Bridge crane control method based on singular perturbation and non-singular terminal sliding mode

Abstract

The application provides a bridge crane control method based on singular perturbation and non-singular terminal sliding mode, and belongs to the technical field of bridge crane control; in order to provide a bridge crane control method which can effectively separate fast and slow dynamics, avoid singular problems of sliding mode, has anti-saturation capability and can realize limited time convergence, an eight-state dynamic model of the bridge crane is established based on the Euler-Lagrange principle, and the generalized coordinate vector of the system is defined; a singular perturbation small parameter is introduced, the dynamic model is decomposed into a slow subsystem and a fast subsystem, and artificial damping is injected into the fast subsystem; a non-singular terminal sliding mode surface for position tracking is designed for the slow subsystem, a non-singular terminal sliding mode surface for swing elimination is designed for the fast subsystem, and a sign preserving algorithm is used for power operation; an approaching law combining constant speed approaching law and exponential approaching law is designed, and equivalent control force is calculated; the application is applied to bridge crane control.

Description

Technical Field

[0001] This invention provides a control method for a bridge crane based on singular perturbation and non-singular terminal sliding mode, belonging to the field of bridge crane control technology. Background Technology

[0002] Bridge cranes are widely used material handling equipment in modern industrial production and logistics transportation. The core control problem is to achieve rapid and accurate positioning of the load while effectively suppressing load sway. Although traditional control methods such as PID control are simple in structure, they are difficult to meet the control requirements of rapid positioning and sway suppression when faced with the nonlinear and strongly coupled characteristics of bridge crane systems.

[0003] Sliding mode control, as a robust control method, is fully robust to system uncertainties and external disturbances and has been widely used in bridge crane control. However, traditional sliding mode control methods have inherent chattering phenomena, and when the system state approaches the sliding surface, singular problems may occur, leading to a decrease in control accuracy or even system instability.

[0004] Existing control methods for bridge cranes fail to effectively separate the rapidly changing dynamics (load swaying) and slowly changing dynamics (trolley position movement) in the bridge crane system, resulting in complex controller design, difficult parameter tuning, and difficulty in handling the coupled effects of system characteristics at different time scales. In addition, in practical applications, the actuator has physical saturation characteristics. When the control input exceeds the actuator limit, traditional controllers will exhibit integral saturation, leading to severe degradation of control performance and even system instability. Meanwhile, although existing terminal sliding mode control methods can achieve finite-time convergence, they are prone to complex or singular cases in the exponential operations of state variables, limiting their application in practical engineering.

[0005] Therefore, there is an urgent need to develop a control method for bridge cranes that can effectively separate fast and slow dynamics, avoid sliding mode singularity problems, have anti-saturation capabilities, and achieve finite-time convergence, which has significant engineering application value. Summary of the Invention

[0006] To address the technical problems existing in the background art, the present invention provides a control method for a bridge crane based on singular perturbation and non-singular terminal sliding mode, comprising the following control steps:

[0007] S1. Based on the Euler-Lagrange principle, establish an eight-state dynamic model of the bridge crane and define the generalized coordinate vector of the system;

[0008] S2. Introduce singular perturbation parameters to decompose the dynamic model into a slow-varying subsystem and a fast-varying subsystem, and inject artificial damping into the fast-varying subsystem;

[0009] S3. Design a non-singular terminal sliding surface for position tracking for the slow-varying subsystem, and a non-singular terminal sliding surface for oscillation elimination for the fast-varying subsystem. Use a sign-preserving algorithm for exponentiation.

[0010] S4. Design a convergence law that combines the constant-rate convergence law and the exponential convergence law, calculate the equivalent control force, and obtain the total control law by weighted combination through coupling coefficients.

[0011] S5. Design an anti-saturation compensator to detect saturation error in real time and update the anti-saturation compensation state. When saturation is detected, conditionally reset the integral term in the sliding surface.

[0012] S6. Numerical stability is enhanced by employing power operation protection, singularity detection and pseudo-inverse processing, acceleration limiting, and NaN / Inf detection measures.

