Method and system for colored noise system identification based on whitening and cross-estimation

Abstract

This invention discloses a method and system for identifying colored noise systems based on whitening processing and interactive estimation. The method includes: constructing an original model of the colored noise system; filtering the original model to obtain a whitened model of the colored noise system; constructing the whitened model based on system information vectors, system parameter vectors, and white noise; splitting the original model to obtain a noise sub-model and a system operation sub-model; constructing the noise sub-model based on noise information vectors and noise parameter vectors; constructing the system operation sub-model based on intermediate information vectors and system parameter vectors; calculating the current system information vector and noise information vector based on historical noise parameter vectors and intermediate information vectors, respectively; determining the current system parameter vector based on the current system information vector; and determining the current noise parameter vector based on the current noise information vector, thereby obtaining the original model without unknowns. This invention has advantages such as wide applicability and high reliability.

Description

Technical Field

[0001] This application relates to the field of system identification technology, and in particular to a method and system for identifying colored noise systems based on whitening processing and interactive estimation. Background Technology

[0002] System identification plays a crucial role in industrial process modeling. Common model structures used to fit the dynamic characteristics of processes include Auto-Regressive with Extra Inputs (ARX), Finite Impulse Response (FIR), and Output-Error (OE) models. Among these, the Output-Error model, which ignores noise terms, is widely used in automatic control due to its structure's similarity to the transfer function. Strictly speaking, ideal linear structures are constrained by various assumptions, and nonlinear phenomena are more prevalent. Therefore, studying the use of nonlinear models to fit the dynamic characteristics of industrial processes is of greater practical significance.

[0003] Parameter identification requires supporting data; however, in industrial production processes, due to factors such as sensor measurement errors, network transmission, and environmental interference, the collected datasets often contain a certain amount of noise. For example, motors, pumps, and compressors in industrial production may generate various vibrations and noises during operation. These noises often have specific spectral characteristics and are classified as colored noise. Certain components in electronic devices, such as antennas and RF filters, may cause a redistribution of the spectrum when white noise passes through, thus forming colored noise. Directly using white noise to describe these errors when fitting the dynamic system characteristics under these conditions is inaccurate and can introduce significant bias.

[0004] CN111427266A discloses a method for identifying nonlinear systems with disturbances, comprising the following steps: A) transforming the industrial control system to be identified into a nonlinear system model with disturbances, wherein the nonlinear system consists of a nonlinear element and a linear element, i.e., a Hammerstein system of output error type; B) decomposing the above nonlinear system model with disturbances into two sub-models: a noiseless output sub-model and a disturbance sub-model; C) updating the system parameters to construct Pζ(k) and eζ(k), updating the parameters by setting k = k + 1, returning to step A, until the cutoff condition is met; D) identifying the parameters and disturbances of the industrial control system. However, this patent considers white noise as the system output noise and does not consider colored noise, and the disturbance in the implementation case is known, failing to solve the parameter estimation problem under colored noise interference. Therefore, the reliability of this patent is low and its practicality is weak.

[0005] CN115981354A discloses a maximum likelihood recursive parameter estimation method for the output error autoregressive model of an underwater vehicle with missing data. Most control strategies are based on mathematical models, and establishing accurate mathematical models plays a crucial role in process control. Underwater vehicles operate in complex marine environments, capable of anticipating underwater conditions and autonomously completing specific tasks. Motion control is a critical aspect of underwater vehicle design; however, difficulties in obtaining observational data from sampling points due to sensor malfunctions, hardware limitations, communication errors, or computer overload, coupled with interference from colored noise, hinder the identification of parameters in the underwater vehicle's motion control system. To achieve online adjustment of the identification model and controller, this patent skips missing data by changing the sampling interval and combines a maximum likelihood recursive least squares algorithm to estimate system parameters. The method includes constructing an output error autoregressive model with missing data, developing a maximum likelihood recursive least squares parameter identification algorithm, and establishing the maximum likelihood recursive least squares algorithm itself. However, the system considered in this patent is linear, and the patent limits its noise characteristics to be fitted through an autoregressive process. Therefore, the patent does not take into account the nonlinear characteristics of the actual dynamic process, and its applicability is limited and its reliability is not high.

[0006] The paper "Parameter Identification of Input Nonlinear Systems Based on Key Term Separation" (Shen Qianyan, Jiangnan University, June 1, 2017) proposes an input nonlinear system consisting of a static, memoryless nonlinear module connected in series with a dynamic linear system. Different forms of nonlinear modules can be constructed to meet the needs of various engineering projects. Addressing the difficulty in parameter identification due to unknown key terms, noise terms, and intermediate terms in the identification model of the input nonlinear system, this paper adopts an auxiliary model identification approach. In the process of implementing the identification algorithm, unknown terms in the information vector are replaced with their estimated values, and the corresponding identification algorithm is used to obtain parameter estimates. Then, the obtained parameter estimates are used to estimate the unknown terms, and this process is repeated until satisfactory identification results are obtained. To address the problem of low efficiency in recursive algorithm identification due to the complex structure, high dimensionality of the parameter space, and coupling between channels in multivariable input nonlinear systems, this paper utilizes model decomposition to decompose the system into two or more subsystems. Combined with hierarchical identification theory, interactive estimation between subsystem parameters is achieved. Finally, the recursive identification algorithm is extended to multivariable input nonlinear systems. Based on the above, and using the least squares principle and gradient search method, the least squares iterative algorithm and gradient iterative algorithm for multivariable input nonlinear systems are derived. This paper only considers the characteristics of multivariable input nonlinear systems, such as complex structure and high parameter space dimensionality, which necessitate artificially decomposing the system into two or more subsystems. The model decomposition strategy employed is uncertain, and its subjective factors lead to uncertainty in the technical effectiveness of this paper. Summary of the Invention

[0007] To address the shortcomings of existing technologies, such as limited applicability, low reliability, and weak practicality, this invention provides a method and system for identifying colored noise systems using whitening processing and interactive estimation.

[0008] The present invention adopts the following technical solution.

[0009] This invention discloses a method for identifying colored noise systems based on whitening processing and interaction estimation, comprising: S1: Based on the output error system transfer function and the colored noise model transfer function, construct the original colored noise system model that reflects the input-output relationship of the dynamic system; S2: Filter the original model of the colored noise system to obtain the whitening model of the colored noise system; the whitening model of the colored noise system is calculated and filtered based on the system information vector, system parameter vector and white noise at each time step; S3: Decompose the original model of the colored noise system to obtain a noise sub-model based on noise information vector and noise parameter vector, and a system operation sub-model based on intermediate information vector and system parameter vector; S4: Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector, respectively; determine the current system parameter vector based on the current system information vector; determine the current noise parameter vector based on the current noise information vector; S5: Determine the current output error system transfer function based on the current system parameter vector, and determine the current colored noise model transfer function based on the current noise parameter vector, to obtain the original colored noise system model without unknowns.

[0010] More preferably, In S1, the original model of the colored noise system is also constructed based on a pre-selected nonlinear function, a column vector of coefficients of the nonlinear function, and white noise; the column vector of coefficients of the nonlinear function includes the coefficients of multiple nonlinear functions.

[0011] More preferably, In S1, the output error system transfer function is constructed based on at least one output error system transfer function denominator polynomial parameter, output error system transfer function numerator polynomial parameter, and displacement operator; The colored noise model transfer function is constructed based on at least one autoregressive process parameter, a moving average process parameter, and a displacement operator.

[0012] More preferably, In S2, before filtering the original model of the colored noise system, a filter term that is the reciprocal of the transfer function of the colored noise model is defined, and the original model of the colored noise system is filtered based on the filter term.

[0013] More preferably, In S2, both the system information vector and the system parameter vector are unknowns; The system information vector includes intermediate filtering variables, filtering inputs, and various filtering nonlinear terms at historical moments. The intermediate filtering variables are calculated based on the output error system transfer function and the filtering inputs. The filtering inputs are calculated based on the colored noise model transfer function, a pre-selected nonlinear function, and the coefficient column vector of the nonlinear function. Each filtering nonlinear term is calculated based on the colored noise model transfer function, the pre-selected nonlinear function corresponding to that term, and the system's measurable inputs. The system parameter vector includes the denominator polynomial parameters of all output error system transfer functions, the numerator polynomial parameters of the output error system transfer functions, and the coefficients of the nonlinear functions.

[0014] More preferably, In S3, the noise sub-model calculates the virtual noise output based on the noise information vector and the noise parameter vector; The noise information vector includes virtual noise output and white noise at historical moments; The noise parameter vector includes all autoregressive process parameters and moving average process parameters.

[0015] More preferably, In S3, the system operation sub-model calculates intermediate variables of system operation based on intermediate information vector and system parameter vector; wherein the intermediate information vector includes intermediate variables of system operation at historical moments, the dynamic input of the colored noise system, and the nonlinear functions of various system input signals; The dynamic input of the colored noise system is calculated based on the nonlinear function of each system input signal and the coefficient column vector of the nonlinear function; The nonlinear function of each system input signal is calculated based on the system's measurable input and the corresponding pre-selected nonlinear function.

