Robust Bayesian identification method for state space model of direct current motor motion system

Abstract

The invention discloses a robust Bayesian identification method for a state space model of a DC motor motion system, and relates to the technical field of industrial automation. The invention aims to solve the problem of poor model parameter estimation precision under the conditions of non-Gaussian noise and unknown system state in the prior art. The method comprises the following steps: establishing a mathematical description of a parameter identification problem of a state space model of a direct current motor motion system to determine to-be-identified parameters of the state space model of the direct current motor motion system; constructing posterior distribution of the to-be-identified parameters; constructing an augmented state space model having the same form as the logarithmic posteriori distribution of the to-be-identified state vector, and optimizing the to-be-identified state vector according to the augmented state space model; and carrying out iterative calculation by utilizing an optimization result and the posterior distribution of the other to-be-identified parameters except the to-be-identified state vector to obtain parameters of the state space model of the direct current motor motion system until a convergence condition is met, and completing parameter identification of the state space model of the direct current motor motion system.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of industrial automation, and particularly relates to system identification. BACKGROUND

[0002] In the new industrial era, the development of intelligent degree brings the improvement of system complexity, the mechanism modeling method cannot be competent for such complex system, and the implementation of control algorithm depends on the accurate system model, which makes the system identification a suitable alternative.

[0003] The existing technology has the problem of poor model parameter estimation accuracy in the complex environment of unknown system state and non-Gaussian noise. This not only limits the application effect of system identification method in actual industrial scene, but also hinders the performance improvement of intelligent system. Therefore, there is an urgent need for a new technical solution that can accurately estimate system model parameters and hidden state variables in a complex noise environment. SUMMARY

[0004] The present application is to solve the problem of poor model parameter estimation accuracy in the existing technology in the case of non-Gaussian noise and unknown system state. The present application provides a robust Bayesian identification method for the state space model of a DC motor motion system, which can effectively avoid the problem of poor model parameter estimation accuracy in the traditional method in the case of non-Gaussian noise and unknown system state, and even completely invalid.

[0005] The present application provides a robust Bayesian identification method for the state space model of a DC motor motion system, which includes:

[0006] Establishing a mathematical description of the parameter identification problem of the state space model of the DC motor motion system to determine the to-be-identified parameters of the state space model of the DC motor motion system;

[0007] Constructing the posterior distribution of the to-be-identified parameters;

[0008] Constructing an augmented state space model with the same form as the logarithmic posterior distribution of the to-be-identified state vector, and optimizing the to-be-identified state vector according to the augmented state space model;

[0009] Using the optimization result and the posterior distribution of the to-be-identified parameters except the to-be-identified state vector to perform iterative calculation to obtain the parameters of the state space model of the DC motor motion system, until the convergence condition is met, and the parameter identification of the state space model of the DC motor motion system is completed.

[0010] In one possible design, the establishment of the mathematical description of the parameter identification problem of the state space model of the DC motor motion system to determine the to-be-identified parameters of the state space model of the DC motor motion system includes:

[0011] Establish a state-space model of the DC motor motion system;

[0012] Within a probabilistic framework, a statistical characterization of the state vector and output vector in the state-space model is established.

[0013] Given an output vector and introducing latent variables, the parameters to be identified are jointly estimated based on a Bayesian framework.

[0014] In one possible design, the state-space model expression of the DC motor motion system is as follows:

[0015] ,

[0016] ,

[0017] in, and They are respectively The state vector and output vector of the DC motor motion system at time t. for The state vector of the DC motor motion system at time t. Sampling time, State noise, To measure noise, For the system matrix, This is the observation matrix.

[0018] In one possible design, within a probabilistic framework, a statistical characterization of the state vectors and measurement outputs in the state-space model is established. Given the output vectors and introducing latent variables, the parameters to be identified are jointly estimated based on a Bayesian framework, including:

[0019] The measured noise It is non-Gaussian white noise and follows a generalized hyperbolic multivariate skewed T-distribution, expressed as:

[0020] ,

[0021] in, This represents a generalized hyperbolic multivariate skewed T-distribution. The mean, Let covariance matrix be the variance matrix. This is the skewness parameter. These are the degrees of freedom parameters;

[0022] exist Introducing latent variables at time Then the generalized hyperbolic multivariate skewed T-distribution function is:

[0023] ,

[0024] ,

[0025] in, To measure noise The conditional prior distribution function, Latent variables The conditional prior distribution function, Represents the gamma distribution. Indicates a Gaussian distribution;

[0026] The state noise It is Gaussian white noise with a mean of 0 and a covariance matrix of... Gaussian distribution, This refers to the state noise accuracy parameter;

[0027] The state vector at time 1 joint probability density function :

[0028] ,

[0029] in, Represents the initial state The probability density function, and These are the mean and variance of the probability density function, respectively.

