CN101873121A - Method for processing signals of non-linear dynamic system on basis of histogram estimation particle filtering algorithm - Google Patents

Abstract

The invention provides a method for processing the signals of a non-linear dynamic system on the basis of the histogram estimation particle filtering algorithm. The state space model of the non-linear dynamic system is as follows: xk=f(xk-1)+vk-1, and zk=h(xk)+nk, wherein xk and zk respectively represent the state value and observed value of the system at the time k; f(xk-1) and h(xk) respectively represent the state transition equation and observation equation of the system; and vk-1 and nk respectively represent the system noise and observation noise. The state space is divided into boxes, the prior probability density and the posterior probability density of the system are estimated upon the division, and the re-sampling process is carried out in the boxes. Accordingly, the method of the invention is capable of better estimating the posterior probability density of the system, ensuring the good diversity of the particles and effectively processing non-linear and non-Gaussian signals.

Description

A kind of method for processing signals of non-linear dynamic system based on the histogram estimation particle filtering algorithm

Technical field

The present invention relates to the signal processing method based on particle filter algorithm, claimed technical scheme belongs to signal processing, artificial intelligence and computer vision field.

Background technology

The state estimation problem of dynamical system relates to a lot of fields, especially signal processing, artificial intelligence and computer vision field.Traditional Kalman filtering only is applicable to linear Gauss system, and the small nonlinearity that EKF also can only answering system.Therefore, the particle filter that is applicable to non-linear, non-Gauss system receives much concern.

Particle filter is a kind of filtering method based on Monte Carlo simulation and recursion Bayesian Estimation.It adopts particle to describe state space, uses the posterior probability density of the heavy particle approximate representation system of one group of cum rights, and realizes the estimation procedure of recursion by model equation and observation information.Common particle filter algorithm comprises SIR particle filter, auxiliary particle filtering (APF), canonical particle filter (RPF), Gaussian particle filtering (GPF) and no mark particle filter (UPF).

Two key technologies in the particle filter algorithm are the algorithms of choosing and resample that suggestion distributes.Obtain easily though suggestion commonly used distributes, easily cause the abrupt degradation of particle weight, make effective population significantly reduce, especially when the likelihood distribution is positioned at the afterbody of prior distribution.Simultaneously, though the resampling algorithm has solved the degenerate problem of particle weight, common resampling algorithm is just to the simple copy and the rejecting of particle, from and caused the deficient problem (the multifarious forfeiture of particle) of particle.

Existing in order to overcome based on decreasing sharply and the particle scarcity in the dynamic system signal processing method of particle filter algorithm owing to the particle weight is easy, can not effectively represent posterior density, thereby can't effectively handle non-linear and deficiency non-Gaussian signal, the method for processing signals of non-linear dynamic system that posterior probability density, the particle diversity that the present invention proposes a kind of estimating system better be good, can effectively handle non-linear and non-Gaussian signal based on the histogram estimation particle filtering algorithm.

For the technical scheme that solves the problems of the technologies described above proposition is:

A kind of method for processing signals of non-linear dynamic system based on the histogram estimation particle filtering algorithm, the state-space model of establishing nonlinear dynamic system is:

x _k＝f(x _k-1)+v _k-1

z _k＝h(x _k)+n _k

Wherein, x _kAnd z _kRepresent that respectively system is at k state and measured value constantly, f (x _K-1) and h (x _k) represent the state transition equation and the observational equation of system, x respectively _K-1The expression system is at k state constantly, x _K-1And n _kRepresent system noise and observation noise respectively;

Described method for processing signals of non-linear dynamic system may further comprise the steps:

The first step is according to k-1 N particle constantly

I=1,2 ..., N obtains k N prediction particle constantly by state transition equation

I=1,2 ..., N;

In second step, determine the occupied state space of prediction particle, and it be divided into case that the number of case is designated as M;

The 3rd step is according to the priori probability density of the case estimating system of dividing, the density d of each case correspondence _iFor:

$d^i=\frac{N^i}{N\×h},i=\mathrm{1,2},...,M$

Wherein, N ⁱExpression falls into the prediction population of corresponding case, and h represents a hyperspatial volume that case is occupied.

The 4th goes on foot, and estimates the posterior probability density w of each case correspondence ⁱ:

w ⁱ＝d ⁱ×p(z _k|x ⁱ)，i＝1，2，...，M

Wherein, x ⁱThe center of representing i case;

In the 5th step, the posterior probability density of each case is carried out normalization:

$w^i=\frac{w^i}{{\mathrm{\Σ}}_{j=1}^M{w}^j},i=\mathrm{1,2}...,M$

The 6th step, a resampling N particle from M case The posterior density of etching system distributes during as k, i=1, and 2 ..., N, wherein, the particle that resamples in the same case carries out uniform sampling in the state space of correspondence;

The 7th step, the estimated value of output system state

$x_k^u\≈{\mathrm{\Σ}}_{i=1}^M{w}^i\×x^i.$

Technical conceive of the present invention is: a recursive process of this algorithm comprises following basic step:

1), obtains k N prediction particle constantly according to k-1 N particle constantly.

2), determine the occupied state space of prediction particle, and it is divided into case.

3), according to the priori probability density of the case estimating system of dividing.

4), estimate the posterior probability density of each case.

5), to the posterior probability density normalization of each case.

6), resample.

7), output.

The present invention has the following advantages:

1, priori and the posterior probability density that adopts the case in the histogram estimation to come estimating system has been avoided the calculating of particle weight, has more effectively represented the posterior probability density of system, and has reduced amount of calculation.

2, in case, resample, avoided the deficient problem of particle.

Description of drawings

Fig. 1 is the flow chart of histogram estimation particle filtering algorithm.

