# The Effect of Coincidence Horizon on Predictive Functional Control

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## Abstract

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## 1. Introduction

- The energy and enthusiasm of key staff alongside the intuitive nature of the approach has enabled effective promotion of the benefits within industry.
- The inclusion of PFC on core curricula [5] for potential plant operators, so that they are familiar with how to tune PFC and, thus, view it as an obvious alternative to PID.
- PFC has costs similar to PID, whereas popular alternative MPC algorithms are neither cheap and/or not embedded in local hardware or software. Moreover, PFC handles constraints and delays much more easily than PID approaches.
- A default PFC set-up handles ramp and parabolic targets, whereas such scenarios have not captured the attention of the more mainstream MPC literature, despite many practical scenarios having such targets.
- Because the concept is so simple, it can easily be applied on non-linear systems, which is certainly not true for most MPC algorithms.

- Can the tuning be systematic?
- Can one give any form of guarantee or expectation of good closed-loop performance?
- How can we make PFC robust against model parameter uncertainty?
- Can one identify scenarios where PFC will work well and, conversely, may not?
- Are there simple adaptations to PFC that would extend the range of scenarios for which it works well?

## 2. Basic Concepts of PFC

#### 2.1. Target Behaviour or Reference Trajectory

_{d}seconds (or m samples), and thus, a key requirement is to identify an input sequence that will achieve this. The target closed-loop behaviour (or reference trajectory) can be expressed as:

^{tr}(s) represents the continuous time or Laplace representation and r

^{tr}(z) the discrete time or z-transform representation; R is taken to be the actual steady-state target. A typical desired step response, with no delay, is plotted in Figure 1, where, here, the discrete pole is set at ρ = 0.8.

_{r}are related through the sampling time T

_{s}as $\rho ={e}^{-\frac{{T}_{s}}{{T}_{r}}}$. Some authors prefer to define a target settling time, such as the time it takes to get within 5% of the target (denoted closed-loop time response (CLTR)) and note that T

_{r}≈ CLTR=3.

#### 2.2. Coincidence Point

_{k}to be the current output and R to be the steady-state target, one wants the future outputs to follow a first order response from the current value to the target, in other words, to take the following values:

_{y}, and ensures that the actual output prediction matches the target response at that point only. Consequently, the PFC control law reduces to (conceptually), enforcing the equality:

_{k}

_{+}

_{n}

_{|}

_{k}means the predicted value of y at sample k + n, where the prediction is made at sample k.

^{tr}is marked with a dotted line. One forces the output prediction (solid line) to “coincide” at a specified horizon (here, 10 samples into the future). The computation is updated at every sample instant, and this introduces feedback.

**Remark 1.**The target behaviour Equation (2) and control law requirement Equation (3) can be written down by inspection. Clearly, Equation (2) is simple. Where the system has a delay of “m” samples, the target behaviour should be modified slightly, as discussed in Section 2.4.

**Remark 2.**One can argue that PFC is intuitive in that one in essence proposes a control move that gets the system output closer (than the current output) to the desired steady state some time in the near future and then continually update this, so in effect, one expects to always be getting closer and closer to the target. However, the main weakness of PFC is the same as its strength. Basing control decision making on a single point will work well if matching that single point implies the rest of the response is also fairly closely matched, as this will imply consistent decision making at each sample. However, conversely, if matching a single point does not imply matching of other parts of the response, then the implied decision making could be seriously flawed, as it could lead to inconsistent decisions from one sample to the next. In simple terms, one needs to be sure that the part of the prediction beyond the coincidence point moves smoothly and monotonically to the steady state.

#### 2.3. Prediction and the PFC Control Law

#### 2.3.1. First Order Models with Open-Loop Dynamics Equal to Target Dynamics

_{1}z

^{−1}= 1−az

^{−1}, that is a = −a

_{1}, as this makes the prediction algebra of Equation (5) easier and more transparent; also, for simplicity, b = b

_{1}. Combining this prediction with the coincidence requirement of Equation (3) results in the following equality or essence, the control law.

**Theorem 1.**If the target pole ρ is chosen equal to the open-loop pole a, that is a = ρ, then the closed-loop pole will match the open-loop pole for any choice of n

_{y}(in the nominal case).

