Fusion estimation method for solving two-step random observation time delay and state time delay system

Abstract

The invention relates to the technical field of multi-source information fusion, in particular to a fusion estimation method for solving a two-step random observation time delay and state time delay system, aiming at uncertain noise variance, multiplicative noise and two-step random observation time delay, two Bernoulli distribution random variables with known distribution are used for describing the random time delay. An original system model is converted into a multi-model multi-sensor system only having an uncertain noise variance by using a model conversion method composed of an augmentation method, a de-randomization method and a virtual noise method. According to a maximum and minimum robust estimation principle, a robust local steady-state Kalman estimator is provided under a unified framework. According to the method, two robust centralized fusion steady-state Kalman estimators are deduced by using a centralized observation fusion algorithm and a centralized state fusion algorithm, and in addition, the robustness of the estimated estimators is proved by using a combined method consisting of an augmented noise method, a positive semidefinite matrix decomposition method, a quadratic matrix representation method and a Lyapunov equation method.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-source information fusion, and particularly relates to a fusion estimation method for solving two-step random observation delay and state time-delay systems. Background Art

[0002] Multi-sensor information fusion, also known as multi-sensor data fusion, focuses on comprehensively considering the local data or information provided by multiple identical or different sensors and information sources to achieve more accurate estimation, evaluation, and inference of the system state. In recent years, due to the advantages of low cost and robustness of networked systems, they have received particular attention. Problems of networked system estimation and control widely appear in many fields, including environmental monitoring, deep space exploration, space development, target tracking, etc. Multi-sensor networked information fusion technology overcomes the defects of single sensors restricted by time and space. In addition, this technology also plays a crucial role in enhancing the robustness of the system, which is particularly important in fields such as target tracking, navigation guidance, signal processing, especially when a certain sensor fails or data is lost, and the system can still operate normally. If a sensor fails or is defective, it may cause the vehicle to fail to correctly identify obstacles or road conditions, leading to accidents. For example, in August 2022, a Tesla Model Y in the United States crashed into a truck parked by the roadside in autonomous driving mode because Tesla's Autopilot system failed to identify the presence of the truck. In the field of industrial automation, estimators need to monitor the production line status and predict equipment maintenance requirements. In February 2018, 7 out of 13 safety hazards at Tianjiayi Company were related to instruments.

[0003] In the past few decades, due to the recursive structure and good performance of the Kalman filter, it has been widely applied to communication, signal processing, satellite positioning, integrated navigation systems, etc. However, the classical Kalman filtering theory requires that the system model is precisely known, that is, the model parameters and noise variances are precisely known. In practical engineering applications, due to the approximation of the model order, the linearization approximation of non-linear systems, and the errors brought by parameter identification, there is a certain error between the obtained mathematical model and the actual system. And due to the changes in the surrounding environment and various unpredictable interferences, the model parameters and noise variances are uncertain. If these uncertainties are not considered in practical applications, it will not only deteriorate the performance of the filter but even cause the filter to diverge. Summary of the Invention

[0004] The object of the present invention is to provide a fusion estimation method for solving two-step stochastic observation time delay and state time-delay systems, construct a linear discrete uncertain multi-sensor network time-delay system and transform it into a multi-model multi-sensor system with uncertain noise variance, and apply the centralized observation fusion and centralized state fusion algorithms to obtain two robust centralized fusion steady-state Kalman estimators.

[0005] To achieve the above object, the present invention provides a fusion estimation method for solving two-step stochastic observation time delay and state time-delay systems, including the following steps:

[0006] Step 1: Establish a linear discrete uncertain multi-sensor network time-delay system;

[0007] Step 2: Use the virtual noise method to transform the model into a form with only uncertain noise variance and construct an augmented multi-model system;

[0008] Step 3: Introduce the Lyapunov equation to obtain a time-invariant multi-model correlated noise system with only uncertain actual virtual noise variance;

[0009] Step 4: Based on the principle of minimax robust estimation, derive a robust local steady-state Kalman predictor, and then obtain a robust local steady-state Kalman filter and smoother;

[0010] Step 5: Combine all the local measurements received by the given sensors to obtain the centralized fusion measurement received by the sensors, and further realize multi-sensor data fusion;

[0011] Step 6: Verify the effectiveness of the estimator through simulation experiments.

