A subspace band-constrained based system identification method

Abstract

The application discloses a system identification method based on subspace band constraint, and steps are as follows: collecting input and output dynamic data of a system and preprocessing; based on the preprocessed input and output dynamic data, a black box state space model containing the input and output dynamic data is identified through a subspace system identification method, and a state space matrix is obtained, the state space matrix comprises a state matrix, an input matrix and an output matrix; a constraint about the state space matrix is constructed through prior knowledge; the state matrix and the input matrix in the state space matrix are kept unchanged, and the output matrix satisfying the constraint is calculated, or the state matrix and the output matrix are kept unchanged, and the input matrix satisfying the constraint is calculated; a step response model of the system is calculated based on the state space matrix satisfying the constraint; and the step response model is fitted into a low-order transfer function model. Through the constraint using the prior knowledge in the subspace identification method, the identification accuracy is improved.

Description

Technical Field

[0001] This invention relates to the field of industrial automation control system simulation technology, and specifically to a system identification method based on subspace constraints. Background Technology

[0002] System identification is a branch of modern control theory that uses the input-output time functions of a system to determine a mathematical model describing the system's behavior. The purpose of establishing a mathematical model through identification is to estimate key parameters characterizing the system's behavior, build a model that mimics the behavior of a real system, predict the future evolution of the system's output using currently measurable input and output data, and design controllers.

[0003] The accuracy of subspace identification largely depends on the measurement data. In many cases, due to minor noise interference or insufficient input stimulation, the obtained sample data and the information it contains may not be accurate enough, potentially leading to a biased identification model. To improve the accuracy of the identification results, the most intuitive approach is to incorporate any known system information, such as stability, static gain, step response, and system structure. This known system information is collectively referred to as prior information, and all prior knowledge can be obtained through system operation experience, basic principles, and historical data analysis.

[0004] Prior knowledge refers to existing knowledge about the laws governing the motion of a system, data, and other aspects. This knowledge plays a crucial role in selecting model structure, designing experiments, and determining identification methods. Models used for different purposes can vary significantly. For example, the model structure can be determined from basic physical laws (Newton's laws, Kirchhoff's laws, the law of conservation of matter, etc.), establishing the relationships between the variables under study. If the knowledge in this area is complete, the model's structure and parameters (at least in principle) can be determined.

[0005] In existing technologies, subspace system identification often employs purely black-box identification methods, relying entirely on historical input and output data. For example, the patent "High-Speed ​​Train Interaction Estimation Method, System, and Readable Medium Based on System Modeling" discloses a model identification method combining parameter estimation and state estimation based on causal constraints. This method obtains a state-space model by mapping the relationship between output data. Such black-box algorithms often forcibly fit the noise or disturbances in the data, leading to significant errors between the identified model and the real system. The model may violate the true physical characteristics of the system, or even exhibit divergence.

[0006] The paper "Improved Closed-loop Subspace Identification with Prior Information" discloses a constrained identification method based on the PARSIM-E (Error Estimation-based) subspace identification method. This method first constructs the corresponding Hankel matrices using the input and output data. Then, it expands the state-space matrix to obtain the equations between the input-output Hankel matrices and the impulse response matrix. Constraint equations are constructed based on the correlation between the impulse response matrix and the system's physical characteristics. The impulse response matrix is ​​then solved using constrained least squares, and SVD decomposition is performed on the impulse response matrix to finally obtain the state-space matrix. This method incorporates physical characteristic constraints, but accuracy loss occurs during SVD decomposition, potentially leading to the final calculated state-space matrix not fully satisfying the constraints.

[0007] The patent "Constrained system identification for incorporation of priori knowledge" discloses a constrained identification method based on an existing state space matrix. This method uses the identified state space matrix and the corresponding step response constraints to construct a constrained optimization equation set, thereby solving for a new state space matrix that satisfies the optimization. However, this method requires pre-determining a relaxation matrix; an inappropriate relaxation matrix may lead to a significant difference between the optimized model and the actual model, and the resulting state space model is relatively abstract and cannot be intuitively correlated with the physical characteristics of the system.

