A high-speed train state acquisition method based on a high-order extended Kalman filter

Abstract

The application provides a high-speed train state acquisition method based on a high-order extended Kalman filter. The method comprises the following steps: establishing a dynamic model of a train in different running states; constructing an extended Kalman filter state estimation error covariance matrix of the train after the dynamic model of the train in different running states is used; establishing a system model of the high-speed train in different running states; and acquiring high-speed train state information according to the extended Kalman filter state estimation error covariance matrix of the train and the system model. Through in-depth dynamic analysis of different running states (such as acceleration, air braking, adhesion braking, etc.) of the train, the application can better capture the complex dynamic behavior of the train in these states, and improve the adaptability and reliability of the positioning system under various running conditions.

Description

Technical Field

[0001] This invention relates to the field of train positioning technology, and in particular to a method for obtaining the state of a high-speed train based on a high-order extended Kalman filter. Background Technology

[0002] In high-speed railway systems, accurate train speed measurement and positioning are key technologies for ensuring safe train operation and improving transportation efficiency. Accurate train dynamics modeling is fundamental to achieving high-precision estimation in train positioning and speed measurement technologies. Most existing research employs linear dynamics models, which are simplifications of train dynamics and are typically used for low-speed or near-uniform speed scenarios. In these scenarios, the train's dynamic behavior can be considered linear, and changes in the system state can be described by linear equations. While linear dynamics models are computationally simple, they struggle to accurately describe the actual dynamic characteristics of trains at high speeds or non-uniform speeds. Some studies utilize multibody dynamics models, treating each component of the train as an independent rigid body and simulating their interactions through connecting elements such as joints, springs, and dampers. This method can describe the complex dynamic behavior of trains under different operating conditions in detail and is one of the most accurate train dynamics modeling methods currently available. However, it suffers from high computational complexity and poor real-time performance.

[0003] The accuracy of train dynamics modeling directly affects the effectiveness of train positioning and speed measurement. Traditional linear or low-order nonlinear models are insufficient to fully capture the complex dynamic behavior of high-speed trains, while existing high-precision modeling methods, such as multibody dynamics models, although highly accurate, have high computational complexity and cannot meet real-time requirements.

[0004] Currently, existing train positioning methods mainly include terrestrial signal-based positioning and satellite-based positioning. However, these methods have some limitations in practical applications: terrestrial signal positioning systems have limited coverage, and positioning accuracy is affected in areas with weak or obstructed signals. Satellite signals are easily blocked in environments such as tunnels and urban canyons, leading to increased positioning errors. Furthermore, satellite positioning has a low update frequency, which is insufficient for the real-time performance and accuracy of high-speed trains. To overcome these problems, researchers have recently begun to employ filtering techniques that fuse information from multiple sensors to improve the accuracy of train speed measurement and positioning. Among these, the Extended Kalman Filter (EKF) is a commonly used method. EKF can improve positioning accuracy to some extent by linearizing the nonlinear system. However, traditional EKF only performs a first-order Taylor expansion on the nonlinear function, and its accuracy remains limited when dealing with high-speed motion and complex dynamic environments. To address this issue, many researchers have proposed improved methods.

[0005] The disadvantages of existing train positioning methods include:

[0006] Limited ability to handle complex dynamic behaviors: The dynamic models of high-speed trains are highly nonlinear and complex, especially during rapid acceleration, deceleration, curvilinear travel, and when subjected to external disturbances (such as wind, gradient changes, etc.). Existing linear or low-order nonlinear models are unable to accurately describe these complex dynamic behaviors, leading to the accumulation of estimation errors, which in turn affects the stability and reliability of the overall system.

[0007] Insufficient positioning accuracy: Traditional Kalman filters for nonlinear systems only perform finite-order Taylor expansions, ignoring higher-order nonlinear information, resulting in low filtering accuracy. This is especially problematic in high-speed motion and complex dynamic environments, where errors accumulate rapidly.

[0008] Poor algorithm robustness: Positioning methods based on ground and satellite signals perform poorly in environments with signal obstruction, reflection, and interference, such as tunnels and urban canyons, where positioning accuracy and reliability are significantly reduced.

[0009] The trade-off between real-time performance and computational complexity: Some improved localization methods, such as particle filtering (PF) or hybrid modeling methods, perform well in handling nonlinear systems, but their computational complexity is high, especially in high-speed train scenarios with high real-time requirements, making it difficult to meet the needs of rapid response. At the same time, the high computational burden of these methods increases hardware requirements and system costs. Summary of the Invention

[0010] Embodiments of the present invention provide a method for obtaining the state of a high-speed train based on a high-order extended Kalman filter, so as to effectively obtain the state of a high-speed train.

[0011] To achieve the above objectives, the present invention adopts the following technical solution.

[0012] A high-speed train positioning method based on a complex nonlinear dynamics model includes:

[0013] Establish dynamic models of trains under different operating conditions;

[0014] Using the dynamic model of the train under different operating conditions, the extended Kalman filter state estimation error covariance matrix of the train is constructed.

[0015] A system model of different operating states during the operation of a high-speed train is established, and the state information of the high-speed train is obtained based on the extended Kalman filter state estimation error covariance matrix of the train and the system model.

[0016] Preferably, the establishment of the dynamic model of the train under different operating conditions includes:

[0017] 1. Train braking model:

[0018]

[0019] Where v(t+1) and v(t) are the train speeds at times t+1 and t, respectively, and the variables are the viscous force and the aerodynamic braking force, respectively. Where C p (v(t),s) represents the running resistance, T represents the sampling time, s represents the current position of the train, and M represents the running resistance. A Let w(t) represent the train passenger mass, and w(t) represent the random disturbance of the train speed caused by external factors.