[0012] Optionally, the expression of the linear discrete uncertain multi-sensor network time-delay system is as follows:

[0013]

[0014] v i (t)=D i w(t)+η i (t), i = 1,..., L

[0015]

[0016] where t is the discrete time, x(t) ∈ R n is the state vector of the system, is the observation value of the i-th sensor, is the measurement value received by the estimator after network transmission, w(t) ∈ R n is the process noise, d is the fixed-length time delay, is the observation noise and is linearly correlated with w(t), βk (t) ∈ R 1 , where k = 1, ..., q are state-dependent multiplicative noises; Φ ∈ R n×n , Φ k ∈ R n×n , Γ ∈ R n×r , and are known constant matrices of appropriate dimensions; q is the number of multiplicative noises and L is the number of sensors;

[0017] When ξ i = 1, y i (t) = z i (t) has no time delay. When , then y i (t) = z i (t - 1), i.e., there is a one-step time delay. When , then , i.e., y i (t) = z i (t - 2) has a two-step time delay.

[0018] Optionally, in the execution process of step 3, specifically, by defining the state error covariance equation and combining the actual state and the conservative state second-order non-central moment, the dynamic equation of the system state error can be obtained. By introducing the Lyapunov equation, the stability of the system state error covariance can be proved, and then by relating it to the matrix inequality, the non-negativity and stability of the virtual noise variance can be proved.

[0019] Optionally, the robust local steady-state Kalman predictor in step 4 is the basis of the robust local steady-state Kalman filter and smoother, providing the initial predicted values and predicted error variances for subsequent filtering and fusion; the filter is used to update the system estimate based on the current observation and provide the most accurate estimation result, and the smoother is used to post-process the estimation of the system state to eliminate errors and improve the estimation accuracy.

[0020] Optionally, in step 5, by combining all the local measurements received by the sensors, the centralized fusion measurement received by the sensors is obtained as follows:

[0021]

[0022] where there are definitions:

[0023]

[0024] And by combining all the local measurements received by the estimator, the centralized fusion measurement received by the estimator is obtained, and the expression is as follows:

[0025]

[0026] There are definitions as follows:

[0027]

[0028] The present invention provides a fusion estimation method for solving two-step random observation time-delay and state time-delay systems. Aiming at uncertain noise variance, multiplicative noise, and two-step random observation time-delay, two Bernoulli distribution random variables with known distributions are used to describe the random time-delay. The model transformation method composed of the augmentation method, the derandomization method, and the virtual noise method is used to transform the original system model into a multi-model multi-sensor system with only uncertain noise variance. According to the minimax robust estimation principle, a robust local steady-state Kalman estimator is given under a unified framework. By applying the centralized observation fusion and centralized state fusion algorithms, two robust centralized fusion steady-state Kalman estimators are derived. The combination method composed of the augmented noise method, the semi-definite matrix decomposition method, the matrix representation method of quadratic form, and the Lyapunov equation method is also used to prove the robustness of the proposed estimator. Finally, through simulation verification, the robust accuracy of the two centralized fusion estimators is higher than that of each local estimator, and there is no difference in accuracy between the robust centralized observation fusion steady-state Kalman estimator and the robust centralized state fusion steady-state Kalman estimator, verifying the applicability and correctness of the proposed robust fusion steady-state estimator. Brief Description of the Drawings

[0029] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0030] Figure 1 It is a schematic flow chart of the steps of a fusion estimation method for solving two-step random observation time-delay and state time-delay systems of the present invention.

[0031] Figure 2 It is a schematic diagram of the state x(t) and its actual local and fusion estimated values in a specific embodiment of the present invention.

[0032] Figure 3 It is the actual centralized state fusion filtering error and three times the standard deviation boundary in a specific embodiment of the present invention as well as the ±3σ (c) (0) boundary schematic diagram.

[0033] Figure 4 It is the robust fusion accuracy P in a specific embodiment of the present invention (c)(-1) With respect to multiplicative noise and Schematic diagram of changes. Specific implementation manner

[0034] The embodiments of the present invention will be described in detail below. Examples of the embodiments are shown in the accompanying drawings, where the same or similar reference numerals denote 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 invention and should not be construed as a limitation of the present invention.

