AI-Driven Control Optimization in Electromagnetic Levitation Train Systems

Material and Methods

Nonlinear Modelling of EDS Maglev Train 

The development of a nonlinear model for an Electrodynamic Suspension (EDS) Maglev train requires the independent formulation of mathematical models for both the electromagnetic and mechanical subsystems. Such a structured approach allows for improved analysis of system dynamics and supports the implementation of advanced nonlinear control techniques (Figure 1).

Figure 1: (a) EDS Model, (b) Single Axis Magnetic Suspension System

Figure 1 shows a single axis magnetic levitation system is used, as well as electromagnetic and mechanical equations. The physical paremeters of Ems Maglev Train shown in Table 1.

Electrical Subsystem Modeling

Applying Kirchhoff’s Voltage Law (KVL) to the electromagnet circuit:

V(t)=Ri(t)+L(x)di(t)dt(1)V(t) = Ri(t) + L(x)\frac{di(t)} {dt} \tag{1}V(t)=Ri(t)+L(x)dtdi(t)​(1)

where:

• V(t)V(t)V(t) = applied voltage

• i(t)i(t)i(t) = coil current
• RRR = resistance
• L(x)L(x)L(x) = position-dependent inductance

Energy Stored in the Inductor

• W=12L(x)i2(t)(2)W = \frac {1}{2} L(x) i^2(t) \tag{2}W=21​L(x)i2(t)(2)

Electromagnetic Force Derivation

The electromagnetic force is obtained from the stored magnetic energy:

fm=−dWdxf_m = -\frac{dW}{dx}fm​=−dxdW​

Substituting equation (2):

fm=−ddx(12L(x)i2) (3)f_m = -\frac{d}{dx} \left (\frac {1}{2} L(x) i^2 \right) \tag {3} fm​=−dxd​(21​L(x)i2) (3) fm=−12i2dL(x)dx (4)f_m = -\frac {1}{2} i^2 \frac{dL(x)} {dx} \tag {4} fm​=−21​i2dxdL(x)​(4)

Inductance Model

The inductance varies with position and is approximated as:

L(x)=Kx(5)L(x) = \frac{K}{x} \tag{5}L(x)=xK​(5)

where:

K=μ0N2A2K = \frac {\mu_0 N^2 A}{2}K=2μ0​N2A​

• μ0\mu_0μ0​ = permeability constant
• NNN = number of coil turns
• AAA = pole area

Table 1: Physical Paremeters of EMS Maglev Train

Final Expression of Electromagnetic Force

Substituting (5) into (4):

fm=Ki2(t)2x2(6)f_m = \frac {K i^2(t)} {2x^2} \tag {6} fm​=2x2Ki2(t)​(6)

Mechanical Dynamics

The total force acting on the train:

f=fm−mgf = f_m - mgf=fm​−mg

Using Newton’s second law:

md2xdt2=fm−mg (7)m \frac {d^2 x} {dt^2} = f_m - mg \tag {7}mdt2d2x​=fm​−mg (7)

where:

• mmm = mass of train
• ggg = gravitational acceleration

State-Space Representation

Define state variables:

x1=x, x2=dxdt, x3=i(8)x_1 = x, \quad x_2 = \frac{dx}{dt}, \quad x_3 = i \tag {8}x1​=x, x2​=dtdx​, x3​=i (8)

General nonlinear affine system:

dxdt=f(x)+g(x)u (9) \frac{dx}{dt} = f(x) + g(x)u \tag {9} dtdx​=f(x)+g(x)u (9)

Model Reference Controller Design

The model reference controller is designed to contain two neural networks: a neural network controller and a neural network plant model, as shown in Figure 2. The plant model is identified first, and then the controller is trained so that the plant output follows the reference model output.

Predictive Controller Design

There are different types of neural network predictive controller that are based on linear model controllers. The proposed neural network predictive controller uses a neural network model of a nonlinear plant to predict future plant performance. The proposed controller then calculates the control input that will optimize plant performance over a specified future time horizon. The primary goal of the model predictive control is to determine

Figure 2: Model Reference Control Architecture

the neural network plant model. Then, the plant model is used by the controller to predict future performance.