Research on the Dynamic Compensation Method of the Shock Wave's Evaluation Pressure Sensor

: Due to the steep and changing fast rising edge of explosion shock wave signal, it asks for good dynamic characteristic of sensor and test system. As far as the development of the sensor level, they are difficult to achieve dynamic characteristic requirement of approximate distortionless transmission, so cause a large dynamic error in the actual test. This paper, aiming at providing theoretical support for technology guarantee and service of the shock wave testing system of national shooting range, conducts a research on the method of dynamic characteristic compensation of pressure sensor facing the power evaluation of the blast shock wave. The research proposes and studies the filter design method of dynamic compensation of pressure sensor based on PFQPSO (the progressive function quantum-behaved particle swarm optimization) algorithm, which belongs to inverse modeling. The dynamic compensation can be realized without knowing the model of sensor, so the extra error caused by dynamic modeling of sensors can be avoided. using 8510 series sensor can be used with lower natural frequency and lower measurement range from Endevco Company, this research has had experiment and modeling on the dynamic characteristics of shock tubes as well as design of dynamic compensation filter. Based on the analysis and confirmation, this study illustrates the possibility and effectiveness of this method. filter of pressure sensor based on PFQPSO is proposed and studied. Taking a sensor with poorer dynamic performance in 8510 series of sensor from Endevco Company as an example, this paper carries out the dynamic performance experiment of shock tube, modelling and the design, analysis and verification of dynamic compensation filter based on PFQPSO. The findings show that this method is feasible and effective. In addition, the dynamic compensation filter or inverse filter is recursive, which can be realized by IIR filter, suitable for real-time processing.


Introduction
Blast shock wave is an important physical quantity to evaluate the power in the blast of ammunition and the damage of weapon system, and the accuracy of its pressure characteristic parameters is a critical indicator of evaluating the explosive power of ammunition and weapon system [1] . The pressure signal of blast shock wave is a dynamic signal, shown by a rapid rising process generally in several microseconds and long duration in millisecond or even hundreds of milliseconds, so it is a kind of instantaneous change signal with broadband. The measurement of blast shock wave puts forward a higher requirement for the dynamic performance of test system. If the land zone of amplitude-frequency characteristic of test system cannot cover the frequency spectrum of signal to be tested, a larger error of dynamic performance will occur, resulting the distortion of waveform observed. Thus, the test data requires follow-up processing and correction to obtain better results [2] . The dynamic characteristic compensation of sensor is an effective method for correcting the dynamic error. In a word, the research on the test technology of shock wave and dynamic characteristic compensation of sensor, and the objective and accurate evaluation of the damage power play a practical role in the development of weapon system and damage theory [3] .
The principle of correcting the dynamic error can be concluded into the studies on the restoration to ideal instrument, the frequency characteristic correction of test instrument, the re-construction of input signal, dynamic error compensation, signal error correction, deconvolution method and inverse filter technique [4]. Studies abroad mainly start from algorithm, first to realize the inverse filter according to priori knowledge of tested signal and adopt lowpass filtering or smoothing for artificial adjustment. Dr. Nahman, the American famous expert on time domain measurement, makes use of frequency domain correction to propose a best single-parameter deconvolution method to estimate tested signal. Its advantages and disadvantages depend on the algorithm of discrete Fourier transformation, suitable for lowpass test system and sensitive to noise [5] ; American Riad and Stafford jointly propose a compensation method to obtain deconvolutional stable solution, strongly inhibiting noise, which suits the deconvolution with bandpass signal; Parruck et al. calculate the deconvolution of pulse response data, derive approximate step waveform by the integral of pulse response and study the mean value and variance of tail [6] ; domestically, there are three basic methods of dynamic characteristic error correction, including frequency domain correction, numerical differentiation and superposition integral. These three methods fail to take the impact of error on the stability of results, which are not suitable for measuring signal [7] .
According to dynamic characteristic compensation of sensor for blast shock wave, this paper aims to study the general form of frequency domain inverse filter based on mathematical inversion, explore the fast recursive fitting modeling and analyze the real-time processing of dynamic characteristic compensation, so as to provide theoretical foundation for the accuracy of power evaluation and provide technological guarantee for the current test system and test system under development.