[0016] More preferably, In S4, the nonlinear output estimate at each time step in the intermediate information vector is calculated based on the system's measurable input at that time step, the estimated values ​​of the coefficients of each nonlinear function, and the pre-selected nonlinear function.

[0017] More preferably, In S4, the current system information vector is calculated based on the historical noise parameter vector and the intermediate information vector, including: Based on the nonlinear output estimate earlier than the time to be calculated, the respective regression process parameter estimates and the moving average process parameter estimates at the time to be calculated, the filtered input estimate at the historical time is calculated. Based on the estimated values ​​of the filtered input earlier than the time to be calculated, the estimated values ​​of the intermediate filtered variables, and the estimated values ​​of the denominator polynomial parameters and the numerator polynomial parameters of the system transfer function of each output error system at the time to be calculated, the estimated values ​​of the intermediate filtered variables at historical time points are calculated. Based on a pre-selected nonlinear function, measurable system inputs earlier than the time to be calculated, the estimated value of the filtered nonlinear term, and the estimated values ​​of the respective regression process parameters and moving average process parameters at the time to be calculated, the estimated value of the filtered nonlinear term at historical time points is calculated.

[0018] More preferably, In S4, the current noise information vector is calculated based on the historical noise parameter vector and the intermediate information vector, including: Based on the system's measurable output, intermediate information vector, and system parameter vector at the time to be calculated, the virtual noise output estimate for historical time points is calculated. Based on the system's measurable output, intermediate information vector, system parameter vector, noise information vector, and noise parameter vector at the time to be calculated, the white noise estimate for historical time points is calculated.

[0019] More preferably, In S4, the current system parameter vector is also determined based on the system parameter vector of the previous time step, the estimated value of the filtered output at the current time step, and the gain matrix of the system operating sub-model at the current time step. The current filtered output estimate is calculated based on the system's measurable output at the current and historical times, the filtered output estimate at the historical times, the respective regression process parameter estimates at the current time, and the respective moving average process parameter estimates at the current time. The gain matrix of the system operating sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the system operating sub-model at the previous moment, as well as the system information vector at the current moment. The covariance matrix of the system operating sub-model at each time step is calculated based on the gain matrix of the system operating sub-model at that time step, the covariance matrix of the system operating sub-model at the previous time step, the system information vector at that time step, and the identity matrix.

[0020] More preferably, In S4, the current noise parameter vector is also determined based on the noise parameter vector of the previous time step, the virtual noise output estimate of the current time step, and the gain matrix of the noise sub-model at the current time step. The gain matrix of the noise sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the noise sub-model at the previous moment, as well as the noise information vector at the current moment. The covariance matrix of the noise sub-model at each time step is calculated based on the gain matrix of the noise sub-model at that time step, the covariance matrix of the noise sub-model at the previous time step, the noise information vector at that time step, and the identity matrix.

[0021] Another aspect of this invention discloses a colored noise system identification system based on a colored noise system identification method, comprising an original model construction module, a whitening model construction module, an original model splitting module, a vector determination module, and an original model unknown quantity determination module: The original model construction module constructs an original model of a colored noise system that reflects the input-output relationship of a dynamic system based on the output error system transfer function and the colored noise model transfer function. The whitening model construction module is used to filter the original model of the colored noise system to obtain a whitening model of the colored noise system; the whitening model of the colored noise system calculates the filtered output based on the system information vector, system parameter vector and white noise at each time step; The original model splitting module is used to split the original model of the colored noise system to obtain a noise sub-model constructed based on noise information vector and noise parameter vector, and a system operation sub-model constructed based on intermediate information vector and system parameter vector. The vector determination module calculates the current system information vector and noise information vector based on the historical noise parameter vector and the intermediate information vector, respectively; determines the current system parameter vector based on the current system information vector; and determines the current noise parameter vector based on the current noise information vector. The original model unknown quantity determination module determines the current output error system transfer function based on the current system parameter vector and the current colored noise model transfer function based on the current noise parameter vector, thus obtaining the original colored noise system model without unknown quantities.

[0022] Another aspect of this application discloses an electronic device, including a processor and a storage medium; characterized in that: The storage medium is used to store instructions; The processor is configured to operate according to the instructions to execute the aforementioned colored noise system identification method.

[0023] This application also discloses a computer-readable storage medium having a computer program stored thereon, characterized in that the program, when executed by a processor, implements the colored noise system identification method.

[0024] The beneficial effects of this invention are compared with those of the prior art: This invention considers the interference of colored noise in dynamic systems. The colored noise model can be fitted using an autoregressive process, or using a moving average or autoregressive moving average process. More generally, this invention uses an autoregressive moving average process to fit the colored noise model, and the resulting algorithm encompasses the other two cases mentioned above, making it more universal. By whitening the colored noise, the noise in the noise sub-model is transformed into white noise, simplifying the parameter estimation problem of the system model after whitening. Simultaneously, this invention also considers the nonlinear characteristics of dynamic systems, taking into account input nonlinearity at the input stage. It uses a linear combination of a set of functions with known basis for the input signal and unknown coefficients to describe general continuous nonlinearity, enabling this invention to describe a wider range of nonlinear dynamic systems that are closer to actual engineering applications, improving system modeling accuracy, reducing model mismatch errors, and enhancing robustness to complex noise and interference. The method disclosed in this invention can simultaneously achieve recursive estimation of the parameters of both the noise sub-model and the system operation sub-model.

[0025] This invention also has outstanding advantages such as wide applicability, high reliability, and strong practicality. Attached Figure Description

[0026] Figure 1 This is a flowchart illustrating the colored noise system identification method based on whitening processing and interactive estimation as described in this application. Figure 2 A schematic diagram of the algorithm architecture based on whitening and hierarchical analysis strategies; Figure 3 noise variance Graph showing the change of estimated time parameters over time; Figure 4 noise variance A comparison of the parameter estimation errors of the two algorithms over time; Figure 5 noise variance A comparison chart of the prediction output errors of the two algorithms. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of this invention. The embodiments described in this application are merely some embodiments of this invention, and not all embodiments. Based on the spirit of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of this invention.

[0028] This application discloses a method for identifying colored noise systems based on whitening processing and interaction estimation, including: S1: Based on the output error system transfer function, the colored noise model transfer function, the pre-selected nonlinear function, the coefficient column vector of the nonlinear function, and white noise, construct the original model of the colored noise system that introduces nonlinear elements and can reflect the measurable input-output relationship of the dynamic system at every moment.

[0029] The coefficient column vectors of the output error system transfer function, the colored noise model transfer function, and the nonlinear function at the current moment are all unknowns and need to be solved in subsequent steps.

[0030] The coefficient column vector of a nonlinear function includes the coefficients of multiple nonlinear functions.

[0031] The original model of the colored noise system is shown in the following equation: ; in, The system's measurable output at time t. The system transfer function is the output error function; and These are the denominator and numerator polynomials of the system transfer function for the output error, respectively. Represents the transfer function of a colored noise model; and These are the denominator and numerator polynomials of the colored noise model transfer function with respect to the unit shift operator, respectively. This represents white noise at time t; It is the input of the colored noise system dynamics at time t, and also the output of the nonlinear mapping function at time t. Its calculation formula is as follows: ; in, It is the measurable input of the system at time t. It is a set of nonlinear functions of the input signal. A column vector representing the coefficients of a nonlinear function; , It is defined as; Represents a real matrix with 1 row and m columns; Indicates about t The first nonlinear function of the system input signal at time t; Indicates about t The second nonlinear function of the system input signal at time; Indicates about t The m-th nonlinear function of the system input signal at time t; This indicates a pre-selected nonlinear function, and the content within the parentheses will serve as the independent variable of that function. The nonlinear function can be selected by those skilled in the art based on the actual situation, and will not be elaborated here. , The coefficients of the first nonlinear function; The coefficients of the second nonlinear function; The coefficients of the m-th nonlinear function; and All are coefficients of nonlinear functions. The preferred value of m is a positive integer greater than or equal to 1; its specific value can be set by those skilled in the art based on actual circumstances, and will not be elaborated here. This invention uses an interactive estimation strategy to jointly estimate the unknown coefficients. Represents an m-dimensional real column vector; This indicates the transpose of the matrix. The output error system transfer function is constructed based on at least one output error system transfer function denominator polynomial parameter, an output error system transfer function numerator polynomial parameter, and a shift operator; the output error system transfer function... middle and The definitions are as follows: ; ; in, This represents a shift operator used to describe the time-domain shift of a signal, satisfying the polynomial: ; This indicates the value of any parameter in this invention at time t; This indicates the value of any parameter in this invention at time t-1; and Let $\mathbf$ and $\mathbf$ represent the orders of the denominator polynomial and the numerator polynomial of the system transfer function for the output error, respectively. Both are determined a priori by those skilled in the art. The first parameter of the denominator polynomial of the system transfer function for the output error; The second parameter of the denominator polynomial of the system transfer function for the output error; The first polynomial of the denominator polynomial of the output error system transfer function Item parameters; The first parameter of the numerator polynomial of the system transfer function for the output error; The second parameter of the numerator polynomial of the system transfer function for the output error; The first polynomial of the numerator polynomial of the output error system transfer function Item parameters; , , All of these are parameters of the denominator polynomial of the system transfer function for output error. , , All of these are parameters of the numerator polynomial of the system transfer function for output error.

