Simultaneous multi-frequency point identification method and system for frequency response matrix of linear time-invariant system

Abstract

The application discloses a method and system for simultaneously identifying frequency response matrix of a linear time-invariant system. In the testing process, an excitation signal containing all target frequency components is simultaneously input into corresponding actuators according to the assigned channels, and the phasor form is filled into each column according to the excitation batch. After the intermediate variable is obtained by orthogonal calculation in the time domain using the excitation signal and the collected response data, the frequency response matrix corresponding to each excitation frequency point can be further obtained by algebraic operation. The method can realize simultaneous identification of multiple frequency response matrices of a multiple-input multiple-output system, has the characteristics of increasing excitation times, simple calculation, strong anti-interference ability and the like, can improve the identification efficiency and precision of the frequency response matrix of the multiple-input multiple-output system in engineering, and provides more valuable basic data for a plurality of application technologies including modal analysis, vibration and sound active control and the like.

Description

Technical Field

[0001] This invention belongs to the field of nonparametric system identification technology, specifically relating to a method and system for simultaneous identification of the frequency response matrix of a linear time-invariant system at multiple frequency points. Background Technology

[0002] For linear time-invariant systems, the frequency response matrix identified by multiple-input multiple-output (MIMO) digital systems plays a fundamental role in many fields. For example, vibration and noise active control technology based on adaptive filters uses the frequency response matrix to determine the iteration direction of filter coefficients; identification errors can lead to slow convergence or even instability in the control system. In engineering, efficient and accurate frequency response matrix identification methods will shorten debugging time, reduce deployment costs, and accelerate product iteration.

[0003] The system under test often contains multiple input channels. If the tests are performed sequentially according to the combination of the frequency to be identified and the input channels, on the one hand, a single excitation impulse may not be able to input enough energy, leading to a decrease in the signal-to-noise ratio of some sampling points; on the other hand, it will obviously take a long time to test, and the effort to improve accuracy through repeated testing will mean doubling the workload. For a class of linear systems that satisfy symmetry, when using the modal decomposition principle to try to recover the complete matrix data by identifying a column of the frequency response matrix (corresponding to different input channels), the degrees of freedom of the excitation may coincide with the nodes of a certain order eigenvector, making it impossible to fully identify the corresponding dynamic characteristics. Therefore, a method that can simultaneously excite all frequencies to be identified across multiple channels will have great engineering application value in terms of data signal-to-noise ratio, testing efficiency, and identification accuracy.

[0004] Currently, the algorithms used in frequency response matrix identification methods mainly include orthogonal correlation algorithms in the continuous time domain (Izerman. Identification of Dynamic Systems - Introduction and Applications [M]. Machinery Industry Press, 2019), adaptive filtering algorithms in the discrete time domain (Li Yan, Yin Tianqi, Zhang Neng, Huang Jin. Research on Real-time Secondary Path Identification Method Against Dense Frequency Linear Spectrum Interference [J], Vibration and Shock, 2025, 44(4): 151-164), direct calculation methods in the frequency domain after Fourier transform, and H estimation methods using power spectrum (Lu Qiuhai, Li Debao. Engineering Vibration Test Analysis [M], Tsinghua University Press, 2015). Apart from frequency domain methods, no other multi-frequency, multi-input, and multi-output simultaneous identification methods using the other two types of algorithms have been found in literature or engineering practice. On the other hand, adaptive filtering algorithms exhibit periodic fluctuations in data under adjacent-channel interference conditions (other excitation signals during simultaneous multi-frequency identification also constitute interference). Accurate elimination of these fluctuations requires precise frequency estimation of the adjacent-channel interference, which increases the workload. Furthermore, algorithm convergence largely depends on manual intervention. Frequency domain methods require Fourier transforms, resulting in relatively high computational costs. Due to limitations in resolution and frequency sampling, the choice of window function and FFT points directly affects the data accuracy at the identified frequency. Especially when adjacent-channel interference exists (or multiple frequencies are simultaneously excited), a long data window is needed to improve resolution, and the number of FFT points must be carefully selected to avoid the picket fence effect. This requires operators to have a high level of theoretical knowledge and also means a greater workload for testing. Both of these methods require the system under test to reach a steady state before calculation or sampling begins, and to achieve convergence or reduce noise, the duration of a single test can be tens of seconds or even longer. Summary of the Invention

[0005] This invention addresses the technical challenge of simultaneous identification of multiple frequencies in multi-input multi-output systems by providing a method and system for simultaneous identification of multiple frequency points in the frequency response matrix of multi-input multi-output linear time-invariant systems.

