Optimal Nash tuning rules for robust PID controllers
Introduction
It is well-known that proportional-integral-derivative (PID) controllers are still the most widespread controllers in the process industry owing to the cost/benefit ratio they can provide, which is often difficult to improve with more advanced control techniques. In fact, even if PID controllers have been employed for almost a century and a lot of experience in their use has been gained, researchers are continuously investigating new design methodologies in order to improve their overall performance, thanks also to the advancement of the computing technology that makes the application of optimization techniques easier.
The research in PID controllers has always been especially focused on the tuning issue, that is, the selection of the PID parameters that are most suitable for a given application. In particular, the development of tuning rules that allow the user to determine the controller gains starting from a simple process model is a topic that has received great attention since the first proposal made by Ziegler and Nichols [47]. Indeed, many tuning rules have been devised [27]. They are related to different controller structures (the PID can be, for example, in interacting or non interacting form), different process models (for example, a first- or second-order plus dead time transfer function), different control tasks they are required to address (for example, set-point following or load disturbance rejection) and different approaches they are based on (for example, empirical, analytical, or optimal). In fact, it has to be recognized that the tuning of the controller has often to be performed by taking into account conflicting requirements.
From the point of view of output performance, we can identify a design trade-off by considering the effects of load disturbances and set-point changes on the feedback control system. On the other side, there is also a trade-off between performance and robustness. It is worth stressing this point because, in the literature, design techniques normally focus on performance in either servo or regulatory mode, see for example [27], [41] for a historical review.
In general, it has to be taken into account that obtaining a fast load disturbance response usually implies increasing the bandwidth of the control system at the expense of a more oscillatory set-point step response [6]. Further, a decrement of the settling time of the response can usually be obtained at the expense of a decrement of the robustness of the closed-loop control system (and, consequently, an increment of the control effort) [1]. It appears therefore that the tuning of the controller is critical if a one-degree-of-freedom PID controller is used and both tracking and regulatory tasks have to be faced.
The issues mentioned before can be considered as a strong motivation for the development of an intermediate tuning that considers the trade-off between servo/regulation operation modes and between performance and robustness. In this context, the use of multi-objective optimization tools can help in determining the most suitable PID parameters for a given application, by taking into account the above mentioned issues [7], [14], [15], [16], [29], [32], [38], [39], [40].
However, it is clear that having a tuning rule is much more desirable in order to keep the simplicity of the use of PID controllers, which is one of their most appreciated features. For this reason, tuning rules that achieve the minimization of integral performance criteria have been proposed in the past, by assuming a first-order-plus-dead-time (FOPDT) process model [46], which is known to capture well the dynamics of many self-regulating processes. In that work, the set-point following and the load disturbance rejection performance have been considered separately, which makes the choice of the tuning difficult if both tasks have to be addressed in a given application, especially if a one-degree-of-freedom control structure is considered. Tuning rules for weighted servo/regulation control operations, namely, tuning rules that balance the optimal tuning in the two modes, have been presented in [3]. Nevertheless, in these cases the robustness of the control system has not been considered and this might imply a significant performance decrease if the dynamics of the process changes and/or the control effort is too high.
Taking this into account, the purpose of this work, which is an extended version of [35], is to provide a new solution to the tuning problem. First, a multi-objective optimization (MOO) procedure is used in order to maximize the performance while, at the same time, achieving a satisfactory level of robustness. In particular, a multi-objective optimization design (MOOD) procedure based on the Normalized Normal Constraint (NNC) algorithm is applied to a family of FOPDT processes, yielding a set of Pareto fronts. Secondly, the Nash solution [18] is calculated for each case and tuning rules are determined by fitting the results obtained for the different PID parameters. A comparison with the non-robust tuning rules proposed in [3] is also performed.
The paper is organized as follows. Section 2 is devoted to the problem statement. Then, in Section 3 the main concepts related to the MOO are briefly reviewed. In Section 4, the Nash solution is presented as a multi-criteria decision making technique, while in Section 5, the optimization results are used to devise the tuning rules for optimal servo/regulation trade-off. Some examples and simulation results are given in Section 6 and conclusions are drawn in Section 7.
Section snippets
Problem statement
In this section, the control framework and the standard evaluation indexes used in this paper to measure the performance and the robustness are introduced.
Multi-objective optimization
This section provides a brief description of MOO and how a bargaining solution can enter at the decision making stage. The definition of the MOP as well as the generation of the potential solutions are presented in a succinct way as much attention will be devoted to the last step of the MOOD procedure where the final solution has to be chosen and where the bargaining option plays a key role.
Bargaining and trade-off solutions selection
In a transaction, when the seller and the buyer value a product differently, a surplus is created. A bargaining solution is then a way in which buyers and sellers agree to divide the surplus. There is an analogous situation regarding a controller design method that is facing two different cost functions for a system. When the controller locates the solution on the disagreement point (D), as shown in Fig. 3, there is a way for the improvement of both cost functions. We can move within the
Optimal tuning and comparison
Returning to the optimization problem described in Eqs. (8)–(9), it has been solved in two different ways:
- 1.
Case 1: by unconstraining the maximum sensitivity Ms in order to compare the proposed method with the intermediate tuning rules presented in [3] where the robustness of the system was not taken into account explicitly.
- 2.
Case 2: by constraining the maximum sensitivity Ms in a range such as 1.4 ≤ Ms ≤ 2.0., therefore guaranteeing that the resulting control system has an acceptable degree of
Simulation examples
In this section, the tuning rules presented in the previous section are evaluated by considering simulation examples with processes with a different dynamics and a case-study example. A comparison with the tuning rules proposed in [3] is also performed. As the proposal in [3] is based on the ISE performance index, for each process, the ISE in both the servo and the regulatory tasks, are computed. In addition, in order to provide a more global comparison framework, the Ms and the total variation
Conclusions
In this paper, a set of tuning rules for one-degree-of-freedom PID controllers based on a multi-objective optimization strategy has been presented. In particular, the tuning allows the minimization of the integrated square error for the set-point following and load disturbance rejection task subject to a constraint on the maximum sensitivity.
This methodology addresses two different trade-offs: that between the performance and robustness and that between the servo and regulatory modes. It has
Acknowledgment
This work was partially supported by the Spanish Ministry of Economy and Competitiveness program under grants DPI2013-47825-C3-1-R, DPI2016-77271-R and PRX16/00271 and by MINECO and FEDER through the project CICYT HARCRICS (DPI2014-58104-R). This work was carried out during the stay of R. Vilanova as Visiting Professor at the University of Brescia.
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2018, Journal of the Franklin InstituteCitation Excerpt :Recent work [15] developed a novel concept of an interval type-2 fractional order fuzzy PID controller, which requires fractional order integrator and fractional order differentiator. For more discussion, the reader may refer to [16–21] and the references therein. During the past decade, optimal control problems of nonlinear systems have been attracting many researchers from various fields in science and engineering, because of their significance in theory and engineering applications [22].