Parameter identification method for data missing output error system
By using the desired maximization algorithm to deal with the data loss problem in the parameter identification method of the output error system, the problem of poor parameter identification accuracy in the prior art is solved, and efficient parameter identification in the case of data loss is achieved.
Patent Information
- Application Number
- CN202411553636.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-02
- Publication Date
- 2025-02-11
AI Technical Summary
In industrial production, due to sensor failure, noise interference, network transmission and other factors, the collected data sets often have data loss. The existing technology mostly uses numerical interpolation methods, resulting in poor parameter identification accuracy.
A parameter identification method for data missing output error system is provided. By defining information vectors and parameter vectors based on the model expression of the output error system, a linear regression equation for the output error system is generated, including white noise. Using the expected maximization algorithm, the collected data set is divided into measurable data sets and hidden data sets, the Q function is constructed, and the estimated values of parameter vectors and noise variances are iteratively calculated until convergence.
When the output data is randomly missing, excellent parameter identification effect is achieved and the accuracy of parameter identification is improved.
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Figure CN119414718A_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the technical field of system identification, and in particular to a parameter identification method for a data missing output error system. Background Art
[0002] System identification plays an important role in process industry. Common structures describing dynamic system models include Auto-Regressive with Extra Inputs (ARX), Finite Impulse Response (FIR), and Output-Error (OE). Among them, the output error model ignoring noise terms is widely used in the field of automatic control because its structure is similar to the transfer function.
[0003] Parameter identification requires collected data as support. However, in the industrial production process, due to sensor failure, noise interference, network transmission and other factors, the collected data sets often have data loss. The existing technology mostly uses numerical interpolation methods to give an estimate of the lost data with the help of data before and after the lost data. However, when the lost data occurs continuously, the accuracy of parameter identification after using the numerical interpolation method is poor. Summary of the invention
[0004] In order to solve the deficiencies in the prior art, the purpose of the present application is to provide a parameter identification method for a data missing output error system, which can achieve excellent parameter identification results when the output data is randomly missing.
[0005] To achieve the above objectives, the present application provides a parameter identification method for a data missing output error system, comprising:
[0006] Based on the model expression of the output error system, an information vector and a parameter vector are defined to generate a linear regression equation of the output error system, wherein the output error system includes white noise;
[0007] According to whether the input and output data pairs are missing, the collected data sets are divided into measurable data sets and hidden data sets;
[0008] Based on the measurable data set and the hidden data set, an expectation maximization algorithm is used to construct a Q function for calculating the conditional expectation of the current parameter vector;
[0009] Based on the linear regression equation and the Q function, estimated values of the parameter vector and the noise variance are iteratively calculated until convergence.
[0010] Furthermore, the specific steps of defining the information vector and the parameter vector based on the model expression of the output error system and generating the linear regression equation of the output error system include:
[0011] Based on the unit shift operator in the model expression, the intermediate variable is defined using the following formula:
[0012]
[0013] Among them, k is the time, u k is the input sequence of the output error system, y k is the output sequence of the output error system, e k The mean is 0 and the variance is σ 2 A(z) and B(z) are polynomials about the unit shift operator, z is the unit backshift operator, is the term coefficient of A(z), n a is the order of A(z), is the term coefficient of B(z), n b is the order of B(z); x k is an intermediate variable;
[0014] Based on the intermediate variable, an information vector and a parameter vector are defined so that the intermediate variable is the product of the information vector and the parameter vector, and the formula is as follows:
[0015]
[0016] Among them, φ k is the information vector defined, θ is the parameter vector defined, j = n a +n b ;
[0017] Based on the defined information vector and parameter vector, a linear regression equation of the output error system is generated, and the formula is as follows:
[0018] x k =φ k θ;
[0019] y k =φ k θ+e k .
