Tuesday , February 7 2023

Performance/Robustness Trade-off Design Framework for 2DoF PI Controllers

Víctor M. ALFARO
Departamento de Automática, Escuela de Ingeniería Eléctrica, Universidad de Costa Rica
San José, 11501-2060 COSTA RICA

Ramon Vilanova
Departament de Telecomunicació i d’Enginyeria de Sistemes, Escola d’Enginyeria, Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, SPAIN


The aim of the paper is to present a design framework for two-degree-of-freedom (2DoF) proportional integral (PI) controllers that allows to deal with the control system performance/robustness trade-off. It is based on the use of a model reference optimization procedure with target servo-control and regulatory control closed-loop transfer functions for first- and second-order-plus-dead-time (FOPDT, SOPDT) models. A smooth servo/regulatory combined performance is obtained by forcing both closed-loop transfer functions to perform as close as possible to target non-oscillatory dynamics. A comparison with other methods shows the effectiveness of the proposed design methodology.


PI controllers, two-degree-of-freedom controllers, model reference control, performance/robustness trade-off.

>>Full text
Victor M. ALFARO, Ramon VILANOVA, Performance/Robustness Trade-off Design Framework for 2DoF PI Controllers, Studies in Informatics and Control, ISSN 1220-1766, vol. 21 (1), pp. 75-83, 2014. https://doi.org/10.24846/v21i1y201209

1. Introduction

Since their introduction in 1940 [1,2] commercial proportional integral derivative (PID) controllers have with no doubt become the best option in industrial control applications. The success is mainly due to its simple structure and the meaning of the corresponding three parameters. This fact makes PID control easier to be understood by more control engineers than advanced control techniques. In addition, the performance of a PI or PID controller is satisfactory in most of industrial applications. See [3,4] as an example.

Since Ziegler and Nichols [5] presented the PID controller tuning rules, a great number of procedures have been developed, from the classic methods of Cohen and Coon [6], López et al. [7], and Rovira et al. [8], and modifications of the original tuning rules [9-11], to a variety of new techniques such as: analytical tuning [12,13]; optimization methods [14,15]; gain and phase margin optimization [14,16].

O’Dwyer [17] presents a collection of tuning rules for PI and PID controllers, which shows their abundance.

Among different approaches, the direct or analytical synthesis constitutes a quite straightforward approach to PI/PID controller design. The controller synthesis presented by Martin [18] made use of zero-pole cancellation techniques. Similar relations were obtained by Rivera et al. [19], applying the IMC concepts [20] to tune PI and PID controllers for low-order process models. A combination of analytical procedures and the IMC tuning can be found in [13, 21-24].

A common characteristic of the analytically deducted tuning methods is that they include a design parameter usually related with the closed-loop control system speed of response. The selection of such design parameter will not only affect the system performance but also its relative stability.

In industrial process control applications, the set-point remains normally constant and a good load-disturbance rejection is required; regulatory control. In addition, due to process operation conditions, the set-point may eventually need to be changed and then a good transient response to this change is required; the so called servo-control. However, because these two demands can not be simultaneously satisfied with a one-degree-of-freedom (1DoF) controller, the use of a two-degree-of-freedom (2DoF) controller allows to tune the controller considering the regulatory control-loop performance and the robustness while using the extra parameter that is provided to improve the servo-control behaviour.

The control system design procedure is usually based on the use of low-order linear models identified at the closed-loop normal operation point. Due to the non-linear characteristics in most of the industrial processes, it is necessary to consider the expected changes in the process characteristics assuming certain relative stability margins, or robustness requirements for the control system. Therefore the design of the closed-loop control system with 2DoF PI controllers must take into account the trade-off between the system performance to load-disturbance and set-point changes and the robustness to variation of the controlled process characteristics [25].

If only the system performance is taken into account, using an integrated error criterion (integrated absolute error (IAE), integrated time-weighted absolute error (ITAE), or integrated squared error (ISE)) or a time response characteristic (overshoot, rise time, or settling time), as in [26,27], the resulting closed-loop control system will probably have very low robustness. On the other hand, if the system is designed to have high robustness, as in [10], and if the performance of the resulting system is not evaluated, the designer would have no idea of the cost involved in operating such a highly robust system. In some previous studies [28,29], the performance and robustness of the system were taken into account for optimizing the IAE or ITAE performance, but only the usual minimum level of robustness could be guaranteed.

