Multi-sensor fusion filtering method for 2-D system with energy harvesting under measurement deletion

Abstract

The invention discloses a multi-sensor fusion filtering method for a 2-D nonlinear system under a measurement deletion and energy harvesting mechanism. The method comprises the following steps: firstly, establishing a Tobit model for a censored measurement value; then, under the condition that the energy of the sensor is limited, the probability that the sensor can successfully transmit the censored measurement value and the conditional expectation and conditional variance of the censored measurement value relative to the system state are calculated; on the basis, a local outlier suppression Tobit Kalman filter is constructed and is used for realizing reliable estimation of the system state under the conditions of measurement deletion and energy limitation. Furthermore, a stochastic analysis method and a 2-D mathematical induction method are utilized to deduce the upper bound of the filtering error variance, and the upper bound is minimized by optimizing the filter gain. And finally, based on the local estimation result of each sensor, a more accurate global fusion estimation value is obtained by adopting a covariance cross fusion method.

Description

Technical Field

[0001] This invention belongs to the field of filtering technology, specifically relating to a multi-sensor fusion filtering method for 2-D systems with energy harvesting under measurement censoring. Background Technology

[0002] As a typical multivariable system, the state of a 2-D system evolves in two independent directions. On the one hand, this characteristic makes 2-D systems widely applicable in practical engineering, such as in heat treatment processes, image data processing, and gas absorption. On the other hand, their complex dynamic behavior also brings many challenges to the study of 2-D systems, because many important conclusions obtained from 1-D models cannot be directly generalized to 2-D systems.

[0003] Measurement censoring is a common phenomenon in engineering, mainly caused by factors such as limited communication, sensor saturation, and obstructed areas. Censoring causes measurement outputs to exhibit nonlinear characteristics, resulting in measurement noise no longer following a Gaussian distribution, thus rendering traditional Kalman filtering methods inapplicable. To date, the most effective approach to address the measurement nonlinearity problem caused by censoring is to construct a Tobit model for the censored measurements and then use a Tobit Kalman filter for estimation.

[0004] With the continuous advancement of information acquisition and transmission technologies, the problem of multi-sensor fusion filtering has gradually attracted widespread attention in academia. This method aims to obtain more accurate state estimates than those from a single sensor by effectively fusing data from multiple sensors. In multi-sensor systems, each sensor consumes a significant amount of energy during signal transmission, making energy supply a crucial challenge in system design. The traditional solution is to equip each sensor with an independent power supply for charging; however, this approach is difficult to sustain long-term due to the limited energy storage capacity of batteries. In recent years, energy-harvesting sensors have emerged due to their self-replenishing energy advantage, effectively extending sensor lifespan.

[0005] In summary, to more accurately reflect the operating environment of the system and achieve reliable state estimation, this paper studies the multi-sensor fusion filtering problem for nonlinear 2-D systems, considering both energy harvesting and measurement censoring, based on the outlier suppression method. Summary of the Invention

[0006] This invention addresses nonlinear 2-D systems in practical engineering that suffer from measurement censoring and energy constraints. Based on outlier suppression, it proposes a multi-sensor fusion filtering method. Specifically, the method involves five steps: constructing a state-space model of the 2-D nonlinear system; establishing a Tobit model for censored measurements; constructing a mathematical model for energy harvesting sensors; designing a Tobit Kalman filter for outlier suppression and deriving the upper bound of the minimum filtering error variance; and obtaining the optimal fused estimate and the corresponding upper bound of the estimation error variance using the CI fusion rule. By employing stochastic analysis and 2-D mathematical induction, the proposed filtering method enables reliable estimation of 2-D nonlinear systems under energy constraints and measurement censoring.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] In a first aspect, the present invention provides a multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring, based on outlier suppression, comprising the following steps:

[0009] S1. Constructing the state-space expression of the 2-D nonlinear system: The dynamics of the studied 2-D nonlinear system evolve along two independent directions, which can more accurately characterize the spatiotemporal coupling characteristics of complex systems. The model consists of a state equation containing nonlinear terms and random noise, and a measurement equation affected by measurement censoring and observation noise;

[0010] S2. Construct a Tobit regression model for censoring measurements: Construct a Tobit regression model for censoring measurements of the 2-D system, and introduce Bernoulli random variables to characterize the censoring phenomenon.

[0011] S3. Establish a mathematical model for the energy harvesting sensor: For the censored signal processed in step S2, construct an energy harvesting constraint model to characterize the energy changes of the sensor during the relay process. Then, calculate the probability that the sensor can successfully transmit the censored measurement value at each time step. Furthermore, under the energy harvesting mechanism, derive the conditional expectation and conditional variance of the censored measurement value with respect to the system state.

[0012] S4. Construct a local outlier suppression Tobit Kalman filter and obtain the upper bound of the minimum filter error variance: Based on the outlier suppression method, a local Tobit Kalman filter is constructed using the received censored measurements to achieve a reliable estimation of the system state; then, the upper bound of the filter error variance is derived by combining stochastic analysis and 2-D mathematical induction, and by designing a suitable filter gain, the upper bound is minimized at each step.

