CN103888100A

CN103888100A - Method for filtering non-Gaussian linear stochastic system based on negentropy

Info

Publication number: CN103888100A
Application number: CN201410124598.5A
Authority: CN
Inventors: 郭雷; 刘云龙; 杨健; 罗建军
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2014-03-29
Filing date: 2014-03-29
Publication date: 2014-06-25
Anticipated expiration: 2034-03-29
Also published as: CN103888100B

Abstract

A non-Gaussian linear stochastic system filtering method based on negentropy. First, a filter is designed for a general linear stochastic system; second, considering that the error equation is a non-Gaussian random process, the index function is selected as a linear combination of the covariance matrix of the estimated error and the negative entropy, so that it can be represented as completely as possible The probability and statistics characteristics of the estimation error; again, use the properties of the characteristic function to obtain the probability density function of the estimation error, so as to obtain the negative entropy expression of the estimation error; finally, solve the filter gain and minimize the index function; this method can Estimating the state of a linear stochastic system affected by non-Gaussian noise can be used in inertial navigation, guidance systems, target tracking, signal processing and other fields.

Description

A Non-Gaussian Linear Stochastic System Filtering Method Based on Negative Entropy

技术领域technical field

本发明涉及一种基于负熵的非高斯线性随机系统滤波方法，特别是一种针对随机输入的概率密度密度函数具有非对称、多峰值等强非高斯特性时的滤波方法，该方法可用于惯性导航、制导系统、目标跟踪、信号处理等领域。The invention relates to a non-Gaussian linear random system filtering method based on negative entropy, especially a filtering method for random input probability density functions with strong non-Gaussian characteristics such as asymmetry and multiple peaks, which can be used for inertial Navigation, guidance system, target tracking, signal processing and other fields.

背景技术Background technique

近年来，随着航空航天技术的发展，对飞行器自主性和快速反应能力的要求越来越高，意味着很多复杂的估计任务必须得到快速有效的解决。而滤波理论，是解决这些问题的关键技术。采用何种滤波方法，从概率统计最优的角度估计系统状态，在现有硬件条件的基础上，提高飞行器的控制和导航的精度具有重要意义，是各类任务成功的重要保证。20世纪60年代，卡尔曼滤波成功解决了美国阿波罗登月中的多导航传感器组合导航难题，引起了工程界的广泛重视。卡尔曼滤波是一种递推滤波算法，只要获知上一时刻的估计值及当前状态的观测值就可以计算出当前状态的估计值，不需要记录观测或者估计的历史信息，满足实时性要求，因此在导航系统中得到普遍应用。卡尔曼滤波方法充分利用了有关信息及被估计量本身的物理特性，优于最小二乘估计法，但是只适用于噪声服从高斯分布的线性系统。由于导航系统工作环境比较复杂，且有高度的不确定性，因此一般工作在非高斯噪声环境下。此时如果仍然采用卡尔曼滤波方法会减低导航精度。In recent years, with the development of aerospace technology, the requirements for the autonomy and rapid response capability of aircraft have become higher and higher, which means that many complex estimation tasks must be solved quickly and effectively. Filtering theory is the key technology to solve these problems. Which filtering method to use, to estimate the system state from the perspective of optimal probability and statistics, based on the existing hardware conditions, is of great significance to improve the accuracy of aircraft control and navigation, and is an important guarantee for the success of various tasks. In the 1960s, Kalman filtering successfully solved the multi-navigation sensor integrated navigation problem in the American Apollo moon landing, which attracted extensive attention from the engineering community. Kalman filtering is a recursive filtering algorithm. As long as the estimated value of the previous moment and the observed value of the current state are known, the estimated value of the current state can be calculated. It does not need to record the historical information of observations or estimates, and meets the real-time requirements. Therefore, it is widely used in navigation systems. The Kalman filter method makes full use of the relevant information and the physical characteristics of the estimated quantity itself, which is superior to the least square estimation method, but it is only suitable for linear systems whose noise obeys Gaussian distribution. Since the working environment of the navigation system is relatively complex and has a high degree of uncertainty, it generally works in a non-Gaussian noise environment. At this time, if the Kalman filter method is still used, the navigation accuracy will be reduced.

