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

Abstract

A method for filtering a non-Gaussian linear stochastic system based on the negentropy comprises the steps that firstly, a filter is designed according to a common linear stochastic system; secondly, due to the fact that an error equation is a non-Gaussian stochastic process, an indicator function is selected as a linear combination of a covariance matrix and the negentropy of an evaluated error, and the probability statistic characteristic of the evaluated error is represented as complete as possible; secondly, a probability density function of the evaluated error is worked out according to the characteristic of a characteristic function, and therefore a negentropy expression of the evaluated error is obtained; finally, a filter gain of the filter is solved and the indicator function is minimized. According to the method, the linear stochastic system state influenced by non-Gaussian noise can be estimated and the method can be applied to the fields of inertial navigation, a guiding system, target tracking, signal processing and the like.

Description

non-Gaussian linear stochastic system filtering method based on negative entropy

Technical Field

The invention relates to a non-Gaussian linear random system filtering method based on negative entropy, in particular to a filtering method when a probability density function for random input has strong non-Gaussian characteristics of asymmetry, multi-peak value and the like.

Background

In recent years, with the development of aerospace technology, the requirements on autonomy and rapid response capability of an aircraft are higher and higher, which means that many complex estimation tasks must be solved quickly and effectively. The filtering theory is a key technology for solving the problems. By adopting the filtering method, the system state is estimated from the perspective of optimal probability statistics, and on the basis of the existing hardware conditions, the method has important significance for improving the control and navigation precision of the aircraft, and is an important guarantee for the success of various tasks. In the 60 s of the 20 th century, the Kalman filtering successfully solves the problem of multi-navigation sensor combined navigation in the American Apollodenum, and draws wide attention of the engineering world. The Kalman filtering is a recursive filtering algorithm, the estimation value of the current state can be calculated as long as the estimation value of the last moment and the observation value of the current state are obtained, the historical information of observation or estimation does not need to be recorded, and the real-time requirement is met, so that the Kalman filtering is generally applied to a navigation system. The Kalman filtering method fully utilizes the relevant information and the physical characteristics of the estimated quantity, is superior to a least square estimation method, but is only suitable for a linear system with noise subjected to Gaussian distribution. Because the navigation system has a complex operating environment and a high degree of uncertainty, it generally operates in a non-gaussian noise environment. In this case, if the kalman filtering method is still used, the navigation accuracy is reduced.

In response to the above problems, many advanced filtering methods are in succession based on kalman filtering theory. In 1993, the Monte Carlo filtering algorithm proposed by Grodon et al was rooted in a Bayesian filtering framework as the Kalman filtering algorithm, but does not require that the posterior distribution satisfy the form of Gaussian distribution, and thus can handle non-Gaussian filtering problems. Condensation, particle filtering, bootstrap filtering and the like are names of Monte Carlo filtering algorithms in different fields, the core idea of the method is that a random sample obtained by sampling a certain probability distribution and corresponding probability distribution are used for representing posterior distribution to be solved, when the number of samples tends to infinity, the algorithm approaches to real distribution infinitely, and the obtained result approaches to an optimal solution infinitely. However, in practice, the optimal sampling distribution is difficult to obtain, so that the monte carlo filtering algorithm is not high in precision, complex, large in calculation amount and poor in real-time performance. In summary, the existing filtering methods either assume that random noise follows gaussian distribution or require a large number of sampling samples, and such processing methods have the following disadvantages: for systems affected by non-gaussian noise, much information is contained within the higher order statistics, and the variance has not been able to fully characterize its probabilistic statistics. The Kalman filtering method takes the minimum variance of estimation errors as a criterion, and the precision is inevitably reduced; the monte carlo filtering algorithm needs a large number of sampling samples to obtain a good result, but because an actual system is subjected to various restrictions, an optimal sampling sample is difficult to obtain, and statistical information of noise is not fully utilized.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: aiming at a non-Gaussian linear random system, the method overcomes the defects of the prior art, fully utilizes the statistical information of random input, provides a filtering method based on negative entropy, solves the filtering problem of the non-Gaussian linear random system, and improves the filtering precision of the system.

The technical solution of the invention is as follows: a non-Gaussian linear stochastic system filtering method based on negative entropy comprises the following implementation 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^s，υ_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>&gamma;</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>&gamma;</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>&gamma;</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+1Of the covariance function, γ e_k+1(x) 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:

<math> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>A</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>G</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&upsi;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>L</mi> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </math>

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

And a characteristic function representing estimation error, wherein t is a time variable and has a value range of (- ∞, infinity) and 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: <math> <mrow> <msub> <mi>&gamma;</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> <mfrac> <mn>1</mn> <msqrt> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> </msqrt> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>x</mi> <mi>T</mi> </msup> <msubsup> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> </mrow> </msup> </mrow> </math>

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.

Compared with the prior art, the invention has the advantages that: the method utilizes the randomly input statistical information, and the selected index function represents the probability statistical characteristic of the estimation error as completely as possible. A traditional filtering method based on the minimum variance is popularized to a filtering method based on the minimum negative entropy and the variance, the application range of Kalman filtering is expanded, and the precision is improved. And the probability density function of the estimation error is obtained through the property of the characteristic function, so that the calculation amount is reduced.

Drawings

FIG. 1 is a design flow chart of a non-Gaussian filtering method based on negative entropy according to the present invention.

