CN113032988B - Design method of high-order extended Kalman filter based on maximum correlation entropy - Google Patents
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Abstract
The invention discloses a design method of a high-order extended Kalman filter based on maximum correlation entropy. According to the method, two one-dimensional random variables are given, and the related entropy of a random variable pair under limited data driving is obtained; then, a state model and a measurement model of the unmanned aerial vehicle are given; the method comprises the steps of enabling a high-order polynomial in a state model to be defined as a hidden variable of a system, enabling the state model of the system to be represented in a pseudo-linear mode, and enabling a measurement model to be represented in a pseudo-linear mode in a similar mode to obtain a linear mode of the state model and the measurement model; for a state model and a measurement model in a linear form, a high-order extended Kalman filter is obtained by utilizing a recursive filter design idea; and designing and obtaining the high-order extended Kalman filter based on the maximum correlation entropy by utilizing the correlation entropy form of the multidimensional independent vector and the obtained high-order extended Kalman filter. The invention can solve the problems of filtering performance decline and divergence under the nonlinear non-Gaussian system condition, and can be applied to the fields of real-time estimation and target tracking.
Description
Technical Field
The invention belongs to the field of filter design, and particularly relates to a novel design method of a high-order extended Kalman filter by utilizing maximum correlation entropy aiming at a nonlinear non-Gaussian system with a polynomial form.
Background
The application of the filter occupies important positions in various fields at home and abroad, and the progress and development of the filter play an important role in national economy construction, especially national defense construction, such as real-time estimation and target tracking. In 1960, kalman et al proposed Kalman filtering under the minimum mean square error criterion for linear systems and rapidly became widely used. In order to solve the nonlinear problem, extended Kalman Filter (EKF), unscented kalman filter (kf), and bulk kalman filter (CKF) are sequentially presented based on kalman filter. But the filtering described above requires its modeling error to be gaussian white noise. However, in practical applications, most dynamic system noise is mostly non-gaussian white noise. Aiming at a nonlinear non-Gaussian system with a polynomial form, the invention provides a novel high-order extended Kalman filter design method (H-MCEKF) by utilizing maximum correlation entropy.
Disclosure of Invention
The invention provides a design method of a high-order extended Kalman filter based on maximum correlation entropy aiming at a nonlinear non-Gaussian dynamic system with a polynomial form.
The invention firstly gives two one-dimensional random variables X, Y epsilon R 1 Their joint distribution function is F XY (X, Y) and then obtaining the related entropy of the random variable pair (X, Y) under the drive of limited data and the related entropy form of the multidimensional independent vector realized on the basis of N times of sampling; then, a state model and a measurement model of unmanned aerial vehicle motion are given, wherein the state model and the measurement model are complex dynamic systems with strong nonlinear characteristics; the method comprises the steps of enabling a high-order polynomial in a state model to be defined as a hidden variable of a system, enabling the state model of the system to be represented in a pseudo-linear mode, and enabling a measurement model to be represented in a pseudo-linear mode in a similar mode to obtain a linear mode of the state model and the measurement model; for a state model and a measurement model in a linear form, a high-order extended Kalman filter is obtained by utilizing a recursive filter design idea; and designing and obtaining the high-order extended Kalman filter based on the maximum correlation entropy by utilizing the correlation entropy form of the multidimensional independent vector and the obtained high-order extended Kalman filter. The invention can solve the problem of filtering under the condition of a nonlinear non-Gaussian systemThe problems of wave performance degradation and divergence can be applied to the fields of real-time estimation and target tracking.
The invention has the beneficial effects that: the invention combines the high-order extended Kalman filtering with the maximum correlation entropy, can be used for solving the estimation problem of a nonlinear non-Gaussian system, can solve the problems of filtering performance degradation and divergence along with the nonlinear improvement under the nonlinear non-Gaussian system condition, and can be applied to the fields of real-time estimation and target tracking.
