By Doug Cooper and Allen Houtz¹

Both "feed forward with feedback trim" and cascade control can provide improved disturbance rejection performance. They have different architectures, however, and choosing between the two depends on our specific control objective and the ability to obtain certain process measurements.

Feed Forward and Feedback Trim are Largely Independent
As illustrated below (click for a larger view), the feed forward with feedback trim architecture is constructed by coupling a feed-forward-only controller to a traditional feedback controller.

The feed forward controller seeks to reject the impact of one specific disturbance, D, that is measured before it reaches our primary process variable, PV, and starts its disruption to stable operation. Typically, this D is one that has been identified as causing repeated and costly upsets, thus justifying the expense of both installing a sensor to measure it, and developing and implementing the feed forward computation element to counteract it.

  
 where:
      CO  = controller output signal
      D    = measured disturbance variable
      e(t) = controller error, SP – PV
      FCE = final control element (e.g., valve, variable speed pump or compressor)
      PV  = measured process variable
      SP  = set point

The feedback controller is designed and tuned like any stand-alone controller from the PID family of algorithms. The only difference is that it must allow for its controller output signal, COfeedback, to be combined with a feed forward controller output signal, COfeedforward, to arrive at a total control action, COtotal.

The process model (CO −› PV) in the feed forward element describes or predicts how each change in CO will impact PV. The disturbance model (D −› PV) describes or predicts how each change in D will impact PV.

In practice, these models can range from simple scaling multipliers (static feed forward) through sophisticated differential equations (dynamic feed forward). Sophisticated dynamic models can better describe actual process and disturbance behaviors, often resulting in improved disturbance rejection performance. Such models can also be challenging to derive and implement, increasing the time and expense of a project.

Dynamic Feed Forward Based on the FOPDT Model
We first develop a general feed forward element using dynamic models (differential equations). Later, we will explore how we can simplify this general construction into a static feed forward element. Static feed forward is widely employed in industrial practice, in part because it can be implemented with an ordinary multiplying relay that scales the disturbance signal.

A dynamic feed forward element accounts for the "how far" gain, the "how fast" time constant and the "how much delay" dead time behavior of both the process (CO −› PV) and disturbance (D −› PV) relationships.

The simplest differential equation that describes such "how far, how fast, and with how much delay" behavior for either the process or disturbance dynamics is the familiar first order plus dead time (FOPDT) model.

Which matches the general FOPDT (first order plus dead time) dynamic model form:

  

where for a change in CO, the FOPDT model parameters are:
  ▪ Kp = process gain (the direction and how far PV will travel)
  ▪ Tp = process time constant (how fast PV moves after it begins its response)
  ▪ Өp = process dead time (how much delay before PV first begins to respond)

This equation describes how each change in CO causes PV to respond (CO −› PV) as time, t, passes. Our past modeling experience also includes developing and documenting FOPDT CO −› PV models for the heat exchanger and jacketed reactor processes.

● The D −› PV Disturbance Model
The procedure used to develop the FOPDT (first order plus dead time) process models referenced above can be used in an identical fashion to develop a dynamic D −› PV disturbance model. While we do not show the graphical calculations at this point, consider the plot below from the gravity drained tanks process (click for a larger view).

Instead of analyzing a plot where a CO step forces a PV response while D remains constant, here we would analyze a D step forcing a PV response while CO remains constant.

We presume that an analogous graphical modeling procedure can be followed to determine the "how far, how fast, and with how much delay" dynamic D −› PV disturbance model:

where for a step change in D:
  ▪ KD = disturbance gain (the direction and how far PV will travel)
  ▪ TD = disturbance time constant (how fast PV moves after it begins its response)
  ▪ ӨD = disturbance dead time (how much delay before PV first begins to respond)

Dynamic Feed Forward as a Thought Experiment
A feed forward element typically performs a model computation every sample time, T, to address any changes in measured disturbance, D. To help us visualize events, we discuss this as if it occurs as a two step "prediction and corrective action" procedure for a single disturbance:

1.

The D −› PV disturbance model receives a change in the measured value of D and predicts an open-loop or uncontrolled “impact profile” on PV. This includes a prediction of how much delay will pass before the disruption first arrives at PV, the direction PV will travel for this particular D once it begins to respond, and how fast and how far PV will travel before it settles out at a predicted new steady state.

2.

The CO −› PV process model then uses this PV impact profile to back-calculate a series of corrective control actions, COfeedforward. These are CO moves sent to the final control element (FCE) to cause an "equal but opposite" response in PV such that it remains at set point, SP, throughout the event. Thus, the CO model seeks to exactly counteract the disruption profile predicted in step 1. The first CO actions are delayed as needed so they meet D upon arrival at PV. A series of CO actions are then deployed to counteract the predicted disruption over the life of the event.

Even sophisticated dynamic models are too simple to precisely describe the behavior of real processes. So although a feed forward element can dramatically reduce the impact of a disturbance on our PV, it will not provide a perfect "prediction and corrective action" disturbance rejection.

To account for model inaccuracies, the feed forward signal is combined with traditional feedback control action, COfeedback, to create a total controller output, COtotal. Whether it be a P-Only, PI, PID or PID w/ CO Filter algorithm, the feedback controller plays the important role of:

▪	minimizing the impact of disturbance variables other than D that can disrupt the PV,
▪	providing set point tracking capability to the overall strategy, and
▪	correcting for the simplifying approximations used in constructing the feed forward computation element that ultimately makes it imperfect in its actions.

The Sign of COfeedforward Requires Careful Attention
Notice in the block diagram above that COfeedforward is added as:

   COtotal = COfeedback + COfeedforward

We write the equation this way because it is consistent with typical vendor documentation and standard piping/process & instrumentation diagrams (P&IDs).

The "plus" sign in the equation above requires our careful attention. To understand the caution, consider a case where the D −› PV disturbance model predicts that D will cause the PV to move up in a certain fashion or pattern over a period of time. According to our thought experiment above, the CO −› PV process model must compute feed forward CO actions that cause the PV to move down in an identical pattern.