Multi-model linear time-varying process industry system identification method

Abstract

The invention discloses a multi-model linear time-varying system identification method, solves the problem that an existing process industry system identification method is difficult to accurately estimate system parameters and switching characteristics, and belongs to the field of process industry system parameter identification. The method comprises the steps that a state space model of a process industry system is established, the process industry system is a multi-model linear time-varying system, and output of the state space model is set value observation generated by a known quantization operator; based on the state space model and measurable scheduling variables, constructing a statistical model taking continuous state variables and sub-model indication variables as hidden variables; based on the statistical model, constructing a complete data log-likelihood function comprising the hidden variables and set value observation; under an expectation maximization framework, constructing a Q function according to conditional expectation of a log-likelihood function about hidden variable posterior distribution; and iteratively executing the step E and the step M, and jointly estimating the to-be-identified parameter until the estimation of the to-be-identified parameter is converged.

Description

Technical Field

[0001] This invention relates to a multi-model linear time-varying process industry system identification method, belonging to the field of process industry system parameter identification. Background Technology

[0002] Process industry systems, such as those in petrochemicals, pharmaceuticals, and metallurgy, typically exhibit complex, multivariable, and nonlinear dynamic systems. Their operating characteristics often change smoothly or frequently with variations in raw materials, loads, and environmental conditions, making multi-model linear time-varying (LPV) systems suitable for description. Accurate identification of these systems is crucial for achieving precise modeling, advanced control, and condition monitoring.

[0003] Traditional system identification methods, such as least squares-based regression, maximum likelihood estimation, and subspace identification, are typically based on the ideal assumption that the system output is continuous and accurately observable. For multi-model or switching systems, existing identification techniques (such as clustering methods, mixture estimation, or hard-switching identification) also mostly require the acquisition of continuous measurements of the system output, or at least require that the observation noise satisfies a specific continuous distribution assumption.

[0004] However, in real-world industrial environments, due to limitations in sensor accuracy, communication bandwidth, or intentional data generalization for security and privacy reasons, system outputs are often only acquired in set-valued form. That is, the observed values ​​are not continuous signals, but rather discrete symbols or levels representing a certain numerical range. This range-based or symbolic observation leads to severe information incompleteness.

[0005] Faced with such set-valued observations, existing system identification methods encounter fundamental challenges: First, traditional criteria based on minimizing continuous errors become directly ineffective due to discontinuous observations; second, the inability to obtain the true values ​​of the continuous states within the system makes it difficult to accurately describe and estimate the dynamic parameters of different sub-models; third, under the condition that the output is only coarse interval information, the implicit operating condition switching behavior of the system becomes extremely ambiguous, making it difficult for traditional methods to effectively identify or explain the switching logic and switching sequence between models. Therefore, given the common reality that process industry systems typically obtain outputs in set-valued form due to factors such as sensor quantization and communication limitations, developing robust identification methods capable of accurately estimating the parameters and switching characteristics of multi-model systems under these conditions has become a core technical challenge that urgently needs to be addressed to improve the level of industrial intelligence. Summary of the Invention

[0006] To address the problem that existing process industry system identification methods struggle to accurately estimate system parameters and switching characteristics when the output of a process industry system is obtained in the form of set values, this invention provides a multi-model linear time-varying process industry system identification method.

[0007] The present invention provides a method for identifying multi-model linear time-varying process industry systems, comprising:

[0008] Establish a state-space model of the process industry system, wherein the output of the state-space model is a set-valued observation generated by a known quantization operator;

[0009] Based on the state-space model and measurable scheduling variables, a statistical model is constructed with continuous state variables and sub-model indicator variables as latent variables.

[0010] Based on the statistical model, a complete log-likelihood function of the data, including the latent variables and set-valued observations, is constructed.

[0011] Within the expectation maximization framework, a Q function is constructed based on the conditional expectation of the log-likelihood function of the complete data with respect to the posterior distribution of the latent variables;

[0012] By iteratively executing the E-step and M-step, the parameters to be identified are jointly estimated: sub-model parameters, noise variance, and sub-model effective width parameter.

[0013] In the E-step, the expected conditional distribution of the latent variable is calculated based on the current value of the parameter to be identified and the set-valued observation, thereby obtaining the Q function; in the M-step, the values ​​of all parameters to be identified are updated by maximizing the Q function.

[0014] The E-step and M-step are executed iteratively until the estimation of the parameters to be identified converges.