[0006] To achieve the above objectives, the present invention provides a method for simultaneous identification of the frequency response matrix at multiple frequency points of a linear time-invariant system, specifically including the following steps:

[0007] 1) Determine the number of test stimuli ,and Not less than the number of input channels of the system under test By balancing noise conditions and testing workload, the number of test excitations was determined. It can be increased or decreased gradually;

[0008] 2) Construct the excitation signal allocated to each input channel of the system under test in each excitation, so that the excitation signal of each channel contains all the frequency components to be identified, and the excitation signals between different excitation batches at each frequency to be identified are uncorrelated.

[0009] 3) Store the constructed excitation signal in a three-dimensional matrix in phasor form. middle, Each dimension corresponds to the input channel, the excitation batch, and the identification frequency, respectively.

[0010] 4) Input the excitation signal into the system under test in batches, so that all input channels are excited simultaneously in each test, and record the time-domain input of each frequency component to be identified. Time domain input orthogonal signals Acquisition time domain output ,in, For discrete time points, For the first One output channel;

[0011] 5) For the data obtained from each test in step 4), use a discrete-time algorithm to obtain the pseudo-frequency response matrix in phasor form. , This indicates the assumption that only the first [unit / item] is excited. The amplitude and phase changes of each output channel at each frequency to be identified when there are multiple input channels;

[0012] 6) Utilize the excitation signal phasor of each test And obtained in step 5) Calculate the averaged output phasor at each frequency point ;

[0013] 7) Calculate the frequency response matrix at each frequency point. ,in The symbol represents the MP pseudoinverse.

[0014] Furthermore, in step 5), the discrete-time algorithm employs an orthogonal correlation algorithm, the specific process of which is as follows:

[0015] 51) Each frequency to be identified Sampling frequency of digital identification devices The ratio, interference frequency Sampling frequency of digital identification devices The ratios are all written in the form of division of positive integers. and make If the value is as small as possible, then the frequency to be identified and interference frequency correspond The least common multiple is the accumulated points. ;

[0016] 52) For each frequency to be identified, calculate the cross-correlation function matrix. , and autocorrelation function vector ;

[0017] 53) Calculate the real part of the pseudo-frequency response matrix based on the cross-correlation function and the autocorrelation function. With the imaginary part .

[0018] The discrete-time algorithm described above can also be implemented using an adaptive filtering algorithm.

[0019] Furthermore, the data records in step 4) are stored in a set. Starting from any point in time, where, To incentivize total duration, To record the total duration.

[0020] Furthermore, the points accumulated in step 51) For each Any common multiple of .

[0021] Furthermore, in step 51), if the interference frequency... For unknown frequencies, candidate periods are calculated using the frequency to be identified, and fluctuations in pseudo-frequency response parameters are observed during testing. The least common multiple at which stability requirements are met is selected to determine the final number of accumulation points. .

[0022] Furthermore, the calculation start time sequence numbers of the cross-correlation function and the autocorrelation function. In the set of integers Choose any one of them, among which This represents the total length of the data sequence recorded in the test.

[0023] A system employing the frequency response matrix multi-frequency point simultaneous identification method described above is also provided, comprising:

[0024] Test stimulus count determination module: Used to determine the number of test stimuli. ,and Not less than the number of input channels of the system under test ;

[0025] Excitation signal construction module: used to construct the excitation signal allocated to each input channel of the system under test in each excitation, so that the excitation signal of each channel contains all the frequency components to be identified, and the excitation signals between different excitation batches at each frequency to be identified are uncorrelated.

[0026] 3D matrix storage module: Used to store the constructed excitation signals in phasor form in a 3D matrix. middle, Each dimension corresponds to the input channel, the excitation batch, and the identification frequency, respectively.

[0027] Excitation module: Used to input excitation signals into the system under test in batches, so that all input channels are simultaneously excited in each test, and to record the time-domain input of each frequency component to be identified. Time domain input orthogonal signals Acquisition time domain output ,in, For discrete time points, For the first One output channel;

[0028] The pseudo-frequency response matrix calculation module calculates the pseudo-frequency response matrix in phasor form using discrete-time algorithms based on the data obtained from each test in the excitation module. , This indicates the assumption that only the first [unit / item] is excited. The amplitude and phase changes of each output channel at each frequency to be identified when there are multiple input channels;

[0029] Frequency response matrix calculation module: Utilizes the phasor of the excitation signal from each test. and pseudo-frequency response matrix Calculate the averaged output phasor at each frequency point Then, the frequency response matrix is ​​calculated at each frequency point. ,in The symbol represents the MP pseudoinverse.