[0008] The patent "Computer Apparatus And Method using Model Structure Information of Model Predictive Control" discloses a constrained identification method based on the state-space matrix. This method transforms the state-space model into a transfer function gain matrix G. The relationship between the gain matrix and the system's prior knowledge, along with the relationship between the state-space matrix and the transfer function gain matrix, simultaneously constitutes a constraint matrix with respect to the state-space matrix. However, this method is complex in constraint design, leading to a smaller solution range and increased difficulty. While it adds constraints through the transfer function gain matrix G, this gain represents the model's gain over infinite time, failing to incorporate time-dimensional information. In industrial settings, the focus is often only on the model gain over a finite time interval, rather than the model's gain over infinite time.

[0009] In summary, the identification models obtained by existing system identification methods may have problems such as large errors with the real system, loss of accuracy, inability of the obtained state-space model to intuitively correspond to the physical characteristics of the system, or inability to incorporate time dimension information, resulting in insufficient accuracy. Summary of the Invention

[0010] The technical problem to be solved by this invention is to provide a system identification method based on subspace constraints, which constructs a simulation model of the system by using historical input and output data and on-site prior knowledge, thereby achieving high-precision simulation of the system.

[0011] To address the aforementioned technical problems, the present invention provides the following technical solution.

[0012] This invention provides a system identification method based on subspace constraints, comprising the following steps:

[0013] S101: Collect the system's input and output dynamic data and preprocess the data;

[0014] S102: Based on the preprocessed input and output dynamic data, a black-box state-space model containing the input and output dynamic data is identified by the subspace system identification method, and a state-space matrix is ​​obtained. The state-space matrix includes a state matrix, an input matrix, an output matrix, and a feedforward matrix.

[0015] S103: Construct constraints on the state space matrix using prior knowledge;

[0016] S104: Keep the state matrix and input matrix in the state space matrix unchanged, and calculate the output matrix that satisfies the constraints; or keep the state matrix and output matrix unchanged, and calculate the input matrix that satisfies the constraints.

[0017] S105: Step response model of a system based on constrained state-space matrix calculation;

[0018] S106: Fit the step response model to a low-order transfer function model using a fitting function.

[0019] Furthermore, in step S101, data is acquired when the system is subjected to a continuous excitation signal. The preprocessing of the data includes filtering the data and denoising the filtered data.

[0020] Furthermore, the state-space model is a MISO-VARX model, wherein the steps for obtaining the state-space matrix include:

[0021] S201: Calculate the Markov parameters of the MISO-VARX model based on input and output data;

[0022] S202: Constructing matrix Φ based on Markov parameters y Φ u ,

[0023] S203: Based on matrix Φ y Φ u The Hankel matrix Y is constructed by iteratively calculating the predicted values. pred ;

[0024] S204: Based on Hankel matrix Y pred Perform SVD decomposition to obtain the state sequence X, and then transpose the state sequence X to obtain the transposed state sequence X. 1,N ;

[0025] S205: Based on the transposed state sequence X 1,N Given the original input and output data u(t) and y(t) of the system, the state-space equation of the system is constructed and the state-space matrix is ​​obtained by solving it.

[0026] Furthermore, step S103 specifically includes:

[0027] Based on the correlation between the impact response matrix and the physical characteristics of the system, the impact response matrix of the system is obtained;

[0028] Express prior knowledge as equations or inequalities concerning the impulse response matrix;

[0029] Based on the relationship between the impact response matrix and the state space matrix, we construct an equality or inequality relationship between prior knowledge and the state space matrix as a constraint.

[0030] The constraints include static gain equality constraints and static gain direction constraints.

[0031] Furthermore, when keeping the state matrix and input matrix in the state space matrix unchanged, and constraining the output matrix, the static gain equality constraint is:

[0032]

[0033] Where K ss The static gain matrix; g i Here, k represents the impact response parameters; k is the number of sampling points within the set steady-state time Tss; l is the output dimension.