[0013] S5. Obtain the fusion estimation result based on local estimation and CI fusion method: Based on the local estimation result obtained in step S4 and its minimum filter error variance upper bound, the CI fusion method is adopted. By optimizing the fusion weight, the global optimal estimation and its corresponding fusion error variance upper bound are further obtained, thereby realizing high-precision fusion estimation of multi-sensor information.

[0014] Furthermore, in step S1, the state-space expression of the constructed 2-D nonlinear system is:

[0015] x(r,s)=g1(r,s-1,x(r,s-1))+g2(r-1,s,x(r-1,s))

[0016] +B1(r,s-1)ω(r,s-1)+B2(r-1,s)ω(r-1,s)

[0017] in ( (where the integer is a known positive integer) Let ω(r,s) be the system state, and let B be a zero-mean Gaussian white noise sequence with variance Q(r,s)≥0. l (r,s)(l=1,2) is a known appropriate-dimensional shear matrix. For l=1,2, the nonlinear function g l (r,s,x(r,s)) satisfies the following constraints:

[0018]

[0019] Where A l (r,s) and a l (r,s)(l=1,2) are known shear matrices and non-negative constants, respectively. The initial states x(r,0) and x(0,s) are set as two white noise sequences, satisfying... and Here, η1(r) and η2(s) are known vectors that satisfy η1(0) = η2(0).

[0020] Assume the system is measured by W sensor nodes, and the uncensored measurement value of the t-th sensor, t∈[1 / W], satisfies the following dynamics:

[0021]

[0022] in For uncensored measurements, m t It is the dimension of the measured value. The variance is Zero-mean Gaussian white noise, Let be a known appropriate dimension shear matrix.

[0023] Further, in step S2, for the censoring measurements of the 2-D system, a Tobit regression model is constructed as follows:

[0024] The Tobit observation model constructed based on the censoring measurements obtained from the sensors is as follows:

[0025]

[0026] Where, for t∈[1 W] and d∈[1 m] t ], For censored measurements The d-th element, τ td The censoring threshold is known.

[0027] To more accurately characterize the Tobit observation model, a set of Bernoulli random variables ξ is used. td (r,s), d∈[1m] t To describe the censoring measurement value

[0028]

[0029] Obviously, when the random variable ξ td When (r,s)=1, the measured value It was not deleted, that is When ξ td When (r,s)=0, the measured value Deletion, i.e. according to Whether something is censored or not, the Tobit model can be rewritten as:

[0030]

[0031] in, It can be written as

[0032]

[0033] here and

[0034]

[0035] in d∈[1 m s [ ] represents a non-negative constant to be determined.

[0036] Further, step S3 specifically includes:

[0037] S31. Constructing a mathematical model for the energy harvesting sensor.

[0038] Each sensor employs energy harvesting technology, enabling it to acquire and store energy from the external environment. Measurements are only taken when the energy stored in each sensor is sufficient to support signal transmission. Only then can the data be successfully transmitted to the corresponding remote filter. In this invention, it is assumed that the sensor consumes one unit of energy each time it transmits a measurement value. First, a double-indexed ordered sequence is defined: symbol (r (1) ,s (1) )<(r (2) ,s (2) )express And the equation (r) (1) ,s (1) )=(r (2) ,s (2) It holds if and only if r (1) =r (2) and s (1) =s (2) Established; for The step before defining (r,s) is (r 0 ,s 0 ), obviously (r 0 ,s 0 The following constraints must be met:

[0039]

[0040] in, condition 1} represents the last time in the ordered sequence that condition 1' is true; at time (r, s), the energy stored in the t-th sensor is denoted as . right z t The evolution of (r,s) satisfies

[0041] z t (r,s)=min{z t (r 0 ,s 0 )+h t (r 0 ,s 0 )-γ t (r 0 ,s 0 ),D t}

[0042] Where (r) 0 ,s 0 ) represents the previous step of (r,s). Maximum storage capacity of the t-th sensor. t The initial condition for (r,s) is set to z. t (0,0)=κt ≤D t , where κ t γ is a known nonnegative constant. t (r,s) is defined as

[0043]

[0044] h t (r,s) represents the amount of energy collected by the t-th sensor at time (r,s). Furthermore, h t (r,s) is assumed to be a random sequence of white numbers, and its probability distribution satisfies

[0045] Prob{h t (r,s)=k}=p t,k k = 0, 1, 2, ..., D t

[0046] Where, p t,k Satisfying 0≤p t,k ≤1 and Since the amount of energy stored in the sensor varies randomly, its ability to successfully transmit a measurement value at each moment is essentially a random event. Let the actual measurement value received by the t-th local filter be... So y t (r,s) can be expressed as:

[0047]

[0048] S32. Calculate the probability that the sensor can successfully transmit the censored measurement signal at each moment.

