Within the distributed control systems (DCS), commonly used
in the hydrocarbon processing industry (HPI), lies a
rarelyused version of the proportionalintegralderivative
(PID) control algorithm. This algorithm, when properly tuned,
can reduce by a factor of four the quantity of offspec
material made during a process upset. As such, it can be the
single largest contributor to the benefits captured by improved
process control.
Basics
Most control engineers are familiar with the conventional
form of the PID algorithm, which determines the controller
output, M, based on the controller error,
E:
(1)
The algorithm has tuning constants controller gain,
K_{c}, integral time, T_{i},
and derivative time, T_{d}. It can also be
written in the Laplace form:
(2)
These forms are usually used by controlsystem vendors to
document the algorithms included within their products.
However, both forms should strictly only be applied to analog
systems. To describe the digital equivalent, let’s first
differentiate in Eq. 1:
(3)
This is often described as the velocity form of the
algorithm, as opposed to the fullposition form
represented by Eq. 1. The discrete form is converted
by replacing dM with ΔM (the change in
controller output) and dt with ts (the
controller scan interval). The change in error, dE, is
given by E_{n} (the current error) less
E_{n–1} (the error at the previous scan).
It is simplified as:
(4)
This incremental form also provides the benefit of
bumpless initialization. It generates the change in
output, rather than the absolute value. So if, when the
controller is switched from manual to automatic, the setpoint
(SP) has been tracking the process variable
(PV) and there will be no change made to the
process.
However, this form is rarely included in the control system
because it is prone to a derivative spike. Imagine
that the controller is in automatic mode and that the process
has been steady at SP for some time. Both the current
error and its recent values will be zero. Consider how the
derivative action then responds to the process operator,
changing SP by ΔSP. Assuming the
error is defined as (PV – SP), the change in
output generated by the derivative action is:
(5)
Since the deadtime of the process is likely to be greater than
ts, it will not have responded to this change before
the next controller scan. So, the error will remain fixed. The
derivative action will cause a change given by:
(6)
At the next scan, the two previous errors will be the same as
the present error, and the derivative action will, therefore,
be zero. This will remain the case until the process deadtime
expires. The net effect is that the derivative action has
generated a spike in the controller output by a insignificant
size.
In this case, ts is typically around one second;
even very modest values of 1 for K_{c} and 0.5
minute for T_{d} will cause a spike that is 30
times larger than ΔSP. Quite easily, this action
could exceed the output range. When the process deadtime does
expire, this will cause a large deviation from the SP,
to which the proportional action will respond, reproducing the
spike as an unnecessary corrective action.
Corrective action
To avoid this problem, controlsystem vendors usually modify
the derivative action so that it is based on PV rather
E:
(7)
Changes in SP will no longer result in derivative
action. If the SP is constant, then changes in
PV will be the same as changes in E (since
E = PV – SP). So, the response of derivative
action to process disturbances will remain unaffected.
Surprisingly, some controlsystem vendors retain the
derivativeonE version of the algorithm, sometimes as
an option. This might explain why derivative action generally
has a poor reputation. For example, if the controller is the
secondary of a cascade, and, thus, experiences frequent
SP changes, the derivative spikes will appear as noise
in the controller output. However, the most misunderstood
algorithm arises from the option that most DCS offer, i.e.,
also basing the proportional action on the PV:
(8)
At first glance, this would appear to significantly undermine
the benefit of the proportional action. With the
proportionalonE version the, change in SP
will cause the proportional action to change the controller
output according to:

(9)
This oneoff proportional kick does much to ensure
that the PV approaches the SP as quickly as
possible. With the proportionalonPV version, only
the integral action responds to the change in
SP—providing more of a ramp function than a step.
This is illustrated in Fig. 1, where the
performances of the two algorithms are compared. This has given
rise to myths such as the proportionalonPV algorithm
should be used if a slow response is required or if the process
is deadtime dominated. In fact both of these situations can be
addressed effectively by the correct tuning of the conventional
proportionalonE algorithm. What engineers overlook
is that the proportionalonPV algorithm can be tuned
to give a performance very similar to that of a welltuned
proportionalonE algorithm, as illustrated in
Fig. 2.

Fig.
1. Relative performance of the
proportionalonPV and
proportionalonE
algorithms in response to a SP
change
(using the same tuning for both). 