针对上述问题，在卡尔曼滤波理论的基础上，许多先进的滤波方法相继出现。1993年，Grodon等人提出的蒙特卡洛滤波算法与卡尔曼滤波算法一样根植于贝叶斯滤波框架，但是却不要求后验分布满足高斯分布的形式，因此可以处理非高斯滤波问题。Condensation，粒子滤波，bootstrap滤波等都是蒙特卡洛滤波算法在不同领域中的名称，它的核心思想是用对某一概率分布采样得到的随机样本及相应的概率分布来表示待求的后验分布，样本数趋于无穷大时，算法无限逼近真实分布，得到的结果将无限接近最优解。但是由于在实际中很难得到最优采样分布，因此蒙特卡洛滤波算法的精度不高而且算法复杂、计算量大、实时性差。综上所述，现有的滤波方法或者假定随机噪声服从高斯分布或者需要大量的采样样本，这样的处理方法主要有以下劣势：对于受非高斯噪声影响的系统，许多信息包含在高阶统计特性里面，方差已不能完全表征其概率统计特性。卡尔曼滤波方法以估计误差的方差最小为准则，势必会降低精度；蒙特卡洛滤波算法需要大量的采样样本才能得到比较好的结果，但是由于实际系统受到各种制约，很难得到最优采样样本，并且没有充分利用噪声的统计信息。In response to the above problems, on the basis of Kalman filtering theory, many advanced filtering methods have emerged one after another. In 1993, the Monte Carlo filtering algorithm proposed by Grodon et al. is rooted in the Bayesian filtering framework like the Kalman filtering algorithm, but it does not require the posterior distribution to satisfy the form of Gaussian distribution, so it can deal with non-Gaussian filtering problems. Condensation, particle filter, bootstrap filter, etc. are the names of Monte Carlo filter algorithms in different fields. Its core idea is to use random samples obtained by sampling a certain probability distribution and the corresponding probability distribution to represent the posterior to be sought. distribution, when the number of samples tends to infinity, the algorithm infinitely approaches the real distribution, and the obtained results will be infinitely close to the optimal solution. However, because it is difficult to obtain the optimal sampling distribution in practice, the precision of the Monte Carlo filtering algorithm is not high, and the algorithm is complex, with a large amount of calculation and poor real-time performance. To sum up, the existing filtering methods either assume that random noise obeys a Gaussian distribution or require a large number of samples. Such processing methods mainly have the following disadvantages: For systems affected by non-Gaussian noise, a lot of information is contained in high-order statistical properties Inside, the variance can no longer fully characterize its probability and statistical properties. The Kalman filtering method is based on the minimum variance of the estimation error, which will inevitably reduce the accuracy; the Monte Carlo filtering algorithm requires a large number of sampling samples to obtain better results, but due to various constraints in the actual system, it is difficult to obtain optimal sampling samples, and does not take full advantage of the statistics of the noise.

发明内容Contents of the invention

本发明的技术解决问题是：针对非高斯线性随机系统，克服现有技术的不足，并充分利用随机输入的统计信息，提供一种基于负熵的滤波方法，解决非高斯线性随机系统的滤波问题，提高系统的滤波精度。The technical solution problem of the present invention is: for non-Gaussian linear random system, overcome the deficiencies in the prior art, and make full use of the statistical information of random input, provide a kind of filtering method based on negative entropy, solve the filtering problem of non-Gaussian linear random system , to improve the filtering accuracy of the system.