[0034] Based on the relationship between the impulse response matrix and the state space matrix, construct the following equality constraints regarding the output matrix:

[0035] A eq C T =b eq

[0036] in:

[0037]

[0038] Where matrix A is the state matrix; B is the input matrix; C is the output matrix; A eq b is the constraint matrix for the constraint equality; eq The constraint vector is the constraint equality vector;

[0039] Static gain direction constraints are used when the gain relationship between the system's input and output is positive and negative. When the gain is negative, an inequality is constructed:

[0040] AC T ≤b

[0041] in:

[0042]

[0043] If the gain is positive, multiply both sides of the inequality by -1 to obtain the new constraint matrix:

[0044]

[0045] Where b is the constraint vector of the constraint inequality.

[0046] Furthermore, in step S104, based on the predicted output Y pred With the transposed state sequence X 1，N The output matrix C satisfying the constraints is obtained from the relationship between Y, where the output matrix C satisfies the following relationship, Y 1，N For Y pred The first to the Nth point,

[0047] constraint equation

[0048] y 1，N =CX 1，N Satisfying A eq C T =b eq

[0049] Constraint inequalities

[0050] y 1，N =CX 1，N Satisfying AC T ≤b

[0051] Then, a new state space matrix that satisfies the prior knowledge constraints is obtained.

[0052] Furthermore, in step S105, based on the state-space model described in the form of innovation, the step response of the system is solved recursively, specifically as follows:

[0053] The state-space model based on the constrained state-space matrix is ​​expressed in innovative form as follows:

[0054]

[0055] Where, x k ∈R n For state vectors, For the input vector, For the output vector, The sequence is a zero-mean white noise sequence; it is assumed that there is no noise influence when solving for the step response, e k =0, the state-space model discards the element containing e. k The term; when x0 = 0, through u k Derivation of all y k and x k y during steady state k This is the step response, from which the step response model of the system is obtained.

[0056] Furthermore, the obtained step response model is smoothed by filtering.

[0057] Furthermore, in step S106, the transfer function of the first-order delay system used as the fitting function is described as follows:

[0058]

[0059] Where s is the Laplace operator, T is the time constant, K is the static gain, U(s) is the deformation of the model input in the Laplace s domain, Y(s) is the deformation of the model output in the Laplace s domain, and τ is the model lag time;

[0060] The system gain K is obtained directly from the steady-state value y(∞) at the last point of the step response, where U is the model input magnitude:

[0061] K=y(∞) / U

[0062] The time constant T and the model lag time τ are solved using data from two observation points (t1, y(t1)) and (t2, y(t2)) in the step response model as follows:

[0063]

[0064]

[0065] Furthermore, the data (t) of multiple observation points are determined from the dynamic part of the step response satisfying the relation (t-τ) / T = c (0 < c < 3). k y(t) kThe system estimates multiple values ​​of T and τ, and then averages them to obtain the final values ​​of T and τ, thereby determining the low-order transfer function model of the system.

[0066] Compared with existing technologies, the system identification method based on subspace constraints provided by this invention has the following advantages:

[0067] This invention identifies a black-box state-space model containing dynamic input and output data using a subspace system identification method, obtaining the state-space matrix. Then, prior knowledge is used to constrain the state-space matrix, and the constrained state-space matrix is ​​recalculated. Based on the constrained state-space matrix, the step response model of the model is calculated. Finally, a low-order transfer function is used to fit the identified high-order transfer function model to a first-order transfer function, achieving the identification of simulation models for industrial systems and obtaining a simple and easy-to-use low-order system. Specific beneficial effects are as follows:

[0068] (1) In this invention, prior knowledge is used to constrain the identification of the black box state space model, thereby improving the accuracy of the identified model.

[0069] (2) When calculating the state space matrix that meets the constraints, this invention only recalculates the output matrix while keeping the state matrix and input matrix unchanged, or only recalculates the input matrix while keeping the state matrix and output matrix unchanged. This invention can achieve prior knowledge constraints on static gain without changing the dynamic characteristics of the model, thereby preserving the system information contained in the input and output data to the maximum extent.

[0070] (3) This invention reflects the relationship between gain and time by relating the parameters of the impulse response matrix to the gain in the time domain, so that the solved model can reach the constraint conditions within the steady-state time defined by the engineer, and implements the constraint on the gain direction through constraint inequalities. The constraint design is simple and the constraint solution has low complexity. At the same time, it realizes the constraint on the gain direction in steady-state time, which increases the applicability of the method.

