CN111737883B - Nonlinear double-rate circuit system robust identification method with output time lag - Google Patents

Nonlinear double-rate circuit system robust identification method with output time lag Download PDF

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CN111737883B
CN111737883B CN202010751921.7A CN202010751921A CN111737883B CN 111737883 B CN111737883 B CN 111737883B CN 202010751921 A CN202010751921 A CN 202010751921A CN 111737883 B CN111737883 B CN 111737883B
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杨宪强
刘新鹏
高会军
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Harbin Institute of Technology
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Abstract

A nonlinear double-rate circuit system robust identification method with output time lag relates to the field of circuit system modeling and system identification. The invention aims to solve the problem of robust identification of a nonlinear double-rate circuit system when time-varying time lag and abnormal value interference exist in output signal data in the prior art, and comprises the following steps: the method comprises the following steps: establishing an input-output model of the nonlinear circuit system based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, and determining the problem to be identified; step two: under a probability framework, establishing a robust identification model of a nonlinear circuit system; step three: based on a variational Bayes framework, deriving to obtain an iterative estimation formula of model parameters and hidden variables; step four: and obtaining an identification result by iteratively updating the robust Kalman filtering algorithm and the variational distribution formula corrected in the step three. The method is suitable for robust identification of the circuit system.

Description

Nonlinear double-rate circuit system robust identification method with output time lag
Technical Field
The invention relates to the field of circuit system modeling and system identification, in particular to a nonlinear double-rate circuit system robust identification method with output time lag.
Background
The circuitry employed in modern industrial processes often exhibit complex non-linear/time-varying characteristics. The linear variable parameter system can effectively describe the nonlinear behavior of the industrial process and has a simple linear model structure, so the linear variable parameter system is widely applied to the field of modeling of complex industrial processes, can avoid the influence of factors such as abnormal values, time delays and the like on the identification algorithm in double-rate sampling and output data of the nonlinear industrial process when applied to a circuit system, and improves the robustness of the identification algorithm.
Due to the complexity of a circuit system in an actual industrial process, identification data are often obtained by double-rate sampling, and time-varying time lag and interference of abnormal values often exist in collected identification output data. If these conditions are not fully considered in the identification algorithm, the identification accuracy of the circuit system is likely to be reduced, and the risk factor of the circuit system is likely to increase.
Disclosure of Invention
The invention aims to solve the problem that in the prior art, when time-varying time lag and abnormal value interference exist in output data, the robust identification rate of a nonlinear double-rate circuit system is low.
The specific process of a nonlinear double-rate circuit system robust identification method with random output time lag is as follows:
step one, two output voltage ports of an arbitrary waveform generator are respectively connected to two voltage input ports of a circuit system; an input voltage port and an output voltage port of the circuit system are respectively connected to a voltage measuring port of the data acquisition card through a buffer register; generating a fast sampling input voltage signal and a reference scheduling voltage signal of any waveform by using an arbitrary waveform generator, and inputting the signals into a circuit system to be tested; a data acquisition card is used for acquiring a fast sampling input signal, a scheduling signal and a generated slow sampling output signal of the circuit system to be tested, namely a slow sampling output value;
wherein the fast sampling input value, the scheduling signal value and the slow sampling output value form an identification data set;
the data acquisition card generates certain noise when acquiring the identification data set, and the generated noise is divided into process noise and observation noise of the circuit system, so that the problem to be identified of the circuit system can be determined;
establishing an input-output model of the nonlinear circuit system based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, and determining the problem to be identified;
secondly, under a probability framework, establishing a robust identification model of the nonlinear double-rate circuit system;
step three, deriving to obtain an iterative estimation formula of model parameters and hidden variables based on a variational Bayesian framework;
and step four, obtaining an identification result by iteratively updating the robust Kalman filtering algorithm and the variational distribution formula corrected in the step three.
The invention has the beneficial effects that:
the invention describes the nonlinearity of the working process of a circuit system through a linear variable parameter double-rate model structure, carries out robust modeling on the identification problem of the circuit system under a probability frame, estimates the loss value existing in the output of a fast sampling process through a corrected robust Kalman filtering algorithm, carries out iterative updating on the model parameter and a hidden variable based on a variational Bayesian algorithm, finally carries out joint estimation to obtain the model parameter, the noiseless process output value and the time-varying time lag existing in the output of the system, and can measure the uncertainty of the model parameter at the same time, thereby effectively ensuring the precision and the reliability of the identification result. The modeling and identification method for the nonlinear circuit system and the practical application in the industry have important values.
