Modeling and estimating method of ammonia desulfurization system based on particle filtering under missing data

Abstract

The invention discloses a modeling and estimating method of an ammonia desulfurization system based on particle filtering under missing data, belongs to the technical field of modeling of the ammonia desulfurization system, and solves the technical problems of modeling the ammonia desulfurization system and accurately estimating unknown parameters and states of the system under the condition of incomplete data sampling. According to the technical scheme, the method comprises the following steps that 1, an ammonia desulfurization nonlinear system model based on a Hammerstein output error autoregression model is constructed, and all information vectors and model description of an identification system are obtained; and 2) constructing a gradient iteration identification model method flow based on particle filtering to accurately estimate unknown parameters of the ammonia process desulfurization nonlinear model under missing data. The method has the beneficial effects that the unknown parameters of the ammonia desulfurization nonlinear system of particle filtering under missing data are accurately estimated through approximation estimation.

Description

Technical Field

[0001] The present invention belongs to the technical field of modeling of ammonia desulfurization systems, and in particular relates to the research and application of a nonlinear system estimation method of ammonia desulfurization using particle filtering under missing data. Background Art

[0002] Ammonia desulfurization technology is one of the desulfurization methods. This process can not only desulfurize efficiently but also partially remove nitrogen oxides in flue gas. The byproduct is ammonium sulfate, which realizes resource recycling. The advantage of this method is that sulfur dioxide in pollutants can be recycled into ammonium sulfate (urea). Urea is a high-value commodity with broad prospects in agricultural production. However, in the current field of ammonia desulfurization, the desulfurization technology based on coke oven flue gas is still in the research stage and is not as mature as that in the field of coal-fired power plants. The main means of controlling pollutants is desulfurization at the end of the coke oven. In the ammonia desulfurization system, the pH value of the absorption liquid can reflect the effect of ammonia desulfurization, which represents the acidity and alkalinity of the absorption liquid in the desulfurization tower. The pH value of the absorption liquid is required to be between 5.5 and 6.5. If the value is too low, the desulfurization efficiency will be reduced. If the value is too high, ammonia will escape, the consumption of ammonia water will increase, and aerosol phenomenon will be triggered, leading to secondary pollution. Since the pH value in the neutralization reaction of ammonia desulfurization has the characteristics of strong nonlinearity, time lag and variable gain, it is difficult to model it.

[0003] With the progress of science, control theory and control engineering have been widely developed and applied. Among them, the application of system identification is becoming more and more extensive. At present, the target of most system identification algorithms is a single-rate system with the same input and output data sampling frequency. However, in most practical systems, due to hardware conditions, it may not be possible to obtain sampling data with the same frequency. Moreover, inconsistent sampling frequencies will lead to the loss of system information. Therefore, the study of missing data systems has important theoretical significance.

[0004] In recent years, parameter estimation of missing data systems has been active and has attracted the attention of many scholars. For example, the idea of ​​the paper Robust control of linear time-invariant plants using periodic compensation is that in the output sampling period, one output and multiple inputs can be obtained, and these multiple inputs form an enhanced input, thereby obtaining a time-invariant system, so that the estimation method can be directly applied to the missing data system. The introduction of lifting technology does not cause the loss of sampled data, but it produces more unknown parameters, resulting in a significant increase in the amount of calculation. The paper Highly efficient identification methods for dual rate Hammerstein systems uses polynomial transformation technology to identify the differential equation model of missing data. However, the model converted by polynomial transformation technology also has the problems of large amount of calculation and insufficient estimation accuracy. The paper An efficient hierarchical identification method for general dual-rate sampled-data systems uses the estimated output of the auxiliary model to replace the missing output of the system, and obtains a more effective parameter estimation. However, in some nonlinear systems, the effectiveness of the auxiliary model may be reduced due to the lack of measurable variables. In summary, more efficient methods need to be proposed to solve the missing data problem in order to estimate parameters more accurately.

[0005] With the improvement of filtering technology, we find that unknown states and variables in the system can also be approximated by particle filtering. Because each particle can adjust its weight according to the currently given data. Based on this feature, the paper Gradient iterative algorithm for dual-rate nonlinear systems based on a novel particle filter applies gradient iteration to dual-rate nonlinear models through a novel particle filter; the paper Particle filtering based parameter estimation for systems with output-errortype model structures derives a recursive least squares method based on particle filtering to identify the output error system. Referring to the above literature, it can be concluded that particle filtering can make full use of sampled data and will not add any unknown parameters. Therefore, the present invention extends it to the ammonia desulfurization nonlinear system with missing data. Therefore, this patent proposes a gradient iterative identification method based on particle filtering, which can directly obtain a more accurate estimation model from an incomplete sampling system, which has important research significance in the field of system identification. Summary of the invention

[0006] The purpose of the present invention is to provide a method for estimating the unknown parameters of an ammonia desulfurization nonlinear system with particle filtering under missing data. First, an ammonia desulfurization nonlinear system based on the Hammerstein output error autoregressive model is established, and then a gradient iterative identification model method based on particle filtering is proposed to accurately estimate the unknown parameters of the ammonia desulfurization nonlinear model. The present invention takes the ammonia desulfurization nonlinear system as the research object, studies a ammonia desulfurization nonlinear system based on the Hammerstein output error autoregressive model, and provides a new solution for estimating the unknown parameters of the ammonia desulfurization nonlinear system with particle filtering under missing data, applies the gradient iterative identification model based on particle filtering to the ammonia desulfurization nonlinear system, and combines the gradient iterative identification of particle filtering with the ammonia desulfurization nonlinear system, and realizes the accurate estimation of the unknown parameters of the ammonia desulfurization nonlinear system with particle filtering under missing data through approximate estimation.