[0020] Furthermore, the specific step of constructing a Q function for calculating the conditional expectation of the current parameter vector using the expectation maximization algorithm based on the measurable data set and the hidden data set adopts the following formula:
[0021]
[0022] Θ={θ,σ 2};
[0023]
[0024] C obs = {U,Y obs};
[0025] U={u1,u2,…,u L};
[0026] C hid = {Y mis};
[0027] Among them, Y is the output data set, Y obs is the measurable output data set, Y mis is the missing output data set, C obs is a testable data set, C hid is the hidden data set, U is the input data set, u L is the Lth input data, t is the sampling time, o p is the pth measurable output moment, m q is the qth missing output moment, s is the iteration variable, Θ is the parameter to be identified, θ is the parameter vector, σ 2 is the variance of white noise.
[0028] Furthermore, the step of iteratively calculating the estimated values of the parameter vector and the noise variance based on the linear regression equation and the Q function until convergence comprises:
[0029] Based on the linear regression equation, simplify the calculation of Q function;
[0030] Based on the simplified Q function, the noise variance and parameter vector are iteratively updated until convergence.
[0031] Furthermore, the simplified calculation of the Q function based on the linear regression equation specifically adopts the following formula:
[0032] ln p(Y,U|Θ)=ln p(Y|U,Θ)p(U|Θ);
[0033] where p(Y|U,Θ) is simplified to:
[0034]
[0035] According to the linear regression equation, it is simplified to:
[0036]
[0037] get:
[0038]
[0039] Rewriting the log-likelihood function is:
[0040]
[0041] Rewrite the Q function as:
[0042]
[0043] in,
[0044]
[0045] Substituting into the rewritten Q function, we get:
[0046]
[0047] Furthermore, the specific steps of iteratively updating the noise variance and parameter vector based on the simplified Q function until convergence include:
[0048] Determine the partial derivatives of the simplified Q function with respect to the noise variance and model parameters, set them to 0, and solve to obtain the iterative formulas of the noise variance and model parameters;
[0049] Initialize the model parameters and intermediate variables, and use the iterative formula to solve the model parameters until the set convergence conditions are met.
[0050] To achieve the above object, the present application also provides an electronic device, including:
[0051] processor;
[0052] a memory having stored thereon one or more computer program instructions executed on the processor;
[0053] When the processor runs the computer instructions, the parameter identification method of the data missing output error system as described above is executed.
[0054] To achieve the above objectives, the present application provides a computer-readable storage medium on which computer instructions are stored. When the computer instructions are executed by a processor, the steps of the parameter identification method for the data missing output error system described above are executed.
[0055] The present application discloses a parameter identification method for a data missing output error system, which divides the collected data set into measurable input and output data and randomly missing output data, and uses the missing data as hidden variables to implement parameter identification of the system model with the help of the framework of the expectation maximization algorithm, thereby achieving good identification effect.
[0056] Other features and advantages of the present application will be set forth in the following description, and in part will become apparent from the description, or may be understood by practicing the present application. BRIEF DESCRIPTION OF THE DRAWINGS
[0057] The accompanying drawings are used to provide a further understanding of the present application and constitute a part of the specification. Together with the embodiments of the present application, they are used to explain the present application and do not constitute a limitation of the present application. In the accompanying drawings:
[0058] Figure 1 A flow chart of a parameter identification method for a data missing output error system of the present application;
[0059] Figure 2 This is a schematic diagram of the model parameter estimates and errors when the missing ratio is 20%;
[0060] Figure 3 This is a schematic diagram of the comparison of model parameter estimates and errors under different missing proportions;
[0061] Figure 4 This is an input-output data diagram when the missing ratio is 20%;
[0062] Figure 5 This is a curve chart showing the change of model parameter estimates with the number of iterations when the missing ratio is 20%;
[0063] Figure 6 This is a curve diagram showing the variation of the noise variance estimate with the number of iterations when the missing ratio is 20%;
[0064] Figure 7 This is a comparison chart of the predicted output and the actual output when the missing ratio is 20%;
[0065] Figure 8 This is a locally enlarged comparison diagram of the predicted output and the actual output when the missing ratio is 20%. DETAILED DESCRIPTION
[0066] The embodiments of the present application will be described in more detail below with reference to the accompanying drawings. Although certain embodiments of the present application are shown in the accompanying drawings, it should be understood that the present application can be implemented in various forms and should not be construed as being limited to the embodiments described herein. Instead, these embodiments are provided to provide a more thorough and measurable understanding of the present application. It should be understood that the drawings and embodiments of the present application are only for exemplary purposes and are not intended to limit the scope of protection of the present application.