[0020] Operating resistance C p (v(t)) consists of the basic resistance C1(v(t)) and the additional resistance C2(s), that is:

[0021] C p (v(t),s)=C1(v(t))+C2(s)

[0022] The basic resistance R1(vt) is positively correlated with the train speed vt, expressed as:

[0023] C1(v(t))=M A ×(c0+c1(t)+c2v 2 (t)×g×10 -3 )

[0024] Where c0 is the rolling resistance coefficient, c1 is other mechanical resistance coefficients that are proportional to the train speed v, C2(s) is the air resistance coefficient that is proportional to the square of the train speed, and c represents the combined resistance of slopes, curves and tunnels.

[0025] A(u) and B r The factors are adhesion braking force and air braking force, respectively; g is the gravitational acceleration coefficient; and the train wheel-rail adhesion coefficient u and train weight M are influenced by the combined effects of adhesion force and braking force A(u). A for:

[0026] A(u)=uM A g

[0027] Air Braking Force B r It is affected by several braking performance parameters, namely:

[0028]

[0029] Where d is the diameter of the brake cylinder, r is the friction radius of the brake disc, and R c N is the diameter of the train wheel. A This represents the total number of brake pads.

[0030] 2. Traction / Cruising / Taking Model:

[0031]

[0032] F Tr (v(t)) is the traction force, and its expression is as follows:

[0033]

[0034] 3. Speed ​​sensor model:

[0035] The train speed measurement system consists of many sensors that generate counting pulses as the gear disk rotates at each pitch, and the gear recording disk M... tacho It consists of a single pitch and a wheel diameter of R. c If in time interval T tacho Internally received p tacho If the pulse is applied, the train speed measured by Doppler is:

[0036]

[0037] Doppler radar measures train speed based on the Doppler effect. The relative speed between the train and the track is calculated based on the frequency difference between the transmitted wave and the reflected wave from the ground. Let N be the number of radar pulses per kilometer. dopp If the train is in time interval T dopp Internally received p dopp If the pulse is applied, the train speed measured by Doppler is:

[0038]

[0039] Based on the train braking model shown in the above equation, the state-space model for wheel Hall sensor speed measurement is expressed as:

[0040]

[0041] Similarly, the state-space model using Doppler radar is as follows:

[0042]

[0043] Where e1(t) and e2(t) are the sensor measurement errors;

[0044] The train operation observation model is as follows:

[0045]

[0046] Where f(·) is the nonlinear state function of the train under different operating modes, and h(·) is the linear measurement function of different speed sensors.

[0047] Preferably, the step of constructing the extended Kalman filter state estimation error covariance matrix after utilizing the dynamic model of the train under different operating states includes:

[0048] The extended Kalman filter method consists of two stages: a prediction stage and an update stage.

[0049] Prediction Phase: Based on the dynamic analysis of the high-speed train and the observation sensor equations, a system model was obtained where the state of the high-speed train during operation is nonlinear and the observation is linear. This system model includes the train braking model, speed sensor model, traction / cruising / tachyon model, and train operation observation model.

[0050] x(k+1)=f(x(k))+w(k)

[0051] y(k+1)=H(k+1)x(k+1)+v(k+1)

[0052] Where f(·) is a nonlinear state function with arbitrary order and continuous differentiability, H(k+1) is the measurement transition matrix; x(k) is an n-dimensional state vector, w(k) is an n-dimensional state model modeling error vector, y(k+1) is an m-dimensional measurement value vector, h(x(k+1) is an m-dimensional measurement function vector, and v(k+1) is an m-dimensional measurement model modeling error vector;

[0053] Given: the estimated value of the original state at time k and the covariance matrix of the estimation error.

[0054]

[0055] in, Let k be the estimated value of the original state at time k. Let be the estimation error covariance matrix of the original state at time k;

[0056] First, obtain the higher-order feature estimation information of the state at time k.

[0057]

[0058] in

[0059]

[0060]

[0061]

[0062] in, Original estimation error The l-order Kronecker set, For order l The estimated value, For order l The estimation error covariance matrix;

[0063] It is the original estimation error The l-th order Kronecker product;

[0064] Establishing a high-precision high-order extended Kalman filter for state estimation of such nonlinear systems based on higher-order features.

[0065]

[0066]

[0067] Where r represents the r-th order Taylor expansion approximation of the nonlinear state model. Let P(k+1|k+1) be the state estimate for the next time step, and let P(k+1|k+1) be the state estimate error covariance matrix for the next time step.

[0068] Assume that a nonlinear state model is used at the point where Performing Taylor expansions of order i = 2, 3, ..., l, we have obtained...

[0069]

[0070] Further recursion yields:

[0071]

[0072] First, there is a linear state model in the process. Taylor expansion of order l

[0073]

[0074] Where, x l (k+1) represents the state value. This is a one-step prediction value. This is the prediction error value for one step.

[0075] One-step prediction estimates based on the l-th order Taylor expansion of the nonlinear state model are obtained.