[0049] First, calculate the energy level z of the t-th sensor. t The probability distribution of (r,s). Let be... in For energy level z t (r,s), whose probability distribution p t (r,s) satisfies the following recurrence relation:

[0050]

[0051] in (r 0 ,s 0 ) represents the previous step of (r,s),

[0052]

[0053] Based on recursive relationship γ is given t The probability distribution of (r,s):

[0054]

[0055] in

[0056]

[0057] In addition, there are

[0058]

[0059] S33. Derive the conditional expectation and conditional variance of the censored measurements with respect to the system state, thus laying the theoretical foundation for filter design and the calculation of filter error variance:

[0060] The actual measurement value y received by the filter td (r,s), t∈[1 W], d∈[1 m] t The conditional expected value and variance of x(r,s) are:

[0061]

[0062] in

[0063]

[0064] Further, step S4 specifically includes:

[0065] S41. Construct a Tobit Kalman filter to suppress outliers:

[0066] For each sensor, with respect to the original system and the measured value, the corresponding nonlinear local Tobit Kalman filter is designed as follows:

[0067]

[0068] in, The predicted value of the system state. This is the state estimate of the t-th filter; and the relevant initial conditions are: and And θ 1,t (0)=θ 2,t (0),

[0069] G t (r,s) represent the filter parameters to be determined. Saturation function. The definition is as follows:

[0070]

[0071] in, d∈[1 m t[Normal amount here]

[0072] Represents a given vector The d-th element. The definition is as follows:

[0073]

[0074] Where, for t∈[1 W] and d∈[1 m] t ],

[0075] and

[0076]

[0077] S42. Derive the upper bound of the filter error variance, and design a suitable filter gain so that the upper bound reaches its minimum value at each step.

[0078] The prediction error variance and estimation error variance for each local filter satisfy the following recursive formula:

[0079]

[0080] Where the matrix sequence {Ω t (r,s)}and{Θ t (r,s)}, It satisfies the following evolution equation:

[0081]

[0082] The initial conditions are satisfied. and Furthermore, for d∈[1 m] t ], l=1,2, have

[0083]

[0084] Here, when d = i, we have

[0085]

[0086] When d≠i, we have:

[0087]

[0088] and And ρ are known normal quantities. And when the local filter gain G... t When (r,s) is designed as follows:

[0089]

[0090] Upper bound Θ of the filter error variance t (r,s) can reach a minimum value and the minimum upper bound satisfies

[0091]

[0092] in

[0093]

[0094] Furthermore, in step S5, based on the local estimation results obtained in step S4 and the upper bound of its minimum filtering error variance, the fused Tobit Kalman filter can be written as follows according to the CI fusion rule:

[0095]

[0096] Where α t (r,s), where t∈[1 W] are weight coefficients, the specific values ​​of which can be obtained by solving the following optimization problem:

[0097]

[0098] Satisfy constraints 0≤α t (r,s)≤1.

[0099] Secondly, the present invention provides a multi-sensor fusion filtering system for a 2-D system with energy harvesting under measurement censoring, comprising:

[0100] State-space expression construction module for 2-D nonlinear systems: Used to construct the state-space expression of 2-D nonlinear systems, which consist of state equations containing nonlinearity and random noise and measurement equations containing random measurement noise;

[0101] Tobit Model Construction Module for Censorship Measurement: Constructs a Tobit model for censorship measurement and introduces Bernoulli random variables to characterize the censorship phenomenon;

[0102] The energy harvesting sensor mathematical model construction module constructs an energy harvesting constraint model to characterize the energy changes of the sensor during the relay process, accurately depicting the energy limitation problem caused by the remote deployment environment of the sensor; then, it calculates the probability that the sensor can successfully transmit censored measurements at each time step. Furthermore, under the energy harvesting mechanism, it derives the conditional expectation and conditional variance of the censored measurements with respect to the system state.

[0103] The filter design and minimum filter error variance construction module: A local outlier suppression filter is designed using censored measurements; furthermore, the upper bound of the filter error variance is obtained using stochastic analysis and 2D mathematical induction, and the filter parameters are designed to minimize this upper bound.

[0104] Fusion filter design module: Uses CI fusion rules to obtain fusion estimates and corresponding upper bounds of fusion filter error variance.

[0105] Furthermore, the present invention also includes a computer storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method described above.

[0106] The present invention also includes an electronic device comprising a memory and one or more processors, the memory being used to store one or more programs; when the one or more programs are executed by the one or more processors, they implement the method described above.