Fig. 2 is the histogram estimation particle filtering algorithm and the average correlation curve schematic diagram of the root-mean-square error of other particle filter algorithms under different populations.

Fig. 3 is histogram estimation particle filtering algorithm and the running time correlation curve schematic diagram of other particle filter algorithms under different populations.

Embodiment

The invention will be further described below in conjunction with drawings and Examples.

With reference to Fig. 1～Fig. 3, a kind of method for processing signals of non-linear dynamic system based on the histogram estimation particle filtering algorithm adopts particle to describe the state space of dynamical system, and the state-space model of establishing nonlinear dynamic system is:

x _k＝f(x _k-1)+v _k-1

z _k＝h(x _k)+n _k

Wherein, x _kAnd z _kRepresent that respectively system is at k state and measured value constantly, f (x _K-1) and h (x _k) represent the state transition equation and the observational equation of system, x respectively _K-1The expression system is at k state constantly, v _K-1And n _kRepresent system noise and observation noise respectively;

Method for processing signals of non-linear dynamic system may further comprise the steps:

The first step is according to k-1 N particle constantly

I=1,2 ..., N obtains k N prediction particle constantly by state transition equation

I=1,2 ..., N;

In second step, determine the occupied state space of prediction particle, and it be divided into case that the number of case is designated as M;

The 3rd step is according to the priori probability density of the case estimating system of dividing, the density d of each case correspondence ⁱFor:

$d^i=\frac{N^i}{N\×h},i=\mathrm{1,2},...,M$

Wherein, N ⁱExpression falls into the prediction population of corresponding case, and h represents a hyperspatial volume that case is occupied.

The 4th goes on foot, and estimates the posterior probability density w of each case correspondence ⁱ:

w ⁱ＝d ⁱ×p(z _k|x ⁱ)，i＝1，2，...，M

Wherein, x ⁱThe center of representing i case;

In the 5th step, the posterior probability density of each case is carried out normalization:

$w^i=\frac{w^i}{{\mathrm{\Σ}}_{j=1}^M{w}^j},i=\mathrm{1,2}...,M$

The 6th step, a resampling N particle from M case

The posterior density of etching system distributes during as k, i=1, and 2 ..., N, wherein, the particle that resamples in the same case carries out uniform sampling in the state space of correspondence;

The 7th step, the estimated value of output system state

$x_k^u\≈{\mathrm{\Σ}}_{i=1}^M{w}^i\×x^i.$

Present embodiment compares the present invention and other several particle filter algorithms by the state estimation of a nonlinear dynamic system.The state-space model of system is as follows:

$x_k=\frac{x_{k-1}}{2}+\frac{{25x}_{k-1}}{1+x_{k-1}^2}+8\cos \left(1.2k\right)+v_{k-1}$

$z_k=\frac{{\mathrm{<figure-callout\; id="2"\; label="x\; k"\; filenames="HDA0000022176020000022.png"\; state="\{\{state\}\}">x</figure-callout>}}_k^{}}{20}+n_k$

Wherein, system noise variance and observation noise variance are taken as 10 and 1 respectively.Setting observation time is 100, number of run is 100, got respectively 100,200,300,400,500,600 o'clock at population N, the root-mean-square error that algorithm and other particle filter algorithms produced (RMSE) average that the present invention proposes and the time of being moved are respectively as shown in Figures 2 and 3.

As can be seen, the RMSE average of algorithm of the present invention under different number of particles obviously is better than other algorithm except that UPF from Fig. 2 and Fig. 3, and also significantly reduce running time.When N=600, running time of algorithm of the present invention has only the about 57% of SIR, about 10% of UPF.

Claims (1)

1. method for processing signals of non-linear dynamic system based on the histogram estimation particle filtering algorithm, the state-space model of establishing nonlinear dynamic system is:

x _k＝f(x _k-1)+v _k-1

z _k＝h(x _k)+n _k

Wherein, x _kAnd z _kRepresent that respectively system is at k state and measured value constantly, f (x _K-1) and h (x _k) represent the state transition equation and the observational equation of system, x respectively _K-1The expression system is at k state constantly, v _K-1And n _kRepresent system noise and observation noise respectively;

Described method for processing signals of non-linear dynamic system may further comprise the steps:

The first step is according to k-1 N particle constantly

I=1,2 ..., N obtains k N prediction particle constantly by state transition equation

I=1,2 ..., N;

In second step, determine the occupied state space of prediction particle, and it be divided into case that the number of case is designated as M;

The 3rd step is according to the priori probability density of the case estimating system of dividing, the density d of each case correspondence ⁱFor:

$d^i=\frac{N^i}{N\&times;h},i=\mathrm{1,2},...,M$

Wherein, N ⁱExpression falls into the prediction population of corresponding case, and h represents a hyperspatial volume that case is occupied;

The 4th goes on foot, and estimates the posterior probability density w of each case correspondence ⁱ:

w ⁱ＝d ⁱ×p(z _k|x ⁱ)，i＝1，2，..，M

Wherein, x ⁱThe center of representing i case;

In the 5th step, the posterior probability density of each case is carried out normalization:

$w^i=\frac{w^i}{{\mathrm{\&Sigma;}}_{j=1}^M{w}^j},i=\mathrm{1,2},...,M$

The 6th step, a resampling N particle from M case

The posterior density of etching system distributes during as k, i=1, and 2 ..., N, wherein, the particle that resamples in the same case carries out uniform sampling in the state space of correspondence;

The 7th step, the estimated value of output system state

$x_k^{\mathrm{\&mu;}}\&ap;{\mathrm{\&Sigma;}}_{i=1}^M{w}^i\&times;x^i.$

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