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High precision positioning technology for a kind of high speed maglev train with an electromagnetic suspension (EMS) system is studied. At first, the basic structure and functions of the position sensor are introduced and some key techniques to enhance the positioning precision are designed. Then, in order to further improve the positioning signal quality and the fault-tolerant ability of the sensor, a new kind of discrete-time tracking differentiator (TD) is proposed based on nonlinear optimal control theory. This new TD has good filtering and differentiating performances and a small calculation load. It is suitable for real-time signal processing. The stability, convergence property and frequency characteristics of the TD are studied and analyzed thoroughly. The delay constant of the TD is figured out and an effective time delay compensation algorithm is proposed. Based on the TD technology, a filtering process is introduced in to improve the positioning signal waveform when the sensor is under bad working conditions, and a two-sensor switching algorithm is designed to eliminate the positioning errors caused by the joint gaps of the long stator. The effectiveness and stability of the sensor and its signal processing algorithms are proved by the experiments on a test train during a long-term test run.

The suspension function of high speed maglev trains is carried out by the electromagnetic attractive force between the electromagnets and the rail, and the train is driven by linear synchronous motor [

In order to reach the most efficient and stable traction performance, the traction system needs to control the current phase of the 3-phased windings to make the traveling magnetic field be synchronized with the magnetic field of the electromagnets. In this process, the precise relative position between the electromagnets and the long stator is a prerequisite. Considering the dimensional accuracy of the tooth-slot structure of the long stator, high precision positioning can be achieved by detecting the tooth-slot structure based on nondestructive detection technology [

This paper researches the system design and the signal processing algorithms of a high precision position sensor of a high speed maglev train. The sections of this paper are organized as follows: in Section 2, the operating principle of the sensor is introduced. Then, a multiple-table lookup algorithm and a suspension gap fluctuation compensation algorithm are designed to improve the positioning precision and enhance the capacity of resisting mechanical disturbances. In Section 3, the reasons for the positioning signal distortion are analyzed. In Section 4, a new kind of time-discrete tracking differentiator (TD) is proposed based on nonlinear optimal control theory. The stability, convergence property and frequency characteristics of the TD are studied and analyzed thoroughly. The delay constant of the TD is figured out, and an effective time delay compensation algorithm is designed. In Section 5, based on the TD technology, a filtering process is introduced in to improve the positioning signal waveform when the sensor is under bad working conditions, and a two-sensor switching algorithm is designed to eliminate the positioning errors caused by the joint gaps of the rails. Section 6 presents the conclusions of this paper. The effectiveness and stability of the sensor and its signal processing algorithms are proven through experiments on a test train during a long-term test run.

There are four same “8”-shaped coils arranged on one side of the sensor facing the long stator [

Taking one of the coils for example, the resonance circuit of the coil is stimulated by a signal source with a constant frequency. Because the electromagnetic characteristics of the long stator are different from those of the air, when the coil moves along the long stator at a certain suspension gap, its equivalent inductance changes periodically with the tooth-slot structure. Thus, the signal amplitude of the resonance circuit changes accordingly to form an amplitude-modulated signal. After demodulation, an approximated sinusoidal wave can be obtained, shown in

According to the relative location between the two coils in one group shown in

A magnetic pole phase period of the 3-phased windings contains six tooth-slot periods as shown in

The data shown in

At first, we choose two proper threshold values _{1} and _{2}. _{1} is slightly bigger than the upper crossing point of the two sampled signals and _{2} is slightly smaller than the lower crossing point. The four signal sections between the thresholds with relatively better linearity and bigger slope rate are used as phase tables, as shown in _{1} denote the sampled value of the difference signal of coil group 1 shown as the continuous line in _{2} is that of coil group 2 shown as the dotted line. If phase _{1} is the index. Otherwise, _{2} is the index. The phase table chosen in the current processing cycle is decided by the phase table chosen in the previous cycle and the relationship between the current sampled values and the threshold values. The phase table switching algorithm is shown in

The four phase tables are calibrated under a certain normal suspension gap. When the train is running, it's impossible for the suspension control system to make the suspension gap be absolutely invariable considering external disturbances such as the topographical relief. But the fluctuation is controlled to be within a certain range. When the suspension gap fluctuates, the amplitude and the DC components of the sampled signals deviate from the calibrated values causing considerable table lookup error. So, before the table-lookup, the sampled values should be normalized.