[0107] Compared with existing technologies, the present invention has the following advantages: It provides a multi-sensor fusion filtering method for 2-D nonlinear systems under censoring and energy harvesting sensors. The proposed method incorporates nonlinearity into the system state when establishing the 2-D state-space model, making its dynamic behavior more complex and general. A Tobit model is constructed for censored measurements, introducing Bernoulli random variables to characterize the censoring phenomenon. When constructing the energy harvesting constraint model, considering that sensors are usually deployed in remote areas with insufficient power supply, an energy constraint model is used to characterize the energy changes of the sensors, making the results more practically meaningful. The probability that the sensor can successfully transmit censored measurements, as well as the conditional expectation and conditional variance of the censored measurements with respect to the system state, are calculated. A local outlier suppression Tobit Kalman filter is constructed. Using stochastic analysis and 2-D mathematical induction, the upper bound of the filter error variance is derived, and by designing appropriate filter parameters, this upper bound is minimized. Using the CI fusion rule, by optimizing the fusion weights, the global optimal estimate and its corresponding upper bound of the fusion error variance are further obtained, thereby achieving high-precision fusion estimation of multi-sensor information. Attached Figure Description

[0108] Figure 1 This is a flowchart of the steps of the method of the present invention;

[0109] Figure 2 This represents the evolution trajectory of the estimation and fusion estimation for each local filter in this embodiment of the invention;

[0110] Figure 3 In the embodiments of the present invention, tr{Θ s(r,s)}, tr{Θ(r,s)}. and tr{Θ s The evolution trajectory of (r,s)-Θ(r,s)} (considering 5 sensor nodes, s∈[1 5]);

[0111] Figure 4 As described in the embodiments of the present invention and Evolutionary trajectory diagram. Detailed Implementation

[0112] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0113] Example

[0114] Step S1: Construct the state-space expression of the 2-D nonlinear system:

[0115] x(r,s)=g1(r,s-1,x(r,s-1))+g2(r-1,s,x(r-1,s))

[0116] +B1(r,s-1)ω(r,s-1)+B2(r-1,s)ω(r-1,s)

[0117] in ( (where the integer is a known positive integer) Let ω(r,s) be the system state, and let B be a zero-mean Gaussian white noise sequence with variance Q(r,s)≥0. l (r,s)(l=1,2) is a known appropriate-dimensional shear matrix. For l=1,2, the nonlinear function g l (r,s,x(r,s)) satisfies the following constraints:

[0118]

[0119] Where A l (r,s) and a l (r,s)(l=1,2) are known shear matrices and non-negative constants, respectively. The initial states x(r,0) and x(0,s) are set as two white noise sequences, satisfying... and Here, η1(r) and η2(s) are known vectors that satisfy η1(0) = η2(0).

[0120] The t-th, t∈[1 W] sensor measurement without censoring satisfies the following dynamics:

[0121]

[0122] in The variance is Zero-mean Gaussian white noise, Let be a known appropriate dimension shear matrix.

[0123] Step S2: Constrained by censored measurements for a 2-D system, construct a Tobit regression model and introduce Bernoulli random variables to characterize the censoring phenomenon. Furthermore, derive the conditional expectation and conditional variance of the censored measurements with respect to the system state.

[0124] First, based on the censoring measurements obtained from the sensors, the Tobit observation model is constructed as follows:

[0125]

[0126] Where, for t∈[1 W] and d∈[1 m] t ], For censored measurements The d-th element, τ td The censoring threshold is known.

[0127] To more accurately characterize the Tobit observation model, a set of Bernoulli variables ξ are used. td (r,s), d∈[1 m] t To describe the censoring measurement value

[0128]

[0129] Obviously, when the random variable ξ td When (r,s)=1, the measured value It was not deleted, that is When ξ td When (r,s)=0, the measured value Deletion, i.e. according to Whether something is censored or not, the Tobit model can be rewritten as:

[0130]

[0131] For censored measurements in 2-D systems, a Tobit regression model is constructed, and Bernoulli random variables are introduced to characterize the censoring phenomenon. Furthermore, the conditional expectation and conditional variance of the censored measurements with respect to the system state are derived.

[0132] To more accurately characterize the Tobit observation model, a set of Bernoulli variables ξ are used. td (r,s), d∈[1 m] t To describe the censoring measurement value

[0133]

[0134] Obviously, when the random variable ξ td When (r,s)=1, the measured value It was not deleted, that is When ξ td When (r,s)=0, the measured value Deletion, i.e. according to Whether something is censored or not, the Tobit model can be rewritten as:

[0135]

[0136] in, It can be written as

[0137]

[0138] here and

[0139]

[0140] in d∈[1 m s [ ] represents a non-negative constant to be determined.

[0141] Step S3: Construct an energy harvesting constraint model to characterize the energy changes of the sensor during the relay process. Then, calculate the probability that the sensor can successfully transmit the censored measurement at each time step. Furthermore, under the energy harvesting mechanism, derive the conditional expectation and conditional variance of the censored measurement with respect to the system state.

