CN107045490A - A kind of method for estimating state of nonlinear system - Google Patents
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Abstract
Present applicant proposes a kind of method for estimating state based on high-order volume Kalman filtering and neutral net, in the implementation process of algorithm, state-space model is set up to nonlinear system first with neutral net, then the weight of neutral net and the state variable of system are combined together as new state variable, and real-time update is carried out to new state using high-order volume Kalman filtering, so as to reach that neutral net is approached and accurate estimation to state value the true of nonlinear system model, and experiment simulation indicates the validity of the inventive method.
Description
Technical field
The application belongs to Aero-Space, intelligent transportation, pattern-recognition and engineering in medicine technical field, specifically, relates to
And a kind of method for estimating state of nonlinear system.
Background technology
In the case of non-linear system status unknown-model, the existing algorithm based on nonlinear filtering and neutral net is to non-
The precision of linear system state estimation is limited.
When system model is unknown, it is a kind of simple and effective side to carry out nonlinear approximation to model using neutral net
Method.Neutral net determines the nonlinear function that network is approached according to real data collection, however, when real system state variable not
When can survey completely, it individually will be unable to set up the model of process using neutral net.Inside shape of the state-space model method to system
Variation relation between state and outside Observable output variable is described, because this method can be to the complicated shape of internal system
State is estimated and predicted, therefore has been widely used among the processing of dynamical system.To the state space mould of system
Type is set up after completion, it is necessary to being estimated using suitable method the state of system.Filtering method is in state space
Data are observed in middle utilization system input and output, and optimal estimation is carried out to system mode.The method of this filtering not only can be to shape
State is estimated, nerve network system parameter can also be recognized.
Nonlinear filtering algorithm has obtained significant progress in parameter Estimation, particularly the parameter identification in neutral net
Aspect, it is regarded the weight coefficient parameter of network as special state and estimated.Existing method utilizes neural network shape
State space model, using network weight coefficient as system state variable, and then based on expanded Kalman filtration algorithm to state become
Amount carries out real-time update.EKF is a kind of non-linear filtering method being most widely used at present, but ought be
The nonlinear degree of system is higher, and first approximation can bring very big truncated error, the estimation of expanded Kalman filtration algorithm
It will can be greatly reduced therewith, or even diverging.Because EKF stability is poor, the low shortcoming of precision, Julier et al.
Unscented kalman filtering is proposed, the non-linear transmission of average and variance is handled using Unscented transform.There are many scholars
The non-linear system status estimation problem of unknown system model is solved using Unscented kalman filtering and neutral net.It is substantial amounts of
The simulation experiment result shows that the neural network algorithm based on Unscented kalman filtering algorithm is substantially better than based on spreading kalman filter
The neural network algorithm of ripple algorithm.But the precision of Unscented kalman filtering is still limited, and when system dimension is higher, its
Estimation performance is substantially reduced.
The content of the invention
In view of this, the application is for the unknown nonlinear system of state model, it is proposed that one kind is based on high-order volume card
The non-linear system status method of estimation of Kalman Filtering algorithm and neutral net, this method is entered using neutral net to system mode
Row modeling, and real-time update estimation is carried out to state using high-order volume Kalman filtering algorithm, the inventive method compensate for existing
There is the deficiency of algorithm.
A kind of method for estimating state of nonlinear system, comprises the following steps：
Step one：The weight coefficient of the state of nonlinear system and neutral net is combined, neural network is utilized
State-space model, is the state variable of augmentation together by the combinations of states of the weight coefficient of network and nonlinear system；
Step 2：Time renewal is carried out to nerve network system state using high-order volume Kalman filtering algorithm, and
Result after being updated according to the time measures renewal, thus realize network weight coefficient adaptive adjustment and state in real time more
Newly.
Further, method as described above, step one includes：
The weight coefficient of the state of system and neutral net is combined, augmented state x is formeda=[xW]T, then can set up
Nonlinear system as follows：
Wherein, fj(xk) it is the mathematical modeling that neutral net is set up to nonlinear system：
G (x) is neutral net Sigmod kernel functions, WkFor the weight coefficient of neutral net, the process noise w of new systemkAnd sight
Survey noise vkIt is independent zero mean Gaussian white noise, and corresponding covariance matrix is respectively Qk, Rk；
xkThe state vector at kth moment is represented,Represent the augmented state vector at kth moment, zkRepresent and represent the kth moment
Observation vector,Represent non-linear observation function.
