1. Introduction
Soft actuators are actuators whose performance tries to mimic the behavior of a biological muscle. A biological muscle performs only two simple motions: contraction and relaxation. Even though this movement seems very simple, the reality is quite the opposite. The basic structure of a biological muscle is a bundle of muscle fibers. Muscle fibers are a compound of myofibrils, of which the basics are sarcomeres. A sarcomere has two parts: myosin filaments and actin filaments. They interact with each other with the help of
ions. The presence of
ions contracts the muscle fiber by
, whereby the duration of the contraction depends on the amount of these ions [
1]. Contraction of the muscle generates F force; the product of contraction and force is
. If A is measured over time, one can obtain power
generated by the contraction of a muscle.
Regarding the activation principle, soft actuators that are most like biological muscles are ionic electroactive polymers [
2,
3]. They deform their principal shape when there is an internal difference in the ionic state. Even though the principle is similar, its deformation differs from biological muscles. They usually exhibit bending movement with a very slow deformation rate. Activation energy to change ionic state is applied by electrical energy. It is a reverse process, but it has a slow recovery rate.
On the other hand, there is another group of soft actuators called dielectric elastomer actuators, also known as DEAs. Their activation energy is also electricity with 1000 times higher potential then in ionic actuators. The principle of the activation of the structure differs from ionic actuators. DEA has a similar structure as a parallel plated capacitor. It has upper and lower conductive electrodes with elastomer used as the dielectric. Elastomer is considered incompressible. Its Poisson number is 0.5. Only direct current can be applied on a DEA. When DC voltage is applied, attractive force is generated between the positive and negative potentials. This is also called the Maxwell stress.
Figure 1 shows the basic structure and principle of activation of a planar DEA. When DC voltage is applied, the elastomer dimensions change from
,
, and
according to
Figure 1. The ratio of change in each direction can be described as
,
, and
, respectively. Since elastomer is incompressible, the following applies:
.
The basic principles of DEAs’ movement are not the same as for biological muscles, but, in comparison with ionic actuators, they move much faster and exhibit larger movements. Under applied voltage they can also hold the tension or position. As such, they can be used as soft actuators with similar behavior as biological muscles.
The introduction of fractional calculus dates back to the 17th century, where Leibniz addressed a letter to L’Hopital asking him about the existence of a half derivative [
4]. This date is marked as the beginning of fractional calculus. Fractional calculus defines integrals and derivatives with non-integer orders, so the name might be a bit confusing, but since then, this term has been generally accepted. Two centuries later the proper definition was developed, with the works of Riemann et al. [
5].
Fractional calculus can be used in different areas. Lately it is being used in describing the bursting of the two pancreatic β-cells working in synchronization. With the help of changing only fractional orders, they are able to model different bursting activities of insulin [
6]. Studying biological conditions with memory characteristics that can be defined with the fractional calculus is another interesting area of use [
7]. Readers interested in areas where fractional calculus is expanding should refer to the work of Sun et al. [
8]. Among others, the main areas of research with fractional calculus are in physics, control, signal and image processing, mechanics and dynamic systems, biology, environmental science, materials, viscoelasticity, and economics [
8]. Fractional calculus is also being used to describe fractional order chaotic systems, systems that are very sensitive to initial conditions [
9]. One such interesting area for using fractional calculus is rheology, part of which is viscoelasticity. The behavior of viscoelastic materials can be simulated using fractional calculus in rheological models, where basic rheological models can be converted into fractional ones [
10]. It can also be used in control theory to convert a classical PID controller into a fractional one [
5,
11,
12]. Expressed as
, the fractional controller has five tuning parameters, which enables a wider fitting range for fine tuning.
Since most dielectrics used in DEA applications perform viscoelastic behavior, it is challenging to achieve satisfactory position control without overshoot. Babič et al. [
13] used a basic Kelvin-Voigt model for material parametrization and self-tuning PID control. They got the first parameters for PID from the Ziegler–Nichols open-loop method. An experiment was conducted with DEA attached to a servo linear motor that maintained a constant force for pre-stretching. They were able to generate displacements up to 2 mm. Position control had 10–20% of overshoot and 0% of stationary error.
In the work of Sarban et al. [
14], they used a Kelvin-Voigt model with three springs in series, where two springs had dampers in parallel connections for material parametrization of an adaptive vibrational isolator in cylindrical form. They used feedforward control, a method based on knowing the mechanical behavior of the plant. The feedforward adaptive control approach was used to predict the mechanical behavior of the isolator. They achieved active vibrational isolation but had no measured step responses.
