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^rd ed. 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.