A train speed control method based on fractional order sliding mode and Kalman filter

Abstract

This invention discloses a train speed control method based on fractional sliding mode and Kalman filtering, belonging to the field of train control technology. The Kalman filtering used can not only eliminate measurement errors caused by early system noise, but also reduce chattering caused by sliding mode control. Considering that the sliding mode of the sliding controller is independent of the system parameters and disturbances, an improved Kalman filter control algorithm based on sliding mode control is proposed. At the same time, fractional calculus is introduced into the sliding mode switching function to further suppress chattering. This invention effectively suppresses the early sensor data measurement error in the train speed control algorithm, thereby improving the actual control accuracy, while maintaining a fast response speed and weak model dependence.

Description

Technical Field

[0001] This invention relates to the field of train control technology, and in particular to a train speed control method based on fractional sliding mode and Kalman filtering. Background Technology

[0002] With the rapid development of urban rail transit, higher demands are being placed on train driving control technology while maintaining high-efficiency operation. The Automatic Train Operation (ATO) system is primarily responsible for ensuring automatic train operation, providing automatic control and adjustment functions, and assisting the driver. It is one of the core systems of the CBTC (Continuous Train Control) system in rail transit. Speed ​​control is the most crucial function of the ATO system. Many scholars have proposed different solutions for speed control, mainly divided into physical modeling and system identification. Physical modeling offers high control accuracy but involves complex mathematical models. System identification simplifies the model complexity but lacks real-time performance. Currently, control algorithms for train applications mainly include fuzzy control, predictive control, adaptive control, neural network control, or combinations of multiple control theories. However, these algorithms are highly dependent on existing models and do not consider the errors in real-time speed and position states acquired due to noise from early sensors.

[0003] In summary, existing train speed control algorithms suffer from problems such as unprocessed early measurement noise, complex controllers, and low control accuracy. Summary of the Invention

[0004] This invention provides a train speed control method based on fractional sliding mode and Kalman filtering, which can solve the problems pointed out in the background art.

[0005] A train speed control method based on fractional sliding mode and Kalman filtering includes the following steps:

[0006] Step 1: Establish a train dynamics model;

[0007] Step 2: Based on the train dynamics model, establish the train operation state space equations;

[0008] Step 3: Calibrate the wheel axle speed sensor error using the Kalman filter algorithm;

[0009] Step 4: Establish a sliding mode controller and introduce fractional calculus to achieve tracking control of the train's reference speed and reference position.

[0010] The train power car model in step one is as follows:

[0011]

[0012] Where: x is displacement; t is running time; v is the real-time speed of the train; u is the traction / braking force of the train; w is the basic resistance of the train; a, b, and c are the rolling mechanical resistance coefficient, frictional resistance coefficient, and air resistance coefficient of the train, respectively; a c This is the actual acceleration of the train; ξ is the velocity derivative, i.e., acceleration; ξ is the train acceleration coefficient; γ is the train wheel rotational mass coefficient.

[0013] Based on the train dynamics model in step one, the following train operation state space equations are established:

[0014]

[0015] Where m is the mass of the train, v(t) is the real-time speed of the train, x(t) is the real-time position of the train, f(t) is the traction / braking force of the train, w(t) is the real-time basic resistance, and d(t) is the additional resistance and external disturbance. This is the velocity derivative.

[0016] The method for calibrating the wheel axle speed sensor error using the Kalman filter algorithm in step three is as follows:

[0017] Based on the slippage, idling, and measurement errors that occur during train operation, the following error formula is established:

[0018] v = v k +d k #(3)

[0019] Where, d k v is the compensation value during train operation in cycle k; k v represents the speed measurement value including measurement error during train operation in cycle k, where v is the actual speed of the train.

