1. Introduction
Theoretical and practical studies have demonstrated the advantages of using fractional calculus in the modelling and control of dynamic systems, mainly from an industrial process control point of view [
1,
2]. For instance, in process modelling, several works (see, for example [
3,
4,
5,
6,
7,
8,
9,
10,
11]) have shown that fractional order models with one or two fractional parameters can represent the process dynamics better than integer transfer functions of low order, such as the well-known first order plus dead time (FOPDT) and second order plus dead time (SOPDT) models. Therefore, a wider range of real-world processes can be modeled and an improved control system design can be achieved.
In particular, this paper considers a fractional first order plus dead time (FFOPDT) model that is able to represent a wide range of dynamics, including non-oscillatory, as the typical first order and oscillatory ones, as those exhibited by under-damped processes. Moreover, the effectiveness of this model has been proved through research (see for example [
12,
13,
14,
15,
16]). This model is useful in practice because it addresses a frequent shortcoming of tuning rules available in the literature, that is, they are based on a specific low-order integer transfer function, such that they can only be applied to a restricted range of dynamics. Thanks to this feature of the FFOPDT model, it is not necessary to change the method used to tune the integer or non-integer controller, because the structure of the model remains the same irrespective of the over- or under-damped nature of the process.
Another interesting application of fractional calculus is the design of the control algorithm. It has been demonstrated that the fractional proportional-integral-derivative (FOPID) controllers provide more flexibility and accuracy in the adjustment of the feedback system. This can be used to guarantee more stringent specifications related to relative stability—phase, gain margins, and maximum sensitivity—and performance—set-point tracking and load-disturbance rejection—in comparison to those achievable with the classical PID controller (see, for instance [
3,
5,
6,
17,
18,
19]).
In the literature, different approaches to tune FOPID controllers have been devised. Some of them take into account the robustness of the control system with respect to process variations and model uncertainty, and therefore specifications such as gain crossover frequency, phase margin, or robustness to variations of the gain are imposed, see [
3,
5,
20,
21]; while others consider the maximum sensitivity index
as a measure of robustness, as in [
6,
16,
22,
23,
24,
25].
Other approaches have applied artificial intelligence based on fuzzy logic control to adjust the FOPID controller parameters [
7,
17,
26], while others focus exclusively on integral performance criteria as in [
27].
Finally, some approaches take into consideration the robustness and the performance of the closed-loop system at the same time; among them, ref [
28] optimizes the load-disturbance rejection, ref [
29] minimises an objective function in the frequency response, and [
18] considers either an optimal performance for set-point tracking task or for load-disturbance rejection.
Despite the wealth of results available in the literature, it is difficult to find a set of rules that consider the trade-off between performance and robustness, and between the set-point tracking task, also known as servo-control operation, and load-disturbance rejection, also known as regulatory-control mode. To overcome these limitations, in this paper we propose a new approach, the FOMCoRoT, to design fractional-order PID and PI controllers. The main novelty of the proposed method is that it explicitly considers the above mentioned trade-offs and uses an FFOPDT model, thereby taking advantage of both a fractional controller and the flexibility of a fractional model. Furthermore, given the considered fractional-order process model, the devised tuning rules are also more general than those found in the literature.
This paper is organized as follows.
Section 2 is devoted to the problem formulation, the description of the process model, as well as of the control algorithm and of the performance and robustness indices.
Section 3 focuses on the design of the
FOMCoRoT method for FOPID and FOPI controllers. An analysis of the robustness and performance is presented in
Section 4 and bounds are established to decide when the use of fractional controllers is recommended to guarantee a minimum improvement of the performance compared to the one obtainable with classical PID/PI controllers. Then, in
Section 5, simulation examples with the corresponding results are presented. Finally, conclusions are drawn in
Section 6.
3. Optimal Tuning
In order to find the optimal parameters for the FOPID controller, the range for the fractional order of the model has been considered. Note that for values of greater than 1.8, the process becomes practically an undamped second order system which is rarely encountered in practical applications.
In the case of FOPI controllers, the fractional order of the model in the range has been established. This is because values of greater than 1.6 leads to extremely low values of the normalized proportional gain , which is in line with the long-held perception that PI regulators are unsuitable to control highly under-damped processes.
In both cases, the fractional order was varied in steps of 0.1 and the normalized fractional dead-time was considered in the range , in steps of 0.1. Note that such a range includes both lag-dominant and dead-time dominant processes for which (FO)PID controllers can achieve a reasonable performance. For larger values of the normalized dead time, more complex control structures, e.g., Smith predictor, should be used.
The optimal parameters for FOPI and FOPID controllers for both
and
were obtained by optimizing the cost function (
8) constrained to
, where the maximum sensitivity
is defined in (
10). To solve the optimization problem, the MATLAB© solver
fminimax and the active-set algorithm were used. Once the optimal parameters were found, different fitting functions were used to obtain simple tuning rules based on the fractional order model (
1).
As an example of how the tuning rule has been obtained, in
Figure 3 the optimal values of the normalized proportional gain
have been plotted for different values of the normalized fractional dead-time
, as well as the corresponding interpolating function in the case of the FOPID controller.