Drawings
FIG. 1 shows the estimation results of model parameters at an abnormal value ratio of 10%;
FIG. 2 shows the estimation result of output time lag at an abnormal value ratio of 10%;
FIG. 3 shows the estimation result of the cross-validation output value at the abnormal value ratio of 10%;
FIG. 4 shows the estimation result of Monte Carlo simulation parameters at an abnormal value ratio of 5%;
FIG. 5 shows the estimation result of Monte Carlo simulation parameters at an abnormal value ratio of 10%;
FIG. 6 shows the results of estimating Monte Carlo simulation parameters at an outlier ratio of 15%;
FIG. 7 is a schematic diagram of an experimental setup for a bandpass filter circuitry;
FIG. 8 shows the result of time lag estimation for the bandpass filter circuitry;
figure 9 shows the band pass filter circuitry output estimation.
Detailed Description
The first embodiment is as follows: the robust identification method for the nonlinear double-rate circuit system with the output time lag comprises the following specific processes:
step one, two output voltage ports of an arbitrary waveform generator are respectively connected to two voltage input ports of a circuit system; an input voltage port and an output voltage port of the circuit are respectively connected to a voltage measuring port of the data acquisition card through a buffer register; generating a fast sampling input voltage signal and a reference scheduling voltage signal of any waveform by using an arbitrary waveform generator, and inputting the signals into a circuit system to be tested; a data acquisition card is used for acquiring a fast sampling input signal, a scheduling signal and a generated slow sampling output signal of the circuit system to be tested, namely a slow sampling output value;
wherein the fast sampling input value, the scheduling signal value and the slow sampling output value form an identification data set;
the data acquisition card generates certain noise when acquiring the identification data set, and the generated noise is divided into process noise and observation noise of the circuit system, so that the problem to be identified of the circuit system can be determined;
establishing an input-output model of the nonlinear circuit system based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, and determining the problem to be identified;
secondly, under a probability framework, establishing a robust identification model of the nonlinear circuit system;
step three, deriving to obtain an iterative estimation formula of model parameters and hidden variables based on a variational Bayesian framework;
and step four, obtaining an identification result by iteratively updating the robust Kalman filtering algorithm and the variational distribution formula corrected in the step three.
The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is: in the first step, an input/output model of the nonlinear circuit system is established based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, so that the problem to be identified is determined, and the specific process is as follows:
the method comprises the following steps: in order to model double-rate sampling and time lag existing in a circuit system, the method is characterized by the following linear variable parameter model structure:
A(wl,z-1)xl＝B(wl,z-1)ul+∈l,l＝1,2,...,L (1)
Figure GDA0003127086780000031
n＝1,2,...,N (2)
wherein u islTo be fastSampling input signal value, wlFor scheduling signal values elIs process noise, xlFor fast sampling the output signal value, L is the total number of fast sampling points,
Figure GDA0003127086780000032
Is TnThe fast sampling process output of the moment is dnThe data after the time delay,
Figure GDA0003127086780000033
For slowly sampling the output signal value,
Figure GDA0003127086780000034
For observing noise, N is the total number of slow samples;
furthermore, polynomial A (w)l,z-1) And B (w)l,z-1) Can be expressed as
Figure GDA0003127086780000035
Figure GDA0003127086780000036
Wherein z is-iAs time shift operator, coefficient a in the above equationi(wl) And bj(wl) Can be expressed as
Figure GDA0003127086780000037
Wherein psir(wl) And gammac(wl) Is a known basis function with respect to the scheduling signal, aiAnd bjFor model parameters, #rAnd gammacIs a basis function;
the first step is: converting the linear variable parameter model into a linear regression form:
by introducing intermediate variables
Figure GDA0003127086780000041
Figure GDA0003127086780000042
Figure GDA0003127086780000043
And according to the formulas (3) to (4), the process output model in the formula (1) can be further converted into a linear regression form:
Figure GDA0003127086780000044
wherein
Figure GDA0003127086780000045
Figure GDA0003127086780000046
Represents a kronecker product ∈lAnd theta is the model parameter of the robust identification system, wherein theta is the process noise.