[0076]

[0077] One-step prediction estimation error value

[0078]

[0079] One-step prediction estimation error value

[0080]

[0081] Where, x l (k+1), and P xx,l The subscript "l" in (k+1|k) indicates that it is based on a nonlinear state model at the point of... It is obtained from the l-th order Taylor expansion;

[0082]

[0083] Furthermore, the nonlinear model is applied to... Perform a (l+1) order Taylor expansion

[0084]

[0085] in

[0086]

[0087] Substituting the above equation into the linear measurement model, we have:

[0088]

[0089] Where y(k+1) is the measured value, A (i) (k) is the Jacobian matrix corresponding to the Taylor expansion; after combining like terms and rearranging, it relates to the identified quantity. The measurement equations are:

[0090]

[0091] in

[0092]

[0093] In the above formula

[0094]

[0095]

[0096] in, For the observed values ​​in the newly constructed observation equation, The estimated value of the newly constructed observation noise, The estimation error covariance matrix of the newly constructed observation noise;

[0097] Applying the under-least-squares method to equation (), we have:

[0098]

[0099] Calculate the expected estimate of the quantity

[0100]

[0101] Calculate the expected estimation error value of the quantity

[0102]

[0103] Calculate the expected estimation error covariance matrix of the variable.

[0104]

[0105] At this point, we have obtained

[0106]

[0107] in, Let l+1 be the state estimation error value. The expected value of the quantity.

[0108] The expected estimation error value of the quantity. The expected estimation error covariance matrix of the variable

[0109] Update phase:

[0110] Based on the r-order Taylor expansion

[0111]

[0112] One-step state prediction estimate

[0113]

[0114] One-step state prediction estimation error value

[0115] One-step state prediction estimation error value covariance matrix

[0116] Based on the measurement equation, a one-step measurement prediction estimate is obtained.

[0117]

[0118] One-step measurement prediction estimation error value

[0119]

[0120] Calculate the state estimation error of the extended Kalman filter.

[0121] and measured values ​​and decomposition

[0122]

[0123] Using the orthogonal principle

[0124]

[0125] Obtain the gain matrix K of the extended Kalman filter. r Solving equation (k+1)

[0126]

[0127] Solving the above equations yields the extended Kalman filter gain matrix K. r The analytical expression for (k+1):

[0128]

[0129] The cross-covariance matrix between the state prediction estimation error and the measurement prediction estimation error is as follows:

[0130]

[0131] The autocovariance matrix of the measurement prediction estimation error is:

[0132]

[0133] Based on formula

[0134]

[0135] Obtain the estimated state of the train at the next moment;

[0136] Calculate the state estimation error covariance matrix of the high-order extended Kalman filter:

[0137]

[0138] The extended Kalman filter state estimation error covariance matrix of the train is obtained as follows:

[0139]

[0140] Preferably, the step of establishing a system model for different operating states during high-speed train operation, and obtaining high-speed train state information based on the extended Kalman filter state estimation error covariance matrix of the train and the system model, includes:

[0141] A dynamic model of the high-speed train is established based on the actual motion state of the high-speed train. The dynamic model is used as the state equation of the entire high-speed train operation system. A mathematical model is established based on the measurement data obtained by the sensors on the high-speed train and along the track. The mathematical model is used as the observation equation of the entire high-speed train operation system. The system model of the high-speed train operation system is obtained based on the state equation and mathematical model of the high-speed train operation system.

[0142] The extended Kalman filter state estimation error covariance matrix is ​​applied to the system model of the high-speed train operation system. By performing Taylor expansion on the nonlinear state equation, higher-order nonlinear information is obtained, and the current state estimation prediction value of the train is obtained. This state estimation prediction value includes the train speed estimate value and the displacement estimate value. The current state estimation prediction value is substituted into the measurement equation, and the state estimation value of the train at the next moment is obtained with the help of the measurement equation. Then, the estimate value at this moment is substituted back into the Kalman filter update formula. The above steps are repeated to obtain the train speed estimate value and displacement estimate value at the next moment after that. This process is repeated to obtain the train speed and position estimate values ​​for the entire process.

[0143] As can be seen from the technical solutions provided by the embodiments of the present invention above, the method of the embodiments of the present invention enhances the ability to handle complex dynamic behaviors: by conducting in-depth dynamic analysis of different operating states of the train (such as acceleration, air braking, adhesion braking, etc.), the model of the present invention can better capture the complex dynamic behaviors of the train in these states, and improve the adaptability and reliability of the positioning system under various operating conditions.

[0144] Improving positioning accuracy: By establishing a nonlinear dynamic model of high-speed trains and combining it with a high-order extended Kalman filter algorithm, the high-order nonlinear information of the system is used to make more accurate state estimation, thereby significantly improving the positioning and speed measurement accuracy of high-speed trains, especially in complex nonlinear environments.

[0145] Enhanced algorithm robustness: By combining the undermeasured least squares algorithm, the robustness of the system in various complex environments is improved, ensuring that high-precision positioning information can still be provided even under harsh external environmental conditions.

[0146] Optimize computational efficiency: While ensuring high accuracy, optimize the algorithm structure, reduce computational complexity, improve the real-time performance of filtering, and meet the real-time positioning requirements of high-speed trains.

[0147] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and will become apparent from the description or may be learned by practice of the invention. Attached Figure Description

[0148] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0149] Figure 1 A flowchart illustrating a high-speed train state acquisition method based on a high-order extended Kalman filter, provided in an embodiment of the present invention.

[0150] Figure 2 This invention provides a displacement variation curve under train dynamics modeling.

