Filtering method for uncertain system with complex multiplicative noise and random time delay

Abstract

The invention relates to the technical field of signal processing, in particular to a filtering method for dealing with a system with complex multiplicative noise and random time delay uncertainty, which designs a reasonable system model of a multi-sensor networked system meeting conditions, and adopts a maximum and minimum robust estimation principle, a de-randomization method and a virtual noise technology. The system is converted into a multi-model multi-sensor system only with uncertain noise variance, an augmented noise method, a non-negative definite matrix factorization method and a Lyapunov equation method are applied, two robust centralized fusion steady-state Kalman estimators are obtained, and the robustness of the estimators is further proved. The method can be used for solving the problem of robust fusion Kalman filtering of multi-sensor single-channel ARMA signals with random parameter matrixes, uncertain noise variance and networked random uncertainty.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of signal processing, and particularly relates to a filtering method for dealing with a system with complex multiplicative noise and random time delay uncertainty. BACKGROUND

[0002] Kalman filter is a basic tool for multi-sensor data fusion estimation, which is an algorithm for obtaining state estimation by using linear system state equation and system input and output observation data, and requires that the model parameters and noise variance of the system are accurately known. In many engineering applications, this condition may not always be true, because in practical applications, there are always some uncertainties in the system due to modeling errors or unmodeled dynamics and random disturbances. When the system model has uncertainties, the performance of the classical Kalman filter will be poor. This has stimulated a lot of research on robust Kalman filter. Designing a corresponding robust Kalman filter is one of the classical methods to solve this uncertainty, and the key to the robustness of the filter is that when all the allowed uncertainties are included, the actual filtering error variance is guaranteed to have a minimum upper limit, i.e. robustness.

[0003] In recent years, due to the advantages of low cost, low power consumption and inherent robustness, the state estimation problem of network systems and sensor networks has gradually developed into a research hotspot. So far, many important results have been reported, which are related to the robust or optimal state estimation of network systems with multiplicative noise, uncertain noise variance, random delay and packet loss.

[0004] The disadvantages and deficiencies of the prior art: for the system with mixed uncertainty, including state-dependent multiplicative noise, two-step random delay, missing measurement and uncertain noise variance, document 1 introduces an enhanced state vector and applies a weighted least squares (WLS) algorithm to obtain a centralized fusion and weighted measurement fusion system, but does not consider the influence of colored noise and noise-dependent noise. For a kind of system, all parameter matrices with the same multiplicative noise, and one-step random observation lag and discontinuous packet loss of multi-sensor networked system, according to the maximum minimum robust estimation principle, document 2 proposes a robust matrix weighted fusion Kalman estimator, but does not consider the packet loss compensation mechanism. For the system including colored noise, the same multiplicative noise in the system parameter matrix, uncertain noise variance and one-step random delay (OSRD) and packet loss, based on the worst-case CF system, a robust centralized steady-state Kalman estimator (predictor, filter and smoother) is proposed. For the uncertain system with random parameter matrix, uncertain noise variance, one-step random time delay, packet loss and missing observation, a robust local and matrix weighted steady-state one-step and multi-step signal predictor is proposed, but the system model does not consider centralized fusion. In addition, the two centralized fusion systems proposed above are not completely the same, and only consider the case that the parameter matrix is with the same multiplicative noise. SUMMARY

[0005] The purpose of the present application is to provide a filtering method for dealing with uncertain systems with complex multiplicative noise and random time delay, and to design two robust centralized fusion steady-state Kalman estimators to solve the state estimation problem of multi-sensor networked systems with multiple uncertainties.

[0006] To achieve the above purpose, the present application provides a filtering method for dealing with uncertain systems with complex multiplicative noise and random time delay, comprising the following steps:

[0007] Step 1: For the multi-sensor networked system that meets the conditions, a reasonable system model is designed, and the variables are defined appropriately. Through the assumption of each noise, initial state and its robustness relationship, two robust centralized fusion steady-state Kalman estimators are designed. The advantages of centralized fusion are high data processing accuracy, flexible algorithm and fast fusion speed; centralized measurement fusion is similar to centralized fusion, and can realize global optimal estimation;