[0142] Sensor devices are typically deployed in remote areas far from city centers to achieve long-distance signal transmission. This deployment method can lead to energy constraints for the sensors, resulting in measurement failures due to insufficient energy. Therefore, each sensor employs energy harvesting technology to acquire and store energy from the external environment. Measurements are only transmitted when the energy stored in each sensor is sufficient to support signal transmission. Only then can the data be successfully transmitted to the corresponding remote filter. In this invention, it is assumed that the sensor consumes one unit of energy each time it transmits a measurement value. First, a double-indexed ordered sequence is defined: symbol (r (1) ,s (1) )<(r (2) ,s (2) )express And the equation (r) (1) ,s (1) )=(r (2),s (2) It holds if and only if r (1) =r (2) and s (1) =s (2) Established; for The step before defining (r,s) is (r 0 ,s 0 ), obviously (r 0 ,s 0 The following constraints must be met:

[0143]

[0144] in, condition 1} represents the last moment in the ordered sequence when condition 1 is true;

[0145] At time (r, s), the energy stored in the t-th sensor is denoted as . right z t The evolution of (r,s) satisfies

[0146] z t (r,s)=min{z t (r 0 ,s 0 )+h t (r 0 ,s 0 )-γ t (r 0 ,s 0 ),D t}

[0147] Where (r) 0 ,s 0 ) represents the previous step of (r,s). Maximum storage capacity of the t-th sensor. t The initial condition for (r,s) is set to z. t (0,0)=κ t ≤D t , where κ t γ is a known nonnegative constant. t (r,s) is defined as

[0148]

[0149] h t (r,s) represents the amount of energy collected by the t-th sensor at time (r,s). Furthermore, h t (r,s) is assumed to be a random sequence of white numbers, and its probability distribution satisfies

[0150] Prob{h t (r,s)=k}=p t,k k = 0, 1, 2, ..., D t

[0151] Where, p t,k Satisfying 0≤p t,k ≤1 and Since the amount of energy stored in the sensor varies randomly, its ability to successfully transmit a measurement value at each moment is essentially a random event. Let the actual measurement value received by the t-th local filter be... So y t (r,s) can be expressed as:

[0152]

[0153] First, calculate the energy level z of the t-th sensor. t The probability distribution of (r,s). Let be... in For energy level z t (r,s), whose probability distribution p t (r,s) satisfies the following recurrence relation:

[0154]

[0155] in (r 0 ,s 0 ) represents the previous step of (r,s),

[0156]

[0157] Based on recursive relationship γ is given t The probability distribution of (r,s):

[0158]

[0159] in

[0160]

[0161] In addition, there are

[0162]

[0163] The actual measurement value y received by the filter td (r,s)(t∈[1 W],d∈[1 m t The conditional expected value and variance of x(r,s) are:

[0164]

[0165] in

[0166]

[0167] Step S4: Based on the outlier suppression method, a local Tobit Kalman filter is constructed using the received censored measurement values ​​to achieve a reliable estimate of the system state. Subsequently, the upper bound of the filter error variance is derived by combining stochastic analysis and 2-D mathematical induction, and by designing a suitable filter gain, the upper bound is minimized at each step.

[0168] First, the outlier suppression Tobit Kalman filter is constructed: For each sensor, with respect to the original system and the measured value, the corresponding nonlinear local Tobit Kalman filter is designed as follows:

[0169]

[0170] in, The predicted value of the system state. This is the state estimate of the t-th filter; and the relevant initial conditions are: and And θ 1,t (0)=θ 2,t (0),

[0171] G t (r,s) represent the filter parameters to be determined. Saturation function. The definition is as follows:

[0172]

[0173] in, d∈[1 m t [Normal amount here]

[0174] Represents a given vector The d-th element. The definition is as follows:

[0175]

[0176] Where, for t∈[1 W] and d∈[1 m] t ],

[0177] and

[0178]

[0179] Then, the upper bound of the filter error variance is derived, and a suitable filter gain is designed to minimize this upper bound at each step. The prediction error variance and estimation error variance of each local filter satisfy the following recursive formula:

[0180]

[0181] Where the matrix sequence {Ω t (r,s)}and{Θ t (r,s)}, It satisfies the following evolution equation:

[0182]

[0183] The initial conditions are satisfied. and Furthermore, for d∈[1 m] t ], l=1,2, have

[0184]

[0185] Here, when d = i, we have

[0186]

[0187] When d≠i, we have:

[0188]

[0189] and And ρ are known normal quantities. Furthermore, when the local filter gain G... t When (r,s) is designed as follows:

[0190]

[0191] Upper bound Θ of the filter error variance t (r,s) can reach a minimum value and the minimum upper bound satisfies

[0192]

[0193] in

[0194]

[0195] Step S5: Based on the local estimation results and the upper bound of the minimum filtering error variance obtained in Step S4, the covariance cross-fusion method is adopted. By optimizing the fusion weights, the global optimal estimate and its corresponding upper bound of the fusion error variance are further obtained, thereby achieving high-precision fusion estimation of multi-sensor information. According to the CI fusion rule, the fused Tobit Kalman filter can be written as:

[0196]

[0197] Where α t (r,s), where t∈[1 W] are weight coefficients, the specific values ​​of which can be obtained by solving the following optimization problem:

[0198]

[0199] Satisfy constraints 0≤α t (r,s)≤1.

[0200] Test case

[0201] To verify the effectiveness of the method proposed in this invention, the following test experiment was conducted: In practical engineering, the evolution of many continuous physical systems (such as air drying, steam heating, and gas absorption processes) can be characterized by the following Darboux equation:

[0202]

[0203] Where u(z,t) is an unknown function in space z∈[0, Z] and time t∈[0, T], and f(z,t) is a known input function that satisfies here

[0204]

[0205] a0, a1, a2, b0, b1, and b2 are known real constants. According to the definition of r(z,t), we have...

