王宏健,徐金龍,李娟,張愛華

(哈爾濱工程大學(xué)自動(dòng)化學(xué)院,黑龍江哈爾濱 150001)

非平穩(wěn)非高斯測(cè)量噪聲條件下改進(jìn)差分粒子濾波算法研究

王宏健,徐金龍,李娟,張愛華

(哈爾濱工程大學(xué)自動(dòng)化學(xué)院,黑龍江哈爾濱 150001)

針對(duì)非平穩(wěn)非高斯測(cè)量噪聲(NSNGN)條件下差分粒子濾波(DDPF)算法狀態(tài)估計(jì)精度低、易發(fā)散的問(wèn)題,提出了一種改進(jìn)DDPF(IDDPF)算法.IDDPF算法采用高斯混合密度函數(shù)近似估計(jì)測(cè)量噪聲,替代傳統(tǒng)算法中測(cè)量噪聲的高斯密度函數(shù)近似估計(jì),采用似然函數(shù)的對(duì)數(shù)最大化法求解高斯混合密度函數(shù)模型參數(shù),并將該模型應(yīng)用于粒子權(quán)值計(jì)算,避免了高斯密度函數(shù)近似估計(jì)噪聲模型所易于導(dǎo)致的粒子退化問(wèn)題;通過(guò)建立水下目標(biāo)純方位角跟蹤系統(tǒng)模型,將IDDPF算法應(yīng)用于閃爍測(cè)量噪聲條件下水下目標(biāo)純方位角跟蹤問(wèn)題的求解。50次Monte Carlo對(duì)比仿真實(shí)驗(yàn)結(jié)果表明:在NSNGN條件下IDDPF算法具有跟蹤響應(yīng)快、估計(jì)精度高、魯棒性較好等優(yōu)點(diǎn)。

控制科學(xué)與技術(shù);非平穩(wěn)非高斯噪聲;差分粒子濾波;高斯混合密度函數(shù);水下目標(biāo)純方位角跟蹤

0 引言

非線性系統(tǒng)狀態(tài)最優(yōu)估計(jì)在目標(biāo)跟蹤、導(dǎo)航制導(dǎo)、信號(hào)處理等多種領(lǐng)域都具有重要的應(yīng)用[1]。對(duì)于線性系統(tǒng),Kalman濾波是解決估計(jì)問(wèn)題的最優(yōu)方案[2]。而對(duì)于非線性系統(tǒng),并沒(méi)有一個(gè)完全最優(yōu)的狀態(tài)估計(jì)方案。為此,人們提出了很多次優(yōu)的近似方法,其中主要由擴(kuò)展Kalman濾波(EKF)、Sigma點(diǎn)Kalman濾波、粒子濾波(PF)及上述方法的改進(jìn)型構(gòu)成。

EKF及其改進(jìn)型(如強(qiáng)跟蹤EKF[3]、迭代EKF (IEKF)[4])的主要缺點(diǎn)是面對(duì)強(qiáng)非線性系統(tǒng)時(shí),估計(jì)精度可能嚴(yán)重降低甚至發(fā)散。Sigma點(diǎn)Kalman濾波算法主要分為無(wú)跡Kalman濾波(UKF)[5]及差分濾波(DDF)[6]。文獻(xiàn)[7-8]針對(duì)噪聲相關(guān)條件下UKF濾波失效的問(wèn)題,提出了一種非線性離散系統(tǒng)的UKF設(shè)計(jì)方法,擴(kuò)展了UKF的應(yīng)用范圍。文獻(xiàn)[9]提出了UKF的新濾波算法,很好地解決了量測(cè)噪聲有色情況下濾波器失效的問(wèn)題。雖然UKF的相關(guān)文獻(xiàn)取得了很多有意義的成果,但是文獻(xiàn)[10-12]指出UKF的濾波精度要低于DDF且計(jì)算復(fù)雜度要略高于DDF.文獻(xiàn)[13]在文獻(xiàn)[6]的基礎(chǔ)上提出了符合加性噪聲的DDF.文獻(xiàn)[14]提出一種迭代差分濾波(IDDF)算法,且仿真驗(yàn)證了在強(qiáng)非線性系統(tǒng)情況下,IDDF比IEKF精度高的優(yōu)點(diǎn)。文獻(xiàn)[15]提出基于極大似然的迭代差分濾波器(MLIDDF),解決了非線性系統(tǒng)狀態(tài)估計(jì)因初始估計(jì)誤差較大且測(cè)量方程具有強(qiáng)非線性所導(dǎo)致的濾波精度低的問(wèn)題。文獻(xiàn)[16]提出了一種具有魯棒性的自適應(yīng)差分濾波算法(ADDF)來(lái)處理具有建模誤差的非線性系統(tǒng)。