Fig.
2. Performance of the
proportionalonPV
algorithm (optimally tuned for SP
changes). 
This begs the question: Why are both made available? To
answer this question, consider how each algorithm responds to
process disturbances (load changes). Subjecting both
algorithms to a load change, with the tuning kept the same as
that used in Fig. 2, shows that the
proportionalonPV version substantially outperforms
the proportionalonE version. From Fig.
3, the duration of the disturbance is typically
halved, as well as the maximum deviation from SP. If
the controller were, even indirectly, controlling a key
variable such as product quality, then the quantity of
offgrade product would be reduced by a factor of at least
four.

Fig.
3. Response to a load change if both
algorithms are tuned for SP changes. 
Tuning vs. algorithm
What should be emphasized is that it is not the choice of
algorithm that has brought about this improvement, but the
choice of tuning. With the same tuning, modifying the
controller to use PV instead of E has no
effect on the way it responds to process disturbances. However,
if the proportionalonE version were installed with
the tuning designed for the proportionalonPV
algorithm, its response to SP changes would be
unacceptably aggressive—particularly in terms of
manipulated variable (MV) movement—as shown in
Fig. 4. The choice of the
proportionalonPV algorithm merely allows installing
the tuning preferred by the engineer without causing problems
when the SP is changed.

Fig.
4. Performance of
proportionalonE
algorithm in response to an SP
change
(using the SP tuning designed for
the
proportionalonPV algorithm). 
Why industry has not generally adopted this algorithm can be
explained by engineers largely assessing controller performance
by its response to SP changes in the belief that load
changes will be handled similarly. Engineers are simply unaware
of what improvements can be achieved by switching algorithms.
This is partly the fault of the system vendors; many of their
own staff fail to appreciate why the algorithm is included in
the system and, therefore, cannot properly explain its purpose
to users. But, since it is the tuning that gives the improved
response, perhaps the major limitation is the lack of an
effective tuning method.
Well over 200 methods have been published.^{1} Most
require that the process dynamics are known—usually as a
firstorder model with process gain, K_{p},
deadtime, θ, and lag, τ. Almost all are designed for
the conventional algorithm as described by Eq. 1. But, even for
this purpose, they often prove inadequate. Designed to make the
controller approach the SP as fast as possible, most
neglect the effect this has on the MV. The recommended
tuning will often cause unacceptably fast
corrections—particularly when the θ/τ ratio is
small. Indeed a means of evaluating a tuning method is to
determine what value it recommends for K_{c}
as θ/τ approaches zero. If K_{c}
approaches infinity, then the method is theoretically correct
but it has little practical use.
Options for controllers
There are a number of methods that allow the user to specify
the aggressiveness of the controller. Commonly used is the IMC
method.^{2} This requires the engineer to specify the
term λ—the time constant of the approach of the
controlled variable to a new SP. However, λ
does not explicitly define the impact on the MV. It is
necessary to adjust λ by trial and error to achieve the
best compromise between a fast return to SP and
acceptable changes to the MV.
Processes
It is unrealistic to expect simple formulabased methods to
generate effective tuning constants for all the situations
where they are likely to be applied. First, there are two
fundamentally different process
types—selfregulating and integrating.
Second, most control systems offer multiple versions of the PID
algorithm. Further, different vendors have modified the
algorithm in different ways—greatly increasing the number
of variations. Third, the consideration given to the changes
made to the MV will be processspecific. Some will
tolerate very fast and large changes, while others will not.
And, finally, the method must take account of the controller
scan interval. Usually, because the interval is often small
compared to the process dynamics, engineers can apply methods
designed for analog control. But for very fast processes (such
as compressor surge protection) or for controllers with very
long scan intervals (such as those based on discontinuous
onstream analyzers), a method that incorporates ts is
needed.
To consider all these factors, an impractically large
catalog of tuning formulas is required. However, applying
trialanderror methods may be the optimum approach. While
timeconsuming and not always properly applied on the real
process, trialanderror methods can be replicated relatively
simply in a computer simulation.^{3} Such tools are
available on the Internet and take account of all the issues
raised in a concise review of published tuning
methods.^{4} Using the software, the points plotted in
Figs. 5–7 were derived for the socalled
ideal proportionalonPV,
derivativeonPV algorithm described by Eq. 8. They
apply to a selfregulating process with dynamics much larger
than the controller scan interval. The tuning criterion was to
minimize integral over time of absolute error (ITAE) subject to
a limit placed on the movement of the MV so that it
did not overshoot the necessary steadystate change by more
than 15%. No limit was placed (or was necessary) on the
PV overshoot. Curve fitting gave these tuning
formulas:

(10)
(11)
(12)

Fig.
5. Tuning chart for controller gain. 

