6. Conclusion

The purpose of this Chapter is to present tuning methods for PID controllers which are based on the Magnitude Optimum (MO) method. The MO method usually results in fast and stable closed-loop responses. However, it is based on demanding criteria in the frequency domain, which requires the reliable estimation of a large number of the process parameters. In practice, such high demands cannot often be satisfied.

It was shown that the same MO criteria can be satisfied by performing simple time-domain experiments on the process (steady-state change of the process). Namely, the process can be parameterised by the moments (areas) which can be simply calculated from the process steady-state change by means of repetitive integrations of time responses. Hence, the method is called the "Magnitude Optimum Multiple Integration" (MOMI) method. The measured moments can be directly used in the calculation of the PID controller parameters without making any error in comparison with the original MO method. Besides this, from the time domain responses, the process moments can also be calculated from the process transfer function (if available). Therefore, the MOMI method can be considered to be a universal method which can be used either with the process model or the process timeresponses.

Magnitude Optimum Techniques for PID Controllers 101

Huba, M. (2006). Constrained pole assignment control. Current Trends in Nonlinear

Kessler, C. (1955). Über die Vorausberechnung optimal abgestimmter Regelkreise Teil III.

Preuss, H. P. (1991). Model-free PID-controller design by means of the method of gain

Rake, H. (1987). Identification: Transient- and frequency-response methods. In M. G. Singh

Strejc, V. (1960). Auswertung der dynamischen Eigenschaften von Regelstrecken bei

Umland, J. W. & M. Safiuddin (1990). Magnitude and symmetric optimum criterion for the

others? IEEE Transactions on Industry Applications, 26 (3), pp. 489-497. Vrančić, D. (1995). A new PI(D) tuning method based on the process reaction curve, Report

Vrančić, D. (2008). MOMI Tuning Method for Integral Processes. Proceedings of the 8Th Portuguese Conference on Automatic Control, Vila Real, July 21-23, pp. 595-600. Vrančić, D. & Huba, M. (2011). Design of feedback control for unstable processes with time

Vrančić, D. Kocijan, J. & Strmčnik, S. (2004a). Simplified Disturbance Rejection Tuning

Vrančić, D., Lieslehto, J. & Strmčnik, S. (2001b). Designing a MIMO PI controller using the

Vrančić, D., Peng, Y. & Strmčnik, S. (1999). A new PID controller tuning method based on multiple integrations. Control Engineering Practice, Vol. 7, pp. 623-633. Vrančić, D., Strmčnik S. & Kocijan J. (2004b). Improving disturbance rejection of PI

Vrančić, D., Strmčnik S., Kocijan J. & Moura Oliveira, P. B. (2010). Improving disturbance

Vrančić, D., Strmčnik S. & Juričić Đ. (2001a). A magnitude optimum multiple integration method for filtered PID controller. Automatica. Vol. 37, pp. 1473-1479. Vrečko, D., Vrančić, D., Juričić Đ. & Strmčnik S. (2001). A new modified Smith predictor: the

concept, design and tuning. ISA Transactions. Vol. 40, pp. 111-121.

Available on http://dsc.ijs.si/Damir.Vrancic/bibliography.html

optimum (in German). Automatisierungstechnik, Vol. 39, pp. 15-22.

Birkhäuser, pp. 163-183.

Pergamon Press.

Regeln, 3(1), pp. 7-10

Regelungstechnik, Jahrg. 3, pp. 40-49.

DP-7298, J. Stefan Institute, Ljubljana.

20-23, Melbourne, pp. 491-496.

Transactions. Vol. 49, pp. 47-56.

468.

pp. 73-84.

17, Tatranska Lomnica, Slovakia, pp. 100-105.

Systems and Control, L. Menini, L. Zaccarian, Ch. T. Abdallah, Edts., Boston:

Die optimale Einstellung des Reglers nach dem Betragsoptimum.

(Ed.), Systems & control encyclopedia; Theory, technology, applications. Oxford:

gemessenen Ein- und Ausgangssignalen allgemeiner Art. Z. Messen, Steuern,

design of linear control systems: what is it and how does it compare with the

delay. Proceedings of the 18th International Conference on Process Control. June 14-

Method for PID Controllers. Proceedings of the 5th Asian Control Conference. July

multiple integration approach. Process Control and Quality, Vol. 11, No. 6, pp. 455-

controllers by means of the magnitude optimum method. ISA Transactions. Vol. 43,

rejection of PID controllers by means of the magnitude optimum method. ISA

The MO (and therefore the MOMI) method optimises the closed-loop tracking performance (from the reference to the process output). This may lead to a degraded disturbancerejection performance, especially for lower-order processes. In order to improve the disturbance-rejection performance, the MO criteria have been modified. The modification was based on optimising the integral of the closed-loop transfer function from the process input (load disturbance) to the process output. Hence, the method is called the "Disturbance-Rejection Magnitude Optimum" (DRMO) method.

The MOMI and the DRMO tuning methods have been tested on several process models and on one hydraulic laboratory setup. The results of the experiments have shown that both methods give stable and fast closed-loop responses. The MOMI method optimises tracking performance while the DRMO method improves disturbance-rejection performance. By using a two-degrees-of-freedom (2-DOF) PID controller structure, the optimal disturbance-rejection and improved tracking performance have been obtained simultaneously.

The MOMI and DRMO methods are not limited to just PID controller structures or stable (self-regulatory) processes. The reader can find more information about different controller structures and types of processes in Vrančić (2008), Vrančić & Huba (2011), Vrečko et al., (2001), Vrančić et al., (2001b) and in the references therein.

The drawback of the MO method (and therefore the MOMI method and, to an extent, the DRMO method) is that stability is not guaranteed if the controller is of a lower-order than the process. Therefore, unstable closed-loop responses may be obtained on some processes containing stronger zeros or else complex poles. Although the time-domain implementation of the method is not very sensitive to high-frequency process noise (due to multiple integrations of the process responses), the method might give sub-optimal results if lowfrequency disturbances are present during the measurement of the process steady-state change.