[0013] The specific method for step S1 is as follows:

[0014] The generalized coordinate vector is defined as:

[0015] ;

[0016] in, and The car is in direction and The position of direction, Indicates the suspension rope is in Projection on a plane and The angle in the negative direction of the axis, Indicates the suspension rope and Angle between planes;

[0017] Based on the Euler-Lagrange principle, the dynamic model of the system in matrix form is obtained, and the expression is:

[0018] ;

[0019] In the formula, It is a symmetric inertia matrix; The centripetal-Cole force matrix; It is the gravity vector; The input matrix; The driving force vector; Let the friction force vector be... For generalized coordinate vectors, The first time derivative of the generalized coordinate system is the velocity. The second time derivative of the generalized coordinates is the acceleration.

[0020] The specific method for step S2 is as follows:

[0021] Introducing singular perturbation small parameters Defined as:

[0022] ;

[0023] in, This refers to the length of the hoisting rope. It is the acceleration due to gravity;

[0024] Converting the dynamic model to the singular perturbation canonical form yields the dynamic model of the slowly varying subsystem, expressed as:

[0025] ;

[0026] in, For the quality of the cable tray, For the mass of the car, For load quality, , For slow-varying subsystems direction and The first-order time derivative of the orientation, position, and state is velocity. , For slow-varying subsystems direction and The second time derivative of the orientation, position, and state is acceleration. , This is the averaged term for the anti-pendulum reaction force. , The control force after anti-saturation compensation. , The damping coefficient;

[0027] Introducing singular perturbation small parameters Subsequently, the dynamic model of the rapidly changing subsystem was derived as follows:

[0028] ;

[0029] in, The artificial damping ratio, , To quickly change control, This represents the small parameter of the singular perturbation. , The loads are respectively , The sway angle of the direction, , The loads are respectively , Angular velocity in the direction, , The loads are respectively , Angular acceleration in the direction;

[0030] When the slow-varying forcing term is ignored, the dynamic model of the fast-varying subsystem is simplified to:

[0031] .

[0032] The specific method for step S3 is as follows:

[0033] The tracking error is defined as follows:

[0034] , , , ;

[0035] in, , For the desired position, for Orientation and position tracking error, for Orientation and position tracking error, for Directional load swing angle error, for Directional load swing angle error;

[0036] For the design of a non-singular terminal sliding surface for a slowly varying subsystem, the expression is:

[0037] ;

[0038] ;

[0039] in, for Directional velocity tracking error; for Directional velocity tracking error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain;

[0040] For the design of a non-singular terminal sliding surface for a fast-changing subsystem, the expression is:

[0041] ;

[0042] ;

[0043] in, for Directional angular velocity error; for Directional angular velocity error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain;

[0044] The symbol preservation algorithm Defined as:

[0045] ;

[0046] in, For the defined sign-preserving algorithm, this algorithm ensures that when the base is... Negative and exponent When the value is a fraction, the calculation result remains a real number.

[0047] The specific method for step S4 is as follows:

[0048] The design convergence law is expressed as follows:

[0049] ;

[0050] ;

[0051] ;

[0052] ;

[0053] in, , , , This is a constant-rate approach gain; , , , For exponentially approaching gain; , , , Boundary layer thickness; , , , This is a system convergence law to ensure system convergence; , , , For the design of non-singular terminal sliding surfaces; It is a saturation function;

[0054] The saturation function is defined as:

[0055] ;

[0056] in, , For sliding surface variables; Boundary layer thickness; It is a symbolic function;

[0057] Calculate equivalent control force:

[0058] Based on the principle that the derivative of the sliding surface equals the reaching law, the slowly varying equivalent control force can be calculated using the following formula:

[0059] ;

[0060] ;

[0061] in, , for , Slowly changing direction equivalent control force; , Inertia matrix The middle corresponds to , Elements of direction; , for , Expected acceleration in the direction; , for , Directional acceleration tracking error; , The model compensation terms include Coriolis force, gravitational components, friction, and pendulum coupling terms, which counteract known nonlinear dynamics.

[0062] The fast-changing equivalent control force is calculated using the following formula:

[0063] ;

[0064] ;

[0065] in , , , For rapidly changing state variables, they are defined as follows: , , , ;

[0066] By weighting and combining the slow-changing equivalent control force with the fast-changing equivalent control force through coupling coefficients, the overall control law is obtained, expressed as:

[0067] ;

[0068] ;

[0069] in, , for , The overall control law of direction, For singular perturbation coupling coefficients.