[0015] As a preferred option, the state-space model for process industry systems is as follows:

[0016]

[0017]

[0018] in, It is a time index. Regression vector , For process industry systems at all times The actual output, For process industry systems at all times Input, For the first The parameter vector of each sub-model Indicates the process industry system at time... noise, noise Follows a mean of 0 and a variance of Gaussian distribution, For known quantization operators, For set-valued observations, This indicates the regression order of the output of a process industry system to its historical outputs. This represents the regression order of the output of a process industry system with respect to its historical inputs. Indicates the number of samples collected.

[0019] Preferably, the statistical model includes a sub-model prior probability model, which is as follows:

[0020]

[0021] Represents the prior probability of the sub-model's identity; Indicates time The implicit variable is used to represent the system at time t. The actual output Sub-model number;

[0022] Indicates time The scheduling variable is used to characterize the system at time [time]. Operating conditions;

[0023] Each sub-model corresponds to a given central operation point. It has a valid width parameter. , Process industry systems include Sub-model;

[0024] This indicates that the scheduling variable is in the process of scheduling. Under the action, the first The sub-model at time... The activated weight coefficients.

[0025] Preferably, the statistical model further includes a system output probability model, which is:

[0026]

[0027] in, The output probability of the system. , , This represents the mean. This represents a Gaussian distribution.

[0028] Preferably, the statistical model further includes a quantitative observation probability model, which is as follows:

[0029]

[0030] in, Indicates the probability of quantified observation. Represents the set of quantization intervals The indicator function on the variable The value falls within the set of quantization intervals When the specified time interval is reached, the indicator function takes a value of 1; otherwise, it takes a value of 0. The quantization interval set... By quantization operator Induced by, used to characterize continuous variables Consistency constraints between its corresponding quantitative observations.

[0031] As a preferred option, the log-likelihood function for the complete data is:

[0032]

[0033] in, This represents the complete data likelihood of the entire data sequence. , Observational data Latent variables , Indicating process industry systems in A continuous real output sequence at each sampling time;

[0034] Indicates at time The parameters corresponding to the activated sub-model. Indicates in scheduling variables Under the influence, at any time The weight coefficients of the activated sub-model.

[0035] As a preferred option, the Q function is:

[0036] Among them, the responsibility of the sub-model , This represents the set of current values ​​for the parameter to be identified. It represents conditional expectation.

[0037] Preferably, the updated sub-model parameter values ​​are:

[0038] .

[0039] Preferably, the updated noise variance value is:

[0040]

[0041] in, Indicates time The observation data in the first The weighted residual statistics formed under the conditions of each sub-model .

[0042] Preferably, the value of the updated sub-model effective width parameter is:

[0043] ;

[0044] This represents the step size in gradient descent.

[0045] Represents the current set of values ​​for the effective width parameter.

[0046] The beneficial effects of this invention are that this method, through explicit probabilistic modeling, describes set-valued observations as the result of continuous states after quantization mapping, thus characterizing the incompleteness of information in a probabilistic sense. This allows the algorithm to directly process set-valued data such as intervals and levels, thereby enabling system identification even in real-world industrial scenarios where only low-precision or discrete observations are available, solving the fundamental problem of traditional methods failing under such conditions. Within a unified framework, this method treats continuous states and sub-model indicator variables as latent variables and introduces scheduling variables to characterize smooth switching. By iteratively inferring latent variables and updating parameters using the Expectation-Maximization (EM) algorithm, without prior knowledge of the model switching sequence, it can simultaneously and accurately identify the dynamic parameters of each sub-model, the statistical characteristics of system noise, and the effective width parameter reflecting the correlation between operating conditions and models, significantly improving the completeness of the description of the dynamic behavior of complex systems. Attached Figure Description

[0047] Figure 1 This is a flowchart illustrating the process of this application;

[0048] Figure 2 A comparison of the responsibility level of sub-model 1 with the indicator variables of the actual activated sub-model;

[0049] Figure 3 A comparison of the responsibility levels of the two sub-models;

[0050] Figure 4 Comparison of responsibility and scheduling variables in sub-model 1;

[0051] Figure 5 This describes the parameter convergence process of sub-model 1.

[0052] Figure 6 This describes the parameter convergence process of sub-model 2;

[0053] Figure 7 This is the noise variance convergence process;

[0054] Figure 8 This describes the convergence process of the width parameter. Detailed Implementation

[0055] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0056] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0057] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0058] The multi-model linear time-varying process industry system identification method of this embodiment includes:

[0059] Step 1: Establish a state-space model of the process industry system. The output of the state-space model is a set-valued observation generated by a known quantization operator.

