Inter-Ing 2005 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC CONFERENCE WITH INTERNATIONAL PARTICIPATION, TG. MUREŞ ROMÂNIA, NOVEMBER 2005.

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1 Inter-Ing 5 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC CONFERENCE WITH INTERNATIONAL PARTICIPATION, TG. MUREŞ ROMÂNIA, 1-11 NOVEMBER 5. FUZZY CONTROL FOR A MAGNETIC LEVITATION SYSTEM. MODELING AND SIMULATION DUKA ADRIAN-VASILE, GRIF HORATIU PETRU MAIOR UNIVERSITY TG. MURES Key words: magnetic levitation system, fuzzy control, modeling simulation Abstract: This paper investigates a fuzzy control design methodology that can be used to construct a fuzzy controller for a one degree of freedom magnetic levitation (Maglev) system for keeping a steel ball suspended in the air. By controlling the current of an electromagnet, at equilibrium, the generated electromagnetic force will counteract the weight of the steel ball. Fuzzy control provides a formal methodology for representing, manipulating, and implementing a human s heuristic knowledge about how to control a system. The performances of the proposed control algorithm are evaluated and shown by means of digital simulation. 1. Introduction Magnetic levitation systems have many uses such as frictionless bearings, suspension of wind tunnel models, high speed passenger trains etc. These electromagnetic suspension systems can be divided into two types: (1) repulsion type which are naturally stable, and therefore easier to control, and () attraction type which are unstable. This paper examines the design methodology of a fuzzy logic controller for an attraction type magnetic levitation device with one degree of freedom. This plant is characterized by a nonlinear dynamic that is open-loop unstable and, as a result, feedback fuzzy control will be used to stabilize it. The design of controllers, using conventional techniques, for plants with nonlinear dynamics and modeling uncertainties can be often quite difficult. Fuzzy control is a practical alternative for a variety of challenging control applications, since it provides a convenient method for constructing nonlinear controllers via the use of heuristic information. The control objective is to keep a ferro-magnetic object (steel ball) suspended in midair by controlling the current through an electromagnet. The basic principle is to use the current to manipulate the electromagnetic force which can counteract the weight of the steel ball and keep it suspended in the air. By measuring the location of the object using a non-contact sensor, and adjusting the current in the electromagnet based on this measurement, the levitated object can be maintained at a predetermined location.. Dynamical model of the plant The dynamic of the magnetic levitation system shown in Figure 1 is described by the following nonlinear equation. d x( m = mg + f ( x, e (1) dt where f e (x, is the electromagnetic force that counteracts the weight of the ball, x is the distance between the electromagnet and the steel ball, m is the mass of the ball and g is the gravitational constant The electromagnetic force produced by current i(, which acts on the steel ball, is found using the magnetic energy equation as follows: 393

2 L( x) i ( W ( i, x, = W ( i, x, i ( dl( x) i( f e ( x, = = = C (3) x dx x( C is a nonlinear electromagnetic parameter which depends on the incremental inductivity caused by the steel ball (L ) and the levitation distance (X ). For a given distance (X ) this parameter can be determined experimentally based on equation (6). L X C = (4) () Fig. 1 Magnetic levitation system Since the position of the ball influences the electromagnet s inductivity (3) and therefore the electromagnetic force, the changes being nonlinear, and the equilibrium point between the electromagnetic force and the gravity is unstable, the linearization of the electromagnetic force is required as a solution to solving this problem. The linearization is done, using Taylor expansion around a predetermined equilibrium point (I, X ), as follows: I CI CI f e ( x, = C( ) ( )( i I ) + ( )( x X ) (5) X X X where I is the current of the electromagnetic coil when the ball is at X. These linearization values were determined experimentally based on equation (6). At equilibrium the electromagnetic force cancels the gravity. At this moment the acceleration is zero and equation (1) takes the following form: mg = C( X I ) Neglecting the higher order terms of the Taylor expansion (5) we get: 3 d x CI CI m = ( ) i+ ( ) x dt X X where: x = x X (8) i = i I Equation (7) represents the linear equation which describes the dynamic of magnetic levitation system (plan. The plant parameters are given in Tabel 1 (6) (7) 394

3 Tabel 1 Plant parameters Parameter Value M.11 kg X.76 m I.41 m C Nm /A 3. Design of the controller Figure shows the fuzzy control structure applied to the magnetic levitation system. Fig. Control system: block diagram The parameters of the system are: X - displacement of the ball from the equilibrium position; V x voltage supplied by the sensor, proportional to X ; V ref reference voltage needed to keep the ball at the desired position X = ; e control error; V supply voltage needed for I ; V - output control voltage; I electromagnet current. The sensor system consists of an infra-red LED and a phototransistor. Its transfer function is given next: K sensor = 3333[ V / m] (1) The transfer function for the current amplifier is given by equation (11) K amp =.1[ A/ V ] (11) The design of a direct fuzzy controller can be resumed to choosing and processing the inputs and outputs of the controller and designing its four component elements (the rule base, the inference engine, the fuzzification and the defuzzification interfaces). We consider the inputs to the fuzzy system the following variables: the error: e(=v r V x (1) and change in error: d c ( = e( (13) dt and the output variable the incremental control voltage V u = V (14) The universe of discourse of the variables (that is, their domain) was normalized to cover a range of [-1, 1] and scaling gains (g e, g c, g u ) were used to normalize. A standard choice for the membership functions was used with five membership functions for the three fuzzy variables (meaning 5 = 5 rules in the rule base) and symmetric, 5% overlapping triangular shaped membership functions (Figure 3), meaning that only 4 (= ) rules at most can be active at any given time. 395