[0151] Figure 3 A speed variation curve under train dynamics modeling is provided as an embodiment of the present invention;

[0152] Figure 4 A flowchart of a high-precision fusion positioning method for high-speed trains based on a complex nonlinear dynamics model, provided in an embodiment of the present invention;

[0153] Figure 5 This invention provides a high-speed train application scenario for a high-precision fusion positioning method based on a complex nonlinear dynamics model using high-order extended Kalman filtering.

[0154] Figure 6 This is a framework diagram of a high-precision fusion positioning method for high-speed trains based on a complex nonlinear dynamics model, provided in an embodiment of the present invention. Detailed Implementation

[0155] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein 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 with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0156] Those skilled in the art will understand that, unless specifically stated otherwise, the singular forms “a,” “an,” “the,” and “the” used herein may also include the plural forms. It should be further understood that the term “comprising” as used in this specification means the presence of the stated features, integers, steps, operations, elements, and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. It should be understood that when this invention refers to an element as being “connected” or “coupled” to another element, it may be directly connected or coupled to the other element, or there may be intermediate elements. Furthermore, “connected” or “coupled” as used herein may include wireless connections or couplings. The term “and / or” as used herein includes any and all combinations of one or more of the associated listed items.

[0157] It will be understood by those skilled in the art that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It should also be understood that terms such as those defined in general dictionaries should be understood to have the same meaning as in the context of the prior art, and should not be interpreted in an idealized or overly formal sense unless defined as herein.

[0158] To facilitate understanding of the embodiments of the present invention, the following will provide further explanation and description with reference to the accompanying drawings and several specific embodiments. These embodiments do not constitute a limitation on the embodiments of the present invention.

[0159] Since estimation algorithms such as the Kalman filter rely on system and observation models to predict and update state estimates, inaccurate models will affect the filter's prediction and update steps, thus reducing estimation accuracy. Therefore, to improve overall estimation accuracy, this invention first establishes system models for different operating states of high-speed trains. The operating states of high-speed trains can be broadly categorized as traction, cruising, coasting, and braking. Traction refers to the speed increase caused by the traction force. During cruising, the train maintains its speed under the influence of traction or braking force and resistance. During coasting, the traction and braking forces are zero, and the train is only affected by running resistance. Finally, braking is the situation where the train's speed decreases only under the action of braking force. Based on this, according to the traction requirements of the train under different operating modes, traction, cruising, and coasting are treated as one model type, and braking as another. Considering the different braking characteristics of trains on dry and wet tracks, the existing braking model is further improved. The influence of adhesion coefficient and adhesion braking force on train braking performance is analyzed.

[0160] The processing flow of a high-speed train state acquisition method based on a high-order extended Kalman filter provided in this embodiment of the invention is as follows: Figure 1 As shown, the processing steps include the following:

[0161] Step S10: Establish dynamic models of the train under different operating conditions.

[0162] Step S20: Using the dynamic models of the train under different operating conditions described above, construct the extended Kalman filter state estimation error covariance matrix.

[0163] Step S30: Establish system models of different operating states during the operation of high-speed trains, and use the state estimation error covariance matrix of the extended Kalman filter mentioned above to perform high-speed train speed measurement and positioning.

[0164] Specifically, step S10 above includes:

[0165] 1. Train braking model:

[0166]

[0167] Where v(t+1) and v(t) are the train speeds at times t+1 and t, respectively, and the variables are the viscous force and the aerodynamic braking force, respectively. Where C p (v(t),s) represents the running resistance, T represents the sampling time, s represents the current position of the train, and M represents the running resistance. A Let w(t) represent the train passenger mass, and w(t) represent the random disturbance of train speed caused by external factors.

[0168] Among them, the running resistance C p (v(t)) consists of the basic resistance C1(v(t)) and the additional resistance C2(s), that is:

[0169] C p (v(t),s)=C1(v(t))+C2(s)

[0170] The basic resistance R1(vt) is positively correlated with the train speed vt, and is usually expressed as:

[0171] C1(v(t))=M A ×(c0+c1v(t)+c2v 2 (t)×g×10 -3 )

[0172] Where c0 is the rolling resistance coefficient, c1 is other mechanical resistance coefficients, which are proportional to the train speed v, and C2(s) is the air resistance coefficient, which is proportional to the square of the train speed. Their values ​​are usually set heuristically. c represents the combined resistance of slopes, curves, and tunnels.

[0173] Where A(u) and B rThese are the adhesive braking force and the air braking force, respectively. The combined influence of adhesion force and braking force A(u) on the train wheel-rail adhesion coefficient u and train weight M... A ,for

[0174] A(u)=uM A g

[0175] Where g is the gravitational acceleration coefficient, 9.8 m / s².

[0176] In the above formula, the wheel-rail adhesion coefficient decreases as the train speed decreases. Their relationship is affected by the track surface. Taking the train braking process as an example, when the train moves from dry rail to wet rail, the adhesion coefficient decreases significantly, and the train system tends to lock up and slip. The air braking force acting on the wheels is insufficient to stop the train. In this situation, the system cannot accurately obtain the real-time status of the train, increasing the risk of train operation.

[0177] Air Braking Force B r It is affected by several braking performance parameters, namely:

[0178]

[0179] Where d is the diameter of the brake cylinder, r is the friction radius of the brake disc, and R c N is the diameter of the train wheel. A The total number of brake pads. These parameters can be directly measured using conventional methods and remain constant during train operation. Therefore, these parameters are considered constants in the calculations of this invention. The additional parameter η in the above formula is the transmission efficiency of the main braking device, and γ... B is the braking ratio. is the friction coefficient of the train brake disc, which depends on the train's operating conditions and the brake disc material. Pre is the air pressure in the brake cylinder. During actual emergency braking of the train, the entire air pressure will be released.

