CN111324852B - Method of CSTR reactor time delay system based on state filtering and parameter estimation - Google Patents
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Abstract
The invention discloses a CSTR reactor time delay system method based on state filtering and parameter estimation, which comprises the following steps: step 1, representing a time delay state space model in a CSTR reactor by using a double-input double-output form; decomposing a dual-input dual-output model into two dual-input single-output subsystems based on the identification model decomposition, wherein each subsystem has smaller dimension and variable, and each subsystem is calculated; step 2, estimating parameters of a system in the CSTR reactor; and 3, estimating the state of the system in the CSTR reactor. The invention can fully utilize all data to generate highly accurate parameter estimation, simplifies the derivation process of the identification model, reduces the calculated amount of the identification of the multivariable system and improves the convergence rate; the model can identify the multivariate/nonlinear model of colored noise interference to obtain the optimal identification parameter, and has practical value for industrial process modeling of processing state delay.
Description
Technical Field
The invention relates to a state filtering and parameter estimation method based on a time delay state space system, and belongs to the field of chemical process modeling and identification.
Background
The CSTR reactor is a common chemical process reaction device, and meanwhile, due to the wide working domain characteristic, the CSTR reactor becomes an important technical and research object for the chemical process. Nowadays, various chemical products can be flexibly produced by operating in a plurality of working areas, so that the competitiveness of the reactor is greatly improved. As a typical chemical reaction process, the process dynamic characteristics of the reaction process are widely researched.
The increasing demand of industrial control intellectualization makes the requirements for realizing the rapidity, stability and accuracy control of system response increasingly strict. However, the actual process inevitably has time delay, for example, in the process industries of chemical industry, electric power, metallurgy and the like, due to the lack of on-line monitoring equipment or poor reliability of the equipment and the like, key variables of some index products can only be obtained through manual sampling and laboratory test analysis, so that the modeling data has time delay; time delays may also be caused by sensors, actuators, network signal transmissions, etc. in the control system. The design of the controller and the analysis of the system stability are greatly difficult due to the factors such as the change and uncertainty of the time delay, and the like, so that the production efficiency is difficult to control and improve effectively, and the cost is reduced. Therefore, it is of great importance to research such system modeling and identification methods with time delay.
Disclosure of Invention
1. Objects of the invention
The invention aims to provide a method for state filtering and parameter estimation of a time delay state space system based on a CSTR reactor, so as to achieve high precision of system parameter and state identification.
2. The technical scheme adopted by the invention
The method comprises the following steps of representing the structure of a time delay state space model in the CSTR reactor by using a double-input double-output form, wherein the specific expression of the model is as follows:
x(t+1)＝Ax(t)+Bx(t-d)+Fu(t), (1)
y(t)＝Cx(t)+v(t). (2)
the present invention contemplates an exothermic reaction in a CSTR wherein x (t) e RnThe method comprises the following steps of (1) selecting an unmeasured state variable, n being a real number, t being time, and taking the flow rate of a cooling liquid in a CSTR reactor as an input variable of a model: u (t) ═ u1(t),u2(t)]T∈R2Product concentration as output variable: y (t) ═ y1(t),y2(t)]T∈R2，v(t)＝[v1(t),v2(t)]T∈R2Is white noise in the CSTR reactor, and the model has time delay d due to the time delay reaction of the sensor in the CSTR reactor, A belongs to the Rn×n，B∈Rn ×n，F∈Rn×2And C ∈ R2×nIs a system parameter matrix to be identified;
since the model is a multivariate system, direct identification is difficult and decomposition of the model is required. Based on the idea of identification model decomposition, a double-input double-output model is decomposed into two double-input single-output subsystems, each subsystem has smaller dimensions and variables, and each subsystem is calculated. Since the models (1) - (2) contain the system's unknown parameter vectors/matrices and unmeasurable state vectors, which are difficult to identify, the present invention represents the state vectors with measurable inputs and outputs. First of all the first sub-system is analyzed,
wherein, y1(t+i),y1(t+n1) At time t + i, t + n1Output of the first subsystem of time, n1Is a real number, e1,A1,A12,B1,F1Are all model parameters, x is the state, u is the input, and v is white noise.