[0035] Please refer to Figure 1 , the present invention provides a fusion estimation method for solving a two-step random observation time delay and state time delay system, including the following steps:

[0036] Step 1: Establish a linear discrete uncertain multi-sensor network time delay system;

[0037] Step 2: Use the virtual noise method to convert the model into a form with only uncertain noise variances, and construct an augmented multi-model system;

[0038] Step 3: Introduce the Lyapunov equation to obtain a time-invariant multi-model related noise system with only uncertain actual virtual noise variances;

[0039] Step 4: Based on the max-min robust estimation principle, derive a robust local steady-state Kalman predictor, and then obtain a robust local steady-state Kalman filter and smoother;

[0040] Step 5: Combine all the local measurement values received by the given sensors to obtain the centralized fusion measurement value received by the sensors, and further realize multi-sensor data fusion;

[0041] Step 6: Verify the effectiveness of the estimator through simulation experiments.

[0042] The following is further described in conjunction with specific implementation steps:

[0043] Step 1: Consider the following discrete uncertain linear state time delay system:

[0044]

[0045]

[0046] v i (t) = D i w(t) + η i (t), i = 1,..., L

[0047]

[0048] where: t is discrete time, and \(x(t)\in\mathbb{R}\) n is the state vector of the system, \(z\) i (t)\(\in\mathbb{R}\) mi is the observation value of the \(i\)-th sensor, is the measurement value received by the estimator after network transmission, \(w(t)\in\mathbb{R}\) n is the process noise, \(d\) is the fixed-length time delay, is the observation noise, and is linearly related to \(w(t)\), \(\beta\) k (t)\(\in\mathbb{R}\) 1 , \(k = 1,\cdots,q\) is the state-dependent multiplicative noise. \(\varPhi\in\mathbb{R}\) n×n , \(\varPhi\) k \(\in\mathbb{R}\) n×n , \(\varGamma\in\mathbb{R}\) n×r , and are known constant matrices with appropriate dimensions; \(q\) is the number of multiplicative noises, and \(L\) is the number of sensors.

[0049] The goal is to describe the random time delays of the networked system, including one-step random time delay and two-step random time delay. When \(\xi\) i \( = 1\), \(y\) i (t)\( = z\) i (t) has no delay situation. When , then \(y\) i (t)\( = z\) i (t - 1) that is, a one-step time delay occurs. When , then i.e., \(y\) i (t)\( = z\) i (t - 2) a two-step time delay occurs. The model establishment part provides the necessary background information and theoretical basis for the subsequent model establishment and the derivation of the robust weighted fusion KALMAN estimator, such as assumption conditions, model transformation, virtual noise, etc. It plays a crucial role and clearly expounds the characteristics and instabilities of the system under study.

[0050] In the following analysis, the derivation is carried out based on the following assumptions:

[0051] Assumption 1. \(L\) are independent Bernoulli random variables, taking values 1 and 0 respectively, and having known probabilities \(\text{prob}(\xi\) i (t)\( = 1)=\xi\) i , \(\text{prob}(\xi\) i (t)\( = 0)=1 - \xi\) i , and also where and \(\xi\) i are both known and satisfy Meanwhile, assume \(\xi\) i(t) and are all uncorrelated with other random signals. Therefore: The random time delays describing the network system include one-step random time delay and two-step random time delay. When ξ i = 1, y i (t) = z i (t) has no delay situation. When , then y i (t) = z i (t - 1) that is, a one-step time delay occurs. When , then y i (t) = z i (t - 2) that is, a two-step time delay occurs.

[0052] Assumption 2. w(t), η i (t), i = 1, ..., L, and β k (t), k = 1, ..., q are all uncorrelated white noises with zero mean and covariance, satisfying:

[0053]

[0054] where and are the uncertain actual variances of the white noises w(t), η i (t) and β k (t) respectively. δ kj represents the Kronecker delta function, δ kk = 1, δ kj = 0 (k ≠ j).

[0055] Assumption 3. The initial state x(0) is uncorrelated with w(t), η i (t) and β k (t), E[x(0)] = μ0, where μ0 is the mean of x(0). In addition, the variance of the initial state satisfies where is the unknown uncertain actual variance.

