Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System

Maglev (Magnetic Levitation) systems are an interesting class of systems since they work without any physical contact and are hence frictionless. Due to this attractive property, such systems have the potential for wide range of applications such as maglev trains. Maglev is non-linear due to magnetic field and unstable that suggest the need of stabilizing controller. An appropriate controller is required to levitate the object at desired position. FOPID (Fractional Order Proportional Integral Derivative) controller and ILC (Iterative learning Control) based FOPID controller are designed in this paper for the levitation of metallic ball with desired reference at minimum transient errors. Since maglev is unstable and ILC is used only for stable systems, FOPID controller is used to stabilize the plant. Non-linear interior point optimization method is used to obtain the parameters of FOPID controller. An ILC is used as a feedforward controller in order to improve the response iteratively. P, PD and PID-ILC control laws are used to update the new control input in ILC based FOPID controller. The overall control scheme is therefore a hybrid combination of ILC and FOPID. The effectiveness of proposed technique is analyzed based on performance indices via simulation. ISE (Integral Square Error) and IAE (Integral Absolute Error) is lesser in case of hybrid PID-ILC as compared to simple FOPID controller.


INTRODUCTION
M aglev is a non-contact system. An object is levitated through this system without any human interaction through magnetic force generated by an electromagnetic coil. This type of process offers many real-world applications, for example: magnetic bearings, artificial blood pumps, suspension of wind models and transportation systems [1]. Because of such applications, these systems have been getting increasing attention [2]. Maglev has many advantages.
These systems are frictionless and work with low noise for accurate positioning [3]. Because of their non-contact nature, Maglev system moderates the cost of maintenance and the energy efficiency of these system is high [4].
An electromagnetic coil is fixed in Maglev on the top of the box and sensor is located on the opposite side of this box which senses the position of the ball. When voltage

Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System
is applied to the electromagnet, the current is induced.
This current magnetizes the coil and applies a magnetic force on the ball. When this magnetic force and force of gravity become equal then this ball is suspended at desired position [5].
In theory, it looks very simple to find out "how much is the force on the ball at a specific point", but it is not possible to levitate the object (ball) at desired position in the absence of an appropriate controller. Maglev is highly non-linear and inherently an unstable plant, described by non-linear differential equations [6]. A real-time controller is necessary to stabilize the system and levitate the object at desired position. This real-time controller will keep the two forces (gravitational force and electromagnetic force) on equilibrium point in order to keep the object at desired position. Several techniques have appeared recently in the literature to control the maglev system.
Hassanzadeh et. al. [7] used GA (Genetic Algorithm) to design controller parameters for unstable Maglev and used xPC target and Simulink to implement proposed algorithm. By Abdel-Hady and Abuelenin [8], indicated a technique based on fuzzy logic to improve the performance of Maglev controlled by PID controller.
Ahmad and Javaid [9] proposed linear and non-linear controllers for desired reference trajectory of Maglev.
Kumar et. al. [6] presented PID controller to stabilize the system and for trajectory tracking of Maglev, LQR (Linear Quadratic Regulator) approach was used to obtain the parameters of PID controller. Sathiyavathi et. al. [10] proposed Hybrid ILC technique for Maglev. They designed ILC with PID controller for reference tracking of ball. With this proposed controller the closed loop system has 5.58sec settling time which needed to be improved.
Huang et. al. [11] used two electromagnets to enhance the stability of Maglev. Duka et. al. [1] proposed IMC (Internal Model Control) based PID controller for the levitation of ball at desired position. Unni et. al. [5] modeled maglev and proposed LQR, LQG (Linear Quadratic Gaussian) and Fuzzy Controller to control the system.
Magaji et. al. [12] proposed Fuzzy Logic based PI and PD Controllers for the Maglev. Hajimani et. al. [13] proposed Neural Adaptive Method for the reference tracking of ball in Maglev. Verma et. al. [14] proposed simple PID controller and FOPID controller design for the levitation of ball in the system. Sgaverdea et. al. [15] proposed the design and implementation of MPC (Model Predictive Controller) for Maglev. With this proposed control strategy, the closed loop system has more than 10% overshoot which needs to be improved. Hypiusova et. al. [16] proposed robust PID Controller for unstable Maglev and D-partition technique is used to obtain the PID controller parameters.
Sahoo et. al. [24] proposed fuzzy logic controller for the control of the position of a ferromagnetic ball in Maglev.
With this proposed technique, the close loop system has 0.4sec settling time, 28.57% steady state error, 0.135sec rise time and 0% overshoot. ISE of controller output is 139.8. Fuzzy logic controller has removed overshoot but steady state error, rise time and ISE needs to be improved.
Yaseen et. al. [25] presented PID controller, LQR controller and lead compensator to stabilize the Maglev system.
With these controllers, settling time and rise time is minimum but peak overshoot with PID controller is 43.6, and 39.3% with lead compensator and 0.505% with LQR.
Although LQR controller has minimized the overshoot but failed to remove the overshoots completely.

Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System
ILC based FOPID controller for Maglev is presented in this study. Hybrid P-type, PD-type and PID-type ILC are used in cascade with FOPID controller. As Maglev is nonlinear, firstly it is linearized at equilibrium point. ILC cannot be used for unstable system directly because ILC gives an open loop control for a system so if a system is unstable the error will never be minimized, and ultimately desired response can never be achieved. Therefore, FOPID controller is used to stabilize the system. After

MATHEMATICAL MODELLING OF MAGLEV
Maglev is a complete laboratory system which is used in many real-world applications. The simple illustration of maglev system free body diagram is shown in Fig. 1 [10].
The force F generated by the magnet is increased up to the extent so that it compensates the gravitational force acting on the metallic sphere.

Nonlinear Mathematical Model
The nonlinear mathematical model of maglev is given by where, The electromagnetic force (F em ) depends on two variables: x 1 and x 3 , where x 1 = Position of the ball from electromagnet, x 2 = ball velocity, and x 3 = Current in magnetic coil.
Electromagnetic Coil

Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System
The parameters of the Maglev used in Equations (1-5) are given in Table 1 [17]:

Linearized Mathematical Model
Maglev is a nonlinear system because of magnetic field and approximate linear model can be achieved around some equilibrium point. As the system is third order, so it can be approximated by three first order differential equations. The general state space model is [17]: where,

Fractional Order PID Controller
Podlubny [18] proposed the concept of FOPID (Fractional Order PID) Controller. It consists of fractional order integrator and fractional order differentiator. The general mathematical form of FOPID Controller is given by: Where k p is proportional gain, k i is integral gain and k d is derivative gain.  and  are the fractional powers of

Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System
The parameters of FOPID controller can be obtained by the minimization of the objective function: By putting the value of s 1 from Equation (18) and extracting the real and imaginary parts, we have By substituting Equations (21)(22) in Equation (19)

ITERATIVE LEARNING CONTROLLER
Simple definition of ILC is "A control methodology in which the controller generates the control signal depending upon the previous control signal plus error during the iterations". ILC works to minimize the error iteratively in such a way that the error of second trial is always less than the error of first trial and so on. ILC is like an intelligent control method which is appropriate for controlled systems in which a given task is repeatedly carried out in a limited interval [20].
ILC is an active control scheme which controls the systems in iteration domain, whereas conventional controllers like PID, LQR or MPC control the system in time domain [21,26].
ILC is used in many applications like chemical batch process, robotics and industrial manipulation [22]. The major problem in a feedback system is that if operator repeats the procedure hundred times then every time Here derivative of error is multiplied by some gain. where, Equation (24)

Iterative Learning Control Based Fractional Order PID Controller for Magnetic Levitation System
PID-ILC uses proportional, derivative and integral function on error signal to generate new control signal. The mathematical form is: PID-ILC gives best results because integrator removes the steady state error.
There are two basic configurations used in ILC, embedded and cascaded. In embedded, ILC is used by making some changes in the actual loop of the system while in cascaded, without disturbing the existing configuration of the system ILC is integrated independently.
In this research work, cascaded ILC is used. Block diagram of cascaded ILC is shown in Fig. 3. Here ILC is used as an external controller without disturbing the existing control process. ILC uses the desired signal and the error stored from the previous cycle to make a new reference trajectory for the existing control process. In this configuration only few commands for input reference signal have to be rewritten and no change in the existing loop has to be done, which is an easy task to carry out practically.
If the error and control signal of previous cycles are used to generate the next control input, then such scheme of learning is called as ILC learning by previous cycle.
Learning through previous cycle does not conceal the disturbance which is present in current cycle. To cover up these disturbances, some kind of feedback mechanism should be used which can be done by making control loop closed in time domain. In this paper, we used FOPID as a feedback controller.

SIMULATION RESULTS
FOPID controller is designed for stabilization of the system. It has several advantages over PID controllers due to two additional parameters  and . It stabilizes the system and helps to track the desired trajectory. The closed loop response of maglev with FOPID controller is shown in Fig. 4.
In Fig. 4, there is 19.3% overshoot in closed loop response with simple FOPID which needs to be improved.
Therefore, hybrid ILC is implemented to improve the transient response of the Maglev.
Hybrid ILC is a combination of FOPID and ILC. ILC generates the updated reference signal for existing loop which allows FOPID to easily track the reference signal.

Implementation of proposed technique is done by using
Simulink and MATLAB. In this research work P, PD and PID control laws are used to design ILC.

FIG. 3. BLOCK DIAGRAM OF CASCADED ILC
P-ILC and PD-ILC based FOPID controller has removed the overshoot in the system but still there is steady state error. We obtained best results with hybrid PID-ILC, in this case steady state error is zero.       Results show that proposed hybrid PID-ILC stabilizes the ball in the desired position.