2. Reactor Control Division, Bhabha Atomic Research Centre, India
In 1914, American inventor Emile Bachelet presented his idea of a magnetically levitated (maglev) vehicle with a display model. In magnetic levitation system (MLS), ferromagnetic object levitate by the electromagnetic force induced due to electric current flowing through coil around a solenoid [1][5]. The MLS is inherently nonlinear and unstable [6][10]. However, the advantage is that, as the suspended object has no mechanical support, there is no friction and noise. This allows us to position it accurately — a major advantage, explored in many applications such as magnetically levitated train, magnetic bearing, conveyor system, etc. [1].
In recent years, various methods have been proposed to improve control in MLSbased applications. In 2006, Chiang et al. proposed the concept of integral variablestructure grey control [2]. Yang et al. introduced the concept of adaptive robust outputfeedback control with Kfilter approach in 2008 [3]. In 2011, Lin et al. developed an adaptive PID controller and a fuzzy compensation controller for MLS [1]. In the same year, Morales et al. proposed generalized proportional integral output feedback controller [4]. Recently in 2014, Lin et al. proposed a functionlink cerebellar model articulation control system design based on the neural network concept [5]. However, in spite of all these developments, there is scope for improving efficiency of the controller. The energy required to achieve and maintain the object's position (in the face of disturbances) form an important part of improving the control action. The aim of this paper is to control and maintain the desired object's position, with lesser controller effort. The controller effort minimization is reported in literature [11][14].
The conventional integer order controllers such as, PD and PID controller have been applied in industry for over halfacentury to control linear and nonlinear systems [15]. Recently, such control schemes have been extended to their generalized form using fractional calculus [16], [17] (differentiation and integration of an arbitrary order). The FOPID controller has fractional differintegrator operations. This makes the controller have memory (i.e., its action will memorize its past states) and avoids instantaneous actions. Using the definition of convolution integral, the expression for the fractional integration (which also is embedded in the fractional differentiation) can be written as the convolution of the function and the power function, which is elaborately explained in [17].
In last few decades, the fractional order approach to represent the plant and its controllers are increasingly used to describe the dynamic process accurately [17]. The fractional order transfer function is approximated by integer order transfer function using various methods [16][20]. The proposed method can achieve the desired accuracy over a much larger bandwidth than has been achieved using earlier methods. In applications, where noninteger order controllers are used for integer order plant, there is more flexibility in adjusting the gain and phase characteristics as compared to integer order controllers. This flexibility makes fractional order control a more versatile tool in designing robust and precise control systems.
This paper presents the control of magnetic levitation system using FOPID controller based on optimal polezero approximation method. An algorithm is developed to realize digital FOdifferentiators and FOintegrators. The proposed design procedure aims to ensure that the performance is within required tolerance bandwidth. Five parameters
This work is organized as follows: Section Ⅱ presents the system description. Design procedure of proposed digital FOPID controller using discrete optimal polezero approximation method and dPSO technique is discussed in Section Ⅲ. In Section Ⅳ simulation and experimental results on MLS are provided to validate effectiveness of the proposed controller. Paper concludes with a summary of the results obtained in Section Ⅴ.
Ⅱ. SYSTEM IDENTIFICATION OF MLS MODELA laboratory scale magnetic levitation system is used to evaluate the performance of proposed controller in a controlled environment. MLS levitates an object (metallic ball with mass
Download:


Fig. 1 Schematic diagram of MLS. 
$ m\ddot x = mg  k \frac{i^2}{x^2} $  (1) 
$ i =k_1 u $  (2) 
where
The nonlinear form of maglev model is linearized for analysis of the system [21]. The linear form of the model is obtained from (1) as follows:
$ \begin{align} \label{eq:3} \ddot{x} = g  f(x, i) \end{align} $  (3) 
where
Equilibrium point is calculated by setting
$ g= f(x, i)_{i_o, x_o}. $  (4) 
Linearization is carried out about the equilibrium point of
$ \begin{align} \label{eq:5} \ddot{x} = \left(\frac {\partial f(i, x)}{\partial i}\Big_{i_o, x_o} \triangle i + \frac {\partial f(i, x)}{\partial x}\Big_{i_o, x_o} \triangle x \right). \end{align} $  (5) 
Application of Laplace transform on (5) simplifies it to (6).
$ \begin{align} \label{eq:6} \frac{\triangle X(s)}{\triangle I(s)}= \frac {K_i}{s^2 + K_x} \end{align} $  (6) 
where
Linearized model transfer function (6) has two poles, one of which is in the right half plane at
System identification is a process for obtaining mathematical model using input and output system response. The identified model response should fit with measured response for input applied to the system model [21]. Usually there are two methods for system identification, least mean square (LMS) method and instrumental variable method. The identification of MLS is generally accomplished via traditional least squares method, and is implemented in MATLAB [21], [22].
As MLS is unstable, it has to be identified with a running, stabilizing controller i.e. closed loop identification. Fig. 2 shows the scheme of unstable system identification. LMS method minimizes error between the model and plant output. The optimal model parameters, for which the square of the error is minimal is the result of identification. In order to carry out identification experiment, a discrete controller has to be applied, in the absence of which, the ball falls down, rendering identification impossible. The reference signal
Download:


Fig. 2 Block diagram of MLS control and close loop system identification. 
Fig. 3 presents the comparison between measured and identified model output. Input and output data is taken from MLS system for realtime identification. The best fit obtained is 90.78% for integer order identification, which gives close loop discrete transfer function as
Download:


Fig. 3 Measured and simulated model output. 
$ \begin{align} \label{eq:7} Y(z^{1})= \frac {G(z^{1})}{1+C(z^{1})G(z^{1})} \end{align} $  (7) 
where
Fractional calculus is a branch of mathematics that studies the possibility of taking real or complex number powers of differential and integral operator. Basic definitions of fractional calculus and approximation of fractional integrator and fractional differentiator are described in [16], [17]. The real order operator is generalized as follows:
$ \begin{align} \label{eq:8} D^\alpha = \begin{cases} \dfrac{d^\alpha}{{d}t}, & \alpha>0 \\[2mm] 1, & \alpha=0 \\[2mm] \int_a^t({d}\tau)^{\alpha}, & \alpha < 0 \end{cases} \end{align} $  (8) 
where
Some popular definitions used for general fractional derivatives/integrals in fractional calculus are:
1) RiemannLiouville (RL) definition is given in (9).
$ \begin{align} \label{eq:9} {_a}D_t^\alpha f(t)= \frac{1}{\Gamma(n\alpha)}\left(\frac{{d}}{{ d}t}\right)^n\int_a^t\frac{f(\tau)}{(t\tau)^{\alphan+1}} { d}\tau \end{align} $  (9) 
for
$ \begin{align} \label{eq:10} L\{{_a}D_t^{\pm\alpha} f(t)\}= s^{\pm\alpha}F(s). \end{align} $  (10) 
2) Another definition is based on the concept of fractional differentiation i.e., GrunewaldLetnikov (GL) definition. It is given in (11)
$ \begin{align} \label{eq:11} {_a}D_t^\alpha f(t)= \lim\limits_{h\to 0}h^{\alpha} \displaystyle\sum\limits_{j=0}^{\big[\frac{ta}{h}\big]}(1)^{j}\binom{\alpha}{j} f(tjh) \end{align} $  (11) 
where
3) One more option for computing fractional derivatives is Caputo fractional derivative, its definition is as follows:
$ \begin{align} \label{eq:12} _a^CD_t^\alpha f(t)= \frac{1}{\Gamma(n\alpha)}\int_a^t\frac{f^{n}(\tau)}{(t\tau)^{\alpha+1n}} {d}\tau \end{align} $  (12) 
where
Initial conditions for Caputo's derivatives are expressed in terms of initial values of integer order derivatives. It is noted that for zero initial conditions RL, GL, and Caputo fractional derivatives coincide. Hence, any of the mentioned methods may be used, using the case of zero initial conditions. That would then eliminate the differences arising due to different initial conditions (amongst the three methods).
B. Digital Realization of Fractional Order Differintegrals With Optimal PoleZero for Phase ShapingThe aim behind the choice of frequency domain rational approximation of FOPID controller is to realize the controller in real time using existing analog/digital filters [16][20], [23][25]. Precise hardware implementation of multidimensional natured of fractional order operator is difficult. However, recent research work revealed that bandlimited implementation of FOPID controllers using higher order integer transfer function approximation of the differintegrals give satisfactory performance [26]. This paper, hence utilizes optimal polezero algorithm to realize fractional differintegrals in the frequency domain.
1) Optimal PoleZero Approximation for Phase Shaping: Any rational transfer function is characterized by its poles and zeros. The Bode magnitude plot of noninteger order transfer function has a slope of
$ \begin{align} \label{eq:13} &{\rm{First~pole, }}~~p_1=10^{[\frac{\phi_{\rm req}+45\log\omega_l}{45}+1]}\notag \\ &{\rm{First~zero, }}~~z_1=10\omega_l\qquad\quad\qquad\notag \\ &{\rm{Second~pole, }}~~p_2=10^{[{\log}(p_1)+2\mu]}\notag \\ &{\rm{Second~zero, }}~~z_2=10^{[{\log}(z_1)+2\mu]}\notag \\ &\qquad\qquad\vdots\notag \\ &{\rm{till}\hspace{0.1 cm}} p_n\geq\omega_h. \end{align} $  (13) 
As a particular case, asymptotic phase plot for fractional order integrator circuit having
Download:


Fig. 4 Asymptotic phase plot with three polezero pairs for 
Download:


Fig. 5 Asymptotic phase plot with seven polezero pairs. 
Download:


Fig. 6 The basic idea of frequency band tightening. 
Generally, three polezero pairs per decade give the phase plot within
2) Design of Digital Fractional Order Integrator: The key point in digital implementation of fractional order controller is discretization of fractional order differintegral [24], [27][29]. Contributions related to the discretization have been reported in literature [30][33]. The polezero pairs obtained by algorithm in the above case are discretized using first order hold (foh), zero order hold (zoh), Tustin operator, impulse invariant, matched polezero, and Tustin with prewarp frequency methods. In Fig. 7, Bode plot for
Download:


Fig. 7 Bode plot of 
$ \begin{align} \label{eq:14} z = e^{sT_s}\approx \frac{1+\frac{sT_s}{2}}{1\frac{sT_s}{2}}. \end{align} $  (14) 
The optimal polezero algorithm for digital fractional integrator of
Digital fractional differentiator is designed along the lines of approach similar to that of digital fractional integrator. The architecture of digital FOPID with digital fractional integrator and digital fractional differentiator is shown in Fig. 8.
Download:


Fig. 8 Digital FOPID controller. 
3) Dynamic Particle Swarm Optimization: Recently, many researchers have focused on fractional order controllers tuning, and have obtained meaningful results [34][47]. In this work, dPSO method is used to tune the gains and orders of the controller. PSO is a method for optimizing hard numerical functions, analogous to social behavior of flocks of birds, schools of fish, etc. Here, each particle in swarm represents a solution to the problem defined by its instantaneous position and velocity [48]. The position vector of each particle is represented by unknown parameters to be ascertained. In present case, five control parameters
$ \begin{align} v_{id}=&\ (f(p_{id})f(x_{id})) \times (p_{id}x_{id}) \times sf_1\notag \\[1mm]&\ + (f(p_{gd})f(x_{id})) \times (p_{gd}x_{id}) \times sf_2\notag \\[1mm] &\ + rand( ) \times signis( )\times sf_3 \end{align} $  (15) 
where,
Population size is taken as 100, maximum iteration is set as 50, lower and higher translation frequencies are taken as
Control of MLS using optimized PD, PID, and FOPID controller is studied by MATLAB simulation. A sinusoidal excitation signal is used to study the effects. The controller generates a compensating control signal (based on the positional error) to achieve desired ball position. Controller parameters are tuned using dPSO method as discussed in Section ⅢB3. Figs. 911 present simulation results of the controlled output of MLS using PD, PID, and FOPID controller respectively. Here, encircled part pointed by an arrow shows deviation between desired and actual ball position.
Download:


Fig. 9 Controlled output result of MLS using PD. 
Download:


Fig. 10 Controlled output result of MLS using PID. 
Download:


Fig. 11 Controlled output result of MLS using FOPID. 
The measured and desired ball positions with PD, PID, and FOPID controllers are quantitatively presented in Table Ⅲ. The simulation results indicate that deviation between measured and desired ball positions by using dPSO tuned FOPID controller, is less as compared to PD or PID controllers.
Error values presented in Table Ⅲ are calculated using (16)
$ \begin{align} Percent \; error =& \\ & \frac{desired \; positionactual \; position}{actual \; position}\times 100 \% \end{align} $  (16) 
From the data presented in Table Ⅲ, it is observed that FOPID controller tracks the desired position more efficiently than PD or PID controllers.
B. Real Time Implementation of ClosedLoop SystemThe MLS used for experimentation is shown in Fig. 12. Due to high nonlinearity and openloop instability, MLS system is a very challenging plant. Assembly of MLS consists of a mechanical unit labeled A in Fig. 12. Analogue control interface unit labeled A is used to transfer control signals between computing system and MLS. Advanced PCI1711 I/O card has been inserted into a PCI computer slot and connected with SCSI adapter box using SCSI cable. Mathworks software tools are used to implement control algorithm. It includes MATLAB control toolbox, real time windows workshop (RTW), real time windows target (RTWT), and visual C as programming environment. The flowchart required to obtain executable file is shown in Fig. 13. RTW builds a C++ source code from the simulink model. C code compiler compiles and links the code to produce executable program. RTWT communicates with executable program acting as the control program and interfaces with hardware through input/output board. The block diagram of MLS close loop control is shown in Fig. 14.
Download:


Fig. 12 Experimental setup. 
Download:


Fig. 13 Control system development flow diagram. 
Download:


Fig. 14 Block diagram of MLS close loop control. 
1) Experimental Results Using a PD Controller: The measured and desired ball positions using real time PD controller is shown in Fig. 15(a) and control signal
Download:


Fig. 15 (a) Controlled output result of MLS using a PD controller. (b) Control signal of PD controller. 
Download:


Fig. 16 Experimental PD controller output and object's trajectory captured on DSO. 
The control effort required by controller to maintain object's position can be observed from the control signal
2) Experimental Results Using a PID Controller: Fig. 17(a) shows measured and desired ball positions using PID controller and output of controller
Download:


Fig. 17 (a) Controlled output result of MLS using a PID controller. (b) Control signal of PID controller. 
Download:


Fig. 18 Experimental PID controller output and object's trajectory captured on DSO. 
3) Experimental Results Using a FOPID Controller: The deviation in ball positions using real time FOPID controller is shown in Fig. 19(a). It depicts that error in desired and actual ball positions has reduced in comparison to both PD or PID control actions. The control signal
Download:


Fig. 19 (a) Controlled output result of MLS using a FOPID controller. (b) Control signal of FOPID controller. 
Download:


Fig. 20 Experimental FOPID controller output and object's trajectory captured on DSO. 
From the data presented in Table Ⅳ it is observed that FOPID controller has improved the position accuracy of MLS compared to PD or PID controllers in real time implementation. Also, the percentage error is least for FOPID controller.
The control effort required by PD, PID, and FOPID controllers is calculated using integral absolute error (IAE), ITAE, and integral square error (ISE). The analysis has been carried out for a period of 100 s and is tabulated in Table Ⅴ. Fig. 21 represents the control effort analysis in pictorial form. The error signal is maximum in the case of PD controller and least in the case of FOPID controller. The control signal also follows the same pattern and is least in case of FOPID controller, leading to inference that the control effort in terms of power required by the FOPID controller to maintain the ball position is least amongst the three controllers.
Download:


Fig. 21 Control effort analysis. 
From the analysis, it infers that PID controller is better than PD controller through performance characteristic. FOPID controller shows slight improvement over PID controller, but the effort required is appreciably less for the same improvement. Thus proving superiority of FOPID over integer order controllers.
4) Disturbance Injection Analysis of Controllers: The effect of disturbance is studied by injecting step input to MLS and effect of increased load is studied by introducing another metallic ball in levitation system as shown in Fig. 22. The step is applied after interval of 25 s on initiation of the input while another ball is introduced manually after 35 s. The measured and desired ball positions using a PD controller are presented in Fig. 23(a) and the control signal of a controller is shown in Fig. 23(b). PD controller output and object's trajectory as captured on DSO is presented in Fig. 24.
Download:


Fig. 22 Levitation of two metallic balls. 
Download:


Fig. 23 (a) Controlled output result of MLS using a PD controller. (b) Control signal of PD controller. 
Download:


Fig. 24 Experimental PD controller output and object's trajectory captured on DSO. 
The instant of step applied in input signal and the instant of the addition of extra load are demonstrated by circles marked on figures. Overshoot is observed at the instant of step and after introducing second ball in levitation system. The deviation in ball position is higher as load is increased and greater amount of effort (power consumption, as indicated by high switching fluctuations in the voltage graph) is required by controller to achieve desired ball position.
Similar analysis for PID and FOPID controllers is presented in Figs. 2528. These figures lead to inference that in case of PID controller, the deviation in ball position is high and greater amount of effort is required by controller to achieve ball position as compared to FOPID controller. Comparison shows that FOPID controller requires lesser effort to levitate the object and effect of disturbance is less as compared to PD or PID controllers.
Download:


Fig. 25 (a) Controlled output result of MLS using a PID controller. (b) Control signal of PID controller. 
Download:


Fig. 26 Experimental PID controller output and object's trajectory captured on DSO. 
Download:


Fig. 27 (a) Controlled output result of MLS using a FOPID controller. (b) Control signal of FOPID controller. 
Download:


Fig. 28 Experimental FOPID controller output and object's trajectory captured on DSO. 
In this paper, digital FOPID controller is applied on MLS to improve the positional accuracy and control effort. A new discrete optimal polezero approximation method is proposed for realization of controller. This method provides the optimal number of polezero pairs to maintain the phase value within the tolerance, of around
[1]  C. M. Lin, M. H. Lin, and C. W. Chen, "SoPCbased adaptive PID control system design for magnetic levitation system, " IEEE Syst. J. , vol. 5, no. 2, pp. 278287, Jun. 2011. doi: 10.1109/JSYST.2011.2134530. http://ieeexplore.ieee.org/document/5746626/ 
[2]  H. K. Chiang, C. A. Chen, and M. Y. Li, "Integral variablestructure grey control for magnetic levitation system, " IEE Proc. Electric Power Appl. , vol. 153, no. 6, pp. 809814, Nov. 2006. doi: 10.1049/ipepa:20060056. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4015867 
[3]  Z. J. Yang, K. Kunitoshi, S. Kanae, and K. Wada, "Adaptive robust outputfeedback control of a magnetic levitation system by Kfilter approach, " IEEE Trans. Industr. Electron. , vol. 55, no. 1, pp. 390399, Jan. 2008. doi: 10.1109/TIE.2007.896488. http://ieeexplore.ieee.org/document/4401205/ 
[4]  R. Morales, V. Feliu, and H. SiraRamírez, "Nonlinear control for magnetic levitation systems based on fast online algebraic identification of the input gain, " IEEE Trans. Control Syst. Technol. , vol. 19, no. 4, pp. 757771, Jul. 2011. doi: 10.1109/TCST.2010.2057511. http://ieeexplore.ieee.org/document/5545451/ 
[5]  C. M. Lin, Y. L. Liu, and H. Y. Li, "SoPCbased functionlink cerebellar model articulation control system design for magnetic ball levitation systems, " IEEE Trans. Industr. Electron. , vol. 61, no. 8, pp. 42654273, Aug. 2014. doi: 10.1109/TIE.2013.2288201. http://ieeexplore.ieee.org/document/6651815/ 
[6]  A. El Hajjaji and M. Ouladsine, "Modeling and nonlinear control of magnetic levitation systems, " IEEE Trans. Industr. Electron. , vol. 48, no. 4, pp. 831838, Aug. 2001. doi: 10.1109/41.937416. 
