Front. Phys. Frontiers in Physics Front. Phys. 2296-424X Frontiers Media S.A. 1347843 10.3389/fphy.2024.1347843 Physics Original Research Maximum correntropy unscented filter based on unbiased minimum-variance estimation for a class of nonlinear systems Zhang et al. 10.3389/fphy.2024.1347843 Zhang Yike Niu Ben Song Xinmin * School of Information Science and Engineering, Shandong Normal University, Jinan, China

Edited by: Teoman Özer, Istanbul Technical University, Türkiye

Reviewed by: Badong Chen, Xi’an Jiaotong University, China

Guici Chen, Wuhan University of Science and Technology, China

 *Correspondence: Xinmin Song, xinminsong@sina.com 10 06 2024 2024 12 1347843 01 12 2023 29 04 2024 Copyright © 2024 Zhang, Niu and Song. 2024 Zhang, Niu and Song

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Introduction: The unscented Kalman filter based on unbiased minimum-variance (UKF-UMV) estimation is usually used to handle the state estimation problem of nonlinear systems with an unknown input. When the nonlinear system is disturbed by non-Gaussian noise, the performance of UKF-UMV will seriously deteriorate.

Methods: A maximum correntropy unscented filter based on the unbiased minimum variance (MCUF-UMV) estimation method is proposed on the basis of the UKF-UMV without the need for estimation of an unknown input and uses the maximum correntropy criterion (MCC) and fixed-point iterative algorithm for state estimation.

Results: When the measurement noise of the nonlinear system is non-Gaussian noise, the algorithm performs well.

Discussion: Our proposed algorithm also does not require estimation of an unknown input, and there is no prior knowledge available about the unknown input or any prior assumptions. The unknown input can be any signal. Finally, a simulation example is used to demonstrate the effectiveness and reliability of the algorithm.

 maximum correntropy criterion unbiased minimum-variance unscented Kalman filter unknown input state estimation section-at-acceptance Mathematical Physics 1 Introduction

The state estimation problem of systems with unknown input is very common in practical applications, such as target tracking and automatic control [1–6]. There have been many studies on state estimation for linear discrete-time systems with unknown input, and the methods for state estimation are mainly summarized into the following three categories: the first method is the augmented state Kalman filter (ASKF), in which the unknown input is considered a part of the state and then estimated; that is, both the state and the unknown input are estimated simultaneously [7]. This method assumes the unknown input as a random process with known statistical characteristics, but in reality, the dynamic disturbance is unknown, so its performance usually does not achieve the desired effect. The second method is the modified Kalman filter (MKF) using the Bayesian method when the input variable of the state equation is not fully observed [8]. The third method is to use unbiased minimum-variance (UMV) state estimation when information on unknown input is not available [9–11]. Compared with ASKF and MKF, which rely on all or part of the knowledge of the unknown input, the UMV filter does not require any prior knowledge or assumption about the unknown input, and the unknown input can be any signal, making it more practical.

Recently, a large number of research studies have emerged on nonlinear systems with an unknown input. [12–14] proposes an unscented Kalman filter-based unbiased minimum-variance (UKF-UMV) estimation, which uses the UMV state estimation framework to develop a new nonlinear filter to handle the unknown input. [15] proposes a robust unscented unbiased minimum-variance (RU-UMV) estimator for nonlinear systems with unknown input, which can effectively handle innovation and observe outliers. [16] proposes a robust unscented M-estimation-based filter (RUMF) for state estimation of nonlinear systems of actual vehicles with unknown input. The proposed algorithm is robust to non-Gaussian process noise and innovation in different maneuvering scenarios. However, this algorithm adopts a complex 7-degree-of-freedom vehicle dynamics model, which is very time-consuming, and considers non-Gaussian process noise, without considering non-Gaussian measurement noise.