[0032] The colored noise model transfer function is constructed based on at least one autoregressive process parameter, a moving average process parameter, and a displacement operator; the colored noise model transfer function Middle denominator Based on at least one autoregressive process parameter and a displacement operator, the molecule is constructed. Constructed based on at least one moving average process parameter and a displacement operator; The definition is as follows: ; in, This is the first parameter in the autoregressive process; This is the second parameter in the autoregressive process; For the first autoregressive process Item parameters; , and All are parameters of an autoregressive process; For the prior model order, the preferred order is... The value of is a positive integer greater than or equal to 1; The value of can be set by those skilled in the art according to the actual situation, and will not be elaborated here.

[0033] The definition is as follows: ; in, This is the first parameter in the moving average process; This is the second parameter of the moving average process; The first moving average process Item parameters; , and All are moving average process parameters; For the prior model order, the preferred order is... The value of is a positive integer greater than or equal to 1; and ; The value of can be set by those skilled in the art according to the actual situation, and will not be elaborated here.

[0034] S2: Determine the filtering terms based on the original model of the colored noise system, and filter the original model of the colored noise system based on the filtering terms to obtain the whitening model of the colored noise system.

[0035] The filter term designed based on the original model of the colored noise system refers to a filter term that is the reciprocal of the transfer function of the colored noise model. .

[0036] The whitening model for the colored noise system calculates the filtered output based on the system information vector, system parameter vector, and white noise at each time step, as shown in the following equation: ; in, This represents the filtered output at time t; This represents the system information vector at time t; Represents the system parameter vector; This represents the white noise at time t.

[0037] The filter output, system information vector, and system parameter vector are all unknowns. The filtered output is calculated based on the filter terms and the measurable output of the dynamic system at each moment (hereinafter referred to as the system measurable output), as shown in the following formula: ; The system information vector includes intermediate filter variables from historical times corresponding to the time of the calculated filter output (when calculating the filter output at time t, the intermediate filter variables from historical times include times from t-1 to t-). The intermediate filter variables at time t), and the filter input at the historical time corresponding to the time when the filtered output is calculated (when calculating the filtered output at time t, the required historical filter input includes times from t-1 to t-1). The filter input at time (and the corresponding nonlinear terms of the calculated filter output at time); specifically... ; express A 3D real-valued column vector. Intermediate filter variables. The result is calculated based on the output error system transfer function and the filtered input, as shown in the following formula; ; Filtered input The formula is obtained based on the colored noise model transfer function, the measurable input of the dynamic system (hereinafter referred to as the system measurable input), the pre-selected nonlinear function, and the coefficient column vector of the nonlinear function, as shown in the following equation: ; Each filtering nonlinear term is calculated based on the colored noise model transfer function, the pre-selected nonlinear function corresponding to the term, the measurable input of the dynamic system at the time to be calculated and the corresponding historical time, and the filtering nonlinear term at the corresponding historical time, as shown in the following formula: ; The system parameter vector This includes the denominator polynomial parameters of all output error system transfer functions, the numerator polynomial parameters of all output error system transfer functions, and the coefficients of all nonlinear functions. Specifically: .

[0038] S3: Decompose the original model of the colored noise system to obtain a noise sub-model based on noise information vector and noise parameter vector, and a system operation sub-model based on intermediate information vector and system parameter vector; The noise sub-model calculates the virtual noise output based on the noise information vector and the noise parameter vector, as shown in the following formula: ; in, This represents the virtual noise output at time t; This represents the noise information vector at time t; Represents the noise parameter vector at time t; Both the noise information vector and the noise parameter vector are unknowns. The noise information vector This includes the virtual noise output at the historical time corresponding to the time when the virtual noise output is calculated (i.e., the virtual noise output from the time before the time corresponding to the time when the virtual noise output is calculated to the time before the time when the virtual noise output is calculated). The virtual noise output at the given time and the white noise at the corresponding historical time (i.e., the time preceding the time corresponding to the calculated virtual noise output) are both calculated virtual noise output timestamps. (white noise at any given moment), i.e. ; in, This represents the virtual noise output at time t-1; This represents the virtual noise output at time t-2; Indicates t- Virtual noise output at any given moment; This represents white noise at time t-1; This represents white noise at time t-2; Indicates t- Time-based white noise; express A 3D real column vector; The noise parameter vector This includes all autoregressive process parameters and moving average process parameters, i.e. .

[0039] The system operation sub-model calculates intermediate variables based on intermediate information vectors and system parameter vectors, as shown in the following formula: ; in, This represents the intermediate variables in the system operation at time t; This represents the transpose of the intermediate information vector at time t; where, the intermediate information vector... This includes the system's intermediate variables at the historical time corresponding to the intermediate variables at the time of calculation (i.e., the time preceding the time corresponding to the intermediate variables at the time of calculation). The system's intermediate variables at the calculated time point, and the colored noise system dynamics input at the corresponding historical time point (i.e., the time preceding the calculated system's intermediate variables). The nonlinear functions of the colored noise system dynamics at time 1 and time 2, and the various system input signals at time 3, which are intermediate variables of the system operation; specifically: ; in, This represents the intermediate variables of the system at time t-1; This represents the intermediate variables of the system at time t-2; Indicates t- Intermediate variables in the system operation at any given time; It is the dynamic input of the colored noise system at time t; It is the dynamic input of the colored noise system at time t-1; It is the dynamic input of the colored noise system at time t-2; It is t- The system constantly receives dynamic inputs of colored noise. Indicates about t The first nonlinear function of the system input signal at time t; Indicates about t The second nonlinear function of the system input signal at time; Indicates about t The m-th nonlinear function of the system input signal at time t is given.

[0040] The dynamic input of the colored noise system is calculated based on the nonlinear function of each system input signal and the coefficient column vector of the nonlinear function; The nonlinear function of each system input signal is calculated based on the measurable input of the dynamic system and the corresponding pre-selected nonlinear function. Specifically, the measurable input of the system is used as the independent variable to input each pre-selected nonlinear function to obtain the nonlinear function of the corresponding system input signal.

[0041] S4: Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector respectively; determine the current system parameter vector based on the current system information vector; determine the current noise parameter vector based on the current noise information vector.

[0042] 4.1 Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector, respectively.

[0043] 4.1.1 Determine the historical intermediate information vector to be calculated based on the historical intermediate information vector earlier than the historical intermediate information vector to be calculated and the system parameter vector.

[0044] The estimated value of the intermediate information vector is used as the intermediate information vector. This includes estimates of intermediate system variables and nonlinear output values ​​at that moment: ; in, This represents the estimated value of intermediate variables in the system operation at time t-1; This represents the estimated value of intermediate variables in the system operation at time t-2; Indicates t- Estimates of intermediate variables during system operation at any given time; This represents the nonlinear output estimate at time t-1; This represents the nonlinear output estimate at time t-2; Indicates t- The nonlinear output estimate at time step 1.

[0045] The estimated value of the intermediate variables of the system at each time step in the intermediate information vector is calculated based on the intermediate information vector and the system parameter vector at that time step, as shown in the following formula: ; in, This represents the estimated value of the intermediate variables of the system at time t; Nonlinear output estimate at each time step in the intermediate information vector It is calculated based on the system's measurable input at that moment, the estimated values ​​of the coefficients of each nonlinear function, and the pre-selected nonlinear function.

