CN107844834A - A kind of nonlinear system modeling method based on mixed model - Google Patents

A kind of nonlinear system modeling method based on mixed model Download PDF

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CN107844834A
CN107844834A CN201710857386.1A CN201710857386A CN107844834A CN 107844834 A CN107844834 A CN 107844834A CN 201710857386 A CN201710857386 A CN 201710857386A CN 107844834 A CN107844834 A CN 107844834A
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彭辉
徐文权
曾小勇
周锋
田晓盈
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Central South University
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Abstract

The invention discloses a kind of nonlinear system modeling method based on mixed model, when carrying out system modelling and parameter Estimation, system is divided into linear segment and non-linear partial, linear segment is estimated by least square method using linear AR models, the residual error that is drawn afterwards will be predicted to linear segment reuses DBN AR models and be fitted.The final predicted value of system is to be added to obtain using both value and value using DBN AR models fittings of linear AR model predictions.The present invention is greatly improved the precision of prediction of nonlinear system, and improves the robustness of system.Such mixed model can be used for data prediction, and the design of controller can be given to provide reliable system modeling method, be with a wide range of applications and practical value.

Description

A kind of nonlinear system modeling method based on mixed model
Technical field
It is particularly a kind of based on the non-linear of mixed model the present invention relates to actual engineering design and parameter optimization field System modeling method.
Background technology
In the research and application of natural phenomena parsing and the industrial processes monitoring etc. of reality, it is widely present and is difficult to The situation for the mechanism mathematical model that can be used for describing system dynamic characteristic is obtained, thus, usually using the modeling side of data-driven Method builds model to describe the dynamic characteristic of system.Linear part and non-thread is usually contained in the dynamic characteristic of real system Property part characteristic, be sometimes difficult to well the linear segment in processing data simultaneously and non-thread using single forecast model Property part.Sampling linear AR models, linear portion that can well in forecasting system data by least square method (LSM) It is intrinsic.Depth belief network (DBN) is a kind of model more commonly used in deep learning algorithm, is had stronger non-linear Ability in feature extraction, and convergence rate is slow when being avoided that traditional neural network parameter optimization and is easily trapped into local optimum Limitation.DBN model has been applied successfully to multiple necks such as forecasting traffic flow, weather forecasting, function approximation and signal transacting Domain, have become a kind of model for being used to portray nonlinear characteristic popular at present.But due to the nature of reality The complexity of phenomenon and industrial process etc., samples single DBN model and describes its nonlinear characteristic and there may come a time when to make The modeling effect of people's satisfaction.The stronger DBN nonlinear function approximation capability of DBN-AR models couplings, and State-Dependent AR Model can more efficiently apply to lead Nonlinear Time Series system modelling etc. to the descriptive power of nonlinear characteristic Domain.Linear AR models are combined with DBN-AR models, first sampling linear AR models, by least square method to data It is modeled and predicts, draw prediction residual, reuses DBN-AR models and residual error portion is fitted, mixed model is most Whole predicted value is to be added to obtain using both the predicted value that linear AR models to obtain and match value of DBN-AR models.It is such a to incite somebody to action The method that linear segment and non-linear partial are separately modeled and predicted, the prediction essence of constructed model system can be greatly improved Degree and robustness.
The content of the invention
The present invention is intended to provide a kind of nonlinear system modeling method based on mixed model, improves the pre- of nonlinear system Precision is surveyed, and improves the robustness of system.
