A Simple Analog Controller for a Magnetic Levitation Kit

A Simple Analog Controller for a Magnetic Levitation Kit

Myung – Gon Yoon

Department of Precision Mechanical Engineering Gangneung-Wonju National University,

South Korea

Jung – Ho Moon Department of Electrical Engineering Gangneung-Wonju National University,

South Korea

Abstract A simple proportional-derivative analog controller is proposed for a stabilization of a magnetic levitation kit which is widely adopted as a control education kit. We present a dynamic model of a magnetic levitation system and systematic procedures for a controller synthesis and an actual implementation. From experiments we confirmed a good performance of our closed loop levitation system.

KeywordsMagnetic Levitation; Controller Design; PD Controller

1. INTRODUCTION

   A magnetic levitation system is one of the most popular systems in control education [1-5]. This is because it clearly and intuitively shows a real control action and thus effectively demonstrates the importance of control engineering.

   As a matter of fact in many control laboratories worldwide a magnetic levitation system is widely chosen as a control demonstration kit or a student project. Moreover, the control problem of a magnetic levitation system is not merely important

   but also the electromagnet (actuator). This results in an intricate dynamics of a whole magnetic levitation system.

   In this paper, we report that a simple PD controller can work for a magnetic levitation system with a Hall sensor.

2. SYSTEM MODEL

   We have designed an analog controller for a commercial electromagnetic levitation kit from Zeltom© [6]. This kit has an electromagnet combined with a linear Hall Effect sensor A1324 from Allegro ©. Originally this kit has a digital controller board including a microcontroller, which was replaced with our simple analog controller.

   A. Mathematical Dynamic Model

   In our system, as shown in Fig. 1, a Hall sensor is attached beneath an electromagnet. Define as a distance between the sensor and the mass center of a floating magnet ball. The force

   between the electromagnet and a magnetic ball is given as

   for control education but there are many industrial applications of magnetic levitation techniques.

   =

   4

   (1)

   When a magnetic levitation system is used for a student project or a lab experiment topic, it is highly desirable that a control circuit is simple enough that students with little experiences in electronics can actually implement the control circuit in a limited time. It is also required that a levitation system is low-cost and easy to maintain.

   A low-cost magnetic levitation kit was proposed in [3]. However the controller circuit in [3] includes a fan- management IC and an H-bridge motor driver IC. The functions of those components in a controller circuit are not easy to understand.

   In addition, in [5], a simple analog PD (proportional and derivative) controller was proposed for a magnetic levitation testbed. The position sensor used for measuring the location of a floating object in [5] was an optical sensor. Even though an optical sensor has a merit of having a simple dynamics and showing good robustness to electrical noises, a levitation system combined with an optical sensor requires an external frame to install/align optical sender/receiver and inevitably its size is bigger and it gets far from being low-cost in general. Furthermore, as a minor disadvantage, an optical sensor frame

   where () denotes the current across the electromagnet and

   is a constant.

   Fig. 1. Magnetic Levitation Plant

   From a force balancing equation, we have

   2 [()] = () , (2)

   hides the view of a levitating object.

   2

   4

   A good alternative to an optical sensor is a Hall Effect sensor which is typically installed underneath of an electromagnet. A main disadvantage of a Hall sensor however is that it is subject to not only the position of a levitating object

   In addition, an electrical dynamics of the electromagnet can be expressed by

   () = () + () , (3)

   where , are resistance and inductance of the electromagnet.

   Now consider the following perturbations

   () = + (),

   TABLE I. SYSTEM PARAMETERS

   () =

   + (),

   (4)

   () = + ().

   Under this perturbation, the dynamics (2) and (3) around an operating point (,, ) can be linearized as

   2

   2

   2

   (d) =

   4

   4

   + 4 ,

   5

   5

   (5)

   	parameter	value	unit
   Sensor		2.92	V
   		0.48	V/A
   Electro -magnet		17.31 × 109	kg5/2A
   	R	2.6
   	L	15.0 × 103	H
   Operating Points		0.41	A
   		27.0	mm

   	parameter	value	unit
   Sensor		2.92	V
   		0.48	V/A
   Electro -magnet		17.31 × 109	kg5/2A
   	R	2.6
   	L	15.0 × 103	H
   Operating Points		0.41	A
   		27.0	mm

   = + 1 .

   By substituting system parameters in Table I into (9), we

   After eliminating in (5) and applying Laplace transforms, we obtain the transfer function from to given as

   finally obtain an explicit representation

   31.94 2 + 1888

   ()() = 3 + 173 2 108.4 1.875 × 104

   (10)

   ()

   () =

   ( + ) (2 4)

   5

   5

   (6)

   Some parameters in Table I are taken from a technical note of Zeltom© [7].

3. CONTROLLER DESIGN

   where () and () denote the Laplace transforms of

   () and (), respectively.