Further, method as described above, step 2, which carries out time renewal to state, to be included：
6) in moment k, it is assumed that the error covariance at k-1 moment is known and is Pk-1|k-1, decompose factor：
Wherein, vectorial Sk-1|k-1For Pk-1|k-1Cholesky decompose；Represent Sk-1|k-1Transposition, T represents transposition
Operation；
7) volume point is calculated
Represent the estimation of the moment of kth -1 augmented state
Value；
Wherein m=2n, vectorial ξiFor
In formula,eiRepresent n dimension unit vectors and its i-th of element is 1；Respectively
8) calculate state equation propagate after volume point (i=1,2 ..., m)
9) a step status predication is calculated
Wherein, weight wiRespectively
10) one-step prediction error co-variance matrix is calculated
Further, method as described above, the result after step 2 updates according to the time measures renewal and included：
10) factor is decomposed：
Sk|k-1Represent matrix decomposition value；
11) the state volume point after updating is calculated
12) the volume point after measurement equation is propagated is calculated
13) calculate the step of k moment one and measure prediction
14) new breath covariance matrix is calculated
15) one-step prediction Cross-covariance is calculated
16) gain matrix is calculated
17) more new state is calculated
18) covariance matrix is calculated
Compared with prior art, the application can be obtained including following technique effect：
The present invention is directed to the nonlinear system of unknown state model, first with neural network state-space model,
It is the state variable of augmentation together by the combinations of states of the weight coefficient of network and system, then using high-order volume Kalman filtering
Algorithm to nerve network system state carry out estimation prediction so that realize network weight coefficient adaptive adjustment and state it is real-time
Update, improve the estimated accuracy of non-linear system status, emulation experiment indicates the validity that the present invention carries algorithm.
Brief description of the drawings
Accompanying drawing described herein is used for providing further understanding of the present application, constitutes the part of the application, this Shen
Schematic description and description please is used to explain the application, does not constitute the improper restriction to the application.In the accompanying drawings：
Fig. 1 is the model structure of neutral net；
Fig. 2 is condition estimating system schematic diagram；
Fig. 3 is the estimation curve of state 1 in emulation experiment；
Fig. 4 is the estimation curve of state 2 in emulation experiment；
Fig. 5 is the error curve of state 1 in emulation experiment；
Fig. 6 is the error curve of state 2 in emulation experiment.
Embodiment
Describe presently filed embodiment in detail below in conjunction with drawings and Examples, thereby how the application is applied
Technological means can fully understand and implement according to this to solve technical problem and reach the implementation process of technology effect.
The state-space model of neutral net
The structural model of neutral net generally can be divided into BP network model, back propagation neural network model and random
Neural network model etc..BP network model is most widely used in industry-by-industry field at present, its state space
Model structure is as shown in figure 1, wherein, x1,x2,...xnFor representing the sample node of input, y1,y2,...ymFor representing defeated
Go out sample node, θ1,θ2,...θqWeight coefficient is represented, the Artificial Neural Network Structures have three node layers, respectively inputted
It is attached between layer, hidden layer and output layer, each layer by weight coefficient, input and output layer is at two ends, the section of middle hidden layer
Points are chosen according to actual requirement.
High-order volume Kalman filter theory
Following Discrete-time Nonlinear Systems are considered first：
xk=f (xk-1)+wk (1)
zk=h (xk)+vk (2)
Wherein, xkThe state vector tieed up for n；zkThe observation vector tieed up for m；Function f, h are known nonlinear function；
{wkAnd { vkIt is independent zero mean Gaussian white noise.
For general nonlinear system, Bayesian Estimation basic theories and arbitrary order volume can be advised under gaussian assumptions
Then it is combined, so as to derive the volume Kalman filtering of high-order.It is identical with Unscented kalman filtering structure, equally it is divided into state
(time renewal) and two steps of measurement updaue are predicted, mainly high-order volume Kalman filtering uses phase footpath volume rule
Solve High Dimensional Systems dimension explosion issues.High-order volume rule is met：
In formula, ejFor n-dimensional space RnUnit vector matrix jth row.WithFor the point set being shown below：
Weight coefficientWithRespectively
Wherein,It is the surface area of unit sphere,According to square
Matching method, as n=2, weights are：
State estimation based on high-order volume Kalman filtering and neutral net
When the unknown-model of system, system model is modeled using neutral net and approached, then be accomplished by solving
Optimal network node weight coefficient, while state is also unknown, however, state is communication with one another with weight coefficient, therefore, this
The state and weight coefficient of system are combined together as new state by invention, by original system equation and weight coefficient equation
Augmented equation is estimated in real time as new system model, and then using high-order volume Kalman filtering algorithm to state and weight coefficient
Meter, specific system principle is as shown in Figure 2.Among Fig. 2, at k-1 moment, the state x of original systemk-1With weight coefficient Wk-1It is combined into
New augmented state vector [xk-1Wk-1], it is input in neutral net, and then high-order volume Kalman filtering is according to augmented state
Time renewal step, the result that last high-order volume Kalman filtering binding time updates are carried out with the output result of neutral net
The system augmented state value [x that time renewal obtains the k moment is carried out with the export structure of measurementk Wk], and it is used as lower subsystem
Input state.Specific algorithm is as follows：
The weight coefficient of the state of system and neutral net is combined first, augmented state x is formeda=[xW]T, then may be used
Set up nonlinear system as follows：
Wherein, fj(xk) it is the mathematical modeling that neutral net is set up to nonlinear system：
G (x) is neutral net Sigmod kernel functions, WkFor the weight coefficient of neutral net, the process noise w of new systemkAnd sight
Survey noise vkIt is independent zero mean Gaussian white noise, and corresponding covariance matrix is respectively Qk, Rk。
Then time renewal is carried out to state：
1) in moment k, it is assumed that the error covariance at k-1 moment is known and is Pk-1|k-1, decompose factor：
Wherein, vectorial Sk-1|k-1For Pk-1|k-1Cholesky decompose.
