Tuning Rules for PID Controllers

This chapter introduces new PID tuning rules, derived from the general PID design method proposed in the previous chapter, for frequently encountered first order plus delay processes and integrating plus delay processes. 

This chapter consists of five sections. Section 7.2 presents the development of the PID controller tuning rules for first order plus delay processes. Sections 7.3 and 7.4 illustrate the new tuning rules using simulation and experimental studies, respectively, and compares the results with those obtained using the IMC-PID tuning rules. Section 7.5 presents the development of the PID tuning rules for integrating plus delay processes. 

Portions of this chapter have been reprinted from lEE Proceedings-Control Theory and Applications 142, L. Wang and W.R. Cluett, Tuning PID controllers for integrating processes , pp. 385-392, 1997, with permission from lEE. 

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A rule based approach to process control has for many years provided an alternative to traditional methods in the form of fuzzy logic control (8,9). Since the advent of expert systems, rulebases have been used for fault diagnosis [10], to advise operators (11) 9 to aid control engineers when installing PID controllers (12), to provide expert on-line tuning for PID controllers (13), and to control processes without the use of fuzzy logic (14,15). 

O Dwyer, A. (2000) 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. 

Since the general IMC method is unnecessarily complicated for processes that are well approximated by first-order dead time or integrator dead-time models, simplified IMC rules were developed by Fmehauf et al. [5] for PID controller tuning (see Table 5.1). 

Comparison of the closed-loop responses obtained using the Z-N tuning rule and the proposed tuning rule for both PI and PID controllers... 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

S. Skogestad, Simple analytic rules for model reduction and PID controller tuning, J. Process Control 13 (2003) 291-309. 

Although Equations (7.11)-(7.25) give analytical solutions for the PID controller parameters, it would be even more convenient for the user to have tuning rules requiring a minimum number of calculations. In order to present a range of desired closed-loop response speeds, we have chosen six different values for the normalized closed-loop time constant id = 4, 2, 1.33, 1, 0.8 and 0.67. The corresponding value of the parameter a can be simply calculated as a = - and the aw tual desired closed-loop time constant is given by... 

Table 7.2 PID controller parameters for Process A using new tuning rules...

Prom Equations (7.43)-(7.45), the normalized PID controller parameters can be calculated for a given set of performance parameters. In this case, tuning rules are more straightforward to derive than for the first order plus delay case because the normalized Y jw) is not a function of the process parameters. 

In the derivation of the rules, the damping factor has been chosen equal to either 0.707 or 1 to produce two sets of tuning rules, with the parameter /3 being used to adjust the closed-loop response speed. The desired closed-loop time constant is t = /3d. /3 has been varied for both cases from 1 to 17 and the corresponding normalized PID controller parameters Kc, ti and td have been calculated. Prom this information, polynomial functions have been fit to produce explicit solutions for the normalized controller parameters as a function of j3. 

With our tuning rules, if we choose C = 1 and 0 = 3, then Kc = 0.466 and f/ = 7.19. Therefore, for this particular choice of closed-loop performance, our tuning rules correspond almost exactly to those presented by Tyreus amd Luyben (1992). The key benefit with our timing rules is that different values for 0, depending on the control objective and desired stability margins, may be selected and used to eaisily calculate the corresponding PID controller parameters. 

Skogestad, S., Simple Analytic Rules for Model Reduction and PID Controller Tuning, J. Process Control, 13,291 (2003). 

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