[0030] The method of this invention consists of several steps, including constructing excitation signals, acquiring response data, calculating intermediate variables, and synthesizing frequency response matrices. During the testing process, the excitation signal containing all target frequency components is simultaneously input to all corresponding actuators according to the assigned channels. After filling the columns with the phasor form of the excitation signals in batches, the resulting matrix must be of full row rank. After obtaining intermediate variables through orthogonal calculations in the time domain using the excitation signals and acquired response data, the frequency response matrix corresponding to each excitation frequency point can be further derived through simple algebraic operations. This method can simultaneously identify multiple frequency response matrices of multi-input multi-output systems, and features the advantages of progressively increasing excitation times, simple calculation, and strong anti-interference capability. It can improve the efficiency and accuracy of frequency response matrix identification for multi-input multi-output systems in engineering, and provide more valuable basic data for various application technologies, including modal analysis and active vibration and acoustic control.

[0031] The principle of simultaneous multi-frequency identification in discrete-time orthogonal correlation algorithm

[0032] Known

[0033]

[0034] in, Indicates taking any consecutive Given a set of integers, consider a sine sequence. and Then, using Euler's formula, we can obtain...

[0035]

[0036] in, For the first One signal ( The normalized circumfrequency of ) For the phase of the corresponding signal, and This corresponds to the frequency of a continuous-time signal. Normally... And since it is a rational number, it can be expressed as In the form of, The period of the discrete-time signal is given by equation (2), and it is always possible to find the period. For the discrete-time sequence represented by equation (2), its minimum period is... and Least Common Multiple .make , , , .

[0037] The following discussion will be divided into different scenarios.

[0038] when , hour,

[0039]

[0040] like , Then, from equation (1), we can know that

[0041]

[0042] And if If , then equation (3) will not be 0.

[0043] when When, it can be seen from equation (1) that equation (2) is 0. This is the orthogonal characteristic between signals of different frequencies that is independent of phase difference.

[0044] Using equation (3) to display the same frequency ( Considering the relationship between sinusoidal sequences and equation (1), the frequency is... Input sine sequence The autocorrelation function in The value at that location is

[0045]

[0046] in, For sequence normalized circumfrequency, The period is [period]. In steady state, the system output is [output]. ,in The frequency response function is the cross-correlation function between the system output and input. exist The value at that location is

[0047]

[0048] in, For the system under test in The phase difference at a given frequency. Similarly, the output is quadrature with the input signal. Cross-correlation function between exist The value at that location is

[0049]

[0050] Obviously,

[0051]

[0052]

[0053] This result shows that the frequency response function of the system can be obtained using the sampled discrete signal, just as it has in the continuous-time case. The above method can be easily extended to the simultaneous identification of multiple frequencies, provided that the different frequencies (i.e.,...) are taken into account. In the case of (the situation where), the response components only need to produce a non-zero cumulative sum for excitations of the same frequency.

[0054] The principle of multi-input multi-output discrete-time orthogonal correlation algorithm

[0055] Considering the case where all actuators are excited at a single frequency, multi-frequency point superposition identification only requires separating each frequency using different excitation signals. In the... In this test, the excitation signals of different actuators were represented in phasor form and arranged into variables in columns. After recording all sensor outputs, the "pseudo-frequency response" of each actuator-sensor combination can be obtained using the orthogonal correlation method. (A set of complex numbers reflecting the input-output relationship in the current test). The number of rows in this matrix represents the number of response sensors. The number of columns represents the number of actuators. Thus, the phasor of the averaged output signal of this test can be calculated.

[0056]

[0057] pass The following matrix can be obtained from this test.

[0058]

[0059] in , , Let be the frequency response matrix to be determined. Clearly, when Then, we can solve equation (11) to get

[0060]

[0061] in The symbol represents the MP pseudoinverse.

[0062] The orthogonal correlation algorithm of this invention in the discrete time domain proves its correctness in terms of no waiting time for data acquisition, low data requirement, and strong anti-interference and noise capabilities through the above-described principles.

[0063] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0064] 1. By compressing the waiting time and duration of data acquisition and providing a mechanism to improve noise immunity by successively increasing the number of test stimuli, the identification efficiency of frequency response matrices in engineering is improved;

[0065] 2. By using orthogonal correlation algorithms, simultaneously exciting all input channels, and using matrix operations to solve the frequency response matrix as a whole, the impact of adjacent-channel interference and noise on the accuracy of the results is greatly avoided, and to a certain extent, problems such as loss of dynamic characteristics caused by the excitation point coinciding with the modal frequency are avoided.