Step one is three: defining the problem to be identified:
the problem to be identified in the present invention can be described as: according to an identification data set W ═ { u, W, T, y } collected by a data acquisition card, a model parameter theta of the circuit system is obtained through robust identification, and a time lag d ═ d1:NAnd outputting the signal estimate. u-u1:LFor fast sampling of the input signal values, w ═ w1:LFor scheduling signal values, T ═ T1:NFor the time of the slow sampling instant,
Figure GDA00031270867800000411
the signal values are output for slow sampling.
Other steps are the same as those in the first embodiment.
The third concrete implementation mode: the present embodiment differs from the first or second embodiment in that: in the second step, under a probability framework, a specific process of establishing a robust identification model of the nonlinear circuit system is as follows:
step two, firstly: modelling the observed noise as a student t distribution, i.e.
Figure GDA0003127086780000047
Wherein xi is the precision parameter of student t distribution, v is the degree of freedom parameter,
Figure GDA0003127086780000048
to observe noise;
step two: the implicit variable tau of the student T distribution is introduced into the formula (6), and the conditional probability distribution p (y | Lambda, tau, xi; T) of the output signal can be decomposed into the following two formulas:
Figure GDA0003127086780000049
Figure GDA00031270867800000410
wherein Λ ═ λ1,…,λNIs the importance matrix of the time lag, τ ═ τ1,…,τNThe t is a hidden variable vector distributed by the students; lambda [ alpha ]nBy a set of binary variables λ representing time lagnmWith the composition (M-1, 2, …, M), i.e. when the real time lag at the nth sampling instant is M-1, the corresponding λnm＝1，
Figure GDA00031270867800000515
Outputting a signal value for slow sampling, v is a degree of freedom parameter, and xi is a precision parameter of student t distribution;
step two and step three: introducing a weight coefficient variable pi-pi with respect to time lag1,…,πMA conditional probability distribution of its Λ over π can be expressed as
Figure GDA0003127086780000051
Step two, four: the conditional probability distribution of the slowly sampled output signal can be expressed as:
Figure GDA0003127086780000052
wherein the content of the first and second substances,
Figure GDA0003127086780000053
the signal data is output for a process having a time lag,
Figure GDA0003127086780000054
is a matrix formed by regression vectors, and theta is a model parameter of the system;
step two and step five: introducing a prior distribution about implicit variables θ, π, v, ξ, δ:
Figure GDA0003127086780000055
Figure GDA0003127086780000056
Figure GDA0003127086780000057
Figure GDA0003127086780000058
Figure GDA0003127086780000059
Figure GDA00031270867800000510
where eta ═ eta1,…,ηpAnd δ are precision parameters of the gaussian distribution. C (alpha)0) In order to be a normalization constant, the method comprises the following steps of,
Figure GDA00031270867800000511
for obeying the hyper-parameter eta of the Gamma distributioniThe shape parameter and the rate parameter of (a),
Figure GDA00031270867800000512
to obey the shape parameter and rate parameter of the hyper-parameter v of the Gamma distribution,
Figure GDA00031270867800000513
to obey the shape parameter and the rate parameter of the hyper-parameter xi of the Gamma distribution,
Figure GDA00031270867800000514
a shape parameter and a rate parameter that are hyper-parameters δ that follow a Gamma distribution.
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the present embodiment differs from the first to third embodiments in that: and step three, deriving to obtain an iterative estimation formula of model parameters and hidden variables based on a variational Bayesian framework, wherein the specific process is as follows:
step three, firstly: the joint probability distribution of the robust identification model of the nonlinear double-rate circuit system is
Figure GDA0003127086780000061
Wherein C is a constant term.