[0020] a c =a k +ε k #(4)

[0021] Where, ε k The error compensation value for train acceleration measurement in period k; a k a is the measurement value from the accelerometer. c This is the actual acceleration of the train;

[0022] The following noise formula is established:

[0023] v k+1 =v k +a k t+ω k #(5)

[0024] Where, ωk For the combined error of acceleration and velocity, a k t is (no explanation needed); v k The speed measurement value includes measurement error during train operation in cycle k; v k+1 The velocity value at the next moment;

[0025] The following observation equation is established:

[0026] X = x k +D k #(6)

[0027] Among them, D k The error disturbance caused by the inherent accuracy error of the wheel axle speed sensor within k periods is the observation noise; x k X represents the measured displacement value within period k; X is the actual displacement.

[0028] The above formula (5) can be expressed in matrix form:

[0029]

[0030] Where X(k+1) is the velocity state quantity of period k+1; B represents system parameters; X(k) represents the velocity state variables in the k-cycle; A(k) represents the system control matrix in the k-cycle; and ΓW(k) represents Gaussian noise.

[0031] Right now:

[0032]

[0033] The above formula (6) can be expressed in matrix form:

[0034] Y(k)=HX(k)+V(k)#(9)

[0035] Where Y(k) represents the displacement observation; H represents the parameters of the measurement system; and V(k) represents the observation noise.

[0036] Right now:

[0037]

[0038] Predict the next system state equation based on the initial input:

[0039]

[0040] in, The estimated value at time k is the value at time k+1; the optimal linear prediction estimate of X(k+1); φ and B are system parameters; A(k-1) is the system control matrix at time k-1; P(k) is the covariance matrix at time k.

[0041] The covariance at this point is:

[0042]

[0043] Where P(k|k-1) is the covariance prediction of time k-1 to time k; P(k-1|k-1) is the optimal covariance at time k-1; and Q(k-1) is the covariance of the system process at time k-1.

[0044] The filter gain equation is:

[0045] K(k)=P(k|k-1)H T [HP(k|k-1)H T +R(k)] -1 #(13)

[0046] Where K(k) is the filter gain; P(k|k-1) is the covariance prediction from time k-1 to time k; H T Let H be the transpose of H; R(k) be the Gaussian noise at time k;

[0047] Based on the above formulas (7), (9), (11), (12), and (13), the recursive filtering estimation equation is:

[0048]

[0049] in, X(k|k-1) represents the optimal velocity estimate at time k; X(k|k-1) represents the predicted state at time k-1; K(k) represents the filter gain at time k; and V(k) represents the observation noise.

[0050] The corresponding covariance is updated as follows:

[0051] P(k|k)=[IK(k)H]P(k|k-1)#(15)

[0052] Where P(k|k) is the optimal covariance at time k, and P(k|k-1) is the covariance prediction at time k-1 for time k; I is the identity matrix;

[0053] The estimated value of the Kalman filter is calculated using the recursive steps above.

[0054] Before constructing the sliding mode controller in step four, the velocity-position error state equation is defined as follows:

[0055]

[0056] Where e represents the train position error; x represents the train speed error; x represents the actual position of the train obtained after passing through the Kalman filter.r v is the reference position; v is the actual velocity obtained after Kalman filtering algorithm; v r For reference speed;

[0057] In step four, fractional calculus is introduced as follows:

[0058]

[0059] Where, d m / dt m This is the traditional differential, where m is the smallest integer not less than fractional a, t is time, and τ is the integration variable; when α < 0, It is a fractional differential, and when α > 0, it is a fractional integral; Γ(x) is the gamma function. m is a fractional integer;

[0060] The sliding surface introduced in step four, which is the fractional calculus, is:

[0061]

[0062] Where λ is the sliding surface gain coefficient, λ>0; e i For train position error, For train speed error, D α-1 It is a fractional operator;

[0063] The sliding mode controller established in step four is as follows:

[0064]

[0065] in, is the estimation term for the basic resistance of the train; k2sgn(S)d is the nonlinear switching control term of the system, used to handle external disturbances and uncertainties, k1 and k2 are the control gains, where k1>0 and k2>0; S is the sliding mode switching function.