[0060] The state-space model is as follows:

[0061]

[0062]

[0063] in, It is a time index. Regression vector , For process industry systems at all times The actual output, For process industry systems at all times Input, For the first The parameter vector of each sub-model Indicates the process industry system at time... noise, noise Follows a mean of 0 and a variance of Gaussian distribution, For known quantization operators, For set-valued observations, This indicates the regression order of the output of a process industry system to its historical outputs. This represents the regression order of the output of a process industry system with respect to its historical inputs. Indicates the number of samples collected.

[0064] The process industry system is described by M local ARX sub-models. At time k, if the i-th sub-model is active, the actual system output satisfies

[0065]

[0066] in: This is the parameter vector (including constant coefficients) of the i-th sub-model. It follows a mean of 0 and a variance of . The Gaussian distribution.

[0067] System actual output The observable quantities that cannot be directly observed are:

[0068]

[0069] in, The quantization operator is known.

[0070] Quantization operators have Each quantitative level is defined as:

[0071]

[0072] in: These are non-overlapping intervals. , For interval The corresponding quantized output value.

[0073] Step 2: Based on the state-space model and measurable scheduling variables, construct a statistical model with continuous state variables and sub-model indicator variables as latent variables; the statistical model of this application includes a sub-model prior probability model, a system output probability model, and a quantitative observation probability model.

[0074] Furthermore, set These are measurable scheduling variables used to characterize the system's operating condition at time k. Each sub-model corresponds to a given central operating point. It has a valid width parameter. Introducing discrete latent variables , used to indicate that the output is generated at time k The sub-model number.

[0075] Given scheduling variables The prior probability of the sub-model is determined by With each sub-model center The Gaussian distance function is normalized to achieve smooth probability switching between sub-models. The prior probability model of the sub-model is:

[0076]

[0077] This indicates that the scheduling variable is in the process of scheduling. Under the action, the first The sub-model at time... The activated weight coefficients.

[0078] Furthermore, this refers to the true continuous state in the system that cannot be directly observed. Provide a precise probabilistic mathematical description, using a system output probability model, specifically as follows: :

[0079] like ,but ,therefore:

[0080]

[0081] Furthermore, the quantitative observation probability model in this application is as follows:

[0082]

[0083] in, Indicates the probability of quantified observation. Represents the set of quantization intervals The indicator function on the above, the quantitative observation probability model of this application describes the set-valued observation process as a deterministic interval mapping, defines the observation conditional probability As an indicator function, when Falling The corresponding set of quantization intervals The probability is 1 if it is within the time frame, and 0 otherwise. The set... By quantization operator Induced by, used to characterize continuous variables Consistency constraints between its corresponding quantized observations. This step will define the continuous state of the system. Sub-model indication Unification is considered as a latent variable to be inferred.

[0084] Step 3: Based on the statistical model, construct the complete data log-likelihood function including the latent variables and set-valued observations; specifically, the complete data likelihood at a single time point combines the three terms:

[0085]

[0086] in .

[0087] Observable data Latent variables , Indicating process industry systems in A continuous sequence of true outputs at each sampling time, wherein each For the system at time The unquantized true output value; this variable cannot be directly observed and can only be observed through quantization. Indirect reflection.

[0088] Then the complete data likelihood of the entire data sequence:

[0089]

[0090] The log-likelihood function for the complete data is:

[0091]

[0092] Indicates at time The parameters corresponding to the activated sub-model. Indicates in scheduling variables Under the influence, at any time The weight coefficients of the activated sub-model.

[0093] Step 4: Within the expectation maximization framework, construct the Q function based on the conditional expectation of the log-likelihood function of the complete data with respect to the posterior distribution of the latent variables;

[0094] Within the EM framework, the Q-function for the r-th iteration is defined as:

[0095]

[0096] Introducing Sub-model Responsibility Therefore, the Q function can be written as:

[0097]

[0098] Quantitative indicators It is always 0 in the conditional distribution, is independent of parameters, and can be ignored. (gating term) Therefore, we get: This part only relates to the effective width parameter. Relevant. Gaussian regression term, taking conditional expectation:

[0099]

[0100] The complete Q function is obtained by merging:

[0101] This represents the set of current values ​​for the parameter to be identified.