Some vectors/matrices are defined:
wherein, Yi(t+n1) To output a vector, Ui(t+n1) For the input vector, X (t-d + n)i) Is a state vector, Vi(t+ni) As a noise vector, Mi,QiMatrix parameters, n, which are modelsiAre real numbers.
From equations (3) - (4) one can derive
Y1(t+n1)＝Tx1(t)+M1X(t-d+n1)+Q1U1(t+n1)+V1(t+n1).
Wherein, Y1(t+n1) Is the output vector of the first subsystem, X (t-d + n)1) Is the state vector of the first subsystem, U1(t+n1) Is the input vector of the first subsystem, V1(t+n1) Is the noise vector of the first subsystem, T is the observable matrix, M1,Q1Is the matrix parameter of the first subsystem.
To obtain constant parameter estimates, an information vector is defined as
The parameter vector is theta1
Wherein,
are all information vectors, θ1,θ11,θ12,θ13Are all parameter vectors, U1(t+n1) For the input vector, X (t-d + n)1) Is a state vector, Y1(t+n1),Y2(t+n2) To output vector, V1(t+n1),V2(t+n2) Is a noise vector, e1,A1,Q1,F1,M1,B1As a system parameter,/1,h1As auxiliary vectors, respectively
Combining equation (4) and the above definition
Wherein, y1(t+n1) Is the system output, n1Is real number, x is system state, u is system input, v is white noise, e1,A1,A12,B1,F1As a result of the parameters of the system,
as information vectors, U1For the input vector, X (t-d + n)1) Is in a stateVector, θ11,θ12,θ13,θ1Is a system parameter.
Substituting t for t-n in equation (5)1It can be simplified to the following regression model,
this is an identification model of the first subsystem of the dual-input dual-output state space with time delay, and the second subsystem is derived in a similar way:
let t be the current time, { u (t), y (t): t ═ 0,1, 2. } is the measurable input-output information, y (t) and
is the current information that is being presented to the user,
is information in the past.
The following gives the parameter estimates for the system in the CSTR reactor:
the basic idea adopted by the invention is to replace the unknown noise term and the unknown state vector by the estimated residual error and the estimated state vector, defining
Is composed of
Estimation at time t. V according to equation (6)iThe estimate of (t) can be calculated as
Thus, minimizing the criterion function according to the least squares principle, when calculating one parameter vector, the remaining vectors are replaced by their estimates, the following algorithm can be derived to calculate the parameters:
wherein,
is an estimate of the parameter theta and,
as vectors of information
Is determined by the estimated value of (c),
in order to be a matrix of gains, the gain matrix,
is a covariance matrix. Because it is a multivariable system, the coupling of the system needs to be analyzed in the decomposition process, thereby realizing the decoupling of the system: a multivariable system that correlates inputs and outputs recognizes that each output is controlled only by the corresponding input.
The state estimation of the system in the CSTR reactor is given below:
Yi(t)＝[yi(t-ni),yi(t-ni+1),...,yi(t-1)]T,
Ui(t)＝[uT(t-ni),uT(t-ni+1),...,uT(t-1)]T,
wherein, Yi(t) is the output vector, Ui(t) is the input vector, and,
in the form of a state vector, the state vector,
in order to be a vector of the noise,
matrix parameters, n, which are modelsiIn the case of a real number,
is a system parameter.
3. Advantageous effects adopted by the present invention
(1) The invention researches a least square identification algorithm based on residual errors to estimate the state and parameters of a dual-input and dual-output system at the same time. Based on the decomposition idea in the identification model, the dual-input dual-output system is decomposed into two dual-input single-output subsystems with smaller dimensionality and variable to identify each subsystem again. To solve the difficulty of the information matrix including the unmeasurable noise terms, the unknown noise terms are replaced by their estimated residuals, which are calculated by the previous parameter estimation. Simulation results show that the algorithm has good effect.
(2) The two-input and two-output time delay state space model has large parameter quantity, complex dimension and coupling, and when the system state is calculated according to the layered identification principle, the identification of the two subsystems is combined. These parameters make the calculation more difficult and, since the recursive algorithm calculates the inverse of the matrix during the calculation, the amount of calculation is relatively large, thereby affecting the recognition accuracy. The invention starts from a bivariate model and researches a recursive least square algorithm based on residual errors, and the proposed algorithm can fully utilize all data to generate highly accurate parameter estimation, thereby simplifying the derivation process of an identification model, reducing the calculated amount of identification of a multivariable system and improving the convergence speed.