Correntropy is a measure of local similarity defined in kernel space. The maximum correntropy criterion (MCC) has been successfully applied in many fields of signal processing and machine learning in recent years to cope with non-Gaussian measurement noise in the system, especially heavy-tailed measurement noise. [17–21] proposes a maximum correntropy Kalman filter (MCKF), which uses the robust MCC as the optimality criterion instead of the minimum mean square error (MMSE) criterion. [22] derives a multi-kernel maximum correntropy Kalman filter (MKMCKF) to deal with the interference of multivariate non-Gaussian noise for the systems with an unknown input. This algorithm makes the assumption that d k+1 = d k when estimating the unknown input, which can indeed be applied in most cases, such as when the unknown input is represented by continuous signal sin or cos cycles, the error is relatively small. However, if the unknown input is discontinuous, such as a pulse square wave function, this simple assumption will reduce the accuracy in a certain sense and become inaccurate. In order to improve the robustness of the unscented Kalman filter (UKF) to impulse noise, [23] proposes a maximum correntropy unscented filter (MCUF) for nonlinear systems, but does not consider nonlinear systems with unknown input.

Based on the analysis of the above research studies, we propose a maximum correntropy unscented filter based on the unbiased minimum-variance (MCUF-UMV) estimation algorithm. When the nonlinear system with unknown input is disturbed by non-Gaussian measurement noise, especially pulse measurement noise, the performance of the algorithm is good. The contributions of this paper are summarized as follows:

1. The MCUF-UMV algorithm is proposed. First, the prior estimate of the state and prior error covariance matrix are obtained through unscented transformation (UT), and then the nonlinear system and measurement equation are transformed into a quasi-linear regression form using statistical linearization technology. A state augmented model is built, and the MCC and fixed-point iterative algorithm are used to estimate the state.

2. Different from [22], we do not use the simple assumption d k+1 = d k . Based on the UKF-UMV form in [14], which does not require unknown input estimation, the MCC is used to estimate the state. There is no need for any prior knowledge or assumptions about the unknown input, and the unknown input can be any signal.

3. We show that for non-Gaussian noise interference, MCUF-UMV is significantly superior to the existing filter in simulation.

The remainder of the paper is structured as follows: section 2 presents preliminary preparation and gives the nonlinear system model and problem statement. Section 3 presents the derivation and equations summary of the MCUF-UMV algorithm. Section 4 demonstrates the excellent performance of the MCUF-UMV algorithm through an illustrative example. Section 5 presents the conclusion.

 2 Preliminary and problem statement 2.1 Maximum correntropy criterion

The correntropy representing the similarity measure is as follows: V X , Y = E ψ X , Y = ∫ ψ x , y d F X Y x , y , where X , Y ∈ R are two random variables, F XY is the joint probability distribution function, and ψ ( x , y ) = G σ ( e ) = e x p ( − e 2 2 σ 2 ) is the shift-invariant Mercer kernel. e = x − y, and σ is the kernel bandwidth. In most practical cases, the correntropy of the Gaussian kernel can be approximated through the sampling estimator: V ̂ X , Y = 1 N ∑ i = 1 N G σ e i , where e(i) = x(i) − y(i) and { x ( i ) , y ( i ) } i = 1 N is the N samples extracted from F XY . Using Taylor series to expand the Gaussian kernel V X , Y = ∑ n = 0 ∞ − 1 n 2 n σ 2 n n ! E X − Y 2 n , the correntropy is the weighted sum of all even moments of the error variable X − Y.

 2.2 System model

The following system model can be used to describe nonlinear discrete-time systems with unknown input: x k + 1 = f x k , u k + 1 + G k d k + q k , z k = h x k , u k + r k , where x k ∈ R n , u k ∈ R l , d k ∈ R p , and z k ∈ R m are, respectively, the state vector, known input vector, unknown input vector, and measurement vector at time k; f(⋅) and h(⋅) are nonlinear functions; G k is a known matrix; the process noise q k is assumed to be zero mean white noise and the process noise q k and the measurement noise r k are uncorrelated. The covariance matrices of process noise and measurement noise are Q k and R k , respectively. In fact, we do not have any prior knowledge about unknown input d k available, nor do we make a prior assumption that unknown input d k can be any signal. Our research is based on this fact.