[0030] The output vector at time 1 joint probability density function :

[0031] ,

[0032] in, As a latent variable, To measure noise accuracy parameters, This is the skewness parameter;

[0033] System Matrix prior distribution function for: , representing the system matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution;

[0034] Observation matrix prior distribution function for: , representing the observation matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution, and All are vector dimensions;

[0035] System matrix accuracy parameters prior distribution function for: System matrix accuracy parameters Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution;

[0036] State noise accuracy parameters prior distribution function for: , representing the state noise accuracy parameter Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution;

[0037] Observation matrix accuracy parameters prior distribution function for: , representing the accuracy parameter of the observation matrix Each element in All conform to shape parameters as And the scale parameter is The gamma distribution;

[0038] Measurement noise accuracy parameters prior distribution function for: This indicates the measurement noise accuracy parameter. Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution;

[0039] Skewness parameter prior distribution function for: , representing the skewness parameter Each element in All follow a mean of 0 and a variance of . Gaussian distribution;

[0040] Precision parameter vector prior distribution function for: , representing the precision parameter vector each element in the set obeys a gamma distribution with shape parameter and scale parameter ;

[0041] the prior distribution function of the degree of freedom parameter is : , which means the degree of freedom parameter obeys a gamma distribution with shape parameter and scale parameter ;

[0042] The to-be-identified parameters of the direct current motor motion system state space model include: , , , , , , , , , and .

[0043] In one possible design, the posterior distribution of the to-be-identified parameters is constructed, including:

[0044] constructing the joint probability density function of all to-be-identified parameters based on the mean field theory;

[0045] using the KL divergence to measure the distance between the approximate posterior distribution and the true posterior distribution of each to-be-identified parameter, and obtaining the log posterior distribution of each to-be-identified parameter by minimizing the distance;

[0046] deriving the log posterior distribution of the state vector , and using the log posterior distribution of the remaining to-be-identified parameters to derive the approximate posterior distribution of the corresponding to-be-identified parameter.

[0047] In one possible design, the joint probability density function of all to-be-identified parameters is approximated as the product of the approximate posterior distributions of each to-be-identified parameter:

[0048] ,

[0049] wherein, , , , , , , , , , ​and are the approximate posterior distributions of , , , , , , , , , and respectively.

[0050] The log posterior distribution of each parameter to be identified is given by

[0051] ,

[0052] where is the th unknown parameter in , denotes the log posterior distribution of , denotes the mathematical expectation of the function inside the angular brackets, which is the mathematical expectation of the remaining unknown parameters after , denotes the mathematical expectation of the variable contained in , denotes the constant term, is the joint probability density function of and .

[0053] In one possible design, the joint probability density function of and is given by

[0054]

[0055] where is the conditional prior distribution function of the latent variable .

[0056] In one possible design, the approximate posterior distribution of the corresponding parameter to be identified is derived using the log posterior distribution of the remaining parameters to be identified except the state vector , including

[0057] The approximate posterior distribution of the system matrix is a Gaussian distribution:

[0058] ,

[0059] where ,​​ , denotes the element in the th row of

[0060] ; the approximate posterior distribution of the observation matrix is Gaussian:

[0061] ,

[0062] where ,

[0063] , denotes the element in the th row of

[0064] ; the approximate posterior distribution of the system matrix precision parameter is Gamma distributed:

[0065] ,

[0066] where , , denotes the element in the th row and th column of the system matrix

[0067] ; the approximate posterior distribution of the observation matrix precision parameter is Gamma distributed:

[0068] ,

[0069] where , , denotes the element in the th row and th column of the observation matrix

[0070] ; the approximate posterior distribution of the state noise precision parameter is Gamma distributed:

[0071] ,

[0072] where , , denotes the total number of sampling time instants;

[0073] the measurement noise precision parameterapproximate posterior distribution of is a gamma distribution:

[0074] ,

[0075] where, , ;

[0076] skewness parameter approximate posterior distribution of is a Gaussian distribution:

[0077] ,

[0078] where, , ;

[0079] degrees of freedom parameter approximate posterior distribution of is a gamma distribution:

[0080] ,

[0081] where, , ;

[0082] precision parameter vector approximate posterior distribution of is a gamma distribution:

[0083] ,

[0084] where, , ;

[0085] latent variable approximate posterior distribution of is a generalized inverse Gaussian distribution:

[0086] ,

[0087] where, , , .