[0007] The idea of ​​the present invention is: since the target of most system identification algorithms is a single-rate system with the same input and output data sampling frequency, however, in most actual systems, due to hardware and other conditions, it may not be possible to obtain sampling data of the same frequency, which leads to the loss of system information and inconsistent sampling frequencies. Multi-rate sampling data cannot be used directly due to the inaccuracy of identification and estimation caused by missing data. Therefore, the present invention uses particle filtering to approximate the unknown states and variables in the multi-rate system. Each particle can adjust its weight according to the currently given data, so a gradient iterative identification model method based on particle filtering is constructed to accurately estimate the unknown parameters of the nonlinear model of ammonia desulfurization under missing data.

[0008] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is specifically: a modeling and estimation method of an ammonia desulfurization system using particle filtering under missing data, comprising the following steps:

[0009] Step 1) construct a model of the missing data ammonia desulfurization system based on the Hammerstein output error autoregressive model, and obtain all information vectors and the model description of the identification system.

[0010] Step 2) A process for accurately estimating the unknown parameters of the missing data ammonia desulfurization model by constructing a gradient iterative identification model based on particle filtering.

[0011] Step 2-1) Initialize the gradient iteration method based on particle filtering, set the iteration initial value k to 1, and the initial time t to 0;

[0012] Step 2-2) collecting measurement data of the input, i.e., ammonia flow rate, and the output, i.e., pH value of the absorption liquid, of the Hammerstein output error autoregressive system of ammonia flue gas desulfurization under missing data;

[0013] Step 2-3) Form an estimate of the kth iteration of the noise-free information vector of the particle filtered system The k-th iteration estimate of the noise information vector after particle filtering The output information vector of the entire missing data ammonia desulfurization system is estimated for the kth iteration

[0014] Step 2-4) Generate particles and calculate the weight of each particle at the kth iteration according to the particle filter method, and calculate the estimate of the noise-free output intermediate variable at the kth iteration based on the particle filter;

[0015] Step 2-5) Calculate the noise variable of the kth iteration, that is, the estimate of the concentration of inlet SO2, and construct the estimated vector of the noise variable And calculate the estimate of the colored noise disturbance variable at the kth iteration

[0016] Step 2-6) Update the estimate of the particle's output vector y(dt) at the kth iteration, calculate the estimate of the output vector y(dt) of the entire system at the kth iteration, obtain the estimate of the noise-free output information vector and the noise information vector at the kth iteration, and then obtain the estimate of the output information vector of the entire system at the kth iteration and the information matrix containing all information vectors The estimate of the kth iteration;

[0017] Step 2-7) Update the estimate of the parameter vector of the entire system at iteration k;

[0018] Step 2-8) Update the estimate of the nonlinear noise-free output and the estimate of the noise output polynomial at iteration k

[0019] Step 2-9) Increase the number of iterations k by 1, and return to step 2-3);

[0020] Step 2-10) When the number of iterations k reaches the maximum number of iterations K, terminate the iteration and output the estimate of the final parameter vector

[0021] The present invention studies the parameter estimation of the Hammerstein output error autoregressive system of ammonia desulfurization under missing data, comprising the following steps:

[0022] (1-1) Construct the structure of an ammonia desulfurization Hammerstein output error autoregressive system model.

[0023] (1-2) Based on this model, the expression of the Hammerstein output error autoregressive model of ammonia desulfurization is constructed as follows:

[0024]

[0025] Among them, t is the time variable, u(t) and y(t) are the input of the system, i.e., the ammonia flow rate, and the output, i.e., the pH value of the absorption liquid, respectively. The nonlinear output is a polynomial function of the input ammonia flow rate u(t), e(t) is a polynomial function with zero mean and variance σ 2 The noise variable is the concentration of inlet SO2, s represents the unit shift operator: s -1 y(t)=y(t-1), A(s) and B(s) are polynomials of the noise-free output with respect to s, and C(s) is a polynomial of the noise output with respect to s, and they satisfy the following form:

[0026]

[0027] are the coefficients of the noise-free output polynomial A(s) with respect to s; are the coefficients of the noise-free output polynomial B(s) with respect to s; are the coefficients of the noise output polynomial C(s) with respect to s;

[0028] Non-linear output It is a known nonlinear basis A linear combination of ),Right now

[0029]

[0030] Among them, h i represents the i-th nonlinear basis function, n a 、n b 、n c 、n α A(s), B(s), C(s) and The number of parameters in , the noise-free output parameter vectors a, b, the nonlinear output parameter vector α, and the noise output parameter vector c are defined as:

[0031]

[0032] Among them, a i represents the i-th parameter in parameter vector a, b i represents the i-th parameter in parameter vector b, c i represents the i-th parameter in the parameter vector c, α i represents the i-th parameter in the parameter vector α, and the superscript T is the transpose of the vector or matrix. The calculation method of the noise-free output intermediate variable ω(t) and the colored noise interference variable γ(t) is as follows:

[0033]

[0034]

[0035] Define the noise-free output parameter vector And the parameter vector θ of the entire ammonia desulfurization system is as follows:

[0036]

[0037] Redefine the noise-free output information vector The noise information vector ψ(t) and the output information vector φ(t) of the entire system are as follows:

[0038]

[0039] ψ(t):=[-γ(t-1),…,-γ(tn c )]T

[0040]

[0041] Among them, ω(t-1),…,ω(tn a ) is the output intermediate variable at t-1,…,tn a The value of the moment, is the nonlinear output at t-1,…,tn b The value of the moment, The nonlinear output h(u(t)) is the nonlinear basis The value of the linear combination of γ(t-1),…,γ(tn c ) is the colored noise disturbance variable at t-1,…,tn c The value of the moment, is a nonlinear basis function.