Step three: estimating the loss value existing in the output of the fast sampling process to further obtain the regression quantity
Figure GDA0003127086780000062
The robust kalman filtering algorithm that can be modified is as follows:
for a fast sampling instant k 1, 2.., L, the following two steps are cycled:
(1) a prediction step:
Figure GDA0003127086780000063
Figure GDA0003127086780000064
wherein
Figure GDA0003127086780000065
Is the model regression at time k, xk+1|k，Pk+1|kRespectively mean and variance, parameters Σ predicted at time k +1, for a given observed value at time kkIs defined as
Figure GDA0003127086780000066
Wherein sigmak|k＝Pk|k,σk|k-i＝-(1-skc)[σk-1|k-i,…,σk-i|k-i,…,σk-i|k-n]θx。
(2) And (3) updating:
Pk|k＝(1-skc)Pk|k-1
xk|k＝xk|k-1+sk(yk-cxk|k-1)
wherein xk|k，Pk|kThe process outputs the mean and variance, P, updated at time k, given the observed signal value at time k, respectivelyk|k-1And xk|k-1Obtained from the predicted value of the previous step, and
Figure GDA0003127086780000071
when no output signal is observed at time k (i.e., data loss occurs during time k), x is maintainedk|kFor the prediction of the previous step, i.e. xk|k＝xk|k-1。
Through the circulation of the prediction step and the updating step, the loss value existing in the output of the fast sampling process can be estimated, and then the regression quantity is obtained
Figure GDA0003127086780000072
And laying a cushion for a subsequent variational updating formula.
Step three: introducing a variational distribution q (h) to approximate the true posterior distribution:
p(h|W)≈q(h)
＝q(x,Λ)q(π)q(τ)q(v)q(θ)q(η)q(ξ)q(δ)
step three and four: updating the variational distribution according to a variational Bayesian formula:
lnqi(hi)＝<lnp(W,h)>j≠i+const
wherein q isi(hi) For the introduced variational distributions (corresponding to q (x), q (Λ), q (π), q (τ), q (v), q (θ), q (η), q (ξ), q (δ)), p (W, h) is the joint probability distribution,<·>i≠jis shown with respect to qj(hj) Expect (where i ≠ j).
Further, an iterative estimation formula for obtaining model parameters and hidden variables is as follows:
(a)
Figure GDA0003127086780000077
has the following distribution form:
Figure GDA0003127086780000073
wherein
Figure GDA0003127086780000074
Figure GDA0003127086780000075
(b) q (Λ) obeys the following distribution:
Figure GDA0003127086780000076
wherein
Figure GDA0003127086780000081
And is
Figure GDA0003127086780000082
Figure GDA0003127086780000083
Wherein tr (-) is a matrix trace-solving operator;
(c) the hidden variable pi obeys the dirichlet distribution:
Figure GDA0003127086780000084
wherein α ═ α (α)1,...,αM) Is a centralized parameter of Dirichlet distribution;
wherein
Figure GDA0003127086780000085
(d) The model parameters obey normal distribution:
Figure GDA0003127086780000086
wherein
Figure GDA0003127086780000087
Figure GDA0003127086780000088
(e) Parameter etaiObeying the gamma distribution:
Figure GDA0003127086780000089
wherein
Figure GDA0003127086780000091
Figure GDA0003127086780000092
(f) Hidden variable tau of student t distributionnThe precision parameter xi and the degree of freedom parameter v both obey gamma distribution:
Figure GDA0003127086780000093
Figure GDA0003127086780000094
Figure GDA0003127086780000095
wherein
Figure GDA0003127086780000096
Figure GDA0003127086780000097
Figure GDA0003127086780000098
Figure GDA0003127086780000099
Figure GDA00031270867800000910
(g) Furthermore, the process noise parameters also follow a gamma distribution:
Figure GDA00031270867800000911
wherein
Figure GDA00031270867800000912
Figure GDA00031270867800000913
Other steps and parameters are the same as those in one of the first to third embodiments.
The fifth concrete implementation mode: the difference between this embodiment and one of the first to fourth embodiments is: the fourth step is to update the robust kalman filtering algorithm and the variational distribution formula corrected in the third step by iteration, and the specific process of finally obtaining the required identification result is as follows:
obtaining the model parameter theta and the uncertainty sigma thereof according to the iterative formula of the model parameter and the hidden variable obtained in the third stepθOutput estimate x for a noiseless processlTime lag d existing in system output;
mu is equal to thetaθ，
Figure GDA0003127086780000101
d＝max(R,2)；
Wherein R is a parameter RnmThe resulting matrix, max (·,2), is the maximum value of the matrix by row.
Other steps and parameters are the same as those in one of the first to fourth embodiments.