[0046] 4.1.2 Calculate the current system information vector based on historical noise parameter vectors and intermediate information vectors, including: Based on the nonlinear output estimate earlier than the time corresponding to the filter input estimate to be calculated (when the time corresponding to the filter input estimate to be calculated is t, the nonlinear output estimate earlier than the time corresponding to the filter input estimate to be calculated includes time t-1 up to t- The nonlinear output estimate at time t), and the filter input estimate at a time earlier than the filter input estimate to be calculated (when the time corresponding to the filter input estimate to be calculated is t, the filter input estimate at a time earlier than the time corresponding to the filter input estimate to be calculated, including time t-1 up to t- The filtered input estimate at time point (i.e., the previously mentioned filtered input estimate) is calculated using the estimated regression process parameters and moving average process parameters corresponding to the filtered input estimate to be calculated at each time point. The filtered input estimate at each time point is then calculated based on the following formula: ; in, It is the filtered input estimate at time t; It is the filtered input estimate at time t-1; It is the filtered input estimate at time t-2; It is t- The filtered input estimate at time t; It is the nonlinear output estimate at time t; It is the nonlinear output estimate at time t-1; It is the nonlinear output estimate at time t-2; It is time t- The nonlinear output estimate; This is the estimated value of the first parameter of the autoregressive process at time t; This is the estimated value of the second parameter of the autoregressive process at time t; The first autoregressive process at time t Parameter estimates; This is the estimated value of the first parameter of the moving average process at time t; This is the estimated value of the second parameter of the moving average process at time t; The first moving average process at time t The estimated values ​​of the parameters are as follows. When t≤0, each value is obtained through prior settings and can be set by those skilled in the art according to the actual situation, which will not be elaborated here; the estimated value of the nonlinear output at each time step is calculated based on the measurable input of the dynamic system at that time step, the estimated value of the coefficients of each nonlinear function, and the pre-selected nonlinear function, as shown in the following formula: ; in, This represents the nonlinear output estimate at time t; , This is an estimate of the coefficient of the first nonlinear function at time t; This is an estimate of the coefficient of the second nonlinear function at time t; The coefficient of the m-th nonlinear function at time t is an estimate, determined by the system parameter vector at time t; Based on the estimated value of the intermediate filter variable to be calculated at the time corresponding to that time and the estimated value of the filter input earlier than that time (when the estimated value of the intermediate filter variable to be calculated corresponds to time t, the estimated value of the filter input earlier than that time includes time t-1 up to t- The estimated value of the filtered input at time t), and the estimated value of the intermediate filtered variable at a time earlier than the estimated value of the intermediate filtered variable to be calculated (when the estimated value of the intermediate filtered variable to be calculated corresponds to time t, the estimated value of the intermediate filtered variable at a time earlier than the estimated value of the intermediate filtered variable to be calculated includes time t-1 up to t- The intermediate filter variable estimates at each time point are calculated using the estimated values ​​of the intermediate filter variables at the time point to be calculated, the estimated values ​​of the denominator polynomial parameters and the estimated values ​​of the numerator polynomial parameters of the output error system transfer function at each time point corresponding to the estimated values ​​of the intermediate filter variables to be calculated, and the intermediate filter variable estimates at each time point are calculated using the following formula: ; in, This represents the estimated value of the intermediate filtered variable at time t; It is t - The filtered input estimate at time t; Let represent the estimated value of the intermediate filtered variable at time t-1; Let represent the estimated value of the intermediate filtered variable at time t-2; Indicates t- The estimated value of the intermediate filtered variable at time t; The first parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The second parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial in the denominator of the system transfer function at time t is the first polynomial. Estimates of the term parameters; The first parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the first term. The second parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial of the numerator polynomial of the system transfer function at time t is the output error system. Estimates of the term parameters; Based on a pre-selected nonlinear function, the measurable inputs of the dynamic system at the time corresponding to the estimated value of the filtered nonlinear term to be calculated and historical times earlier than that time (when the estimated value of the filtered nonlinear term to be calculated corresponds to time t, the measurable inputs of the system at the corresponding historical times include time t-1 up to t-1). The system's measurable input at time t), and the estimated value of the filter nonlinear term earlier than the estimated value of the filter nonlinear term to be calculated at time t (when the estimated value of the filter nonlinear term to be calculated corresponds to time t, the estimated value of the filter nonlinear term earlier than the estimated value of the filter nonlinear term to be calculated at time t-1, including time t-1 up to t- The estimated value of the filtered nonlinear term at time point (i.e., the estimated value of the filtered nonlinear term to be calculated) is calculated using the estimated values ​​of the regression process parameters and the estimated values ​​of the moving average process parameters at the corresponding time points. The estimated value of the filtered nonlinear term at each historical time point (i.e., the aforementioned estimated value of the filtered nonlinear term to be calculated) is then calculated based on the following formula: ; in, This is the estimated value of the j-th nonlinear term of the filter at time t; This is the estimated value of the j-th filter nonlinear term at time t-1; This is the estimated value of the j-th nonlinear term of the filter at time t-2; For t- The estimated value of the j-th filtering nonlinear term at time j.

[0047] The current system information vector estimate is used as the current system information vector. The current system information vector estimate includes:

[0048] in, It is the estimated value of the j-th nonlinear term of the filter at time t; The estimated value of the second nonlinear term of the filter at time t; The estimated value of the m-th filter nonlinear term at time t; It is the estimated value of the system information vector at time t.

[0049] 4.1.3 Calculate the current noise information vector based on the historical system parameter vector, noise information vector, noise parameter vector, and intermediate information vector, including: Based on the measurable output, intermediate information vector, and system parameter vector of the dynamic system at historical moments corresponding to the virtual noise output estimate to be calculated, the virtual noise output estimate at historical moments (i.e., the aforementioned virtual noise output estimate to be calculated) is calculated. The virtual noise output estimate at each moment is calculated based on the following formula: ; in, This represents the virtual noise output estimate at time t; This represents the estimated value of the transpose of the intermediate information vector at time t; This represents the estimated value of the system parameter vector.

[0050] Based on the measurable output, intermediate information vector, system parameter vector, noise information vector, and noise parameter vector of the dynamic system at the corresponding historical time point of the white noise estimate to be calculated, the white noise estimate at the corresponding historical time point (i.e., the aforementioned white noise estimate to be calculated) is calculated based on the following formula: ; in, This represents the white noise estimate at time t; This represents the estimated value of the transpose of the noise information vector at time t; This represents the estimated value of the noise parameter vector.

[0051] The current noise information vector estimate is used as the current noise information vector. include: .

[0052] in, This represents the virtual noise output estimate at time t-1; This represents the virtual noise output estimate at time t-2; Indicates t- The estimated virtual noise output value at time; This represents the estimated white noise value at time t-1; This represents the white noise estimate at time t-2; Indicates t- The white noise estimate at time 1.

[0053] 4.2 Determine the current system parameter vector based on the current system information vector, the current filtered output estimate, and the gain matrix of the current system operating sub-model.

[0054] The estimated values ​​of the system parameter vector are determined according to the following formula: ; in, This is an estimate of the system parameter vector at time t, based on the first to second digits of the denominator polynomial of the system transfer function at time t. The estimated values ​​of the term parameters, and the first to second terms of the numerator polynomial of the system transfer function at time t. The estimated values ​​of the parameters are used to construct the system parameter vector at time t, which is then used as the system parameter vector at time t. This represents the estimated value of the system parameter vector at time t-1; This represents the gain matrix of the sub-model of the system operating at time t; This represents the transpose of the estimated system information vector at time t; Represents the filtered output estimate at time t; the system parameter vector estimate at time t. ; The first parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The second parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial in the denominator of the system transfer function at time t is the first polynomial. Estimates of the term parameters; The first parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the first term. The second parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial of the numerator polynomial of the system transfer function at time t is the output error system. Estimates of the term parameters; This is an estimate of the coefficient of the first nonlinear function at time t; This is an estimate of the coefficient of the second nonlinear function at time t; Let be the estimated value of the coefficient of the m-th nonlinear function at time t.

[0055] The current filtered output estimate is calculated based on the system's measurable output at the current and historical times, the filtered output estimate at historical times, the respective regression process parameter estimates at the current time, and the respective moving average process parameter estimates at the current time, as shown in the following formula: ; in, This represents the filtered output estimate at time t; This represents the filtered output estimate at time t-1; This represents the filtered output estimate at time t-2; Indicates t- The filtered output estimate at time t; The system's measurable output at time t; The system's measurable output at time t-1; The system's measurable output at time t-2; For t- The time system's measurable output.

[0056] The gain matrix of the system operating sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the system operating sub-model at the previous moment, as well as the system information vector at the current moment, as shown in the following formula: ; in, This represents the covariance matrix of the sub-model of the system at time t-1; Represents the identity matrix.

[0057] The covariance matrix of the system operating sub-model at each time step is calculated based on the gain matrix of the system operating sub-model at that time step, the covariance matrix of the system operating sub-model at the previous time step, the system information vector at that time step, and the identity matrix, as shown in the following formula: ; in, Let the covariance matrix of the sub-model of the system at time t be denoted as . OK A matrix of columns.

[0058] When performing the above recursion, it is necessary to first set , , and The initial value at t≤0, specifically, is , and When setting initial values ​​for each parameter, you can choose from the preferred range (0, 10). -2 Select a random number from ) The initial value is set to a large symmetric positive definite matrix, preferably 10. 6 I.

[0059] 4.3 Determine the current noise parameter vector based on the current noise information vector, the estimated virtual noise output at the current time, and the gain matrix of the noise sub-model at the current time.

[0060] The formula for determining the current noise parameter vector is as follows: ; in, This represents the noise parameter vector estimate at time t, which is based on the first to second iterations of the autoregressive process at time t. The estimated values ​​of the parameters, and the first to second values ​​of the moving average process at time t. The parameter estimates are constructed by taking the noise parameter vector estimate at time t as the noise parameter vector at time t. This represents the estimated value of the noise parameter vector at time t-1; Let represent the gain matrix of the noise sub-model at time t; This represents the virtual noise output estimate at time t; It is the transpose of the noise information vector estimate; the noise parameter vector estimate at time t. ; This is the estimated value of the first parameter of the autoregressive process at time t; This is the estimated value of the second parameter of the autoregressive process at time t; The first autoregressive process at time t Parameter estimates; This is the estimated value of the first parameter of the moving average process at time t; This is the estimated value of the second parameter of the moving average process at time t; The first moving average process at time t Parameter estimates.