Problem statement: Based on systems 1, 2, this paper first uses UT to obtain the prior estimate of the state and prior error covariance matrix, then uses the statistical linearization technique to transform the nonlinear system and measurement equation into the quasi-linear regression form, and finally uses the MCC and fixed-point iterative algorithm for state estimation, which can effectively solve the interference of non-Gaussian measurement noise on nonlinear systems with unknown input.

 3 MCUF-UMV algorithm derivation 3.1 Statistical linear regression

First, the one-step prediction is calculated. Given the mean x ̂ k − 1 ∣ k − 1 and covariance P k − 1 ∣ k − 1 x x , 2n + 1 sigma points χ k − 1 ∣ k − 1 i and corresponding weight values w can be obtained through UT, which gives the formula χ k − 1 ∣ k − 1 i χ k − 1 ∣ k − 1 i = x ̂ k − 1 ∣ k − 1 , i = 0 x ̂ k − 1 ∣ k − 1 + n + λ P k − 1 ∣ k − 1 x x i , i = 1 , … , n x ̂ k − 1 ∣ k − 1 − n + λ P k − 1 ∣ k − 1 x x i − n , i = n + 1 , … , 2 n , where n refers to the dimension of the state and ( P ) i represents the i-th column of the matrix root. The corresponding weights of these sampling points are calculated as follows: w m 0 = λ n + λ , w c 0 = λ n + λ + 1 − α 2 + β , w m i = w c i = 1 2 n + λ , i = 1 , … , 2 n ,

where the subscript m represents the mean, c represents the covariance, and the superscript represents the i-th sampling point. The parameter λ = α 2(n + κ) − n is a scaling parameter. The selection of α controls the distribution state of the sampling points, and κ is the parameter to be selected, whose value should generally ensure that the matrix ( n + λ ) P k − 1 ∣ k − 1 x x is a positive semi-definite matrix. The selected parameter β is a non-negative weight coefficient that can merge the motion errors of higher-order terms in the equation. Using nonlinear process function f(⋅) transformation for each sigma point, we obtain χ k ∣ k − 1 i = f χ k − 1 ∣ k − 1 i , u k , The predicted mean and covariance matrix of the state are x ̂ k ∣ k − 1 = ∑ i = 0 2 n w m i χ k ∣ k − 1 i , P k ∣ k − 1 x x = ∑ i = 0 2 n w c i χ k ∣ k − 1 i − x ̂ k ∣ k − 1 χ k ∣ k − 1 i − x ̂ k ∣ k − 1 T + Q k − 1 .

New sigma points are generated based on one-step prediction χ k ∣ k − 1 * i = x ̂ k ∣ k − 1 , i = 0 x ̂ k ∣ k − 1 + n + λ P k ∣ k − 1 x x i , i = 1 , … , n x ̂ k ∣ k − 1 − n + λ P k ∣ k − 1 x x i − n , i = n + 1 , … , 2 n , using nonlinear measurement function h(⋅) transform for newly generated sigma points Z k ∣ k − 1 i = h χ k ∣ k − 1 * i , u k . and the prediction of the measurement vector is z ̂ k ∣ k − 1 = ∑ i = 0 2 n w m i Z k ∣ k − 1 i . The innovation and cross covariance matrices are P k ∣ k − 1 z z = ∑ i = 0 2 n w c i Z k ∣ k − 1 i − z ̂ k ∣ k − 1 Z k ∣ k − 1 i − z ̂ k ∣ k − 1 T + R k P k ∣ k − 1 x z = ∑ i = 0 2 n w c i χ k ∣ k − 1 * i − x ̂ k ∣ k − 1 Z k ∣ k − 1 i − z ̂ k ∣ k − 1 T .

Before introducing the proposed algorithm, we transform the nonlinear measurement equation into the linear form using the statistical linearization technique as follows: z k = H k x k − x ̂ k ∣ k − 1 + z ̂ k ∣ k − 1 + θ k , where H k is the measurement slope matrix. H k = P k ∣ k − 1 x z T P k ∣ k − 1 x x − 1 . The covariance of θ k is Φ k = P k ∣ k − 1 z z − P k ∣ k − 1 x z T P k ∣ k − 1 x x − 1 P k ∣ k − 1 x z = P k ∣ k − 1 z z − H k P k ∣ k − 1 x x H k T .