The Gent model is also commonly used for the parametrization of viscoelastic materials, and it has one spring and a pair of spring dampers in parallel connection. It was used in the work of Li et al. [
15] for DEA parametrization. They used PID control on the proposed model on a sine wave reference signal. They were able to capture amplitude and frequency, but with a phase shift of 90°.
In the work of Ngujen et al. [
16], they used the Mooney–Rivlin model for the material parametrization of cylindrically configured DEA. They used a PWM-PID control that did not eliminate overshoot and still performed small oscillation around the reference point.
In the work of Mustaza et al. [
17], they used a modified Voigt model for the material parametrization of a DEA compound of three equally spaced cylinders disposed at 120° in a symmetrical radial configuration. The modified Voigt model had been already introduced by Yeoh [
18]. Displacement control was implemented with the help of model-based control. The performance of the control depended on the accuracy of the material parametrization. Experiments show that although control is possible, overshoot and static error are not eliminated.
A worm-like soft actuator was created as well as a controller in the work of Cao et al. [
19]. For material parametrization they used a multi-layer Kelvin-Voigt model. They used feedforward plus feedback control schemes. They used a PI control in the feedback loop. Its parameters were gained with the help of the Nichols–Ziegler method. Simulations were tested in MATLAB/Simulink, where step responses were given without overshoot or static errors. Once tested on a real model, the performance of the proposed control algorithm showed poor performance regarding overshoot and static error.
In the work of Rizzello et al. [
20,
21], DEA in the shape of an annular membrane was used for position control. It was pre-stretched on the disc, so there was no need for additional load. However, a spring and an additional load were attached to the actuator. The model was derived with the help of Helmholtz free energy and a nonlinear version of a standard linear solid model. Position control was performed with PID, with the linearization of the model with PID, and with a robust
control algorithm. Although PID control gave satisfactory results, overshoot was not eliminated. On the other hand, linearized PID and
control gave excellent performance with little to no overshoot.
As can be seen, it is quite challenging to eliminate overshoot when dealing with position control on viscoelastic materials.
The use of fractional calculus in mathematical modeling and position control of the DEA is presented for the first time to improve the step response of the DEA and to eliminate the overshoot. The planar structure of DEA is used where pre-stretch is maintained with the help of appropriate weight. Since the performance of the PID control did not give satisfactory results, the control was upgraded to fractional control with the help of the
. The step response of the model was compared to the step response of the first order plus death time (FOPDT) function. The parameters for the PID controller were obtained with the help of FOPDT and the Wang–Juang–Chan algorithm. These parameters and
are starting parameters for a
controller. Nevertheless, the FOMCON toolbox was used in the search for the optimal parameters for the
controller [
22]. With its help, different starting parameters were chosen to find the optimal parameters. The step responses of optimal parameters were tested in Laplace, time, and discrete domains, where responses were, surprisingly, not the same. Consequently, all optimized parameters needed to be validated with a real experiment. For the implementation of
to a real controller, its fractional order of integration and derivation needs to be approximated with an Oustaloup filter. The new transfer function of
with fractional orders approximated can be transformed into a discrete domain and implemented into the microcontroller. Implementing the standard PID Simulink block on a microcontroller provides an anti-windup method for limiting the output of the integrator. However, the possibility of integral windup is not eliminated by implementing
in the form of an approximated discrete transfer function. Consequently, a quasi-anti-windup method was developed and implemented on the microcontroller. Real experiments showed the advantages of using
over a PID controller regarding response time, settling time, and overshoot.
In the Materials and Methods section, a summary is given of the fractional calculus applied to the Kelvin-Voigt model and on parameter identification. Some basics are also presented for the use of fractional calculus in control theory. A Laplace transformation was employed to transform the fractional Kelvin-Voigt model into the
s-domain. The step response was analyzed and compared with the step response of the first order plus death time (FOPDT) function. After the parameters of the FOPDT function were identified, the Wang–Juang–Chan algorithm was introduced to identify parameters of the PID controller [
23]. The identified parameters represented the starting points for parameter identification of the
controller. The optimization of the
controller was explained using the FOMCON toolbox in Matlab [
22]. The
controller needed to be transformed into the Laplace and discrete domain from the time domain. Also presented is the implementation of the
controller to a microcomputer. A quasi-anti-windup method was introduced to limit windup output of the approximated controller in the real experiment.