**Proof.**The PFC control Equation (6) is summarised as follows when ρ = a:

_{y}. Clearly if the input selection is a constant, open-loop dynamics will follow.

#### 2.3.2. First Order Models with Open-Loop Dynamics Different from Target Closed-Loop Dynamics

**Theorem 2.**For first order models, the link between the desired pole ρ and the actual closed-loop pole is weak and is effective only when n

_{y}= 1.

**Proof.**The PFC control and system is summarised as follows when ρ ≠ a:

_{k}, one finds that the closed-loop dynamic is given as:

_{y}increases, the deviation of the pole from a reduces. However, for a coincidence horizon of one, the resulting pole is given as:

_{y}= 1.

_{y}, but are close to ρ iff n

_{y}is small.

**Corollary 1.**The coincidence horizon and parameter ρ only have a noticeable impact on the control law if the coincidence horizon is small. Irrespective of the choice of ρ, the closed-loop pole tends to a as the coincidence horizon tends to infinity. This is a natural consequence of Equation (10), because:

#### 2.4. First Order Model with Delay

- The replacement of ${\rho}^{{n}_{y}}$ by ${\rho}^{{n}_{y}-m}$ to take into account the fact that the response cannot begin until after m samples.
- The replacement of process measurement y
_{k}by a predicted value y_{k}_{+}_{m}_{|}_{k}; this second change is essential as the target first order response is expected to begin m samples ahead, so the initial error from the target must be based on that sample.

#### 2.5. PFC for Higher Order Models

_{y}step ahead prediction [8]. It can be shown that for a model of the form (usually, we assume a

_{0}= 1):

_{y}step ahead prediction, with the future inputs fixed, takes the form:

_{y}.

**Theorem 3.**As the coincidence horizon approaches infinity, the PFC control law will reduce to giving open-loop characteristics for a stable open-loop process, irrespective of the choice of ρ.

**Proof.**For a stable open-loop process, it is easy to show that the prediction parameters of Equation (15) have the following asymptotic properties:

**Corollary 2.**The PFC code for large coincidence horizons can be reduced to a few trivial lines. This is self-evident from Equation (18).

#### 2.6. Model Uncertainty, Offset Free Tracking and Disturbances

_{m}) in parallel with the process G

_{p}; the model is used for prediction. The difference d = y

_{p}−y

_{m}between the model output y

_{m}and process output y

_{p}is taken as an estimate of the system disturbance, although, in fact, it can also correct the offset for parameter uncertainty. The PFC control law that ensures offset free tracking reduces to:

**Remark 3.**Where a system has real roots only, several PFC practitioners (e.g., [4,9,10]) favour using partial fractions to write the model G

_{m}as a sum of first order models. This has the advantage that the prediction can be written down explicitly in forms like Equation (5) and, thus, with much simpler coding. However, this issue is not central to the current paper.

#### 2.7. Constraint Handling

#### 2.8. Summary of PFC Tuning

## 3. Numerical Illustrations for First Order Systems

_{y}.

#### 3.1. Concepts of Well-Posed Decision Making

- The implied PFC prediction at a given sample (denoted up; yp for the input and output, respectively).
- The actual closed-loop behaviour that results (denoted us; ys for the input and output, respectively).

#### 3.2. PFC Responses with Target Behaviour Equal to Open-Loop Behaviour

^{+1})y

_{k}= bz

^{−1}u

_{k}; without loss of generality, we will use a = 0:8; b = 0:25 for the simulation, as the observations here are based around selecting ρ = a. Plot the closed-loop responses for various choices of coincidence horizon and overlay the predictions made at the first sample. The results are given in Figure 4. What is immediately clear is that the predictions and closed-loop behaviour overlap (ys = yp) precisely (as expected from Theorem 1) in the nominal case, so in this case, PFC gives well-posed decision making.

#### 3.3. PFC Responses with Target Behaviour Different from Open-Loop Behaviour

_{y}, as discussed in Section 2.3.2, but here, we are also interested in the consistency of the decision making, so the figures overlay the implied predictions utilised by PFC and, of course, the target dynamic.