[0067] It should be understood that the various steps described in the method implementation of the present application can be performed in different orders and / or performed in parallel. In addition, the method implementation may include additional steps and / or omit the steps shown. The scope of the present application is not limited in this respect.
[0068] The term "including" and its variations used herein are open inclusions, i.e., "including but not limited to". The term "based on" means "based at least in part on". The term "one embodiment" means "at least one embodiment"; the term "another embodiment" means "at least one additional embodiment"; the term "some embodiments" means "at least some embodiments". The relevant definitions of other terms will be given in the following description.
[0069] It should be noted that the modifications of "one" and "plurality" mentioned in this application are illustrative rather than restrictive, and those skilled in the art should understand that unless otherwise clearly indicated in the context, it should be understood as "one or more". "Plurality" should be understood as two or more.
[0070] Hereinafter, embodiments of the present application will be described in detail with reference to the accompanying drawings.
[0071] Example 1
[0072] An embodiment of the present application provides a parameter identification method for a data missing output error system. Figure 1 The parameter identification method of the data missing output error system of the present application is described in detail.
[0073] Step S101: Based on the model expression of the output error system, define the information vector and the parameter vector, and generate a linear regression equation of the output error system, wherein the output error system contains white noise;
[0074] Based on the unit shift operator in the model expression, the intermediate variable is defined using the following formula:
[0075]
[0076] Among them, k is the time, u k is the input sequence of the output error system, y k is the output sequence of the output error system, e k The mean is 0 and the variance is σ 2 A(z) and B(z) are polynomials about the unit shift operator, z is the unit backshift operator, is the term coefficient of A(z), n a is the order of A(z), is the term coefficient of B(z), n b is the order of B(z); x k is an intermediate variable;
[0077] Based on the intermediate variable, an information vector and a parameter vector are defined so that the intermediate variable is the product of the information vector and the parameter vector, and the formula is as follows:
[0078] Multiply both sides of the expression by A(z) and then move the terms around to get;
[0079]
[0080] In which, let j = n a +n b , define the information vector φ k , define the parameter vector θ,
[0081]
[0082] Based on the defined information vector and parameter vector, a linear regression equation of the output error system is generated, and the formula is as follows:
[0083] x k =φ k θ;
[0084] y k =φ k θ+e k .
[0085] Step S102: dividing the collected data set into a measurable data set and a hidden data set according to whether the input and output data pairs are missing;
[0086]
[0087] C obs = {U,Y obs};
[0088] U={u1,u2,…,u L};
[0089] C hid = {Y mis}
[0090] Among them, Y is the output data set, Y obs is the measurable output data set, Y mis is the missing output data set, C obs is a testable data set, C hid is the hidden data set, U is the input data set, u L is the Lth input data, t is the time, o p is the pth measurable output moment, m q is the qth missing output moment.
[0091] Step S103: Based on the measurable data set and the hidden data set, an expectation maximization algorithm is used to construct a Q function for calculating the conditional expectation of the current parameter vector.
[0092] The Expectation-Maximization Algorithm (EM algorithm) is an iterative optimization algorithm, which is mainly used to estimate the parameters of probability models with hidden variables. It has a wide range of applications in machine learning and statistics, including but not limited to Gaussian Mixture Model (GMM), Hidden Markov Model (HMM), and various clustering and classification problems. Its probability model is an iterative algorithm used to describe the relationship between observable data and potential structure to solve the maximum likelihood estimation (MLE) problem (incomplete data, with hidden variables).