(3) The invention effectively avoids the problem of model comprehensive performance reduction caused by overlarge model parameters possibly caused by ill-condition problems in the process of identifying the model parameters and the states through a decomposition technology, thereby greatly improving the identification precision and the robustness of the bivariate model, providing a reliable system identification method for data prediction and controller design based on the model, and having higher practical value and better application prospect. The model in the invention can be extended to a multivariate/nonlinear model, which can identify the multivariate/nonlinear model of colored noise interference in a practical system to obtain the optimal identification parameters.
(4) The CSTR reactor is used as a process modeling simulation example under the condition of state time delay, and parameters and states of a state space model in the CSTR reactor are accurately estimated while a process model is established. Simulation results show that the method has good identification effect and has very practical value for modeling and identifying the industrial process for processing the state delay.
Drawings
FIG. 1 is a CSTR reactor of the present invention;
FIG. 2 is a parameter estimation of the present invention;
FIG. 3 shows the inventionState x of input dual-output time delay state space system1(t) estimation;
FIG. 4 shows state x of the dual-input dual-output delay state space system of the present invention2(t) estimation;
FIG. 5 shows state x of the dual-input dual-output delay state space system of the present invention3(t) estimation;
FIG. 6 shows state x of the dual-input dual-output delay state space system of the present invention4(t) estimation.
Detailed Description
The technical solutions in the examples of the present invention are clearly and completely described below with reference to the drawings in the examples of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without inventive step, are within the scope of the present invention.
The present invention will be described in further detail with reference to the accompanying drawings.
Examples
The method comprises the following steps of representing the structure of a time delay state space model in the CSTR reactor by using a double-input double-output form, wherein the specific expression of the model is as follows:
x(t+1)＝Ax(t)+Bx(t-d)+Fu(t), (1)
y(t)＝Cx(t)+v(t). (2)
the present invention contemplates an exothermic reaction in a CSTR wherein x (t) e RnThe method comprises the following steps of (1) selecting an unmeasured state variable, n being a real number, t being time, and taking the flow rate of a cooling liquid in a CSTR reactor as an input variable of a model: u (t) ═ u1(t),u2(t)]T∈R2Product concentration as output variable: y (t) ═ y1(t),y2(t)]T∈R2，v(t)＝[v1(t),v2(t)]T∈R2Is white noise in the CSTR reactor, and the model has time delay d due to the time delay reaction of the sensor in the CSTR reactor, A belongs to the Rn×n，B∈Rn×n，F∈Rn ×2And C ∈ R2×nIs a system parameter matrix to be identified;
since the model is a multivariate system, direct identification is difficult and decomposition of the model is required. Based on the idea of identification model decomposition, a double-input double-output model is decomposed into two double-input single-output subsystems, each subsystem has smaller dimensions and variables, and each subsystem is calculated. Since the models (1) - (2) contain the system's unknown parameter vectors/matrices and unmeasurable state vectors, which are difficult to identify, the present invention represents the state vectors with measurable inputs and outputs. First of all the first sub-system is analyzed,
wherein, y1(t+i),y1(t+n1) At time t + i, t + n1Output of the first subsystem of time, n1Is a real number, e1,A1,A12,B1,F1Are all model parameters, x is the state, u is the input, and v is white noise.
Some vectors/matrices are defined:
wherein, Yi(t+n1) To output a vector, Ui(t+n1) For the input vector, X (t-d + n)i) Is a state vector, Vi(t+ni) As a noise vector, Mi,QiMatrix parameters, n, which are modelsiAre real numbers.
From equations (3) - (4) one can derive
Y1(t+n1)＝Tx1(t)+M1X(t-d+n1)+Q1U1(t+n1)+V1(t+n1).
Wherein, Y1(t+n1) Is the output vector of the first subsystem, X (t-d + n)1) Is the state vector of the first subsystem, U1(t+n1) Is the input vector of the first subsystem, V1(t+n1) Is the noise vector of the first subsystem, T is the observable matrix, M1,Q1Is the matrix parameter of the first subsystem.