4 Fig. 3 Membership functions for the fuzzy controller The fuzzy controller implements a rule base made of a set of IF-THEN type of rules. These rules were determined heuristically based on the knowledge of the plant. An example of IF-THEN rule is the following: IF e is negative big (NB) and c is negative big (NB) THEN u is positive big (PB) This rule quantifies the situation where the steel ball is far down of the equilibrium position (X ) and it is movingdownwards (it is falling), hence a large upward electromagnetic force is needed to counteract the movement of the steel ball so that it s gravity will be balanced. To get this large electromagnetic force an increase in the current flowing through the coil is required. This current is obtained by a large (positive big) control voltage supplied by the fuzzy controller. Tabel Rule table for the fuzzy controller d control voltage change in error e( u dt NB NS Z PS PB NB PB PB PB PS Z NS PB PB PS Z NS error Z PB PS Z NS NB e PS PS Z NS NB NB PB Z NS NB NB NB The min-max inference engine was chosen, which for the premises, uses maximum for the OR operator and minimum for the AND operator. The conclusion of each rule, introduced by THEN, is also done by minimum. The final conclusion for the active rules is obtained by the maximum of the considered fuzzy sets. To obtain the crisp output, the centre of gravity (COG) defuzzification method is employed. This crisp value is the resulting controller output 4. Simulation results In order to test the fuzzy controller a Simulink model of the plant was developed. Fig. 4 - Model of the plant 396

5 Using this model (Figure 4) the unstable character of the plant is shown by the step response presented in figure 5. Fig. 5 - Step response The performances of the fuzzy controller were tested on a Simulink model of the entire control scheme, as shown in the following figure. Fig. 6 Fuzzy control structure Based on this model several simulations have been performed. Some interesting results are shown in the following figures. At first the suspension of the steel ball at a predetermined distance has been tested. This distance represents the air gap between the electromagnet and the ball, and the linearizing value X has been chosen for the first simulation. The second simulation tests the controller for a square trajectory around the same linearizing value X ± x. In both figures the evolution of the ball s position and the controller output are shown. Fig. 7 Simulation results The controller s input and output gains used with these simulations were determined heuristically. They were tuned to the following values: g e =.1, g c = 1, g u = 5 which seemed to perform the best. 397

6 5. Conclusions This paper presents a simple method for controlling an electromagnetic levitation system using a simple fuzzy controller. The direct fuzzy controller we designed allowed the use of heuristics (which model the way a human would control the process) via the use of the rule table. Since we generally know and we are able to describe the way to keep the steel ball suspended in the air, the heuristics we have chosen in the design of the controller accomplished the required task successfully. The main goal of this control problem was to assure a stable working condition to keep the steel ball suspended in the air. Due to the particularities of the system (nonlinear, unstable, very fast varying dynamic response, the nonlinear parameter C) and to the uncertainties of the model caused by the experimental identification of the plant parameters further studies would demand a way to automatically tune the fuzzy controller so that it can adapt to different operating conditions. The simulations we performed provided some interesting results which make the fuzzy controller a suitable solution for this control problem. However, we have to consider that there is a deficiency in the plant model, due to the use of the nonlinear parameter C, which we have considered constant. This choice doesn t affect that much the way to control the plant and the results we get as long as the operating conditions are kept around the linearizing values (I and X ) for which parameter C has been determined based on equation (6) References [1] Barie, Walter and Chiasson, John Linear and nonlinear state-space controllers for magnetic levitation, International Journal of Systems Science, Vol. 7, No. 11, 1996, pp [] Chindriş Mircea, Cziker Andrei Utilizarea logicii fuzzy în energetică, Casa cărţii de ştiinţă Cluj Napoca, 4 [3] Craig, K.C. Mechatronics System Design at Rensselaer, Workshop on Mechatronics Education, Stanford University, July 1994 [4] Duka, Adrian-Vasile Studiul metodelor moderne de control automat pentru sistemul de levitaţie magnetică, proiect de diplomă, Universitatea Petru Maior Tg. Mureş, 4 [5] Duka, Adrian-Vasile Implementarea metodelor de control adaptive inteligente, lucrare de disertaţie, Universitatea Petru Maior Tg. Mureş, 5 [6] Franklin, Gene F., Powel, J. David, Emami-Naeini, Abbas - Feedback Control of Dynamic Systems, 4th edition, Prentice Hall, [7] Mathew L. Moore, John T. Musacchio, Kevin M. Passino - Genetic Adaptive Control for an Inverted Wedge: Experiments and Comparative Analyses, Engineering Applications of Artificial Intelligence, Vol. 14, No. 1, Feb. 1, pp [8] Kevin M. Passino, Stephen Yurkovich - Fuzzy Control, Addison Wesley Longman Inc., [9] Shiao, Ying-Shing - Design and Implementation of a Controller for a Magnetic Levitation System, Proc. Natl. Sci. Counc., 1 [1] Taghirad, H.D., Abrishamchian, M. and Ghabcheloo, R., Electromagnetic levitation system: An experimental approach, Proceedings of the 7th International Conference on Electrical Engineering, Power System Vol, pp 19-6, May 1998, Tehran 398