3.2 Existing UKF-UMV without unknown input estimation

The MCUF-UMV algorithm we derived in Section 3.3 is based on the UKF-UMV in [14] that does not require estimation of unknown input. Therefore, this section provides an introduction and summary of UKF-UMV without unknown input estimation. According to Eq. 3, the innovation Δz k is represented as Δ z k = z k − z ̂ k ∣ k − 1 = H k x k − x ̂ k ∣ k − 1 + θ k . According to Eqs 1, 4 Δ z k = H k G k − 1 d k − 1 + η k , where ζ k = f x k − 1 , u k − x ̂ k ∣ k − 1 + q k − 1 , η k = H k ζ k + θ k ,

and E ζ k ζ k T = P k ∣ k − 1 x x , R ̃ k = E η k η k T = H k P k ∣ k − 1 x x H k T + Φ k = P k ∣ k − 1 z z .

The existing UKF-UMV without unknown input estimation is summarized as follows: K k = P k ∣ k − 1 x x H k T R ̃ k − 1 = P k ∣ k − 1 x z P k ∣ k − 1 z z − 1 , M k = G k T H k T R ̃ k − 1 H k G k − 1 G k T H k T R ̃ k − 1 , L k = K k + I − K k H k G k M k , x ̂ k ∣ k = x ̂ k ∣ k − 1 + L k z k − z ̂ k ∣ k − 1 , P k ∣ k x x = I − L k H k P k ∣ k − 1 x x I − L k H k T + L k Φ k L k T .

3.3 State estimation

From the nonlinear model described above, the augmented model is given as follows: x ̂ k ∣ k − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 = I H k x k + v k , where I is the dimension of the n × n identity matrix and v k can be expressed as v k = − x k − x ̂ k ∣ k − 1 θ k , with E v k v k T = P k ∣ k − 1 x x 0 0 Φ k = B k ∣ k − 1 p B k ∣ k − 1 p T 0 0 B k Φ B k Φ T = B k B k T , where B k ∣ k − 1 p , B k Φ and B k is obtained by using Cholesky decomposition. Then, multiplying both sides of Eq. 5 by B k − 1 , the following formula is obtained: D k = W k x k + e k , where D k = B k − 1 x ̂ k ∣ k − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 , W k = B k − 1 I H k , e k = B k − 1 v k .

Then, the cost function based on the MCC can be obtained as follows: J L x k = 1 L ∑ i = 1 L G σ D k i − W k i x k , where the dimension of D k is expressed in L and L = n + m. D k i is the ith element of D k , W k i is the ith row of W k , and σ is the kernel bandwidth of correntropy. Then, the optimal estimate of x k is x ̂ k = a r g max x k 1 L ∑ i = 1 L G σ e k i , where e k i is the ith element of e k : e k i = D k i − W k i x k .

Let ∂ J L x k ∂ x k = 0 , The optimal solution is given as x k = ∑ i = 1 L G σ e k i W k i T W k i − 1 × ∑ i = 1 L G σ e k i W k i T D k i .

It can be seen that the optimal solution is a fixed-point x k equation, which can also be rewritten as x k = W k T C k W k − 1 W k T C k D k , where C k = C k x 0 0 C k z , with C k x = d i a g G σ e k 1 , … , G σ e k n , C k z = d i a g G σ e k n + 1 , … , G σ e k n + m .

Eq. 6 can be further expressed as follows: x k = x ̂ k ∣ k − 1 + K ̃ k z k − z ̂ k ∣ k − 1 , where K ̃ k = P ̃ k ∣ k − 1 x x H k T H k P ̃ k ∣ k − 1 x x H k T + Φ ̃ k − 1 , P ̃ k ∣ k − 1 x x = B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T , Φ ̃ k = B k Φ C k z − 1 B k Φ T .

The detailed derivation process of Eq. 7 is in the Appendix.