The Improved method
The quantum-behaved particle swarm optimization (QPSO) is a new PSO (the particle swarm optimization) algorithm model proposed from the perspective of quantum mechanical theory. It changes PSO algorithm evolution search strategy and has no more need of velocity vector, so that the form of evolution equation is simpler and easier to control. Different from standard PSO system, there is also prematurity in QPSO system. After each iteration of QPSO, the search space of individual particle is all the feasible solution space of problems [8] . However, due to the aggregation of particles in bound state, the decline of population diversity is unavoidable. In order to improve the global and local searching ability of quantum-behaved particle swarm, the rate of convergence and computational accuracy, this paper integrates new optimization algorithm on the basis of QPSO algorithm, and proposes PFQPSO based on QPSO of progress function.
In the process of system identification, QPSO algorithm is adopted for global searching. Whereas, the accuracy of identification is lower. Thus, the result searched by QPSO is taken as the input value of progress function. The local optimal solution can searched by learning the progress function to enhance the identification accuracy of system parameter [9] .
The sensor system can be expressed with univariate difference equation, shown in Equation In Equation (1) (2) In Equation (2), is progress factor and  is progress rate.
In the system identification by PSO algorithm, each particle is the system parameter to be identified. According to the results of identification, the square of the difference value between predicated output and actual output can be calculated and considered as fitness value, and the identification of system parameters can be judged and optimized. In order to verify the system identification ability of PSO algorithm, through the simulation of the known system model, according to its simulation input and output signal, standard PSO, QPSO and PFQPSO algorithms are adopted respectively for system identification.
The simulation object is shown in Equation (5), The population size of standard PSO, QPSO and PFQPSO algorithms is 50, and the maximum number of iterations is 500.
The setting of standard PSO algorithm is presented. The inertia weight ω is decreased linearly from 0.9 to 0.4, c1 and c2 are set to be 2, and the value range of particle is ai，bi∈ [-2， 2].
The setting of QPSO algorithm is given. a is linearly decreased from 1 to 0.5, and the value range of particle is ai，bi∈[-2，2].
The setting of improved QPSO algorithm is provided. a is linearly decreased from 1 to 0.5, 0.1   , and the value range of particle is ai，bi∈[-2，2]. Different algorithms are executed for 50 times, and the minimum error is selected. The simulation results are shown in Table 1. Table 1 shows the simulation results of different algorithms. It can be seen that the error of system identification by QPSO algorithm is far less than that by standard PSO algorithm. Whereas, the error of PFQPSO algorithm is far less than that of QPSO algorithm, and its results is nearly close to truth value.  Figure 2 show the fitness value of 500 iterations of system identification by standard PSO, QPSO and PFQPSO algorithms and the change trajectory of parameters, respectively. It can be seen that the convergence speeds of standard PSO, QPSO and PFQPSO algorithms is increasing successively. PFQPSO can reach the convergence after 310 times of iterations, and the average error of PFQPSO on training set is far less than that of the other two algorithms. It can be concluded from the optimal parameters obtained from simulation that, the accuracy of parameters from PFQPSO is the best, which nearly approximates to truth value when noise exists. Therefore, the simulation experiment shows that PFQPSO has a more rapid convergence speed and higher accuracy in system identification.

Experiments
The principle of sensor dynamic error correction based on PFQPSO is to make use of intelligent optimization algorithm to perform the inverse identification of rating data of sensor, taking the obtained sensor inverse model as the sensor dynamic compensator [10] . Its basic principle is shown in Figure 3.  Figure 4 The design schematic diagram of sensor dynamic compensation filter based on PFQPSO Figure 4 shows the general principle of sensor dynamic error correction based on PFQPSO, wherein,   yt is sensor output signal,   yt  is the output signal of reference model,   yt  is the output signal after compensation, and ｍis the highest order of dynamic compensator. First of all, the dynamic calibration experiment is conducted on sensor to obtain its response to step signal. Then the output signal of sensor is taken as the input signal of compensator, and the step signal is taken as the output signal of compensator. Next, the inverse identification is performed on the parameters of compensation filter by PFQPSO, so as to obtain the transfer function of compensation filter and realize the compensation and correction of sensor dynamic error.