Without considering the exception of [13,21,22] the analytically deducted and the IMC-PID tuning rules normally do not take into account the performance/robustness trade-off or provide a recommendation for the design parameter selection.

An alternative way for designing 2DoF PI controllers is presented in this case. What is presented in this paper is a design framework that allows considering all the previously commented aspects at once. The design is build up on a constrained model matching model reference optimization that allows resolving the performance/robustness trade-off with the selection of an appropriate design parameter for first- and second-order-plus-dead-time controlled process models.

As additional contribution the approach also provides a framework where different tuning rules can be evaluated and compared. An original way of establishing such comparison is addressed.

This paper is organized in the following way: the transfer functions of the controlled process model, the controller, and the control system are presented in Section 2; the proposed optimization procedure is described in Section 3; the optimization procedure is summarized in Section 4 and a comparison with other tuning methods is shown in Section 5. The paper ends with some conclusions.


  1. BABB, M., Pneumatic Instruments Gave Birth to Automatic Control, Control Engineering, vol. 37, no. 12, 1990, pp. 20-22.
  2. BENNETT, S., The Past of PID Controllers, in IFAC Digital control, Past, Present and Future of PID Control, Terrassa, Spain, 2000
  3. ABDEL-HADY, F, S. ABUELENIM, Design and Simulation of a Fuzzy-Supervised PID Controller for a Magnetic Levitation System, Studies in Informatics and Control, vol. 17 (3), 2008, pp. 315-328.
  4. MARABA, V. A., A. E. KUZUCUOGLU, Speed Control of an Asynchronous Motor Using PID Neural Network, Studies in Informatics and Control, vol. 20 (3), 2011, pp.199-208.
  5. ZIEGLER, J. G., N. B. NICHOLS, Optimum settings for Automatic Controllers, ASME Transactions, vol. 64, 1942, pp. 759-768.
  6. COHEN, G. H., G. A. COON, Theoretical Considerations of Retarded Control, ASME Transactions, vol. 75, July, 1953.
  7. LÓPEZ, A. M., J. A. MILLER, C. L. SMITH, P. W. MURRILL, Tuning Controllers with Error-Integral Criteria, Instrumentation Technology, vol. 14, 1967, pp. 57-62.
  8. ROVIRA, A., P. W. MURRILL, C. L. SMITH, Tuning Controllers for Setpoint Changes, Instrumentation & Control Systems, vol. 42, 1969, pp. 67-69
  9. CHIEN, I., J. HRONES, J. RESWICK, On the automatic Control of generalized passive systems, Trans. ASME, 1952, pp. 175-185.
  10. HÄGGLUND, T., K. J. ASTRÖM, Revisiting the Ziegler-Nichols Tuning Rules for PI Control, Asian Journal of Control, vol. 4, 2002, pp. 354-380.
  11. HANG, C., K. J. ASTRÖM, W. HO, Refinement of the Ziegler-Nichols formula, IEE Proceedings, Part D, 138, 1991, pp. 111-118.
  12. HWANG, S. H., H. C. CHANG, H.C., Theoretical Examination of Closed-loop Properties and Tuning Methods of Single Loop PI Controllers, Chem. Eng. Sci, vol. 42, 1987, pp. 2395-2415.
  13. SKOGESTAD, S., Simple Analytic Rules for Model Reduction and PID Controller Tuning, Journal of Process Control, vol. 13, 2003, pp. 291-309.
  14. ASTRÖM, K. J., T. HÄGGLUND, Advanced PID Control, ISA – The Instrumentation, Systems, and Automation Society, Research Triangle Park, NC, 2006.
  15. ASTRÖM, K. J., H. PANAGOPOULOS, T. HÄGGLUND, Design of PI Controllers Based on Non-convex Optimization, Automatica, vol. 34, 1998, pp. 585-601.
  16. FUNG, H., Q. WANG, Q., T. LEE, T., PI tuning in Terms of Gain and Phase Margins, Automatica, vol. 34, 1998, pp. 1145-1149.