[0102] Step 5: By iteratively executing the E-step and M-step, jointly estimate the parameters to be identified: sub-model parameters, noise variance, and effective width parameter of the sub-model;

[0103] In the E-step, the expected conditional distribution of the latent variable is calculated based on the current value of the parameter to be identified and the set-valued observation, thereby obtaining the Q function; in the M-step, the values ​​of all parameters to be identified are updated by maximizing the Q function.

[0104] The E-step and M-step are performed iteratively until the parameter estimates to be identified converge. The convergence condition can be set to the number of iterations reaching a set upper limit, or the relative change of the current parameter estimate compared to the parameter estimate obtained in the previous iteration being less than an arbitrarily small constant. The derivation in this step... The update formula, the specific process is as follows:

[0105] make ,get:

[0106]

[0107] Solving for:

[0108]

[0109] make ,get:

[0110]

[0111] Summarized as follows:

[0112]

[0113] Since for each k, we have Therefore, the denominator is simplified to N, and we finally get:

[0114]

[0115] make ,get:

[0116]

[0117] Since this equation has no analytical solution, Gradient update

[0118]

[0119] Right now:

[0120]

[0121] in, The step size.

[0122] This application discloses a method for identifying multi-model linear time-varying systems (LPVs) under set-valued observation conditions. It explicitly introduces quantization operators at the system modeling level, models extreme value observations as the result of continuous hidden states after quantization mapping, and characterizes the incompleteness of information introduced by quantization in a probabilistic sense. It introduces sub-models to indicate hidden variables to describe the operating mode of the multi-model system, and uses scheduling variables to construct a smooth gating mechanism to characterize the model switching behavior of the system under different operating conditions.

[0123] This application, based on the Expectation-Maximization (EM) inference framework, treats the continuous state of the system and sub-model indicator variables as latent variables. It weights the contributions of different local models through sub-model responsibility, achieving joint estimation of sub-model dynamic parameters, system noise statistics, and model effective width parameters. This method does not require prior knowledge of the model switching sequence and can effectively identify multi-model LPV systems using only set-valued observation information.

[0124] Simulation results show that, even with significant quantization effects, the method proposed in this invention can still achieve stable and accurate parameter estimation, exhibiting good numerical stability and robustness. This invention is applicable to industrial process control, complex equipment condition monitoring, and intelligent system modeling scenarios where sensor accuracy is limited or communication bandwidth is constrained.

[0125] Example:

[0126] Consider an LPV system consisting of two sub-models, whose state evolution is as follows:

[0127]

[0128] in Let represent the sub-model activated at time k, and let noise 'a' be zero-mean Gaussian white noise.

[0129] The sub-model parameters are set as follows:

[0130] Sub-model 1:

[0131] Sub-model 2:

[0132] Scheduling variables Given a known signal, the activation probability of the sub-model is determined using a Gaussian gating function. The system noise variance is set to... The system output state is processed by a uniform quantizer with a quantization step size of [value missing]. Quantized observation sequence obtained The data length is set to N=300.

[0133] Experimental results:

[0134] 1. Parameter estimation results

[0135] Table 1 shows the comparison between the actual system parameters and the final estimated parameters of the proposed method. It can be observed that the estimated values ​​of the parameters of each sub-model are quite close to the actual parameters, indicating that even with only quantitative observations, the proposed method can still effectively recover the dynamic characteristics of the LPV system.

[0136] Table 1 Comparison of experimental results

[0137]

[0138] Table 2 Comparison of Noise Variance

[0139]

[0140] Meanwhile, noise variance The estimated results are on the same order of magnitude as the true values, with no significant deviation, indicating that the method in this application is reasonable for modeling the uncertainty of quantitative observations.

[0141] 2. Relationship between sub-model responsibility and real-world switching

[0142] Figure 2 Demonstrates the responsibility of sub-model 1 Indicator variables of the real activator model The comparison shows that when the real system generates data from sub-model 1, Most of the time it is close to 1; when the system switches to sub-model 2... The value dropped significantly to near zero. This indicates that the EM algorithm, within the soft-switching framework, can accurately identify which sub-model the data at each time step is more likely to originate from. Furthermore, in Figure 3 Two sets of responsibility can be observed. and They exhibit a clear complementary relationship and meet the probability normalization constraint. .

[0143] 3. The relationship between responsibility and scheduling variables

[0144] exist Figure 4 In the middle, the responsibility level of sub-model 1 was also plotted. With scheduling variables It can be observed that when the scheduling variable... When approaching the center position of sub-model 1 Significantly increased; when When deviating from this area, The corresponding decrease indicates that the gating mechanism based on scheduling variables can be correctly learned by the EM algorithm and reflects the impact of scheduling variables on model switching behavior at the responsibility level.