[0008] Step 2: Apply the augmentation method, de-randomization method and virtual noise technology to convert the considered system into a multi-model multi-sensor system containing only uncertain noise variance;

[0009] Step 3: Based on the principle of Minimax robust estimation, the standard Kalman filtering algorithm is applied to obtain a locally optimal time-varying Kalman predictor, and then the robustness of its estimator is obtained by applying the corresponding assumptions in Step 1. Based on the robust locally fused time-varying Kalman predictor, a robust locally fused time-varying Kalman filter and smoother are designed respectively;

[0010] Step 4: Two different centralized fusion rules are adopted, and two corresponding centralized fusion Kalman estimators are designed according to different centralized fusion methods.

[0011] Step 5: Robustness verification is considered for multi-sensor single-channel ARMA signals with multiple noises and multiple uncertainties.

[0012] Optionally, in Step 1, the system model expression of the multi-sensor networked system meeting the conditions is designed as follows:

[0013]

[0014] y i (t)=ξ i (t)z i (t)+(1-ξ i (t))ζ i (t)z i (t-1)+(1-ξ i (t))(1-ζ i (t))y i (t-1),i=1,…,L

[0015] where t is a discrete time, x(t)∈R n is the estimated state, is the observation value received by the sensor, is the observation value received by the estimator, w(t)∈R r is the process noise α k (t)∈R 1 ,δ k (t)∈R 1 ,β k (t)∈R 1 ,Φ∈R n×n ,Φ αk ∈R n×n ,Φ δk ∈R n×n ,Γ∈R n×r ,Γ k ∈R n×r , and is a constant matrix with appropriate dimension, and αk ,Φ δk ,Γ k ,H ik is called the disturbance orientation matrix, n is the corresponding number of noise, and L is the number of sensors.

[0016] Optionally, the augmentation method in step 2 includes a state augmentation and an interacting multiple model method, and an augmented noise and an augmented state vector are introduced.

[0017] Augmented noise:

[0018] Augmented state vector: x ai (t) = [x T (t)z i T (t-1)z i T (t-2)] T .

[0019] Optionally, the centralized fusion rule in step 4 includes centralized fusion and centralized measurement fusion, the centralized fusion refers to directly transmitting original data of all sensors to a central processor, and unified processing and comprehensive analysis are performed in a fusion center; the centralized measurement fusion is a simplified form of the centralized fusion; the sensors perform preprocessing on the original data locally, and then transmit the processed measurement data to the fusion center for fusion.

[0020] Optionally, the multi-sensor single-channel ARMA signal is a signal observed by each sensor in a multi-sensor system, which can be modeled as an autoregressive moving average process.

[0021] The application provides a filtering method for dealing with a system with complex multiplicative noise and random time delay uncertainty, a reasonable system model of a multi-sensor networked system meeting a condition is designed, a maximum minimum robust estimation principle, a de-randomization method and a virtual noise technology are applied to convert the system into a multi-model multi-sensor system with only uncertain noise variance, an augmented noise method, a non-negative matrix decomposition method and a Lyapunov equation method are applied to obtain two robust centralized fusion steady-state Kalman estimators, and the robustness of the estimators is further proved. BRIEF DESCRIPTION OF DRAWINGS

[0022] In order to make the technical solutions of the embodiments of the present application or the prior art clearer, the accompanying drawings needed in the embodiments or prior art description will be briefly introduced. Obviously, the accompanying drawings in the following description only show some embodiments of the present application, and for those skilled in the art, other drawings can be obtained without creative effort based on these drawings.

[0023] Figure 1 is a centralized fusion process schematic diagram of the present application.

[0024] Figure 2 is a centralized measurement fusion process schematic diagram of the present application.

[0025] Figure 3 is a signal instance application process schematic diagram in a specific embodiment of the present application.

[0026] Figure 4 is a signal s(t) and its actual local and centralized fusion two-step smoother schematic diagram in a specific embodiment of the present application.

[0027] Figure 5 is an actual first robust centralized fusion one-step smoothing error curve and and boundary schematic diagram.

[0028] Figure 6 is a trace trP c2 of the second robust centralized fusion filtering error variance with the change of λ1 and λ2. DETAILED DESCRIPTION

[0029] The embodiments of the present application are described in detail below, and examples of the embodiments are shown in the accompanying drawings, in which the same or similar notations represent the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the accompanying drawings are exemplary and are intended to explain the present application, and cannot be understood as a limitation of the present application.