Drawings
FIG. 1 is a flow chart of the steps of the method for designing a high-order extended Kalman filter based on maximum correlation entropy of the present invention;
FIG. 2 is a flow chart of step 4 of the present invention;
FIG. 3 is an estimated value of displacement in case 1 of example 1;
FIG. 4 is an estimate of speed for case 1 of example 1;
fig. 5 is an estimation error of case 1 of embodiment 1;
FIG. 6 is an estimated value of displacement for case 2 of example 1;
FIG. 7 is an estimate of speed for case 2 of example 1;
fig. 8 is an estimation error of case 2 of embodiment 1;
FIG. 9 is an estimated value of displacement in case 1 of example 2;
FIG. 10 is an estimated value of the velocity in case 1 of example 2;
fig. 11 is an estimation error of case 1 of embodiment 2;
FIG. 12 is an estimated value of displacement in case 2 of example 2;
FIG. 13 is an estimated value of the velocity in case 2 of example 2;
fig. 14 is an estimation error of case 2 of embodiment 2.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The invention provides a novel high-order extended Kalman filter design method (H-MCEKF) by utilizing maximum correlation entropy based on target tracking aiming at a nonlinear non-Gaussian system with a polynomial form.
As shown in fig. 1, the present invention includes the steps of:
step (1) related entropy description:
the correlation entropy is a generalized similarity measure describing the difference between two random variables. Given two one-dimensional random variables X, Y E R 1 Their joint distribution function is F XY (x, y), then the relative entropy is defined as
V(X,Y)＝E[κ(X,Y)]＝∫κ(X,Y)dF XY (x,y) (1)
Where E is the desired operator, and κ (. Cndot. ) is the translation-invariant Mercer kernel. In this context, without particular emphasis, this kernel function is a gaussian kernel, defined as follows:
in the above equation (2), e=x-y, σ >0 represents the core bandwidth.
Taylor series expansion of the expression (2) can be obtained
The relative entropy of formula (1) has the following expression:
wherein E { (X-Y) 2k }＝∫(x-y) 2k dF XY (X, Y) is the 2k truncated moment statistic of the random variable X, Y ε R.
However, in most practical cases, the joint distribution F XY Is generally unknown, there is often a finite implementation for the random variable pair (X, Y), (X) (j) ,y (j) ) j=1, 2, …, N; in these cases, the anisotropy may be estimated using a sample mean estimator.
Then the associated entropy expression of the random variable pair (X, Y) under finite data driving
When X, Y epsilon R n In the case that the components in the vector e=x-Y are mutually independent, the multidimensional correlation entropy realized based on N times of sampling is as follows
Step (2) giving a state model and a measurement model of unmanned aerial vehicle motion, which are complex dynamic systems with strong nonlinear characteristics, defining a high-order polynomial in the state model as a hidden variable of the system, and pseudo-linearizing and linearizing the state model to represent the state model and the measurement model; state model and measurement model of the system:
x(k+1)＝f(x(k))+w(k) (8)
y(k+1)＝h(x(k+1))+v(k+1) (9)
wherein x (k) ∈R n For an n-dimensional state vector, y (k) ∈R m Representing an m-dimensional measurement vector;
and->Respectively representing a state transfer function and a measurement function; w (k) and v (k) are modeling errors for non-gaussian systems.
Step (2-1) pseudo-linearizing the state model representation for a given unmanned aerial vehicle motion system:
for convenience of description and understanding, let m=n=2 be exemplified, an exemplary description is made. Two state variables x 1 、x 2 Representing the displacement and velocity of the target, respectively.
Let state transition function f i (x (k)) has the following higher order polynomial form
Wherein,is the sum of all first-order tensors; />Representing the weights corresponding to the tensors of each order.
Definition 1:is a set of l-order hidden variables.
Definition 2:and a weight vector corresponding to the l-order hidden variable.
According to definition 1 and definition 2, the pseudo-linear dimension expansion form of formula (13) is as follows
And (3) making:
the matrix form of formula (13) is
Also, assume that the measurement function h i (x (k+1)) has the following higher order polynomial
According to definition 1 and definition 2, formulas (12) and (13), a matrix form of formula (15) can be obtained
Step (2-2) converting the pseudo-linearized state model into a truly linear form, and equivalently rewriting the nonlinear measurement model into a linear form with states and parameters as variables:
in order to convert the pseudo-linear model established in section 2-1 into a truly linear form, a dynamic relationship between the l-order hidden variables and the u-order hidden variables needs to be established.
Wherein,the identification can be performed according to the input information of the original state. Without any a priori information, it is set as follows
In combination with definition 1 and definition 2, equation (13) and equation (14), the state model (8) has the following linear matrix form
Order the
X(k)＝[(x (1) (k)) T ,(x (2) (k)) T ,…,(x (l) (k)) T ,…,(x (r) (k)) T ] T
Then formula (19) has the following linearization form
X (k+1)＝A(k+1,k)X(k)+W(k) (20)
Where W (k) is the modeling error.
