Survey Study of Factional Order Controllers

This is a survey study that presents recent researches concerning factional controllers. It presents several types of fractional order controllers, which are extensions to their integer order counterparts. The fractional order PID controller has a dominant importance, so thirty-one paper are presented for this controller. The remaining types of controllers are presented according to the number of papers that handle them; they are fractional order sliding mode controller (nine papers), fuzzy fractional order sliding mode controller (five papers), fractional order lag-lead compensator (three papers), fractional order state feedback controller (three papers), fractional order fuzzy logic controller (three papers). Finally, several conclusions were drawn from the results that were given in these papers.


INTRODUCTION
Fractional control has become a research topic in control theory and application recently.
Fractional control relies on fractional calculus, which extends the meaning of the derivative d α f(t) dt α to every α, real or complex. In fractional framework, the change from derivation to integration is continuous; thus, a unique operator for differentiation and integration can be defined. This operator is called differintegrator, and is given by if α ∈ ℤ − where α ∈ ℝ is the order of differentiation or integration and c and t are the lower and upper limits, respectively. The three most commonly used definitions for the general fractional differintegral are Riemann-Liouville (RL), Caputo, and Grunwald-Letnikov (GL) definitions (Petras, 2011), (Padula, 2015), and (Zhou, 2017). A fractional controller has fractional order differential equation dynamics. Fractional control adds a degree of freedom by extending the domain of the differentiation and/or integration to real or complex numbers such that it best fits the required specifications. In the past decade, research efforts related to applying fractional calculus to control theory increased. There are many types of fractional order controllers used to control different systems according to the requirements of these system. Some of these types of fractional order controllers with a survey of related papers are given in the following sections.

FRACTIONAL ORDER PID CONTROLLER
The most common fractional order controller is the fractional order PID controller (also called PI λ D μ controller), proposed by (Podlubny, 1994) and (Podlubny, 1999). It is a generalization of the conventional PID controller, where the integer order derivative and integral actions are replaced by fractional order derivative and integral actions. Some of the recent works that utilize this controller are:  Sadati et al. (2007) presented a novel approach to design an optimal PI D controller using Particle Swarm Optimization (PSO) algorithm. In this paper, a new performance criterion in the time domain was proposed. This performance criterion includes overshoot, rising time, settling time, steady state error, and Integral Absolute Error (IAE). The PSO algorithm was employed to search for the optimal PI D controller for Linear Time Invariant (LTI) SISO and MIMO systems. The proposed approach was applied to an electromagnetic suspension system (unstable LTI system). Simulation results showed that this system is more robust and it outperforms the PID control system.  levitation system (unstable nonlinear system). Stabilizing PI D controllers were designed by linearizing the nonlinear system around an operating point. Then these controllers were evaluated using the IAE performance index, and those with best performance were selected to enhance the performance of the closed loop system.  Bucanovic (2014) designed a PI D controller for a cryogenic air separation process (nonlinear MIMO system). After linearizing and decoupling the system, the controller parameters were obtained by minimizing a performance index that involves the IAE, the overshoot, and the rise time. Simulation results showed that the PI D controller enhances the transient response and that it is more robust than the PID controller against external disturbances.  Tajbakhsh et al (2014) presented a robust PI D controller to control the speed of a DC motor (LTI system) with parameter uncertainty. The controller was designed by specifying phase margin, gain crossover frequency, ISO damping property, noise rejection, and disturbance rejection. Simulation results revealed that the proposed controller enhances the performance of the control system and is more robust for model uncertainty.  Bhisikar et al. (2014) presented an approach to design a PD controller for unstable and integrated systems. The controller was designed by using Bode's ideal transfer function to stabilize the system and to improve its performance. The proposed controller outperformed the PD and PID controllers with respect to rise time, percentage overshoot, and settling time and the improvements in these quantities are significant.  Ozkan (2014) designed a PI D controller for an electromechanical actuation system (LTI system). The controller was designed by placing the poles of the closed loop control system on the complex plane by specifying bandwidth and damping ratio values.
Simulations showed that the proposed controller enhances the system stability and robustness against the disturbance.  Divya et al. (1014) designed a PI controller and a PI controller for two interacting tank level process (nonlinear system). The performance of the control system investigated with the Integral Squared Error (ISE) performance index. Simulation results showed that the PI controller outperforms the PI controller in terms of reference input tracking, robustness against variations in the plant parameters, and disturbance rejection.  Junyi (2015) proposed a PI D controller for hydroturbine governing system (nonlinear system with time varying and non-minimum phase characteristics). Investigation of the control system demonstrated that the PI D controller outperforms the PID controller in terms of reducing the oscillations and reducing the settling time.  Sharma (2015) designed a PI D controller for a two-link planar rigid robotic manipulator (coupled and highly nonlinear MIMO system) for trajectory tracking task. The performance of the proposed controller was compared with that of a PID controller. The robustness of the control system was tested for model uncertainties, payload variations with time, external disturbance and random noise; simulation results revealed that the proposed controller is more robust than the PID controller. [PD] controllers, where these methods can be applied to any LTI system, integer or fractional. In a similar manner to the [PI] controller, in the [PD] controller, the sum of the proportional term and derivative term is raised to the fractional order .  Zhong et al. (2015) designed PI D and PID controllers and applied them to stabilize a solid core magnetic bearing system (LTI FO system). Simulations and experiments were implemented to compare the performance of the PID and the PI D controllers. The results showed that the PI D control system outperforms the PID control system by giving smaller overshoot, less oscillation, and less settling time. Also, the PI D controller achieves larger stability margin, higher closed loop bandwidth, and better robustness for gain variation.