[0030] The present application provides a filtering method for a system with complex multiplicative noise and random time delay uncertainty, comprising the following steps:

[0031] Step 1: design a reasonable system model for a multi-sensor networked system meeting the conditions, and make appropriate assumptions and definitions for the variables therein, design two robust centralized fusion steady-state Kalman estimators by assuming the relationship of the noise, initial state and its robustness.

[0032] Step 2: The considered system is transformed into a multi-model multi-sensor system with only uncertain noise variances by using the augmentation method, the de-randomization method and the virtual noise technique;

[0033] Step 3: Based on the principle of max-min robust estimation, the standard Kalman filtering algorithm is applied to obtain the locally optimal time-varying Kalman predictor, and the robustness of its estimator is obtained by using the corresponding assumptions in step 1. Based on the robust locally fused time-varying Kalman predictor, the robust locally fused time-varying Kalman filter and smoother are designed respectively;

[0034] Step 4: Two different centralized fusion rules are adopted, and two corresponding centralized fusion Kalman estimators are designed according to different centralized fusion methods;

[0035] Step 5: The robustness is verified by considering multi-sensor single-channel ARMA signals with multiple noises and multiple uncertainties.

[0036] The following is further described in combination with the execution steps:

[0037] The reasonable system model of the multi-sensor networked system designed in step 1 is

[0038]

[0039] y i (t)=ξ i (t)z i (t)+(1-ξ i (t))ζ i (t)z i (t-1)+(1-ξ i (t))(1-ζ i (t))y i (t-1),i=1,…,L (3)

[0040] Where: t is a discrete time, x(t)∈R n is the estimated state, is the observation value received by the sensor, is the observation value received by the estimator, w(t)∈R r is the process noise α k (t)∈R 1 ,δ k (t)∈R 1 , k=1,…,q is the state-dependent multiplicative noise, β k (t)∈R 1 , k=1,…,q is the noise-dependent multiplicative noise. Φ∈R n×n ,Φ αk ∈Rn×n ,Φ δk ∈R n×n ,Γ∈R n×r ,Γ k ∈R n×r , and are constant matrices of appropriate dimensions. Φ αk ,Φ δk ,Γ k ,H ik are called perturbation orientation matrices. n is the number of corresponding noises, and L is the number of sensors.

[0041] For the results to be presented, the following assumptions are made:

[0042] Assumption 1 ξ i (t)∈R 1 , i = 1,..., L are mutually independent Bernoulli white noises with values of 0 or 1, and have known probabilities Prob(ξ i (t) = 1) = π i , Prob(ξ i (t) = 0) = 1 - π i , where π i is known, and 0 ≤ π i ≤ 1. Furthermore, the present invention assumes that ξ i (t) is uncorrelated with other random signals.

[0043] Assumption 2 w(t), η i (t), α k (t), δ k (t), and β k (t) are mutually independent zero-mean white noise sequences with covariances

[0044]

[0045] Assumption 3 The initial state x(0) is uncorrelated with the random signals w(t), η i (t), α k (t), δ k (t), β k (t), and ξ i (t), and where μ0is the known mean of x(0), is the unknown actual variance of uncertainty.

[0046] Assumption 4 Q, and P0are known conservative upper bounds of and , respectively, i.e., satisfy

[0047]

[0048] From the above assumptions, the problem to be solved by the present technology is derived, that is, to design a robust centralized fusion steady-state Kalman estimator for the state x(t) in an uncertain multi-sensor networked system such that for all admissible uncertainties, their actual steady-state estimation error variances have a corresponding minimum upper bound P c (N), that is The subscript "c = i, c1, c2" represents the i-th local robust estimator, and the two centralized fusion methods, N = -1, N = 0, N > 0 represent the predictor, filter and smoother, respectively.

[0049] The augmentation method in step 2 generally includes two kinds: state augmentation and interacting multiple model (IMM) method.