[0056] Assumption 4. and and P0 are and and of the known conservative upper bounds, respectively satisfying:

[0057]

[0058] Assumptions 2 and 4 respectively obtain that the actual variance and conservative variance of v i (t) are respectively

[0059]

[0060] and v i (t) and v j The actual and conservative cross-covariances of (t) are respectively

[0061]

[0062] and define and R vii = R vi . Subtract from R vi to get to obtain

[0063]

[0064] A system with unknown uncertain actual variance and as well as is called an actual system, while a worst-case conservative system with known uncertain noise variance and initial state variance upper bounds and and P0 is called a worst-case conservative system. The so-called "worst-case" system refers to a system with the largest noise and initial state variances. The "minimum" variance estimator for the "worst-case" system is called a minimax robust estimator, and this is the minimax robust estimation principle.

[0065] The object of the present invention is to design local and fusion robust steady-state Kalman estimators (predictors, filters, and smoothers) for the state x(t - d) in the uncertain multi-sensor network time-delay system (1)-(4). Such that for all admissible uncertainties, their actual steady-state estimation error variances have corresponding minimum upper bounds, that is, satisfying:

[0066]

[0067] where i, c, (c) respectively represent the ith local robust estimator, the centralized observation fusion estimator, and the centralized state fusion estimator. For N = -1, N = 0, and N > 0, they are respectively called one-step predictors, filters, and smoothers.

[0068] Step 2: Use the virtual noise method to convert the model into a form with only uncertain noise variance, and construct an augmented multi-model system

[0069] The system can be converted into the following augmented system with multi-models:

[0070] x ai (t + 1) = Φ ai (t)x ai (t) + Γai w ai (t)

[0071] y i (t) = Η ai (t)x ai (t) + G ai (t)v i (t)

[0072] wherein it is defined that and are respectively the average matrices of Φ ai (t), Γ ai (t), H ai (t) and D ai (t),

[0073]

[0074]

[0075]

[0076] Let and Statistical properties can be obtained from Hypothesis 1. Therefore, it can be easily proven that ξ iz (t), and γ iz (t) are all uncorrelated white noises with zero mean.

[0077] In addition, the following equivalent multi-model system with constant parameter matrices and multiplicative noise can be obtained:

[0078]

[0079]

[0080] It can be obtained that

[0081]

[0082] Thus, the actual variance and conservative variance of w ai (t) can be obtained as follows:

[0083]

[0084] and the actual and conservative cross-covariances of the white noises w ai (t) and w aj (t) respectively satisfy the following formulas:

[0085]

[0086] Let \(M\) be an \(m\times m\) positive semi - definite matrix, i.e., \(M\geq0\). Then for any \(C\in\mathbb{R}\) r×m , it can be obtained that \(CMC\) T is also a positive semi - definite matrix, i.e., \(CMC\) T \(\geq0\). Thus, we have:

[0087]

[0088] Step 3: By defining the state error covariance equation and combining the actual and conservative state second - order non - central moments, the dynamic equation of the system state error can be obtained. Introducing the Lyapunov equation, the stability of the system state error covariance can be proved. Then, by relating it to matrix inequalities, the non - negativity and stability of the virtual noise variance can be proved. This helps to verify the effectiveness of the virtual noise method.

[0089] For the augmented state \(x\) ai (t), define the actual second - order non - central moment as and the actual state \(x\) ai (t), and then obtain the actual noise variance from the actual augmented state variance Combining Assumption 2 and the uncorrelation of \(x\) ai (t) and \(w\) ai (t), we can get which satisfies the generalized Lyapunov equation as follows:

[0090]

[0091] Similarly, the conservative second - order non - central moment of the conservative augmented state \(x\) ai (t) is recursively calculated as follows:

[0092]

[0093] Let \(R\) i be an \(m\) i \(\times m\) i positive semi - definite matrix, i.e., \(R\) i \(\geq0\). Then the block - diagonal matrix \(R\) δ \(=\text{diag}(R_1,K,R\) L ) is also semi - positive definite, i.e., \(R\) δ \(=\text{diag}(R_1,K,R\) L )\(\geq0\).