[0071] (4) After solving the state space matrix with constraints, the present invention does not need to perform matrix decomposition on the obtained state space model, ensuring that there is no loss of constraint accuracy after identification.

[0072] (5) The present invention fits the identified high-order transfer function model into a first-order transfer function by using a low-order transfer function, which reduces the impact of noise and disturbance in the data on the model, while improving the readability of the model and making it easier for on-site engineers to understand the identified model.

[0073] (6) This invention fits the inverse part of the step response curve in a non-minimum phase system as a hysteresis by fitting a low-order transfer function, which can effectively improve the control performance of the controller using the model. Attached Figure Description

[0074] Figure 1 A flowchart illustrating a subspace-based constrained system identification method provided in an embodiment of the present invention;

[0075] Figure 2 This is a flowchart of the PBSID subspace identification method in an embodiment of the present invention. Detailed Implementation

[0076] To clearly illustrate the objectives, technical solutions, and advantages of this invention, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of this invention. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0077] like Figure 1 As shown, the system identification method based on subspace constraints of the present invention includes the following steps:

[0078] S101. Collect the system's input and output dynamic data and preprocess the data;

[0079] To achieve better identification results and obtain prior knowledge of the system, data acquisition from actual industrial systems requires that the system be subjected to continuous excitation signals. For example, generalized binary noise signals (GBNS) or pseudo-binary random signals (PRBS) can be used to obtain dynamic process data of the system. Where u(t) and y(t) are the input and output data of the dynamic process, respectively, N is the data volume, ts is the time period of uniform sampling, and the dimensions of the system input and output are n. u ,n y .

[0080] The data preprocessing includes data filtering and noise reduction. During data filtering, the excitation signal should be retained, while longer steady-state data segments should be removed. After selecting suitable data segments, noise reduction is performed using methods such as mean filtering and wavelet filtering to reduce noise in the data.

[0081] S102. Based on the preprocessed input and output dynamic data, the black-box state-space model containing the input and output dynamic data is identified by the subspace system identification method, and the state-space matrix is ​​obtained.

[0082] Subspace identification methods include the Canonical Variable Analysis-based Subspace Algorithm (CVA), N4SID, and the Predictor-Based Subspace Identification (PBSID) method. In this embodiment, the Predictor-Based Subspace Identification (PBSID) method is used for identification.

[0083] Based on the input-output dynamic data collected in step S101, a black-box state-space model containing the input-output dynamic data is identified using the PBSID subspace identification method. In this embodiment, the black-box state-space model adopts the MISO-VARX model in the ARX model. This state-space model contains the system characteristics exhibited in the input-output data. During identification, the output is split and identified separately. MISO (Multiple-Input Single-Output) is used to identify all outputs of the MIMO (Multiple-Input Multiple-Output) system separately, avoiding mutual interference between all outputs when using MIMO (Multiple-Input Multiple-Output) identification, thereby improving the accuracy of identification and reducing the influence of other outputs.

[0084] See Figure 2 The specific implementation method of the PBSID subspace identification method is as follows:

[0085] S201: Calculate the Markov parameters of the MISO-VARX model based on input and output data;

[0086] The system can also be described using the MISO VARX (controlled autoregressive) model:

[0087] A(z)y k =B(z)u k +e k

[0088] z -1 It is a unit shift operator, and each output y is approximately equal to these n. y One output, n u A linear combination of historical data from the past p time points of input, where

[0089]

[0090] The Markov parameters a of the MISO-VARX model can be obtained using the least squares method. i b i .

[0091] S202: Construct matrix Φ based on the Markov parameters obtained in S201 y Φ u ;

[0092] Constructing matrix Φ based on Markov parameters y Φ u for:

[0093]

[0094] S203: Based on matrix Φ y Φ u The Hankel matrix Y is constructed by iteratively calculating the predicted values. pred ;

[0095] Based on matrix Φ y Φ u The nth moment in the future y Each output value can be approximated by the MISO-VARX model structure prediction, and assuming that the past time domain p is appropriately selected, it can be extended to N time samples to obtain:

[0096]

[0097] in,

[0098] This allows for iterative prediction of the output value at each of the next p time steps, and the construction of the Hankel matrix Y from these values. pred .