[0088] In one possible design, the log posterior distribution is derived for the state vector :

[0089] ,

[0090] ,​ ,

[0091] , .

[0092] In one possible design, the construction is an augmented state space model with the same form as the log-posterior distribution of the state vector to be recognized, including:

[0093] The augmented state space model expression is:

[0094] , ,

[0095] , ,

[0096] Wherein: , denotes the mathematical expectation of the variable contained in , is the state transition disturbance, , , denotes the connection of the input parameter in the vertical direction, and are zero vectors of and dimensions respectively, and are the dimensions of and respectively, is the observation disturbance, , and are identity matrices of and dimensions respectively,

[0097] , , and denote the Cholesky decomposition of and respectively,

[0098] , ,

[0099] , ,

[0100] When , is a zero matrix of dimensions.

[0101] Advantages of the present application:

[0102] The present application firstly establishes a linear state space model for the DC motor motion system; then in the probability framework, assuming that all unknown parameters are random variables, and selecting a suitable prior distribution for each parameter in the system matrix, introducing a generalized hyperbolic multivariate skew-t distribution to model the measurement noise, establishing the statistical characterization of the state and output of the linear state space model, and summarizing the mathematical description of the modeling problem; according to the mean field theory, constructing an approximate decomposition form of the joint probability density function of the unknown parameters, and under the framework of variational Bayesian estimation, deriving the joint estimation algorithm of the model parameters and the posterior distribution of the system state by maximizing the variational lower bound with respect to the KL divergence. The method of the present application can effectively suppress the adverse effects of non-Gaussian noise and unknown system state on model identification, improve the identification accuracy of the DC motor motion system, and has important significance for ensuring the motion control accuracy of the DC motor motion system. BRIEF DESCRIPTION OF DRAWINGS

[0103] Figure 1 Fig. 1 is a schematic diagram of a DC motor motion system;

[0104] Figure 2 Fig. 2 is a curve diagram of experimental data;

[0105] Figure 3 Fig. 3 is a comparison diagram of the identification results of two identification methods and the true value;

[0106] Figure 4 Fig. 4 is a comparison diagram of the state estimation values obtained by two identification methods and the true value;

[0107] Figure 5 Fig. 5 is a flowchart of the robust Bayesian identification method of the state space model of the DC motor motion system according to the embodiment. DETAILED DESCRIPTION

[0108] The technical solutions in the embodiments of the present application will be described clearly and completely below in combination with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application. It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict.

[0109] Traditional system identification methods mainly involve two sub-problems: estimation of unknown parameters and determination of hidden states. In the case of unknown hidden state variables, the least squares method, gradient-based algorithm, maximum likelihood estimation algorithm, expectation maximization algorithm and other methods are usually used to estimate unknown parameters. These methods can directly use input and output data to estimate the corresponding model parameters without explicitly estimating intermediate state information, thereby reducing the complexity of modeling and computational complexity to some extent. However, this simplification also has drawbacks, namely ignoring the key dynamic characteristics contained in the intermediate state variables. With the increase of the complexity of the internal model, this neglect will lead to the decline of the accuracy of parameter estimation. In addition, the above-mentioned identification methods are limited to point estimation and cannot describe the uncertainty of the estimated parameters.

[0110] Further analysis shows that the accuracy of parameter estimation is closely related to the number and quality of the collected input and output data. In actual application, due to the influence of various factors such as sensor failure, environmental interference, data transmission error, etc., the observed data collected usually contains outliers. These outliers, which are outliers significantly deviating from other data points, will have a negative impact on the accuracy of parameter estimation. Traditionally, it is assumed that the noise follows a Gaussian distribution, but this assumption cannot accurately represent the true noise characteristics. The least squares method, maximum likelihood estimation and other methods are derived based on the assumption of Gaussian noise. When facing non-Gaussian noise, these assumptions are no longer valid, thus inevitably leading to a decrease in algorithm performance. Although potential outliers can be screened and corrected by pre-designed standards, it is often very difficult to determine the appropriate threshold, which will again lead to the problem of bias estimation.