本发明的技术解决方案为：一种基于负熵的非高斯线性随机系统滤波方法，其实现步骤如下：The technical solution of the present invention is: a kind of non-Gaussian linear random system filtering method based on negative entropy, and its realization steps are as follows:

第一步，设计滤波器：The first step is to design the filter:

针对一般的离散时间线性非高斯随机系统：For general discrete-time linear non-Gaussian stochastic systems:

x_k+1＝A_kx_k+G_kω_k x _k+1 ＝A _k x _k +G _k ω _k

y_k＝C_kx_k+υ_k y _k ＝C _k x _k +υ _k

构造如下形式的滤波器：Construct a filter of the form:

${\overset{^^}{x x}}_{k k + + 11} = = {A A}_{k k} {\overset{^^}{x x}}_{k k} + + {L L}_{k k} (({y the y}_{k k + + 11} - - {C C}_{k k + + 11} {A A}_{k k} {\overset{^^}{x x}}_{k k}))$

其中，x_k∈Rⁿ为系统状态，y_k∈R^m为量测输出；A_k∈R^n×n为已知的状态转移矩阵，G_k∈R^n×s为已知的干扰转移矩阵，C_k∈R^m×n为已知的输出矩阵；

为系统状态x_k的估计值，L_k∈R^n×m为要确定的滤波器增益；ω_k∈R^s，υ_k∈R^m分别为零均值、有界、相互独立且概率密度函数已知的非高斯过程噪声和量测噪声，其概率密度函数分别为

初始向量x₀与ω_k，υ_k相互独立，概率密度函数为

α₁，β₁，α₂，β₂为已知的实数；R表示实数域；Among them, x _k ∈ R ⁿ is the system state, y _k ∈ R ^m is the measurement output; A _k ∈ ^{R n×n} is the known state transition matrix, G _k ∈ ^{R n×s} is the known interference transfer matrix , C _k ∈ ^{R m×n} is a known output matrix;

is the ^estimated value _of the system state x _k , L _k ∈ R ^n×m ^is the filter gain _to be determined; Known non-Gaussian process noise and measurement noise, their probability density functions are

The initial vector x ₀ is independent of ω _k , υ _k , and the probability density function is

α ₁ , β ₁ , α ₂ , β ₂ are known real numbers; R represents the field of real numbers;

定义估计误差

则误差方程满足：Define estimation error

Then the error equation satisfies:

e_k+1＝(I-L_kC_k+1)(A_ke_k-G_kω_k)+L_kυ_k+1 e _k+1 ＝(IL _k C _k+1 )(A _k e _k -G _k ω _k )+L _k υ _k+1

其中I为n×n维单位矩阵；Wherein I is an n×n dimensional identity matrix;

第二步，指标函数选为：In the second step, the indicator function is selected as:

${V V}_{k k + + 11} (({L L}_{k k})) = = {R R}_{11 k k} &Integral; &Integral; {γ γ}_{{e e}_{k k + + 11}} ((x x,, {L L}_{k k})) log log (({γ γ}_{{e e}_{k k + + 11}} ((x x,, {L L}_{k k})) / / {γ γ}_{k k + + 11} ((x x,, {L L}_{k k})))) + + {R R}_{22 k k} {c c}_{k k + + 11}$

其中，c_k+1表示估计误差e_k+1的协方差函数，γe_k+1(x)和γ_k+1(x)分别表示估计误差e_k+1的概率密度函数以及与e_k+1具有相同协方差矩阵的高斯随机变量的概率密度函数；R_1k，R_2k为已知的权重矩阵；Among them, c _k+1 represents the covariance function of the estimated error e _k+1 , γe _k+1 (x) and γ _k+1 (x) respectively represent the probability density function of the estimated error e _k+1 and the relationship with e _{k+ 1} Probability density function of Gaussian random variables with the same covariance matrix; R _1k and R _2k are known weight matrices;

第三步，估计误差协方差阵和概率密度函数的求取：The third step is to estimate the error covariance matrix and probability density function:

误差方程的协方差阵及特征函数分别满足如下两式：The covariance matrix and characteristic function of the error equation satisfy the following two formulas respectively:

$c c ((k k + + 11)) = = ((I I - - {L L}_{k k} {C C}_{k k + + 11})) {A A}_{k k} c c ((k k)) {A A}_{k k}^{T T} {((I I - - {L L}_{k k} {C C}_{k k + + 11}))}^{T T} + + ((I I - - {L L}_{k k} {C C}_{k k + + 11})) {G G}_{k k} {c c}_{{ω ω}_{k k}} {G G}_{k k}^{T T} {((I I - - {L L}_{k k} {C C}_{k k + + 11}))}^{T T} + + {L L}_{k k} {c c}_{{&upsi; &upsi;}_{k k}} {L L}_{k k}^{T T}$