Fig.
6. Tuning chart for integral time. 

Fig.
7. Tuning chart for derivative time. 
The controller used as illustration in this article was tuned
using these formulas (Eqs. 10–12). It should be emphasized
that they are not applicable to all situations. They cannot be
applied to integrating processes. Their accuracy would be
suspect if applied to any systemspecific modification of the
control algorithm or if the process dynamics are of similar
magnitude to the scan interval. And, while an effective
ruleofthumb, the 15% MV overshoot limit might be
relaxed on processes that would tolerate more aggressive
corrective action. However any of the assumptions made in
developing the formulas can be modified by reverting to the
simulation tool, using it to generate either other sets of
curves or tuning for any specific case.
How the engineer specifies the use of the
proportionalonPV algorithm varies from vendor to
vendor and even between different systems provided by the same
vendor. Careful review of the system documentation is required.
Some vendors offer the choice of control equations. Others
provide additional parameters in a single equation, for
example, as:
(13)
This is sometimes described as the two degrees of freedom
controller. Setting α and β to 1 will result in
the controller described in Eq. 4, while setting them both to 0
will result in the version described by Eq. 8.
We should consider under what circumstances we should apply
the proportionalonPV algorithm. Its omission of any
proportional action when the SP is changed will always
make it less effective than the proportionalonE
version, even if tuned specifically to handle this disturbance.
However, as shown in Fig. 2, the difference in
performance would probably be indistinguishable in practice.
Standalone controllers are generally subject to frequent
process disturbances and relatively few SP changes;
thus, the decision to select the proportionalonPV
algorithm is clear.
Less clear is when the controller is the secondary of a
cascade. Under these circumstances, SP changes will be
frequent—as indeed they would be if the SP is the
MV of a higher level multivariable predictive
controller (MPC). However, as shown in Fig. 8,
while selecting the proportionalonE will slightly
reduce the ITAE for SP changes, and it will
substantially worsen it for load changes. Given that secondary
controllers will usually be on processes with a very low
θ/τ ratio, the approximately 10% improvement in
response to SP changes will be at the expense of a
600% degradation in the response to load changes. The secondary
would have to be subject to very few process disturbances to
justify selection of the proportionalonE
algorithm.

Fig.
8. Impact of switching from the
proportionalonPV to the
proportionalonE algorithm. 
Most controllers in industry are configured as the
proportionalonE type. Modification is not trivial,
since they then require retuning. There is no simple formula
for converting the tuning from one algorithm to that for
another. Plant testing would be necessary to obtain the process
dynamics. Further, some controllers could generate far greater
redesign. For example, those on a process which has MPC
installed could require repetition of the steptesting
performed for MPC design.
Control engineers must address the question as to which
controllers would show sufficient improvement to justify
modification. Certainly, improving a simple flow controller so
that it handles disturbances in two seconds, where before it
took around five seconds, would probably go unnoticed. But
halving the time it takes a relatively slow temperature
controller so that it recovers from a disturbance in five
minutes rather than 10 minutes can substantially improve
process profitability. On new installations, there is much to
be said for adopting the proportionalonPV algorithm
throughout. HP
LITERATURE
CITED
^{1} O’Dwyer, A., “A summary of PI and PID
controller tuning rules for processes with time delay,”
IFAC Digital Control: Past, Present and Future of PID Control,
Terrassa, Spain, 2000.
^{2} Chien, IL. and P. S. Fruehauf, “Consider IMC
tuning to improve controller performance,” Chemical
Engineering Progress, June 1990, pp. 33–41.
^{3}
http://www.whitehouseconsulting.com/tune.htm.
^{4} King, M., Process Control: A Practical
Approach, Wiley, Chichester, 2011, pp. 51–77.
The author
Myke King is the author of Process
Control: A Practical Approach. He is the director
of Whitehouse Consulting and is responsible for
consultancy services, assisting clients with
improvements to basic controls, and with the
development and execution of advanced control projects. Mr. King has 35
years of experience in such projects, working with many
of the world’s leading oil and petrochemical companies. He
holds an MS degree in chemical engineering from
Cambridge University, and is a Fellow of the Institute
of Chemical Engineers.