[0042] (1-3) Substituting the parameter vector and information vector into (2) and (3), it can be simplified to:

[0043]

[0044] Substitute (4) and (5) into (1), and replace t in the formula with dt, where d is the sampling interval of the ammonia flue gas desulfurization output and is a positive integer greater than or equal to 2. Therefore, the measurable input is u(t), the measurable multi-rate output is y(dt), ω(dt) is the missing data noise-free output intermediate variable, γ(dt) is the missing data colored noise interference variable, and e(dt) is the missing data noise variable. The model description of the ammonia flue gas desulfurization system under missing data can be obtained as follows:

[0045]

[0046] And the missing data ammonia desulfurization system noise-free output information vector The noise information vector ψ(dt) and the output information vector φ(dt) of the entire system are defined as

[0047]

[0048] Among them, ω(dt-1),…,ω(dt-n a ) is the dual-rate output intermediate variable at dt-1,…,dt-n a The value of the moment, is a dual-rate nonlinear output at dt-1,…,dt-n b The value of the moment, The dual-rate nonlinear output h(u(dt)) is based on the nonlinear basis The values ​​of γ(dt-1),…,γ(dt-nc ) is the double-rate colored noise interference variable at dt-1,…,dt-n c The value of the moment.

[0049] (2-1) In order to solve the estimation of the parameter vector, a gradient iterative identification method based on particle filtering is constructed as follows:

[0050] Define the stacked output vector Y(Qd), the stacked noise vector V(Qd) and the stacked information matrix Φ(Qd) as

[0051] Y(Qd):=[y(d),y(2d),…,y(Qd)] T

[0052] E(Qd):=[e(d),e(2d),…,e(Qd)] T

[0053] Φ(Qd):=[φ(d),φ(2d),…,φ(Qd)] T

[0054] Where Q is the data length, y(d), y(2d), …, y(Qd) are the output values ​​at time d, 2d, …, Qd respectively, e(d), e(2d), …, e(Qd) are the noise values ​​at time d, 2d, …, Qd respectively, and φ(d), φ(2d), …, φ(Qd) are the information vector values ​​at time d, 2d, …, Qd respectively.

[0055] Therefore, formula (6) can be rewritten as:

[0056] Y(Qd)=Φ(Qd)θ+E(Qd)

[0057] The parameter information vector θ of the entire system should be estimated by minimizing the criterion function F(θ):

[0058]

[0059] ||·|| represents the magnitude of a vector, Represents the gradient estimate of the parameter vector.

[0060] (2-2) Then, the gradient iterative estimation algorithm is derived based on F(θ). Let k be the iteration variable, and the estimation of the parameter vector is defined as follows:

[0061]

[0062] definition is the estimation of the parameter vector θ of the entire ammonia desulfurization system at the kth iteration, is the noise-free output parameter vector The estimate at k iterations. is the estimate of the noise output parameter vector c at the kth iteration, where is the estimate of the first parameter of c at iteration k, is the nth c The estimation of the parameters at k iterations. is the estimate of the noise-free output parameter vector a at k iterations, where is the estimate of the first parameter of a at k iterations, is the nth a The estimation of the parameters at k iterations. is the estimate of the noise-free output parameter vector b at k iterations, where is the estimate of the first parameter of b at iteration k, is the nth b The estimation of the parameters at k iterations. is the estimate of the parameter vector α of the nonlinear output at k iterations, where is the estimate of the first parameter of α at the kth iteration, is the nth α The estimation of the parameters at k iterations.

[0063] At iteration k, the estimated values ​​of the polynomials A(s), B(s) about the noise-free output of s and the polynomial C(s) about the noise output of s are calculated as follows:

[0064]

[0065] in, are the estimated values ​​of polynomials A(s), B(s) and C(s) at k iterations respectively.

[0066] (2-3) Order is the gradient based on the objective function (8), is the mathematical symbol for partial derivative, is the mathematical symbol for gradients, and they are calculated as follows:

[0067]

[0068] is the partial derivative of the objective function F(θ) with respect to the parameter vector θ of the entire ammonia desulfurization system, is the partial derivative of the stacked noise vector E(Qd) with respect to the parameter vector θ, represents the estimated value of the parameter vector θ at iteration k.

[0069] Define the filtering information vector of the missing data ammonia desulfurization system after particle filtering in the kth iteration as:

[0070]

[0071] φ h,k (dt) is the estimate of the output information vector φ(dt) of the entire system at the kth iteration, where the noise-free output filter information vector The estimated and noise filtered information vector ψ at k iterations h,k The estimate of (dt) at iteration k is derived as:

[0072]

[0073] in, is the partial derivative of the missing data error e(dt) with respect to the parameter vector θ of the entire system, Represents the parameter vector The estimated value at k iterations, is the partial derivative of the missing data error e(dt) with respect to the parameter vector a, is the partial derivative of the missing data error e(dt) with respect to the parameter vector b, is the partial derivative of the missing data error e(dt) with respect to the nonlinear parameter vector α. is the estimated value of the i-th parameter of parameter vector b at the k-th iteration, u(dt-i) is the value of the input at time dt-i, is a nonlinear function of its corresponding order. is the estimated value of the i-th parameter of parameter vector a at the k-th iteration, y(dt-1),…,y(dt-n c ) is the output at dt-1,…,dt-n c The value at the moment, ω(dt-i-1),…,ω(dt-in c ) is the noise-free output intermediate variable at dt-i-1,…,dt-in c The value of the moment, For nonlinear output at dt-i-1,…,dt-in c The value of the moment.

[0074] The information matrix Φ contains the filtering information vectors at all times h,k Defined as:

[0075] Φ h,k (Qd):=[φ h,k (d),φ h,k (2d),…,φ h,k (Qd)] T (13)

[0076] The simplified gradient is calculated according to formula (9):

[0077]

[0078] (2-4)λ is the forgetting factor, The parameter vector of the entire missing data ammonia desulfurization system in the previous iteration is estimated, and the update formula of the gradient iterative identification algorithm is:

[0079]

[0080] However, there are unknown variables ω(dt-i) in the above equation. Therefore, the present invention applies particle filtering (PF) to obtain these unknown variables. Its core is to approximate the posterior probability density function based on discrete random sampling data. Then, the minimum variance estimate of the state variable is obtained, and the unknown intermediate term ω(dt-i) can be regarded as an unknown state variable. Therefore, the unknown intermediate term ω(dt-i) can be obtained by measuring the data u(t), y(dt) and the estimated value The estimate of the previous iteration To estimate.