[7]  C. A. Kluever, Dynamic Systems: Modeling, Simulation, and Control. Hoboken, NJ, USA: John Wiley and Sons, 2015. 
[8]  G. F. Franklin, J. D. Powell, and A. EmamiNaeni, Feedback Control of Dynamic Systems. 3rd ed.. Reading, MA, USA: AddisonWesley, 1994. 
[9]  T. H. Wong, "Design of a magnetic levitation control system: an undergraduate project, " IEEE Trans. Educ. , vol. E29, no. 4, pp. 196200, Nov. 1986. doi: 10.1109/TE.1986.5570565. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5570565 
[10]  R. Sinha and M. L. Nagurka, "Analog and labviewbased control of a maglev system with NIELVIS, " in Proc. ASME Int. Mechanical Engineering Congr. Expo. , Orlando, Florida, USA, 2005, pp. 741746. 
[11]  S. Saha, S. Das, R. Ghosh, B. Goswami, R. Balasubramanian, A. K. Chandra, and A. Gupta, "Design of a fractional order phase shaper for Isodamped control of a PHWR under stepback condition, " IEEE Trans. Nucl. Sci. , vol. 57, no. 3, pp. 16021612, Jun. 2010. doi: 10.1109/TNS.2010.2047405. http://ieeexplore.ieee.org/document/5485191/ 
[12]  S. Das, S. Das, and A. Gupta, "Fractional order modeling of a PHWR under stepback condition and control of its global power with a robust PI^{λ}D^{μ} controller, " IEEE Trans. Nucl. Sci. , vol. 58, no. 5, pp. 24312441, Oct. 2011. doi: 10.1109/TNS.2011.2164422. http://ieeexplore.ieee.org/document/6025228/ 
[13]  S. Saha, S. Das, R. Ghosh, B. Goswami, R. Balasubramanian, A. K. Chandra, S. Das, and A. Gupta, "Fractional order phase shaper design with Bodeś integral for isodamped control system, " ISA Trans. , vol. 49, no. 2, pp. 196206, Apr. 2010. doi: 10.1016/j.isatra.2009.12.001. http://www.sciencedirect.com/science/article/pii/S0019057809001013 
[14]  S. Das, "Fuel efficient nuclear reactor control, " in Proc. Int. Conf. Nuclear Engineering, Beijing, China, 2005. 
[15]  K. J. Aström and T. Hägglund, PID Controllers: Theory, Design and Tuning. 2nd ed. Triangle Park, NC, USA: Instrument Society of America, 1995. 
[16]  I. Podlubny, Fractional Differential Equations. New York, NY, USA: Academic Press, 1999. 
[17]  S. Das, Functional Fractional Calculus for System Identification and Controls, Berlin, Heidelberg, Germany: Springer Science and Business Media, 2008. doi: 10.1007/9783540727033. 
[18]  I. Podlubny, "Fractionalorder systems and PI^{λ}D^{μ} controllers, " IEEE Trans. Automat. Control, vol. 44, no. 1, pp. 208214, Jan. 1999. doi: 10.1109/9.739144. 
[19]  A. Charef, H. H. Sun, Y. Y. Tsao, and B. Onaral, "Fractal system as represented by singularity function, " IEEE Trans. Automat. Control, vol. 37, no. 9, pp. 14651470, Sep. 1992. doi: 10.1109/9.159595. 
[20]  A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, "Frequencyband complex noninteger differentiator: characterization and synthesis, " IEEE Trans. Circuits Syst. I Fundam. Theory Appl. , vol. 47, no. 1, pp. 2539, Jan. 2000. doi: 10.1109/81.817385. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=817385 
[21]  Feedback Instruments Ltd., "Magnetic levitation control experiments, " East Susses, UK, Feedback Part No. 116033942S, 2006. 
[22]  R. Pintelon and J. Schoukens, System Identification: A Frequency Domain Approach. New York, NY, USA: WileyIEEE Press, 2012. 
[23]  C. Yeroglu and N. Tan, "Note on fractionalorder proportionalintegraldifferential controller design, " IET Control Theory Appl. , vol. 5, no. 17, pp. 19781989, Nov. 2011. doi: 10.1049/ietcta.2010.0746. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6044595 