其中，

分别表示随机噪声ω_k，υ_k的协方差阵，

分别表示随机噪声ω_k，υ_k的特征函数且

表示估计误差的特征函数，t为时间变量，取值范围为(-∞,∞)，∞代表无穷。in,

represent the covariance matrix of random noise ω _k and υ _k respectively,

represent the characteristic functions of random noise ω _k , υ _k respectively and

Represents the characteristic function of the estimation error, t is the time variable, the value range is (-∞,∞), and ∞ represents infinity.

由于特征函数与概率密度函数相互唯一确定，估计误差e_k+1的概率密度函数可通过如下的傅里叶逆变换求得：Since the characteristic function and the probability density function are mutually uniquely determined, the probability density function of the estimated error e _k+1 can be obtained by the following inverse Fourier transform:

与估计误差e_k+1具有相同协方差阵的高斯随机变量的概率密度函数为： $γ_{k + 1} (x, L_{k}) = \frac{1}{\sqrt{{(2 π)}^{n} | c_{k + 1} |}} e^{- \frac{1}{2} x^{T} c_{k + 1}^{- 1} x}$ The probability density function of a Gaussian random variable with the same covariance matrix as the estimated error e _k+1 is: $γ_{k + 1} (x, L_{k}) = \frac{1}{\sqrt{{(2 π)}^{no} | c_{k + 1} |}} e^{- \frac{1}{2} x^{T} c_{k + 1}^{- 1} x}$

第四步，求解滤波器增益：The fourth step is to solve the filter gain:

由于指标函数为非线性函数，一般情况下不可能通过求解

得到L_k的解析表达式，因此这里滤波器增益通过如下梯度算法求得：Since the index function is a nonlinear function, it is generally impossible to solve

The analytical expression of L _k is obtained, so here the filter gain is obtained by the following gradient algorithm:

L_k＝L_k-1+λ_kd_k L _k ＝L _k-1 +λ _k d _k

其中d_k是从L_k-1出发的搜索方向，即

λ_k是从L_k-1出发的沿方向d_k进行的一维搜索的步长。where d _k is the search direction starting from L _k-1 , namely

λ _k is the step size of the one-dimensional search along the direction d _k starting from L _k-1 .

本发明与现有技术相比的优点在于：本发明利用了随机输入的统计信息，所选指标函数尽可能完备的表征了估计误差的概率统计特性。将基于最小方差的传统的滤波方法推广到基于最小负熵与方差的滤波方法，扩展了卡尔曼滤波的应用范围并提高了精度。且通过特征函数的性质来求取估计误差的概率密度函数，减小了计算量。The advantage of the present invention compared with the prior art is that the present invention utilizes randomly input statistical information, and the selected index function characterizes the probability and statistical characteristics of the estimation error as completely as possible. The traditional filtering method based on the minimum variance is extended to the filtering method based on the minimum negative entropy and variance, which expands the application range of Kalman filtering and improves the accuracy. Moreover, the probability density function of the estimation error is obtained through the properties of the characteristic function, which reduces the amount of calculation.

附图说明Description of drawings

图1为本发明一种基于负熵的非高斯滤波方法的设计流程图。Fig. 1 is a design flowchart of a non-Gaussian filtering method based on negentropy in the present invention.