[0099] S204: Based on the constructed Hankel matrix Y pred Performing Singular Value Decomposition (SVD) yields the state sequence X, as follows:

[0100] For the transposed Hankel matrix Y pred Perform SVD decomposition.

[0101]

[0102] Where U∈R N×N , n is the order of the system and also the number of large singular values, U n ∈R N ×n , ∑ n ∈R n×n It is a diagonal matrix. The state sequence X is calculated as follows:

[0103] X = Y pred V n ∈R N×n

[0104] Transpose the state sequence X to obtain the transposed state sequence X. 1，N ,

[0105] X1,N =X T ∈R n×N

[0106] S205: Based on the state sequence X and the original data u(t) and y(t) of the system input and output, construct the state space equation of the system and solve the state space matrix;

[0107] The system can be described by a state-space model in the form of innovations as follows:

[0108]

[0109] Where A∈R n×n , These are the state space matrices of corresponding dimensions: A is the state matrix; B is the input matrix, controlling the interaction between the input and the state matrix; C is the output matrix, controlling the interaction between the state matrix and the output; D is the feedforward matrix; K is the Kalman gain matrix; x k ∈R n For state vectors, For the input vector, For the output vector, The sequence is a zero-mean white noise sequence, and ny is 1 for MISO identification.

[0110] The transposed state sequence X is obtained by transposing the state sequence X. 1,N Given the original input and output data u(t) and y(t) of the system, the state-space equation of the system is constructed as follows:

[0111]

[0112] The equation is solved by methods such as least squares, QR decomposition, or singular value decomposition to obtain the identified state space matrix A, B, C, D, K, where matrix D is the feedforward matrix. In this embodiment, it is assumed that the system has no direct path from input to output, so this matrix is ​​the zero matrix.

[0113] S103: Construct constraints on the state space matrix using prior knowledge;

[0114] Based on the state-space model of the innovation form in S205 above, the extended state-space matrix is ​​obtained recursively:

[0115]

[0116]

[0117] The system's state-space matrix and impact response matrix are related, and the impact response matrix can be described in a form related to the state-space matrix as follows:

[0118]

[0119] Where H f Let g be the impact response matrix. i These are the impact response parameters.

[0120] The system's impact response matrix is ​​related to its physical characteristics. Based on field analysis of the system mechanism or long-term operational experience, process engineers can obtain information about the system's static gain K. ss With the time T to reach that gain ss The prior knowledge can be described as equations or inequalities concerning the impact response matrix; the impact response matrix can be described in a form related to the state space matrix, which can be further extended to equations or inequalities between prior knowledge and the state space matrix. This embodiment directly constructs constraints through the relationship between the impact response matrix and prior knowledge, simplifying constraint design and reducing the difficulty of solving the problem.

[0121] In this model, matrices A and B primarily determine the dynamic part of the identified state-space model. These dynamics are mainly determined by the input and output data, and prior knowledge is difficult to ascertain. Therefore, when solving for constraints, matrices A and B remain unchanged, and only the parameters of matrix C are solved. Specific constraints can take the following forms:

[0122] Static gain equality constraints:

[0123] According to existing research (Alenany et al., 2011), information related to the low-frequency characteristics of a system is often not fully revealed in the available input-output data. Therefore, prior knowledge of the static gain is crucial for improving model accuracy; the static gain of the system is equal to the sum of the impulse response parameters. Assume the system's steady-state time is set to T. ss The number of internal sampling points is k, f is the number of predicted impulse response parameters, f should be equal to 1.1 times k, and 1 is the output dimension. The constraint can be expressed as:

[0124]

[0125] Where K ss The static gain matrix is:

[0126]

[0127] Static gain can be combined with the physical characteristics of the system, and the physical characteristics of the system can represent the system gain. This allows us to describe the relationship between the system gain and the impulse response parameters. The state-space matrix of the system is related to the impulse response matrix, thus deriving the relationship between the static gain and the state-space matrix. In other words, from the relationship between the impulse response matrix parameters and the state-space matrix, we can construct equality constraints about matrix C:

[0128] A eq C T =b eq

[0129] Among them, A eq b is the constraint matrix for the constraint equality; eq The constraint vector is the constraint equality vector;

[0130]

[0131] By relating the parameters of the impulse response matrix to the system gain in the time domain, the relationship between system gain and time is demonstrated, enabling the solved model to meet the constraints within the steady-state time defined by the engineers.