[0066] 3. Parameter adjustment and experimental design are simple and easy to implement, making it more suitable for engineering practice scenarios. This invention addresses the technical challenge of simultaneous multi-frequency identification in multi-input multi-output systems by providing a method for simultaneous multi-frequency identification of the frequency response matrix of multi-input multi-output linear time-invariant systems. Attached Figure Description

[0067] Figure 1 Output diagram of the system to be identified, which includes noise and interference;

[0068] Figure 2 This is an image showing the change of pseudo-frequency response parameters over time, obtained using the method described in this invention.

[0069] Figure 3 This is a graph showing the frequency response parameters changing over time, obtained using an adaptive filtering method. Detailed Implementation

[0070] The technical solutions (including preferred technical solutions) of the present invention will be further described in detail below with reference to the accompanying drawings and by way of listing some optional embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0071] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0072] The example demonstrates a system identification process conducted in a simulation environment. The system to be identified is a 146th-order discrete-time system with two inputs and two outputs. Its state-space equation parameter matrix is ​​generated by connecting a continuous-time controlled object model to sensors, anti-aliasing filtering, sampling, zero-order hold, and multi-rate conversion. The two frequencies to be identified are randomly selected below 100Hz, with a difference of 1Hz between them. The acquired system outputs are subject to adjacent-channel interference and white noise pollution. To reduce noise impact, the minimum possible number of acquisitions is performed. Relevant parameters are shown in Table 1. Taking any test as an example, the two-channel output is as follows... Figure 1 As shown.

[0073] Table 1 Test Parameters

[0074]

[0075] Identification was performed using an adaptive filtering method that does not calculate the pseudo-frequency response matrix, a frequency domain H1 estimation method for harmonic excitation, and the method described in this invention. The step size used in the adaptive filtering method was the minimum value obtainable after several attempts.

[0076] Taking the time-varying calculated value of the imaginary part of the frequency response (or pseudo-frequency response) parameter between system input channel 2 and output channel 2 at 53Hz in the first test as an example, the advantages of the method described in this invention in terms of parameter stability compared to the adaptive filtering method are shown. Figure 2 and Figure 3 As shown.

[0077] As can be seen, the method described in this invention can obtain pseudo-frequency response data with high accuracy after 2 seconds, and data with extremely high accuracy after 5 seconds. In contrast, the adaptive filtering method requires a longer convergence time, and the parameters fluctuate greatly due to the presence of adjacent channel interference, thus requiring further averaging to obtain usable results, resulting in a significantly longer testing time than the method described in this invention.

[0078] Due to the frequency resolution requirements, the H1 estimation method requires at least 2 seconds of acquisition time to distinguish dense frequency interference of 0.5 Hz, and an additional 5 seconds or so to wait for the system to enter a steady state.

[0079] On the other hand, using adaptive filtering methods requires setting an appropriate step size to balance the algorithm's convergence and efficiency, and misjudging the convergence status will worsen the test results, which often requires multiple trials and a certain amount of experience. The H1 estimation method, on the other hand, requires determining complex parameters such as waiting time, acquisition time, windowing type, and FFT points, which places high demands on the testers' professional skills and is more prone to errors in actual work.

[0080] The final identification performance comparison is shown in Table 2. It can be seen that the multi-frequency identification method for multiple-input multiple-output systems described in this invention has advantages in accuracy, efficiency, and ease of use. Therefore, the method described in this invention is expected to demonstrate even more significant advantages in larger-scale identification tasks (number of frequency points, number of channels).

[0081] Table 2 Performance Comparison of Different Identification Methods

[0082]

[0083] The system of the frequency response matrix multi-frequency point simultaneous identification method of the present invention includes:

[0084] Test stimulus count determination module: Used to determine the number of test stimuli. ,and Not less than the number of input channels of the system under test ;

[0085] Excitation signal construction module: used to construct the excitation signal allocated to each input channel of the system under test in each excitation, so that the excitation signal of each channel contains all the frequency components to be identified, and the excitation signals between different excitation batches at each frequency to be identified are uncorrelated.

[0086] 3D matrix storage module: Used to store the constructed excitation signals in phasor form in a 3D matrix. middle, Each dimension corresponds to the input channel, the excitation batch, and the identification frequency, respectively.

