instrumentation, all control modes should be proportionately adjusted. This is automatically done if the controller uses the Ideal or ISA PID equation, or if the controller uses the Series or Interacting PID equation. All that needs to be done to an Ideal-equation controller in order to compensate for a change in process gain is to change that controller’s proportional (P) constant setting. Since this constant directly affects all terms of the equation, the other control modes (I and D) will be adjusted along with the proportional term. If the controller happens to be executing the Parallel PID equation, you will have to manually alter all three constants (P, I, and D) in order to compensate for a change in process gain.

A very important aspect of process gain is how consistent it is over the entire measurement range. It is entirely possible (and in fact very likely) that a process may be more responsive (have higher gain) in some areas of control than in others. Take for instance this hypothetical trend showing process response to a series of manual-mode step-changes:

Steady-state process example - 3

Note how the PV changes about 5% for the first 5% step-change in output, corresponding to a process gain of 1. Then, the PV changes about 7.5% for the next 5% output step-change, for a process gain of 1.5. The final increasing 5% step-change yields a PV change of about 12.5%, a process gain of 2.5. Clearly, the process being controlled here is not equally responsive throughout the measurement range. This is a concern to us in tuning the PID controller because any set of tuning constants that work well to control the process around a certain setpoint may not work as well if the setpoint is changed to a different value, simply because the process may be more or less responsive at that different process variable value.

Inconsistent process gain is a problem inherent to many different process types, which means it is something you will need to be aware of when investigating a process prior to tuning the controller. The best way to reveal inconsistent process gain is to perform a series of step-changes to the controller output while in manual mode, “exploring” the process response throughout the safe range of operation.

Compensating for inconsistent process gain is much more difficult than merely detecting its presence. If the gain of the process continuously grows from one end of the range to the other (e.g. low gain at low output values and high gain at high output values, or vice-versa), a control valve with a different characteristic may be applied to counter-act the process gain.

If the process gain follows some pattern more closely related to PV value rather than controller output value, the best solution is a type of controller known as an adaptive gain controller. In an adaptive gain controller, the proportional setting is made to vary in a particular way as the process changes, rather than be a fixed constant set by a human technician or engineer.

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