[0070] The specific method for designing the anti-saturation compensator in step S5 is as follows:

[0071] Define the saturation error function as:

[0072] ;

[0073] ;

[0074] in, , for , Directional saturation error function; , The desired control force calculated for the controller; , The actual control force output by the actuator satisfies the amplitude limiting constraint. , , , for , The maximum permissible output force of the directional actuator;

[0075] Establish an anti-saturation compensation dynamic system, with the following expression:

[0076] ;

[0077] ;

[0078] in, , for , Directional anti-saturation compensation state; , for , The first derivative of the directional anti-saturation compensation state; For anti-saturation feedback gain;

[0079] The expression for the compensated control quantity is:

[0080] ;

[0081] ;

[0082] in, , for , Control quantity after directional compensation;

[0083] Define a saturation flag, expressed as:

[0084] ;

[0085] in, A saturation flag is defined.

[0086] When saturation occurs The integral term in the sliding surface decays exponentially, and its expression is:

[0087] ;

[0088] in, For the next moment The value of the integral term; For time; For the current moment The value of the integral term; It is an exponential decay factor; The attenuation coefficient controls how quickly the integral term is reset.

[0089] The specific method for step S6 is as follows:

[0090] The following numerical stability enhancement measures are adopted:

[0091] In progress When setting a protection threshold ,when season:

[0092] ;

[0093] in, To protect the threshold;

[0094] The specific method for using singularity detection and pseudo-inverse processing is as follows:

[0095] Real-time calculation of matrix condition number ,when The Moore-Penrose pseudo-inverse is used to solve the problem, where ;

[0096] The calculated acceleration is limited, and the expression is:

[0097] ;

[0098] ;

[0099] Among them, , , , .

[0100] The advantages of this invention compared to existing technologies are as follows: This invention provides a bridge crane control method based on singular perturbation and non-singular terminal sliding mode. By decomposing the complex coupled system into two subsystems, fast and slow, the complexity of controller design is greatly reduced, and the physical meaning of parameters is clear. This invention adopts non-singular terminal sliding mode control, which fundamentally avoids the singularity problem of traditional terminal sliding mode and achieves finite-time convergence of the system state. This invention introduces an observer-based anti-saturation compensation mechanism, which effectively solves the integral saturation problem when the actuator is saturated through saturation error feedback and integrator condition reset, and designs complete numerical stability enhancement measures to ensure reliable operation of the algorithm under various working conditions. Attached Figure Description

[0101] The present invention will be further described below with reference to the accompanying drawings:

[0102] Figure 1 This is a flowchart of the steps of the bridge crane control method based on singular perturbation and non-singular terminal sliding mode of the present invention;

[0103] Figure 2 This is a simplified three-dimensional model schematic diagram of the bridge crane system of the present invention;

[0104] Figure 3 This is a diagram showing the position response curve of the cable tray in an embodiment of the present invention;

[0105] Figure 4 This is a diagram showing the position response curve of the vehicle in an embodiment of the present invention;

[0106] Figure 5 This is a graph showing the load's swing angle response in the X direction in an embodiment of the present invention.

[0107] Figure 6 This is a graph showing the load's swing angle response in the Y direction in an embodiment of the present invention. Detailed Implementation

[0108] like Figure 1 As shown, the present invention provides a control method for a bridge crane based on singular perturbation and non-singular terminal sliding mode, comprising the following control steps:

[0109] S1. Based on the Euler-Lagrange principle, establish an eight-state dynamic model of the bridge crane and define the generalized coordinate vector of the system;

[0110] S2. Introduce singular perturbation parameters to decompose the dynamic model into a slow-varying subsystem and a fast-varying subsystem, and inject artificial damping into the fast-varying subsystem;

[0111] S3. Design a non-singular terminal sliding surface for position tracking for the slow-varying subsystem, and a non-singular terminal sliding surface for oscillation elimination for the fast-varying subsystem. Use a sign-preserving algorithm for exponentiation.

[0112] S4. Design a convergence law that combines the constant-rate convergence law and the exponential convergence law, calculate the equivalent control force, and obtain the total control law by weighted combination through coupling coefficients.

[0113] S5. Design an anti-saturation compensator to detect saturation error in real time and update the anti-saturation compensation state. When saturation is detected, conditionally reset the integral term in the sliding surface.

[0114] S6. Numerical stability is enhanced by employing power operation protection, singularity detection and pseudo-inverse processing, acceleration limiting, and NaN / Inf detection measures.