[0050] State Augmentation Method: This method adds the delay measurement or other uncertainty factors that need to be considered as additional state variables to the state vector, thereby converting the original Kalman filtering problem into an augmented state space problem. The augmented method can fuse data from multiple sensors or information sources, improving the accuracy and reliability of measurements. It is suitable for dynamic systems and can update state estimation in real time to adapt to changes in the system. This method preserves the state space representation, simplifying the extension of the delay processing method to other methods such as mobile horizon estimation (MHE), nonlinear dynamic data reconciliation (NDDR), unscented Kalman filter (UKF), particle filter, etc.

[0051] Interacting Multiple Model (IMM) Method: IMM method is a robust filtering method for handling multiple model problems, which handles model uncertainty by considering the interaction between multiple different models. IMM method can have both adaptability to model uncertainty and robustness to non-Gaussian measurement noise. This method handles multiple model problems by introducing the concept of interaction to achieve model error adaptability and measurement error robustness in real-time applications.

[0052] Moreover, introducing augmented noise and augmented state vector can also be considered as an augmentation method, especially when dealing with robustness issues. This method considers model uncertainty and external disturbances by introducing additional noise terms in the model, thereby improving the robustness of the filter. Because in practical applications, the dynamic model of the system may have uncertainties, such as parameter changes, external disturbances, etc. The introduction of augmented noise can help the model to more accurately describe these uncertainties, and improve the robustness of the filter. Augmented state vector allows the system to consider more uncertain factors, such as observation errors or external disturbances, thereby providing more accurate state estimation.

[0053] Augmented noise δ fi (t) is

[0054]

[0055] According to the calculation of the actual and conservative cross-covariance of w fi (t) and v fi (t), the augmented noise δ fi (t) and the actual and conservative cross-covariance of δ fj (t) are obtained as

[0056]

[0057] where:

[0058]

[0059] In the above formula, the actual steady-state variance corresponding to the virtual noise is obtained as and The actual steady-state cross-covariance corresponding to the virtual noise is obtained as and

[0060] The actual variance and conservative variance of the augmented noise δ fi (t) are defined as and fi (t) = Λ fii (t).

[0061] For all allowable uncertain actual variances and there are

[0062] By augmenting the state vector, the number of system state variable estimates that need to be performed simultaneously during the identification process can be reduced, thereby reducing the computational workload and improving the computational efficiency. The augmented state vector x g ​​(t) is expressed as

[0063]

[0064] its actual and conservative second-order non-central moments and X g (t) are respectively

[0065]

[0066] Second-order non-central moments are a characteristic number that describes the characteristics of random variables, which provides information about the shape of the distribution of random variables, especially the distribution characteristics beyond the mean and variance. Calculating the actual and conservative second-order non-central moments can help to understand the statistical characteristics of random variables more comprehensively. The actual second-order non-central moments of the augmented state vector can be used to analyze the stability of the system. By considering the non-central moments, the behavior of the system in the face of uncertainty and external disturbances can be more accurately evaluated, thereby improving the robustness of the system.

[0067] For all admissible uncertainties actual variance and there are

[0068] Specifically, the purpose of step 2 is to transform the original system model into a multi-model multi-sensor system containing only uncertain noise variance, mainly including:

[0069] Substituting equation (2) into equation (3) and combining with equation (1), the original system (1)-(3) can be converted into the following augmented system with multi-model:

[0070]

[0071] To better describe the system model, define

[0072]

[0073] The following three noises ξ iz (t) = ξ i (t) - π i , γ iz (t) = ξ iz (t) ζ iz (t) It is easy to prove that ξ iz (t), ζ iz (t) and γ iz (t) are each independent white noise with zero mean. Further, the system (10) can be converted into the following equivalent multi-model system with constant parameter matrix and multiplicative noise:

[0074]

[0075] Further explanation of the augmented multi-model system in equation (12):

[0076]

[0077]

[0078] Further for augmented state x ai (t) to calculate its actual and conservative second-order non-central distance, introduce virtual process w fi (t) and measurement noise v fi (t),

[0079]

[0080] On the basis of the above discussion, the original multi-sensor system (1)-(3) is transformed into a time-invariant augmented multi-model correlated noise system with only uncertain actual virtual noise variance.