To obtain constant parameter estimates, an information vector is defined as
The parameter vector is theta1
Wherein,
are all information vectors, θ1,θ11,θ12,θ13Are all parameter vectors, U1(t+n1) For the input vector, X (t-d + n)1) Is a state vector, Y1(t+n1),Y2(t+n2) To output vector, V1(t+n1),V2(t+n2) Is a noise vector, e1,A1,Q1,F1,M1,B1As a system parameter,/1,h1As auxiliary vectors, respectively
Combining equation (4) and the above definition
Wherein, y1(t+n1) Is the system output, n1Is real number, x is system state, u is system input, v is white noise, e1,A1,A12,B1,F1As a result of the parameters of the system,
as information vectors, U1For the input vector, X (t-d + n)1) Is a state vector, θ11,θ12,θ13,θ1Is a system parameter.
Substituting t for t-n in equation (5)1It can be simplified to the following regression model,
this is an identification model of the first subsystem of the dual-input dual-output state space with time delay, and the second subsystem is derived in a similar way:
let t be the current time, { u (t), y (t): t ═ 0,1, 2. } is the measurable input-output information, y (t) and
is the current information that is being presented to the user,
is information in the past.
The following gives the parameter estimates for the system in the CSTR reactor:
the basic idea adopted by the invention is to replace the unknown noise term and the unknown state vector by the estimated residual error and the estimated state vector, defining
Is composed of
Estimation at time t. V according to equation (6)iThe estimate of (t) can be calculated as
Thus, minimizing the criterion function according to the least squares principle, when calculating one parameter vector, the remaining vectors are replaced by their estimates, the following algorithm can be derived to calculate the parameters:
wherein,
is an estimate of the parameter theta and,
as vectors of information
Is determined by the estimated value of (c),
in order to be a matrix of gains, the gain matrix,
is a covariance matrix. Because it is a multivariable system, the coupling of the system needs to be analyzed in the decomposition process, thereby realizing the decoupling of the system: make inputMultivariable systems that correlate outputs recognize that each output is controlled only by a corresponding input.
The state estimation of the system in the CSTR reactor is given below:
Yi(t)＝[yi(t-ni),yi(t-ni+1),...,yi(t-1)]T,
Ui(t)＝[uT(t-ni),uT(t-ni+1),...,uT(t-1)]T,
wherein, Yi(t) is the output vector, Ui(t) is the input vector, and,
in the form of a state vector, the state vector,
in order to be a vector of the noise,
matrix parameters, n, which are modelsiIn the case of a real number,
is a system parameter.
The state vector is represented by measurable input and output variables according to the equation of state at different times t, and a discriminative model of the system is derived. A single-input single-output model algorithm is popularized, a corresponding enhanced least square algorithm based on residual errors is deduced, and the estimated parameters are used for calculating the state of the system.