[0087] Excitation module: Used to input excitation signals into the system under test in batches, so that all input channels are simultaneously excited in each test, and to record the time-domain input of each frequency component to be identified. Time domain input orthogonal signals Acquisition time domain output ,in, For discrete time points, For the first One output channel;

[0088] The pseudo-frequency response matrix calculation module calculates the pseudo-frequency response matrix in phasor form using discrete-time algorithms based on the data obtained from each test in the excitation module. , This indicates the assumption that only the first [unit / item] is excited. The amplitude and phase changes of each output channel at each frequency to be identified when there are multiple input channels;

[0089] Frequency response matrix calculation module: Utilizes the phasor of the excitation signal from each test. and pseudo-frequency response matrix Calculate the averaged output phasor at each frequency point Then, the frequency response matrix is ​​calculated at each frequency point. ,in The symbol represents the MP pseudoinverse.

Claims

1. A method for simultaneous multi-frequency point identification of a frequency response matrix of a linear time-invariant system, characterized in that, The method specifically comprises the following steps: 1) determine the number of test stimuli , and not less than the number of input channels of the system under test ; 2) constructing the excitation signals allocated to each input channel of the system under test in each excitation batch, so that the excitation signals of each channel contain all the frequency components to be identified, and the excitation signals of different excitation batches are irrelevant at each frequency to be identified; 3) store the constructed excitation signal in a three-dimensional matrix in the form of phasors , Each dimension of the three-dimensional matrix corresponds to an input channel, an excitation batch, and a frequency to be recognized, respectively. 4) input the excitation signal to the system under test by excitation batch, so that all input channels are simultaneously excited in each test, and record the time-domain input of each frequency component to be identified , time-domain input , the quadrature signal of the time-domain input , collect the time-domain output , wherein, is a discrete time point, is the th output channel; 5) for the data obtained for each test in step 4), a pseudo-frequency response matrix is obtained in phasor form using a discrete-time algorithm , represents the amplitude and phase variations of each output channel at each frequency to be identified, assuming that only the i-th input channel is excited ​ 6) the excitation signal phasor of each test and the one obtained in step 5) the averaged output phasor at each frequency point ; 7) Compute the frequency response matrix at each frequency bin where The notation M-P pseudo-inverse is used. In the step 5), the discrete-time algorithm adopts a quadrature correlation algorithm, and the specific process is as follows: 51) Each frequency to be identified Sampling frequency of digital identification devices The ratio, interference frequency Sampling frequency of digital identification devices The ratios are all written in the form of division of positive integers. and make If the value is as small as possible, then the frequency to be identified and interference frequency correspond The least common multiple is the accumulated points. ; 52) for each frequency to be recognized, calculating a cross-correlation function matrix , and an autocorrelation function vector ; 53) Calculate real part of pseudo-frequency response matrix from cross- and auto-correlation functions and imaginary part ; The step 51) several disturbance frequencies Unknown, using the to-be-identified frequency to calculate the alternative period, and observing the fluctuation of the pseudo frequency response parameters in the test process, selecting the common multiple when the stability requirement is reached to determine the final accumulation point number .

2. The method of claim 1, wherein, The data records in step 4) are recorded in the set at any point in time, wherein is the total duration of the incentive, is the total duration of the recording.

3. The method of claim 1, wherein, The points accumulated in the step 51 For each Any common multiple of the 4. The method of claim 1, wherein, The start time index of the calculation of the cross-correlation function and the autocorrelation function In the integer set is arbitrarily selected, wherein is the total length of the data sequence of the test record.

5. A system for using the multi-frequency simultaneous identification method of frequency response matrix according to any one of claims 1 to 4, characterized in that: The method comprises: Test excitation number determination module: for determining the test excitation number , and Not less than the input channel number of the system under test ; An excitation signal construction module is configured to construct the excitation signals allocated to each input channel of the system under test in each excitation batch, so that the excitation signals of each channel contain all the frequency components to be identified, and the excitation signals of different excitation batches are irrelevant at each frequency to be identified; Three-dimensional matrix storage module: for storing the constructed excitation signal in the form of phasor in a three-dimensional matrix In other words, Each dimension of the three-dimensional matrix corresponds to an input channel, an excitation batch, and a frequency to be recognized, respectively. Excitation module: Used to input excitation signals into the system under test in batches, so that all input channels are simultaneously excited in each test, and to record the time-domain input of each frequency component to be identified. Time domain input orthogonal signals Acquisition time domain output ,in, For discrete time points, For the first One output channel; Pseudo-frequency response matrix calculation module: for the data obtained in each test of the excitation module, a pseudo-frequency response matrix in the form of a phasor is obtained by using a discrete-time algorithm , represents the amplitude and phase changes of each output channel at each frequency to be identified, assuming that only the first input channel is excited. Frequency response matrix computation module: uses the excitation signal phasor of each test and the pseudo frequency response matrix , computes the averaged output phasor at each frequency ; then computes the frequency response matrix at each frequency where The notation M-P pseudo-inverse is used.

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