In order to solve the above technical problems, the technical solution adopted in the present invention is：It is a kind of based on the non-linear of mixed model System modeling method, comprise the following steps：
1) linear AR models are used, the linear segment of data are modeled using least square method, according to AIC criterion It is determined that the order of linear AR models is p, linear AR models are obtained；
2) predicted value of the system based on linear AR models is drawn using least square method, actual value and predicted value is subtracted each other Obtain cutting edge aligned AR modeling residual error e (t)；
3) the linear AR modelings residual error is fitted using DBN-AR models；Linear DBN- is determined according to AIC criterion The order of AR models is q；
4) the expectation target value of each DBN modules in the DBN-AR models in pre-training stage is determined；
5) according to the expectation target value of each DBN modules of acquisition, pre-training is carried out to the parameter of all DBN modules；
If 6) if pre-training meets actual requirement, deconditioning, otherwise change each DBN modules in DBN-AR models The implicit number of plies, each hidden layer neuron number, learning rate；Return to step 5) re -training；Until meeting DBN-AR models Pre-training demands, obtain the prediction output valve of DBN-AR models
7) whole DBN-AR models are finely adjusted using back-propagation algorithm, by making following object function for most It is small, obtain the optimized parameter of model：Wherein：N is data length, and θ is institute in DBN-AR models There is parameter to be optimized, q is the input signal order of non-linear DBN-AR models；
8) according to the parameter picked outCalculate the final predicted value of DBN-AR models
9) by predicted value and the final predicted value of DBN-AR models based on linear AR modelsIt is added, obtains based on mixed The output predicted value of matched moulds type.
In step 1), the expression formula of linear AR models is as follows：
Wherein, y (t) is the data of system t, and { y (t-1), y (t-2) ..., y (t-p) } is that system goes over p moment Historical data, p be linear AR models order, αiIt is for linear AR models parameter to be identified, i=0,1 ..., p, e (t) Linear AR model modeling residual errors；
In step 2), AR modeling residual error e (t) expression formula is as follows：Wherein, The predicted value of the system linear part obtained for linear AR model predictions.In step 3), DBN-AR model expressions are as follows：
Wherein：For the modeling residual error of the linear AR models of t；For the modeling that linear AR models are last Error；Q is the input signal order of non-linear DBN-AR models；NhFor the hidden layer number of plies of DBN modules；wstFor visible layer with Connection weight between hidden layer；DBN activation primitive isM is visible layer nodes；N is hidden layer node Number；φ0(X (t-1)) is the output valve of the 1st DBN module in DBN-AR models；φj(X (t-1)) is jth in DBN-AR models The output valve of+1 DBN module；For the output of d-th of hidden layer of the 1st DBN module in DBN-AR models Value；For the output valve of d-th of hidden layer of+1 DBN module of jth in DBN-AR models；For DBN-AR moulds The biasing of d-th of hidden layer of the 1st DBN module in type；D-th for+1 DBN module of jth in DBN-AR models is hidden Biasing containing layer；{ e (t-1) e (t-2) ... e (t-q) } is the history value that modeling residual error goes over q moment；X (t-1) is DBN- The input value of AR models.
In step 4), the expectation target value of each DBN modules in DBN-AR models is determined using following formula：Φ=Ψ+Τ； Wherein：
Wherein, e (q+a), a=1,2 ..., N are the value at AR model residual error q+a moment, and e (N-b), b=1,2 ..., N-1 are Value of the AR models residual error at the N-1 moment；Τ is the desired output of DBN-AR models；X (r)=(e (r), e (r-1) ..., e ), (r-q+1) r=q, q+1 ..., N-1 is input value of the DBN-AR models at the r moment；Ψ is the history of mission nonlinear part Value；Φ is the desired output of each DBN modules in DBN-AR models； Φc(X (r)), c=0,1 ..., q is at the r moment The output valve of each DBN modules when DBN-AR mode inputs value is X (r)；Ψ+For Ψ pseudo inverse matrix；N is data length.
In step 5), the prediction output valve of DBN-AR models
In step 7), the final predicted value of DBN-AR modelsExpression formula be：
Wherein,For the final output of each DBN modules in the DBN-AR models finally obtained Value.