[0081] (3-1) Apply the particle filter method to solve the above formula to estimate the noise-free output intermediate variable at the kth iteration The core of particle filtering is to approximate the posterior probability density function based on discrete random sampling data. Then, the minimum variance estimate of the state variable is obtained. Therefore, ω(dt-i) can be regarded as an unknown state variable, is the estimated value of θ for the kth iteration. The particle filter method uses the measurable data {u(t), y(dt)} and the estimated value of the previous iteration To estimate the noise-free output intermediate variable ω(dt-i).

[0082] (3-2) According to the Bayesian filtering technology, the estimation problem of the unknown noise-free output intermediate variable ω(dt-i) is transformed into processing the posterior probability density function p:

[0083]

[0084] Among them, y(d), y(2d), …, y(Qd) are the output values ​​at time d, 2d, …, Qd, respectively, and u(1), …, u(Qd) are the input values ​​at time 1, 2, …, Qd, respectively. Based on the maximum a posteriori criterion, is the noise-free output intermediate variable ω(dt-i) with the maximum posterior probability density, which can be calculated through the intermediate term:

[0085]

[0086] Among them, arg(max(ω(dt-i))) is the maximum posterior probability density.

[0087] (3-3) Since the Bayesian posterior probability contains integrals, the closed analytical expression is difficult to calculate. Therefore, a Monte Carlo-based particle filter algorithm is proposed. Then, according to the parameter vector estimation The posterior probability density function of the calculated noise-free output intermediate variable ω(dt-i) is approximated as

[0088]

[0089] Where N represents the number of particles; represents the normalized weight of the jth particle in the kth iteration; Δ represents the Dirac delta function; is the posterior probability density function at time dt-i and iteration k. After obtaining each particle and its corresponding weight, the estimated value of the intermediate term at iteration k can be calculated As shown below:

[0090]

[0091] in, represents the normalized weight, which is calculated as

[0092] (3-4) Since it is difficult to extract samples directly from the true posterior probability density function, in order to generate new sampling particles, an importance probability density function d(·) is introduced to replace the true posterior probability density function. At the same time, d(·) should be known and easy to sample, so it can be selected as:

[0093]

[0094] Based on (4) and (18), new particles can be generated, and their corresponding weights It can be calculated recursively:

[0095]

[0096] in, is the posterior probability density function at time dt and iteration k The jth particle extracted from .

[0097] The formula for calculating the probability density function is:

[0098]

[0099] Where exp is the natural number e, and σ is the square root of the variance. Substituting (19) into (20), we get:

[0100]

[0101] (3-5) The output cannot be obtained at time dt-1, dt-2, dt-3, ..., dt-d+1. Therefore, from the importance probability density function Particles are extracted from the To remain unchanged, equal to its value at the previous moment Right now

[0102]

[0103] The updated weights After normalization, we get

[0104]

[0105] and the estimated values ​​of the state variables become

[0106]

[0107] (3-6) After some iterations of the particle filter method, the particles began to degenerate. To avoid this problem, the resampling method was introduced into the particle filter. During the resampling process, a large number of high-weight particles were copied, while low-weight particles were eliminated. Based on the above theory, some particles are replicated and some particles are normalized according to their weights is eliminated, and the total number of particles remains unchanged. After resampling, the weights of the particles become the same, that is, Therefore, updating the weights in (24) becomes

[0108]

[0109] (3-7) After obtaining the estimated value of the intermediate term After that, the parameter vector θ of the entire missing data ammonia desulfurization system can be finally determined by the gradient iterative identification algorithm based on particle filtering:

[0110]

[0111] in, is the k-th iteration estimate of the parameter vector of the entire system, The result of the previous iteration estimation, is the k-th iteration estimate of the information matrix containing all information vectors at all times, Q is the length of the information vector, λ is the genetic factor, and T represents the matrix transposition. is the estimate of the noise variable at the kth iteration at time Qd.

[0112] Compared with the prior art, the present invention has the following beneficial effects:

[0113] 1. The present invention establishes a modeling and estimation method for an ammonia desulfurization system with particle filtering under missing data. The system includes two types of data with different sampling frequencies. Acquiring unknown states and variables in the missing data system without adding unknown parameters is the key technology of the present invention.

[0114] 2. In order to solve the problem of not adding unknown parameters, the particle filter method is applied to estimate the unknown states and variables in the system. The characteristic that each particle can adjust its weight according to the current given data is used to derive the gradient iteration method based on particle filtering to approximate the unknown parameters, avoiding the addition of more unknown parameters and reducing the amount of calculation. Then the algorithm is used to estimate the unknown parameters and states in the ammonia desulfurization nonlinear system. Therefore, the technology of the present invention avoids adding more unknown parameters, reduces the calculation burden in the prior art, and greatly improves the calculation speed of the identification method.

[0115] 3. The present invention includes two types of data with different sampling frequencies. Without adding unknown parameters, the system data is fully utilized, and the particle filter method is applied to obtain the unknown states and variables in the missing data system. The gradient iteration method based on particle filtering is derived, and then the algorithm is used to identify the unknown parameters in the ammonia desulfurization nonlinear system. This method can effectively estimate the nonlinear model under missing data, and the algorithm has a small amount of calculation and the parameter estimation is more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. The simulations of the present invention are used to explain the present invention and do not constitute a limitation of the present invention.

[0117] Figure 1 It is a schematic diagram of the ammonia desulfurization reaction in the present invention.

[0118] Figure 2 It is a structural diagram of the missing data ammonia desulfurization Hammerstein output error autoregressive system in the present invention.

[0119] Figure 3 It is a flow chart of estimating unknown parameters of a model using a gradient iterative identification method based on particle filtering in the present invention.

[0120] Figure 4 This is a graph showing the variation of parameter estimation errors of the ammonia desulfurization model using the gradient iterative identification method based on particle filtering under different noise variances in Example 1.