[0145] 4. Convergence analysis of parameters

[0146] Figure 5 and Figure 6 The trajectories of the parameters of the two sub-models as a function of the number of EM iterations are given (the convergence criterion was reached in the seventh iteration). Figure 7 Demonstrates noise variance Changes during the EM iteration process. Figure 8 The convergence process of the gating width parameter is given.

[0147] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for identifying multi-model linear time-varying process industry systems, characterized in that, include: Establish a state-space model of the process industry system, wherein the output of the state-space model is a set-valued observation generated by a known quantization operator; Based on the state-space model and measurable scheduling variables, a statistical model is constructed with continuous state variables and sub-model indicator variables as latent variables. Based on the statistical model, a complete log-likelihood function of the data, including the latent variables and set-valued observations, is constructed. Within the expectation maximization framework, a Q function is constructed based on the conditional expectation of the log-likelihood function of the complete data with respect to the posterior distribution of the latent variables; By iteratively executing the E-step and M-step, the parameters to be identified are jointly estimated: sub-model parameters, noise variance, and sub-model effective width parameter. In this step, the E-step calculates the conditional distribution expectation of the latent variable based on the current value of the parameter to be identified and the set-valued observation, thereby obtaining the Q function; The M-step updates the values ​​of all parameters to be identified by maximizing the Q-function; The E-step and M-step are executed iteratively until the estimation of the parameters to be identified converges.

2. The multi-model linear time-varying process industry system identification method according to claim 1, characterized in that, The state-space model of a process industry system is as follows: in, It is a time index; Regression vector ; For process industry systems at all times The actual output, For process industry systems at all times Input, For the first The parameter vector of each sub-model Indicates the process industry system at time... noise, noise Follows a mean of 0 and a variance of Gaussian distribution; For known quantization operators, For set-valued observations; This indicates the regression order of the output of a process industry system to its historical outputs. This indicates the regression order of the output of a process industry system to its historical inputs; Indicates the number of samples collected.

3. The multi-model linear time-varying process industry system identification method according to claim 2, characterized in that, The statistical model includes a sub-model prior probability model, which is as follows: Represents the prior probability of the sub-model's identity; Indicates time The implicit variable is used to represent the system at time t. The actual output Sub-model number; Indicates time The scheduling variable is used to characterize the system at time [time]. Operating conditions; Each sub-model corresponds to a given central operation point. It has a valid width parameter. , Process industry systems include Sub-model; This indicates that the scheduling variable is in the process of scheduling. Under the action, the first The sub-model at time... The activated weight coefficients.

4. The multi-model linear time-varying process industry system identification method according to claim 3, characterized in that, The statistical model also includes a system output probability model, which is: in, The output probability of the system. , , This represents the mean. This indicates a Gaussian distribution.

5. The multi-model linear time-varying process industry system identification method according to claim 4, characterized in that, The statistical model also includes a quantitative observation probability model, which is as follows: in, Indicates the probability of quantified observation; Represents the set of quantization intervals The indicator function on the variable The value falls within the set of quantization intervals When the time is within the specified range, the indicator function takes a value of 1; otherwise, it takes a value of 0. Quantization interval set By quantization operator Induced by, used to characterize continuous variables Consistency constraints between its corresponding quantitative observations.

6. The multi-model linear time-varying process industry system identification method according to claim 5, characterized in that, The log-likelihood function for the complete data is: in, This represents the complete data likelihood of the entire data sequence; , Observational data Latent variables ; Indicating process industry systems in A continuous real output sequence at each sampling time; Indicates at time The parameters corresponding to the activated sub-model. Indicates in scheduling variables Under the influence of, at any time The weight coefficients of the activated sub-model.

7. The multi-model linear time-varying process industry system identification method according to claim 6, characterized in that, The Q function is: Among them, the responsibility of the sub-model , This represents the set of current values ​​for the parameter to be identified. It represents conditional expectation.

8. The multi-model linear time-varying process industry system identification method according to claim 7, characterized in that, The updated sub-model parameter values ​​are: 。 9. The multi-model linear time-varying process industry system identification method according to claim 7, characterized in that, The updated noise variance value is: in, Indicates time The observation data in the first The weighted residual statistics formed under the conditions of each sub-model .

10. The multi-model linear time-varying process industry system identification method according to claim 7, characterized in that, The value of the updated sub-model effective width parameter is: ； This represents the step size in gradient descent. Represents the current set of values ​​for the effective width parameter.

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