[0115] In step S1, as Figure 2 As shown, the bridge crane system of the present invention includes a bridge frame (trolley), a trolley, lifting ropes, and a load. Among them, The quality of the bridge frame, representing the bridge crane system, For the mass of the car, For the quality of the hoisting rope, For load quality, This refers to the length of the suspension rope. The axis is the direction of movement of the cable tray. The axis is the direction of the trolley's movement. The axis indicates the direction of load lifting and lowering. Indicates the suspension rope is in Projection on a plane and The angle in the negative direction of the axis, Indicates the suspension rope and Angle between two planes.

[0116] The generalized coordinate vector of the system is defined as follows:

[0117] ;

[0118] In the formula, , The car is in direction and The position of direction; , The loads are respectively direction and The directional swing angle.

[0119] Based on the Euler-Lagrange principle, the dynamic equations of the system in matrix form can be obtained:

[0120] ;

[0121] In the formula, It is a symmetric inertia matrix; The centripetal-Cole force matrix; It is the gravity vector; The input matrix; The driving force vector; This is the friction force vector; It is a generalized coordinate vector; The first time derivative of the generalized coordinates, i.e., velocity; Let be the second-order time derivative of the generalized coordinates, i.e., acceleration; the specific expressions for each matrix and vector are:

[0122] ;

[0123] ;

[0124] ;

[0125] ;

[0126] ;

[0127] ;

[0128] in:

[0129] ;

[0130] ;

[0131] , Respectively represent and , The coefficient of friction related to motion, , is the equivalent damping coefficient for the swing angle.

[0132] The system parameter settings are shown in the table below:

[0133]

[0134] In step S2, a singular perturbation small parameter is introduced. Defined as:

[0135] ;

[0136] In the formula, For singular perturbation small parameters; This refers to the length of the suspension rope; It is the acceleration due to gravity; The value directly reflects the magnitude of the load oscillation period. The smaller, The smaller the value, the faster the oscillation, the more obvious the time scale separation of the fast and slow subsystems, and the better the applicability of the singular perturbation method.

[0137] In order to apply singular perturbation theory, the original dynamic model is transformed to a "slow time scale" and the fast dynamics are explicitly separated. Through mathematical derivation, the dynamic model of the fast-changing subsystem is obtained:

[0138] ;

[0139] in, The artificial damping ratio, , To quickly change control, This represents the small parameter of the singular perturbation. , The loads are respectively , The sway angle of the direction, , The loads are respectively , Angular velocity in the direction, , The loads are respectively , Angular acceleration in the direction.

[0140] When the slowly varying forced terms are ignored, it simplifies to:

[0141] ;

[0142] The ideal boundary layer system derived above is undamped. To ensure the asymptotic stability of the rapidly changing subsystem and actively suppress load oscillations, this embodiment injects artificial damping into the rapidly changing subsystem. After introducing artificial damping, the dynamics of the rapidly changing subsystem become a second-order system with the desired damping characteristics, enabling rapid convergence.

[0143] Converting the dynamic model to the singular perturbation canonical form yields the dynamic model of the slowly varying subsystem, expressed as:

[0144] ;

[0145] in, For the quality of the cable tray, For the mass of the car, For load quality, , For slow-varying subsystems direction and The first time derivative of the orientation, position, and state (velocity). , For slow-varying subsystems direction and The second time derivative (acceleration) of the orientation, position, and state. , This is the averaged term for the anti-pendulum reaction force. , The control force after anti-saturation compensation. , is the damping coefficient.

[0146] According to Tikhonov's singular perturbation theorem, for sufficiently small The true trajectory of the entire original system can be approximated as the trajectory of the slowly varying subsystem plus a boundary layer (fast-changing) correction term. This theoretical basis guarantees the effectiveness of the combined control law.

[0147] ;

[0148] in, , They are respectively direction and The overall control law (overall control force) for direction; , The equivalent control force of the slow-varying subsystem (position tracking) guides the position tracking of the vehicle and ensures positioning accuracy; , To provide additional anti-sway torque for the equivalent control force of the fast-changing subsystem (anti-sway), thus suppressing load sway; This is the singular perturbation coupling coefficient, ranging from 0.1 to 0.5, used for priority in balancing positioning and eliminating sway. When load sway is suppressed ( , The system degenerates into a purely slow-varying subsystem control problem, achieving smooth and accurate positioning.