The latest peak value _{0} and _{0} denote the amplitude and the DC component under the normal suspension gap and _{0} = (_{0}/_{0}. The influence of gap fluctuation is thus eliminated effectively by using the normalized value as the index.

The sensor designed above operates well under normal working conditions, but there are still some special cases need to be considered:

The compensation algorithm supposes that the fluctuation frequency is low enough to ignore the gap change in a tooth-slot period, so the compensation is invalid for high frequency vibration, although this rarely happens. Besides, when suspension gap is too large, the amplitude of the sample signals reduces considerably. In this case, the gap fluctuation compensation method may amplify noises and errors. Further more, considering comparator hysteresis, its impossible for a jumping moment of a square wave to be exactly at the time when the sampled signal of the other coil group reaches its peaks or troughs. This also introduces errors.

There are two kinds of gaps with different size shown in

The situations shown in

Consider the system:
_{1} and _{2} are the system states,

Driving by input

And then, move to the original point along the switching line, shown in

Substituting _{1}(_{1}(

For digital signal processing applications, a discrete form of

Based on

By properly introduce in sampling switch and zero-order hold, a discrete system corresponding to _{1}(0), _{2}(0)). In order to make the state trajectory of _{a}

Substituting (_{1}(0), _{2}(0)) into

If the initial states satisfy:

Let:

Then, the system states can move to the original point in one step. Where _{1}(1), _{2}(1)) = (0, 0).

On the other hand, to make the states of _{1}(0), _{2}(0)) into _{1}(2), _{2}(2)) can be obtained. Substituting the expression into:

We have:

According to

If the initial values satisfy

Furthermore, denoting _{a}

Although, _{1}(_{2}(_{1}(_{1}(_{0}

State _{1}(_{2}(_{0}(≥1) is called as filtering factor. The bigger _{0} is, the smoother _{1}(_{2}(

The characteristic

It can be proved that the roots of _{i}

_{0} = 1.

For convenience, the analysis is carried out in continuous-time field. Consider the system:
_{1}, _{2}) = −_{1} − 3_{2}/2. Choose the Lyapunov function as:

Its easy to prove that there exists a constant

Because the partial derivatives of _{i}

On the other hand, consider the continuous form of _{0}^{(}^{i}^{)}(_{i}^{(i − 1)}(_{i}

Let _{1}(_{1}(_{2}(_{2}(

Choose Lyapnov function as (^{+}(_{1} and _{2} are the upper bounds of the first derivative and second derivative of _{0} > 0 satisfying [_{1}/(_{1}+2), 1/2)). Let ^{1/ρ1}, 1) and _{0}

Thus:

Because _{1}(_{1} + 2), we have: _{1}(1 − _{1}_{1}

The block diagram of

Its transfer function is:

Substitute z = e^{jωT}

To avoid frequency aliasing, according to Shannon's sampling theorem, it should be satisfied that _{0} is assigned to different values. It can be seen that the system is a low pass filter. The bigger _{0} is, the lower the pass band is. According to the amplitude-frequency characteristic diagram, to make the amplitude of the filtered signal be approximate to that of the original signal, it should be satisfied that _{0}

The phase-frequency characteristic of _{f}_{f}_{0}_{m}_{e}

Thus:

Considering that a phase lag of

_{0}

Maglev trains' acceleration and change rate are limited to a relatively small range, so they can be approximately treated as constants in time span _{0} = 5.