PF是由Hammersley等于20世紀(jì)50年代末首先提出基于貝葉斯采樣估計(jì)的順序重要采樣(SIS)濾波思想,并由Gordon等于1993年提出新的基于SIS的Bootstrap非線性濾波算法[17-19]。文獻(xiàn)[20-21]提出采用DDF算法來(lái)產(chǎn)生粒子建議分布的新算法,即差分粒子濾波(DDPF)算法,改善了PF算法的估計(jì)精度。文獻(xiàn)[22-26]中將PF及其各種改進(jìn)算法分別應(yīng)用于目標(biāo)跟蹤、定位及機(jī)器人導(dǎo)航等領(lǐng)域,并通過(guò)仿真實(shí)驗(yàn)表明PF及其改進(jìn)算法具有較好的魯棒性及準(zhǔn)確性。

針對(duì)PF算法的各種改進(jìn)方法尚不能完全有效地解決粒子的退化及貧化問(wèn)題,在處理含有非平穩(wěn)非高斯噪聲(NSNGN)系統(tǒng)時(shí),上述濾波方法的精度不高甚至可能發(fā)散。其根本原因在于計(jì)算粒子權(quán)值時(shí)所用的似然函數(shù)無(wú)法得到準(zhǔn)確描述。文獻(xiàn)[27]中實(shí)驗(yàn)結(jié)果證明了似然函數(shù)對(duì)PF算法估計(jì)精度的影響,但未給出似然函數(shù)模型建模的詳細(xì)過(guò)程。

為此,本文基于 DDPF算法原理[20-21],提出NSNGN測(cè)量噪聲條件下改進(jìn)DDPF(IDDPF)算法,針對(duì)NSNGN條件下DDPF算法存在似然函數(shù)計(jì)算不正確從而導(dǎo)致粒子退化及貧化較為嚴(yán)重,使得狀態(tài)精度不高甚至發(fā)散問(wèn)題,改進(jìn)設(shè)計(jì)了似然函數(shù)近似計(jì)算方法,有效降低了粒子的退化現(xiàn)象,通過(guò)Monte Carlo仿真實(shí)驗(yàn)對(duì)比驗(yàn)證該濾波算法的估計(jì)精度和魯棒性。

1 IDDPF方法設(shè)計(jì)

本節(jié)首先給出DDPF算法,并給出該算法存在的問(wèn)題,尤其是在處理NSNGN噪聲時(shí)的問(wèn)題,進(jìn)而提出IDDPF算法。

1.1 DDPF算法原理

在DDPF算法中,由(17)式、(18)式可知, p(yk|x(i)k)～p(yk;h(xk),Rk),即假設(shè)測(cè)量噪聲νk服從均值為0、協(xié)方差為Rk的高斯白噪聲,而實(shí)際上測(cè)量噪聲νk為NSNGN,這種測(cè)量噪聲假設(shè)極易導(dǎo)致DDPF算法中粒子權(quán)值嚴(yán)重退化,從而使得狀態(tài)估計(jì)過(guò)程中出現(xiàn)較大誤差而造成濾波算法發(fā)散。

對(duì)于測(cè)量噪聲為NSNGN,其密度函數(shù)p(ν(i)k)無(wú)法用精確的數(shù)學(xué)函數(shù)來(lái)表達(dá),只能用近似方法求解。下面首先給出NSNGN測(cè)量噪聲的參數(shù)近似估計(jì)方法,再基于該參數(shù)近似估計(jì)方法,提出IDDPF算法。

1.2 NSNGN噪聲參數(shù)估計(jì)

假設(shè)噪聲αk為NSNGN噪聲序列,并采用高斯混合密度函數(shù)加以近似表達(dá),如(25)式所示,式中各相關(guān)參數(shù)在每一時(shí)刻通過(guò)N次測(cè)量數(shù)據(jù)的似然函數(shù)對(duì)數(shù)最大化方法求得。