[0180] 2. Traction / Cruising / Taking Model:

[0181]

[0182] Among them, F Tr (v(t)) represents the traction force, which can be adjusted in real time according to the actual operating conditions. The specific operating mode of the model is determined by the traction force, and its expression is as follows:

[0183]

[0184] 3. Speed ​​sensor model:

[0185] Train speed measurement systems consist of numerous sensors, such as inertial navigation systems (INS), tachometers, Doppler radar, and global positioning systems (GPS). Taking the CTCS3.300T train control system as an example, the current multi-sensor system architecture comprises two tachometers and a combination of two tachometers. Specifically, as the gear disk rotates at each pitch, it generates counting pulses. The gear recording disk M... tacho It consists of a single pitch and a wheel diameter of R. c If in time interval T tacho Internally received p tacho If the pulse is applied, the train speed measured by Doppler is:

[0186]

[0187] It should be noted that the tachometer is affected by the adhesion of the track surface. When the train is idling or slipping, the tachometer's measurement accuracy is greatly affected.

[0188] Therefore, Doppler radar measures train speed based on the Doppler effect. The relative speed between the train and the track is calculated based on the frequency difference between the transmitted wave and the reflected wave from the ground. Let N be the number of radar pulses per kilometer. dopp If in time interval T dopp Internally received p dopp If the pulse is applied, the train speed measured by Doppler is:

[0189]

[0190] Doppler radar is unaffected by wheel spin, slippage, or changes in wheel diameter. However, it is generally more susceptible to interference than a tachometer.

[0191] Combining the train braking model shown in the above equation, the state-space model for wheel Hall sensor speed measurement can be expressed as:

[0192]

[0193] Similarly, the state-space model using Doppler radar is as follows:

[0194]

[0195] Where e1(t) and e2(t) are the sensor measurement errors.

[0196] In summary, the train operation observation model can be uniformly rewritten as follows:

[0197]

[0198] Where f(·) is the nonlinear state function of the train under different operating modes, and h(·) is the linear measurement function of different speed sensors.

[0199] Specifically, step S20 above includes:

[0200] The extended Kalman filter filtering method of this invention mainly includes two stages: prediction stage and update stage.

[0201] Prediction Phase: Assuming that this invention has obtained a system model of the high-speed train's nonlinear state and linear observation during operation based on the dynamic analysis and observation sensor equations of the high-speed train, this system model includes the train braking model, speed sensor model, traction / cruising / gliding model, and train operation observation model obtained in step S10 above:

[0202] x(k+1)=f(x(k))+w(k)

[0203] y(k+1)=H(k+1)x(k+1)+v(k+1)

[0204] Where f(·) is a nonlinear state function with arbitrary order and continuous differentiability, H(k+1) is the measurement transition matrix; x(k) is an n-dimensional state vector, w(k) is an n-dimensional state model modeling error vector, y(k+1) is an m-dimensional measurement value vector, h(x(k+1) is an m-dimensional measurement function vector, and v(k+1) is an m-dimensional measurement model modeling error vector.

[0205] Given: the estimated value of the original state at time k and the covariance matrix of the estimation error.

[0206]

[0207] in, Let k be the estimated value of the original state at time k. Let be the estimation error covariance matrix of the original state at time k.

[0208] The objective of this invention is to first obtain the higher-order feature estimation information of the state at time k.

[0209]

[0210] in

[0211]

[0212]

[0213]

[0214] in, Original estimation error The l-order Kronecker set, For order l The estimated value, For order l The estimated error covariance matrix.

[0215] It is the original estimation error The l-th order Kronek product.

[0216] Establishing a high-precision high-order extended Kalman filter for state estimation of such nonlinear systems based on higher-order features.

[0217]

[0218] Where r represents the r-th order Taylor expansion approximation of the nonlinear state model. Let P(k+1|k+1) be the state estimate for the next time step, and let P(k+1|k+1) be the state estimate error covariance matrix for the next time step.

[0219] Assume that a nonlinear state model is used at the point where Performing Taylor expansions of order i = 2, 3, ..., l, we have obtained...

[0220]

[0221] Further recursive acquisition

[0222]

[0223] Therefore, this invention first addresses the issue based on a linear state model. Taylor expansion of order l

[0224]

[0225] Where, x l (k+1) represents the state value. This is a one-step prediction value. This represents the prediction error value for one step.

[0226] One-step prediction estimates based on the l-th order Taylor expansion of the nonlinear state model are obtained.

[0227]

[0228] One-step prediction estimation error value

[0229]

[0230] One-step prediction estimation error value

[0231]

[0232] Where, xl (k+1), and P xx,l The subscript "l" in (k+1|k) indicates that it is based on a nonlinear state model at the point of... It is obtained from the l-th order Taylor expansion.

[0233]

[0234] Furthermore, the nonlinear model is applied to... Perform a (l+1) order Taylor expansion

[0235]

[0236] in

[0237]

[0238] Substituting the above equation into the linear measurement model, we have:

[0239]

[0240] Where y(k+1) is the measured value, A (i) (k) is the Jacobian matrix corresponding to the Taylor expansion. After combining like terms and rearranging, it relates to the identified quantity. The measurement equations are:

[0241]

[0242] in

[0243]

[0244]

[0245]

[0246] In the above formula

[0247]

[0248]

[0249] in, For the observed values ​​in the newly constructed observation equation, The estimated value of the newly constructed observation noise, The estimation error covariance matrix of the newly constructed observation noise.