[0081] Further, the following is a coherent introduction to the content of step 3 and step 4:

[0082] Two kinds of fusion rules are introduced as centralized fusion and centralized measurement fusion, please refer to Figure 1 and Figure 2 , respectively, as two kinds of fusion algorithm flow chart.

[0083] First, centralized fusion refers to transmitting all sensor raw data directly to a central processor (fusion center), and performing unified processing and comprehensive analysis in the fusion center. Its characteristics are as follows:

[0084] Data integrity: the fusion center obtains all sensor raw data, with less information loss.

[0085] High computational complexity: since the fusion center needs to process a large amount of raw data, the computational burden is large, and the requirements for hardware and algorithm are high.

[0086] Heavy communication burden: all sensors need to transmit raw data to the fusion center, with high communication cost.

[0087] High fusion accuracy: due to the completeness of data, the accuracy of fusion results is usually high.

[0088] Second, centralized measurement fusion is a simplified form of centralized fusion. The sensor pre-processes the raw data locally (such as filtering, noise reduction, etc.), and then transmits the processed measurement data to the fusion center for fusion. Its characteristics are as follows:

[0089] Reduced data volume: the sensor pre-processes the data locally, reducing the amount of data transmitted to the fusion center.

[0090] Computational efficiency improvement: The fusion center processes preprocessed data, reducing the computational burden.

[0091] Reduced communication burden: The amount of data transmitted is reduced, reducing communication costs.

[0092] Possible loss of information: Since the sensor performs preprocessing locally, some information may be lost, which may affect the fusion accuracy.

[0093] Application scenarios: Centralized fusion is suitable for scenarios with fewer sensors, sufficient communication bandwidth, and abundant computing resources, such as unmanned driving, aerospace, etc. Centralized measurement fusion is suitable for scenarios with a large number of sensors, limited communication resources, and limited computing resources, such as the Internet of Things, smart home, etc.

[0094] The centralized fusion system is:

[0095]

[0096] Where:

[0097] The augmented centralized fusion system can be obtained by applying the centralized fusion system equation:

[0098] x a (t+1) = Φ a (t) x a (t) + Γ a (t) w a (t)

[0099] y (c1) (t) = H a (t) x a (t) + C a (t) w a (t)

[0100] Where:

[0101]

[0102] Centralized measurement fusion equation:

[0103] y (c2) (t) = H (c2) x g (t) + v (c2) (t) (15)

[0104] Where:

[0105] The robust local time-varying Kalman estimator is given as

[0106] The predictor is

[0107]

[0108] The filter and smoother are

[0109]

[0110] The robust local steady-state Kalman estimator is Robust, i.e., for all admissible uncertainties, we have and P i (N) is The actual and conservative estimation error variances are defined as

[0111] Secondly, the centralized fusion robust Kalman estimator is introduced. According to the robust centralized fusion algorithm introduced in the above steps, the virtual noise is introduced as follows:

[0112]

[0113] The conservative steady-state variance cross-covariance of the two is

[0114]

[0115] For the worst-case time-invariant augmented centralized fusion system with conservative noise statistics, similar to the derivation in the above local estimator, the corresponding centralized fusion steady-state Kalman estimator and its actual and conservative fusion steady-state error variances and are obtained, and for all admissible uncertainties, the two are robust, i.e., Further, according to a (t) = [x T (t) z (c1)T (t - 1) y (c1)T (t - 1)] T The centralized fusion steady-state Kalman estimator is robust for all admissible uncertainties, i.e.,

[0116]

[0117] Then, for the centralized measurement fusion system, the augmented state vector is introduced to obtain the actual and conservative second-order non-central moments and g (t) of x g (t), and the Xg (t) is of the form

[0118]

[0119] In formula (23) replace X and by g (t), X ai (t), Q b and gives the actual and conservative second order non-central moments of x g (t) where the definitions are as follows:

[0120]

[0121] Define and as the actual and conservative steady state variances of the white noise w g (t), respectively, to obtain the expression for

[0122]

[0123] Define and R c as the actual and conservative steady state variances of v (c2) (t), respectively, to obtain the expression for R c

[0124] Similar to the derivation in the local estimators above, the corresponding centralized measurement fusion steady state Kalman estimator

[0125] and its actual and conservative fusion steady state error variances

[0126] and are obtained and both are robust for all admissible uncertainties, i.e.,

[0127] From and it follows that the robust centralized fusion steady state Kalman estimator is robust for all admissible uncertainties, i.e.