式中:αlk表示k時(shí)刻數(shù)據(jù)集合中(假設(shè)數(shù)據(jù)集合中共有N個(gè)數(shù)據(jù))的第l個(gè)數(shù)據(jù)。

由(27)式可知,αlk的第i個(gè)高斯組件(表示為fi(αlk))與事件Ai,k相關(guān)。由(28)式可得(30)式成立:

k時(shí)刻的參數(shù)pi,k、μi,k及σi,k通過(guò)似然函數(shù)的對(duì)數(shù)最大化方法來(lái)求得。這里之所以采用似然函數(shù)的對(duì)數(shù)最大化而不直接采用似然函數(shù)的最大化,主要是因?yàn)樗迫缓瘮?shù)的對(duì)數(shù)最大化方法不改變似然函數(shù)的本身固有單調(diào)屬性,而表達(dá)形式更加簡(jiǎn)單。由文獻(xiàn)[29]可知似然函數(shù)的對(duì)數(shù)形式如(34)式所示:

1.3 IDDPF

IDDPF算法的流程如下所示:

利用DDPF算法計(jì)算各粒子信息,即

基于(38)式、(41)式、(42)式求解相關(guān)參數(shù);

end.

歸一化粒子權(quán)重及粒子重采樣過(guò)程如(17)式～(20)式所示。

在IDDPF算法中,計(jì)算粒子權(quán)值時(shí)所用到的噪聲密度函數(shù)p(ν(i)k)用高斯混合密度函數(shù)而不是高斯函數(shù)來(lái)近似描述,能夠更加真實(shí)地描述噪聲特性,從而一定程度上避免了粒子的退化。

2 仿真實(shí)驗(yàn)及結(jié)果分析

為了驗(yàn)證所提出IDDPF算法的有效性,本節(jié)針對(duì)水下目標(biāo)純方位角跟蹤這一典型的非線性系統(tǒng)進(jìn)行Monte Carlo仿真與分析,并將IDDPF與DDPF在仿真初始條件相同情況下進(jìn)行了性能對(duì)比。由于實(shí)際應(yīng)用中聲納進(jìn)行目標(biāo)跟蹤時(shí)受到水下復(fù)雜環(huán)境的影響,其測(cè)量噪聲通常不能滿足高斯噪聲屬性,而研究中發(fā)現(xiàn)可以使用具有長(zhǎng)尾部概率密度的閃爍噪聲來(lái)很好地近似表示水下聲納的測(cè)量噪聲[30]。

2.1 水下目標(biāo)純方位角跟蹤系統(tǒng)模型

系統(tǒng)的相對(duì)運(yùn)動(dòng)狀態(tài)方程為

式中:ε=0.1;κ=1 000;協(xié)方差Rk的變換范圍從(0.1×π/180)2到(0.2×π/180)2,如圖1所示。

圖1 閃爍噪聲νk特征曲線Fig.1 Characteristic curve of glint noise

2.2 Monte Carlo仿真實(shí)驗(yàn)結(jié)果與分析

假設(shè)跟蹤對(duì)象為勻速直線運(yùn)動(dòng)的水下目標(biāo),聲納平臺(tái)在觀測(cè)過(guò)程中先做轉(zhuǎn)向機(jī)動(dòng),再做勻速直線運(yùn)動(dòng),目標(biāo)及聲納平臺(tái)的運(yùn)動(dòng)狀態(tài)如表1所示。

表1 目標(biāo)及平臺(tái)的運(yùn)動(dòng)狀態(tài)設(shè)置Tab.1 State settings of target and platform

在濾波器的初值設(shè)定時(shí),假定初始時(shí)刻目標(biāo)與觀測(cè)平臺(tái)的相對(duì)距離r0已知,目標(biāo)速度未知,初始時(shí)刻聲納測(cè)得目標(biāo)方位角為y0,令M=r20,則濾波器狀態(tài)初值 x0及協(xié)方差矩陣 P0分別如(50)式、(51)式所示:

在上述初始條件下,將本文所提出的IDDPF與DDPF算法進(jìn)行50次Monte Carlo仿真實(shí)驗(yàn)對(duì)比,仿真時(shí)間為1 000 s.圖2為X-Y平面坐標(biāo)系下的位置跟蹤效果,圖3為相對(duì)距離跟蹤及誤差效果曲線,圖4～圖7分別給出了各方向上位置、速度及其誤差曲線。由圖4及圖6的位置誤差曲線可以看出,DDPF位置跟蹤偏差較大,圖5及圖7中DDPF速度跟蹤未能收斂于目標(biāo)的真實(shí)速度值,而IDDPF算法則能夠準(zhǔn)確地跟蹤目標(biāo),其位置及速度跟蹤誤差均較小。