[0121] Figure 5 It is the fitting of the real output and the estimated output when the gradient iterative identification algorithm based on particle filtering in Example 1 estimates the ammonia desulfurization model.

[0122] Figure 6 This is a graph showing the parameter estimation error variation of the ammonia desulfurization model estimated by the gradient iterative identification algorithm based on particle filtering in Example 2.

[0123] Figure 7 It is a schematic diagram of the change of parameter estimation values ​​of the gradient iteration method based on particle filtering in Example 2. DETAILED DESCRIPTION

[0124] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0125] The system parameters are set as follows: Figure 1 , the actual operation data of the DCS system on site of a certain enterprise was used for preprocessing, and the dynamic process data of the continuous period of the ammonia desulfurization device of this enterprise under different steady-state conditions were selected. The input variable was the ammonia flow data, the output variable was the corresponding time series absorption liquid pH value, and the inlet SO2 concentration was used as the system disturbance. 1000 groups of data were collected as samples, and the data sampling interval was 2s. Matlab software and the gradient iterative identification method based on particle filtering were used for simulation.

[0126] See also Figures 2 to 7 During the simulation process, the basic structure of the ammonia desulfurization system was first established, and then the estimation model of the Hammerstein output error autoregressive system of the ammonia desulfurization under missing data was established. The parameters of the ammonia desulfurization model were estimated using the gradient iterative identification method based on particle filtering. Finally, the estimated results were compared with the actual results. It was determined that the method of the present invention can effectively estimate the unknown parameters in the ammonia desulfurization system under missing data with a small error.

[0127] Example 1

[0128] The missing data to be estimated is described as follows:

[0129]

[0130] A(s)=1+a1s -1 =1-0.34s -1 ,

[0131] B(s)=1+b1s -1 β1+5.94s -1 ,

[0132] C(s)=1+c1s -1 =1+0.04s -1 .

[0133] The parameter vector is as follows:

[0134] a=[a1]=[-0.34],

[0135] b=[b1]=[5.94],

[0136] c=[c1]=[0.04],

[0137] α=[α1,α2] T =[-0.20,0.10] T .

[0138] The true value of the parameter vector is

[0139] θ=[a T ,b T ,α T ,c T ] T

[0140] =[-0.34,5.94,-0.20,0.10,0.04] T .

[0141] Where u(t) and y(t) are the input of the system, i.e., the ammonia flow rate, and the output, i.e., the pH value of the absorption liquid. The output sampling interval is set to 2, and e(t) is the variance σ 2 The noise variable is the inlet SO2 concentration.

[0142] The gradient iterative identification method based on particle filtering is used to estimate the parameters of this ammonia desulfurization system model. The estimation error δ is defined as follows:

[0143]

[0144] Example 2

[0145] The missing data to be estimated for the ammonia desulfurization system 2 are described as follows:

[0146]

[0147] θ=[-0.24,0.16,0.014,-0.85,2.71,-2.14,0.93] T .

[0148] In the simulation, the noise (inlet SO2 concentration) e(t) has a mean of zero and a variance of σ 2 =0.60 2 The output sampling interval d is set to 3.

[0149] The modeling and estimation method of the ammonia desulfurization system using particle filtering under missing data in Examples 1 and 2 comprises the following steps:

[0150] (1) An ammonia desulfurization system based on Hammerstein output error autoregression is constructed. The specific steps are as follows:

[0151] Step 1: Construct a nonlinear system model structure diagram of ammonia desulfurization based on the Hammerstein output error autoregressive model, see Figure 2 ;

[0152] Step 2: Based on this model, the expression of the nonlinear model of the Hammerstein output error autoregressive system of ammonia desulfurization is constructed as follows:

[0153]

[0154] Where t is the time variable, u(t) and y(t) are the input of the system, i.e., the ammonia flow rate, and the output, i.e., the pH value of the absorption liquid, respectively, and e(t) is a variable with zero mean and variance σ 2 The noise variable is the inlet SO2 concentration, and s represents the unit post-shift operator:

[0155] s -1 y(t)=y(t-1), nonlinear output is a polynomial function of the input, i.e., the ammonia flow rate u(t), A(s) and B(s) are polynomials of the noise-free output about s, and C(s) is a polynomial of the noise output about s, and satisfies the following form:

[0156]

[0157] Among them, n a 、n b 、n c is the number of coefficients in A(s), B(s), and C(s), are the coefficients of the noise-free output polynomial A(s) with respect to s; are the coefficients of the noise-free output polynomial B(s) with respect to s; are the coefficients of the noise output polynomial C(s) with respect to s;

[0158] Non-linear output It is a known nonlinear basis The linear combination of Right now:

[0159]

[0160] The noise-free output parameter vectors a, b, the nonlinear output parameter vector α, and the noise output parameter vector c are defined as:

[0161]

[0162] Among them, a i represents the i-th parameter in parameter vector a, b i represents the i-th parameter in parameter vector b, c i represents the i-th parameter in the parameter vector c, α i represents the i-th parameter in the parameter vector α, n α for The number of parameters in the equation, and the superscript T is the transpose of the vector or matrix. The noise-free output intermediate variable ω(t) and the colored noise interference variable γ(t) are calculated as follows:

[0163]

[0164] Define the noise-free output parameter vector And the parameter vector θ of the entire ammonia desulfurization system is as follows:

[0165]

[0166] Redefine the noise-free output information vector The noise information vector ψ(t) and the output information vector φ(t) of the entire system are as follows:

[0167]

[0168] ψ(t):=[-γ(t-1),…,-γ(tn c )] T

[0169]

[0170] Among them, ω(dt-1),…,ω(dt-n a ) is the output intermediate variable in dt-1,…,dt-n a The value of the moment, For nonlinear output at dt-1,…,dt-n b The value of the moment, The nonlinear output h(u(dt)) is the nonlinear basis The values ​​of γ(dt-1),…,γ(dt-n c ) is the colored noise disturbance variable at dt-1,…,dt-n c The value of the moment.