2) volume point is calculated
Wherein m=2n, vectorial ξiFor
In formula,eiRepresent n dimension unit vectors and its i-th of element is 1.Respectively
3) calculate state equation propagate after volume point (i=1,2 ..., m)
4) a step status predication is calculated
Wherein, weight wiRespectively
5) one-step prediction error co-variance matrix is calculated
Finally measure renewal：
19) factor is decomposed：
20) the state volume point after updating is calculated
21) the volume point after measurement equation is propagated is calculated
22) calculate the step of k moment one and measure prediction
23) new breath covariance matrix is calculated
24) one-step prediction Cross-covariance is calculated
25) gain matrix is calculated
26) more new state is calculated
27) covariance matrix is calculated
For the known nonlinear system described by formula (9) (10), given state primary conditionP0|0, you can root
Updated according to the above-mentioned time and two steps of measurement updaue carry out high-order volume Kalman filtering, obtain the state vector of augmentation
Value.
Emulation experiment
Consider following Nonlinear Systems ' Discrete model
Y (k)=x1(k)+x2(k)+v(k) (30)
Wherein, process noise w (k) and observation noise v (k) are separate zero mean Gaussian white noises, and variance
RespectivelyWith R (k)=0.1, original state x0=[10 0.6]T, Initial state estimation value isAnd original state error co-variance matrix isNeutral net the number of hidden nodes is 10,
Selection of kernel function Sigmod types, initial weight is set to the random noise that variance is 0.3, and drift variance matrix is Qw=0.022I40×40。
In order to contrast conveniently, the present invention carries out following simple marking to following algorithm：
Algorithm 1：Algorithm for estimating based on high-order volume Kalman filtering and neutral net
Algorithm 2：Algorithm for estimating based on Unscented kalman filtering and neutral net
Simulation result is as shown in Fig. 1-Fig. 6 and table 1.
The evaluated error of table 1 is contrasted
| Average absolute evaluated error | Algorithm 1 | Algorithm 2 |
| State 1 | 0.1835 | 0.2830 |
| State 2 | 0.2202 | 0.5663 |
From the point of view of Fig. 3 and Fig. 4 estimation curve, algorithm 1 and algorithm 2 can carry out preferably tracking to reset condition and estimate
Meter, it is all effective to illustrate two kinds of algorithms, and from the point of view of Fig. 4 and Fig. 5 evaluated error curve, the error of two kinds of algorithms tends to quickly
It is stable, and the error of algorithm 1 is substantially less than the error of algorithm, from the point of view of the statistics of table 1, the state estimation essence of algorithm 1
Degree is much higher than algorithm 2, and particularly in the estimated accuracy to state 2, the evaluated error of algorithm 2 is the evaluated error of algorithm 1
More than twice, this estimated accuracy for being primarily due to high-order volume Kalman filtering algorithm is higher than Unscented kalman filtering algorithm, from
And illustrate the validity of the algorithm for estimating based on high-order volume Kalman filtering and neutral net.
Some preferred embodiments of the application have shown and described in described above, but as previously described, it should be understood that the application
Presently disclosed form is not limited to, the exclusion to other embodiment is not to be taken as, and available for various other groups
Close, change and environment, and the technology or knowledge of above-mentioned teaching or association area in the application contemplated scope, can be passed through and carry out
Change., then all should be in the application institute and the change and change that those skilled in the art are carried out do not depart from spirit and scope
In attached scope of the claims.