圖2 IDDPF及DDPF的位置跟蹤曲線Fig.2 Position tracking for IDDPF and DDPF

圖3 相對(duì)距離跟蹤及其誤差曲線Fig.3 Relative distance and tracking error

圖4 X軸位置跟蹤及其誤差曲線Fig.4 Position and tracking error along X-axis

為了深入對(duì)比兩種濾波算法的性能,分別給出X、Y方向上位置與速度的均方根誤差(RMSE)仿真曲線,分別如圖8、圖9所示。明顯看出,相同的初始仿真條件下,IDDPF改進(jìn)方法的RMSE誤差更小、算法精度更高。而且,IDDPF的RMSE曲線波動(dòng)較小,說(shuō)明其魯棒性要好于DDPF.

3 結(jié)論

本文提出了一種IDDPF算法。該算法基于高斯混合密度函數(shù)對(duì)非平穩(wěn)非高斯噪聲建模,并基于似然函數(shù)的對(duì)數(shù)最大化方法對(duì)模型參數(shù)進(jìn)行求解,避免了由于似然函數(shù)建模不正確所導(dǎo)致的粒子退化問(wèn)題。仿真實(shí)驗(yàn)中針對(duì)閃爍測(cè)量噪聲背景條件下水下目標(biāo)純方位角跟蹤問(wèn)題,將IDDPF算法與DDPF算法作對(duì)比,50次Monte Carlo仿真實(shí)驗(yàn)結(jié)果表明, IDDPF算法具有更快的跟蹤響應(yīng),RMSE統(tǒng)計(jì)分析同樣驗(yàn)證了所提算法具有較高的估計(jì)精度及一定的魯棒性。

圖5 X軸速度跟蹤及其誤差曲線Fig.5 Velocity and tracking error along X-axis

圖6 Y軸位置跟蹤及其誤差曲線Fig.6 Position and tracking error along Y-axis

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Research on Improved Divided Difference Particle Filter under Non-stationary Non-Gaussian Noise Background

WANG Hong-jian,XU Jin-long,LI Juan,ZHANG Ai-hua
(College of Automation,Harbin Engineering University,Harbin 150001,Heilongjiang,China)

An improved divided difference particle filter(IDDPF)is proposed to improve the low accuracy of state estimation and divergent tend problem which may be caused by divided difference particle filter (DDPF)in the condition of the non-stationary non-Gaussian measurement noise(NSNGN).The IDDPF algorithm adopts Gaussian mixture density function to approximately estimate the measurement noise,instead of the Gaussian density function usually adopted in DDPF.The noise parameters are estimated by maximizing the log likelihood function of the measurement noise model.The model is then used to calculate the particle weight,which avoids the particle degeneracy problem.The IDDPF algorithm is tested by establishing bearing-only tracking of underwater target under the glint measurement noise background. The results of 50 Monte-Carlo simulation experiments show that the IDDPF algorithm has the advantages of fast tracking response,high estimated precision and robustness,etc.under NSNGN background.

control science and technology;non-stationary non-Gaussian noise;divided difference parti-cle filter;Gaussian mixture density function;underwater target bearing-only tracking

TB566

A

1000-1093(2014)07-1032-08

10.3969/j.issn.1000-1093.2014.07.015

2013-08-06

國(guó)家自然科學(xué)基金項(xiàng)目(E091002/50979017);教育部高等學(xué)校博士學(xué)科點(diǎn)專項(xiàng)科研基金項(xiàng)目(20092304110008);中央高校基本科研業(yè)務(wù)費(fèi)專項(xiàng)(HEUCFZ 1026);哈爾濱市科技創(chuàng)新人才(優(yōu)秀學(xué)科帶頭人)研究專項(xiàng)(2012RFXXG083);教育部新世紀(jì)優(yōu)秀人才支持計(jì)劃項(xiàng)目(NCET-10-0053)

王宏健(1971—),女,教授,博士生導(dǎo)師。E-mail:cctime99@163.com