[0171] Step 3: Construct the expression of the nonlinear intermediate variable of the Hammerstein output error autoregressive system of missing data ammonia desulfurization as follows:

[0172]

[0173] Substitute (4) and (5) into (1), and replace t in the formula with dt, where d is the sampling interval of the ammonia desulfurization output and is a positive integer greater than or equal to 2, the measurable input is u(t), the measurable multi-rate output is y(dt), ω(dt) is the missing data noise-free output intermediate variable, γ(dt) is the missing data colored noise interference variable, and e(dt) is the missing data noise variable.

[0174] Step 4: Derive and construct the output of the entire system of the Hammerstein output error autoregressive system for ammonia desulfurization under missing data:

[0175]

[0176] Among them, the noise-free output information vector of the missing data ammonia desulfurization system is The noise information vector ψ(dt) and the output information vector φ(dt) of the entire system are defined as

[0177]

[0178] Among them, φ(dt) is the output information vector of the entire missing data ammonia desulfurization system, θ is the parameter vector containing the noise-free output The parameter vector of the entire system is y(dt), and the noise parameter vector c.

[0179] (2) Construct a process for accurately estimating the unknown parameters of the nonlinear model of ammonia desulfurization using a gradient iterative identification method based on particle filtering:

[0180] Step 1: Set the initial iteration value k to 1 and the initial time t to 0;

[0181] Step 2: Collect the measurement data of the input of the Hammerstein output error autoregressive system of ammonia desulfurization under missing data, i.e., the flow rate of ammonia water, and the output, i.e., the pH value of the absorption liquid;

[0182] Step 3: Form an estimate of the kth iteration of the noise-free information vector of the particle filtered system The k-th iteration estimate of the noise information vector after particle filtering The output information vector of the entire missing data ammonia desulfurization system is estimated at the kth iteration

[0183] Step 4: Generate particles and calculate the weight of each particle at the kth iteration based on the particle filter method, and calculate the estimate of the noise-free output intermediate variable at the kth iteration based on the particle filter;

[0184] Step 5: Calculate the noise variable of the kth iteration, that is, the estimate of the inlet SO2 concentration, and construct the estimated vector of the noise variable And calculate the estimate of the colored noise disturbance variable at the kth iteration;

[0185] Step 6: Update the estimate of the particle output vector y(dt) at the kth iteration, obtain the estimate of the noise-free output information vector and the noise information vector at the kth iteration, and then obtain the estimate of the output information vector of the entire desulfurization system at the kth iteration and the estimate of the information matrix containing all information vectors at the kth iteration.

[0186] Step 7: Update the estimate of the parameter vector of the entire system at iteration k;

[0187] Step 8: Update the estimate of the nonlinear noise-free output and the estimate of the noise output polynomial at iteration k

[0188] Step 9: Increase the number of iterations k by 1 and return to step 2-3);

[0189] Step 10: When the number of iterations k reaches the maximum number of iterations K, terminate the iteration and output the estimate of the final parameter vector (3) According to the process of accurately estimating the unknown parameters of the nonlinear model of ammonia desulfurization by particle filtering by gradient iterative identification in (2), the parameter estimation method of the ammonia desulfurization nonlinear system of particle filtering under missing data is constructed as follows:

[0190]

[0191]

[0192] See also Figure 3 , the specific steps of the above method are:

[0193] (1) Set the initial value of the number of iterations k to 1, the initial time t to 0, and set the initial iteration estimate of the parameter vector θ of the entire system is 1 n / p0, the initial iteration estimate of the noise-free output intermediate variable ω(t) 1 / p0, nonlinear output The initial iteration estimate is 1 / p0, the estimate of the initial iteration of the colored noise disturbance variable γ(t) = 1 / p0. n represents a vector with dimension n and all elements are 1, p0 is 10 6 .

[0194] (2) The measurement data of the Hammerstein output error autoregressive system of ammonia flue gas desulfurization with the ammonia flow rate as input and the pH value of the absorption liquid as output are collected. The input is: {u(t), t = 1, 2, ..., Qd}, and the output is: {y(dt), d = 1, 2, ..., Q}, where Q is the data length and Qd is the total measurement time that can be divided by d.

[0195] (3) The k-th iteration estimate of the noise-free information vector after particle filtering is formed by equations (15), (16) and (14): The k-th iteration estimate of the noise information vector after particle filtering And the output information vector of the entire ammonia desulfurization system is estimated at the kth iteration

[0196] (4) Generate N particles and calculate the weight of the jth particle in the kth iteration according to the particle filter method The estimation of the noise-free output intermediate variable at the kth iteration is calculated by equation (11) based on particle filtering. i=1,2,…,d-1;

[0197] (5) The noise variable of the kth iteration, i.e., the estimated concentration at the inlet, is calculated by equation (12): The estimated vector of noise variables is constructed by formula (9): And the estimated value of the colored noise disturbance variable γ(dt) at the kth iteration is calculated by equation (13):

[0198] (6) The output vector y(dt) of the entire system is estimated at the kth iteration by using equation (17): The noise-free output information vector is obtained by equations (19) and (20): and the estimate of the noise information vector ψ(dt) at the kth iteration and Then, the output information vector of the entire ammonia desulfurization system is estimated at the kth iteration by equations (18) and (10): and the estimate of the kth iteration of the information matrix containing the information vectors at all times

[0199] (7) The parameter vector of the entire system at iteration k is updated by equation (8)

[0200] (8) The estimate of the nonlinear noise-free output at iteration k is updated by equations (21) and (22) and the estimation of the noise output polynomial

[0201] (9) Increase the number of iterations by 1 and return to step (3);

[0202] (10) When the number of iterations k reaches the maximum number of iterations K, the iteration is terminated and the final estimate of the parameter vector of the entire system is output

[0203] The definitions of the variables are as follows:

[0204] Define t as the time variable, u(t) and y(t) as the input of the system, i.e., the ammonia flow rate, and the output, i.e., the pH value of the absorption liquid. is a polynomial function of the input ammonia flow rate u(t), e(t) is a polynomial function with zero mean and variance σ 2 The noise variable is the concentration of inlet SO2, s represents the unit shift operator: s -1 y(t)=y(t-1), A(s) and B(s) are polynomials of the noise-free output with respect to s, and C(s) is a polynomial of the noise output with respect to s. are the coefficients of the noise-free output polynomial A(s) with respect to s; are the coefficients of the noise-free output polynomial B(s) with respect to s; are the coefficients of the noise output polynomial C(s) about s; nonlinear output It is a known nonlinear basis A linear combination of ), n a 、n b 、n c 、n α A(s), B(s), C(s) and The number of parameters in . Define a, b as the noise-free output parameter vector, α as the nonlinear output parameter vector, and c as the noise output parameter vector. i represents the i-th parameter in parameter vector a, b i represents the i-th parameter in parameter vector b, c i represents the i-th parameter in the parameter vector c, u i Represents the i-th parameter in the parameter vector u.

[0205] Define ω(t) as the noise-free output intermediate variable, γ(t) as the colored noise interference variable, is the noise-free output parameter vector, θ is the parameter vector of the entire ammonia desulfurization system, is the noise-free output information vector, ψ(t) is the noise information vector, and φ(t) is the output information vector of the entire system. d is the sampling interval of the ammonia desulfurization output, and is a positive integer greater than or equal to 2. The measurable input is u(t), y(dt) is the measurable multi-rate output, ω(dt) is the missing data noise-free output intermediate variable, γ(dt) is the missing data colored noise interference variable, and e(dt) is the missing data noise variable. The missing data ammonia desulfurization system is the noise-free output information vector, ψ(dt) is the noise information vector, and φ(dt) is the output information vector of the entire system.

[0206] Define F(θ) as the criterion function, Q as the data length, Qd as the total measurement time that is divisible by d, define Y(Qd) as the stacked output vector, E(Qd) as the stacked noise vector, Φ(Qd) as the stacked information matrix, ||·|| represents the modulus of the vector, represents the gradient estimate of the parameter vector, defined is the estimation of the parameter vector θ of the entire ammonia desulfurization system at the kth iteration, is the noise-free output parameter vector The estimate at k iterations. is the estimate of the noise output parameter vector c at the kth iteration, is the estimate of the first parameter of c at iteration k, is the nth c The estimation of the parameters at k iterations. is the estimate of the noise-free output parameter vector a at the kth iteration, is the estimate of the first parameter of a at k iterations, is the nth a The estimation of the parameters at k iterations. is the estimate of the noise-free output parameter vector b at the kth iteration, is the estimate of the first parameter of b at iteration k, is the nth b The estimation of the parameters at k iterations. is the estimate of the parameter vector α of the nonlinear output at k iterations, is the estimate of the first parameter of α at the kth iteration, is the nth α The estimation of the parameters at k iterations. and are the estimates of polynomials A(s), B(s) and C(s) at k iterations respectively.

[0207] definition is the gradient based on the objective function, is the mathematical symbol for partial derivative, is the mathematical symbol for gradient, Represents the parameter vector The estimated value at k iterations, is the partial derivative of the stacked noise vector E(Qd) with respect to the parameter vector θ, ω(dt-1),…,ω(dt-n a ) is the dual-rate output intermediate variable at dt-1,…,dt-n a The value of the moment, is a dual-rate nonlinear output at dt-1,…,dt-n b The value of the moment, The dual-rate nonlinear output h(u(dt)) is based on the nonlinear basis The values ​​of γ(dt-1),…,γ(dt-n c ) is the double-rate colored noise interference variable at dt-1,…,dt-n c The value of the moment, represents the estimated value of the parameter vector θ at k iterations, and defines φ h,k (dt) is the estimate of the output information vector φ(dt) of the entire system at the kth iteration, is the partial derivative of the missing data error e(dt) with respect to the parameter vector θ of the entire system, is the partial derivative of the missing data error e(dt) with respect to the parameter vector a, is the partial derivative of the missing data error e(dt) with respect to the parameter vector b, is the partial derivative of the missing data error e(dt) with respect to the nonlinear parameter vector α. is the estimated value of the i-th parameter of parameter vector b at the k-th iteration, u(dt-i) is the value of the input at time dt-i, is a nonlinear function of its corresponding order. is the estimated value of the i-th parameter of parameter vector a at the k-th iteration, y(dt-1),…,y(dt-n c ) is the output at dt-1,…,dt-n c The value at the moment, ω(dt-i-1),…,ω(dt-in c ) is the noise-free output intermediate variable at dt-i-1,…,dt-in c The value of the moment, For nonlinear output at dt-i-1,…,dt-in c The value at the moment. Define λ as the forgetting factor, is the estimated value of the parameter vector of the entire missing data ammonia desulfurization system in the previous iteration.

[0208] definition is the noise-free output intermediate variable ω(dt-i) with the maximum posterior probability density, where N represents the number of particles, represents the normalized weight of the jth particle in the kth iteration, Δ represents the Dirac delta function, is the posterior probability density function at time dt-i and iteration k. The jth particle extracted from represents the normalized weight, d(·) is the probability density function, exp is the natural number e, and σ is the square root of the variance.

[0209] The recognition effect of the recognition method designed by this embodiment can be seen in Figure 3 , Figure 4 , Figure 5 , Figure 6 and Figure 7 The identification results show that the method of the present invention has the following advantages in terms of identification error, estimation accuracy and convergence speed: as the number of iterations k increases, the parameter estimation error becomes smaller and smaller, and the final error tends to be less than 5%. 2 As the value increases, the estimation error increases and the estimation accuracy decreases, but this effect is very small and within the range allowed by the accuracy. Therefore, the proposed gradient iteration identification method based on particle filtering can effectively estimate the unknown parameters in the multi-rate ammonia desulfurization system. The final fitting output is close to the true output, and the parameter estimation value also converges to the true value. Therefore, this estimation method can effectively realize the parameter and state estimation of the ammonia desulfurization model, solve the problem of missing data, and is fully suitable for the modeling of the ammonia desulfurization nonlinear system of particle filtering under missing data.