  17. O’DWYER, A., Handbook of PI and PID Controller Tuning Rules, Imperial College Press, London, UK, 2nd edition, 2006.
  18. MARTIN, J., C. L. SMITH, A. B. CORRIPIO, Controller Tuning from Simple Process Models, Instrumentation Technology, vol. 22, no. 12, 1975, pp. 39-44.
  19. RIVERA, D. E., M. MORARI, S. SKOGESTAD, Internal Model Control. 4. PID Controller Design, Industrial & Engineering Chemistry Process Design and Development, vol. 25, 1986, pp. 252-265.
  20. GARCÍA, C. E., M. MORARI, Internal Model Control. 1. a unifying review and some new results, Industrial & Engineering Chemistry Process Design and Development, vol. 21, 1982, pp. 308-323.
  21. ALFARO, V. M., R. VILANOVA, O. ARRIETA, Analytical Robust Tuning of PI Controllers for First-Order-Plus-Dead-Time Processes, In 13th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA2008). 15-18 September, 2008, Hamburg, Germany.
  22. ALFARO, V. M., R. VILANOVA, O. ARRIETA, A Single-Parameter Robust Tuning Approach for Two-Degree-of-Freedom PID Controllers, In European Control Conference, August 23-26, 2009, Budapest, Hungary.
  23. ISAKSSON, A. J., S. F. GRAEBE, Analytical PID Parameter Expressions for Higher Order Systems, Automatica, vol. 35, 1999, pp. 1121-1130.
  24. KAYA, I., Tuning PI Controllers for Stable Process with Specifications on Gain and Phase Margings, ISA Transactions, vol. 43, 2004, pp. 297-304.
  25. ALFARO, V. M., R. VILANOVA, Sintonización de los controladores PID de 2GdL: desempeño, robustez y fragilidad, XIV Congreso Latinoamericano de Control Automático, Santiago, Chile, 24-27 Agosto, 2010, pp. 267-272 (in Spanish).
  26. HUANG, H. P., J. C. JENG, Monitoring and Assessment of Control Performance for Single Loop Analysis, Industrial & Engineering Chemistry Research, vol. 41, 2002, pp. 1297-1309.
  27. TAVAKOLI, S., M. TAVAKOLI, Optimal Tuning of PID Controllers for First Order Plus Time Delay Models using Dimensional Analysis, in The Fourth International Conference on Control and automation (ICCA’03), Montreal, Toronto, Canada, (2003).
  28. SHEN, J. C., New Tuning Method for PID Controller, ISA Transactions, vol. 41, 2002, pp. 473-484.
  29. TAVAKOLI, S., I. GRIFFIN, P. J. FLEMING, Robust PI Controller for Load Disturbance Rejection and Set-point Regulation, In IEEE Conference on Control Applications. 2005, Toronto, Canada.
  30. ASTRÖM, K. J., T. HÄGGLUND, PID Controllers: Theory, Design and Tuning, Instrument Society of America, Research Triangle Park, NC, USA, 1995.
  31. ALFARO, V. M., Low-Order Models Identification from the Process Reaction Curve, Ciencia y Tecnología (Costa Rica), vol. 24, no. 2, pp. 197-216 (in Spanish), at http://revista-ciencia-tecnologia.ucr.ac.cr/index.php/ciencia-tecnologia/article/view/35.
  32. ALFARO, V. M., R. VILANOVA, O. ARRIETA, NORT: a Non-Oscillatory Robust Tuning Approach for 2-DoF PI Controllers, In 18th IEEE International Conference on Control Applications, Saint Petersburg, Russia, July 8-10, 2009, pp. 1003-1008.
  33. ASTRÖM, K. J., T. HÄGGLUND, Revisiting the Ziegler-Nichols step response method for PID control, Journal of Process Control, vol. 14, 2004, pp. 635-650.
  34. TAGUCHI, H., M. ARAKI, Two-Degree-of-Freedom PID controllers – Their functions and optimal tuning, In IFAC Digital Control: Past, Present and Future of PID Control. 2000, Terrassa, Spain.
  35. ASTRÖM, K. J., T. HÄGGLUND, Benchmark Systems for PID Control, In IFAC Digital Control: Past, Present and Future of PID Control (PID’00). 5-7 April, 2000, Terrasa, Spain.
  36. CHIEN, I. L., P. S. FRUEHAUF, Consider IMC Tuning to Improve Controller Performance, Chemical Engineering Progress, October, 1990, pp. 22-41.