Magnetic Levitation System Modeling

Introduction

Magnetic levitation is a remarkable technique that utilizes the power of magnetic fields to defy gravity and suspend objects in mid-air. A prominent illustration of its practical implementation is the maglev train, which capitalizes on this levitating effect to minimize friction and achieve remarkable speeds.

The magnetic levitation system is represented by the understudied system depicted below.

By analyzing the aforementioned system, the equations are written as follows.

• $F_m=c\frac{I}{1-y}$
• $m\ddot{y}=-mg-f_v\dot{y}+F_m$
• $V=RI+L\dot{I}$

R	L	g	c	M	fv
$5[Ω]$	$0.02[H]$	$9.84 [\frac{m}{s^2}]$	$0.3 [\frac{N \cdot m}{A^2}]$	$106[g]$	$0.02[\frac{N \cdot s}{m}]$

The state space is depicted below.

$$x(t) = \left(\begin{array}{cc} x_1(t)\\ x_2(t)\\ x_3(t) \end{array}\right)=\left(\begin{array}{cc} y(t)\\ \dot{y}(t)\\ I(t) \end{array}\right)$$

The linearization around y_d as the setpoint is accomplished with the following result.

$$ \left(\begin{array}{cc} \dot{x_1}(t)\\ \dot{x_2}(t)\\ \dot{x_3}(t) \end{array}\right)= \left(\begin{array}{cc} 0 & 1 & 0\\ \frac{9.84}{1-y_d} & -0.188 & 10.55\sqrt{\frac{1}{1-y_d}}\\ 0 & 0 & -250 \end{array}\right) \left(\begin{array}{cc} x_1(t)\\ x_2(t)\\ x_3(t) \end{array}\right)+ \left(\begin{array}{cc} 0\\ 0\\ 250 \end{array}\right)V $$

The open-loop transfer function is obtained by assuming $y_d = 0.306[m]$ and following this procedure.

$$G(s)=\frac{X_1(s)}{V(s)}=\frac{633}{s^3+250.19s^2+32.83s-3542.5}$$

Time Analysis

The closed-loop transfer function with a negative unit feedback is written as follows.

$$H(s)=\frac{kG(s)}{1+kG(s)}=\frac{633k}{s^3+250.19s^2+32.83s+633k-3542.5}$$

The root locus diagram of the given closed-loop transfer function is displayed below. However, since the diagram lies on the right side of the imaginary axis, it indicates instability.

Please find the zoomed version of the root locus diagram displayed below.

To stabilize this system, a PID controller is designed with the following desired attributes for a step input:

1. The maximum step response should be within 2 seconds.
2. The steady-state error should be zero.
3. The settling time should not exceed 2 seconds.
4. The maximum overshoot should be limited to 35%.

The PID controller: $G_c = \frac{6.046(s + 4)(s + 6)}{s}$

After incorporating the PID controller into the system, its behavior is depicted as follows:

Root Locus	Step Response	Step Info

Now this controller is added to the main non-linear system and its ability to control it is as follows.

Frequency_Analysis

The Bode and Nyquist diagrams of this uncontrolled system are depicted below.

Bode	Nyquist

The following conclusions can be drawn from the provided diagrams:

1. As the magnitude never reaches 0 dB, the phase margin is considered to be infinite.
2. The gain margin can be calculated as $k=\frac{1}{G(0)}$.
3. At a frequency of 3 Hz, the gain decreases by 3 dB, indicating a bandwidth of 3 Hz.
4. The system is unstable according to the Nyquist Stability Criterion because there is a pole in the right half of the imaginary axis and N=0, violating the condition $Z=N+P=0$.

To stabilize this system, a PI controller is designed with the following desired attributes for a step input:

1. The maximum step response should be within 2 seconds.
2. The steady-state error should be zero.
3. The maximum overshoot should be limited to 35%. The PI controller: $G_c = \frac{(s + 2.371)}{s}$

Nyquist	Bode	Step Response	Step Info

The system satisfies the Nyquist Stability Criterion as it possesses a pole in the right half of the imaginary axis, and N equals -1, meeting the condition $Z = N + P = 0$, thus establishing stability.

Furthermore, the system currently exhibits a phase margin of approximately 50 degrees.

Now this controller is added to the main non-linear system and its ability to control it is as follows.

Course Description

• Course: Linear Control Systems [ECE 224]
• Semester: Fall 2022
• Institution: School of Electrical & Computer Engineering, College of Engineering, University of Tehran
• Instructor: Dr. Yaghmaea
• Contributors: Fardin Abbasi, Amirhossein Yousefvand