[0206]

[0207] To discretize this partial differential equation, the following approximation equation is used:

[0208]

[0209] Where Δz and Δt are suitable step sizes. Definition

[0210] Therefore, the partial differential equation can be transformed into x(r,s)=g1(r,s-1,x(r,s-1))+g2(r-1,s,x(r-1,s)), where g l (r,s,x(r,s))=D l (r,s)x(r,s)+E l (r,s)cos(b1x(r,s))(l=1,2), and

[0211]

[0212] here

[0213] In this example, the parameters of the shear system are chosen as follows: Δz = Δt = 0.2, a0 = -2.6825 + 1.1cos(0.25(r+s)), a1 = -2.15, a2 = -2.55, b0 = 1.1, b1 = 0.11, b2 = -0.05. Clearly, the above nonlinear function g... l (r,s,x(r,s)) satisfies the conditional local Lipschitz condition, where A l (r,s)=D l (r,s) and a l (r,s)=b1‖E l (r,s)‖(l=1,2). Other system parameters are selected as follows:

[0214] B1(r,s)=[0.16,0.18+exp(-3r)] T ,B2(r,s)=[0.17+0.1sin(s),0.17] T

[0215] C1(r,s)=[1+0.1sin(r+s),-0.6],C2(r,s)=[0.99,-0.63+exp(-2s)]

[0216] C3(r,s)=[1.1+0.1cos(r+s),-0.62], C4(r,s)=[1.13,-0.58], C5(r,s)=[0.98,-0.65]

[0217] For simulation purposes, the initial conditions for the original system and the filter are set as follows:

[0218] η1(r)=η2(s)≡[0,0] T X(r,0)=X(0,s)≡0.01I

[0219] θ 1,1 (r)=θ 2,1 (s)≡[0.05,0.05] T ,

[0220] θ 1,2 (r)=θ 2,2 (s)≡[0.3,0.3] T ,

[0221] θ 1,3 (r)=θ2,3 (s)≡[0.2,0.2] T ,

[0222] θ 1,4 (r)=θ 2,4 (s)≡[0.12,0.12] T ,

[0223] θ 1,5 (r)=θ 2,5 (s)≡[0.25,0.25] T , Process noise ω(r,s) and measurement noise ν t Let the variances of (r,s) be Q(r,s) = 0.012I, R1(r,s) = 0.38, R2(r,s) = 0.12, R3(r,s) = 0.64, R4(r,s) = 0.25, and R5(r,s) = 0.46, respectively.

[0224] Furthermore, the deletion thresholds are set to τ1 = 0, τ2 = 0.2, τ3 = -0.2, τ4 = 0.12, and τ5 = -0.12, and the saturation series is set to... t∈[1 5]. During the energy harvesting process, the maximum energy storage capacity of the sensor is set to D1=3, D2=4, D3=5, D4=5, D5=3, and the initial stored energy of the sensor is κ1=3, κ2=4, κ3=5, κ4=5, and κ5=3. Assume that the energy collected by the sensor at each time step is h. t (r,s) follows the following distribution:

[0225] p 1,0 =0.1,p 1,1 =0.2,p 1,2 =0.3,p 1,3 =0.4

[0226] p 2,0 =0.1,p 2,1 =0.1,p 2,2 =0.2,p 2,3 =0.3,p 2,4 =0.3

[0227] p 3,0 =0.1,p 3,1 =0.1,p 3,2 =0.1,p 3,3 =0.1,p 3,4 =0.3,p 3,5 =0.3

[0228] p 4,0 =0.1,p4,1 =0.1,p 4,2 =0.1,p 4,3 =0.1,p 4,4 =0.1,p 4,5 =0.5

[0229] p 5,0 =0.2,p 5,1 =0.1,p 5,2 =0.3,p 5,3 =0.4.

[0230] Therefore, it is possible to calculate The measured outliers consist of a set of values ​​with a variance of 3 × 10⁻⁶. 5 Characterized by a zero-mean Gaussian random variable. Furthermore, it is assumed that outliers occur at times satisfying mod(r+s,5)=0. For all t∈[1 5] and k∈

[510] , other adjustable parameters are set to ρ=1.

[0231] Under the above parameter settings, the filter gain G t (r,s) can be obtained. Based on the state estimate and the upper bound of the minimum filter error variance, the fusion estimate can be obtained according to the CI fusion rule. And the upper bound of the estimated error variance Θ(r,s), where, for t∈[1 5], the parameter α t (r,s) can be obtained by solving an optimization problem.

[0232] In the finite time domain The simulation results are as follows: Figure 2-3 shown. Specifically, Figure 2 Let x represent the system state, the estimate of each local filter, and the evolution trajectories of the first and second components of the fusion estimate, where, for l = 1, 2 and t ∈ [1 5], x (l) (r,s), and Let x(r,s) represent r and s respectively. and The l-th element. Figure 3 Describes tr{Θ t (r,s)}, tr{Θ(r,s)}, and tr{Θ t The evolution trajectory of (r,s)-Θ(r,s)}, t∈[1 5] shows that the upper bound Θ(r,s) of the fusion estimation error variance reaches the minimum value.