As the final result, the following general structure for the normalized FOPID controller parameters has been devised:
Values of the constants are presented in
Table 1 for
and in
Table 2 for
.
In the case of the FOPI controller, the following general structure was devised:
Values of the constants are presented in
Table 3 for
and in
Table 4 for
. It is important to highlight that when the fractional order
of the model was also equal to one, the optimal value for the fractional term
of the integral mode was also equal to one. The same happens for values of the normalized fractional dead-time
greater than
and
and therefore, in those cases, there is not any advantage in using a more complex structure for the controller as the FOPI. For that reason, the fitting functions presented in Equations (
16)–(
18) were limited to the mentioned range.
5. Simulation Examples
In order to demonstrate the effectiveness of the designed tuning rule
FOMCoRoT, a high order process
studied in [
18] and shown in Equation (
21), as well as a fractional order process
studied in [
16] and shown in Equation (
22), are considered. All numerical simulations have been performed using MATLAB©.
Consider the process
. For the purpose of comparison, and in order to apply different tuning methods, two models were identified: a fractional order model
, by using the IDFOM tool [
41], and the integer order model
that was considered in [
18].
The accuracy of the model is evaluated by measuring the integral of the absolute value of the difference between the step response of the actual system and the model, or ( index). For the index is equal to 0.6031 and for it is equal to 0.5938 and therefore, in this example, the advantages of the fractional calculus in the robustness and performance of the closed-loop system due the controller’s algorithm will mainly be quantified.
Three methods were applied to tune the parameters of different structures of FOPID controllers. The technique proposed in [
18] (referred here as P.&V. SP for set-point tracking or P.&V. LD for load-disturbance rejection) which considers a FOPID controller in series form and a FOPDT model, and aims at minimizing the integrated absolute error when a step change in the set-point or in the load-disturbance appears, with a constraint on the maximum sensitivity index. The second method proposed in [
16], (referred here as H-F) uses series FOPID controller and a fractional order model as the one considered in this work, which aims for a trade-off performance in both set-point tracking and load-disturbance rejection, also considering the maximum sensitivity index as measure of robustness. The third approach is the
FOMCoRoT method developed in this work. The results for performance and robustness are presented in
Table 5 for
and
. As was mentioned above, in this work the derivative mode was only applied to the feedback signal in order to avoid extreme changes in the controller output when a step change in the set-point value is applied. However, for the sake of fair comparison, the tracking performance when the derivative action is applied to the error signal is also evaluated. To this end, the performance indexes
and
are defined, similarly to
and
, respectively, but replacing
with
.
Figure 10 shows the closed-loop responses for both tasks: a step change in the set-point value and in the load-disturbance signal. It can be noted in
Table 5 that, when the constraint on robustness is given by
, the best performance is achieved with the closed-loop system designed using the
FOMCoRoT method (
only has practically the same value in comparison with the one obtained by applying P.&V. LD). When the robustness is given by a nominal value
, the best performance in the feedback system is again provided by the
FOMCoRoT method (only
is better with the H-F 2.0 method, but this can be attributed to a greater value of the maximum sensitivity index,
). If the analysis is focused on the overall performance given by the indices
or
, the best performance is obtained by applying the
FOMCoRoT method.
For the fractional order process
two models were identified: a fractional order model
by applying the methodology proposed by [
16] and an integer order model
obtained by applying the
three points 123c identification method presented in [
42].
The model accuracy was also evaluated through the index. For the index is equal to 0.7557 and for it is equal to 0.2566. It can be noted that the fractional model provides a lower accuracy than the integer model. This is because, despite the increased flexibility of the fractional model, different identification methods yield different results. In any case, the fractional model is sufficiently accurate to be used effectively with the FOMCoRoT tuning rules.
In
Table 6, the performance indexes when step changes appear in the set-point and in the load-disturbance values are presented. It can be noted that the best performance for each operation mode (measured with the
or
index) and for the overall performance (
index) is obtained when the controller parameters are tuned with the
FOMCoRoT method. This can be noted also in
Figure 11.
6. Conclusions
This paper deals with the design of a closed-loop control system using fractional order models and controllers. The procedure used to develop the FOMCoRoT method guarantees a suitable performance in both set-point tracking and load-disturbance rejection, because it has been designed to minimize a combined performance index that takes into account both operation modes. Moreover, the tuning rule designed in this work provides guaranteed stability margins due to a constraint in the control system robustness obtained by imposing suitable values for the maximum sensitivity index .
Through performance analysis and simulation results, it can be concluded that FOPID and FOPI controllers provide a major impact in the closed-loop system performance when a high degree of robustness is required.
Two simulation examples have shown the effectiveness of the FOMCoRoT method in combination with a FFOPDT model, and this work will allow the user to have a systematic method to decide whether it is suitable to use the fractional calculus to design the control system for a given application or not.
Finally, it is worth pointing out that the proposed method deals with the design of industrial controllers, but other research lines such as controllability, observability, optimal control, robust control, etc. should be pursued in fractional order control systems in order to have a complete theoretical framework for these systems, and in particular an understanding of how the fractional order impacts the above-mentioned structural properties of a system.