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
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.
CITE THIS PAPER AS:
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
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  presented the PID controller tuning rules, a great number of procedures have been developed, from the classic methods of Cohen and Coon , López et al. , and Rovira et al. , 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  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  made use of zero-pole cancellation techniques. Similar relations were obtained by Rivera et al. , applying the IMC concepts  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 .
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 , 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.
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