[0093] In this embodiment, the randomly lost output data is regarded as the hidden data in the expectation maximization algorithm, the recorded input and measurable data sets are regarded as the observed data, the model parameter θ is regarded as the estimated parameter, and the expectation maximization algorithm is used to construct the Q function. The formula is as follows:
[0094]
[0095] Among them, s is the iteration variable, Θ is the parameter to be identified, θ is the parameter vector, σ 2 is the variance of white noise
[0096] Step s104: Based on the linear regression equation and the Q function, iteratively calculate the estimated values of the parameter vector and the noise variance until convergence.
[0097] First, calculate the log-likelihood function:
[0098] ln p(Y,U|Θ)=ln p(Y|U,Θ)p(U|Θ);
[0099] Since the input signal U is measurable and independent of the parameter to be identified Θ, the probability p(U|Θ) can be considered as a constant C. Then the probability density p(Y|U,Θ) can be further decomposed into:
[0100]
[0101] According to the linear regression equation, the output y k Depends on the parameters to be identified Θ and the input sequence {u 1:k-1}, it can be simplified:
[0102]
[0103] Then, the log-likelihood function can be simplified to:
[0104]
[0105] Considering the missing output data set, the log-likelihood function is further expressed as:
[0106]
[0107] Then we get the simplified Q function:
[0108]
[0109] Then, according to the linear regression equation and the properties of white noise, we can know that:
[0110]
[0111] According to the output data y k The probability density of , calculate the integral term in the above formula:
[0112]
[0113] Then we get:
[0114]
[0115] It can be understood that the above steps correspond to the E-step of the expectation-maximization algorithm.
[0116] Based on the simplified Q function, the noise variance and parameter vector are iteratively updated until convergence.
[0117] Specifically, calculate Q(Θ|Θ s ) for the unknown parameter noise variance σ 2 , the partial derivative of the model parameter θ, and set it to zero, as follows:
[0118]
[0119]
[0120] Solve to get the noise variance σ 2 , iterative update formula of model parameters θ:
[0121]
[0122] It can be understood that the intermediate variable x in the information vector in the iterative update formula k , we can use the parameter estimate θ obtained at each iteration s+1 Calculated.
[0123] Initialize the model parameters and intermediate variables, and use the iterative formula to solve the model parameters until the set convergence conditions are met.
[0124] In this embodiment, the model parameters are initialized θ0=0.01I, I represents a matrix of all 1s;
[0125] initialization Set the iteration condition ∈, set the number of iterations S = 1, start iteration, compare and like Let s = s + 1 and continue iterating; otherwise, the convergence condition is met and the output is Get the estimated values of model parameters and end the iteration.
[0126] It can be understood that the above steps correspond to the M-step of the expectation-maximization algorithm.
[0127] For example, an output error system:
[0128]
[0129] A(z)=1+a1z -1 +a2z -2 =0.13z -1 -0.1z -2 ;
[0130] B(z)=b1z -1 +b2z -2 =0.4z -1 +1.5z -2 ;
[0131] θ=[a1,a2,b1,b2] T =[0.13,-0.1,0.4,1.5] T ;
[0132] During the simulation, a set of unrelated measurable random signal sequences {u(t)} is set as the input signal, and the mean of the sequence is zero and the variance is 1. The noise signal has a mean of 0 and a variance of σ 2 =0.10 2 The parameter identification method of this application is used to estimate the model parameters under different proportions of missing data. The parameter estimation results and error values δ are shown in Figure 2 , 3 shown.
[0133] Among them, the error value
[0134] See also Figure 2, parameter estimates and errors obtained using the EM iterative algorithm when 20% of the data are randomly missing. Figure 2 It can be seen that the proposed algorithm can better estimate the parameters of the model when data is randomly missing. After algorithm iteration, the model parameters gradually approach the true values.
[0135] See also Figure 3 ,Although the missing proportion of output data increases, the ,precision of parameter estimation is still high, which proves the ,effectiveness of the proposed method.
[0136] See also Figure 4 , Figure 4 The input and output data are obtained when 20% of the data is randomly lost. In the output data, the missing data sampling points are reset to zero for the convenience of calculation and drawing. Figure 4 In the lower part of the figure, the output value y k = 0 is the point where the output sample is lost.