Similarly, a linear matrix form of the measurement model can be obtained
Further, the linearization form of the measurement model is as follows
Y(k+1)＝H(k+1)X(k+1)+V(k+1) (22)
Where V (k+1) is the modeling error.
And (3) obtaining a high-order extended Kalman filter by using a state model and a measurement model in a linear form and utilizing a recursive filter design idea:
for the linear models (20) and (22), a KF-based filter is given. Given an initial value X (0), when W (k) and V (k+1) are both zero-mean Gaussian white noise, and the variances are respectively noted as Q 1 And R, and extend modeled W 1 (k) Also zero mean, variance is Q 1 White noise of (a) is provided.
Design idea of recursive filter:
P(k+1|k)＝A(k+1,k)P(k|k)A T (k+1,k)+Q(k) (24)
K(k+1)＝(P(k+1|k)H T (k+1))(H(k+1)P(k+1|k)H T (k+1)+R(k+1)) -1 (25)
P(k+1|k+1)＝(I-K(k+1)H(k+1))P(k+1|k) (27)
wherein Q (k) =diag { Q (1) (k)…Q (r) (k)}, w (1) (k) Is non-Gaussian noise with limited sampling w (1,j) (k) Realization, w (l) (k) Gaussian white noise with 0 mean and variance Q (l) ,l＝2,3,…,r
Step (4) a high-order extended kalman filter design based on maximum correlation entropy, see fig. 2:
step (4-1) obtaining an estimated value of the system state variable X (k) based on the k momentAnd estimating an error covariance P (k|k), predicting based on a filtering formula (26) to obtain a one-step predicted estimate of the system state variable X (k+1)>And a corresponding one-step prediction error covariance matrix +.>
And (3) recording: the one-step predictive estimation error of the system state variable X (k+1) is
And can be equivalently rewritten as a measurement model for the system state variable X (k+1)
Will beConsidered as a measure of the system state variable X (k+1),. Sup.>Consider as measurement error; combining with the actual measurement equation (22), and synthesizing to obtain the combination of the combined measurement models
Wherein I is a unit array of corresponding dimension,and has
Here, the one-step prediction error covariance matrix of the system state variable X (k+1) is obtained as per equation (28)
In the above-mentioned method, the step of,the definition is as follows:
wherein Q is (2) (k),…,Q (r) (k) Each higher-order hidden variable x (2) (k),…,x (r) (k) Random error direction during dynamic modelingQuantity w (2) (k),…,w (r) (k) Is a covariance matrix of (a);is the non-Gaussian modeling error w in the original system state variable model (1) (k) After obtaining limited times of sampling, second order statistics are calculated
In (31)Is the non-Gaussian modeling error v (k+1) in the original system measurement state model, and the second order statistic is calculated after finite sampling is obtained
Wherein v is (j) (k+1) is the jth realization vector of the non-gaussian random noise vector v (k+1).
Step (4-2) a statistical independence process of components in a non-Gaussian modeling error vector u (k+1) in a measurement model combination:
vector u (k+1) in measurement model combination (30) is of dimensionIs non-statistically independent between the components. In order to utilize the correlated entropy form of the multidimensional independent vector shown by equation (7), a statistically independent transformation is required for the L-dimensional non-gaussian vector u (k+1). For this purpose, formula (31) is further represented as
Wherein B is X (k+1) and B Y (k+1) are respectivelyAnd->Cholesky factorization matrix of (a).
Will B -1 (k+1) acts on both sides of the formula (30) respectively to obtain
The above can be further simplified into
D(k+1)＝S(k+1)X(k+1)+e(k+1) (36)
Wherein the method comprises the steps of
e(k+1)＝B -1 (k+1)u(k+1)
Due to
E{e(k)e T (k)}＝E{[B -1 (k+1)u(k+1)][B -1 (k+1)u(k+1)] T }
＝B -1 (k+1)E{u(k+1)u T (k+1)}(B -1 (k+1)) T
＝B -1 (k+1)B(k+1)B T (k+1)(B -1 (k+1)) T
＝I (37)
Thus, the non-Gaussian modeling error random variable u (k+1) passes through matrix B -1 After the (k+1) equivalent transformation, the components of the obtained random e (k+1) become statistically independent.