[0250] Applying the under-least-squares method to equation (), we have:

[0251]

[0252] Calculate the expected estimate of the quantity

[0253]

[0254] Calculate the expected estimation error value of the quantity

[0255]

[0256] Calculate the expected estimation error covariance matrix of the variable.

[0257]

[0258] Thus, the present invention has been obtained.

[0259]

[0260] in, Let l+1 be the state estimation error value. The expected value of the quantity.

[0261] The expected estimation error value of the quantity. The expected estimation error covariance matrix of the variable

[0262] Update phase:

[0263] Based on the r-order Taylor expansion

[0264]

[0265] One-step state prediction estimate

[0266]

[0267] One-step state prediction estimation error value

[0268] One-step state prediction estimation error value covariance matrix

[0269] Based on the measurement equation, a one-step measurement prediction estimate is obtained.

[0270]

[0271] One-step measurement prediction estimation error value

[0272]

[0273] Calculate the high-precision EKF state estimation error

[0274] and measured values ​​and decomposition

[0275]

[0276] Using the orthogonal principle

[0277]

[0278] This invention obtains information about the EKF gain matrix K. r Solving equation (k+1)

[0279]

[0280] Solving the above equation yields the EKF gain matrix K. r The analytical expression of (k+1)

[0281]

[0282] The cross-covariance matrix between the state prediction estimation error and the measurement prediction estimation error is:

[0283]

[0284] The autocovariance matrix of the measurement prediction estimation error is

[0285]

[0286] Based on formula

[0287]

[0288] This allows us to obtain the state estimate for the next moment.

[0289] Finally, the state estimation error covariance matrix of the higher-order extended Kalman filter is calculated:

[0290]

[0291] Therefore, the high-precision EKF state estimation error covariance matrix is ​​obtained.

[0292]

[0293] Thus, this invention completes the design of a high-precision EKF based on higher-order features for estimating nonlinear states and linear measurement systems.

[0294] Specifically, step S30 includes: establishing a dynamic model of the high-speed train based on its actual motion state, using this dynamic model as the state equation of the entire high-speed train operation system; establishing a mathematical model based on the measurement data acquired by sensors on the high-speed train and along the track, using this mathematical model as the observation equation of the entire high-speed train operation system. Based on the state equation and mathematical model of the high-speed train operation system, a system model of the high-speed train operation system is obtained. Applying the extended Kalman filter state estimation error covariance matrix proposed in this invention to the system model of the high-speed train operation system yields state estimates of the high-speed train at different times, namely, displacement estimates and velocity estimates. This allows for accurate estimation of the position and velocity of the high-speed train under external disturbance conditions.

[0295] The state of a high-speed train includes two indicators: displacement and velocity. The train's state can be represented as a state vector, which includes two physical quantities: displacement and velocity. By modeling the dynamics of the high-speed train, we obtain the equations for how its state changes over time. However, due to various noises and interferences in the train's environment, the modeling accuracy of these equations is inaccurate. Therefore, we employ Kalman filtering to reduce the interference of environmental noise on the train's state estimation. Specifically, we perform a Taylor expansion of the nonlinear state equation to obtain higher-order nonlinear information, resulting in a more accurate one-step prediction of the state. This prediction is then substituted into the measurement equation to obtain the state estimate for the next time step, i.e., the train's velocity and displacement estimates. These estimates are then substituted back into the Kalman filter update formula. Repeating this process yields the train's velocity and displacement estimates for the next two time steps. By continuously refining this process, we can obtain the estimated train velocity and position for the entire process.

[0296] Figure 2 and Figure 3 The state change curve of the high-speed train after dynamic modeling is derived from... Figure 2 and Figure 3 As can be seen, train operation is weakly nonlinear. This invention takes into account the influence of multiple resistances such as friction, air resistance, and additional forces on curves during train operation, and obtains a more accurate train operation state curve. This can greatly reduce the error of the estimation algorithm in the cyclic process, thereby achieving higher accuracy.

[0297] Figure 4 The flowchart of the high-order extended Kalman filter algorithm of this invention is shown below. Figure 4As shown, after inputting the system state model and initial state into the algorithm, the nonlinear state equation in the system is first expanded using Taylor. Then, it is substituted into the observation equation, and the known and unknown terms are separated and rearranged to form a new observation equation. The undermeasured least squares algorithm is applied to the observation equation to calculate the estimated values ​​of the higher-order terms of the Taylor expansion. Then, the nonlinear state equation is expanded to a higher order again, and the above operation is repeated to obtain the theoretically infinite-order information of the nonlinear function, thereby obtaining higher estimation accuracy.

[0298] Figure 5 This diagram illustrates an application scenario of the high-order extended Kalman filter algorithm of this invention. It assumes a high-speed train is running on a straight high-speed railway, with multiple observation devices such as radio frequency detectors and sensors arranged in a zigzag pattern along the trackside. The train can exchange information with the radio frequency detectors and Doppler radar via communication equipment. By inputting the observation information from different devices and the train's status information into the high-order extended Kalman filter algorithm, real-time estimates of the train's position and speed can be obtained.

[0299] Figure 6 The diagram illustrates the framework for applying the high-order extended Kalman filter algorithm to a high-speed train system. High-speed trains have multiple operating states, so the train is modeled separately according to different motion states. Then, different observation instruments such as sensors, Doppler radar, and train tachometers are modeled to obtain an accurate high-speed train operating system model, in which the high-order extended Kalman filter algorithm can be applied.