[0149] In step S3, the tracking error is defined as follows:

[0150] , , , ;

[0151] in, for Orientation and position tracking error; for Orientation and position tracking error; for Directional load swing angle error; for Directional load swing angle error; , The desired position.

[0152] For the design of a non-singular terminal sliding surface for a slowly varying subsystem, the expression is:

[0153] ;

[0154] ;

[0155] in, for Directional velocity tracking error; for Directional velocity tracking error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain.

[0156] For the design of a non-singular terminal sliding surface for a fast-changing subsystem, the expression is:

[0157] ;

[0158] ;

[0159] in, for Directional angular velocity error; for Directional angular velocity error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain.

[0160] The symbol preservation algorithm Defined as:

[0161] ;

[0162] in, For the defined sign-preserving algorithm; For input variables; It is a power exponent; this algorithm ensures that when the base is... Negative and exponent When the value is a fraction, the calculation result remains a real number.

[0163] In exponentiation, when the base is negative, the traditional calculation method... Complex results may be generated. This invention employs a sign-preserving algorithm to ensure that the calculation result is always a real number. Numerical protection: When (Pick When ), directly order To avoid numerical sensitivity issues.

[0164] In step S4, the approach law is designed as follows:

[0165] ;

[0166] ;

[0167] ;

[0168] ;

[0169] in, , , , This is a constant-rate approach gain; , , , For exponentially approaching gain; , , , Boundary layer thickness; , , , This is a system convergence law to ensure system convergence; , , , For the design of non-singular terminal sliding surfaces; It is a saturation function.

[0170] The saturation function is defined as:

[0171] ;

[0172] in, , For sliding surface variables; Boundary layer thickness; It is a symbolic function.

[0173] Equivalent control refers to the control quantity required to maintain the system state when the system state is located on the sliding surface. According to sliding mode control theory, let... The equivalent control can be obtained.

[0174] Calculate equivalent control force:

[0175] By applying the principle that the derivative of the sliding surface equals the reaching law, the slowly varying equivalent control force can be obtained:

[0176] ;

[0177] ;

[0178] in, , for , Slowly changing direction equivalent control force; , Inertia matrix The middle corresponds to , Elements of direction; , for , Expected acceleration in the direction; , for , Directional acceleration tracking error; , The model compensation terms include Coriolis force, gravitational components, friction, and pendulum coupling terms, which counteract known nonlinear dynamics.

[0179] The solution quickly becomes equivalent to control force:

[0180] ;

[0181] ;

[0182] in , , , For rapidly changing state variables, they are defined as follows: , , , .

[0183] By weighting and combining the slow-changing equivalent control force with the fast-changing equivalent control force through coupling coefficients, the overall control law is obtained, expressed as:

[0184] ;

[0185] ;

[0186] in, , for , The overall control law of direction, For singular perturbation coupling coefficients.

[0187] Define Lyapunov functions Differentiate it:

[0188] ;

[0189] Scenario 1: When hour, ,but:

[0190] ;

[0191] Scenario 2: When hour, ,but:

[0192] .

[0193] In conclusion, The condition holds true consistently, meaning the system state approaches the sliding surface within a finite time and remains bounded within the boundary layer. Due to the use of a non-singular terminal sliding surface, the system's tracking error will converge to zero within a finite time, with the upper bound of the convergence time being:

[0194] ;

[0195] In step S5, the anti-saturation compensator is designed as follows:

[0196] Define the saturation error function as:

[0197] ;

[0198] ;

[0199] in, , for , Directional saturation error function; , The desired control force calculated for the controller; , The actual control force output by the actuator satisfies the amplitude limiting constraint. , , , for , The maximum permissible output force of the directional actuator (drive unit).

[0200] The actual control quantity after actuator saturation is:

[0201] ;

[0202] ;

[0203] Establish an anti-saturation compensation dynamic system, with the following expression:

[0204] ;

[0205] ;

[0206] in, , for , Directional anti-saturation compensation state; , for , The first derivative of the directional anti-saturation compensation state; This is for anti-saturation feedback gain.

[0207] The expression for the compensated control quantity is:

[0208] ;

[0209] ;

[0210] in, , for , Control quantity after directional compensation.