The simulation results were validated by experiments and deviations in different domain models are presented in the Results section.
The Discussion section highlights conclusions and outlines proposals for future work.
3. Results
The step responses of optimized
parameters on average had the same rise time, the settling time was approximately 5.5 times faster, and overshoot was nearly eliminated in comparison to its PID counterpart, as can be seen in
Table 1. However, the Laplace domain can easily handle fractional orders since derivative is translated into multiplication and integration into division. Even more, the output of the regulator in the Laplace domain is not limited, which means its output can be infinite. Configuring the model in Simulink where the output of the controller must be limited is vital to testing optimized parameters before performing the real experiment. The most promising parameters for the real experiment are highlighted with yellow and the best parameters are highlighted with green in
Table 1.
The step responses of the continuous and the discrete Simulink model were compared with the step responses of the Laplace domain model for all optimized parameters.
Table 2 shows the results of the step responses of the same controller plant model built and tested in different domains with optimized parameters.
The best results were obviously obtained in the Laplace domain. There are more explanations for this. Fractional derivatives and integrals can be transformed easily into the Laplace domain. This could be understood as having an exact plant model with a controller. The second reason is that the output of the controller in the Laplace domain is not saturated between 0 and 1.5 N to limit its output to the maximum force that could be applied on the DEA. Not only is its output not limited, but it also enables the output to be negative. Negative output from the controller means negative force acting on the DEA model, which reduced the settling time and the overshoot. On the other hand, the continuous Simulink model uses a high order Oustaloup filter for approximation since direct use is not possible in Simulink. The output of the approximated controller was saturated between 0 and 1.5, which was responsible for the longer settling time and the overshoot of the DEA model. For the simulation of the continuous Simulink model, a variable simulation step was chosen. The simulation step in the discrete model was set to , which was the same as the step used in the microprocessor. When a grater step was chosen, discrete simulation in Simulink did not converge successfully or it had large settling time. If a small enough step size is chosen for the discrete simulation then the differences between the continuous and the discrete Simulink models are significantly lowered. Otherwise, the differences can be obvious.
Optimal parameters for the model built in the Laplace domain were also optimal for the continuous and discrete Simulink model. The best parameters for all domains are highlighted in green, and the worst parameters used for PID control in all domains are highlighted in red in
Table 2.
After the DEA was set up for the experimental part it was then mounted on the platform in a vertical position and loaded with 315 g weight. According to previously conducted experiments, this is the appropriate weight to hold the DEA in the desired position, otherwise its internal polymers’ connections would bring it back to the initial unloaded state [
29]. Laser position sensor Wenglor YP05MGV80 [
39] was mounted on the DEA. A laser was used since it offers contactless measurements. A high voltage supply was generated from DC to the HVDC converter E1-P600/Y [
40].
Figure 10 shows the experimental setup for the real DEA.
3.1. Step Responses of the Real DEA with PID and Control
The step responses of the DEA were separated into step responses with PID control and step responses with
control.
Figure 11 shows the step responses of individual sets of PID parameters. Eighteen experiments were performed, where different parameters were used. Promising parameters were experimented multiple times. Response time and settling time were measured as well as overshoot (
P[%]) and steady-state error.
Table 3 shows the results of the step responses of the DEA with PID control. It can be seen from
Figure 11 that not all successful step responses performed in the Laplace domain and in the Simulink model gave good results with the real DEA. The best results were obtained with the experiments of numbers 4, 6, 8, 10, and 17. The average rise time was 1.9 s. The average settling time was 10.4 s. The average overshoot was 8.1% of the desired value. Some experiments also produced a steady-state error. The best performers in
Table 3 are marked with the same color as in
Figure 11a.
Figure 11b shows the step responses of the
control of the real DEA. An integral windup problem can be seen at the first four experiment numbers. In some experiments (4 to 9) the integral windup problem did not occur, but this did not guarantee the stability of the controller. Experiments with numbers 10 to 15 show step responses with the quasi-anti-windup method applied. The results were greatly improved, as rise times and settling times were reduced, and overshoots were reduced to an average 2.6% of the desired value.
Table 4 shows the parameters used for individual experiments, as well as the results. The best results are highlighted with the same color as in
Figure 11b.