[0111] Therefore, the embodiments of the present application provide a robust Bayesian identification method for state space model of DC motor motion system to solve the above problems. The following will be combined with the accompanying drawings to describe the embodiments of the present application in detail. Figure 5 The scheme of the embodiments of the present application will be described in detail.

[0112] The robust Bayesian identification method for state space model of DC motor motion system described in the embodiments includes the following steps:

[0113] Step 1: For the DC motor motion system, a linear state space model is established, and the specific process is as follows:

[0114] The linear state space model of the DC motor motion system can be described as:

[0115] ,

[0116] ,

[0117] wherein, and are the state vector and observation vector of the DC motor motion system at time t, is the sampling time, is the total sampling time, are column vectors of dimension is the state noise, is the system matrix,

[0118] Second step: in the probability framework, the statistical description of the state and output in the linear state space model is established, and the specific process is as follows:

[0119] Step 2.1: assume that the measurement noise is a non-Gaussian white noise, which obeys a generalized hyperbolic multivariate skew-t distribution, and the expression is as follows:

[0120]

[0121] wherein represents the generalized hyperbolic multivariate skew-t distribution, is the mean, is the covariance matrix, is the skewness parameter, is the degree of freedom parameter.

[0122] At time t, introduce the latent variable , then the above generalized hyperbolic multivariate skew-t distribution function can be rewritten as:

[0123]

[0124]

[0125] wherein is the conditional prior distribution function of the measurement noise , and obeys a Gaussian distribution with the mean and the covariance ;

[0126] is the conditional prior distribution function of the latent variable , and obeys a gamma distribution with the representation parameter ;

[0127] represents the gamma distribution; represents the Gaussian distribution.​​​​​​​​​​​

[0128] Step 2.2: Assume state noise It is Gaussian white noise with a mean of 0 and a covariance matrix of... Gaussian distribution:

[0129] ,

[0130] This is the state noise accuracy parameter.

[0131] Step 2.3: Utilize express The state data at each moment, i.e. ,but joint probability density function Represented as:

[0132] ,

[0133] in, Initial state The probability density function, i.e., the mean is And the covariance is The Gaussian distribution.

[0134] Step 2.4: Utilize express The output vector at each time step, i.e. And let the mean of the measurement noise be 0, the covariance matrix be... ,but The joint probability density function is expressed as follows:

[0135] .

[0136] Step 2.5: The probability distributions of the state-space model parameters and hyperparameters can be described as follows:

[0137] System Matrix prior distribution function for: , representing the system matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution;

[0138] Observation matrix prior distribution function for: , representing the observation matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution, and are vector dimensions;

[0139] system matrix precision parameters prior distribution function is: system matrix precision parameters each element of follows a gamma distribution with shape parameter and scale parameter

[0140] state noise precision parameters prior distribution function is: state noise precision parameters each element of follows a gamma distribution with shape parameter and scale parameter

[0141] observation matrix precision parameters prior distribution function is: observation matrix precision parameters each element of follows a gamma distribution with shape parameter and scale parameter

[0142] measurement noise precision parameters prior distribution function is: measurement noise precision parameters each element of follows a gamma distribution with shape parameter and scale parameter

[0143] skewness parameters prior distribution function is: skewness parameters each element of follows a Gaussian distribution with mean 0 and variance

[0144] precision parameter vector prior distribution function is: precision parameter vector each element of follows a gamma distribution with shape parameter ​​​​​and the scale parameter is a gamma distribution with shape parameter

[0145] a degree-of-freedom parameter a prior distribution function is: , indicating that the degree-of-freedom parameter obeys a gamma distribution with shape parameter and scale parameter .

[0146] The third step is to establish a mathematical description of the model identification problem, and the specific process is as follows:

[0147] The mathematical description of the model identification problem in this embodiment can be expressed as:

[0148] Given the measured output , and introducing the latent variable , the unknown parameters are jointly estimated based on the Bayesian framework, including: the system matrix , the observation matrix , the system state , and , , , , , , and .

[0149] The fourth step is to obtain the approximate posterior distribution of the unknown parameters by using the mean field theory, and the specific process is as follows:

[0150] All unknown parameters can be marked as , and there are . Based on the mean field theory, the joint probability density function of all unknown parameters can be approximately expressed as:

[0151] ,

[0152] wherein , , , , , , , , , and are , , , , , , 、 、 、 and the approximate posterior distribution.