Compared with prior art, the advantageous effect of present invention is that：The present invention is greatly improved nonlinear system Precision of prediction, and improve the robustness of system.Such mixed model can be used for data prediction, and can give the design of controller Reliable system modeling method is provided, is with a wide range of applications and practical value.
Brief description of the drawings
Fig. 1 is DBN-AR identification of Model Parameters flow charts；
Fig. 2 is present system structured flowchart.
Embodiment
1) present invention is modeled and predicted to data linear parts using linear AR models, using least square method, Determine that the order of linear AR models is p according to AIC (Akaike information criterion, AIC) criterion；Obtain and be The predicted value of system is
Wherein αj(j=0,1 ..., p) is parameter to be identified,For the output of system, e (t) is linear AR models Modeling error；
2) the modeling residual values drawn after the linear AR model predictions obtained using Least Square Method are
3) residual error is fitted using DBN-AR models；Select order q, the DBN-AR model structure of the model as follows It is shown：
Wherein：For the modeling residual error of the linear AR models of t；For the modeling that linear AR models are last Error；Q is the input signal order of non-linear DBN-AR models；NhFor the hidden layer number of plies of DBN modules；wstFor visible layer with Connection weight between hidden layer；DBN activation primitive isM is visible layer nodes；N is hidden layer node Number；φ0(X (t-1)) is the output valve of the 1st DBN module in DBN-AR models；φj(X (t-1)) is jth in DBN-AR models The output valve of+1 DBN module；For the output of d-th of hidden layer of the 1st DBN module in DBN-AR models Value；For the output valve of d-th of hidden layer of+1 DBN module of jth in DBN-AR models；For DBN-AR models In the 1st DBN module d-th of hidden layer biasing；D-th for+1 DBN module of jth in DBN-AR models is implicit The biasing of layer；{ e (t-1) e (t-2) ... e (t-q) } is the history value that modeling residual error goes over q moment；X (t-1) is DBN-AR The input value of model.
4) the expectation target value of each DBN modules in the DBN-AR models in pre-training stage is determined
Wherein, e (q+a), a=1,2 ..., N are the value at AR model residual error q+a moment, and e (N-b), b=1,2 ..., N-1 are Value of the AR models residual error at the N-1 moment；Τ is the desired output of DBN-AR models；X (r)=(e (r), e (r-1) ..., e (r-q+1)) (r=q, q+1 ..., N-1) is the input value of DBN-AR models；Ψ is the history value of mission nonlinear part；Φ For the desired output of each DBN modules in DBN-AR models； Φc(X (r)) (c=0,1 ..., q) it is DBN-AR mode inputs Be worth for X (r) when each DBN modules output valve； Ψ+For Ψ pseudo inverse matrix；N is data length.
5) according to the expectation target value of each DBN modules of acquisition, pre-training is carried out to the parameter of all DBN modules, if Meet index, then deconditioning, otherwise return step 4), readjust the hidden layer of each DBN modules in DBN-AR models The parameters such as number, each hidden layer neuron number, learning rate；Pre-training demands until meeting DBN-AR models；And obtain The prediction output valve of DBN-AR models：
6) whole DBN-AR models are finely adjusted using back-propagation algorithm, by making following object function for most It is small, obtain the optimized parameter of model
Wherein：N is data length, and θ is all parameters to be optimized in DBN-AR models；For in specification step (5) Calculated DBN-AR model predication values.
7) finally according to the parameter picked outThe final predicted value for calculating DBN-AR models is
The different order q values of selection in specification step 3)；Repeat specification book step 3) is to 7)；AIC is calculated respectively (Akaike information criterion, AIC) value；The order in the case that AIC values are minimum is selected as DBN- The final order of AR models；
8) according to the DBN-AR mode input orders finally determined.Step 3) is performed to step 7), is obtained by last estimation DBN-AR models calculate the predicted value of residual error.
9) predicted value of General Linear part and non-linear partial, the predicted value that both are added the system that draws are
The present invention illustrates a kind of nonlinear system modeling side based on mixed model by taking following nonlinear system as an example The embodiment of method.The present invention can be combined with different embodiments and be carried out and use, in without departing from the present invention Various modifications or alterations are carried out under the premise of appearance spirit.
