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In this section, for levitation height control design, first define, the nonlinear state fictitious controllers. What we want is

Figure 46 In this section, for levitation height control design, first define, the nonlinear state fictitious controllers. What we want is

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Figure 1.1 EMS and EDS System Concept Figure 1.1 illustrates the concept of EMS and EDS briefly.
Figure 1.1 EMS and EDS System Concept Figure 1.1 illustrates the concept of EMS and EDS briefly.
Figure 2.1 Permanent Magnet with Magnetization Orientation
Figure 2.1 Permanent Magnet with Magnetization Orientation
Figure 2.3 Halbach Array Field Concentrations position and the others are located at each side. These are mounted on each side at a 45° angle.
Figure 2.3 Halbach Array Field Concentrations position and the others are located at each side. These are mounted on each side at a 45° angle.
These enhance the stability of the carriage in both the vertical and lateral directions
These enhance the stability of the carriage in both the vertical and lateral directions
which is given by the right hand rule convention. Figure 3.2 Vector Product The direction of the vector @xb is perpendicular to the plane of the vectors a and b
which is given by the right hand rule convention. Figure 3.2 Vector Product The direction of the vector @xb is perpendicular to the plane of the vectors a and b
Figure 3.3 Ideal Halbach Array Configuration (a), Practical (b) Figure 3.4 shows the geometry of a permanent magnet sheet with a thickness of d.
Figure 3.3 Ideal Halbach Array Configuration (a), Practical (b) Figure 3.4 shows the geometry of a permanent magnet sheet with a thickness of d.
For the ideal Halbach array, the magnetizations are given by
For the ideal Halbach array, the magnetizations are given by
Suppose there is another ideal array with a different spatial period and wave number.
Suppose there is another ideal array with a different spatial period and wave number.
If there is a pair magnetization of m, and m-,, these can be decomposed into the sine and cosine pairs as follows,
If there is a pair magnetization of m, and m-,, these can be decomposed into the sine and cosine pairs as follows,
3.1 show the configuration for the modeling of Halbach array configuration. defined, via FEMLAB’s CAD like interface, as 1.25 cm by 1.25 cm square. Figure 3.5 and Table left behind in a medium after an external magnetic field is removed. The bar cross-sections are
3.1 show the configuration for the modeling of Halbach array configuration. defined, via FEMLAB’s CAD like interface, as 1.25 cm by 1.25 cm square. Figure 3.5 and Table left behind in a medium after an external magnetic field is removed. The bar cross-sections are
Table 3.1 Magnetization Orientations and Values for Each Region Another region is defined to surround the Halbach array and serves as the outer boundary
Table 3.1 Magnetization Orientations and Values for Each Region Another region is defined to surround the Halbach array and serves as the outer boundary
To show the magnetic field shape of the four piece Halbach magnet array, respect to
To show the magnetic field shape of the four piece Halbach magnet array, respect to
X-axis Array (width): 0.625, 1.25, 0.625, 1.25, 0.625, Y (height): 1.25
X-axis Array (width): 0.625, 1.25, 0.625, 1.25, 0.625, Y (height): 1.25
Figure 3.10 Case 2, Center Magnet Width is 25% X-axis Array (width): 0.625, 1.25, 0.3125, 1.25, 0.625, Y (height): 1.2!
Figure 3.10 Case 2, Center Magnet Width is 25% X-axis Array (width): 0.625, 1.25, 0.3125, 1.25, 0.625, Y (height): 1.2!