[0300] In summary, the embodiments of the present invention have the following advantages in high-speed train speed measurement and positioning:

[0301] Enhancing the ability to handle complex dynamic behaviors: Existing train dynamics models are typically simplified linear or low-order nonlinear models, which struggle to accurately capture the complex dynamic behaviors of high-speed trains under different operating conditions, especially when track irregularities and external disturbances are present, affecting positioning accuracy. This invention establishes a more accurate nonlinear dynamic model, enabling comprehensive capture of the motion characteristics of high-speed trains under various dynamic conditions. Combined with the HEKF algorithm, this model significantly improves the system's ability to handle complex dynamic behaviors, thereby enhancing positioning accuracy and system robustness.

[0302] Significantly Improved Positioning Accuracy: Traditional Extended Kalman Filter (EKF) algorithms, when dealing with nonlinear systems, rely solely on first-order Taylor expansions for linearization, resulting in significant estimation errors in complex nonlinear environments, making it difficult to meet the high-precision positioning requirements of high-speed trains. This invention introduces a higher-order Extended Kalman Filter (HEKF) algorithm, which utilizes higher-order nonlinear information of the system, significantly improving positioning and speed measurement accuracy. This improvement enables the invention to maintain high-precision state estimation under various operating conditions (such as acceleration, deceleration, and constant speed), especially under high-speed or complex nonlinear conditions.

[0303] Overall Performance Advantages: Existing technologies often involve trade-offs between accuracy, real-time performance, and complexity. For example, high-precision particle filter algorithms suffer from computational complexity bottlenecks, while simple linearization methods, although computationally fast, lack sufficient accuracy. This invention, through an innovative high-order extended Kalman filter algorithm and nonlinear dynamic modeling, demonstrates outstanding performance in positioning accuracy, real-time performance, computational efficiency, and environmental adaptability, achieving comprehensive performance optimization and meeting the multiple requirements of high-speed train positioning.

[0304] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of one embodiment, and the modules or processes shown in the drawings are not necessarily essential for implementing the present invention.

[0305] As can be seen from the above description of the embodiments, those skilled in the art can clearly understand that the present invention can be implemented by means of software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in various embodiments or some parts of the embodiments of the present invention.

[0306] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, for apparatus or system embodiments, since they are basically similar to method embodiments, the description is relatively simple; relevant parts can be referred to the descriptions in the method embodiments. The apparatus and system embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0307] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A high-speed train positioning method based on a complex nonlinear dynamics model, characterized in that, include: Establish dynamic models of trains under different operating conditions; Using the dynamic model of the train under different operating conditions, the extended Kalman filter state estimation error covariance matrix of the train is constructed. A system model of different operating states during the operation of a high-speed train is established, and the state information of the high-speed train is obtained based on the extended Kalman filter state estimation error covariance matrix of the train and the system model. The construction of the extended Kalman filter state estimation error covariance matrix using the dynamic model of the train under different operating states includes: The extended Kalman filter method consists of two stages: a prediction stage and an update stage. Prediction Phase: Based on the dynamic analysis of the high-speed train and the observation sensor equations, a system model was obtained where the state of the high-speed train during operation is nonlinear and the observation is linear. This system model includes the train braking model, speed sensor model, traction / cruising / tachyon model, and train operation observation model. x(k+1)=f(x(k))+w(k) y(k+1)=H(k+1)x(k+1)+v(k+1) Where f(·) is a nonlinear state function with arbitrary order and continuous differentiability, H(k+1) is the measurement transition matrix; x(k) is an n-dimensional state vector, w(k) is an n-dimensional state model modeling error vector, y(k+1) is an m-dimensional measurement value vector, h(x(k+1) is an m-dimensional measurement function vector, and v(k+1) is an m-dimensional measurement model modeling error vector; Given: the estimated value of the original state at time k and the covariance matrix of the estimation error. in, Let k be the estimated value of the original state at time k. Let be the estimation error covariance matrix of the original state at time k; First, obtain the higher-order feature estimation information of the state at time k. in in, Original estimation error The l-order Kronecker set, For order l The estimated value, For order l The estimation error covariance matrix; It is the original estimation error The l-th order Kronecker product; Establishing a high-precision high-order extended Kalman filter for state estimation of such nonlinear systems based on higher-order features. Where r represents the r-th order Taylor expansion approximation of the nonlinear state model. Let P(k+1|k+1) be the state estimate for the next time step, and let P(k+1|k+1) be the state estimate error covariance matrix for the next time step. Assume that a nonlinear state model is used at the point where Perform i=2 , Taylor expansions of order 3, ..., l have been obtained. Further recursion yields: First, there is a linear state model in the process. Taylor expansion of order l Where, x l (k+1) represents the state value. This is a one-step prediction value. This is the prediction error value for one step. One-step prediction estimates based on the l-th order Taylor expansion of the nonlinear state model are obtained. One-step prediction estimation error value One-step prediction estimation error covariance matrix Where, x l (k+1) , and P xx,l The subscript "l" in (k+1|k) indicates that the nonlinear state model is used at the point where... It is obtained from the l-th order Taylor expansion; Furthermore, the nonlinear model is applied to... Perform a (l+1) order Taylor expansion in Substituting the above equation into the linear measurement model, we have: Where y(k+1) is the measured value, A (i) (k) is the Jacobian matrix corresponding to the Taylor expansion; after combining like terms and rearranging, it relates to the identified quantity. The measurement equations are: in In the above formula in, For the observed values ​​in the newly constructed observation equation, The estimated value of the newly constructed observation noise, The estimation error covariance matrix of the newly constructed observation noise; Applying the under-least-squares method to equation (), we have: Calculate the expected estimate of the quantity Calculate the expected estimation error value of the quantity Calculate the expected estimation error covariance matrix of the variable. At this point, we have obtained in, Let l+1 be the state estimation error value. The expected value of the quantity. The expected estimation error value of the quantity. The expected estimation error covariance matrix of the variable Update phase: Based on the r-order Taylor expansion One-step state prediction estimate One-step state prediction estimation error value One-step state prediction estimation error value covariance matrix Based on the measurement equation, a one-step measurement prediction estimate is obtained. One-step measurement prediction estimation error value Calculate the state estimation error of the extended Kalman filter. and measured values ​​and decomposition Using the orthogonal principle Obtain the gain matrix K of the extended Kalman filter. r Solving equation (k+1) Solving the above equations yields the extended Kalman filter gain matrix K. r The analytical expression for (k+1): The cross-covariance matrix between the state prediction estimation error and the measurement prediction estimation error is as follows: The measurement prediction estimation error autocovariance matrix is: Based on formula Obtain the estimated state of the train at the next moment; Calculate the state estimation error covariance matrix of the high-order extended Kalman filter: The extended Kalman filter state estimation error covariance matrix of the train is obtained as follows:

2. The method according to claim 1, characterized in that, The establishment of dynamic models for trains under different operating conditions includes:

1. Train braking model: Where v(t+1) and v(t) are the train speeds at times t+1 and t, respectively, and the variables are the viscous force and the aerodynamic braking force, respectively. C p (v(t),s) represents the running resistance, T represents the sampling time, s represents the current position of the train, and M represents the running resistance. A Let w(t) represent the train passenger mass, and w(t) represent the random disturbance of the train speed caused by external factors. Operating resistance C p (v(t)) consists of the basic resistance C1(v(t)) and the additional resistance C2(s), that is: C p (v(t),s)=C1(v(t))+C2(s) The basic resistance R1(vt) is positively correlated with the train speed vt, expressed as: C1(v(t))=M A ×(c0+c1v(t)+c2v 2 (t)×g×10 ―3 ) Where c0 is the rolling resistance coefficient, c1 is other mechanical resistance coefficients that are proportional to the train speed v, C2(s) is the air resistance coefficient that is proportional to the square of the train speed, and c represents the combined resistance of slopes, curves and tunnels. A(u) and B r The factors are adhesion braking force and air braking force, respectively; g is the gravitational acceleration coefficient; and the train wheel-rail adhesion coefficient u and train weight M are influenced by the combined effects of adhesion force and braking force A(u). A for: A(u)=uM A g Air Braking Force B r It is affected by several braking performance parameters, namely: Where d is the diameter of the brake cylinder, r is the friction radius of the brake disc, and R c N is the diameter of the train wheel. A This represents the total number of brake pads.

2. Traction / Cruising / Taking Model: F Tr (v(t)) is the traction force, and its expression is as follows:

3. Speed ​​sensor model: The train speed measurement system consists of many sensors that generate counting pulses as the gear disk rotates at each pitch, and the gear recording disk M... tacho It consists of a single pitch and a wheel diameter of R. c If in time interval T tacho Internally received p tacho If the pulse is applied, the train speed measured by Doppler is: Doppler radar measures train speed based on the Doppler effect. The relative speed between the train and the track is calculated based on the frequency difference between the transmitted wave and the reflected wave from the ground. Let N be the number of radar pulses per kilometer. dopp If the train is in time interval T dopp Internally received p dopp If the pulse is applied, the train speed measured by Doppler is: Based on the train braking model shown in the above equation, the state-space model for wheel Hall sensor speed measurement is expressed as: Similarly, the state-space model using Doppler radar is as follows: Where e1(t) and e2(t) are the sensor measurement errors; The train operation observation model is as follows: Where f(·) is the nonlinear state function of the train under different operating modes, and h(·) is the linear measurement function of different speed sensors.

3. The method according to claim 1, characterized in that, The establishment of a system model for different operating states during high-speed train operation, and the acquisition of high-speed train state information based on the extended Kalman filter state estimation error covariance matrix and the system model, includes: A dynamic model of the high-speed train is established based on the actual motion state of the high-speed train. The dynamic model is used as the state equation of the entire high-speed train operation system. A mathematical model is established based on the measurement data obtained by the sensors on the high-speed train and along the track. The mathematical model is used as the observation equation of the entire high-speed train operation system. The system model of the high-speed train operation system is obtained based on the state equation and mathematical model of the high-speed train operation system. The extended Kalman filter state estimation error covariance matrix is ​​applied to the system model of the high-speed train operation system. By performing Taylor expansion on the nonlinear state equation, higher-order nonlinear information is obtained, and the current state estimation prediction value of the train is obtained. This state estimation prediction value includes the train speed estimate value and the displacement estimate value. The current state estimation prediction value is substituted into the measurement equation, and the state estimation value of the train at the next moment is obtained with the help of the measurement equation. Then, the estimate value at this moment is substituted back into the Kalman filter update formula. The above steps are repeated to obtain the train speed estimate value and displacement estimate value at the next moment after that. This process is repeated to obtain the train speed and position estimate values ​​for the entire process.

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