[0066] The tracking error obtained by the sliding mode controller is output to the train ATO system until the train reaches its destination.

[0067] Compared with the prior art, the beneficial effects of the present invention are:

[0068] To improve the accuracy of train speed control in urban rail transit, this invention proposes a Kalman filter algorithm based on fractional sliding mode control. For speed control algorithms, Kalman filtering can not only eliminate measurement errors caused by early system noise but also reduce chattering caused by sliding mode control. Considering that the sliding mode of the sliding controller is independent of system parameters and disturbances, an improved Kalman filter control algorithm based on sliding mode control is proposed. Furthermore, fractional calculus is introduced into the sliding mode switching function to further suppress chattering.

[0069] This invention effectively suppresses the measurement error of sensor data in the early stage of the train speed control algorithm, thereby improving the actual control accuracy, while maintaining a fast response speed and weak model dependency effect. Attached Figure Description

[0070] Figure 1 This is a flowchart of the present invention;

[0071] Figure 2 This is a structural block diagram of the present invention. Detailed Implementation

[0072] The following detailed description of a specific embodiment of the present invention is provided in conjunction with the accompanying drawings. However, it should be understood that the scope of protection of the present invention is not limited to the specific embodiment.

[0073] like Figures 1 to 2 As shown in the figure, an embodiment of the present invention provides a train speed control method based on fractional sliding mode and Kalman filtering, which includes the following steps:

[0074] Step 1: Establish a train dynamics model;

[0075] Step 2: Based on the train dynamics model, establish the train operation state space equations;

[0076] Step 3: Calibrate the wheel axle speed sensor error using the Kalman filter algorithm;

[0077] Step 4: Establish a sliding mode controller and introduce fractional calculus to achieve tracking control of the train's reference speed and reference position;

[0078] The controlled object of this invention is a single-track high-speed train. During the entire operation, the biggest challenge for the controller lies in the accurate tracking of the speed curve. Secondly, the complexity of the entire line's resource environment dictates that the train's speed, position, and status data will inevitably contain noise and measurement errors. Therefore, if the controller itself has good disturbance suppression capabilities, the tracking accuracy during operation will be guaranteed, which is beneficial for realizing the train's status. In the controller design process, the uncertainty of system model parameters and the error of sensor data must be considered. Therefore, a Kalman filter model was designed at the beginning to eliminate noise and data measurement errors. The designed controller should have good robustness, enabling it to overcome unknown external interference and the uncertainty of measurement data, thereby achieving fast and stable online control, ensuring high-precision speed tracking and smooth control input during train tracking operation.

[0079] The train power car model in step one is as follows:

[0080]

[0081] Where: x is displacement; t is running time; v is the real-time speed of the train; u is the traction / braking force of the train; w is the basic resistance of the train; a, b, and c are the rolling mechanical resistance coefficient, frictional resistance coefficient, and air resistance coefficient of the train, respectively; a c This is the actual acceleration of the train; ξ is the velocity derivative, i.e., acceleration; ξ is the train acceleration coefficient; γ is the train wheel rotational mass coefficient.

[0082] In actual operation, the resistance of a train consists of two parts: additional resistance and basic resistance. The basic resistance is affected by train speed and mechanical wear, but the additional resistance only occurs in the fixed part of the track. This paper will combine the additional resistance with the uncertain disturbance part for processing.

[0083] Based on the train dynamics model in step one, the following train operation state space equations are established:

[0084]

[0085] Where m is the mass of the train, v(t) is the real-time speed of the train, x(t) is the real-time position of the train, f(t) is the traction / braking force of the train, w(t) is the real-time basic resistance, and d(t) is the additional resistance and external disturbance. The velocity derivative;

[0086] According to high-speed railway technical regulations, the basic running resistance of a train is affected by environmental factors such as wind speed and track surface conditions. In engineering, various coefficients are obtained through multiple tests and fitting. Therefore, the formulas used in the model inevitably contain factors caused by external environmental factors during actual train operation. The parameter structure of the basic running resistance w is w = a + bv + cv. 2 .