[24]  D. Valerio and J. S. da Costa, "Introduction to singleinput, singleoutput fractional control, " IET Control Theory Appl. , vol. 5, no. 8, pp. 10331057, May 2011. doi: 10.1049/ietcta.2010.0332. 
[25]  D. L. Chen, Y. Q. Chen, and D. Y. Xue, "Digital fractional order SavitzkyGolay differentiator, " IEEE Trans. Circuits Syst. Ⅱ Express Briefs, vol. 58, no. 11, pp. 758762, Nov. 2011. doi: 10.1109/TCSⅡ.2011.2168022. http://ieeexplore.ieee.org/document/6080716/ 
[26]  M. Ö. Efe, "Fractional order systems in industrial automation: a survey, " IEEE Trans. Industr. Inf. , vol. 7, no. 4, pp. 582591, Nov. 2011. doi: 10.1109/TⅡ.2011.2166775. http://ieeexplore.ieee.org/document/6011685/ 
[27]  J. A. T. Machado, "Discretetime fractionalorder controllers, " Fract. Calc. Appl. Anal. , vol. 4, no. 1, pp. 4766, Jan. 2001. 
[28]  A. S. Dhabale, R. Dive, M. V. Aware, and S. Das, "A new method for getting rational approximation for fractional order differintegrals, " Asian J. Control, vol. 17, no. 6, pp. 21432152, Nov. 2015. doi: 10.1002/asjc.1148. 
[29]  Y. Q. Chen and K. L. Moore, "Discretization schemes for fractionalorder differentiators and integrators, " IEEE Trans. Circuits Syst. I Fundam. Theory Appl. , vol. 49, no. 3, pp. 363367, Mar. 2002. doi: 10.1109/81.989172. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=989172 
[30]  I. Pan and S. Das, "Gain and order scheduling for fractional order controllers, " in Intelligent Fractional Order Systems and Control, I. Pan and S. Das, Eds. Berlin, Heidelberg, Germany: Springer, 2013, pp. 147157. doi: 10.1007/97836423154976. 
[31]  I. Petráš, "Fractionalorder feedback control of a DC motor, " J. Electr. Eng. , vol. 60, no. 3, pp. 117128, Mar. 2009. 
[32]  S. Cuoghi and L. Ntogramatzidis, "Direct and exact methods for the synthesis of discretetime proportionalintegralderivative controllers, " IET Control Theory Appl. , vol. 7, no. 18, pp. 21642171, Dec. 2013. doi: 10.1049/ietcta.2013.0064. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6670358 
[33]  Y. Q. Chen, I. Petras, and D. Y. Xue, "Fractional order controla tutorial, " in Proc. American Control Conf. , St. Louis, MO, USA, 2009, pp. 13971411. doi: 10.1109/ACC.2009.5160719. 
[34]  Y. Jin, Y. Q. Chen, and D. Xue, "Timeconstant robust analysis of a fractional order[proportional derivative] controller, " IET Control Theory Appl. , vol. 5, no. 1, pp. 164172, Jan. 2011. doi: 10.1049/ietcta.2009.0543. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5708228 
[35]  C. A. Monje, B. M. Vinagre, V. Feliu, and Y. Q. Chen, "Tuning and autotuning of fractional order controllers for industry applications, " Control Eng. Pract. , vol. 16, no. 7, pp. 798812, Jul. 2008. doi: 10.1016/j.conengprac.2007.08.006. http://www.sciencedirect.com/science/article/pii/S0967066107001566 
[36]  J. P. Zhong and L. C. Li, "Tuning fractionalorder PI^{λ}D^{μ} controllers for a solidcore magnetic bearing system, " IEEE Trans. Control Syst. Technol. , vol. 23, no. 4, pp. 16481656, Jul. 2015. doi: 10.1109/TCST.2014.2382642. 
[37]  F. Padula and A. Visioli, "Optimal tuning rules for proportionalintegralderivative and fractionalorder proportionalintegralderivative controllers for integral and unstable processes, " IET Control Theory Appl. , vol. 6, no. 6, pp. 776786, Apr. 2012. doi: 10.1049/ietcta.2011.0419. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6218256 