具体实施方式Detailed ways

如图1所示，本发明具体实现步骤如下（以某一沿直线运动的运动体为例来说明方法的具体实现）：As shown in Figure 1, the specific implementation steps of the present invention are as follows (taking a moving body moving along a straight line as an example to illustrate the specific implementation of the method):

1、设计滤波器1. Design filter

当运动体作直线运动时，可以得到如下的动力学方程：When the moving body moves in a straight line, the following dynamic equation can be obtained:

$\{\begin{matrix} {s the s}_{k k} = = {s the s}_{k k - - 11} + + {Tv Tv}_{k k - - 11} + + \frac{{T T}^{22}}{22} {a a}_{k k - - 11} \\ {v v}_{k k} = = {v v}_{k k - - 11} + + {Ta Ta}_{k k - - 11} \\ {a a}_{k k} = = {a a}_{k k - - 11} + + {Tω Tω}_{k k - - 11} \end{matrix}$

位置观测值为：The position observations are:

y_k＝s_k+υ_k y _k = s _k +υ _k

其中，s_k，v_k，a_k分别为运动体在k时刻的位移、速度、加速度，T为采样周期；y_k为观测输出；ω_k为加加速度，对运动体的跟踪者来说，加加速度ω_k是随机量，此处假设其在区间[-0.5,0.5]上服从均匀分布；υ_k为观测噪声，其概率密度函数是：Among them, s _k , v _k , a _k are the displacement, velocity, and acceleration of the moving body at time k respectively, T is the sampling period; y _k is the observation output; ω _k is the jerk, for the tracker of the moving body, Jerk ω _k is a random quantity, here it is assumed that it obeys a uniform distribution on the interval [-0.5,0.5]; υ _k is observation noise, and its probability density function is:

${γ γ}_{{&upsi; &upsi;}_{k k}} ((x x)) = = \{\begin{matrix} - - {66 x x}^{22} + + 33 / / 22 & x x &Element; &Element; [[- - 0.5,0.5 0.5,0.5]] \\ 00 & x x &Element; &Element; ((- - \infty \infty,, 0.5 0.5)) \cup \cup ((0.5 0.5,, \infty \infty)) \end{matrix}$

取状态向量 $x_{k} = [\begin{matrix} s_{k} \\ v_{k} \\ a_{k} \end{matrix}],$ 则运动体作直线运动的动力学方程可转化为如下形式：Take the state vector $x_{k} = [\begin{matrix} {the s}_{k} \\ v_{k} \\ a_{k} \end{matrix}],$ Then the dynamic equation of the moving body moving in a straight line can be transformed into the following form:

x_k+1＝Ax_k+Gω_k x _k+1 ＝Ax _k +Gω _k

y_k＝Cx_k+υ_k y _k ＝Cx _k +υ _k

其中：in:

$A A = = [\begin{matrix} 11 & T T & {T T}^{22} / / 22 \\ 00 & 11 & T T \\ 00 & 00 & 11 \end{matrix}],, G G [\begin{matrix} 00 \\ 00 \\ T T \end{matrix}],, C C = = [\begin{matrix} 11 & 00 & 00 \end{matrix}],,$

ω_k，υ_k为相互独立的非高斯随机变量，概率密度函数已知，其特征函数可简单求得，分别为

状态初值取为x₀＝a₀，则其特征函数为

ω _k , υ _k are independent non-Gaussian random variables, the probability density function is known, and its characteristic function can be obtained simply, respectively

The initial value of the state is taken as x ₀ ＝a ₀ , then its characteristic function is

针对上述离散时间线性非高斯随机系统，构造如下形式的滤波器：For the above discrete-time linear non-Gaussian random system, construct a filter of the following form:

其中，

为xk的估计值，Lk为要确定的滤波器增益，通过后续步骤4求得。定义估计误差则误差方程满足：in,

is the estimated value of xk, and Lk is the filter gain to be determined, obtained through the subsequent step 4. Define estimation error Then the error equation satisfies:

${e e}_{k k + + 11} = = {\overset{^^}{x x}}_{k k + + 11} - - {x x}_{k k + + 11} = = ((I I - - {L L}_{k k} C C)) (({Ae Ae}_{k k} - - {Gω Gω}_{k k})) + + {L L}_{k k} {&upsi; &upsi;}_{k k + + 11}$

其中I为3×3维单位矩阵。where I is a 3×3 dimensional identity matrix.