[0087] Using wheel and axle speed sensors can reduce data processing and communication burdens, but during train operation, there are errors such as slippage, idling, and measurement errors.

[0088] The method for calibrating the wheel axle speed sensor error using the Kalman filter algorithm in step three is as follows:

[0089] Based on the slippage, idling, and measurement errors that occur during train operation, the following error formula is established:

[0090] v = v k +d k #(3)

[0091] Where, d k v is the compensation value during train operation in cycle k; k v is the true speed of the train during cycle k without measurement error, where v is the actual speed of the train.

[0092] a c =a k +ε k #(4)

[0093] Where, ε k The error compensation value for train acceleration measurement in period k; a k a is the measurement value from the accelerometer. c This is the actual acceleration of the train;

[0094] First, discretize the train's operation process, assuming a sampling time of T and a sufficiently small sampling interval of t. Because the sampling interval is sufficiently small, the motion between two sampling points can be considered as uniformly accelerated motion. Therefore, based on the kinematic equations, the following noise formula can be obtained:

[0095] v k+1 =v k +a k t+ω k #(5)

[0096] Where, ω k For the combined error of acceleration and velocity, a k t is (no explanation needed); v k The speed measurement value includes measurement error during train operation in cycle k; v k+1The velocity value at the next moment;

[0097] For the train's position, using the position calculation value from the wheel axle speed sensor as the observation simplifies the procedure. Therefore, the displacement equation of the wheel axle speed sensor is used as the observation equation here, as follows:

[0098] X = x k +D k #(6)

[0099] Among them, D k The error disturbance caused by the inherent accuracy error of the wheel axle speed sensor within k periods is the observation noise; x k X represents the measured displacement over period k; X is the actual displacement.

[0100] The above formula (5) can be expressed in matrix form:

[0101]

[0102] Where X(k+1) is the velocity state quantity of period k+1; B represents system parameters; X(k) represents the velocity state variables in the K-cycle; A(k) represents the control matrix of the k-cycle system; ΓW(k) represents Gaussian noise;

[0103] Right now:

[0104]

[0105] The above formula (6) can be expressed in matrix form:

[0106] Y(k)=HX(k)+V(k)#(9)

[0107] Where Y(k) represents the displacement observation; H represents the parameters of the measurement system; and V(k) represents the observation noise.

[0108] Right now:

[0109]

[0110] Predict the next system state equation based on the initial input:

[0111]

[0112] in, The estimated value at time k is the value at time k+1; the optimal linear prediction estimate of X(k+1); φ and B are system parameters; A(k-1) is the system control matrix at time k-1; P(k) is the covariance matrix at time k.

[0113] The covariance at this point is:

[0114]

[0115] Where P(k|k-1) is the covariance prediction of time k-1 to time k; P(k-1|k-1) is the optimal covariance at time k-1; and Q(k-1) is the covariance of the system process at time k-1.

[0116] The filter gain equation is:

[0117] K(k)=P(k|k-1)H T [HP(k|k-1)H T +R(k)] -1 #(13)

[0118] Where K(k) is the filter gain; P(k|k-1) is the covariance prediction from time k-1 to time k; H T Let H be the transpose of H; R(k) be the Gaussian noise at time k;

[0119] Based on the above formulas (7), (9), (11), (12), and (13), the recursive filtering estimation equation is:

[0120]

[0121] in, X(k|k-1) represents the optimal velocity estimate at time k; X(k|k-1) represents the predicted state at time k-1; K(k) represents the filter gain at time k; and V(k) represents the observation noise.

[0122] The corresponding covariance is updated as follows:

[0123] P(k|k)=[IK(k)H]P(k|k-1)#(15)

[0124] Where P(k|k) is the optimal covariance at time k, and P(k|k-1) is the covariance prediction at time k-1 for time k; I is the identity matrix;

[0125] The estimated value of the Kalman filter is calculated using the recursive steps above.

