(190c) Beyond Foptd Models in Tuning PI Controllers

The
first-order plus time delay (FOPTD) model is widely used to design proportional-integral
(PI) controllers.^1-2 The FOPTD model

has three parameters (process gain, time
constant, and time delay) and many PI controller tuning rules based on these FOPTD
model parameters are available.

When
high-order process models are available, their FOPTD models can be obtained by
applying model reduction techniques. Otherwise, they can be obtained from
process responses. Traditional simple methods to obtain FOPTD models are based
on the several measurements of process step responses as shown in Fig. 1.³

Fig. 1. A graphical method to
identify the FOPTD model from open-loop step response.

For
some overdamped processes, even though their step
responses are very similar to that in Fig. 1, FOPTD models fail to provide
acceptable PI controller parameters. For such processes, a half-order plus time
delay (HOPTD) model

can be used to approximate their
dynamics and tune PI controllers.⁴ Field data for the three model
parameters of G_H(s) are
just the same as those for the FOPTD model, i.e., only two point data with the
final steady state are the field data required.

For
processes showing overshoots in step responses as shown in Fig. 2, FOPTD models
also fail to describe step responses and provide acceptable PI controller
parameters. For such processes, instead
of finding high order models that fit step responses, step responses are
divided into two parts. FOPTD models for two regions are obtained and they are
combined as

Here the first two terms are for G_I(s) and G_II(s) in Fig. 2 and the last
term is to cancel the effects of G_I(s)
after t_p. Each model of G_I(s) and G_II(s) can describe well each
part of the response in Fig. 2. This
combined model (dFOPTD model) can be used to tune PI
controller.⁵

Fig. 2. Typical processes where dFOPTD
models are applied.

For
the above two types of processes, FOPTD models suffer from approximating step
responses due to the structural limitations and fail to provide acceptable PI
controller parameters. Here merits and demerits of two models of HOPTD and dFOPTD are studied.

References

[1] Seborg DE,
Mellichamp DA, Edgar TF, Doyle FJ. Process Dynamics and Control, 3^rded. New York: Wiley, 2010.

[2] O'Dwyer A.
Handbook of PI and PID Controller Tuning Rules, 3^rd ed. London,
UK: Imperial College Press, 2009.

[3] Liu T, Wang QG, Huang HP,
A tutorial review on process identification from step or relay feedback test. J.
Process Control, 2013, 23, 1597-1623.

[4]
Lee J, Lee Y, Yang DR, Edgar TF, Half Order Plus Time
Delay (HOPTD) Models to Tune PI Controllers, submitted to AIChE J., 2016.

[5] Lee Y, Yang DR, Lee J, Edgar
TF, Double First Order Plus Time Delay Models to Tune
PI Controllers, submitted to I&EC
Research, 2016.