If high order compensation is needed, multiple tracking differentiators should be combined. Reference [

Inverting the matrix in

Thus:

The block diagram of compensation

_{1} and _{2} are compensation results of

It can be seen that the performance of

The compensation method proposed in [_{1} and _{2} are compensation results of

The experiments are carried out on the 1.5 km test line in Shanghai. The position sensors designed are installed in the box girder of a maglev train shown in

The onboard computer can save and monitor the original data, midcourse data and final data on-line, and the saved data can be analyzed off-line to guide the algorithm modification and program debugging. In the debugging phase, the train is pulled by a rail test truck. After the performance of the positioning system is verified to some extent, traction system can be introduced in to drive the train and further test the positioning system. Generally, the performance of the positioning system can be verified in several ways:

The slope edges of the saw-tooth phase signal wave is always fine and smooth in any case as shown in

There are flags located on certain points on the rail encode with data indicating the precise position of these points. When the train passes a flag, a special onboard instrument can read out the precise position information encoded on the flag and the send it to the onboard computer. With this information, the computer can check the validity of the position sensor.

If the phase signal is incorrect, the traction efficiency will be reduced. The traction efficiency can be estimated according to the traction current and the train's velocity and acceleration.

However, considering the text length, only the experiment results when the positioning system is under certain bad working conditions are given in this paper.

Taking the data shown in

Because of the slight time difference between the jumping edges of the phase signal and those of the tooth-slot number signal, there are spike pulses in signal _{a}_{0} = 100, _{a}_{1} is the filtered signal.

It can be seen that the smoothing effect is good, but the time delay is serious which will result in considerable positioning errors. _{2} is the compensated signal. It can be seen, the phase waveform is improved obviously after the procedure.

Denote the compensated differential signal and filtered signal of _{f}_{a}_{f}

_{1} is the combined signal of sensor 1 and sensor 2 though the switching algorithm, and _{2} is the filtered signal. It can be seen that, the algorithm guaranties a normal signal when the sensors are passing a large joint gap, and tooth-slot period counting loss is avoided.

High precision positioning technology for a high speed maglev train is studied. At first, the operating principle of the position sensor is introduced. A multiple-table lookup algorithm is proposed to improve the linearity of the phase table and the table-lookup precision. A suspension gap fluctuation compensation method is designed to enhance the sensor's capacity of resisting mechanical disturbance. Then, in order to further improve the signal waveform and enhance the reliability of the sensor, a new kind of discrete tracking differentiator is proposed to do some further signal processing based on optimal control theory. The new TD has good filtering and differentiating performances and relatively small calculation load. The dynamic characteristics of the TD are studied thoroughly. The delay constant of the TD is figured out and an effective time delay compensation algorithm is proposed. The TD is use to filter the phase signal, obviously improving the waveform. Finally, a two-sensor switching algorithm is designed based on the TD to avoid the phase signal distortion and tooth-slot period counting loss caused by the large joint gap. The designed sensor performed well during long-term testing runs.

This work was performed at the Engineering Research Center of Maglev Technology at National University of Defense Technology with funding from National Natural Science Foundation of China under grant No. 60874015.

(

(

Operating principle of the sensor.

The relationship between the magnetic pole phase and the tooth-slot structure.

The sampled values of the difference signals in a tooth-slot period.

Phase waveform requirement near joint gaps.

Phase data under different working conditions: (

The state trajectory of

The tracking and differentiating performances of the TD designed.

Block diagram of

Amplitude-frequency characteristic of

Nyquist diagram of

Phase-frequency characteristic of

(_{1}(

(_{2}(

Block diagram of compensation

Compensation effect diagram for _{1}(

Compensation effect diagram for _{1}(

Compensation effect diagram for _{1}(

Filtering effect of the phase signal.

(

(

Switching experiment result.

Phase table switching algorithm.

Phase Table 1 or 2 | _{1} > _{1} |
Phase Table 4 |

_{1} < _{2} |
Phase Table 3 | |

_{2} ≤ _{1} < _{1} |
No switching | |

Phase Table 3 or 4 | _{2} > _{1} |
Phase Table 1 |

_{2} < _{2} |
Phase Table 2 | |

_{2} ≤ _{2} < _{1} |
No switching |