[0126] The estimated value of the Kalman filter can be calculated using the recursive steps above. In the steps above, I represents the identity matrix. The covariance matrix Q is determined by ΓW(k), and the covariance matrix R is determined by V(k). V(k) represents the observation noise;

[0127] Sliding mode control exhibits strong robustness, enabling the system state to converge to the desired trajectory within a finite time. It also possesses strong parameter adaptive processing capabilities, ensuring that the system does not experience discontinuous switching when parameter uncertainties exist, thus avoiding adverse effects on the system.

[0128] Before constructing the sliding mode controller in step four, the velocity-position error state equation is defined as:

[0129]

[0130] Where e represents the train position error; x represents the train speed error; x represents the actual position of the train obtained after passing through the Kalman filter. r v is the reference position; v is the actual velocity obtained after Kalman filtering algorithm; v r For reference speed;

[0131] Since sliding mode control inevitably exhibits switching phenomena, fractional calculus is introduced to mitigate the resulting chattering. Based on the advantages of fractional calculus in smoothing discontinuous switching, the Caputo form of fractional calculus used in this invention is defined as follows:

[0132] Step four introduces fractional calculus as follows:

[0133]

[0134] Where, d m / dt m This is the traditional differential, where m is the smallest integer not less than fractional a, t is time, and τ is the integration variable; when α < 0, It is a fractional differential, and when α > 0, it is a fractional integral; Γ(x) is the gamma function. m is a fractional integer;

[0135] Fractional sliding mode control enables faster convergence of the error system, improves control accuracy, and makes the control process smoother. The performance of the adjustment process varies under different fractional orders. Based on the actual operating conditions, an appropriate fractional-order calculus operator should be selected to ensure the system meets different dynamic and static performance requirements. Since integer-order derivatives are a special case, fractional-order calculus parameters have a wider range of applicability and greater flexibility compared to integer-order derivatives, resulting in better dynamic processing performance.

[0136] During train tracking operations, it is necessary to achieve accurate tracking of both the reference position and the reference speed curve simultaneously. Therefore, the designed sliding hyperplane needs to incorporate the train position error e. i and train speed error To ensure rapid and synchronous convergence of errors; the sliding surface of the fractional calculus introduced in step four is:

[0137]

[0138] Where λ is the sliding surface gain coefficient, λ>0; e i For train position error, For train speed error, D α-1 It is a fractional operator;

[0139] To achieve online tracking of the train's reference speed and reference position, the sliding mode controller established in step four is as follows:

[0140]

[0141] in, is the estimation term for the basic resistance of the train; k2sgn(S)d is the nonlinear switching control term of the system, used to handle external disturbances and uncertainties, k1 and k2 are the control gains, where k1>0 and k2>0; S is the sliding mode switching function.

[0142] The tracking error obtained by the sliding mode controller is output to the train ATO system until the train reaches its destination.

[0143] This invention employs a Kalman filter algorithm to eliminate various abrupt data, observation noise, and measurement errors at the front end, thereby providing the controller with optimized control data.

[0144] Introducing fractional calculus into the sliding mode switching function further improves the adjustment accuracy and also suppresses the frequency and amplitude of chattering.

[0145] Verification has shown that the Kalman filter speed control algorithm based on fractional sliding mode can effectively overcome the chattering phenomenon caused by sliding mode control and achieve the requirement of high-precision speed control.