[0128] From the above results, the corresponding accuracy relations are given as follows

[0129] P (c) (N) ≤ P​​i (N),c=c1,c2,i=1,...,L (22)

[0130] In Step 5, the multi-sensor single-channel ARMA signal refers to the signal observed by each sensor in a multi-sensor system, which can be modeled as an autoregressive moving average (ARMA) process. The ARMA model is one of the commonly used models in time series analysis, which combines the characteristics of autoregressive (AR) model and moving average (MA) model, and can describe the dynamic characteristics of time series data and the influence of random disturbances. In a multi-sensor system, the ARMA model can be used for information fusion, which can improve the estimation accuracy by matrix weighting, diagonal matrix weighting and scalar weighting, and unify the processing of information fusion, filtering and smoothing. The specific design mainly includes the following steps:

[0131] Firstly, the order of the ARMA model (i.e. the order of the AR part p and the order of the MA part q) needs to be determined. This can be achieved by analyzing the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots of the time series. The ACF plot shows a tail (i.e. the autocorrelation value slowly decreases to zero) which may indicate the applicability of the AR model, while the PACF plot is truncated after a certain order which may indicate the applicability of the MA model.

[0132] Secondly, after determining the order of the model, the next step is to estimate the model parameters. This can be done by methods such as maximum likelihood estimation (MLE) or ordinary least squares (OLS). The purpose of parameter estimation is to find the best (autoregressive coefficients) and (moving average coefficients).

[0133] Finally, after model fitting, it needs to be tested to ensure its applicability. Once the model passes the test, it can be used to predict future values. The ARMA model usually performs well in short-term prediction of time series, especially when the data has significant autocorrelation and random fluctuations.

[0134] Consider the following multi-sensor single-channel ARMA signal with colored noise and multiple uncertainties

[0135] A t (q -1 )s(t) = B t (q -1 )w(t)

[0136]

[0137] θ i (t+1) = G i θ i (t) + η i (t)

[0138] At (q -1 ) = 1 + a1(t - 1)q -1 +... + a n (t - n)q -n , a k (t) = a k + a k (t), k = 1,..., n

[0139] B t (q -1 ) = 1 + b1(t - 1)q -1 +... + b n (t - n)q -n , b k (t) = b k + b k (t), k = 1,..., n

[0140] P t (q -1 ) = 1 + p1(t - 1)q -1 +... + p n (t - n)q -n , p k (t) = p k + p k (t), k = 1,..., n(23)

[0141] where s(t) e R 1 is the estimated single-channel signal, w(t) e R r is the input noise, m i (t) e R 1 is the measurement of the i-th sensor, ξ i (t) e R 1 , ζ i (t) e R 1 and γ i (t) e R 1 , i = 1,..., L are mutually independent scalar Bernoulli white noises satisfying Assumption 1, θ i (t) e R 1 is the colored measurement noise, η i (t) e R 1 is the white noise. A t (q -1 ), B t (q -1 ) and P t (q -1 ) are n-th order polynomials of q -1 , q -1 is the unit lag operator, i.e., q -1 s(t) = s(t - 1), ak (t), k = 1,..., n are scalar stochastic parameters with known mean a k and random disturbance a k (t), b k (t), k = 1,..., n are scalar stochastic parameters with known mean b k and random disturbance b k (t), p k (t), k = 1,..., n are scalar stochastic parameters with known mean p k and random disturbance p k (t), w(t), η i (t), a k (t), b k (t), and p k (t) are mutually uncorrelated zero-mean white noises, and are w(t), η i (t), a k (t), b k (t), and p k (t) are unknown and uncertain actual (true) variances, Q, R ηi , and are known conservative upper bounds of and , respectively.