[0233] To illustrate the impact of the censoring threshold on the performance of the designed filter, we compared the magnitudes of the minimum upper bound Θ1(r,s) by selecting different censoring thresholds. In addition to the previously provided censoring threshold τ...11 =0, and for τ1, choose two other values: -0.5 and 0.5, and denote the corresponding minimum upper bounds as 0, -0.5, -0.5, and 0.5, respectively. and Figure 4 Described and The evolutionary trajectory shows that as the censoring threshold τ increases... 11 As the threshold increases, the trace of the minimum upper bound Θ1(r,s) gradually increases, meaning that the filtering performance gradually deteriorates as the censoring threshold increases, which is consistent with actual engineering practice.

Claims

1. A multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring, characterized in that, The outlier suppression method specifically includes the following steps: S1. Constructing the state-space expression of the 2-D nonlinear system: The 2-D nonlinear system evolves dynamically along two independent directions. Its state-space model consists of a state equation containing nonlinear terms and random noise, and a measurement equation affected by measurement censoring and observation noise. S2. Construct a Tobit regression model for censoring measurements: Construct a Tobit regression model for censoring measurements of the 2-D system, and introduce Bernoulli random variables to characterize the censoring phenomenon; S3. Establish a mathematical model for the energy harvesting sensor: For the censored signal processed in step S2, construct an energy harvesting constraint model to characterize the energy change of the sensor during the relay process; then, calculate the probability that the sensor can successfully transmit the censored measurement value at each time step. Furthermore, under the energy harvesting mechanism, the conditional expectation and conditional variance of the censored measurements with respect to the system state are derived. S4. Construct a local outlier suppression Tobit Kalman filter and obtain the upper bound of the minimum filter error variance: Based on the outlier suppression method, a local Tobit Kalman filter is constructed using the received censored measurements to achieve a reliable estimation of the system state; then, the upper bound of the filter error variance is derived by combining stochastic analysis and 2-D mathematical induction, and by designing a suitable filter gain, the upper bound is minimized at each step. S5. Obtain the fusion estimation result based on local estimation and CI fusion method: Based on the local estimation result obtained in step S4 and its minimum filter error variance upper bound, the CI fusion method is adopted. By optimizing the fusion weight, the global optimal estimation and its corresponding fusion error variance upper bound are further obtained, thereby realizing high-precision fusion estimation of multi-sensor information.

2. The multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring as described in claim 1, characterized in that, Step S1 specifically involves: x(r,s)=g1(r,s-1,x(r,s-1))+g2(r-1,s,x(r-1,s))+B1(r,s-1)ω(r,s-1)+B2(r-1,s)ω(r-1,s) in Given a positive integer, Let ω(r,s) be the system state, and let B be a zero-mean Gaussian white noise sequence with variance Q(r,s)≥0. l (r,s) is a known shear matrix of appropriate dimension; for l = 1, 2, the nonlinear function g l (r,s,x(r,s)) satisfies the following constraints: Where A l (r,s) and a l (r,s) are known shear matrices and non-negative constants, respectively; the initial states x(r,0) and x(0,s) are set as two white noise sequences, satisfying... and Here, η1(r) and η2(s) are known vectors that satisfy η1(0) = η2(0); Assume the system is measured by W sensor nodes, and the uncensored measurement value of the t-th sensor satisfies the following dynamics, t∈[1 W]: in For uncensored measurements, m t To measure the dimension of the output, The variance is Zero-mean Gaussian white noise, Let be a known appropriate dimension shear matrix.

3. The multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring as described in claim 1, characterized in that, Step S2 specifically involves: The Tobit observation model constructed based on the censoring measurements obtained from the sensors is as follows: Where, for t∈[1 W] and d∈[1 m] t ], For censored measurements The d-th element, τ td The known censoring threshold; To more accurately characterize the Tobit observation model, a set of Bernoulli variables ξ are used. td (r,s) is used to describe the censoring measurement. When random variable ξ td When (r,s)=1, the measured value It was not deleted, that is When ξ td When (r,s)=0, the measured value Deletion, i.e. according to Whether something has been censored, the Tobit model rephrases it as follows: in, Written as here and in Let d be a nonnegative constant to be determined, d∈[1 m] s ].