[0094] Define and to obtain the Lyapunov equation for \(\Delta X\) ai (t):

[0095]

[0096] Obtained From the positive semi - definiteness of the variance matrix, it can be concluded that and Q ai ≥ 0. From Q ai ≥ 0 and X ai (0) ≥ 0, through iteration, it can be obtained that Therefore, it can be obtained that It can be obtained that Subtracting from X ai (0) gives It can be obtained that ΔX ai (0) ≥ 0. Therefore, through further iteration, it can be obtained that Q.E.D.

[0097] Introduce the matrices and which are defined as:

[0098]

[0099]

[0100] where For the multi - sensor network system, assume that the spectral radius of is less than 1, that is Then, under any initial condition the solutions of the generalized Lyapunov equation and X ai (t) converge to the unique positive semi - definite solutions X of the following generalized steady - state Lyapunov equation ai and

[0101] Similarly, if the spectral radius of is less than 1, that is Then, under any initial condition X aij ≥ 0, the solutions of the generalized Lyapunov equation and X aij (t) converge to the unique positive semi - definite solutions X of the following generalized steady - state Lyapunov equation aij and

[0102] Introduce the virtual process noise as follows

[0103]

[0104] The state equation can be rewritten as:

[0105]

[0106] It can be clearly seen that w fi(t) is white noise with zero mean.

[0107] Define and Q fi as the actual steady-state variance and the conservative steady-state variance of the fictitious noise w fi (t) respectively. The actual steady-state and conservative steady-state cross-covariances of w fi (t) and w fj (t) can be obtained. Introduce the fictitious measurement noise. Then, the measurement equation becomes:

[0108]

[0109] Define and as the actual steady-state variance and the conservative steady-state variance of the fictitious noise v fi (t) respectively. Define and S fi as the actual and conservative steady-state correlation matrices of w fi (t) and v fi (t) respectively. Based on the above discussion, a time-invariant multi-model correlated noise system with only uncertain actual virtual noise variances is obtained. It should be noted that ρ(Φ i ) < 1. Under the condition of ρ(Φ i ) < 1, there is This shows that is a stable matrix and is a completely detectable pair.

[0110] Assumption 5. Assume that is a completely detectable pair, and is a completely stabilizable pair. Then, among them

[0111] Design a robust local steady-state Kalman predictor in step 4. The predictor is used to predict the evolution of the system state and provide an initial state estimate

[0112] For the worst-case time-invariant augmented multi-model correlated noise system with known conservative noise statistics Q fi , R fi and S fi , under the conditions of Assumptions 1 to 5, based on the minimax robust estimation principle, applying the standard Kalman filtering algorithm, the following conservative local optimal (linear minimum variance) steady-state Kalman predictor is obtained:

[0113]

[0114] The conservative local steady-state prediction error variance P ai (-1) satisfies the following steady-state Riccati equation:

[0115]

[0116] With a conservative upper bound generated by the worst-case system And the local measurement y of P0 i (t) is called a conservative local measurement value which is unknown. While the measurement value y And With actual variance i (t) generated by the actual system is called the actual measurement value which is known and obtained through sensors. In the given conservative local steady-state Kalman predictor, the actual local measurement value y i (t) is used to replace the conservative local measurement value y i (t) to obtain the actual local steady-state Kalman predictor.

[0117] Define the local steady-state prediction error as Where the augmented noise λ fi (t), i = 1, K, L is defined as follows: The actual and conservative steady-state cross-covariances λ fi (t) and λ fj (t) of the augmented noise can be obtained and respectively satisfy the following equations

[0118]

[0119] The actual and conservative steady-state variances λ fi (t) of the augmented noise are respectively defined as In addition, the actual and conservative local steady-state prediction error cross-covariances respectively satisfy the following Lyapunov equations:

[0120]

[0121] The actual and conservative local steady-state prediction error variances are:

[0122] Robust local steady-state Kalman filters and smoothers. The filter is used to update the system estimate according to the current observation value and provide the most accurate estimation result. The smoother is used to post-process the estimation of the system state to eliminate errors and improve the estimation accuracy.