Seneor system
The dynamic compensation filter model is determined by the minimum mean squared error and the overshoot value of compensated data relative to ideal step signal, by making use of the response data of pressure sensor under the excitation of step signal in the movable-scale experiment of shock tube [11] . Figure 5 and Figure 6 is the relationship between the compensator order of 8510B sensor from American Endevco Company and error of mean square, as well as overshoot. Order Overshoot Figure 6 The relationship between the compensator order of 8510B sensor From Figure 5 and Figure 6, the curves for error of mean square of compensated data and overshoot gradually decline along with the increase of the order. The orders of sensor compensator filter begin to decline slowly from the dramatical decline after 5-th order. Due to some noise of system and sensor structure, a high-order system can be applied to describe more accurately the pressure sensor system, establish the compensator of corresponding order and obtain better compensation result of compensation from sensor output. However, the higher order of compensation filter, the harder the realization of its hardware, which is hard to be inserted in test system. After balancing the compensation effect and the difficulty to realize the hardware of filter, the order of 8510B compensation filter from American Endevco Company is selected to be 5.
With the improved QPSO, the dynamic compensation filter of sensor is built by inverse modeling, which has the advantages of simple algorithm and good convergence. This paper carries out the modeling of dynamic compensation filter on 8510B piezoresistive pressure sensor from American Endevco Company, and analyzes the indicators of its frequency domain and time domain performance to judge the compensation effect of dynamic compensation filter built by this algorithm. The indicator of frequency domain performance is mainly to analyze its effective bandwidth (the working band when the amplitude error is±3dB); the time domain performance is mainly to analyze its overshoot, rise time, peak time and steady-state response time (±5%).
The dynamic compensation filter model of 8510B piezoresistive pressure sensor from Endevco Company is shown in Equation (7).
The coefficient values are given.
The model of equivalent system after compensation is shown in Equation (8).
The coefficient values are provided.  Figure 7 The amplitude-frequency characteristic of sensor before and after compensation by PFQPSO for 8510B sensor The verification of time domain performance In order to analyze the time domain performance of sensor system before and after the compensation, the built dynamic compensation filter can be applied to obtain the step response of sensor system after compensation [13] , and to compare various performance indicators of time domain.
The step response output of 8510B piezoresistive pressure sensor from Endevco Company before and after compensation by PFQPSO is shown in Figure 8.  Figure 8 The comparison of step response output of 8510B piezoresistive pressure sensor before and after compensation by PFQPSO In order to explain the repeatability of compensator with the optimal order for the same sensor, the compensator with the optimal order is adopted for 10 times of shock tube data rated by one sensor, to examine the consistency of compensation characteristic of time domain and frequency domain. The result of time domain indicator is shown in Table 2, and the result of frequency indicator is shown in Table 3. From the data, it can be concluded that, there is a large difference between different sensors for original overshoot, which is caused by the larger difference in the characteristics of sensor (mainly inherent frequency).
Whereas, the original overshoot of ten times of shock wave data of 8510B from Endevco Company is averaged to 154.9%, with the highest value of 185.2%. The mean value of overshoot after compensation is 3.58%. It can be seen from Table 4 that the frequency domain indicator of ten times of repetitive shock tube experiments for the same one sensor is consistent. There is a good repeatability in the compensation effect of the compensator with the optimal order for the same one sensor. At the same time, the compensation effect of compensator obtained by PFQPSO is further verified, suitable for practical overpressure measurement of shock wave.

Conclusion
In this paper, the design of dynamic compensation filter of pressure sensor based on PFQPSO is proposed and studied. Taking a sensor with poorer dynamic performance in 8510 series of sensor from Endevco Company as an example, this paper carries out the dynamic performance experiment of shock tube, modelling and the design, analysis and verification of dynamic compensation filter based on PFQPSO. The findings show that this method is feasible and effective. In addition, the dynamic compensation filter or inverse filter is recursive, which can be realized by IIR filter, suitable for real-time processing.

List of Abbreviations
QPSO: the quantum-behaved particle swarm optimization PSO: the particle swarm optimization PFQPSO: the progressive function quantum-behaved particle swarm optimization

Declarations
Ethical Approval and Consent to participate: Approved. Consent for publication: Approved. Availability of supporting data: We can provide the data.

Competing interests
These no potential competing interests in our paper. And all authors have seen the manuscript and approved to submit to your journal. We confirm that the content of the manuscript has not been published or submitted for publication elsewhere.

Funding
This work was supported by the National Natural Science Foundation of China (Approved Nos. 51575499 and 61473267) and the Natural Science Foundation of Shanxi Province of China (Approved Nos. 201801D121161) and Sponsored by the Fund for Shanxi '1331 K SC'.

Author's contributions
The author takes part in the discussion of the work described in this paper. The author contributed to this work and should be considered the first author.
the Ph.D degree in Instrument and electronics from the North University of China, in 2016. Now, he works in School of Instrument and Electronics, North University of China. His scientific research interests are concerned with digital signal processing in dynamic measurement and uncertainty of measurement systems. Figure 1 The comparison of convergence curve of standard PSO, QPSO and PFQPSO algorithms Figure 2 The comparison of parameter estimation of standard PSO, QPSO and PFQPSO algorithms Figure 3 The basic functional block diagram of inverse modeling Figure 4 The design schematic diagram of sensor dynamic compensation filter based on PFQPSO Figure 5 The relationship between the compensator order of 8510B sensor and error of mean square Figure 6 The relationship between the compensator order of 8510B sensor Figure 7 The amplitude-frequency characteristic of sensor before and after compensation by PFQPSO for 8510B sensor Figure 8 The comparison of step response output of 8510B piezoresistive pressure sensor before and after compensation by PFQPSO