Y (t)=2 [sin (y (t-1))] cos (y (t-2))+2 [sin (y (t-2))] cos (y (t-5))+ξ (t)
WhereinFor in the output of t system, ξ (t) is white Gaussian noise signal；
Assuming that produced in t white Gaussian noise signal xi (t) by pseudo-random signal.
N=1000 continuous time series datas are produced using above formula；
1) first by the linear AR models of least squares identification, model order is determined according to AIC criterion；Assuming that use { y (t-1), y (t-2), y (t-3), y (t-4), y (t-5) } prediction y (t), selects p=5, obtaining cutting edge aligned AR model predication values isResidual error is
2) residual error is fitted using DBN-AR models, according to the order of AIC criterion preference pattern；In this embodiment party Q=5 is selected in formula；
DBN-AR models are
Wherein：For the modeling residual error of the linear AR models of t；For the modeling that linear AR models are last Error；Q is the input signal order of non-linear DBN-AR models；NhFor the hidden layer number of plies of DBN modules；wstFor visible layer with Connection weight between hidden layer；DBN activation primitive isM is visible layer nodes；N is hidden layer node Number；φ0(X (t-1)) is the output valve of the 1st DBN module in DBN-AR models；φj(X (t-1)) is jth in DBN-AR models The output valve of+1 DBN module；For the output of d-th of hidden layer of the 1st DBN module in DBN-AR models Value；For the output valve of d-th of hidden layer of+1 DBN module of jth in DBN-AR models；For DBN-AR models In the 1st DBN module d-th of hidden layer biasing；D-th for+1 DBN module of jth in DBN-AR models is implicit The biasing of layer；{ e (t-1) e (t-2) ... e (t-5) } is the history value that modeling residual error goes over 5 moment；X (t-1) is DBN-AR The input value of model.
3) the expectation target value of each DBN modules in the DBN-AR models in pre-training stage is determined
Φ=Ψ+Τ
Wherein：
Wherein, e (q+a) (a=1,2 ..., N) be the AR model residual error q+a moment value, e (N-b) (b=1,2 ..., N-1) For AR models residual error the N-1 moment value；Τ is the desired output of DBN-AR models；X (r)=(e (r), e (r-1) ..., E (r-q+1)) (r=q, q+1 ..., N-1) be DBN-AR models input value；Ψ is the history value of mission nonlinear part；Φ For the desired output of each DBN modules in DBN-AR models； Φc(X (r)) (c=0,1 ..., q) it is DBN-AR mode inputs Be worth for X (r) when each DBN modules output valve； Ψ+For Ψ pseudo inverse matrix；N is data length.
4) the implicit number of plies of each DBN modules in DBN-AR models, each hidden layer neuron number, learning rate etc. are determined Parameter；
5) pre-training is carried out to each DBN modules in DBN-AR models, if meeting pre-training index, i.e. pre-training stage Mean square deviation meets actual requirement, then deconditioning, otherwise returns step 4), each parameter in readjusting 4), until Meet the pre-training stage index request of DBN-AR models；The predicted value that DBN-AR models are obtained in the pre-training stage is
6) whole DBN-AR models are finely adjusted using back-propagation algorithm, by making following object function for most It is small, obtain the optimized parameter of model
Wherein：N is data length, and θ is all parameters to be optimized in DBN-AR models；For according to step 4) and step It is rapid 5) calculated by DBN-AR model predication values.
If 7) step 6) reaches specified final index request, the i.e. mean square deviation of terminal stage, then stop circulation, otherwise Book step 4) and step 6) are repeated, until meeting final requirement；
8) parameter picked out according to specification step 6)The predicted value for calculating DBN-AR models is
Wherein,For the final output of each DBN modules in the DBN-AR models finally obtained Value；
9) the final predicted value of data is linear segment predicted valueWith non-linear partial predicted valueBoth are added Arrive；
It is worth noting that data are predicted using a kind of nonlinear system modeling method based on mixed model, its In used pseudo inverse matrix in the pre-training stage of non-linear partial, DBN-AR models to determine the expectation of each DBN modules Desired value, reuse back-propagation algorithm and whole DBN-AR models are finely adjusted, improve parameter search efficiency, improve The predictive ability of DBN-AR models.The present invention is applied to data prediction and controller design based on such mixed model, has Higher practical value and wide application prospect.