[0142] The equivalent state space model of the ARMA signal model (23) for the stochastic parameters is

[0143] x s (t+1) = Φ s (t) x s (t) + Γ s (t) w(t) (24)

[0144] s(t) = H s x s (t) (25)

[0145] The equivalent state space model with colored noise is

[0146] x m (t+1) = Φ m (t) x m (t) + Γ m s(t) (26)

[0147] m i (t) = H m (t) x m (t) + θ i (t) (27)

[0148] The above equivalent state space model is explained as follows:

[0149]

[0150] Through the above transformation, the augmented method is applied, and the corresponding augmented multi-model system is further obtained, so that the ARMA signal model is converted into a state space model with multiplicative noise, one-step random time delay, data packet loss and uncertain noise variance, which is regarded as a special case of the original system.

[0151] The following specific examples are proposed in combination with step 5 to further prove the applicability and accuracy of the technical method. The process of applying the signal example is shown in Figure 3 :

[0152] It is known here that the signal model used can be transformed into a special case of the original system model. Further, in combination with the process of transforming the signal model into a qualified original system model, s(t) = H s x s (t) = [H s (0) 1×n ]x(t), so that the robust signal estimation problem can be solved by the corresponding robust local state estimator. The formula of the robust local signal estimator is The estimation error of the signal s(t) is

[0153] The actual and conservative local estimation error variances of the signal s(t) are given as

[0154]

[0155] Further, the cross-covariance of the actual and conservative local estimation errors of the signal is given as

[0156]

[0157] Based on the robust local signal estimator Similar to the derivation given in sections 5 and 6, we can obtain two centralized fusion robust signal estimators and their actual and conservative fusion steady-state estimation error variances and

[0158] Through the proof in steps 2, 3 and 4 above, it is easy to obtain that the error variance of the signal s(t) is also robust. And there is the following relationship:

[0159]

[0160] With simulation examples, let n = 2, L = 3, and define

[0161]

[0162] pi1= 0.85, pi2= 0.85,

[0163] pi3= 0.9, ai1= -0.85, ai2= -0.045, bi1= -0.1, bi2= -0.27,

[0164] pi1= 0.8, pi2= -0.0375

[0165] The simulation results are shown in

[0166] Since the signal s(t) e R 1 , the trace of the estimation error variance is equal to the corresponding estimation error variance value. Table 1 gives the comparison of the actual and conservative local and centralized fusion steady-state estimation error variances, which verifies the steady-state accuracy relationship given by equation (29). In Table 1, N = -1 represents the predictor, N = 0 represents the filter, N = 1 represents the one-step smoother, and N = 2 represents the two-step smoother.

[0167] Figure 4 The tracking performance of the robust local and two centralized fusion steady-state two-step smoothers and is given, and it is seen from Figure 4 that the tracking performance of the robust two centralized fusion steady-state two-step smoothers is better, and the two centralized fusion estimators have the same effect within a certain allowable error range.

[0168] To illustrate the robustness of the first centralized fusion signal one-step smoother , three different actual noise variances

[0169]

[0170] are randomly selected and satisfy the robustness, for example

[0171]

[0172] According to the one-step smoothing tracking diagram similar to Figure 4 , the corresponding three robust centralized fusion signal one-step smoothers and the corresponding actual and conservative robust scalar weighted fusion time-varying one-step smoothing error variances and P c1 (t|t+1) are easily obtained, where the actual error curves of the three fusion smoothers and their 3 times standard deviation bounds are shown in Figure 5The actual fusion smoothing error curves are shown by solid lines, The boundary is shown by a dashed line, The actual fusion smoothing error curves are shown by solid lines, Figure 5 It can be seen that for each error curve, more than 99% of the fusion smoothing error values are located between and , and also between and . This verifies the robustness of the robust scalar-weighted fusion one-step smoothing of the first component and the correctness of the actual standard deviation , which increases when the actual noise variance increases.

[0173] To illustrate the influence of the state-dependent multiplicative noise k (t) and k (t), k = 1, 2, and the noise-dependent multiplicative noise k (t), k = 1, 2, on the robust centralized fusion steady-state Kalman signal filter robust accuracy, let λ1 and λ2 increase from 0.01 to 0.1 in steps of 0.01, and the changes of the robust accuracy trP c2 (t|t) with respect to λ1 and λ2 are shown in Figure 6 It can be seen that the value of trP c2 (t|t) increases when the variances of the two types of noise increase, which indicates that the robust accuracy of the robust fusion signal predictor decreases.