[0123] For a worst-case time-invariant augmented multi-model multi-sensor system with known conservative noise statistics Q fi , R fi And S fi , according to the actual local steady-state Kalman predictor is The actual local steady-state Kalman filter (N = 0) and smoother is:

[0124]

[0125]

[0126]

[0127] According to the definition, x ai (t)=[x(t) x(t - 1) ... x(t - (d - 1)) x(t - d) z i (t - 1) z i (t - 2)] T , the robust local steady-state Kalman estimator of each original subsystem can be expressed as where is the robust local steady-state Kalman estimator of each augmented subsystem.

[0128] Step 5: Combine all the local measurements received by the given sensors to obtain the centralized fusion measurement received by the sensors, and further implement multi-sensor data fusion;

[0129] Combining all the local measurements received by the given sensors, the centralized fusion measurement received by the sensors is obtained as follows:

[0130]

[0131] And combining all the local measurements received by the given estimators, the centralized fusion measurement received by the estimators is obtained as follows:

[0132]

[0133] For the worst-case time-invariant augmented centralized fusion system with conservative noise statistics Q f , R f and S f , the corresponding robust centralized fusion steady-state Kalman estimator can be obtained as well as their actual and conservative fusion steady-state error variances and P ac (N), which are robust for all admissible uncertainties, i.e.:

[0134]

[0135] And P ac (N) is the least upper bound.

[0136] From x a(t) = [x(t) x(t - 1) … x(t - (d - 1)) x(t - d) z (c) (t - 1) z (c) (t - 2)] T It can be seen that the robust centralized fusion steady-state Kalman estimator of the original system can be obtained through And their actual and conservative fusion steady-state error variances respectively satisfy the following formulas:

[0137]

[0138] Robust centralized fusion steady-state Kalman estimator Is robust in the following sense: for all admissible uncertainties there are

[0139]

[0140] And P c (N) is The least upper bound of.

[0141] Furthermore, the robust centralized state fusion steady-state Kalman estimator realizes multi-sensor data fusion.

[0142] First, the augmented state vector x g (t) is introduced, and its definition is as follows:

[0143]

[0144] Then the global Lyapunov equations for And X g (t) are respectively given as follows

[0145]

[0146]

[0147] Combining all local measurement equations, the following centralized fusion measurement equation is obtained:

[0148] y (c) (t) = H (c) x g (t) + v (c) (t)

[0149] It can be seen that introducing the augmented state vector x g (t) results in a centralized fusion (CF) system, which is constructed based on multi-model multi-sensors, and its system state is the augmented state vector x g(t). This method is completely different from the method for constructing the centralized fusion system given in the previous one, where the previous centralized fusion system was constructed based on the original single-model multi-sensor system.

[0150] Hypothesis 6. Assume that (Φ g , H (c) ) is completely detectable, and is completely stabilizable, then among them

[0151] for the worst-case time-invariant measurement fusion system with known conservative constant noise statistics and S (c) , under the conditions of Hypotheses 1 - 5, applying the standard Kalman prediction algorithm, the conservative optimal centralized fusion steady-state one-step Kalman predictor is as follows:

[0152]

[0153]

[0154] Ψ (c) = Φ g - K (c) H (c)

[0155]

[0156]

[0157] Based on the actual centralized fusion steady-state Kalman one-step predictor The actual centralized fusion steady-state Kalman filter (N = 0) and smoother are given as follows:

[0158]

[0159]

[0160]

[0161] By definition

[0162]

[0163] and it can be obtained that:

[0164] x(t) = Z g x g (t),

[0165]

[0166] Therefore, the robust centralized fusion steady-state Kalman estimator of the original system can be obtained as follows:

[0167]

[0168] And their actual and conservative fusion steady-state estimation error variances are respectively expressed as:

[0169]

[0170] From the above results, the following relevant accuracy relationships can be obtained:

[0171]

[0172] Accuracy relationships related to matrix inequalities:

[0173] P c (N) = P (c) (N) ≤ P i (N), i = 1,..., L.

[0174] Taking the trace, the following accuracy relationships related to matrix inequalities are obtained:

[0175]

[0176] trP c (N) = trP (c) (N) ≤ trP i (N), i = 1,..., L.