[38]  S. Das, S. Saha, S. Das, and A. Gupta, "On the selection of tuning methodology of FOPID controllers for the control of higher order processes, " ISA Trans. , vol. 50, no. 3, pp. 376388, Jul. 2011. doi: 10.1016/j.isatra.2011.02.003. http://www.ncbi.nlm.nih.gov/pubmed/21420085 
[39]  S. Das, I. Pan, S. Das, and A. Gupta, "A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices, " Eng. Appl. Artif. Intell. , vol. 25, no. 2, pp. 430442, Mar. 2012. doi: 10.1016/j.engappai.2011.10.004. 
[40]  S. Das, I. Pan, S. Das, and A. Gupta, "Improved model reduction and tuning of fractionalorder PI^{λ}D^{μ} controllers for analytical rule extraction with genetic programming, " ISA Trans. , vol. 51, no. 2, pp. 237261, Mar. 2012. doi: 10.1016/j.isatra.2011.10.004. http://www.sciencedirect.com/science/article/pii/S0019057811001194 
[41]  S. Das, I. Pan, K. Halder, S. Das, and A. Gupta, "LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index, " Appl. Math. Model. , vol. 37, no. 6, pp. 42534268, Mar. 2013. doi: 10.1016/j.apm.2012.09.022. http://www.sciencedirect.com/science/article/pii/S0307904X12005197 
[42]  S. Das, I. Pan, and S. Das, "Performance comparison of optimal fractional order hybrid fuzzy PID controllers for handling oscillatory fractional order processes with dead time, " ISA Trans. , vol. 52, no. 4, pp. 550566, Jul. 2013. doi: 10.1016/j.isatra.2013.03.004. http://www.ncbi.nlm.nih.gov/pubmed/23664205 
[43]  S. Das, I. Pan, S. Das, and A. Gupta, "Masterslave chaos synchronization via optimal fractional order PI^{λ}D^{μ} controller with bacterial foraging algorithm, " Nonlinear Dyn. , vol. 69, no. 4, pp. 21932206, Sep. 2012. doi: 10.1007/s110710120419x. 
[44]  S. Saha, S. Das, S. Das, and A. Gupta, "A conformal mapping based fractional order approach for suboptimal tuning of PID controllers with guaranteed dominant pole placement, " Commun. Nonlinear Sci. Numer. Simul. , vol. 17, no. 9, pp. 36283642, Sep. 2012. doi: 10.1016/j.cnsns.2012.01.007. http://www.sciencedirect.com/science/article/pii/S100757041200010X 
[45]  S. Das, I. Pan, K. Halder, S. Das, and A. Gupta, "Impact of fractional order integral performance indices in LQR based PID controller design via optimum selection of weighting matrices, " in Proc. 2012 IEEE Int. Conf. Computer Communication and Informatics, Coimbatore, India, 2012, pp. 16. doi: 10.1109/ICCCI.2012.6158892. 
[46]  S. Das, I. Pan, S. Das, and A. Gupta, "Genetic algorithm based improved suboptimal model reduction in nyquist plane for optimal tuning rule extraction of PID and PI^{λ}D^{μ} controllers via genetic programming, " in Proc. 2011 IEEE Int. Conf. Process Automation, Control and Computing, Coimbatore, India, 2011, pp. 16. doi: 10.1109/PACC.2011.5978962. 
[47]  A. Rajasekhar, S. Das, and A. Abraham, "Fractional order PID controller design for speed control of chopper fed DC motor drive using artificial bee colony algorithm, " in Proc. 2013 World Congr. Nature and Biologically Inspired Computing, Fargo, ND, USA, 2013, pp. 269266. doi: 10.1109/NaBIC.2013.6617873. 
[48]  R. R. Song and Z. L. Chen, "Design of PID controller for maglev system based on an improved PSO with mixed inertia weight, " J. Netw. , vol. 9, no. 6, pp. 15091517, Jan. 2014. doi: 10.4304/jnw.9.6.15091517. 
[49]  A. Q. H. Badar, B. S. Umre, and A. S. Junghare, "Reactive power control using dynamic particle swarm optimization for real power loss minimization, " Int. J. Electr. Power Energy Syst. , vol. 41, no. 1, pp. 133136, Oct. 2012. doi: 10.1016/j.ijepes.2012.03.030. http://www.sciencedirect.com/science/article/pii/S0142061512000853 