[0153] The distance between the approximate posterior distribution and the true posterior distribution is measured by the KL divergence, and the log posterior distribution formula of any unknown parameter can be obtained by minimizing the distance. Specifically as follows:

[0154] The th unknown parameter in the is denoted as , then the log posterior distribution of can be obtained by minimizing the distance as:

[0155] ,

[0156] where denotes the mathematical expectation of all variables in except in the function inside the angle brackets, denotes the constant term.

[0157] The joint probability density function of the output and all unknown parameters is denoted as:

[0158] where is the conditional prior distribution function of the latent variable .

[0159] Step 5: Derive the approximate posterior distribution of the system matrix and the observation matrix , the specific process is as follows:

[0160] According to the log posterior distribution formula of any unknown parameter given in Step 4 , the posterior distribution of the system matrix can be derived as a Gaussian distribution:

[0161] ,

[0162] where the covariance matrix is: ,

[0163] The mean vector is: ,

[0164] This indicates the expected value of the function enclosed in angle brackets with respect to the variables it contains. express The Middle Row element.

[0165] Similarly, the measurement matrix can be derived. posterior distribution It follows a Gaussian distribution:

[0166] ,

[0167] The covariance matrix is: ,

[0168] The mean vector is: ,

[0169] for The Middle Row element.

[0170] Step 6: Derivation The approximate posterior distribution is obtained through the following process:

[0171] The system matrix accuracy parameter can be derived from the formula for calculating the logarithmic posterior distribution of any unknown parameter given in step four. posterior distribution Gamma distribution:

[0172] ,

[0173] in: , , For the system matrix The Middle Line number Column elements, for The Middle Each element.

[0174] Similarly, the accuracy parameter of the observation matrix can be derived. posterior distribution Gamma distribution:

[0175] ,

[0176] in: , , Observation matrix The Middle Line number Column elements, for The Middle element.

[0177] Similarly, the posterior distribution of the state noise precision parameter is a gamma distribution:

[0178] ,

[0179] where: , .

[0180] Similarly, the approximate posterior distribution of the measurement noise precision parameter is a gamma distribution:

[0181] ,

[0182] where: , .

[0183] Similarly, the posterior distribution of the skewness parameter is a Gaussian distribution:

[0184] ,

[0185] where: , .

[0186] Similarly, the posterior distribution of the latent variable is a generalized inverse Gaussian distribution:

[0187] ,

[0188] where: , , , denotes a generalized inverse Gaussian distribution.

[0189] Similarly, the posterior distribution of the degrees of freedom parameter is a gamma distribution:

[0190] ,

[0191] where: , .

[0192] Similarly, the posterior distribution of the precision parameter vector is a multivariate gamma distribution: ​​​​​Gamma distribution:

[0193] ,

[0194] where: , .

[0195] Step 7: Derive the approximate posterior distribution of state The procedure is as follows:

[0196] The log posterior distribution of state can be derived from the formula of the log posterior distribution of any unknown parameter given in Step 4:

[0197] ,

[0198] , ,

[0199] , .

[0200] Step 8: Construct the augmented state space model with deterministic parameters that have the same form of log posterior distribution:

[0201] , ,

[0202] , ,

[0203] where: , , , is the state transition disturbance, is the observation disturbance.

[0204] , , ,

[0205] and are the identity matrices of dimension and , and are the zero vectors of dimension and , is the zero matrix of dimension , denotes the concatenation of input parameters in the vertical direction, and denote the and Cholesky decomposition (square root method).

[0206] For the augmented state space model, Kalman filter and Kalman smoother are used to estimate the state and the covariance matrix is used as the uncertainty measure of the state estimate. The posterior distribution of the state and other unknown parameters is obtained. In summary, the iterative update formula of all unknown parameters has been derived.

[0207] Step 9: Repeat steps 5 to 8 until the algorithm meets the convergence condition. The convergence condition can be set as the number of iterations reaching a set upper limit, or the relative change of the current parameter estimate with respect to the parameter estimate obtained in the previous iteration being less than an arbitrarily small constant.

[0208] To further introduce the scheme of the embodiments of the present application, a direct current motor motion system is used to verify the effectiveness of the present application. The experimental device diagram of the system is shown in Figure 1 . The main components include: (1) a direct current motor equipped with an incremental encoder, which uses a PD controller to realize position control; (2) a star-shaped high-precision low-friction ball screw transmission positioning unit; (3) a load. The dynamic characteristics of the direct current motor motion system can be described by a linear state space model, where the system matrix is:

[0209] .