[0177] In the above remarks, and are called the actual accuracy of the corresponding robust Kalman estimator, while trP c (N), trP (c) (N) and trP i (N) are called their global accuracy (robust accuracy). The smaller the trace value, the higher the accuracy. The above remarks indicate that for all allowable uncertainties, the actual accuracy corresponding to each estimator is higher than or equal to its robust accuracy, and the robust accuracy is the lowest actual accuracy. The robust accuracy of centralized state fusion and centralized observation fusion is the same. The robust accuracy of all fusion estimators is higher than that of each local estimator. The robust centralized fusion estimator has the highest robust accuracy, while the robust local estimator has the lowest robust accuracy.

[0178] Furthermore, please refer to Figures 2 to 4 , and in the present invention, simulation experiments are verified through specific embodiments:

[0179] Step 6: In the simulation experiment, to verify the effectiveness of the estimator proposed in the present invention, the time-delay system (where d = 1) given is as follows:

[0180]

[0181] z i (t) = ([0.85 0.045] + β1(t)[-1 0] + β2(t)[0 -1])x(t) + v i (t), i = 1,..., L

[0182] v i (t) = D i w(t) + η i (t), i = 1,..., L

[0183]

[0184] where q = 2, L = 3

[0185] Take

[0186] R η3 = 0.3, ξ1 = 0.85, ξ2 = 0.85 ξ3 = 0.9, D1 = 0.97, D2 = 0.90, D3 = 0.92. Based on these parameters, important simulation results are obtained.

[0187] Since x(t) ∈ R 1 , so 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, verifying the steady-state accuracy relationship given above. 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.

[0188] Table 1 Comparison of steady-state robustness and actual accuracy

[0189]

[0190] To prove is robust, three arbitrary groups of different actual noise variances are selected satisfying, specifically as follows:

[0191] (1)

[0192]

[0193] (2)

[0194]

[0195] (3)

[0196]

[0197] Thus, three corresponding groups of robust fusion steady-state Kalman filters can be easily obtained as well as the corresponding actual and conservative fusion steady-state filtering error variances and P (c) (0). Figure 3 Shows the three actual filtering error curves of the corresponding robust fusion filters and their three-standard-deviation bounds, where the curves of the actual fusion filtering errors are represented by solid lines, the bounds are represented by dotted lines, and the ±3σ (c) (0) bounds are represented by dashed lines.

[0198] In the figure, A is the actual filtering error curve of the fusion filter and the corresponding and ±3σ (c) (0) bounds for group (1). B is the actual filtering error curve of the fusion filter and the corresponding and ±3σ (c) (0) bounds for group (2). C is the actual filtering error curve of the fusion filter and the corresponding and ±3σ (c) (0) bounds for group (3).

[0199] According to probability theory, the actual standard deviation can be obtained the robust standard deviation From Figure 3 it can be seen that for each fusion error curve, more than 99% of the fusion filtering error values are within the bounds and also between ±3σ (c) (0). This verifies the robustness of the robust fusion steady-state filter and the correctness of the actual standard deviation . Also, when the actual noise variance increases, the actual standard deviation also increases.

[0200] To illustrate the influence of the multiplicative noise β h (t), h = 1, 2 on the robust accuracy of the robust fusion steady-state Kalman predictor , let the conservative multiplicative noise variance increase from 0.003 to 0.03 in steps of 0.001 and increase from 0.002 to 0.02. The robust accuracy P increase from 0.002 to 0.02. The robust accuracy P (c)(-1),(trP (c) (-1)) For and the changes are as Figure 4 shown.

[0201] From Figure 4 it can be seen that when the conservative multiplicative noise variance increases, the value of P (c) (-1) also increases. This indicates that as the conservative multiplicative noise variance increases, the robust accuracy of the fusion steady-state Kalman predictor will decrease.

[0202] In summary, compared with the prior art, the beneficial effects of the present invention are:

[0203] 1. For the first time, the state-dependent multiplicative noise in the state transition matrix and the observation matrix, the two-step random observation time delay, and the uncertain noise variance in the state-delay system are taken into consideration, making the model closer to the actual application scenarios, such as environmental monitoring, target tracking, etc. Using two Bernoulli-distributed random variables to describe the random time delay can more accurately reflect the actual situation.