X-axis Array (width): 0.625, 1.25, 0.125, 1.25, 0.625, Y (height): 1.25
X-axis Array (width): 0.625, 1.25, 0.125, 1.25, 0.625, Y (height): 1.25
Figure 3.12 Case 4, Consider Width is 13 and Height 1 (unit is not considered) X-axis Array (width): 2, 3, 3, 3,2, Y (height): 1
Figure 3.12 Case 4, Consider Width is 13 and Height 1 (unit is not considered) X-axis Array (width): 2, 3, 3, 3,2, Y (height): 1
Table 3.2 Configuration Used in Halbach Array Modeling 3.4. Wavelet Analysis
Table 3.2 Configuration Used in Halbach Array Modeling 3.4. Wavelet Analysis
lower frequency otherwise; second signal has a high resolution so this signal is higher frequency Figure 3.15 Low Frequency is First, High Frequency is Next
lower frequency otherwise; second signal has a high resolution so this signal is higher frequency Figure 3.15 Low Frequency is First, High Frequency is Next
To make it clear, figure 3.17 is shown that the high frequency portion first, and then the
To make it clear, figure 3.17 is shown that the high frequency portion first, and then the
resolution so this signal is lower frequency. Figure 3.17 High Frequency is First, Low Frequency is Nex
resolution so this signal is lower frequency. Figure 3.17 High Frequency is First, Low Frequency is Nex
thickness d; is used in this analysis is shown as figure 3.18 and figure 3.19.
thickness d; is used in this analysis is shown as figure 3.18 and figure 3.19.
Figure 3.20 Magnetic Field Distribution of Four-piece Halbach Array From chapter 3.2, if there is a pair magnetization of m, and m-., it can be decomposed into
Figure 3.20 Magnetic Field Distribution of Four-piece Halbach Array From chapter 3.2, if there is a pair magnetization of m, and m-., it can be decomposed into
d=d_, as shown in Figure 3.21 that shows ideal magnetization of Halbach array both vertical and Similar to Fourier analysis, wavelet analysis deals with the expansion of functions in
d=d_, as shown in Figure 3.21 that shows ideal magnetization of Halbach array both vertical and Similar to Fourier analysis, wavelet analysis deals with the expansion of functions in
The Haar Wavelet is described as a step function f(x) with Figure 3.22 Haar Wavelet wavelet due to its similarity in shape and its simple computation. The Haar wavelet is the first
The Haar Wavelet is described as a step function f(x) with Figure 3.22 Haar Wavelet wavelet due to its similarity in shape and its simple computation. The Haar wavelet is the first
field intensity given by [11]. Figure 3.23 Haar Wavelet Level 7 Decomposition, (a) Vertical; (b) Horizontal the wavelet. The calculated wavelet coefficient is applied to estimate the Halbach array magnetic
field intensity given by [11]. Figure 3.23 Haar Wavelet Level 7 Decomposition, (a) Vertical; (b) Horizontal the wavelet. The calculated wavelet coefficient is applied to estimate the Halbach array magnetic
Wavelet Transform, Vertical (top) and Horizontal (bottom) Figure 3.25 Enhanced Harmonic Components at Distance 0.1d for One Period,
Wavelet Transform, Vertical (top) and Horizontal (bottom) Figure 3.25 Enhanced Harmonic Components at Distance 0.1d for One Period,
Figure 3.26 Wavelet Analysis of Enhanced Horizontal Components
Figure 3.26 Wavelet Analysis of Enhanced Horizontal Components
at Different Distances for One Period Figure 3.27 Enhanced Horizontal Components at Different Distances
at Different Distances for One Period Figure 3.27 Enhanced Horizontal Components at Different Distances
Figure 4.1 Classified Stages for the Controller Design
Figure 4.1 Classified Stages for the Controller Design
Let define first the coordination for body fixed and inertial coordinate system. 4.2. Dynamics Analysis Theory
Let define first the coordination for body fixed and inertial coordinate system. 4.2. Dynamics Analysis Theory
Equation 4.1 consists of two parts, translational and rotational, which can be rewritten as
Equation 4.1 consists of two parts, translational and rotational, which can be rewritten as
The equation 4.7 can be rewritten as I,,=0. This reduces the rigid body inertia tensor to: If the vehicle is symmetric around both the x-z and y-z planes, it is implied that L,, = 1. =
The equation 4.7 can be rewritten as I,,=0. This reduces the rigid body inertia tensor to: If the vehicle is symmetric around both the x-z and y-z planes, it is implied that L,, = 1. =
of travel as the x-axis, the lateral across the track as y-axis and the vertical direction as z-axis.