[0137] See also Figure 5 , Figure 5 The figure shows how the estimated values of model parameters change with the number of iterations when the random missing ratio is 20%. From the curve in the figure, we can see that the estimated values of the parameters can stabilize near the true value after several iterations.
[0138] See also Figure 6 , Figure 6 The figure shows how the noise variance estimate changes with the number of iterations when the random missing proportion is 20%.
[0139] See also Figure 7 and Figure 8 , Figure 7 and Figure 8 The comparison between the system output prediction value generated by the parameter estimation value obtained by the parameter identification method of the present application and the actual value is shown. Figure 8 It can be clearly seen in the local enlarged figure that at the point where the output sampling value is lost (that is, the blue output value is set to zero point in the figure), the predicted value can fit the output curve well.
[0140] Example 2
[0141] In this embodiment, an electronic device is also provided, including a processor and a memory. The memory is used to store non-temporary computer-readable instructions. The processor is used to run the non-temporary computer-readable instructions, and when the non-temporary computer-readable instructions are run by the processor, one or more steps of the parameter identification method of the data missing output error system described above can be executed. The memory and the processor can be interconnected through a bus system and / or other forms of connection mechanisms.
[0142] For example, the processor can be a central processing unit (CPU), a digital signal processor (DSP), or other forms of processing units with data processing capabilities and / or program execution capabilities, such as a field programmable gate array (FPGA); for example, the central processing unit (CPU) can be an X86 or ARM architecture, etc.
[0143] For example, the memory may include any combination of one or more computer program products, and the computer program product may include various forms of computer-readable storage media, such as volatile memory and / or non-volatile memory. Volatile memory may include, for example, random access memory (RAM) and / or cache memory (cache), etc. Non-volatile memory may include, for example, read-only memory (ROM), hard disk, erasable programmable read-only memory (EPROM), portable compact disk read-only memory (CD-ROM), USB memory, flash memory, etc. One or more computer program modules may be stored on the computer-readable storage medium, and the processor may run one or more computer program modules to implement various functions of the electronic device. Various applications and various data, as well as various data used and / or generated by the application, etc. may also be stored in the computer-readable storage medium.
[0144] It should be noted that in the embodiments of the present application, the specific functions and technical effects of the electronic device can refer to the description of the parameter identification method of the data missing output error system in the above text, which will not be repeated here.
[0145] Example 3
[0146] In this embodiment, a computer-readable storage medium is also provided, and the storage medium is used to store non-transitory computer-readable instructions. For example, when the non-transitory computer-readable instructions are executed by a computer, one or more steps in the parameter identification method of the data missing output error system described above can be executed.
[0147] For example, the storage medium can be applied to the above-mentioned electronic device. For example, the storage medium can be the memory in the electronic device of Example 2. For example, the relevant description of the storage medium can refer to the corresponding description of the memory in the electronic device of Example 2, which will not be repeated here.
[0148] The above description is only a partial embodiment of the present application and an explanation of the technical principles used. Those skilled in the art should understand that the scope of disclosure involved in the present application is not limited to the technical solution formed by a specific combination of the above technical features, but should also cover other technical solutions formed by any combination of the above technical features or their equivalent features without departing from the above disclosed concept. For example, the above features are replaced with the technical features with similar functions disclosed in this application (but not limited to) by each other to form a technical solution.
[0149] In addition, although each operation is described in a specific order, this should not be construed as requiring these operations to be performed in the specific order shown or in a sequential order. Under certain circumstances, multitasking and parallel processing may be advantageous. Similarly, although some specific implementation details are included in the above discussion, these should not be construed as limiting the scope of the application. Some features described in the context of a separate embodiment can also be implemented in a single embodiment in combination. On the contrary, the various features described in the context of a single embodiment can also be implemented in multiple embodiments individually or in any suitable sub-combination mode.
[0150] Although the subject matter has been described in language specific to structural features and / or methodological logical actions, it should be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or actions described above. On the contrary, the specific features and actions described above are merely example forms of implementing the claims.