[0061] The gain matrix of the noise sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the noise sub-model at the previous moment, as well as the noise information vector at the current moment, as shown in the following formula: ; in, Let represent the covariance matrix of the noise submodel at time t-1; The covariance matrix of the noise sub-model at each time step is calculated based on the gain matrix of the noise sub-model at that time step, the covariance matrix of the noise sub-model at the previous time step, the noise information vector at that time step, and the identity matrix, as shown in the following formula: ; in, Let represent the covariance matrix of the noise sub-model at time t.

[0062] S5: Determine the current output error system transfer function based on the current system parameter vector, and determine the current colored noise model transfer function based on the current noise parameter vector, to obtain the original colored noise system model at the current moment without unknowns.

[0063] Those skilled in the art should know how to determine the output error system transfer function based on the current system parameter vector, and how to determine the current colored noise model transfer function based on the current noise parameter vector; these will not be elaborated upon here.

[0064] This application also discloses a colored noise system identification system based on a colored noise system identification method, including an original model construction module, a whitening model construction module, an original model splitting module, a vector determination module, and an original model unknown quantity determination module.

[0065] The original model construction module constructs an original model of a colored noise system that reflects the input-output relationship of a dynamic system, based on the output error system transfer function and the colored noise model transfer function.

[0066] The whitening model construction module is used to filter the original model of the colored noise system to obtain the whitening model of the colored noise system; the whitening model of the colored noise system calculates the filtered output based on the system information vector, system parameter vector and white noise at each time step.

[0067] The original model splitting module is used to split the original model of the colored noise system to obtain a noise sub-model constructed based on noise information vector and noise parameter vector, and a system operation sub-model constructed based on intermediate information vector and system parameter vector.

[0068] The vector determination module calculates the current system information vector and noise information vector based on the historical noise parameter vector and the intermediate information vector, respectively; determines the current system parameter vector based on the current system information vector; and determines the current noise parameter vector based on the current noise information vector.

[0069] The original model unknown quantity determination module determines the current output error system transfer function based on the current system parameter vector and the current colored noise model transfer function based on the current noise parameter vector, thus obtaining the original colored noise system model without unknown quantities.

[0070] Example 1 A method for identifying colored noise systems using whitening processing and interactive estimation.

[0071] Considering the nonlinear characteristics of the input stage of a dynamic system, the measured values ​​of the dynamic system are affected by colored noise (hereinafter, a dynamic system affected by colored noise will be referred to as a colored noise system). By defining a filter term that is the reciprocal of the noise model's transfer function, and multiplying both sides of the system equation by the filter term, the colored noise term is rewritten as white noise, simplifying the parameter estimation problem of the filtered system model. To solve the identification of the defined unknown filter, a strategy of decomposition and interactive estimation is adopted. The model of the input-output nonlinear error system, which introduces nonlinear elements in the original model, is decomposed into two sub-models: a system operation sub-model and a noise sub-model. Two virtual intermediate variables are introduced to represent the output of system operation and the output of colored noise, respectively. Combined with the interactive estimation method, the estimated values ​​of all unknown parameters in the original dynamic system are obtained. Figure 2 This paper demonstrates a strategy based on whitening and stepwise analysis, which decomposes the original dynamic system into two sub-models for separate processing. By defining two criterion functions, the paper achieves efficient estimation of unknown parameters.

[0072] Furthermore, a method for identifying colored noise systems using whitening processing and interactive estimation, such as... Figure 1 As shown, it includes: S1: Based on the output error system transfer function, the colored noise model transfer function, the pre-selected nonlinear function, the coefficient column vector of the nonlinear function, and white noise, construct the original model of the colored noise system that introduces nonlinear elements and can reflect the measurable input-output relationship of the dynamic system at every moment.

[0073] Consider a typical input-nonlinear output error system under noise interference, that is, to mathematically model a dynamic system with nonlinear input data and colored noise superimposed on the output based on the output error model OE: ; ; in, The system's measurable output at time t. The disturbance of the system at time t is the colored noise at time t; It is the measurable input of the system at time t. It is a set of nonlinear functions of the input signal. The column vector of coefficients representing a nonlinear function. The input to the colored noise system dynamics at time t is also the output of the nonlinear mapping function at time t. The nonlinear mapping function is a key component of the input stage of the nonlinear output error system; its function is to convert the input signal u(t) into a nonlinear form, which then serves as the input to the dynamic part of the system. The output of the nonlinear mapping function is the direct input to the dynamic part of the system, transmitted through the linear output error system transfer function. Together with colored noise, they determine the final output of the system.

[0074] and The whole can be viewed as the original model of a colored noise system; among which, Here is a model of an input nonlinear output error system, where This invention introduces a nonlinear element.

[0075] and These are the denominator and numerator polynomials of the system transfer function for the output error, respectively. This represents a shift operator used to describe the time-domain shift of a signal, satisfying the polynomial: : ; ; Among them, the orders of the denominator polynomial and the numerator polynomial of the output error system transfer function. and It is a priori set to satisfy the requirements of system stability, that is, the order of the system is assumed to be known. , All are positive integers greater than or equal to 1; and Those skilled in the art can set it according to the actual situation, which will not be elaborated here. The first parameter of the denominator polynomial of the system transfer function for the output error; The second parameter of the denominator polynomial of the system transfer function for the output error; The first polynomial of the denominator polynomial of the output error system transfer function Item parameters; The first parameter of the numerator polynomial of the system transfer function for the output error; The second parameter of the numerator polynomial of the system transfer function for the output error; The first polynomial of the numerator polynomial of the output error system transfer function Item parameters.

[0076] Because the algorithm of this invention is recursive, and each step of the calculation requires historical data, in actual identification experiments, a certain amount of historical data is usually sampled before identification. Therefore, at times t < 0, and It is known. Furthermore, when , set up ,in It is white noise at time t, and zero mean and variance can be chosen. Random white noise.

[0077] If historical data is not collected, you can set it to be collected when... , , , .

[0078] Different systems exhibit varying nonlinear characteristics, and there is no single, universally applicable formula. Without loss of generality, basis functions can be used for continuous nonlinearity. The linear combination of these functions describes a nonlinear function with respect to the input signal. The basis functions can be power functions, exponential functions, trigonometric functions, polynomials, typical nonlinear functions, etc., which can be selected by those skilled in the art based on system characteristics or data-driven approaches; further details are omitted here. That is, assume the output of the nonlinear module is a set of nonlinear functions (functions with known basis) with respect to the input signal. and unknown coefficients Linear combination: , in, This represents the column vector of coefficients for a nonlinear function. These coefficients are unknown and need to be estimated from the input and output data using a parameter identification algorithm. , The coefficients of the first nonlinear function; The coefficients of the second nonlinear function; Let m be the coefficient of the m-th nonlinear function. The preferred value of m is a positive integer greater than or equal to 1; its specific value can be set by those skilled in the art based on actual conditions, and will not be elaborated here. This invention uses an interactive estimation strategy to jointly estimate the unknown coefficients. Represents an m-dimensional real column vector. This indicates that the coefficients of each term form an m-dimensional real column vector. It is a set of nonlinear functions of the input signal. , Represents a real matrix with 1 row and m columns; This represents the first nonlinear function of the system input signal at time t; This represents the second nonlinear function with respect to the system input signal at time t; Let m be the m-th nonlinear function of the system input signal at time t. This indicates that the various nonlinear functions form a real matrix with 1 row and m columns.

[0079] Those skilled in the art should know how to distinguish between colored noise and white noise, and for systems with colored noise interference, they should know how to select different models to fit the noise based on its different characteristics. The noise fitting methods proposed in the embodiments of the invention are merely preferred embodiments and are not necessarily a limitation on the implementation of the invention. The three preferred noise fitting methods of the present invention are as follows: (1) An autoregressive process can be used to fit colored noise. : ; in, The transfer function of the colored noise model with respect to the unit shift operator The denominator polynomial.

[0080] (2) A moving average process can be used to fit colored noise. : ; in, The transfer function of the colored noise model with respect to the unit shift operator The molecular polynomial.

[0081] (3) An autoregressive moving average process can be used to fit colored noise. : ; in, It is white noise at time t, i.e., zero mean and variance. Random white noise (i.e., randomly generated white noise with a mean of 0 and a variance of σ) 2 (white noise) and These are the colored noise model transfer functions with respect to the unit shift operator. The denominator and numerator polynomials: ; ; in, This is the first parameter in the autoregressive process; This is the second parameter in the autoregressive process; For the first autoregressive process Item parameters; This is the first parameter of the moving average process; This is the second parameter in the moving average process; The first moving average process Item parameters; where and The prior model order, satisfying the system stability requirement, is known; the preferred order is... The value of is a positive integer greater than or equal to 1; The value of is a positive integer greater than or equal to 1; and ; and The values ​​of can be set by those skilled in the art according to the actual situation, and will not be elaborated here.

[0082] This invention will describe in detail the case where the noise is an autoregressive moving average process. The proposed algorithm is also applicable to cases where the noise is a moving average process and an autoregressive process. This invention will utilize measured input and output data, combined with whitening and interactive estimation strategies, to identify unknown parameters in the system model.