[0146] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit and essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0147] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A train speed control method based on fractional sliding mode and Kalman filtering, characterized in that, Includes the following steps: Step 1: Establish a train dynamics model; Step 2: Based on the train dynamics model, establish the train operation state space equations; Step 3: Calibrate the wheel axle speed sensor error using the Kalman filter algorithm; Step 4: Establish a sliding mode controller and introduce fractional calculus to achieve tracking control of the train's reference speed and reference position; The method for calibrating the wheel axle speed sensor error using the Kalman filter algorithm in step three is as follows: Based on the slippage, idling, and measurement errors that occur during train operation, the following error formula is established: in, for Compensation values ​​during periodic train operation; for Speed ​​measurements during periodic train operation include measurement errors. This refers to the actual speed of the train. in, for Error compensation value for periodic train acceleration measurement; for Periodic train acceleration; and The measured value is from the accelerometer. This is the actual acceleration of the train; The following noise formula is established: in, The error is the resultant error of the acceleration and velocity. The speed measurement value includes measurement errors during train operation in cycle k. The velocity value at the next moment; The following observation equation is established: in, This refers to the error disturbance caused by the inherent accuracy error of the wheel axle speed sensor within k periods, i.e., the observation noise. The measured displacement is within period k; This is the actual amount of displacement; The above formula (5) can be expressed in matrix form: in, For the k+1 period velocity state quantity; and B For system parameters; For K-period velocity state variables; For a k-period system control matrix; It is Gaussian noise; Right now: The above formula (6) can be expressed in matrix form: in, For displacement observation; To measure the parameters of the system; To observe noise; Right now: Predict the next system state equation based on the initial input: in, express Always The estimated value at time; The optimal linear prediction estimate; and B For system parameters; The system control matrix for period k-1; for Time-varying covariance matrix; The covariance at this point is: in, This is the prediction of the covariance at time k-1 to time k. The optimal covariance at time k-1; Let be the covariance of the system process at time k-1; The filter gain equation is: in, This is the filter gain; This is the prediction of the covariance at time k-1 to time k. Let H be the transpose of H; The noise is Gaussian at time k; Based on the above formulas (7), (9), (11), (12), and (13), the recursive filtering estimation equation is as follows: in, The optimal velocity estimate at time k; The predicted state at time k-1 is the state predicted for time k. Let be the filter gain at time k; To observe noise; The corresponding covariance is updated as follows: in, Let be the optimal covariance at time k. This is the prediction of the covariance at time k-1 to time k. It is the identity matrix; The estimated value of the Kalman filter is calculated using the recursive steps above. ; In step four, fractional calculus is introduced as follows: （17） in, For differentials in the traditional sense, where Not less than fractional order The smallest integer, For time; For integration variables; when hour, For fractional derivatives, when When it is a fractional integral; For gamma function, ; The fractional order is limited to integers; The sliding surface introduced in step four, which is the fractional calculus, is: in, The gain coefficient of the sliding surface. ; For train position error, For train speed error, It is a fractional operator; The sliding mode controller established in step four is as follows: （19） in, This is an estimate of the train's basic resistance. This is the system's nonlinear switching control term, used to handle external disturbances and uncertainties. , To control the gain, where , ; This is the sliding mode switching function; The tracking error obtained by the sliding mode controller is output to the train ATO system until the train reaches its destination.

2. The train speed control method based on fractional sliding mode and Kalman filtering as described in claim 1, characterized in that, The train power car model in step one is as follows: Wherein: For displacement; For runtime; The real-time speed of the train; The traction / braking force acting on the train; The basic resistance experienced by the train; These are the rolling mechanical resistance coefficient, frictional resistance coefficient, and air resistance coefficient of the train, respectively. This is the actual acceleration of the train; This is the derivative of velocity, i.e., acceleration; This refers to the train acceleration coefficient. This is the wheel rotational mass coefficient of the train.

3. The train speed control method based on fractional sliding mode and Kalman filtering as described in claim 2, characterized in that, Based on the train dynamics model in step one, the following train operation state space equations are established: in, For the quality of the train, For the train's real-time speed, This is the train's real-time location. For the traction / braking force of the train, For the real-time fundamental resistance, This is to mitigate additional resistance and external disturbances. This is the velocity derivative.

4. In the train speed control method based on fractional sliding mode and Kalman filtering as described in claim 1, before constructing the sliding mode controller in step four, the speed-position error state equation is defined as: in, This refers to the train's position error; For train speed error; To obtain the actual position of the train after passing through a Kalman filter, For reference position; The actual speed obtained after applying the Kalman filter algorithm; For reference speed.

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