of travel as the x-axis, the lateral across the track as y-axis and the vertical direction as z-axis.
shown in figure 4.2. Each labeling is a Halbach magnet array of the reference system [57] Figure 4.3 Carriage Axis Arrangements
shown in figure 4.2. Each labeling is a Halbach magnet array of the reference system [57] Figure 4.3 Carriage Axis Arrangements
the current for levitation, R[Q] is the resistance, L[H] is the inductance, v; [V] is the levitation
the current for levitation, R[Q] is the resistance, L[H] is the inductance, v; [V] is the levitation
(refer to figure 4.3) where [p,g,7r] are the body-fixed angular velocity (rad/s), J, are the permeability, and Az [m] is the perturbation. inertial terms, and [¢,9,y] are the inertial orientation (Euler Angles) of the vehicle (rad) and are thickness of the rails, up is the desired linear speed, m[Kg] is the vehicle mass, to is the governed by the following set of differential equations:
(refer to figure 4.3) where [p,g,7r] are the body-fixed angular velocity (rad/s), J, are the permeability, and Az [m] is the perturbation. inertial terms, and [¢,9,y] are the inertial orientation (Euler Angles) of the vehicle (rad) and are thickness of the rails, up is the desired linear speed, m[Kg] is the vehicle mass, to is the governed by the following set of differential equations:
From equation 4.34 to equation 4.36, L,, Ly, and L, are the distances between center of
From equation 4.34 to equation 4.36, L,, Ly, and L, are the distances between center of
Table 4.1 Simulation Parameters
Table 4.1 Simulation Parameters
the performance of the levitation trajectory and controllers. The parameters used in th
the performance of the levitation trajectory and controllers. The parameters used in th
Figure 4.4 Ideal System, without Disturbance Figure 4.5 with Disturbance at 5[sec]
Figure 4.4 Ideal System, without Disturbance Figure 4.5 with Disturbance at 5[sec]
Figure 4.8 Trajectory of Voltage and Magnification of Left Figure
Figure 4.8 Trajectory of Voltage and Magnification of Left Figure
The system equation with control is
The system equation with control is
equations some constraints can be given to get optimized solutions. To minimize the mear square of control forces, it can be solved with Lagrange multiplier optimization method. For a
equations some constraints can be given to get optimized solutions. To minimize the mear square of control forces, it can be solved with Lagrange multiplier optimization method. For a
The performance index is given by
The performance index is given by
The solutions are given as: As an alternative, linear quadratic regulator (LQR) control is investigated. For a system
The solutions are given as: As an alternative, linear quadratic regulator (LQR) control is investigated. For a system
Figure 4.9 LQR Optimal Control Block Diagram R is a positive scalar and yields a matrix of optimal gains K for the state feedback.
Figure 4.9 LQR Optimal Control Block Diagram R is a positive scalar and yields a matrix of optimal gains K for the state feedback.
solving it and select the positive values, the results are given substituting equation 4.56 into equation 4.54, we get equation
solving it and select the positive values, the results are given substituting equation 4.56 into equation 4.54, we get equation
Figure 4.10 System Response Under No Control, Levitation Height maximum oscillation region is 0.4cm.
Figure 4.10 System Response Under No Control, Levitation Height maximum oscillation region is 0.4cm.
Figure 4.11 System Response Under No Control, Pitch
Figure 4.11 System Response Under No Control, Pitch
Figure 4.16 System Response Under Optimized Damping, Lateral Position
Figure 4.16 System Response Under Optimized Damping, Lateral Position
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EngineeringElectrical EngineeringAutomotive EngineeringMagnetic LevitationElectrical and Computer Engineering
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