[0204] 2. Based on the min-max robust estimation principle, robust local and fusion Kalman estimators are designed under a unified framework, and this framework is applied to the above-mentioned uncertain networked time-delay system, which has generality and flexibility.

[0205] 3. Combining the combination method proves that the proposed fusion estimator has robustness, and this robustness ensures that the estimator has good performance for the admissible uncertainties.

[0206] 4. Analyzing the accuracy relationship between the robust local and fusion steady-state Kalman estimators helps to optimize the performance of the fusion system.

[0207] The accuracy relationship and robustness between the two fusion estimators and the local fusion estimator are verified through simulation experiments, and the simulation results verify the practicability and correctness of the proposed method.

[0208] What is disclosed above is only one or more preferred embodiments of the present invention. Of course, the scope of the rights of the present invention cannot be limited thereby. Those of ordinary skill in the art can understand all or part of the processes of implementing the above embodiments, and the equivalent changes made according to the claims of the present invention still fall within the scope covered by the invention.

Claims

1. A fusion estimation method for solving two-step stochastic observation delay and state time-delay systems, characterized in that, It includes the following steps: Step 1: Establish a linear discrete uncertain multi-sensor network time-delay system; Step 2: Use the virtual noise method to convert the model into a form with only uncertain noise variances, and construct an augmented multi-model system; Step 3: Introduce the Lyapunov equation to obtain a time-invariant multi-model correlated noise system with only uncertain actual virtual noise variances; Step 4: Based on the minimax robust estimation principle, derive a robust local steady-state Kalman predictor, and then obtain a robust local steady-state Kalman filter and smoother; Step 5: Combine all the local measurement values received by the given sensors to obtain the centralized fusion measurement values received by the sensors, and further realize multi-sensor data fusion; Step 6: Verify the effectiveness of the estimator through simulation experiments.

2. The fusion estimation method for solving the two-step random observation time delay and state time delay system according to claim 1, wherein the expression of the linear discrete uncertain multi-sensor network time-delay system is as follows: v i v(t) = D i w(t) + η i v_i(t), i = 1, ..., L where \(t\) is discrete time, \(x(t)\in\mathbb{R}\) n is the state vector of the system, is the observation value of the \(i\) - th sensor, is the measurement value received by the estimator after network transmission, \(w(t)\in\mathbb{R}\) n is the process noise, \(d\) is the fixed - length time delay, is the observation noise, and is linearly related to \(w(t)\), \(\beta\) k (t)\in\mathbb{R} 1 , \(k = 1,\cdots,q\) are state - dependent multiplicative noises; \(\varPhi\in\mathbb{R}\) n×n , \(\varPhi\) k \in\mathbb{R} n×n , \(\Gamma\in\mathbb{R}\) n×r , and are known constant matrices with appropriate dimensions; \(q\) is the number of multiplicative noises, \(L\) is the number of sensors; When ξ i = 1, y i (t) = z i (t) has no delay. When , then y i (t) = z i (t - 1) has a one-step time delay. When , then i.e., y i (t) = z i (t - 2) has a two-step time delay.

3. The fusion estimation method for solving the two-step random observation time delay and state time delay system according to claim 2, wherein the execution process of Step 3 is specifically to obtain the dynamic equation of the system state error by defining the state error covariance equation and combining the actual and conservative state second-order non-central moments of the state, introduce the Lyapunov equation, prove the stability of the system state error covariance, and then relate it to the matrix inequality to prove the non-negativity and stability of the virtual noise variance.

4. The fusion estimation method for solving the two-step random observation time delay and state time delay system according to claim 3, wherein the robust local steady-state Kalman predictor in Step 4 is the basis for the robust local steady-state Kalman filter and smoother, providing the initial prediction value and prediction error variance for subsequent filtering and fusion; the filter is used to update the system estimation value according to the current observation value and provide the most accurate estimation result, and the smoother is used to post-process the estimation of the system state to eliminate errors and improve the estimation accuracy.

5. The fusion estimation method for solving the two-step random observation time delay and state time delay system according to claim 4, wherein in Step 5, combine all the local measurement values received by the given sensors to obtain the centralized fusion measurement values received by the sensors, as follows: Where: And combine all the local measurement values received by the given estimator to obtain the centralized fusion measurement values received by the estimator, and the expression is as follows: Where:

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