The model can be applied to a reaction kettle system (CSTR), influences of various input and output interferences on the feeding and discharging of the CSTR system, and verifies the convergence and the effectiveness of the proposed method on the phenomena of data loss and uncertain time delay in the continuous reaction process of the system. From fig. 1-5, the following conclusions can be drawn: the parameter estimation error generally becomes smaller as t increases; under the condition that the zero mean square error is the same, the parameter estimation precision is improved along with the increase of the data length t; when the noise variance is lower, the data convergence is faster; as time t increases, the state estimate approaches its true value.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (1)
1. A CSTR reactor delay system method based on state filtering and parameter estimation is characterized in that:
step 1, representing a time delay state space model in a CSTR reactor by using a double-input double-output form
x(t+1)＝Ax(t)+Bx(t-d)+Fu(t), (1)
y(t)＝Cx(t)+v(t). (2)
Consider an exothermic reaction in a CSTR, where x (t) e RnIs an unmeasurable state variable, n is a real number, t is time, cooling in a CSTR reactor is selectedLiquid flow rate as input variable for the model: u (t) ═ u1(t),u2(t)]T∈R2Product concentration as output variable: y (t) ═ y1(t),y2(t)]T∈R2，v(t)＝[v1(t),v2(t)]T∈R2Is white noise in the CSTR reactor, and the model has time delay d due to the time delay reaction of the sensor in the CSTR reactor, A belongs to the Rn×n，B∈Rn×n，F∈Rn×2And C ∈ R2×nIs a system parameter matrix to be identified;
decomposing a dual-input dual-output model into two dual-input single-output subsystems based on the identification model decomposition, wherein each subsystem has smaller dimension and variable, and each subsystem is calculated, and the method specifically comprises the following steps:
representing the state vector with measurable inputs and outputs; first of all the first sub-system is analyzed,
wherein, y1(t+i),y1(t+n1) At time t + i, t + n1Output of the first subsystem of time, n1Is a real number, e1,A1,A12,B1,F1Are all parameters of the model, x is the state, u is the input, v is white noise;
some vectors/matrices are defined:
wherein, Yi(t+n1) To output a vector, Ui(t+n1) For the input vector, X (t-d + n)i) Is a state vector, Vi(t+ni) As a noise vector, Mi,QiMatrix parameters, n, which are modelsiIs a real number;
from equations (3) - (4) one can derive
Y1(t+n1)＝Tx1(t)+M1X(t-d+n1)+Q1U1(t+n1)+V1(t+n1).
Wherein, Y1(t+n1) Is the output vector of the first subsystem, X (t-d + n)1) Is the state vector of the first subsystem, U1(t+n1) Is the input vector of the first subsystem, V1(t+n1) Is the noise vector of the first subsystem, T is the observable matrix, M1,Q1The matrix parameters of the first subsystem;
to obtain constant parameter estimates, information is defined intoMeasured as
Integral vector of parameters is theta1
Wherein,
are all information vectors, θ1As a whole parameter vector, θ11,θ12,θ13Are all parameter components, U1(t+n1) For the input vector, X (t-d + n)1) Is a state vector, Y1(t+n1),Y2(t+n2) To output vector, V1(t+n1),V2(t+n2) Is a noise vector, e1,A1,Q1,F1,M1,B1As a system parameter,/1,h1As auxiliary vectors, respectively
Combining equation (4) and the above definition
Wherein, y1(t+n1) Is the system output, n1Is real number, x is system state, u is system input, v is white noise, e1,A1,A12,B1,F1As a result of the parameters of the system,
as information vectors, U1For the input vector, X (t-d + n)1) Is a state vector, θ1As a whole parameter vector, θ11,θ12,θ13Are all parameter components;
substituting t for t-n in equation (5)1It can be simplified to the following regression model,
this is an identification model of the first subsystem of the dual-input dual-output state space with time delay, and the second subsystem is derived in a similar way:
let t be the current time, { u (t), y (t)) T is measurable input/output information, y (t) and
is the current information that is being presented to the user,
is past information;
step 2, the parameter estimation step of the system in the CSTR reactor is specifically as follows:
defining by replacing the unknown noise term and the unknown state vector by the estimated residual and the estimated state vector
Is composed of
An estimate at time t; v according to equation (6)iThe estimate of (t) can be calculated as
Thus, when one parameter vector is calculated, the remaining vectors are replaced with estimates, according to the least squares principle minimizing the criteria function, resulting in the following algorithm calculated parameters:
wherein,
in order to select an estimate of the parameter theta,
as vectors of information
Is determined by the estimated value of (c),
in order to be a matrix of gains, the gain matrix,
is a covariance matrix; because it is a multivariable system, the coupling of the system needs to be analyzed in the decomposition process, thereby realizing the decoupling of the system: a multivariable system that correlates inputs and outputs to realize that each output is controlled only by the corresponding input;
step 3, specifically, the state estimation step of the system in the CSTR reactor is as follows;
Yi(t)＝[yi(t-ni),yi(t-ni+1),...,yi(t-1)]T,
Ui(t)＝[uT(t-ni),uT(t-ni+1),...,uT(t-1)]T,
wherein, Yi(t) is the output vector, Ui(t) is the input vector, and,
in the form of a state vector, the state vector,
in order to be a vector of the noise,
matrix parameters, n, which are modelsiIs a real number, ei,
Is a system parameter.
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