 3.4 Summary of MCUF-UMV equations

This section presents the summary of the MCUF-UMV algorithm. Given the mean x ̂ k − 1 ∣ k − 1 and covariance P k − 1 ∣ k − 1 x x , when 2n + 1 sigma points χ k − 1 ∣ k − 1 i and the corresponding weights w m i , w c i are obtained through UT, and χ k ∣ k − 1 i is obtained through nonlinear process function f(⋅).

1. Time update

x ̂ k ∣ k − 1 = ∑ i = 0 2 n w m i χ k ∣ k − 1 i , P k ∣ k − 1 x x = ∑ i = 0 2 n w c i χ k ∣ k − 1 i − x ̂ k ∣ k − 1 χ k ∣ k − 1 i − x ̂ k ∣ k − 1 T + Q k − 1 .

Based on x ̂ k ∣ k − 1 and covariance P k ∣ k − 1 x x , new sigma points χ k ∣ k − 1 * i are obtained by UT, and then Z k ∣ k − 1 i is obtained by the nonlinear measurement function h(⋅). Then

z ̂ k ∣ k − 1 = ∑ i = 0 2 n w m i Z k ∣ k − 1 i , P k ∣ k − 1 z z = ∑ i = 0 2 n w c i Z k ∣ k − 1 i − z ̂ k ∣ k − 1 Z k ∣ k − 1 i − z ̂ k ∣ k − 1 T + R k , P k ∣ k − 1 x z = ∑ i = 0 2 n w c i χ k ∣ k − 1 * i − x ̂ k ∣ k − 1 Z k ∣ k − 1 i − z ̂ k ∣ k − 1 T .

2. Measurement update

Given a suitable kernel bandwidth σ and small constant ϵ for state estimation, let t = 1 and x ̂ ( k ∣ k ) 0 = x ̂ k ∣ k − 1 , where x ̂ ( k ∣ k ) t represents the state estimation during fixed-point iteration t. e k i = D k i − W k i x ̂ k ∣ k t − 1 , C k x = d i a g G σ e k 1 , … , G σ e k n , C k z = d i a g G σ e k n + 1 , … , G σ e k n + m , P ̃ k ∣ k − 1 x x = B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T , Φ ̃ k = B k Φ C k z − 1 B k Φ T , H k = P k ∣ k − 1 x z T P k ∣ k − 1 x x − 1 , K ̃ k = P ̃ k ∣ k − 1 x x H k T H k P ̃ k ∣ k − 1 x x H k T + Φ ̃ k − 1 , x ̂ k ∣ k t = x ̂ k ∣ k − 1 + K ̃ k z k − z ̂ k ∣ k − 1 .

The following inequality is given as ‖ x ̂ k ∣ k t − x ̂ k ∣ k t − 1 ‖ ‖ x ̂ k ∣ k t − 1 ‖ ≤ ϵ , where x ̂ k ∣ k = x ̂ ( k ∣ k ) t . If the above inequality is true, continue to the next step; otherwise, return to iterative steps in the measurement update again.

Finally, the covariance P k ∣ k x x of the state measurement update error is obtained: P k ∣ k x x = I − K ̃ k H k P k ∣ k − 1 x x I − K ̃ k H k T + K ̃ k Φ k K ̃ k T .

4 Illustrative example

In this section, an example of uniformly accelerating linear motion target tracking in [24] with minor modification is used to demonstrate the effectiveness and reliability of the MCUF-UMV algorithm by comparing its performance with that of UKF-UMV in nonlinear systems with unknown input. We only use mixed Gaussian noise as an example to illustrate the performance of the algorithm in the presence of non-Gaussian measurement noise interference. In this example, by manually switching between mixed Gaussian noise and Gaussian noise, we demonstrate the performance of the algorithm under mixed Gaussian noise and Gaussian noise interference, respectively.