Claims (7)

  1. A kind of 1. nonlinear system modeling method based on mixed model, it is characterised in that comprise the following steps：
    1) sampling linear AR models, the linear segment of data is modeled using least square method, line is determined according to AIC criterion The order of property AR models is p, obtains linear AR models；
    2) predicted value of the system based on linear AR models is drawn using least square method, actual value and predicted value is subtracted each other into obtain outlet Property AR modeling residual error e (t)；
    3) the linear AR modelings residual error is fitted using DBN-AR models；Linear DBN-AR moulds are determined according to AIC criterion The order of type is q；
    4) the expectation target value of each DBN modules in the DBN-AR models in pre-training stage is determined；
    5) according to the expectation target value of each DBN modules of acquisition, pre-training is carried out to the parameter of all DBN modules；
    If 6) pre-training meets actual requirement, deconditioning, otherwise change the hidden layer of each DBN modules in DBN-AR models Several, each hidden layer neuron number, learning rate；Return to step 5) re -training；Pre-training rank until meeting DBN-AR models Duan Yaoqiu, obtain the prediction output valve of DBN-AR models
    7) whole DBN-AR models are finely adjusted using back-propagation algorithm, by making following object function be obtained for minimum Obtain the optimized parameter of model：Wherein：N is data length, and θ is excellent by needing in DBN-AR models The parameter of change, q are the input signal order of non-linear DBN-AR models；
    8) according to the parameter picked outCalculate the final predicted value of DBN-AR models
    9) by predicted value and the final predicted value of DBN-AR models based on linear AR modelsIt is added, obtains being based on hybrid guided mode The output predicted value of type.
  2. 2. the nonlinear system modeling method according to claim 1 based on mixed model, it is characterised in that step 1) In, the expression formula of linear AR models is as follows：
    Wherein, y (t) is the data of system t, and { y (t-1), y (t-2) ..., y (t-p) } is that system goes over going through for p moment History data, p be linear AR models order, αiFor linear AR models parameter to be identified, i=0,1 ..., p, e (t) is linear AR model modeling residual errors；
  3. 3. the nonlinear system modeling method according to claim 2 based on mixed model, it is characterised in that step 2) In, AR modeling residual error e (t) expression formula is as follows：Wherein, For linear AR The predicted value for the system linear part that model prediction obtains.
  4. 4. the nonlinear system modeling method according to claim 3 based on mixed model, it is characterised in that step 3) In, DBN-AR model expressions are as follows：
    Wherein：For the modeling residual error of the linear AR models of t；For the modeling error that linear AR models are last；q For the input signal order of non-linear DBN-AR models；NhFor the hidden layer number of plies of DBN modules；wstFor visible layer and hidden layer it Between connection weight；DBN activation primitive isM is visible layer nodes；N is node in hidden layer；φ0(X (t-1) it is) output valve of the 1st DBN module in DBN-AR models；φj(X (t-1)) is+1 DBN of jth in DBN-AR models The output valve of module；For the output valve of d-th of hidden layer of the 1st DBN module in DBN-AR models；For the output valve of d-th of hidden layer of+1 DBN module of jth in DBN-AR models；For in DBN-AR models The biasing of d-th of hidden layer of 1 DBN module；For+1 DBN module of jth in DBN-AR models d-th of hidden layer it is inclined Put；{ e (t-1) e (t-2) ... e (t-q) } is the history value that modeling residual error goes over q moment；X (t-1) is the defeated of DBN-AR models Enter value.
  5. 5. the nonlinear system modeling method according to claim 4 based on mixed model, it is characterised in that step 4) In, the expectation target values of each DBN modules in DBN-AR models is determined using following formula：Φ=Ψ+Τ；Wherein：
    Wherein, e (q+a), a=1,2 ..., N are the value at AR model residual error q+a moment, and e (N-b), b=1,2 ..., N-1 are AR moulds Value of the type residual error at the N-1 moment；Τ is the desired output of DBN-AR models；X (r)=(e (r), e (r-1) ..., e (r-q+ ), 1) r=q, q+1 ..., N-1 is input value of the DBN-AR models at the r moment；Ψ is the history value of mission nonlinear part；Φ For the desired output of each DBN modules in DBN-AR models；Φc(X (r)), c=0,1 ..., q is in r moment DBN-AR moulds The output valve of each DBN modules when type input value is X (r)；Ψ+For Ψ pseudo inverse matrix；N is data length.
  6. 6. the nonlinear system modeling method according to claim 5 based on mixed model, it is characterised in that step 5) In,
    The prediction output valve of DBN-AR models
  7. 7. the nonlinear system modeling method according to claim 6 based on mixed model, it is characterised in that step 7) In, the final predicted value of DBN-AR modelsExpression formula be：
    Wherein,For the final output value of each DBN modules in the DBN-AR models finally obtained.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108960496A (en) * 2018-06-26 2018-12-07 浙江工业大学 A kind of deep learning traffic flow forecasting method based on improvement learning rate
CN111612226A (en) * 2020-05-12 2020-09-01 中国电子科技集团公司电子科学研究院 Group daily average arrival number prediction method and device based on hybrid model

Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103473477A (en) * 2013-09-29 2013-12-25 哈尔滨工业大学 Variable parameter iterative estimation method based on improved Kalman filtering

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103473477A (en) * 2013-09-29 2013-12-25 哈尔滨工业大学 Variable parameter iterative estimation method based on improved Kalman filtering

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108960496A (en) * 2018-06-26 2018-12-07 浙江工业大学 A kind of deep learning traffic flow forecasting method based on improvement learning rate
CN108960496B (en) * 2018-06-26 2021-07-23 浙江工业大学 Deep learning traffic flow prediction method based on improved learning rate
CN111612226A (en) * 2020-05-12 2020-09-01 中国电子科技集团公司电子科学研究院 Group daily average arrival number prediction method and device based on hybrid model

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Application publication date: 20180327