Claims
1. A parameter identification method for a data missing output error system, characterized in that: include: Based on the model expression of the output error system, an information vector and a parameter vector are defined to generate a linear regression equation of the output error system, wherein the output error system includes white noise; According to whether the input and output data pairs are missing, the collected data sets are divided into measurable data sets and hidden data sets; Based on the measurable data set and the hidden data set, an expectation maximization algorithm is used to construct a Q function for calculating the conditional expectation of the current parameter vector; Based on the linear regression equation and the Q function, estimated values of the parameter vector and the noise variance are iteratively calculated until convergence.
2. The parameter identification method of the data missing output error system according to claim 1, characterized in that: The specific steps of defining the information vector and the parameter vector based on the model expression of the output error system and generating the linear regression equation of the output error system include: Based on the unit shift operator in the model expression, the intermediate variable is defined using the following formula: Among them, k is the time, u k is the input sequence of the output error system, y k is the output sequence of the output error system, e k The mean is 0 and the variance is σ 2 A(z) and B(z) are polynomials about the unit shift operator, z is the unit backshift operator, is the term coefficient of A(z), n a is the order of A(z), is the term coefficient of B(z), n b is the order of B(z); x k is an intermediate variable; Based on the intermediate variable, an information vector and a parameter vector are defined so that the intermediate variable is the product of the information vector and the parameter vector, and the formula is as follows: Among them, φ k is the information vector defined, θ is the parameter vector defined, j = n a +n b ; Based on the defined information vector and parameter vector, a linear regression equation of the output error system is generated, and the formula is as follows: x k =φ k I; y k =φ k θ+e k 。 3. The parameter identification method of the data missing output error system according to claim 2, characterized in that: The specific step of constructing a Q function for calculating the conditional expectation of the current parameter vector based on the measurable data set and the hidden data set using the expectation maximization algorithm is as follows: Θ={θ,σ 2 }; C obs ={U,Y obs }; U={u1,u2,…,u L }; C hid ={Y mis }; Among them, Y is the output data set, Y obs is the measurable output data set, Y mis is the missing output data set, C obs is a testable data set, C hid is the hidden data set, U is the input data set, u L is the Lth input data, t is the sampling time, o p is the pth measurable output moment, m q is the qth missing output moment, s is the iteration variable, Θ is the parameter to be identified, θ is the parameter vector, σ 2 is the variance of white noise.
4. The parameter identification method of the data missing output error system according to claim 3, characterized in that: The step of iteratively calculating the estimated values of the parameter vector and the noise variance based on the linear regression equation and the Q function until convergence comprises: Based on the linear regression equation, simplify the calculation of Q function; Based on the simplified Q function, the noise variance and parameter vector are iteratively updated until convergence.
5. The parameter identification method of the data missing output error system according to claim 4, characterized in that: The simplified calculation of the Q function based on the linear regression equation is specifically based on the following formula: ln p(Y,U|Θ)=ln p(Y|U,Θ)p(U|Θ); where p(Y|U,Θ) is simplified to: According to the linear regression equation, it is simplified to: get: Rewriting the log-likelihood function is: Rewrite the Q function as: in, Substituting into the rewritten Q function, we get:
6. The parameter identification method of the data missing output error system according to claim 5, characterized in that: The specific steps of iteratively updating the noise variance and parameter vector until convergence based on the simplified Q function include: Determine the partial derivatives of the simplified Q function with respect to the noise variance and model parameters, set them to 0, and solve to obtain the iterative formulas of the noise variance and model parameters; Initialize the model parameters and intermediate variables, and use the iterative formula to solve the model parameters until the set convergence conditions are met.
7. An electronic device, characterized in that: include: processor; a memory having stored thereon one or more computer program instructions executed on the processor; Wherein, when the processor runs the computer instructions, it executes the parameter identification method of the data missing output error system described in any one of claims 1-6.
8. A computer-readable storage medium, characterized in that: Computer instructions are stored thereon, and when the computer instructions are executed by the processor, the steps of the parameter identification method for the data missing output error system described in any one of claims 1-6 are executed.