4.1 Mixed Gaussian noise

Consider a particle M moving in the two-dimensional plane, whose position, velocity, and acceleration at a certain moment k can be represented by the vector x k = [ x ̄ k , y ̄ k , x ̄ ̇ k , y ̄ ̇ k , x ̄ ̈ k , y ̄ ̈ k ] T . Assuming that M undergoes approximately uniformly accelerated linear motion in the x-axis direction and also approximately uniformly accelerated linear motion in the y-axis direction, the equation of motion for this particle in Cartesian coordinates is x k + 1 = 1 0 T 0 T 2 2 0 0 1 0 T 0 T 2 2 0 0 1 0 T 0 0 0 0 1 0 T 0 0 0 0 1 0 0 0 0 0 0 1 x k + 1 − 1 0.4 − 0.2 0.5 − 0.5 d k + q k , where d k is the unknown input, simulated with d k = 0.1cos(0.2k). q k is the process noise. Assuming that the radar with the coordinate position ( x ̄ 0 , y ̄ 0 ) tracks particle M, the distance l k between the radar and particle M and the angle ϕ k of particle M relative to the radar can be obtained. In actual measurements, the radar has noise r k . In a coordinate system centered on the radar, the measurement equation is z k = h x k + r k = l k + r k l ϕ k + r k ϕ = x ̄ k − x ̄ 0 2 + y ̄ k − y ̄ 0 2 + r k l a r c t a n y ̄ k − y ̄ 0 x ̄ k − x ̄ 0 + r k ϕ , where r k l is the measurement noise regarding the distance l k between the radar and particle M and r k ϕ is the measurement noise regarding the angle ϕ k between the radar and particle M relative to the radar. In the Cartesian coordinate system, the state equation of the model is linear, while the measurement equation is nonlinear. In the simulation, the covariance matrix Q k of system noise q k and the measurement noise r k , which are mixed Gaussian noise, are as follows: Q k = d i a g 1,1,0 . 1 2 , 0 . 1 2 , 0.0 1 2 , 0.0 1 2 , r k ∽ 0.9 N 0,0.01 + 0.1 N 0,100 . Initial state x 0 = [1000,5000,10,50,2,−4] T . Measurement number N = 50 and sampling time T = 0.5s.

The generated motion trajectory diagram is shown in Figure 1, and the tracking position, velocity, and acceleration mean square errors (MSEs) are shown in Figure 2. Table 1 shows the comparison of MSEs of two algorithms from ten independent experiments. From the experimental results, it can be seen that the MCUF-UMV algorithm performs significantly better under mixed Gaussian noise interference.

Motion trajectory map.

Tracking error chart with mixed Gaussian noise.

MSEs of 10 independent experiments of position, velocity, and acceleration in mixed Gaussian noise.

Experiment MCUF-UMV position UKF-UMV position MCUF-UMV velocity UKF-UMV velocity MCUF-UMV acceleration UKF-UMV acceleration
1 28.208414 84.579935 2.711820 44.612122 0.424037 5.516357
2 13.989319 18.533778 1.181625 12.586217 0.281214 6.309897
3 11.871045 21.010794 2.730424 19.616551 0.179617 5.849185
4 40.149114 62.241404 3.809605 23.279794 0.147400 7.892510
5 31.350351 84.129213 2.618486 8.262821 0.373872 4.021028
6 18.848458 44.971707 1.678344 12.760293 0.305540 10.316087
7 44.078077 72.514440 2.660592 23.560852 0.182067 14.191144
8 30.835332 68.140025 3.255109 20.098220 0.412753 11.119720
9 65.427237 89.223493 5.985490 7.390882 0.418364 15.480138
10 12.654580 35.408031 1.455289 12.159764 0.399467 5.044264
4.2 Gaussian noise

Using the same model as given in Section 4.1, the measurement noise r k is replaced with Gaussian noise, and its covariance matrix R k is represented as follows: R k = d i a g 1 0 2 , 0.00 1 2 .

To ensure that the entire system model is in the Gaussian environment, the unknown input is set to a random number. The tracking position, velocity, and acceleration MSEs are shown in Figure 3. Table 2 shows the comparison of the MSEs of two algorithms from ten independent experiments. From the data, it can be seen that when the measurement noise is Gaussian noise, the performance of MCUF-UMV is not as good as compared to that of UKF-UMV.

Tracking error chart with Gaussian noise.

MSEs of 10 independent experiments of position, velocity, and acceleration in Gaussian noise.