[0132] Static gain direction constraint:

[0133] In real-world systems, there are certain inputs and outputs where the exact proportional relationship cannot be determined. The gain relationship between input and output is either positive or negative. If the relationship is positive, the model gain is positive (increased input leads to increased output); if the relationship is negative, the model gain is negative (increased input leads to decreased output). However, during black-box model identification, data perturbations and noise can cause the final identified model to exhibit completely opposite results for such inputs and outputs. For this type of input-output data, constraints can be implemented by constructing constraint inequalities using the impulse response matrix. The implementation method is as follows: When the positive and negative effects of the system input and output are determined, the impulse response parameter g... i The sign of the gain can also be determined. Inequality constraints can be implemented by using the sign of the impact response parameters. For example, an inequality can be constructed when the gain is negative:

[0134] AC T ≤b

[0135] Where b is the constraint vector of the constraint inequality.

[0136]

[0137] If the gain is positive, multiply both sides of the inequality by -1 to obtain a new constraint matrix:

[0138]

[0139] By constructing inequality constraints, gain direction constraints can be achieved. The prior knowledge corresponding to such constraints is often easier to determine, and the demand is greater in practical applications, thus increasing the applicability of the method.

[0140] S104: Keeping the state space matrices A and B unchanged, calculate the C matrix that satisfies the constraints;

[0141] Based on the constraint matrix constructed in S103, the predicted output Y can be determined. pred With state sequence X 1，N The relationship between the states is determined using constrained least squares to obtain the constrained matrix C. Only the values ​​of the state-space matrix C are changed, without altering the values ​​of A and B. This achieves prior knowledge constraints on the static gain without changing the dynamic characteristics of the identified model, thus preserving the system information contained in the input and output data to the maximum extent. C satisfies the following relationship:

[0142] constraint equation

[0143] y 1，N =CX 1，N Satisfying A eq C T =b eq

[0144] Constraint inequalities

[0145] y 1，N =CX 1，N Satisfying AC T ≤b

[0146] Among them, Y 1，N For Y pred The first to the Nth points.

[0147] The above constrained relational expression satisfies the general form of the constrained least squares algorithm. Matrix C can be directly obtained using the constrained least squares algorithm, or other constrained solution methods, such as quadratic programming, can be used. This yields a new state-space matrix. This model satisfies the constraints of prior knowledge and also includes dynamic information from the input and output data. Furthermore, subsequent steps do not require matrix decomposition of the state-space model, ensuring no loss of constraint accuracy after identification.

[0148] It should be noted that in this step, keeping A and C unchanged and only solving for the parameters of matrix B can achieve the same effect. This is because matrix C controls the relationship between the state matrix and the output, while matrix B controls the relationship between the input and the state matrix. Both have similar effects on the system, so recalculating the parameters of matrix C or matrix B through constraints can achieve the same purpose.

[0149] S105: Step response model of a system based on constrained state-space matrix calculation;

[0150] The state-space model can be described in terms of innovation, and the step response of the system can be solved recursively, specifically as follows:

[0151] The innovation form of the state-space model based on the state-space matrix satisfying the constraints is:

[0152]

[0153] When solving for the step response, it is assumed that there is no noise influence, and the state-space model is e. k The model with =0 is equivalent to discarding the part containing e. k The term, when initially x0 = 0, is passed through u k All y can be recursively derived k and x k y during steady state k This is the step response, from which we obtain the step response model of the system. The length of the step response is the pre-given constrained steady-state time Tss.

[0154] It should be noted that, since the subspace will inevitably fit some noise when identifying the black box model, the step response calculated by the black box model will often have slight jitter in the early dynamic part, which is not conducive to the use of control and simulation models. Therefore, after calculating the original step response, the step response model is smoothed by mean filtering. The smoothing factor is selected by default as 3. The smoothed step response model is closer to the real system.