[0210] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. Modeling and estimation method of ammonia desulfurization system with particle filtering under missing data, characterized in that: The steps include: Step 1) constructing an ammonia desulfurization nonlinear system model based on the Hammerstein output error autoregressive model, and obtaining all information vectors and a model description of the identification system; Step 2) construct a gradient iterative identification model method process based on particle filtering to accurately estimate the unknown parameters of the ammonia desulfurization nonlinear model.

2. The modeling and estimation method of the ammonia desulfurization system with particle filtering under missing data according to claim 1 is characterized in that: The step 1) comprises the following steps: (1-1) Construct a nonlinear system of ammonia desulfurization based on Hammerstein output error autoregressive model; (1-2) Based on this model, the expression of the nonlinear model of the Hammerstein output error autoregressive system of ammonia desulfurization is as follows: y(t)=ω(t)+γ(t) (4) Among them, t is the time variable, u(t) and y(t) are the input of the system, that is, the ammonia flow rate, and the output, that is, the pH value of the absorption liquid, respectively. is a polynomial function of the input ammonia flow rate u(t), e(t) is a polynomial function with zero mean and variance σ 2 The noise variable is the inlet SO2 concentration, ω(t) is the noise-free output intermediate variable, γ(t) is the colored noise interference variable, h(·) is the nonlinear structure of the system, and is given by the known nonlinear basis function Composition, s represents the unit shift operator: s -1 y(t)=y(t-1), A(s) and B(s) are polynomials of the noise-free output of s, and C(s) is a polynomial of the noise output of s; are the coefficients of the noise-free output polynomial A(s) with respect to s; are the coefficients of the noise-free output polynomial B(s) with respect to s; are the coefficients of the noise output polynomial C(s) with respect to s; is the nonlinear output polynomial The coefficients of the noise-free output parameter vectors a and b are defined as and The nonlinear polynomial output parameter vector α is The noise output parameter vector c is a i represents the i-th parameter in parameter vector a, b i represents the i-th parameter in parameter vector b, c i represents the i-th parameter in the parameter vector c, where the symbol T represents the transposition of a vector or matrix; (1-3) The expression of the nonlinear intermediate variable of the Hammerstein output error autoregressive system of ammonia desulfurization with missing data is constructed as follows: γ(dt)=ψ T (dt)c+e(dt) (6) Among them, the set time interval between each set of input and output and the next set of input and output during sampling is different. Let d be the interval of multi-rate sampling and a positive integer greater than or equal to 2. The measured input is u(t), and the measurable multi-rate output is y(dt). Therefore, there is missing data in the system. ω(dt) is the multi-rate noise-free output intermediate variable, γ(dt) is the multi-rate colored noise interference variable, and e(dt) is the multi-rate noise variable. is the noise-free output parameter vector containing parameter vectors a, b and α, c is the noise output parameter vector, is the noise-free output information vector of the ammonia desulfurization system under missing data, ψ(dt) is the noise information vector of the ammonia desulfurization system under missing data; (1-4) The output of the entire system of the Hammerstein output error autoregressive system for ammonia desulfurization under missing data is derived as follows: y(dt)=φ T (dt)θ+e(dt) (7) Among them, φ(dt) is the output information vector of the entire missing data ammonia desulfurization system, θ is the parameter vector containing the noise-free output The parameter vector of the entire system is y(dt), and the noise parameter vector c.

3. The modeling and estimation method of the ammonia desulfurization system with particle filtering under missing data according to claim 1 is characterized in that: The step 2) comprises the following steps: (2-1) Set the initial value of the number of iterations k to 1, the initial time t to 0, and set the initial iteration estimate of the parameter vector θ of the entire system is 1 n / p0, the initial iteration estimate of the noise-free output intermediate variable ω(t) 1 / p0, nonlinear output The initial iteration estimate is 1 / p0, the estimate of the initial iteration of the colored noise disturbance variable γ(t) is 1 / p0; where 1 n represents a vector with dimension n and all elements are 1, p0 is 10 6 ; (2-2) Collect the measurement data of the ammonia flow rate as input and the pH value of the absorption liquid as output of the Hammerstein output error autoregressive system of ammonia desulfurization, input: {u(t), t = 1, 2, ..., Qd}, output: {y(dt), d = 1, 2, ..., Q}, Q is the data length, Qd is the total measurement time that can be divided by d; (2-3) Form the k-th iteration estimate of the noise-free information vector after particle filtering The k-th iteration estimate of the noise information vector after particle filtering And the output information vector of the entire ammonia desulfurization system is estimated at the kth iteration (2-4) Generate N particles and calculate the weight of the jth particle in the kth iteration according to the particle filter method The estimation of the noise-free output intermediate variable at the kth iteration based on particle filtering i=1,2,…,d-1; (2-5) Calculate the noise variable of the kth iteration, i.e., the estimated concentration of inlet SO2 Construct an estimation vector of noise variables And calculate the estimate of the colored noise disturbance variable γ(dt) at the kth iteration (2-6) Calculate the estimate of the output vector y(dt) of the entire system at the kth iteration Get the noise-free output information vector and the estimate of the noise information vector ψ(dt) at the kth iteration and Then the output information vector of the entire ammonia desulfurization system is estimated at the kth iteration and the estimate of the kth iteration of the information matrix containing the information vectors at all times; (2-7) Update the parameter vector of the entire system at iteration k (2-8) Update the estimate of the nonlinear noise-free output at iteration k and the estimation of the noise output polynomial (2-9) Increase the number of iterations by 1 and return to step (2-3); (2-10) When the number of iterations k reaches the maximum number of iterations K, the iteration is terminated and the final estimate of the parameter vector of the entire system is output

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