Experiment MCUF-UMV position UKF-UMV position MCUF-UMV velocity UKF-UMV velocity MCUF-UMV acceleration UKF-UMV acceleration
1 27.097099 6.109113 19.308420 3.477807 5.757299 0.146286
2 21.433896 4.069176 11.316910 4.889262 0.976532 0.789605
3 49.476510 8.287119 15.185673 2.686767 4.090000 3.784623
4 10.998669 6.884131 8.983939 4.569899 2.818557 0.807319
5 27.518734 4.968016 12.597530 0.949519 2.566491 1.745874
6 12.703843 5.485189 6.776403 3.807437 0.910638 0.226301
7 36.631151 2.006545 17.828915 3.368043 3.205974 1.296504
8 15.728420 8.731089 10.039608 5.350884 2.113577 1.014578
9 11.137634 1.403972 9.009863 0.873842 1.399724 0.435985
10 32.226642 5.820621 9.914894 5.363854 3.458389 1.978120
5 Conclusion

We have proposed the MCUF-UMV algorithm for the nonlinear discrete-time system with the unknown input when the system is disturbed by non-Gaussian noise, especially heavy-tailed impulse noise. First, the prior estimation and prior error covariance of the state are obtained by UT. By using statistical linearization techniques, nonlinear system and measurement equation are transformed into quasi-linear regression forms. Based on the UKF-UMV form that does not require unknown input estimation, the MCC and fixed-point iterative algorithm are used to estimate the state. We do not have any prior knowledge or assumptions about the unknown input, and the unknown input can be any signal. Finally, a simulation experiment has been conducted to demonstrate the effectiveness and reliability of the MCUF-UMV algorithm under non-Gaussian noise interference. In future work, we will further apply this algorithm to specific practical applications.

 Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

 Author contributions

YZ: conceptualization, data curation, formal analysis, methodology, software, and writing–original draft. BN: writing–review and editing. XS: resources, supervision, and writing–review and editing.

 Funding

The authors declare that financial support was received for the research, authorship, and/or publication of this article. This research was funded by the National Natural Science Foundation of China (61873152 and 61876101).

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Publisher’s note

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NOTATION: Throughout the entire paper, “superscript “−1” and” superscript “T” represent the inverse and transpose of matrices, respectively. “E” represents the mathematical expectation factor. “I” represents the identity matrix. “ R n ” represents the n-dimensional Euclidean space.