Claims (4)
1. a kind of method for estimating state of nonlinear system, it is characterised in that comprise the following steps：
Step one：The weight coefficient of the state of nonlinear system and neutral net is combined, neural network state is utilized
Spatial model, is the state variable of augmentation together by the combinations of states of the weight coefficient of network and nonlinear system；
Step 2：Using high-order volume Kalman filtering algorithm to the progress time renewal of nerve network system state, and according to
Result after time renewal measures renewal, so as to realize adaptive adjustment and the real-time update of state of network weight coefficient.
2. according to the method described in claim 1, it is characterised in that step one includes：
The weight coefficient of the state of system and neutral net is combined, augmented state x is formeda=[x W]T, then can set up as follows
Shown nonlinear system：
Wherein, fj(xk) it is the mathematical modeling that neutral net is set up to nonlinear system：
G (x) is neutral net Sigmod kernel functions, WkFor the weight coefficient of neutral net, the process noise w of new systemkMade an uproar with observation
Sound vkIt is independent zero mean Gaussian white noise, and corresponding covariance matrix is respectively Qk, Rk；
xkThe state vector at kth moment is represented,Represent the augmented state vector at kth moment, zkRepresent the sight for representing the kth moment
Direction finding amount,Represent non-linear observation function.
3. according to the method described in claim 1, it is characterised in that step 2, which carries out time renewal to state, to be included：
1) in moment k, it is assumed that the error covariance at k-1 moment is known and is Pk-1|k-1, decompose factor：
Wherein, vectorial Sk-1|k-1For Pk-1|k-1Cholesky decompose；Represent Sk-1|k-1Transposition, T represent transposition operation；
2) volume point is calculated
Represent the estimation of the moment of kth -1 augmented state
Value；
Wherein m=2n, vectorial ξiFor
In formula,eiRepresent n dimension unit vectors and its i-th of element is 1；Respectively
3) calculate state equation propagate after volume point (i=1,2 ..., m)
4) a step status predication is calculated
Wherein, weight wiRespectively
5) one-step prediction error co-variance matrix is calculated
。
4. according to the method described in claim 3, it is characterised in that the result after step 2 updates according to the time measures renewal
Including：
1) factor is decomposed：
Sk|k-1Represent matrix decomposition value；
2) the state volume point after updating is calculated
3) the volume point after measurement equation is propagated is calculated
4) calculate the step of k moment one and measure prediction
5) new breath covariance matrix is calculated
6) one-step prediction Cross-covariance is calculated
7) gain matrix is calculated
8) more new state is calculated
9) covariance matrix is calculated
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| CN107992877A (en) * | 2017-10-11 | 2018-05-04 | 衢州学院 | Two benches high-order volume information filtering method |
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| CN107992877B (en) * | 2017-10-11 | 2020-05-19 | 衢州学院 | Two-stage high-order volume information filtering method |
| CN107992877A (en) * | 2017-10-11 | 2018-05-04 | 衢州学院 | Two benches high-order volume information filtering method |
| CN108008632A (en) * | 2017-12-11 | 2018-05-08 | 东北石油大学 | A kind of method for estimating state and system of the time lag Markov system based on agreement |
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| CN109088749A (en) * | 2018-07-23 | 2018-12-25 | 哈尔滨理工大学 | The method for estimating state of complex network under a kind of random communication agreement |
| CN109088749B (en) * | 2018-07-23 | 2021-06-29 | 哈尔滨理工大学 | State estimation method of complex network under random communication protocol |
| CN109828211A (en) * | 2018-12-25 | 2019-05-31 | 宁波飞拓电器有限公司 | A kind of emergency light battery SOC estimation method based on neural network adaptive-filtering |
| CN109459040A (en) * | 2019-01-14 | 2019-03-12 | 哈尔滨工程大学 | More AUV co-located methods based on RBF neural auxiliary volume Kalman filtering |
| CN109612470A (en) * | 2019-01-14 | 2019-04-12 | 广东工业大学 | A kind of single station passive navigation method based on fuzzy volume Kalman filtering |
| CN109459040B (en) * | 2019-01-14 | 2021-06-18 | 哈尔滨工程大学 | Multi-AUV (autonomous Underwater vehicle) cooperative positioning method based on RBF (radial basis function) neural network assisted volume Kalman filtering |
| CN109920514A (en) * | 2019-03-11 | 2019-06-21 | 重庆科技学院 | A kind of self-closing disease based on Kalman filtering neural network embraces body and tests evaluation method and system |
| CN111193528A (en) * | 2019-12-30 | 2020-05-22 | 哈尔滨工业大学 | Gaussian filtering method based on non-linear network system under non-ideal condition |
| CN111381498A (en) * | 2020-03-09 | 2020-07-07 | 常熟理工学院 | Expectation maximization identification method of multi-sensor based on multi-rate variable time-lag state space model |
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Application publication date: 20170815 |