The first embodiment is as follows:
the effectiveness of the present invention was verified by using a band pass filter circuitry (Electronic band pass filter system). The experimental setup of the system is shown in FIG. 7, where reference is made to the input signal urefAnd a reference scheduling signal prefGenerated by an arbitrary waveform generator (model HP E1445A), and a data acquisition card (model HP E1430A) is used for acquiring identification data including input signal measurement umScheduling signal measurement pmAnd the measured value y of the output signalm。
In this embodiment, 4500 samples of the input/scheduling signal are collected, and 1500 observations of the output signal are collected. In addition, 5%, 10%, and 15% of outliers uniformly distributed in [ -0.5,0.5], and an output time lag with a weight coefficient of [0.5,0.3,0.2] are added to the output signal values, respectively.
The initial time lag number of the algorithm is set to be 5, and the corresponding time lag weight coefficients are set to be 1/5. Based on the algorithm of the invention, the output estimation error REE ═ var (y)t-ye)/var(yt) X 100%, skew estimation accuracy and skew weight coefficient estimation results are shown in the following table (the latter two skew weight coefficient estimation results are less than 10)-3Therefore, it is approximately 0).
TABLE 1 estimation results for different abnormal value ratios of bandpass filter circuitry
Figure GDA0003127086780000102
The time lag estimation result and the output cross validation result when 10% abnormal values exist are shown in fig. 8 and 9, respectively. The graphs show that the algorithm can more robustly estimate the unknown time lag, the time lag weight coefficient and the noise-free output of the circuit system of the band-pass filter, and the effectiveness of the algorithm is verified.
In order to verify the accuracy of the algorithm parameter estimation, the method adopts the following linear variable parameter model structure verification with known parameters for further verification:
A(wl,z-1)xl＝B(wl,z-1)ul+∈l,l＝1,2,...,L
Figure GDA0003127086780000111
n＝1,2,...,N
wherein
A(wl,z-1)＝1+a1(wl)z-1+a2(wl)z-2
B(wl,z-1)＝b1(wl)z-1+b2(wl)z-2
Figure GDA0003127086780000112
Figure GDA0003127086780000113
Figure GDA0003127086780000114
Figure GDA0003127086780000115
The system fast sample input is selected to be [ -1,1 [)]Uniformly distributed noise, and the scheduling variable is selected as wl0.5sin (0.05 pi l) +0.5, the total number of fast sampling points is set to 4500, the total number of slow sampling points is set to 750, and the system output time lags are distributed in an integer interval [0,3 ] according to probabilities of 0.4,0.3,0.2 and 0.1]In the meantime. In addition, the process noise and the output noise are mean 0 and variance 0.0152White gaussian noise. To verify the robust performance of the algorithm, different proportions of uniform distributions of [ -0.5,0.5] can be added to the system output]The anomaly point of (1).
First, a single simulation experiment is performedIt is proved that 10% of abnormal points are added in the observation data, the number of the algorithm initial time lags is set to be 5, and the corresponding time lag weight coefficients are set to be 1/5. Based on the algorithm of the present invention, the estimated model parameters are distributed as shown in FIG. 1, and the relative parameter estimation error (RPEE, defined as | | θ - θ, θ) is showntrue||2/‖θtrue‖2 X 100% where θtrueIs a true parameter, θ is an estimated parameter) is 2.87%. Fig. 2 shows the time lag estimation result, and the time lag estimation accuracy is 89.60%. The estimated time-lag weight coefficient is pi ═ 0.3946,0.2871,0.1957,0.1210,0.0016]It can be seen that the first four weight coefficients are close to the true weights, and the last weight coefficient is smaller, so that it can be considered as a time lag that does not exist in the system. The result of cross-validation output estimation (enlarged view) is shown in fig. 3, which shows that the output estimation value is more consistent with the true noise-free output value, and the validity of the algorithm provided by the invention is verified.
Under different abnormal value ratios (5%, 10% and 15%), 100 Monte Carlo simulation experiments were performed respectively, and different noise sequences and abnormal point sequences were used for each simulation. The model parameter estimation bin plots are shown in FIGS. 4-6, respectively, where the upper, middle and lower lines of each bin represent the 25 th, middle and 75 th percentiles of the Monte Carlo simulation parameter estimates, respectively, and "+" represents outliers away from the estimates. As can be seen from fig. 4-6, the algorithm of the present invention can estimate the true value of the system parameter more stably under different abnormal value ratios, and as the data quality improves (i.e. the abnormal value ratio decreases), the estimation deviation of the algorithm parameter decreases.