Detailed Description

As shown in fig. 1, the implementation steps of the present invention are as follows (taking a moving body moving along a straight line as an example to illustrate the implementation of the method):

1. designing filters

When the moving body moves linearly, the following kinetic equation can be obtained:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Tv</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <msup> <mi>T</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> <msub> <mi>a</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Ta</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>T&omega;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </math>

the position observations are:

y_k＝s_k+υ_k

wherein s is_k，v_k，a_kRespectively representing the displacement, the speed and the acceleration of the moving body at the moment k, wherein T is a sampling period; y is_kOutputting for observation; omega_kFor jerk, jerk ω is given to a tracker of a moving body_kIs a random quantity, which is assumed here to be in the interval [ -0.5,0.5 [ ]]Uniformly distributing the upper layer; upsilon is_kTo observe noise, the probability density function is:

<math> <mrow> <msub> <mi>&gamma;</mi> <msub> <mi>&upsi;</mi> <mi>k</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mo>-</mo> <msup> <mrow> <mn>6</mn> <mi>x</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mtd> <mtd> <mi>x</mi> <mo>&Element;</mo> <mo>[</mo> <mo>-</mo> <mn>0.5,0.5</mn> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>&Element;</mo> <mrow> <mo>(</mo> <mo>-</mo> <mo>&infin;</mo> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mo>&cup;</mo> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mo>&infin;</mo> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

taking state vectors $x_k=\left[\begin{array}{c}{s}_k\\ v_k\\ a_k\end{array}\right],$ The kinetic equation of the linear motion of the moving body can be converted into the following form:

x_k+1＝Ax_k+Gω_k

y_k＝Cx_k+υ_k

wherein:

$A=\left[\begin{array}{ccc}1& T& T^2/2\\ 0& 1& T\\ 0& 0& 1\end{array}\right],G\left[\begin{array}{c}0\\ 0\\ T\end{array}\right],C=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],$

ω_k，υ_kfor mutually independent non-Gaussian random variables, the probability density function is known, and the characteristic functions can be simply obtained, namely

The initial value of the state is taken as x₀＝a₀Then its characteristic function is

For the discrete-time linear non-gaussian random system, a filter of the following form is constructed:

wherein,

as an estimate of xk, Lk is the filter gain to be determined, which is found in the following step 4. Defining estimation errorThe error equation satisfies:

<math> <mrow> <msub> <mi>e</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <mi>C</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>Ae</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>G&omega;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>&upsi;</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math>

where I is a 3X 3 dimensional identity matrix.

2. The index function is selected as:

<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>&gamma;</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>&gamma;</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>&gamma;</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 matrix of (a) is determined,

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

3. And (3) solving an estimation error covariance matrix and a probability density function:

since the index function is a function of the probability density function and the covariance of the estimation error, the covariance of the estimation error and the expression of the probability density function need to be obtained first.

Due to the fact that

$e_x={\hat{x}}_0-x_0$

e₁＝(I-L₀C)Ae₀-(I-L₀C)Gω₀+L₀υ₁

<math> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <msub> <mi>Ae</mi> <mn>1</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <mi>G</mi> <msub> <mi>&omega;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>&upsi;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <mi>A</mi> <mo>[</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>0</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <msub> <mi>Ae</mi> <mn>0</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>0</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <msub> <mi>G&omega;</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>0</mn> </msub> <msub> <mi>&upsi;</mi> <mn>1</mn> </msub> <mo>]</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <msub> <mi>G&omega;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>&upsi;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>&CenterDot;</mo> </mtd> </mtr> <mtr> <mtd> <mo>&CenterDot;</mo> </mtd> </mtr> <mtr> <mtd> <mo>&CenterDot;</mo> </mtd> </mtr> </mtable> </mfenced> </math>

e_k+1＝(I-L_kC)(Ae_k-Gω_k)+L_kυ_k+1

Can see e_kIs formed by e₀；ω₀，ω₁，…，ω_k-1；υ₁，…，υ_kThe equal linear combination is carried out; at the same time, because e₀，ω_i，υ_i+1(i =0, 1.. k) are independent of each other, so e_k，ω_k，υ_k+1Are independent of each other. Thus estimating the errorThe covariance function can be obtained by the following equation:

<math> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>A</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>G</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&upsi;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>L</mi> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </math>

wherein,

respectively represent random noise omega_k，υ_kThe covariance matrix of (2).

Since the feature function and the probability density function are uniquely determined from each other, the probability density function can be obtained by using the feature function of the estimation error. For mutually independent random variables, the characteristic function expression of the sum is as follows:

wherein

Respectively representing random noise omega_k，υ_kThe characteristic function of (2). It is assumed here that

Initial value of estimation error

Knowing the expression of the characteristic function of the estimation error, the probability density function can be found by the following inverse fourier transform:

and the estimated error e_k+1The probability density function of a gaussian distribution with the same covariance is:

<math> <mrow> <msub> <mi>&gamma;</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> <mfrac> <mn>1</mn> <msqrt> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> </msqrt> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>x</mi> <mi>T</mi> </msup> <msubsup> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> </mrow> </msup> </mrow> </math>

4. 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.

Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (1)

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>&gamma;</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>&gamma;</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>&gamma;</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:

<math> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mi>c</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>A</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>G</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>L</mi> <mi>k</mi> </msub> <msub> <mi>c</mi> <msub> <mi>&upsi;</mi> <mi>k</mi> </msub> </msub> <msubsup> <mi>L</mi> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </math>

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:

<math> <mrow> <msub> <mi>&gamma;</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> <mfrac> <mn>1</mn> <msqrt> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> </msqrt> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>x</mi> <mi>T</mi> </msup> <msubsup> <mi>c</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> </mrow> </msup> </mrow> </math>

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.

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