   Our Hall sensor has a voltage output of the form

   () = + + () (7)

   2

   where , , are constant sensor parameters. A linearization of

   (7) around z(t)= + results in

   3

   3

   = 2 + . (8)

   Applying Laplace transform to (8) and using () =

   ()/( + ) from (3) and the representation in (6), we obtain a relation between the electromagnet voltage () and a sensor voltage perturbation () as follows;

   (2 4) + 2

   Firstly we note from the root locus plot in Fig. 3 that our open loop transfer function G()() cannot be stabilized with a pure proportional controller () = in Fig. 2.

   Motivated by this fact and, at the same time, in order to minimize the complexities of a controller synthesis and its implementation, we prefer the simplest sabilizing controller. It turns out that a standard PD (proportional and derivative) controller can do the work. The same controller structure was also proposed in [5].

   ()

   () =

   5 ( + ) (2

   7

   5

   5

   4)

   (9)

   From (6) and (9), we can describe our levitation system as a block diagram shown in Fig. 2 where Ref denotes a reference sensor voltage and an open loop transfer function is given as

   (2 4) + 2

   ()() =

   ()

   () =

   5 ( + ) (2

   7

   5

   5

   4)

   Fig. 3. Open Loop Root Locus

   Fig. 2. Block Diagram

   Fig. 4. PD controller

   Our PD controller has the structure shown in Fig. 4 where

   () denote the Laplace transform of an error signal between a reference voltage Ref and an actual sensor signal ().

   The transfer function of the PD controller is given as

   () () = ( + 1) (11)

   ()

   The minus sign in the controller (11) is to be cancelled out because our differential circuit to obtain () also has a minus sign.

   When = 5 and = 10 , for an example, the controller (11) becomes

   () (0.05 + 1). (12) The controller gain is a tuning parameter determined by

   in (11). The root locus of a transfer function (0.05 +

   1)()() is shown in Fig. 5 for 1 . It follow from Fig. 5 that the closed loop system is stable provided that

   8.94 holds.

   A circuit diagram of our analog controller is given in Fig. 6 and a breadboard implementation is shown in Fig. 7. Note that we have ignored the dynamic behaviors of a Darlington transistor TIP102 in the above controller design as we have experimentally verified that the voltage across the electromagnet is almost same as the controller voltage.

   Fig. 5. Closed Loop Root Locus

   Fig. 6. Overall Circuit Diagram

   Fig. 7. Analog Controller Implementation

4. EXPERIMENTS

   By changing controller parameters with different values of resistor and capacitor in (11), we have verified the performance of our closed loop levitation system.

   In overall, for wide ranges of and , our magnetic levitation system remains stable as theoretically predicted from Fig. 5. An example with of = 118 and = 10 is shown in Fig. 8. The closed system in this case has a gain margin of infinity and a phase margin of 31.6° (deg).

   With a fixed = 118 , the magnitude of capacitor can be increased up to 470 , not breaking a stable levitation. Similarly, with a fixed = 10 , the resistance can be increased up to several hundred .

   A more interesting observation is that, as decreasing the capacitor down to zero, the levitating magnet ball shows larger vibrations and when = 0 a stable levitation could not achieved.

   Fig. 8. A Levitation Test

   These observations are well consistent with our theoretical predictions from the root locus in Fig. 5 of the previous sections.

   Finally, note that by changing the reference voltage with a potentiometer (trimmer) in Fig. 6 and Fig. 7, the equilibrium position of a levitating magnet ball can be raised or lowered.

5. CONCLUSION

We have shown that a simple PD analog controller can stabilize an electromagnetic levitation kit equipped with a Hall Effect sensor. Systematic procedures for system modeling, a controller synthesis and an implementation with an operational amplifier are discussed. From experiments we have confirmed that our simple controller can provide a good performance as theoretically predicted.

REFERENCES

1. W. Yu and X. Li, A Magnetic Levitation System for Advanced Control Education, 19th IFAC World Congress, Cape Town, South Africa,

   August 24-29, 2014

2. T. H. Wong, Design of a Magnetic Levitation Control System-An Undergraduate Project, IEEE Transaction on Education, Vol E-29, No. 4, Nov. 1986

3. Lundberg, Kent H., Katie A. Lilienkamp, and Guy Marsden. "Low-Cost Magnetic Levitation Project Kits." IEEE Control Systems Magazine (2004): 65

4. Milica B. Naumovi, Boban R. Veseli, Magnetic Levitation System in Control Engineering Education, Automatic Control and Robotics Vol. 7, No 1, 2008, pp. 151 160

5. K. Craig, T. Kurfess and M. Naguka, Magnetic Levitation Testbed for Controls Education, Proceedings of the ASME Dynamic Systems and Control Division, Vol 64, 1988

6. Zeltom http://zeltom.com/product/magneticlevitation

7. Zeltom, Electromagnetic Levitation System Mathematical Model, private communication, 2009