 Tepljakov et al (2015a)
proposed a PI D controller for a FFOPTD system. The PI D controller was designed by first designing a conventional PID controller, sweeping the fractional order parameters and within a specified interval, and then choosing the best controller that minimizes a performance index that is a linear combination of phase margin, gain margin, and ISO damping. controller for satellite attitude system (nonlinear system). This controller was designed by specifying gain crossover frequency, phase margin, and ISO damping property. A traditional integer order lead controller is also designed for comparison purposes. The [PD] control system gave a larger bandwidth, a larger phase margin, and a faster response than the integer order lead control system.  Fola et al. (2016) designed an I D 1− controller an unstable nonlinear system. The controller was designed by a linearization method. The advantage of the proposed design procedure is that the controller parameters are computed directly without optimization. Experimental results showed that the closed loop system is stable at different operating points and is robust to plant uncertainties.  Nankar (2016) designed a PI D controller to control the speed of a DC motor (LTI system). Results showed that the PI D controller outperforms the PID controller in giving better control effect. The values of and were determined by selecting certain values and ranges for them, evaluating the unit step performance of the control system, and then choosing the combination that minimizes the percentage overshoot, peak time, and settling time. Comparisons showed that the PI D controller highly outperforms the PID controller in terms of steady state error, settling time, ISE, and IAE. However, in order to carry out a fair comparison, both controllers must be designed by the same approach; thus, in this paper both controllers should have been designed using PSO algorithm. Aerodynamic System (TRAS). PID and PID controllers were designed to stabilize this system. The controller parameters were tuned using PSO algorithm and ISE performance index. The PID controller outperforms the PID controller by faster response, smaller overshoot and smaller errors. Also, the PID controller reduces the strong effect of the input and output cross coupling.  Mishra et al. (2014) proposed a method to design optimal PID and PID controllers for the TRAS. The controller parameters were tuned using PSO algorithm and ISE performance index. Both controllers stabilized the TRAS successfully by tracking the desired reference angle, but the PI D control system has much less error and less control effort than the PID control system. This method utilized both the analytic and numeric approach to determine the controller parameters. The control design specifications were gain crossover frequency, phase margin, and peak magnitude at the resonant frequency. As a case study, a third order linear time invariant system was taken to be controlled, and the resultant control system exactly fulfilled the control design specification.