[0083] Observe the input-output nonlinear error system under discussion and construct the original model of the colored noise system: (1) The difficulty in identification lies in the fact that unknown parameters exist in different stages, with unknowns in both the noise sub-model and the system operation sub-model. To simplify the difficulty of parameter identification caused by colored noise interference terms, we consider multiplying both sides of the system equation by a filter term to transform the colored noise terms into white noise terms.

[0084] S2: Determine the filtering terms based on the original model of the colored noise system, and filter the original model of the colored noise system based on the filtering terms to obtain the whitening model of the colored noise system.

[0085] This step involves defining a filter term that is the reciprocal of the noise model transfer function, multiplying both sides of the system equation by the filter term, and rewriting the colored noise term in the original system model as a white noise term.

[0086] Define a linear filter that is the reciprocal of the noise model. (Hereinafter referred to as the filter term) to filter the input-output data, the model structure described by equation (1) can be transformed into an input nonlinear output error model under white noise interference. Multiplying both sides of equation (1) by the filter term... get: ; Define the filtered output at time t and the filtered input at time t : (2) (3) polynomial and Substituting into equations (2) and (3) respectively, we further obtain: (4) in, The system's measurable output at time t-1; The system's measurable output at time t-2; For t- The time system's measurable output; This is the filtered output at time t-1; This is the filtered output at time t-2; For t- The filtered output at any given time.

[0087] (5) in, This represents the first nonlinear term of the filter at time t; This represents the second nonlinear filtering term at time t; This represents the m-th nonlinear filtering term at time t; ,in, This represents the j-th nonlinear term in the filter at time t; Indicates about t The j-th nonlinear function of the system input signal at time t is given.

[0088] polynomial and Substitute respectively We can obtain:

[0089] in, Let j be the j-th nonlinear function relating to the system input signal at time t-1; Let j be the j-th nonlinear function relating to the system input signal at time t-2; Indicates about t- The j-th nonlinear function of the system input signal at time t; This represents the j-th nonlinear filtering term at time t-1; This represents the j-th nonlinear term of the filter at time t-2; Indicates t The j-th nonlinear term of the filter at time j.

[0090] Then equation (1) can be rewritten as: (6) Observing the right side of equation (6), we define an intermediate filter variable at time t. :

[0091] polynomial and Substitute them separately We can obtain:

[0092] in, This represents the filtered input at time t-1; This represents the filtered input at time t-2; Indicates t- Filtered input at time; This represents the intermediate filter variable at time t-1; This represents the intermediate filter variable at time t-2; Indicates t- Intermediate filter variables at time points.

[0093] make Define the system information vector at time t. and system parameter vector : ; ; in, express A column vector of real numbers.

[0094] Then the intermediate filter variable at time t It can be written as: (7) According to equation (6), the filtered output at time t is... for: (8) Formula (8) is denoted as the whitening model of the colored noise system.

[0095] S3: The specific steps for decomposing the original model of the colored noise system into two parts—a system operation sub-model and a noise sub-model—using a step-by-step analysis strategy include: The system equations contain a large number of parameters to be identified, and a single processing step would involve the calculation of high-dimensional matrices. To simplify computational complexity, a phased decomposition approach is adopted, splitting the parameters to be identified into two parts: one part contains the parameters of system operation, denoted as the system operation sub-model, and the other part contains the parameters of colored noise, denoted as the noise sub-model. In the whitened system equation (8), only the system operation parameters are included. Observing the system equation (1), it can be seen that the noise part also contains unknown parameters. A virtual noise output at time t is defined. : (9) polynomial and Substitute them separately We can obtain: , in, This represents the virtual noise output at time t-1; This represents the virtual noise output at time t-2; Indicates t- Virtual noise output at any given moment; This represents white noise at time t-1; This represents white noise at time t-2; Indicates t- White noise at any given moment.

[0096] make Define the noise information vector at time t. and noise parameter vector : ;

[0097] in, express A column vector of real numbers.

[0098] The noise output at time t can then be written as: (10) Let formula (10) be denoted as the noise sub-model; Define intermediate variables of the system at time t. : ; polynomial and Substitute them separately We can obtain:

[0099] in, This represents the intermediate variables of the system at time t-1; This represents the intermediate variables of the system at time t-2; Indicates t- Intermediate variables in the system operation at any given time; It is the dynamic input of the colored noise system at time t; It is the dynamic input of the colored noise system at time t-1; It is the dynamic input of the colored noise system at time t-2; It is t- The system receives dynamic inputs of colored noise at all times.

[0100] Define the intermediate information vector at time t : ; Further, the system operation sub-model can be obtained: .

[0101] S4: Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector respectively; determine the current system parameter vector based on the current system information vector; determine the current noise parameter vector based on the current noise information vector.

[0102] 4.1 Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector, respectively.

[0103] In equations (8) and (10), it can be seen that in the system model after the whitening operation, the system operation parameters and the colored noise parameters are split into two parameter vectors. The new equations (8) and (10) can be regarded as pseudo-linear regression forms with respect to the unknown parameters. Using the least squares principle, two error criterion functions are defined respectively:

[0104]

[0105] Where k represents the time index, k∈[1,t], and t represents the current sampling time; This represents the filtered output at time k; Represents the system parameter vector at time t The value of ; This represents the virtual noise output at time k; Represents the noise parameter vector at time t The value of ; The error criterion function between the filtered output and the model prediction at time t is called the system error criterion function. The noise error criterion function is defined as the error criterion function between the noise output at time t and its model prediction. This represents the system information vector at time k, including intermediate filter variables. Filtered input and its historical data; This represents the noise information vector at time k, including historical data of the virtual noise output.

[0106] By minimizing the criterion function and setting the first-order partial derivatives of the criterion function to be calculated to zero, a least-squares algorithm for estimating the parameters is obtained. Specifically: Criterion function For parameter vectors Find the partial derivative and set its partial derivative to zero:

[0107] in, Representation of criterion function For parameter vectors Find the first-order partial derivative; This represents the system information vector at time k. The system parameter vector at time t is obtained using the recursive least squares algorithm. The estimated value at time t The recursive form: (11) in, Let be the system parameter vector at time t. The estimated value represents the parameter to be estimated in the sub-model of system operation. ; The first parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The second parameter of the denominator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial in the denominator of the system transfer function at time t is the first polynomial. Estimates of the term parameters; The first parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the first term. The second parameter of the numerator polynomial of the system transfer function at time t is the estimated value of the parameter at time t. The first polynomial of the numerator polynomial of the system transfer function at time t is the output error system. Estimates of the term parameters; This is an estimate of the coefficient of the first nonlinear function at time t; This is an estimate of the coefficient of the second nonlinear function at time t; Let be the estimated value of the coefficient of the m-th nonlinear function at time t; This represents the estimated value of the system parameter vector at time t-1; This represents the gain matrix of the sub-model of the system operating at time t; This represents the transpose of the system information vector at time t.

[0108] Calculate the gain matrix of the sub-model of the system at time t. : (12) in, I represents the covariance matrix of the sub-model of the system at time t-1; I represents the identity matrix. This indicates finding the inverse of the matrix within the square brackets.

[0109] Update the covariance matrix of the system running sub-model at time t. : (13) in, This represents the system parameter vector at time t. The estimated value represents the parameter to be estimated in the sub-model of system operation; This represents the estimated value of the system parameter vector at time t-1, which is the initial value for recursive updates; Let represent the gain matrix of the sub-model of the system at time t. A matrix with 1 row and 1 column is used to adjust the parameter update step size to ensure convergence; It represents the covariance matrix of the sub-model of the system at time t-1, characterizing the uncertainty of parameter estimation, and is updated recursively over time; Let the covariance matrix of the sub-model of the system at time t be denoted as . OK A matrix of columns; I represents the identity matrix, used for identity operations in matrix operations.

[0110] When performing the above recursion, it is necessary to first set , , and The initial value at t≤0, specifically, is , and When setting initial values ​​for each parameter, you can choose from the preferred range (0, 10). -2 Select a random number from ) The initial value is set to a large symmetric positive definite matrix, preferably 10. 6 I.

[0111] Similarly, the criterion function For parameter vectors Find the partial derivative and set its partial derivative to zero: The recursive calculation expression for the parameter vector to be estimated in the noise sub-model can be derived:

[0112] in, Let represent the estimated noise parameter vector at time t, and represent the parameters to be estimated in the noise sub-model. ; This is the estimated value of the first parameter of the autoregressive process at time t; This is the estimated value of the second parameter of the autoregressive process at time t; The first autoregressive process at time t Parameter estimates; This is the estimated value of the first parameter of the moving average process at time t; This is the estimated value of the second parameter of the moving average process at time t; The first moving average process at time t Parameter estimates; This represents the virtual noise output at time t; Let represent the noise information vector at time t. This represents the noise information vector at time t after transposition. This represents the estimated value of the noise parameter vector at time t-1, which is the initial value for recursive updates; This represents the gain matrix of the noise sub-model at time t, used to adjust the parameter update step size and ensure convergence; Let represent the covariance matrix of the noise submodel at time t-1; Let represent the covariance matrix of the noise submodel at time t; The algorithms obtained in (11)-(16) still conceal an unresolved problem: the information vector... , , , , , It is unknown. Here, we use the ideas of interactive estimation and estimated value substitution. During the cyclic execution of the algorithm, the unknown value at the current moment is replaced by the corresponding estimated value.