Step (4-3) of establishing a related entropy objective function for solving the estimated value of the system state variable X (k+1) based on the measurement model combination:
establishing a solution state variable estimate based on N samples of L-dimensional observation information of a system state variable X (k+1) according to equation (30)Is the related entropy objective function of (2)
Wherein after independent transformation, the method comprises the steps of,a jth realization of an ith component in the L-dimensional vector D (k+1) in equation (36); s is(s) i (k+1) is the ith row of the matrix S (k+1) of formula (36).
Therefore, by solving for the maximum solution of the correlation entropy objective function (38), an optimal estimate of the system state variable X (k+1) under the maximum correlation entropy criterion can be obtained
Step (4-4) of solving the stationary point based on the system state variable estimated value of the maximum correlation entropy objective function, and solving the iteration value of the stationary point:
for kernel functionsAnd (3) for the system state variable X (k+1), providing an invariant point iterative numerical solution process for solving the system state variable estimated value according to the related entropy objective function (38).
For objective function J (j) (X (k+1) solving for its first partial derivative with respect to the system state variable and letting
Then there is
Due toAlso comprises a state variable X (k+1) of the system to be solved, and the state variable is equivalently transformed into an motionless point equation in the following form so as to facilitate iterative numerical solution
X(k+1)＝f (X(k+1)) (42)
Wherein the method comprises the steps of
In the above
Thus, the equation (42) can be used to implement the iterative numerical solution of the dead point of the estimated value of the system state variable based on the related entropy objective function.
Step (4-5) equivalent conversion to Kalman filtering by using a stationary point solving equation:
the equations in equation (36) and the well-known matrix inversion theory are utilized, and the stationary point solving equations (44) and (45) can be equivalently transformed as follows
Wherein,
in the further derivation process, the gain matrix is utilizedProperties of (C)
In this way, the stationary point numerical solution equation (41) can be equivalently rewritten as
Thereby completing the equivalent conversion process from the equation solving from the active point to the Kalman filter solving.
Step (4-6) solving state estimation values through online iterative Kalman filtering
Given an initial value of iterationSolving state estimation value by online iterative Kalman filtering>The implementation process of (1) is
Wherein the method comprises the steps of
In the above
Obtaining the t-th iteration solution value of the system state variable X (k+1)The iterative algorithm ends at step r (k+1). The choice of r (k+1) depends both on the required estimation accuracy and on the calculation time.
When the iteration ends at the (k+1) th step, the estimated value and the estimated error covariance matrix of the X (k+1) are obtained respectively
Annotation: c of formula (56) x (k+1) t In (a) and (b)By->Associated with X (k+1);
c of formula (57) y (k+1) t In (a) and (b)In this case, X (k+1) is associated with Y (k+1).
To verify the effectiveness of the method of the invention, comparative simulation experiments were performed with two examples: wherein two state variables x 1 、x 2 Representing the displacement and the speed of an object at the time k respectively, wherein the first embodiment is that the state equation is a high-order polynomial and the measurement model is linear; and the second case is that the state and measurement equation are both in the form of higher order polynomials. In contrast to embodiment one, the measurement equation of embodiment two is a higher order polynomial.
Embodiment 1 considers that the state equation is a high order polynomial and the measurement equation is a linear model of the unmanned plane motion system
Wherein w (k+1) to 0.8N (0,0.01) +0.2N (0,0.1), v (k+1) to N (0,0.01) and an initial value x (0) = [1 1)] T Initial estimateInitial estimation error covariance P (0|0) =0.01×diag (1, 1).
The state variable is estimated under two conditions by using the MCEKF and the H-MCEKF, and the estimated values of the displacement and the speed and the estimated errors of the displacement and the speed are compared. And respectively calculating the accuracy of the MCEKF and the H-MCEKF for comparison.
Example 2 unmanned aerial vehicle motion System with both the State equation and the measurement equation in the form of higher order polynomials
Wherein w is 1 (k)～N(0,0.01)，w 2 (k)～N(0,0.01)，w 1 (k) And w 2 (k) V (k+1) -0.8N (0,0.01) +0.2N (0,0.1), initial value x (0) = [1 1), for uncorrelated Gaussian white noise sequences] T Initial estimate Initial estimation error covariance P (0|0) =0.01×diag (1, 1).
And estimating the target state variable under two conditions by using the MCEKF and the H-MCEKF, and comparing the estimated values of the displacement and the speed and the estimated errors of the displacement and the speed. And respectively calculating the accuracy of the MCEKF and the H-MCEKF for comparison.