From the above results, the algorithm of the invention can better avoid the influence of factors such as double-rate sampling and abnormal values, time lag and the like in output data on the identification algorithm in the working process of the nonlinear circuit system, can effectively improve the robust performance of the identification algorithm, and has certain theoretical and practical application values.

Claims (6)

1. A robust identification method for a nonlinear double-rate circuit system with output time lag is characterized by comprising the following specific processes:
step one, two output voltage ports of an arbitrary waveform generator are respectively connected to two voltage input ports of a circuit system; an input voltage port and an output voltage port of the circuit system are respectively connected to a voltage measuring port of the data acquisition card through a buffer register; generating a fast sampling input voltage signal and a reference scheduling voltage signal of any waveform by using an arbitrary waveform generator, and inputting the signals into a circuit system to be tested; a data acquisition card is used for acquiring a fast sampling input signal, a scheduling signal and a generated slow sampling output signal of the circuit system to be tested, namely a slow sampling output value;
wherein the fast sampling input value, the scheduling signal value and the slow sampling output value form an identification data set;
the data acquisition card generates certain noise when acquiring the identification data set, and the generated noise is divided into process noise and observation noise of a circuit system;
establishing an input-output model of the nonlinear circuit system based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, and determining the problem to be identified;
secondly, under a probability framework, establishing a robust identification model of the nonlinear circuit system;
step three, deriving to obtain an iterative estimation formula of model parameters and hidden variables based on a variational Bayesian framework:
step three, obtaining the joint probability distribution of the robust identification model of the nonlinear double-rate circuit system;
step two, the fast sampling time is circulated through the prediction and updating processes, the loss value existing in the output of the fast sampling process is estimated, and then the regression quantity is obtained
Figure FDA0003127086770000012
A corrected robust Kalman filtering algorithm can be obtained;
thirdly, introducing variation distribution q (h) to approximate real posterior distribution;
step three and four: updating the variation distribution according to a variation Bayes formula;
and step four, obtaining an identification result by iteratively updating the robust Kalman filtering algorithm and the variational distribution formula corrected in the step three.
2. The robust identification method of nonlinear dual rate circuitry with output time lag as claimed in claim 1, wherein: in the first step, an input/output model of the nonlinear circuit system is established based on the fast sampling input voltage signal, the reference scheduling voltage signal, the slow sampling output value, the process noise and the observation noise, so that the problem to be identified is determined, and the specific process is as follows:
step one, in order to model double-rate sampling and time lag existing in a circuit system, the following linear variable parameter model structure is used for depicting:
A(wl,z-1)xl＝B(wl,z-1)ul+∈l,l＝1,2,...,L (1)
Figure FDA0003127086770000011
wherein u islFor fast sampling of the input signal value, wlFor scheduling signal value, elIs process noise, xlFor fast sampling of the output signal value, L is the total number of fast sampling points,
Figure FDA0003127086770000021
is TnThe fast sampling process output of the moment is dnThe data after the time delay is processed,
Figure FDA0003127086770000022
in order to slowly sample the output signal value,
Figure FDA0003127086770000023
for observing noise, N is the total number of slow samples;
polynomial A (w)l,z-1) And B (w)l,z-1) Can be expressed as
Figure FDA0003127086770000024
Figure FDA0003127086770000025
Wherein z is-iAs a time shift operator, ai(wl) And bj(wl) Can be expressed as
Figure FDA0003127086770000026
Wherein psir(wl) And gammac(wl) Is a known basis function with respect to the scheduling signal, aiIs a sum of bjFor model parameters, #rAnd gammacIs a basis function;
step one and two, introducing intermediate variable
Figure FDA0003127086770000027
Figure FDA0003127086770000028
Figure FDA0003127086770000029
And according to the formulas (3) to (4), the process output model in the formula (1) can be further converted into a linear regression form:
Figure FDA00031270867700000210
wherein the content of the first and second substances,
Figure FDA00031270867700000211
Figure FDA00031270867700000212
represents a kronecker product ∈lTheta is process noise and is a model parameter of the system obtained by robust identification;
step one and three, determining the problem to be identified
According to an identification data set W ═ { u, W, T, y } collected by a data acquisition card, a model parameter theta of the system is obtained through robust identification, and a time lag d ═ d1:NAnd a process output estimate; u-u1:LFor fast sampling of the input signal values, w ═ w1:LFor scheduling signal values, T ═ T1:NFor the moment of sampling the time of the sample,
Figure FDA00031270867700000213
the signal values are output for slow sampling.
