Mathematical Modelling: Magnetic Levitation System
(Image accessed from paksc.org )
I. INTRODUCTION
The article present a simple model of a Magnetic Levitation System as shown in F ig.1 and F ig.2. This is visually represented in the image above. From these representations, we obtain the mathematical model of its dynamics. In this text, we present in details the process of modelling a system using differential equation and linearize the equation to obtain its state space representation.
II. DESCIPTION OF DYNAMICAL SYSTEMS AND MODELS
A. Electrical Model
A series RL circuit describes the basic electrical model of a magnetic levitation system as shown in Fig.1. We apply Kirchhoff's voltage law (KVL) to obtain the first-order differential equation for the system.
This differential equation (1) describe the dynamics of the electrical model of the magnetic levitation system. In 1986, Wong [2] approximates the inductance L(x) in a magnetic levitation system. Wong approximates the inductances L(x) as
where L0 and x0 are constant and arbitrary to levitation distance. When we substitute (2) to the first-order DE (1), we yield
The first-order DE (3) approximates the electrical model in reference to an arbitrary levitation distance for the magnetic levitation system.
B. Mechanical Model
The ball levitates when the electromagnetic force in the coil is equal to magnetic force F(x, t) exerted by the coil to its mass m. We apply Newton’s Law to yield the dynamics of the system. We get
The energy needed to sustain the levitation in the ball at a distance x is same amount of energy induced by the coil. We obtain
where k1 is a relational constant between energy in the coil and energy experienced by the ball. The inductance L is approximated as 
. If we consider l varies directly with x, we define
where 
and k3 is a constant relating l and x. We substitute (6) to (5) in order to define the magnetic force acting on the ball.
where 
. Then, we substitute (7) to the first order DE (4). We get
III. STATE SPACE MODEL
A. Nonlinear Models
In this section, we define the state variables
We transform first-order DE (3, 8) to state space equations using the state variables (9). We obtain
Considering x0 is a close distance with x, We approximate L(x) as a constant L. We define
We relates the constants for both electrical and mechanical model by considering L and L0 has minimal difference. We substitute 
and 
to equation (5). We get
where k = k4. We simply the state space equation (11) by substituting L0x0 = 2K. We obtain
For first-order DE (9), we have a state space equation:
B. Linearization
Earlier, we derive the state space equations:
We define the equilibrium points:
We use the equilibrium points (16) and partial differentiation to linearize state space equations (15). We obtain the following state space equations:
The state space equations (17, 18, 19) represent the linear approximations of the dynamics of a magnetic levitation system.
C. Vector Matrix
We can express the state space equations (17, 18, 19) to its vector matrix form. We define state space matrices
where A, B, C, and D is the system , input, output and the feedforward matrix respectively.
We obtain the state space system representation
IV. CONCLUSION
In the previous sections, the step by step derivation of the mathematical model (time-domain and state space) was presented. The system represented in Fig. 1. and Fig. 2. yield to a nonlinear equation which is difficult to implement in a controller. These nonlinear equations were linearized using partial differentiation. This is to ensure that the state space model is linear and easier to implement. An important point to remember in linearization is to establish the equilibrium point. In general, the mathematical model of the magnetic levitation system is nonlinear and have two degrees of freedom. This makes the system difficult to develop a suitable controller using the nonlinear system model. Thus, a linearized state space equations were derived to make an easily interpretable controllers.
V. REFERENCES
[3] Norman S. Nise. Control Systems Engineering (7th. ed.). 2000. John Wiley & Sons, Inc., USA.
(Note: All images and formulas in the text is created by the author, except for those with separate citation or links. The formula in the text is formatted in Latex.)
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