Example 1 was performed as case one σ=0.5, ε=10, respectively -1 And case di sigma=3, epsilon=10 -4 For example, the estimation results of the motion state of the unmanned aerial vehicle by the MCEKF and the H-MCEKF filtering methods are given, and the estimation results are shown in fig. 3 to 8. Table 1 summarizes the estimated errors of displacement and velocity for different σ and epsilon conditions.
Table 1 Estimation errors with different filters
/>
Example 2 was performed as case one σ=0.5, ε=10, respectively -1 And case di sigma=10, epsilon=10 -1 For example, MCEKF and H-MC are givenThe estimated results of the motion state of the unmanned aerial vehicle by the EKF two filtering methods are shown in fig. 9 to 14. Table 2 summarizes the estimated errors of displacement and velocity for different σ and epsilon conditions.
Table 2 Estimation errors with different filters
/>
From the above two embodiments, it can be seen that the filtering performance of the H-MCEKF is slightly worse than that of the MCEKF when the measurement noise is in a non-gaussian distribution, given a too small kernel bandwidth σ. However, given a suitable kernel bandwidth σ, the filtering performance of the H-MCEKF may be close to or even better than that of the MCEKF. In example 2, although the MCEKF can achieve a small estimation error for the estimation of the velocity, the error for the displacement and the entire state variable is large. Furthermore, as can be seen from tables 1 and 2, the threshold epsilon has little influence on the filtering performance compared to the kernel bandwidth sigma. The above data analysis verifies the validity of the proposed method.
Claims (1)
1. The design method of the high-order extended Kalman filter based on the maximum correlation entropy is characterized by comprising the following steps of: comprising the following steps:
step (1) two one-dimensional random variables X, Y E R are given 1 Their joint distribution function is F XY (X, Y) obtaining the related entropy of the random variable pair (X, Y) under the drive of limited data and the related entropy form of the multidimensional independent vector based on N times of sampling;
step (2) giving a state model and a measurement model of unmanned aerial vehicle motion, which are complex dynamic systems with strong nonlinear characteristics, and pseudo-linearizing the state model of the system by defining a higher-order polynomial in the state model as a hidden variable of the system, and pseudo-linearizing the measurement model to obtain a linear form of the state model and the measurement model;
step (3) a state model and a measurement model in a linear form are utilized to obtain a high-order extended Kalman filter by utilizing a recursive filter design idea;
step (4) obtaining a high-order extended Kalman filter based on the maximum correlation entropy by utilizing the correlation entropy form in the step (1) and the high-order extended Kalman filter obtained in the step (3);
the state model and the measurement model in the step (2) are respectively expressed as:
x(k+1)＝f(x(k))+w(k)
y(k+1)＝h(x(k+1))+v(k+1)
wherein x (k) ∈R n For an n-dimensional state vector, y (k) ∈R m Representing an m-dimensional measurement vector;
and->
Respectively representing a state transfer function and a measurement function; w (k) and v (k) are modeling errors for non-gaussian systems;
the step (2) comprises:
step (2-1) pseudo-linearizing the state model representation for a given unmanned aerial vehicle motion system: equivalently rewriting a state model into a pseudo-linear model based on the combination of an original variable and a hidden variable;
step (2-2) converting the pseudo-linearized state model into a true linear form, and similarly, equivalently rewriting the nonlinear measurement model into a linear form taking states and parameters as variables;
the step (4) comprises:
step (4-1) obtaining a measurement model related to a system state variable X (k+1) by using a high-order extended Kalman filter, and combining the measurement model with an actual measurement model after system linearization into a measurement model combination
Step (4-2) transforming the statistical independence of each component in the non-Gaussian modeling error vector u (k+1) in the measurement model combination into a relevant entropy form of the multidimensional independent vector in the step (1);
step (4-3) establishing a state variable estimated value of the solving system based on the measurement model combinationBy solving the maximum value solution of the related entropy objective function, the optimal estimated value +.o of the system state variable X (k+1) under the maximum related entropy criterion is obtained>
Step (4-4) solving the stationary point based on the system state variable estimated value of the maximum correlation entropy objective function, and solving the iteration value of the stationary point;
step (4-5), solving an equation to perform Kalman filtering equivalent conversion by using the fixed point;
step (4-6) solving system state variable estimated value through online iterative Kalman filtering
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