- When ρ ≠ a, for small n
_{y}, there is clearly some inconsistency from one sample to the next, as the input prediction up at the first sample is different from the closed-loop evolution us that results. For example, when ρ = 0:7; a = 0:8; n_{y}= 1, the input change from Sample 1 to Sample 2 is from 1.2 to 1.08. - Despite the inconsistency between implied predictions yp and actual behaviour ys, with n
_{y}= 1, the target behaviour r has been achieved. - As n
_{y}increases, the choice of ρ becomes less and less significant (the output yp does not track the target r), but there is much better consistency between predictions yp and behaviour ys.

**Remark 4.**There is an interesting inconsistency noted here. If the coincidence horizon is small, the design parameter ρ is effective in changing the closed-loop dynamics, but the implied long-term predictions are a poor indicator of final behaviour. Conversely, if the coincidence horizon is large, the design parameter ρ has no affect, but the predictions are consistent. Of course, readers may note that PFC never claimed to take any account of long-term predictions within the decision making process, and therefore, it was not implicit that one would keep the input constant at future samples!

## 4. Tuning PFC for Higher Order Systems

#### 4.1. Coincidence Horizon Issues with Higher Order Models

_{3}, along with a higher coincidence horizon (n

_{y}= 12 is marked in this figure) to get sensible answers! Indeed, it can be shown that using n

_{y}= 1 can imply a controller using “plant inversion”, which is not robust.

_{2}and G) by having over actuation during transients, so the implied steady state within the prediction is well above the target. Readers are also reminded, however, of Theorem 3 and that using too large n

_{y}PFC essentially reduces to choosing the steady-state input, so the tuning parameter ρ would have a smaller affect.

_{y}is based on a point where the step response is increasing, and changes relative to the steady state are close to monotonic thereafter.

**Conjecture 1.**For systems with close to monotonically convergent behaviour after immediate transients, a good choice of coincidence horizon is one where the open-loop step response has risen to around 40% to 80% of the steady state and the gradient is significant (formal proofs and details constitute future work).

**Remark 5.**A manual for training engineers [4,5] on PFC suggests that a suitable value for n

_{y}might also be determined by identifying the so-called inflexion point on the step response curve, that is, where the gradient is maximum. However, care must be exercised, as the maximum gradient may not be enough on its own, especially with non-minimum phase systems, as noted briefly with system G

_{4}.

#### 4.2. PFC Tuning for Systems with Relatively Simple Dynamics

- Following Conjecture 1 and due to the lag in the process (the step response takes a few samples to start moving significantly), it is not reasonable to use n
_{y}< 5. Similarly, n_{y}> 15 will put too much emphasis on the steady state. - It is clear that for ρ = 0:9, an intercept exists between the step response and target, and hence, n
_{y}= 8 would be a logical choice of coincidence point. - For smaller and larger choices of ρ, there is no intercept in the range 5 ≤ n
_{y}≤ 15, and hence, to force coincidence (essentially to move the step response up or down), PFC will either have to under- or over-actuate using these choices of ρ. This is what we expect, as a faster ρ requires over-actuation and a slower ρ requires under-actuation. - Limits on the actuation permitted will be a guide as to what range of ρ is permissible to request and, indeed, the appropriate n
_{y}. For example, ρ = 0:7 requires an input over actuation of around 2.5 to get coincidence at n_{y}= 5 or around 1.8 if n_{y}= 8.

_{y}has resulted in the choice of ρ being an effective tuning parameter, that is the closed-loop outputs converge roughly at the same speed as the target.

#### 4.3. PFC Tuning for a Non-Minimum Phase System

_{y}≤ 12. Potential choices of coincidence points are marked on the figure. Readers may also note that the inflexion point is around n

_{y}= 5, and clearly, as the magnitude of the step response is still very small here, such a design choice for n

_{y}is inappropriate and can be shown to give very poor closed-loop performance!

_{y}is well chosen, the choice of ρ has been effective in affecting the closed-loop settling time and input activity, and thus, ρ is an effective tuning parameter.

#### 4.4. PFC Tuning for an under-Damped System

_{y}≤ 6. Potential choices of the coincidence point are marked on the figure.

#### 4.5. Summary

_{y}. One can, of course, do an iterative design, underpinned by simulations with various n

_{y}, to determine the final choice.