3. The robust identification method of nonlinear dual rate circuitry with output time lag as claimed in claim 2 wherein: in the second step, under a probability framework, a robust identification model of the nonlinear circuit system is established, and the specific process is as follows:
modeling observation noise as student t distribution:
Figure FDA0003127086770000031
wherein xi is the precision parameter of student t distribution, v is the degree of freedom parameter,
Figure FDA0003127086770000032
to observe noise;
step two, introducing an implicit variable tau of student T distribution into a formula (6), and decomposing a conditional probability distribution p (y | Lambda, tau, xi; T) of an output signal into the following two formulas:
Figure FDA0003127086770000033
Figure FDA0003127086770000034
wherein Λ ═ λ1,…,λNIs the importance matrix of the time lag, τ ═ τ1,…,τNThe t is a hidden variable vector distributed by the students; lambda [ alpha ]nBy a set of binary variables λ representing time lagnmWith the composition (M-1, 2, …, M), i.e. when the real time lag at the nth sampling instant is M-1, the corresponding λnm＝1，yTnOutputting a signal value for slow sampling, v is a degree of freedom parameter, and xi is a precision parameter of student t distribution;
step two and step three, introducing a weight coefficient variable pi ═ pi related to time lag1,…,πMA conditional probability distribution of its Λ over π can be expressed as
Figure FDA0003127086770000035
Wherein the content of the first and second substances,
Figure FDA0003127086770000036
the data is output for a process having a time lag,
Figure FDA0003127086770000037
a matrix composed of regression vectors;
step two and four, the conditional probability distribution of the slowly sampled output signal can be expressed as
Figure FDA0003127086770000038
Where θ is the model parameter of the system, λnmIs a binary variable representing time lag;
step two, introducing prior distribution of hidden variables theta, pi, v, xi and delta
Figure FDA0003127086770000039
Wherein η ═ η1,…,ηpThe precision parameter of Gaussian distribution is obtained;
Figure FDA0003127086770000041
Figure FDA0003127086770000042
Figure FDA0003127086770000043
Figure FDA0003127086770000044
Figure FDA0003127086770000045
wherein, C (alpha)0) In order to be a normalization constant, the method comprises the following steps of,
Figure FDA0003127086770000046
for obeying the hyper-parameter eta of the Gamma distributioniThe shape parameter and the rate parameter of (a),
Figure FDA0003127086770000047
to obey the shape parameter and rate parameter of the hyper-parameter v of the Gamma distribution,
Figure FDA0003127086770000048
to obey the shape parameter and the rate parameter of the hyper-parameter xi of the Gamma distribution,
Figure FDA0003127086770000049
a shape parameter and a rate parameter that are hyper-parameters δ that follow a Gamma distribution.