4. The multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring as described in claim 1, characterized in that, Step S3 specifically involves: S31. Constructing a mathematical model for the energy harvesting sensor. Each sensor employs energy harvesting technology, enabling it to acquire and store energy from the external environment; measurements are only taken when the energy stored in each sensor is sufficient to support signal transmission. Only then can the data be successfully transmitted to the corresponding remote filter; Assume the sensor consumes one unit of energy per measurement transmitted; first, define a double-indexed ordered sequence: symbol (r (1) ,s (1) )<(r (2) ,s (2) )express And the equation (r) (1) ,s (1) )=(r (2) ,s (2) It holds if and only if r (1) =r (2) and s (1) =s (2) Established; for The step before defining (r,s) is (r 0 ,s 0 ), obviously (r 0 ,s 0 The following constraints must be met: in, This represents the moment when the last condition 1 in the ordered sequence is true. At time (r, s), the energy stored in the t-th sensor is denoted as . right z t The evolution of (r,s) satisfies z t (r,s)=min{z t (r 0 ,s 0 )+h t (r 0 ,s 0 )-γ t (r 0 ,s 0 ),D t } Where (r) 0 ,s 0 ) represents the previous step of (r,s). Maximum storage capacity of the t-th sensor; z t The initial condition for (r,s) is set to z. t (0,0)=κ t ≤D t , where κ t Given a nonnegative constant; variable γ t (r,s) is defined as h t (r,s) represents the amount of energy collected by the t-th sensor at time (r,s); furthermore, h t (r,s) is assumed to be a random sequence of white numbers, and its probability distribution satisfies Prob{h t (r,s)=k}=p t,k ,k=0,1,2,…,D t Where, p t,k Satisfying 0≤p t,k ≤1 and Since the amount of energy stored in the sensor varies randomly, its ability to successfully transmit a measurement value at each moment is essentially a random event; let the actual measurement value received by the t-th local filter be... So y t (r,s) is expressed as: S32. Calculate the probability that the sensor can successfully transmit the censored measurement signal at each moment. First, calculate the energy level z of the t-th sensor. t The probability distribution of (r,s); let in For energy level z t (r,s), whose probability distribution p t (r,s) satisfies the following recurrence relation: in (r 0 ,s 0 ) represents the previous step of (r,s), Based on recursive relationship Given γ t The probability distribution of (r,s): in In addition, there are S33. Derive the conditional expectation and conditional variance of the censored measurements with respect to the system state, thus laying the theoretical foundation for filter design and the calculation of filter error variance: The actual measurement value y received by the filter td (r,s), t∈[1 W], d∈[1 m] t The conditional expected value and variance of x(r,s) are: in 5. The multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring according to claim 1, characterized in that, Step S4 specifically involves: S41. Construct a Tobit Kalman filter to suppress local outliers. For each sensor, with respect to the original system and the measured value, the corresponding nonlinear local Tobit Kalman filter is designed as follows: in, The predicted value of the system state. This is the state estimate of the t-th filter; and the relevant initial conditions are: and and G t (r,s) are the filter parameters to be determined; saturation function σ t (·): The definition is as follows: in, Normal amount Represents a given vector The d-th element; The definition is as follows: Where, for t∈[1 W] and d∈[1 m] t ], and S42. Derive the upper bound of the filter error variance, and design a suitable filter gain so that the upper bound reaches its minimum value at each step. The prediction error variance and estimation error variance for each local filter satisfy the following recursive formula: Where the matrix sequence It satisfies the following evolution equation: The initial conditions are satisfied. and Furthermore, for d∈[1 m] t ], l=1,2, have Here, when d = i, we have When d≠i, we have: and And ρ are known normal quantities; and when the local filter gain G t When (r,s) is designed as follows: Upper bound Θ of the filter error variance t (r,s) reaches its minimum value and the minimum upper bound satisfies in 6. The multi-sensor fusion filtering method for a 2-D system with energy harvesting under measurement censoring as described in claim 5, characterized in that, Step S5 specifically involves: According to the CI fusion rules, the fused Tobit Kalman filter is written as: Where α t (r,s) are weight coefficients, t∈[1 W], and their specific values ​​are obtained by solving the following optimization problem: Satisfy constraints 7. A multi-sensor fusion filtering system for a 2-D system with energy harvesting under measurement censoring, characterized in that: For implementing the method as described in any one of claims 1-6, comprising: State-space expression construction module for 2-D nonlinear systems: Used to construct the state-space expression of 2-D nonlinear systems, which consist of state equations containing nonlinearity and random noise and measurement equations containing random measurement noise; Tobit Model Construction Module for Censorship Measurement: Constructs a Tobit model for censorship measurement and introduces Bernoulli random variables to characterize the censorship phenomenon; The energy harvesting sensor mathematical model construction module constructs an energy harvesting constraint model to characterize the energy changes of the sensor during the relay process, so as to accurately describe the energy limitation problem caused by the remote deployment environment of the sensor; then, it calculates the probability that the sensor can successfully transmit the censored measurement value at each time step, and derives the conditional expectation and conditional variance of the censored measurement value with respect to the system state under the energy harvesting mechanism. The filter design and minimum filter error variance construction module: Design a local outlier suppression filter using censored measurements; Obtain the upper bound of the filter error variance using stochastic analysis and 2-D mathematical induction; and design filter parameters to minimize this upper bound. Fusion filter design module: Uses CI fusion rules to obtain fusion estimates and corresponding upper bounds of fusion filter error variance.

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