## 5. PFC for Processes with Difficult Dynamics

#### 5.1. Second Order Model with Oscillatory Dynamics

- PFC gives good results with ρ = 0:7; n
_{y}= 2; 3 (fails totally for n_{y}= 1), but it is difficult to argue that this is more than just a fluke, given that the shapes of the closed-loop input trajectory are not close to a constant. - For large horizons, the open-loop dynamic dominates and the behaviour is unacceptable.

#### 5.2. Model with Open-Loop Unstable Dynamics

- PFC fails for both n
_{y}= 1 and large n_{y}. - In this case, reasonable behaviour results for n
_{y}= 3, but it would be difficult to argue that this is a systematic design; and certainly, the parameter ρ has not been effective.

#### 5.3. Summary

_{y}, but without a clear expectation of whether this will give a good result or not. Intuitively, the core failing is the desire to follow a first order trajectory, and using a constant future input as this is clearly unrealistic for systems with complex dynamics. Therefore, to extend the benefits of PFC, one needs to extend the target behaviours and/or input parametrisations. It should be emphasised, however, that PFC practitioners have work-arounds, and modifications for these scenarios and future work will investigate those proposals in depth, alongside the consideration of alternative modifications.

- A reasonable conjecture is that one should begin from a target trajectory that is realistic, as then, ensuring the predictions are close to this trajectory would give a high expectation of good behaviour.
- For systems with oscillatory open-loop dynamics, it is reasonable to conjecture that the future control input should be shaped so that the impact of this oscillation is minimised.
- For unstable open-loop processes, it is well known that one should parametrise the future decisions in such a way that the predictions are all guaranteed convergent [7], preferably to the correct steady state.

_{k}as the control decision and assuming that all future inputs are constant, one needs alternative parametrisations, which ensure that the process has underlying good behaviour and, within that parametrisation, include some flexibility that can be used to influence the speed of response.

## 6. Conclusions

#### 6.1. Key Points

_{y}, the PFC control law reduces to estimating the process steady state input; thus, ρ has no impact at all.

_{y}are not consistent and vary from one example to another depending on the underlying process dynamics. Consequently, some suggestions were made as to how one could determine an appropriate range of n

_{y}, ρ before any fine tuning. The methodology was demonstrated on several examples and found to be effective and, moreover, is similar, but not identical, to the empirical suggestions in [4].

#### 6.2. Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Conceptual illustration of PFC control law definition matching prediction to target trajectory at a single point.

**Figure 4.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for a = ρ; here, ys = yp; us = up.

**Figure 5.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for a = ρ; here, ys = yp; us = up for system with a three-sample delay.

**Figure 6.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for a > ρ.

**Figure 7.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for a < ρ.

**Figure 8.**Target first order step response with poles of 0.8 and 0.9 overlaid with a step response of a second order system.

**Figure 9.**Open-loop step response overlaid with several possible targets and potential coincidence points.

**Figure 10.**Closed-loop step responses for a second order system G

_{2}with various ρ; n

_{y}overlaid with the underlying target trajectory.

**Figure 11.**Open-loop step response of G

_{4}overlaid with several possible targets and potential coincidence points for a non-minimum phase system.

**Figure 12.**Closed-loop step responses for a third order system G

_{4}with various ρ; n

_{y}overlaid with the underlying target trajectory.

**Figure 13.**Open-loop step response of G

_{5}overlaid with several possible targets and potential coincidence points for an under-damped system.

**Figure 14.**Closed-loop step responses for an under-damped system G

_{5}with various ρ; n

_{y}overlaid with the underlying target trajectory.

**Figure 15.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for an oscillatory system G

_{6}.

**Figure 16.**Closed-loop PFC input and output us; ys responses overlaid with predictions up; yp made at the first sample for unstable process G

_{7}.

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Rossiter, J.A.; Haber, R.
The Effect of Coincidence Horizon on Predictive Functional Control. *Processes* **2015**, *3*, 25-45.
https://doi.org/10.3390/pr3010025

**AMA Style**

Rossiter JA, Haber R.
The Effect of Coincidence Horizon on Predictive Functional Control. *Processes*. 2015; 3(1):25-45.
https://doi.org/10.3390/pr3010025

**Chicago/Turabian Style**

Rossiter, John Anthony, and Robert Haber.
2015. "The Effect of Coincidence Horizon on Predictive Functional Control" *Processes* 3, no. 1: 25-45.
https://doi.org/10.3390/pr3010025