 References Nikoukhah R . Innovations generation in the presence of unknown inputs: application to robust failure detection. Automatica (1994) 30:1851–67. 10.1016/0005-1098(94)90047-7 Li Z Liu X Ren W Xie L . Distributed tracking control for linear multiagent systems with a leader of bounded unknown input. IEEE Trans Automatic Control (2013) 58:518–23. 10.1109/TAC.2012.2208295 Song X Zheng WX . Linear estimation for discrete-time periodic systems with unknown measurement input and missing measurements. ISA Trans (2019) 95:164–72. 10.1016/j.isatra.2018.11.013 Wei F Chen G Wang W . Finite-time synchronization of memristor neural networks via interval matrix method. Neural Networks (2020) 127:7–18. 10.1016/j.neunet.2020.04.003 Mortensen K Flyvbjerg H Pedersen J . Confined brownian motion tracked with motion blur: estimating diffusion coefficient and size of confining space. Front Phys (2021) 8. 10.3389/fphy.2020.583202 Chen B Dang L Zheng N Principe JC . Kalman filtering under information theoretic criteria (2023). p. 89–126. 10.1007/978-3-031-33764-2_4 Ding B Zhang T Fang H . Infinity augmented state kalman filter and its application in unknown input and state estimation. J Franklin Inst (2023) 360:11916–31. 10.1016/j.jfranklin.2023.08.034 Li B . State estimation with partially observed inputs: a unified kalman filtering approach. Automatica (2013) 49:816–20. 10.1016/j.automatica.2012.12.007 Kitanidis PK . Unbiased minimum-variance linear state estimation. Automatica (1987) 23:775–8. 10.1016/0005-1098(87)90037-9 Darouach M Zasadzinski M . Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica (1997) 33:717–9. 10.1016/S0005-1098(96)00217-8 Gillijns S De Moor B . Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica (2007) 43:111–6. 10.1016/j.automatica.2006.08.002 Khan M Khan S Chu Y . New estimates for the jensen gap using s-convexity with applications. Front Phys (2020) 8. 10.3389/fphy.2020.00313 Qiao J Lu Z Lin B Song J Xiao Z Wang Z A survey of gnss interference monitoring technologies. Front Phys (2023) 11. 10.3389/fphy.2023.1133316 Zheng Z Zhao J Mili L Liu Z Wang S . Unscented kalman filter-based unbiased minimum-variance estimation for nonlinear systems with unknown inputs. IEEE Signal Process. Lett (2019) 26:1162–6. 10.1109/LSP.2019.2922620 Zheng Z Zhao J Mili L Liu Z . Robust unscented unbiased minimum-variance estimator for nonlinear system dynamic state estimation with unknown inputs. IEEE Signal Process. Lett (2020) 27:376–80. 10.1109/LSP.2020.2973116 Xue Z Cheng S Li L Zhong Z Mu H . A robust unscented m-estimation-based filter for vehicle state estimation with unknown input. IEEE Trans Vehicular Tech (2022) 71:6119–30. 10.1109/TVT.2022.3163207 Ehrich J Sivak DA . Energy and information flows in autonomous systems. Front Phys (2023) 11. 10.3389/fphy.2023.1108357 Song H Ding D Dong H Yi X . Distributed filtering based on cauchy-kernel-based maximum correntropy subject to randomly occurring cyber-attacks. Automatica (2022) 135:110004. 10.1016/j.automatica.2021.110004 Fu X Song X . Distributed maximum correntropy kalman filter with state equality constraints in a sensor network with packet drops. Signal Process. (2023) 213:109218. 10.1016/j.sigpro.2023.109218 Zhang M Zheng WX Song X Yuan H . Two efficient kalman filter algorithms for measurement packet dropping systems under maximum correntropy criterion. Syst Control Lett (2023) 175:105515. 10.1016/j.sysconle.2023.105515 Chen B Liu X Zhao H Principe JC . Maximum correntropy kalman filter. Automatica (2017) 76:70–7. 10.1016/j.automatica.2016.10.004 Li S Shi D Zou W Shi L . Multi-kernel maximum correntropy kalman filter. IEEE Control Syst Lett (2022) 6:1490–5. 10.1109/LCSYS.2021.3114137 Liu X Chen B Xu B Wu Z Honeine P . Maximum correntropy unscented filter. Int J Syst Sci (2017) 48:1607–15. 10.1080/00207721.2016.1277407 Huang X Wang Y . Kalman filter principle and application: matlab simulation. Beijing: Electronic Industry Press (2015). Appendix A

W k = B k − 1 I H k = B k ∣ k − 1 p − 1 0 0 B k Φ − 1 I H k = B k ∣ k − 1 p − 1 B k Φ − 1 H k , C k = C k x 0 0 C k z , D k = B k − 1 x ̂ k ∣ k − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 = B k ∣ k − 1 p − 1 0 0 B k Φ − 1 x ̂ k ∣ k − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 = B k ∣ k − 1 p − 1 x ̂ k ∣ k − 1 B k Φ − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 . By (A.1) and (A.2), we have W k T C k W k = B k ∣ k − 1 p − 1 T C k x B k ∣ k − 1 p − 1 + H k T B k Φ − 1 T C k z B k Φ − 1 H k . Next, the matrix inverse lemma was used to obtain: W k T C k W k − 1 = B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T − B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T H k T H k B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T H k T + B k Φ C k z − 1 B k Φ T − 1 H k B k ∣ k − 1 p C k x − 1 B k ∣ k − 1 p T , and by (A.1)–(A.3), we have W k T C k D k = B k ∣ k − 1 p − 1 T C k x B k ∣ k − 1 p − 1 + H k T B k Φ − 1 T C k z B k Φ − 1 z k − z ̂ k ∣ k − 1 + H k x ̂ k ∣ k − 1 . Combining (A.5) and (A.6), we have Eq. 7.