2、指标函数选为：2. The indicator function is selected as:

其中，c_k+1表示估计误差e_k+1的协方差矩阵，

和γ_k+1(x)分别表示估计误差e_k+1的概率密度函数以及与e_k+1具有相同协方差矩阵的高斯分布的概率密度函数；R_1k，R_2k为已知的权重矩阵。Among them, c _k+1 represents the covariance matrix of the estimated error e _k+1 ,

and γ _k+1 (x) represent the probability density function of the estimated error e _k+1 and the probability density function of the Gaussian distribution with the same covariance matrix as e _k+1 respectively; R _1k and R _2k are the known weight matrices .

3、估计误差协方差阵和概率密度函数的求取：3. Calculation of estimated error covariance matrix and probability density function:

由于指标函数为估计误差概率密度函数与协方差的函数，因此需要首先求取估计误差的协方差及概率密度函数的表达式。Since the indicator function is a function of the estimation error probability density function and covariance, it is necessary to obtain the expression of the estimation error covariance and probability density function first.

由于because

${e e}_{x x} = = {\overset{^^}{x x}}_{00} - - {x x}_{00}$

e₁＝(I-L₀C)Ae₀-(I-L₀C)Gω₀+L₀υ₁ e ₁ ＝(IL ₀ C)Ae ₀ -(IL ₀ C)Gω ₀ +L ₀ υ ₁

$\begin{matrix} {e e}_{22} = = ((I I - - {L L}_{11} C C)) {Ae Ae}_{11} - - ((I I - - {L L}_{11} C C)) G G {ω ω}_{11} + + {L L}_{11} {&upsi; &upsi;}_{22} \\ = = ((I I - - {L L}_{11} C C)) A A [[((I I - - {L L}_{00} C C)) {Ae Ae}_{00} - - ((I I - - {L L}_{00} C C)) {Gω Gω}_{00} + + {L L}_{00} {&upsi; &upsi;}_{11}]] - - ((I I - - {L L}_{11} C C)) {Gω Gω}_{11} + + {L L}_{11} {&upsi; &upsi;}_{22} \\ \cdot &Center Dot; \\ \cdot &Center Dot; \\ \cdot &Center Dot; \end{matrix}$

e_k+1＝(I-L_kC)(Ae_k-Gω_k)+L_kυ_k+1 e _k+1 ＝(IL _k C)(Ae _k -Gω _k )+L _k υ _k+1

可以看出e_k是由e₀；ω₀，ω₁，…，ω_k-1；υ₁，…，υ_k等线性组合而成；同时，因为e₀，ω_i，υ_i+1(i=0,1,...,k)相互独立，所以e_k，ω_k，υ_k+1相互独立。因此估计误差的协方差函数可以由下式得到： _It can be seen that e _k is formed by e ₀ ; ω ₀ , _ω ₁ ,..., ω _k _- ₁ _; i=0,1,...,k) are independent of each other, so e _k , ω _k , and υ _k+1 are independent of each other. Therefore, the covariance function of the estimation error can be obtained by the following formula:

其中，

分别代表随机噪声ω_k，υ_k的协方差矩阵。in,

represent the covariance matrix of random noise ω _k and υ _k respectively.

由于特征函数与概率密度函数相互唯一确定，因此可以利用估计误差的特征函数来求取其概率密度函数。对于相互独立的随机变量，其和的特征函数表达式为：Since the characteristic function and the probability density function are mutually uniquely determined, the probability density function can be obtained by using the characteristic function of the estimation error. For independent random variables, the characteristic function expression of the sum is:

其中

分别表示随机噪声ω_k，υ_k的特征函数。这里假设

估计误差初值

in

represent the characteristic functions of random noise ω _k and υ _k respectively. Assume here

Estimated error initial value

已知估计误差的特征函数表达式，其概率密度函数可以通过如下的傅里叶逆变换求得：The characteristic function expression of the estimation error is known, and its probability density function can be obtained by the following inverse Fourier transform:

与估计误差e_k+1具有相同协方差的高斯分布的概率密度函数为：The probability density function of a Gaussian distribution with the same covariance as the estimated error e _k+1 is:

${γ γ}_{k k + + 11} ((x x,, {L L}_{k k})) = = \frac{11}{\sqrt{{((22 π π))}^{n no} | | {c c}_{k k + + 11} | |}} {e e}^{- - \frac{11}{22} {x x}^{T T} {c c}_{k k + + 11}^{- - 11} x x}$

4、求解滤波器增益：4. Find the filter gain:

由于指标函数为非线性函数，一般情况下不可能通过求解

L_k＝L_k-1+λ_kd_k L _k ＝L _k-1 +λ _k d _k

其中d_k是从L_k-1出发的搜索方向，即

本发明说明书中未作详细描述的内容属于本领域专业技术人员公知的现有技术。The contents not described in detail in the description of the present invention belong to the prior art known to those skilled in the art.

Claims

1. A non-Gaussian linear stochastic system filtering method based on negative entropy is characterized by comprising the following steps: firstly, aiming at a general discrete time linear random system, a filter is designed; secondly, selecting the index function as a linear combination of a covariance function of an estimation error and a negative entropy; thirdly, the probability density function of the estimation error is obtained by utilizing the property of the characteristic function; finally, solving the gain of the filter and minimizing an index function; the method comprises the following specific steps:

firstly, designing a filter:

for a general discrete-time linear non-gaussian random system:

x_k+1＝A_kx_k+G_kω_k

y_k＝C_kx_k+υ_k

a filter of the form:

{\hat{x}}_{k + 1} = A_{k} {\hat{x}}_{k} + L_{k} (y_{k + 1} - C_{k + 1} A_{k} {\hat{x}}_{k})

wherein x is_k∈RⁿIs the system state, y_k∈R^mOutputting for measurement; a. the_k∈R^n×nFor a known state transition matrix, G_k∈R^n×sFor a known interference transfer matrix, C_k∈R^m×nIs a known output matrix;

is the system state x_kEstimated value of, L_k∈R^n×mIs the filter gain to be determined; omega_k∈R^sAnd upsilon_k∈R^mnon-Gaussian process noise and measurement noise which are respectively zero mean, bounded, mutually independent and known by probability density functions which are respectively

Initial vector x₀And omega_k，υ_kIndependently of each other, a probability density function of

α₁，β₁，α₂，β₂Is a known real number; r represents a real number domain;

defining estimation error

The error equation satisfies:

e_k+1＝(I-L_kC_k+1)(A_ke_k-G_kω_k)+L_kυ_k+1

wherein I is an nxn dimensional identity matrix;

secondly, selecting an index function as follows:

<math> <mrow> <msub> <mi>V</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mo>&Integral;</mo> <msub> <mi>γ</mi> <msub> <mi>e</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>log</mi> <mrow> <mo>(</mo> <msub> <mi>γ</mi> <msub> <mi>e</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>γ</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math>

wherein, c_k+1Representing the estimation error e_k+1The covariance function of (a) of (b),

and gamma_k+1(x) Respectively representing the estimation errors e_k+1And e_k+1Probability density functions of gaussian random variables having the same covariance matrix; r_1k，R_2kIs a known weight matrix;

thirdly, solving an estimation error covariance matrix and a probability density function:

the covariance matrix and the characteristic function of the error equation respectively satisfy the following two formulas:

wherein,

respectively representing random noise omega_k，υ_kThe covariance matrix of (a) is obtained,

respectively representing random noise omega_k，υ_kIs a characteristic function of

A characteristic function representing estimation error, wherein t is a time variable, and the value range is (- ∞, infinity), and infinity is represented by infinity;

since the characteristic function and the probability density function are uniquely determined to each other, the estimation error e_k+1The probability density function of (a) can be found by inverse fourier transformation as follows:

and the estimated error e_k+1The probability density function of gaussian random variables with the same covariance matrix is:

step four, solving the gain of the filter:

since the index function is a non-linear function, it is generally impossible to solve the function by solving it

To obtain L_kSo here the filter gain is found by the gradient algorithm:

L_k＝L_k-1+λ_kd_k

wherein d is_kIs from L_k-1Search direction of departure, i.e.

λ_kIs from L_k-1Starting in direction d_kStep size of the one-dimensional search performed.