[0155] S106: Fit the step response model to a low-order transfer function model by fitting the function;

[0156] The step response model obtained in S105 is fitted into a low-order transfer function model using a fitting function. The specific implementation steps are as follows:

[0157] The transfer function of a first-order delay system used as a fitting function is described as follows:

[0158]

[0159] Where s is the Laplace operator, T is the time constant, K is the static gain, U(s) is the deformation of the model input in the Laplace s domain, and Y(s) is the deformation of the model output in the Laplace s domain;

[0160] The parameters in the transfer function of the first-order delay system above are solved using data from two observation points (t1, y(t1)) and (t2, y(t2)) in the step response model. The specific solution process is as follows:

[0161] The step response of a first-order delayed system is described as follows:

[0162] y(t)=KU(1-e -(t-τ) / T )

[0163] Where τ is the model lag time, U is the model input amplitude, and the system gain K is directly obtained from the steady-state value of the last point of the step response:

[0164] K=y(∞) / U

[0165] The system's step response can be converted into a dimensionless form, resulting in the dimensionless step response:

[0166]

[0167] Where t is greater than τ.

[0168] Data from two observation points (t1, y(t1)) and (t2, y(t2)) are collected and substituted into the dimensionless step response equation to obtain:

[0169]

[0170]

[0171] Solving the system of equations yields the following parameter estimates for the transfer function of the first-order delayed system:

[0172]

[0173]

[0174] Following the same method, and to ensure that the selected observation points remain within the dynamic part of the step response, generally, when (t-τ) / T > 4, the system has entered the steady-state phase. At this point, the selected points all have the same output, making such observation points useless for the fitting algorithm. Therefore, to ensure that the selected points are within the dynamic part of the step response, the response observation data y(t) corresponding to the observation time tk (k = 1, 2, ..., L) that satisfies the relationship (t-τ) / T = c (0 < c < 3) is selected. k That is, determining data (t) from multiple observation points. k y(t) k Then, by using the formulas for estimating T and τ obtained above, multiple estimates of T and τ are obtained, and the average of these estimates is taken to finally obtain the values ​​of T and τ, thereby determining the low-order transfer function model of the system.

[0175] It should be noted that the reason for using a fitting function to reduce the order of the step response model is as follows: First, the state space matrix obtained through subspace identification has a high order. A higher system order will fit noise and disturbances in the data, resulting in an uneven step response curve. Furthermore, the matrix model of the state space model is not easily understood by field engineers. By fitting the identified high-order transfer function model to a first-order transfer function using a low-order transfer function fitting function, the impact of noise and disturbances in the data on the model is reduced, while improving the model's readability and making it easier for field engineers to understand. Second, models identified by traditional subspace identification methods often exhibit non-minimum phase states, leading to a decrease in the control performance of controllers using this model. By fitting the inverse part of the step response curve in a non-minimum phase system to a hysteresis state using a fitting function, the control performance of controllers using this model can be effectively improved. In summary, this invention uses a fitting function to achieve noise reduction and removal of non-minimum phase states from the identified model, thereby improving the model's usability.

[0176] The above description is merely a preferred embodiment of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and various obvious changes, readjustments, and substitutions can be made without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments, and may include many other equivalent embodiments without departing from the concept of the present invention, the scope of which is determined by the scope of the appended claims.