[0113] use , , , , , Replace the unknown in the information vector , , , , , , where i is an integer representing the number of time points; t represents time t; This represents the intermediate filter variable at time ti; This represents the filtered input at time ti; This represents the j-th nonlinear term of the filter at time ti; This represents the filtered output at time ti; This represents the virtual noise output at time ti; Indicates t- i Time-based white noise; This represents the estimated value of the intermediate filtered variable at time ti; This represents the filtered input estimate at time ti; This represents the estimated value of the j-th filter nonlinear term at time ti; This represents the filtered output estimate at time ti; This represents the virtual noise output estimate at time ti; Indicates t- i The white noise estimate at the current time. Based on the estimate at the current time, the estimates of each unknown variable at the current time can be calculated for use in the next algorithm update loop.

[0114] The estimated values ​​of each unknown variable at the current time are calculated as follows: ; in, It is the filtered input estimate at time t; It is the filtered input estimate at time t-1; It is the filtered input estimate at time t-2; It is t- The filtered input estimate at time t; It is the nonlinear output estimate at time t; It is the nonlinear output estimate at time t-1; It is the nonlinear output estimate at time t-2; It is time t- The nonlinear output estimate. When t≤0, all values ​​are obtained through prior settings and can be set by those skilled in the art according to the actual situation, which will not be elaborated here.

[0115] ; in, This represents the filtered output estimate at time t; This represents the filtered output estimate at time t-1; This represents the filtered output estimate at time t-2; Indicates t- The filtered output estimate at time t;

[0116] in, This represents the estimated value of the intermediate filtered variable at time t; It is t - The filtered input estimate at time t; Let represent the estimated value of the intermediate filtered variable at time t-1; Let represent the estimated value of the intermediate filtered variable at time t-2; Indicates t- The estimated value of the intermediate filtered variable at time t; ; in, This is the estimated value of the j-th nonlinear term of the filter at time t; This is the estimated value of the j-th filter nonlinear term at time t-1; This is the estimated value of the j-th nonlinear term of the filter at time t-2; For t- The estimated value of the j-th filter nonlinear term at time t; The filtered output estimate at time t is a weighted combination of the current measured value y(t), the historical measured value y(ti), and the historical filtered output estimate. The feedback structure and weighting coefficients and Updated online using parameter estimation algorithms; The intermediate filtered variable estimate at time t is the filtered input at time t. The coefficients are obtained from the dynamic weighted combination and feedback calculation of historical states. and Adjust in real time using a recursive algorithm; This is the estimated value of the j-th filter nonlinear term at time t. Its calculation incorporates a weighted term of the current and historical nonlinear function values, along with feedback from historical estimates. The weighting coefficients are... and Updated synchronously with filter parameters.

[0117] Because the parameter estimation algorithm is recursive, the initial value at time t=0 needs to be set based on prior knowledge. With the initial value at t=0, the estimated parameter vector at time t=1 can be obtained according to the recursive algorithm. Then, based on the obtained estimated values ​​of the unknown parameters, the values ​​of each intermediate variable can be calculated, preparing for the next recursive operation. When performing the above recursion, it is necessary to first set... , , and The initial value at t≤0, specifically, is , and When setting initial values ​​for each parameter, you can choose from the preferred range (0, 10). -2 Select a random number from the list; The initial value is set to a large symmetric positive definite matrix, preferably 10. 6 I.

[0118] For unknown variables , To solve this, we need to re-examine the system equation (1), from which we can see that: ; ; Similarly, we can use the estimated value from the previous time step to replace the unknown value at the current time step: , ; in, This represents the virtual noise output estimate at time t; This represents the estimated value of the transpose of the intermediate information vector at time t; This represents the estimated value of the system parameter vector; This represents the white noise estimate at time t; This represents the estimated value of the transpose of the noise information vector at time t; This represents the estimated value of the noise parameter vector; ; in, This represents the estimated value of intermediate variables in the system operation at time t-1; This represents the estimated value of intermediate variables in the system operation at time t-2; Indicates t- The estimated intermediate variables of the system at each time point are as follows: ; in, This represents the estimated value of intermediate variables in the system at time t.

[0119] In the estimated value of the intermediate information vector at time t As shown in the following formula: ; Then, information vector estimates for the algorithm loop are constructed. and :

[0120] in, It is the estimated value of the j-th nonlinear term of the filter at time t; The estimated value of the second nonlinear term of the filter at time t; The estimated value of the m-th filter nonlinear term at time t; It is the estimated value of the system information vector.

[0121] ; in, It is the noise information vector estimate.

[0122] 4.2 Calculate the current system parameter vector based on the current system information vector and the system parameter vector of the previous time step; calculate the current noise parameter vector based on the current noise information vector and the noise parameter vector of the previous time step; Using information vectors and Instead of (11)-(16) and A recursive least squares algorithm for estimating system model parameters based on whitening processing and interactive estimation strategies was obtained: (17) (18) (19) (20) (twenty one) (twenty two) S5: Determine the output error system transfer function based on the current system parameter vector, and determine the current colored noise model transfer function based on the current noise parameter vector, to obtain the current original colored noise system model without unknowns.

[0123] Those skilled in the art should know how to determine the output error system transfer function based on the current system parameter vector, and how to determine the current colored noise model transfer function based on the current noise parameter vector; these will not be elaborated upon here.

[0124] Example 2 One embodiment of this application provides a method for identifying colored noise systems using whitening processing and interactive estimation.

[0125] Consider the following input-nonlinear output error system, where the colored noise component is fitted by an autoregressive moving average model:

[0126] Assumption , , , , , , All are unknown; the method of this invention is used for system identification: During numerical simulation, assume It is a measurable random sequence with a mean of zero and a variance of 1. Using a mean of 0 and a variance of An uncorrelated normally distributed signal sequence. The sampled data length is... The first 10,000 sets were used for algorithm training, and the last 5,000 sets were used for cross-validation. The algorithm proposed in this invention was used to estimate the model parameters. The parameter estimation results and errors are shown in Tables 1 and 2. The parameter estimation values ​​change with the recursive time step as shown in the curves. Figure 3 As shown, based on the foregoing content, Figure 3 middle The coefficients of the first nonlinear function; The coefficients of the second nonlinear function; The first parameter of the denominator polynomial of the system transfer function for the output error; The second parameter of the denominator polynomial of the system transfer function for the output error; The first parameter of the numerator polynomial of the system transfer function for the output error; The second parameter of the numerator polynomial of the system transfer function for the output error; This is the first parameter in the autoregressive process; The first parameter of the moving average process; where This represents the relative error between the estimated parameters and the true parameters:

[0127] in, These are the model parameters for any estimate; yes The corresponding actual parameters; This represents the l2 norm.

[0128] Table 1. Parameter estimation and error of the filter-based recursive least squares algorithm ( ):

[0129] Table 2. Parameter estimation and error of the filter-based recursive least squares algorithm ( ):

[0130] From Table 1, Table 2 and Figure 3 It can be seen that the algorithm of this invention can obtain an effective estimate of the unknown parameters in the system model, and the estimation error gradually decreases as the recursion time increases.

[0131] In contrast, the classic recursive least squares (RLS) algorithm without filtering and splitting was also used to estimate the model parameters. Figure 4 The graphs showing the estimation error versus recursive time for the two algorithms are presented. It can be seen that the algorithm proposed in this invention has a smaller estimation error.

[0132] In this implementation, the root mean square error (RMSE) is used to measure the difference between the predicted output value and the true value. RMSE is calculated as follows: ; Cross-validation was used to test the predictive ability of the method in this application. The cross-validation dataset was of length [length missing]. The sampled data is to . Figure 5 Demonstrates the variance of noise At that time, the prediction output errors of the algorithm of this invention and the RLS algorithm are compared. Figure 5The red continuous curve represents the error generated by the RLS prediction output, while the black curve represents the error generated by the algorithm of this invention. It can be seen that although the red curve corresponding to the RLS algorithm's prediction error is close to the black curve corresponding to the prediction error of the proposed method, the fluctuation of the red curve is significantly stronger than that of the black curve, indicating that the error generated by the algorithm of this invention is smaller. Table 3 lists the root mean square error (RMSE) of the prediction output obtained by the two algorithms under four different noise variances. It can be seen that the RMSE value of the prediction output obtained by the algorithm proposed in this invention is smaller.

[0133] Table 3. Comparison of RMSE values ​​under different noise variances:

[0134] This disclosure can be a system, method, and / or computer program product. A computer program product may include a computer-readable storage medium having computer-readable program instructions loaded thereon for causing a processor to implement various aspects of this disclosure.