[0174] Table 1 Comparison of the steady-state robust and actual accuracy of the signal s(t)

[0175]

[0176] In summary, compared with the prior art, the present application has the following advantages:

[0177] 1. In the system model studied, two types of state-dependent noise, one type of noise-dependent multiplicative noise, one-step random delay, packet loss, and uncertain noise variance are considered simultaneously for the first time, which better reflects the real situation.

[0178] 2. Based on the maximum-minimum robust estimation principle, a robust centralized fusion steady-state Kalman estimator of two different fusion methods is proposed.

[0179] 3. The robustness of the proposed centralized fusion and centralized measurement fusion estimators is proved by using the augmentation method, the non-negative matrix factorization method, and the Lyapunov equation method.

[0180] ​The above disclosure only shows one or more preferred embodiments of the present application, of course, cannot limit the scope of the present application, those skilled in the art can understand that the implementation of all or part of the above processes, and the equivalent changes made by the present application claims, still belong to the scope covered by the present application.

Claims

1. A filtering method for a system with complex multiplicative noise and random time-delay uncertainty, characterized in that, The method comprises the following steps: Step 1: design a reasonable system model for a qualified multi-sensor networked system, and make appropriate assumptions and definitions for variables in the system, design two robust centralized fusion steady-state Kalman estimators by making assumptions about noise, initial state and its robustness relationship; Step 2: apply the augmentation method, de-randomization method and virtual noise technology to convert the considered system into a multi-model multi-sensor system containing only uncertain noise variance; Step 3: based on the maximum-minimum robust estimation principle, apply the standard Kalman filtering algorithm to obtain a locally optimal time-varying Kalman predictor, and then apply the corresponding assumptions in step 1 to obtain the robustness of the estimator, and based on the robust local fusion time-varying Kalman predictor, design a robust local fusion time-varying Kalman filter and smoother; Step 4: two different centralized fusion rules are used to design two corresponding centralized fusion Kalman estimators according to different centralized fusion methods; Step 5: consider using a multi-sensor single-channel ARMA signal with multiple noises and multiple uncertainties for robustness verification.

2. The filtering method for dealing with complex multiplicative noise and random time delay uncertain systems according to claim 1, wherein the expression of the reasonable system model of the qualified multi-sensor networked system in step 1 is as follows:

3. The filtering method for dealing with complex multiplicative noise and random time delay uncertain systems according to claim 1, wherein the augmented method in step 2 includes state augmentation and interactive multi-model method, and introduces augmented noise and augmented state vector; y i (t) = ξ i (t)z i (t) + (1 - ξ i (t))ζ i (t)z i (t - 1) + (1 - ξ i (t))(1 - ζ i (t))y i (t - 1), i = 1,..., L where t is discrete time, x(t) e R n is the estimated state, is the observation received by the sensor, is the observation received by the estimator, w(t) e R r is the process noise a k (t) e R 1 , δ k (t) e R 1 , k = 1,..., q is the state-dependent multiplicative noise, β k (t) e R 1 , k = 1,..., q is the noise-dependent multiplicative noise, Φ e R n×n , Φ αk e R n×n , Φ δk e R n×n , Γ e R n×r , Γ k e R n×r , and are constant matrices of appropriate dimensions, Φ αk , Φ δk , Γ k , H ik is called the disturbance orientation matrix, n is the corresponding number of noises, and L is the number of sensors. Augmented state vector: Augmented noise:

4. The filtering method for dealing with complex multiplicative noise and random time delay uncertain systems according to claim 1, wherein the centralized fusion rule in step 4 includes centralized fusion and centralized measurement fusion, the centralized fusion refers to transmitting the original data of all sensors to a central processor for unified processing and comprehensive analysis in the fusion center; the centralized measurement fusion is a simplified form of centralized fusion; the sensor pre-processes the original data locally, and then transmits the processed measurement data to the fusion center for fusion. x ai (t) = [x T (t)z i T (t-1)z i T (t-2)] T 5. The filtering method for dealing with complex multiplicative noise and random time delay uncertain systems according to claim 1, wherein the multi-sensor single-channel ARMA signal is a signal observed by each sensor in a multi-sensor system, which can be modeled as an autoregressive moving average process. ​ ​ ​ ​

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