Claims

1. A system identification method based on subspace constraints, characterized in that, Includes the following steps: S101: Collect the system's input and output dynamic data and preprocess the data; S102: Based on the preprocessed input and output dynamic data, a black-box state-space model containing the input and output dynamic data is identified using the subspace system identification method, and a state-space matrix is ​​obtained. The state-space matrix includes a state matrix, an input matrix, an output matrix, and a feedforward matrix. The steps for obtaining the state-space matrix include: calculating the Markov parameters of the MISO-VARX model based on the input and output data; constructing a matrix based on the Markov parameters. , Based on matrix , The Hankel matrix Y is constructed by iteratively calculating the predicted values. pred Based on the Hankel matrix Perform SVD decomposition to obtain the state sequence X, and then transpose the state sequence X to obtain the transposed state sequence. Based on transposed state sequence and the raw data of system input and output. , The state-space equations of the system are constructed and the state-space matrix is ​​obtained by solving them. S103: Construct constraints on the state space matrix through prior knowledge, specifically including obtaining the system's impact response matrix based on the correlation between the impact response matrix and the system's physical characteristics; expressing prior knowledge as an equation or inequality about the impact response matrix; and constructing an equation or inequality relationship between prior knowledge and the state space matrix as a constraint based on the correlation between the impact response matrix and the state space matrix. S104: Keep the state matrix and input matrix in the state space matrix unchanged, and calculate the output matrix that satisfies the constraints; or keep the state matrix and output matrix unchanged, and calculate the input matrix that satisfies the constraints. S105: Calculate the step response model of the system based on the state space matrix that satisfies the constraints. Specifically, it includes solving the step response of the system by recursion based on the state space model described in the form of innovation, and then obtaining the step response model of the system. S106: Fit the step response model to a low-order transfer function model using a fitting function.

2. The system identification method based on subspace constraints according to claim 1, characterized in that, In step S101, data is acquired when the system is subjected to a continuous excitation signal. The preprocessing of the data includes filtering the data and denoising the filtered data.

3. The system identification method based on subspace constraints according to claim 1, characterized in that, The constraints include static gain equality constraints and static gain direction constraints.

4. The system identification method based on subspace constraints according to claim 3, characterized in that, When keeping the state matrix and input matrix in the state space matrix unchanged, and constraining the output matrix, the static gain equality constraint is: Where K ss The static gain matrix; g i Here, k represents the impact response parameters; k is the number of sampling points within the set steady-state time Tss; l is the output dimension. Based on the relationship between the impulse response matrix and the state space matrix, construct the following equality constraints regarding the output matrix: in: Where matrix A is the state matrix; B is the input matrix; C is the output matrix; A eq b is the constraint matrix for the constraint equality; eq The constraint vector is the constraint equality vector; Static gain direction constraints are used when the gain relationship between the system's input and output is positive and negative. When the gain is negative, an inequality is constructed: in: If the gain is positive, multiply both sides of the inequality by -1 to obtain the new constraint matrix: Where b is the constraint vector of the constraint inequality.

5. The system identification method based on subspace constraints according to claim 4, characterized in that, In step S104, based on the predicted output Y pred With transposed state sequence The output matrix C satisfying the constraints is obtained from the relationship between Y, where the output matrix C satisfies the following relationship, Y 1,N For Y pred The first to the Nth point; constraint equation Constraint inequalities Then, a new state space matrix that satisfies the prior knowledge constraints is obtained.

6. The system identification method based on subspace constraints according to claim 5, characterized in that, Step S105 specifically includes: The state-space model based on the constrained state-space matrix is ​​expressed in innovative form as follows: in, For state vectors, For the input vector, For the output vector, The sequence is a zero-mean white noise sequence; it is assumed that there is no noise influence when solving for the step response, e k =0, the state-space model discards the element e. k The term; when x0=0, through u k Derivation of all y k and x k y during steady state k This is the step response, from which the step response model of the system is obtained.

7. The system identification method based on subspace constraints according to claim 6, characterized in that, The obtained step response model is smoothed by filtering.

8. The system identification method based on subspace constraints according to claim 1, characterized in that, In step S106, the transfer function of the first-order delay system used as the fitting function is described as follows: Where s is the Laplace operator, T is the time constant, K is the static gain, U(s) is the deformation of the model input in the Laplace s domain, Y(s) is the deformation of the model output in the Laplace s domain, and τ is the model lag time; The system gain K is obtained directly from the steady-state value y(∞) at the last point of the step response, where U is the model input magnitude: The time constant T and the model lag time τ are solved using data from two observation points (t1, y(t1)) and (t2, y(t2)) in the step response model as follows: 。 9. The system identification method based on subspace constraints according to claim 8, characterized in that, Determine the data of multiple observation points for the dynamic part of the step response that satisfies the relationship (t - τ) / T = c (0 < c < 3) (t k , y(t k )). Estimate multiple T and τ estimates, and take their average to finally obtain the values of T and τ, so as to determine the low-order transfer function model of the system.

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