[0135] Computer-readable storage media can be tangible devices capable of holding and storing instructions for use by an instruction execution device. Computer-readable storage media can be, for example—but not limited to—electrical storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of computer-readable storage media include: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital multifunction disc (DVD), memory sticks, floppy disks, mechanical encoding devices, such as punch cards or recessed protrusions storing instructions thereon, and any suitable combination of the foregoing. The computer-readable storage media used herein are not to be construed as transient signals themselves, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmission media (e.g., light pulses through fiber optic cables), or electrical signals transmitted through wires.

[0136] The computer-readable program instructions described herein can be downloaded from computer-readable storage media to various computing / processing devices, or downloaded via a network, such as the Internet, local area network, wide area network, and / or wireless network, to an external computer or external storage device. The network may include copper transmission cables, fiber optic transmission, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards them to the computer-readable storage media in the respective computing / processing device.

[0137] Computer program instructions used to perform the operations of this disclosure may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, status setting data, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, etc., and conventional procedural programming languages ​​such as the "C" language or similar programming languages. The computer-readable program instructions may execute entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving a remote computer, the remote computer may be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or may be connected to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, electronic circuitry, such as programmable logic circuitry, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs), is personalized by utilizing the status information of the computer-readable program instructions to implement various aspects of this disclosure.

[0138] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the protection scope of the claims of the present invention.

Claims

1. A method for identifying colored noise systems based on whitening processing and interaction estimation, characterized in that, include: S1: Based on the output error system transfer function and the colored noise model transfer function, construct the original colored noise system model that reflects the input-output relationship of the dynamic system; S2: Filter the original model of the colored noise system to obtain the whitening model of the colored noise system; the whitening model of the colored noise system is calculated and filtered based on the system information vector, system parameter vector and white noise at each time step; S3: Decompose the original model of the colored noise system to obtain a noise sub-model based on noise information vector and noise parameter vector, and a system operation sub-model based on intermediate information vector and system parameter vector; S4: Calculate the current system information vector and noise information vector based on the historical noise parameter vector and intermediate information vector, respectively; determine the current system parameter vector based on the current system information vector; determine the current noise parameter vector based on the current noise information vector; S5: Determine the current output error system transfer function based on the current system parameter vector, and determine the current colored noise model transfer function based on the current noise parameter vector, to obtain the original colored noise system model without unknowns.

2. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, characterized in that: In S1, the original model of the colored noise system is also constructed based on a pre-selected nonlinear function, a column vector of coefficients of the nonlinear function, and white noise; the column vector of coefficients of the nonlinear function includes the coefficients of multiple nonlinear functions.

3. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 2, characterized in that: In S1, the output error system transfer function is constructed based on at least one output error system transfer function denominator polynomial parameter, output error system transfer function numerator polynomial parameter, and displacement operator; The colored noise model transfer function is constructed based on at least one autoregressive process parameter, a moving average process parameter, and a displacement operator.

4. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, characterized in that: In S2, before filtering the original model of the colored noise system, a filter term that is the reciprocal of the transfer function of the colored noise model is defined, and the original model of the colored noise system is filtered based on the filter term.

5. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 3, characterized in that: In S2, both the system information vector and the system parameter vector are unknowns; The system information vector includes intermediate filter variables, filter inputs, and various filter nonlinear terms at historical moments; The intermediate filtering variables are calculated based on the output error system transfer function and the filtering input; the filtering input is calculated based on the colored noise model transfer function, a pre-selected nonlinear function, and the coefficient column vector of the nonlinear function; each filtering nonlinear term is calculated based on the colored noise model transfer function, the pre-selected nonlinear function corresponding to that term, and the system's measurable input. The system parameter vector includes the denominator polynomial parameters of all output error system transfer functions, the numerator polynomial parameters of the output error system transfer functions, and the coefficients of the nonlinear functions.

6. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 3, characterized in that: In S3, the noise sub-model calculates the virtual noise output based on the noise information vector and the noise parameter vector; The noise information vector includes virtual noise output and white noise at historical moments; The noise parameter vector includes all autoregressive process parameters and moving average process parameters.

7. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 2, characterized in that: In S3, the system operation sub-model calculates intermediate variables of system operation based on intermediate information vector and system parameter vector; The intermediate information vector includes intermediate variables of system operation at historical moments, dynamic inputs of the colored noise system, and nonlinear functions of various system input signals; The dynamic input of the colored noise system is calculated based on the nonlinear function of each system input signal and the coefficient column vector of the nonlinear function; The nonlinear function of each system input signal is calculated based on the system's measurable input and the corresponding pre-selected nonlinear function.

8. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, 5, 6, or 7, characterized in that: In S4, the nonlinear output estimate at each time step in the intermediate information vector is calculated based on the system's measurable input at that time step, the estimated values ​​of the coefficients of each nonlinear function, and the pre-selected nonlinear function.

9. The method for identifying colored noise systems based on whitening processing and interaction estimation according to claim 8, characterized in that: In S4, the current system information vector is calculated based on the historical noise parameter vector and the intermediate information vector, including: Based on the nonlinear output estimate earlier than the time to be calculated, the respective regression process parameter estimates and the moving average process parameter estimates at the time to be calculated, the filtered input estimate at the historical time is calculated. Based on the estimated values ​​of the filtered input earlier than the time to be calculated, the estimated values ​​of the intermediate filtered variables, and the estimated values ​​of the denominator polynomial parameters and the numerator polynomial parameters of the system transfer function of each output error system at the time to be calculated, the estimated values ​​of the intermediate filtered variables at historical time points are calculated. Based on a pre-selected nonlinear function, measurable system inputs earlier than the time to be calculated, the estimated value of the filtered nonlinear term, and the estimated values ​​of the respective regression process parameters and moving average process parameters at the time to be calculated, the estimated value of the filtered nonlinear term at historical time points is calculated.

10. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, 5, 6, or 7, characterized in that: In S4, the current noise information vector is calculated based on the historical noise parameter vector and the intermediate information vector, including: Based on the system's measurable output, intermediate information vector, and system parameter vector at the time to be calculated, the virtual noise output estimate for historical time points is calculated. Based on the system's measurable output, intermediate information vector, system parameter vector, noise information vector, and noise parameter vector at the time to be calculated, the white noise estimate for historical time points is calculated.

11. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, 5, 6, or 7, characterized in that: In S4, the current system parameter vector is also determined based on the system parameter vector of the previous time step, the estimated value of the filtered output at the current time step, and the gain matrix of the system operating sub-model at the current time step. The current filtered output estimate is calculated based on the system's measurable output at the current and historical times, the filtered output estimate at the historical times, the respective regression process parameter estimates at the current time, and the respective moving average process parameter estimates at the current time. The gain matrix of the system operating sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the system operating sub-model at the previous moment, as well as the system information vector at the current moment. The covariance matrix of the system operating sub-model at each time step is calculated based on the gain matrix of the system operating sub-model at that time step, the covariance matrix of the system operating sub-model at the previous time step, the system information vector at that time step, and the identity matrix.

12. The method for identifying colored noise systems based on whitening processing and interactive estimation according to claim 1, 5, 6, 7, or 10, characterized in that: In S4, the current noise parameter vector is also determined based on the noise parameter vector of the previous time step, the virtual noise output estimate of the current time step, and the gain matrix of the noise sub-model at the current time step. The gain matrix of the noise sub-model at the current moment is calculated based on the covariance matrix and identity matrix of the noise sub-model at the previous moment, as well as the noise information vector at the current moment. The covariance matrix of the noise sub-model at each time step is calculated based on the gain matrix of the noise sub-model at that time step, the covariance matrix of the noise sub-model at the previous time step, the noise information vector at that time step, and the identity matrix.

13. A colored noise system identification system utilizing the colored noise system identification method according to any one of claims 1-12, characterized in that, It includes modules for building the original model, building the whitened model, splitting the original model, determining the vectors, and determining the unknowns of the original model. The original model construction module constructs an original model of a colored noise system that reflects the input-output relationship of a dynamic system based on the output error system transfer function and the colored noise model transfer function. The whitening model construction module is used to filter the original model of the colored noise system to obtain a whitening model of the colored noise system; the whitening model of the colored noise system calculates the filtered output based on the system information vector, system parameter vector and white noise at each time step; The original model splitting module is used to split the original model of the colored noise system to obtain a noise sub-model constructed based on noise information vector and noise parameter vector, and a system operation sub-model constructed based on intermediate information vector and system parameter vector. The vector determination module calculates the current system information vector and noise information vector based on the historical noise parameter vector and the intermediate information vector, respectively; determines the current system parameter vector based on the current system information vector; and determines the current noise parameter vector based on the current noise information vector. The original model unknown quantity determination module determines the current output error system transfer function based on the current system parameter vector and the current colored noise model transfer function based on the current noise parameter vector, thus obtaining the original colored noise system model without unknown quantities.

14. A terminal, comprising a processor and a storage medium; characterized in that: The storage medium is used to store instructions; The processor is configured to operate according to the instructions to perform the steps of the colored noise system identification method according to any one of claims 1-12.

15. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the program implements the steps of the colored noise system identification method according to any one of claims 1-12.

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