3.2. Comparing Results of Step Responses for Both Controllers and the Best Parameters in All Domains
Figure 12 shows the comparison of the step responses of both controllers for a real DEA. PID controller results are marked with experiment numbers 0 to 4, and
controller results are marked with experiment numbers 5 to 9. The best parameters for PID control were obtained by the Wuan–Juang–Chan algorithm. Average rise time among step responses with the PID controller was 1.7 s, the average settling time was 10.4 s, and the average overshoot was 8% of the desired value. On the other hand, these gain parameters were also the best for the
control where integration and derivation orders were optimized with help of the FOMCON toolbox.
Table 5 shows the comparison of the results of the step responses obtained with both controllers. It can be seen that the
control improved step responses compared to the PID. For the
control the average rise time was reduced to 1 s, settling time to 5.4 s, and overshoot to 4.3%. With the implementation of the
control a small steady-state error was introduced, which was not greater than 1% and did not affect the controller’s performance significantly. When the order of integration in the
controller was less than 1, steady-state error was introduced to the step response of the plant.
Figure 12b shows the comparison of step responses with the same parameters of PID and
control in different domains. These parameters were
,
0.03,
, and
. The best results were obtained for experiments 0 and 1, representing the Laplace domain, where the rise and settling times were the smallest and there was no overshoot. The reason is that the output of the controller was not saturated between 0 and 1.5 as it was in the Simulink models. The discrete Simulink models had slightly greater response and settling time as well as overshoot, as seen for experiments 2 and 3. Experiments 4 and 5 show the experimental results of the
and the PID control on the real DEA, respectively.
Table 6 shows the response time, settling time, overshoot, and steady-state error of the step responses of controllers in different domains.
4. Discussion
The employment of fractional calculus in control theory can improve the control of the fractional plant. The main objective was to eliminate the overshoot and make the response as fast as possible for the DEA, which was modeled as fractional plant. The easiest way to test fractional plant is in the Laplace domain. With the help of the Laplace transform, fractional derivates are transformed to fractional multiplication and fractional integrals into fractional division. Step responses performed in the Laplace domain gave the best results. However, the output of the fractional controller in the Laplace domain represents the force acting on the DEA plant. If the output is not saturated, then this output can also be negative, which helps reduce the settling time and overshoot of the DEA plant. But this is not the case in real application. When performing the experiment, the only force acting on retracting the DEA to its initial displacement were its internal polymer chains. That is the reason why continuous and discrete models in Simulink must be saturated between 0 and 1.5, which represent the force in N. The introduction of the saturated output of the controller had some negative effects on the step response. It reduced the response time and the settling time and increased the overshoot in the Simulink simulations.
Approximation of the controller with high order Oustaloup filter is needed to perform a Simulink simulation. Continuous simulation was performed with a variable step solver in Simulink. For discrete simulation, a fixed step was chosen, corresponding to the microprocessor step. The size of the fixed step for discrete Simulink simulation must be properly chosen. Greater step size caused larger response time, settling time, and overshoots in discrete Simulink simulation.
To implement the to the microprocessor it must be approximated with an Oustaloup filter in transfer function form and transformed into the z-domain with the same step as defined in the microprocessor. The first position control experiments of the DEA actuator resulted in large overshoot with a large steady-state error. This implied an integral windup problem. A quasi-anti-windup method was introduced to reset the transfer function of the when the output exceeded a threshold value. As shown by the results, successful implementation and realization of the position control was achieved for the selected DEA. Rise time was reduced by nearly half to around 1 s, as was settling time, which was reduced to around 5 s. Overshoot was also much lower, averaging around 4% of the final value. The main disadvantage of the controller was the appearance of a steady-state error, even though the error was smaller than 1% of the desired value and did not influence the performance of the controller.
Observing the transition from the Laplace to the discrete domain, one can conclude that overshoot in step response in the Laplace domain predicts even greater overshoot in step response in continuous and discrete domains for the DEA plant. Controller output in the Laplace domain is not limited as it is in continuous and discrete domains between 0 and 1.5 and can also have negative values, which reduces the overshoot. If overshoot cannot be eliminated with unlimited output of the controller in the Laplace domain, then one can expect overshoot in continuous and discrete domains.
In future work, an anti-windup method with a transfer function for approximation is going to be realized, separated into individual blocks of the z-transform. Each block will be limited with the output. Hopefully this will bring in even better results for position control.