4. The robust identification method of nonlinear dual rate circuitry with output time lag as claimed in claim 3 wherein: and step three, deriving to obtain an iterative estimation formula of model parameters and hidden variables based on a variational Bayesian framework, wherein the specific process is as follows:
step three, one, the joint probability distribution of the robust identification model of the nonlinear double-rate circuit system is
Figure FDA00031270867700000410
Wherein C is a constant term;
step two, the fast sampling time is circulated through the prediction and updating processes, the loss value existing in the output of the fast sampling process is estimated, and then the regression quantity is obtained
Figure FDA00031270867700000411
A corrected robust Kalman filtering algorithm can be obtained;
step three, introducing variation distribution q (h) to approximate the real posterior distribution:
p(h|W)≈q(h)＝q(x,Λ)q(π)q(τ)q(v)q(θ)q(η)q(ξ)q(δ)
step three and four: updating the variational distribution according to a variational Bayesian formula:
lnqi(hi)＝<lnp(W,h)>j≠i+const
wherein q isi(hi) For the introduced variation distribution, p (W, h) is a joint probability distribution,<·>i≠jis shown with respect to qj(hj) Calculating expectation;
further, an iterative estimation formula for obtaining model parameters and hidden variables is as follows:
(a)
Figure FDA0003127086770000051
has the following distribution form:
Figure FDA0003127086770000052
wherein
Figure FDA0003127086770000053
Figure FDA0003127086770000054
(b) q (Λ) obeys the following distribution:
Figure FDA0003127086770000055
wherein
Figure FDA0003127086770000056
And is
Figure FDA0003127086770000057
Figure FDA0003127086770000058
Wherein tr (-) is a matrix trace-solving operator;
(c) the hidden variable pi obeys the dirichlet distribution:
Figure FDA0003127086770000061
wherein α ═ α (α)1,...,αM) Is a centralized parameter of Dirichlet distribution;
wherein
Figure FDA0003127086770000062
(d) The model parameters obey normal distribution:
Figure FDA0003127086770000063
wherein
Figure FDA0003127086770000064
Figure FDA0003127086770000065
(e) Parameter etaiObeying the gamma distribution:
Figure FDA0003127086770000066
wherein
Figure FDA0003127086770000067
Figure FDA0003127086770000068
(f) Hidden variable tau of student t distributionnThe precision parameter xi and the degree of freedom parameter v both obey gamma distribution:
Figure FDA0003127086770000069
Figure FDA00031270867700000610
Figure FDA00031270867700000611
wherein
Figure FDA00031270867700000612
Figure FDA00031270867700000613
Figure FDA0003127086770000071
Figure FDA0003127086770000072
Figure FDA0003127086770000073
Figure FDA0003127086770000074
(g) Furthermore, the process noise parameters also follow a gamma distribution:
Figure FDA0003127086770000075
wherein
Figure FDA0003127086770000076
Figure FDA0003127086770000077
5. The robust identification method of nonlinear dual rate circuitry with output time lag as claimed in claim 4 wherein: step three or two, the fast sampling time is circulated through the prediction and updating processes, the loss value existing in the output of the fast sampling process is estimated, and then the regression quantity is obtained
Figure FDA0003127086770000078
The corrected robust Kalman filtering algorithm can be obtained, and the specific process is as follows:
for a fast sampling instant k 1, 2.. L, the following two steps are performed by a prediction step and an update step loop:
(1) a prediction step:
Figure FDA0003127086770000079
Figure FDA00031270867700000710
wherein
Figure FDA00031270867700000711
Is the model regression at time k, xk+1|k，Pk+1|kRespectively for a given k time observationIn the case of (1), the process outputs the mean and variance, parameters Σ, predicted at the time k +1kIs defined as
Figure FDA00031270867700000712
Wherein σk|k＝Pk|k,σk|k-i＝-(1-skc)[σk-1|k-i,…,σk-i|k-i,…,σk-i|k-n]θx；
(2) And (3) updating:
Pk|k＝(1-skc)Pk|k-1
xk|k＝xk|k-1+sk(yk-cxk|k-1)
wherein x isk|k，Pk|kRespectively, mean and variance updated at time k, P, given the observed value at time kk|k-1And xk|k-1Obtained from the predicted value of the previous step, and
Figure FDA0003127086770000081
when no output signal data is observed at the kth moment, x is keptk|kFor the prediction of the previous step, i.e. xk|k＝xk|k-1；
Through the circulation of the prediction step and the updating step, the loss value existing in the output of the fast sampling process can be estimated, and then the regression quantity is obtained
Figure FDA0003127086770000082
6. The robust identification method of the nonlinear dual-rate circuit system with the output time lag as claimed in claim 4 or 5, wherein: the fourth step is to update the robust kalman filtering algorithm and the variational distribution formula corrected in the third step by iteration, and the specific process of finally obtaining the required identification result is as follows:
according to the model obtained in the third stepThe iterative formula of the parameters and the hidden variables can obtain the model parameters theta and the uncertainty sigma thereofθOutput estimate x for a noiseless processlTime lag d existing in system output;
mu is equal to thetaθ，
Figure FDA0003127086770000083
d＝max(R,2)；
Wherein R is a parameter RnmThe resulting matrix, max (·,2), is the maximum value of the matrix by row.
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