FRACTIONAL ORDER SLIDING MODE CONTROLLER
Fractional calculus can be utilized to extend the Integer Order Sliding Mode Controller (IOSMC) to be a fractional controller, named Fractional Order Sliding Mode Controller (FOSMC). This is achieved when the sliding surface contains fractional order derivative and/or integral of the state variables. According to how the fractional order derivative/integral appears in the sliding surface equation and similar to the fractional order PID controller terminology, FOSMC can be classified to PI D SMC, PI SMC, and PD SMC. Some of the recent works that utilize this controller are:  Zhang et al. (2012) proposed a PD FOSMC for a servo control system (LTI system). The parameters of the sliding surface were designed by specifying crossover frequency and phase margin. Furthermore, the switching gain was determined using fuzzy logic system. Simulations and experiments revealed that the proposed FOSMC outperforms the IOSMC and is robust against external disturbances.  Gao (2013) et al. proposed a PI D FOSMC for a gun control system (LTI system). The performances of the FOSMC system that consist of chattering suppression, positioning accuracy and robustness are investigated and compared with that of the IOSMC system. Simulation results showed that the FOSMC can reduce the chattering effects of the IOSMC system and can give more accurate positioning and better robustness. Simulation results demonstrated that the FOSMC gives fast response and eliminates the chattering compared to IOSMC.  Tang (2013) et al. designed a PD FOSMC for an antilock braking system, which is nonlinear and which includes variation and uncertainties in its parameters due to change in vehicle loading and/or road condition. Experimental results showed that the proposed FOSMC outperforms the IOSMC in giving less slip tracking time, less braking time, and less braking distance, and being more robust against changes in the road conditions.  Tianyi (2015) et al. designed a PD FOSMC for a spacecraft attitude system (nonlinear system).
Simulation results demonstrated that this controller makes the spacecraft attitude system have good performance. However, this paper did not compare its proposed control strategy with the other ones. Quadratic Regulator (LQR) for uncertain nonlinear systems. Input/output feedback linearization was used to linearize the nonlinear system and decouple tracking error dynamics, an LQR was designed to stabilize the system so that the tracking error converges to zero as soon as possible.
A PD FOSMC was designed to achieve system robustness. Then the two outputs produced by the two controllers were added. Simulation results showed that the proposed controller achieves high performance and robustness with system uncertainties and that this controller reduces the approach that was adopted in (Bouarroudj et al., 2013) was also used in this paper and similar results were obtained for these two papers. The same design approach that was adopted in (Bouarroudj et al., 2013) and (Bouarroudj et al., 2014) was also used in this paper and similar results were obtained for these three papers.  Long et al. (2015) proposed a fuzzy PD β SMC for a vehicle clutch driving system (nonlinear system). In this paper, the input to the fuzzy system is the sliding surface variable and the output is the switch gain. Theoretical analysis and numerical simulations demonstrated that FFOSMC outperforms the FIOSMC in position control and is more robust against load disturbance and other uncertainties.

FRACTIONAL ORDER LAG/LEAD COMPENSATOR
A Fractional Order Lead-Lag Compensator (FOLLC) is a generalization of the classical lead-lag compensator. In this thesis, the fractional order lead lag compensator is classified into two types according to the way the fractional order derivative/integral is introduced to the classical compensator. If the whole transfer function of the classical compensator is raised to a fractional power, this is called type 1, while if only the complex frequency variable is raised to the fractional power (the same power in both the numerator and denominator), this is called type 2. Some of the introduced simple formulas to design type 2 FOLLC to fulfill the specified magnitude and phase at a specified frequency for LTI system. The phase-magnitude plane regions that can be accessed by these compensators were determined. Numerical results showed that these compensators can be applied in control system design.  N. Sayyaf et al. (2015) presented a type 3 fractional-order compensators to achieve the required magnitudes and phases at two given frequencies (for example, to achieve desired phase and gain margins with adjustable cross frequencies). In this generalization, at first some basic analysis of the phase behavior of this introduced type of fractional order compensators was presented. Also, exact formulas were found for designing this family of compensators in order to provide the aforementioned control objective. Finally, a numerical example was presented to confirm the effectiveness of the proposed design method in control systems. Moreover, by a numerical example it was shown that the introduced compensator can be used in control system design for obtaining desired phase and gain margins with adjustable cross-over frequencies.  Jadhav et al. (2017) proposed a generalized analytical method to design robust type 2 Fractional-Order Lead Compensator (FOLC) for LTI system. The proposed compensator adjusts the system's Bode phase curve to achieve the required phase margin at a given frequency. This compensator satisfies the specifications on static error constant, gain crossover frequency and phase margin. Simulation results showed that the proposed type 2 FOLC gives robust and stable performance compared to type 1 FOLC and Integer Order Lead Compensator (IOLC).

FRACTIONAL ORDER STATE FEEDBACK CONTROLLER
Fractional calculus can be utilized in a state feedback controller so that the control law is a feedback of the fractional derivative of the states. This is called Fractional Order State Feedback Controller (FOSFC). Some of the recent works that utilize this controller are: proposed a FOSFC that feedbacks only one state variable to stabilize an unstable fractional order nonlinear system. By this fractional order controller, the unstable equilibrium points in the fractional order system could be asymptotically stable. However, this paper did not compare its proposed control strategy with the other ones.  Abdulwahhab and Abbas (2018) designed a fractional order state feedback controller for the DDMR. Three trajectories are taken as case studies for the DDMR to track: Circle, Lemniscate of Bernoulli, and Bowditch. The controller was designed by minimizing the ITSE performance index. A stability analysis method (that utilizes indirect Lyapunov theorem) was developed to show that this controller stabilizes the DDMR. Simulation results demonstrated that the fractional order state feedback controller enhances the ITAE, ITSE, ISE, and IAE performance indices.

Fractional Order Fuzzy Logic Controller
Fractional calculus can be utilized to extend the traditional Integer Order Fuzzy logic controller (IOFLC) to be Fractional Order Fuzzy Logic Controller (FOFLC). This is achieved when the