[0210] The observation matrix is a unit matrix. The experimental data is collected, and the length is selected as 500. 10% of the abnormal values uniformly distributed in [-100, 100] are introduced to simulate non-Gaussian noise. The data after adding abnormal values is shown in Figure 2 . The linear state space model identification of the direct current motor motion system is performed by using the embodiments, and compared with the identification method under the assumption of Gaussian noise. The system matrix parameters identified by the two identification methods are shown in Figure 3 , where RVB-LSSM represents the method of the embodiments, Regular represents the identification method under the assumption of Gaussian noise, and True represents the true parameter value. As can be seen from Figure 3 , the 16 system matrix parameter values identified by the method of the embodiments all approximate the true value, while the system matrix parameter values identified by the identification method under the assumption of Gaussian noise deviate from the true value due to the influence of abnormal values. The comparison of the state estimate values obtained by the two identification methods with the true value is shown in Figure 4 . As can be seen from Figure 4It can be seen that the method of the embodiment estimates the state to approach the true value, and the recognition method under the Gaussian noise assumption estimates the state to seriously deviate from the true trajectory in the time period with abnormal values. The verification result shows that the method of the embodiment can effectively process the non-Gaussian noise and unknown state situation, and effectively ensure the model recognition accuracy.

[0211] While the application has been described with reference to particular embodiments thereof, it is to be understood that these embodiments are merely illustrative of the principles and applications of the application. It will thus be appreciated that numerous modifications can be made to the illustrative embodiments and that other arrangements can be devised without departing from the spirit and scope of the application as defined by the appended claims. It will be understood that the features of the various embodiments can be combined with each other, where appropriate. It will also be understood that features described with reference to a single embodiment can be used in other embodiments.

Claims

1. A robust Bayesian identification method for the state-space model of a DC motor motion system, characterized in that, include: A mathematical description of the parameter identification problem of the state-space model of the DC motor motion system is established to determine the parameters to be identified in the state-space model of the DC motor motion system. Construct the posterior distribution of the parameter to be identified; Construct an augmented state space model that has the same form as the log-posterior distribution of the state vector to be identified, and optimize the state vector to be identified based on the augmented state space model. The parameters of the state space model of the DC motor motion system are obtained by iterative calculation using the optimization results and the posterior distribution of the parameters to be identified other than the state vector to be identified, until the convergence condition is met, thus completing the parameter identification of the state space model of the DC motor motion system.

2. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 1, characterized in that, The mathematical description of the problem of establishing the parameter identification of the state-space model of the DC motor motion system is used to determine the parameters to be identified in the state-space model of the DC motor motion system, including: Establish a state-space model of the DC motor motion system; Within a probabilistic framework, a statistical characterization of the state vector and output vector in the state-space model is established. Given an output vector and introducing latent variables, the parameters to be identified are jointly estimated based on a Bayesian framework.

3. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 2, characterized in that, The state-space model expression of the DC motor motion system is as follows: ， ， in, and They are respectively The state vector and output vector of the DC motor motion system at time t. for The state vector of the DC motor motion system at time t. Sampling time, State noise, To measure noise, For the system matrix, This is the observation matrix.

4. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 3, characterized in that, Within the probabilistic framework, a statistical characterization of the state vector and measurement output in the state-space model is established. Given the output vector and introducing latent variables, the parameters to be identified are jointly estimated based on a Bayesian framework, including: The measured noise It is non-Gaussian white noise and follows a generalized hyperbolic multivariate skewed T-distribution, expressed as: ， in, This represents a generalized hyperbolic multivariate skewed T-distribution. The mean, Let covariance matrix be the variance matrix. This is the skewness parameter. These are the degrees of freedom parameters; exist Introducing latent variables at time Then the generalized hyperbolic multivariate skewed T-distribution function is: ， ， in, To measure noise The conditional prior distribution function, Latent variables The conditional prior distribution function, Represents the gamma distribution. Indicates a Gaussian distribution; The state noise It is Gaussian white noise with a mean of 0 and a covariance matrix of... Gaussian distribution, This refers to the state noise accuracy parameter; The state vector at time 1 joint probability density function : ， in, Represents the initial state The probability density function, and These are the mean and variance of the probability density function, respectively. The output vector at time 1 joint probability density function : ， in, As a latent variable, To measure noise accuracy parameters, This is the skewness parameter; System Matrix prior distribution function for: , representing the system matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution; Observation matrix prior distribution function for: , representing the observation matrix each line All follow a pattern with a mean of 0 and a covariance of . Gaussian distribution, and All are vector dimensions; System matrix accuracy parameters prior distribution function for: System matrix accuracy parameters Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution; State noise accuracy parameters prior distribution function for: , representing the state noise accuracy parameter Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution; Observation matrix accuracy parameters prior distribution function for: , representing the accuracy parameter of the observation matrix Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution; Measurement noise accuracy parameters prior distribution function for: This indicates the measurement noise accuracy parameter. Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution; Skewness parameter prior distribution function for: , representing the skewness parameter Each element in All follow a mean of 0 and a variance of . Gaussian distribution; Precision parameter vector prior distribution function for: , representing the precision parameter vector Each element in All conform to the shape parameter as And the scale parameter is The gamma distribution; Degrees of freedom parameters prior distribution function for: , representing the degree of freedom parameter Obeying shape parameters And the scale parameter is The gamma distribution; The parameters to be identified in the state-space model of the DC motor motion system include: , , , , , , , , , and .

5. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 4, characterized in that, The construction of the posterior distribution of the parameters to be identified includes: Construct the joint probability density function of all parameters to be identified based on mean-field theory; KL divergence is used to measure the distance between the approximate posterior distribution and the true posterior distribution of each parameter to be identified. The log-posterior distribution of each parameter to be identified is obtained by minimizing the distance. Derivation of the state vector The log-posterior distribution of the remaining parameters to be identified is used to derive the approximate posterior distribution of the corresponding parameters to be identified.

6. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 5, characterized in that, All parameters to be identified joint probability density function It is approximately in the form of the product of the approximate posterior distributions of each parameter to be identified: ， in, , , , , , , , , , and They are respectively , , , , , , , , , and The approximate posterior distribution; The log-posterior distribution expression of each parameter to be identified is as follows: ， in, for The first in One unknown parameter, express The log-posterior distribution, This indicates finding the expected value of the function enclosed in angle brackets, where the expected value is the sum of the expected values ​​after removing the expected value from the expected value. The expected value of the remaining unknown parameters. Expressing the request The mathematical expectation of the variables is included. Represents a constant term. for and The joint probability density function.

7. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 6, characterized in that, The and joint probability density function The expression is as follows: in, Latent variables The conditional prior distribution function.

8. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 6, characterized in that, Using the state vector The derivation of the approximate posterior distributions of the other parameters to be identified includes: System Matrix Approximate posterior distribution It follows a Gaussian distribution: ， in, , , express The Middle row element; Observation matrix Approximate posterior distribution It follows a Gaussian distribution: ， in, , , for The Middle row element; System matrix accuracy parameters Approximate posterior distribution Gamma distribution: ， in, , , For the system matrix The Middle Line number Column elements; Observation matrix accuracy parameters Approximate posterior distribution Gamma distribution: ， in, , , Observation matrix The Middle Line number Column elements; State noise accuracy parameters Approximate posterior distribution Gamma distribution: ， in, , , This represents the total number of sampling times. Measurement noise accuracy parameters Approximate posterior distribution Gamma distribution: ， in, , ; Skewness parameter Approximate posterior distribution It follows a Gaussian distribution: ， in, , ; Degrees of freedom parameters Approximate posterior distribution Gamma distribution: ， in, , ; Precision parameter vector Approximate posterior distribution Gamma distribution: ， in, , ; Latent variables Approximate posterior distribution It is a generalized inverse Gaussian distribution: ， in, , , .

9. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 6, characterized in that, Using the log-posterior distribution Derivation of the state vector log-posterior distribution : ， ， ， ， 。 10. The robust Bayesian identification method for the state-space model of a DC motor motion system according to claim 4, characterized in that, The constructed augmented state-space model has the same form as the log-posterior distribution of the state vector to be identified, including: The augmented state-space model expression is as follows: ， ， ， ， in: , Expressing the request The mathematical expectation of the variables is included. This is a state transition perturbation. , , This indicates the vertical connection of the input parameters. and They are respectively and The zero vector of dimension, and They are respectively and Dimensions To observe the disturbance, , and They are respectively and An identity matrix of dimension 1 , , and They represent and Cholesky decomposition, ， ， ， ， hour , for Zero-dimensional matrix.

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