[0211] The saturation flag is defined as:

[0212] ;

[0213] in, The saturation flag is defined.

[0214] When saturation occurs The integral term in the sliding surface decays exponentially, and its expression is:

[0215] ;

[0216] in, For the next moment The value of the integral term; For time; For the current moment The value of the integral term; It is an exponential decay factor; The attenuation coefficient controls how quickly the integral term is reset.

[0217] When saturation disappears ( The integrator returns to normal integration mode, and the expression is:

[0218] ;

[0219] In step S6, the following numerical stability enhancement measures are adopted:

[0220] In progress When setting a protection threshold ,when season:

[0221] ;

[0222] in To protect the threshold.

[0223] The specific method for using singularity detection and pseudo-inverse processing is as follows:

[0224] Real-time calculation of matrix condition number ,when The Moore-Penrose pseudo-inverse is used to solve the problem, where ;

[0225] The calculated acceleration is limited, and the expression is:

[0226] ;

[0227] ;

[0228] in, , , , .

[0229] In an embodiment of the present invention, in order to verify the effectiveness of the designed anti-saturation controller based on singular perturbation and non-singular terminal sliding mode, simulation is performed using theoretical and numerical methods based on the parameters of an actual bridge crane platform, which can verify the feasibility of the proposed control scheme from a kinematic perspective.

[0230] The simulation environment used was MATLAB / Simulink, the solver was ode45, the simulation duration was 20 seconds, and the desired location was set to... , .

[0231] The simulation results of the car's position are as follows: Figure 3 , Figure 4 , Figure 5 , Figure 6 As shown. By Figure 3 It can be seen that the X direction enters the 2% error band in about 5 seconds. ),Depend on Figure 4 It can be seen that the Y-direction enters the 2% error band in about 4 seconds, and the steady-state error is less than 0.002m. This indicates that the designed controller can effectively track the predetermined target and has good dynamic response characteristics.

[0232] Figure 5 , Figure 6 Displays the load swing angle and The graph shows the changes in the swing angle. As can be seen from the graph, the maximum swing angle occurs during the initial stage. , After the system reaches steady state, there is no residual sway angle, indicating that the controller has good anti-sway performance.

[0233] Simulation results demonstrate that the proposed method can effectively suppress load swing angle while achieving positioning, and the system exhibits good anti-interference capability and robustness, providing an effective solution for the control problem of this type of underactuated system.

[0234] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (devices), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0235] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0236] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0237] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A control method for a bridge crane based on singular perturbation and non-singular terminal sliding mode, characterized in that: The control steps include the following: S1. Based on the Euler-Lagrange principle, establish an eight-state dynamic model of the bridge crane and define the generalized coordinate vector of the system; S2. Introduce singular perturbation parameters to decompose the dynamic model into a slow-varying subsystem and a fast-varying subsystem, and inject artificial damping into the fast-varying subsystem; S3. Design a non-singular terminal sliding surface for position tracking for the slow-varying subsystem, and a non-singular terminal sliding surface for oscillation elimination for the fast-varying subsystem. Use a sign-preserving algorithm for exponentiation. S4. Design a convergence law that combines the constant-rate convergence law and the exponential convergence law, calculate the equivalent control force, and obtain the total control law by weighted combination through coupling coefficients. S5. Design an anti-saturation compensator to detect saturation error in real time and update the anti-saturation compensation state. When saturation is detected, conditionally reset the integral term in the sliding surface. S6. Numerical stability is enhanced by employing power operation protection, singularity detection and pseudo-inverse processing, acceleration limiting, and NaN / Inf detection measures.

2. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 1, characterized in that: The specific method for step S1 is as follows: The generalized coordinate vector is defined as: ； in, and The car is in direction and The position of direction, Indicates the suspension rope is in Projection on a plane and The angle in the negative direction of the axis, Indicates the suspension rope and Angle between planes; Based on the Euler-Lagrange principle, the dynamic model of the system in matrix form is obtained, and the expression is: ； In the formula, It is a symmetric inertia matrix; The centripetal-Cole force matrix; It is the gravity vector; The input matrix; The driving force vector; Let the friction force vector be... For generalized coordinate vectors, The first time derivative of the generalized coordinate system is the velocity. The second time derivative of the generalized coordinates is the acceleration.

3. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 2, characterized in that: The specific method for step S2 is as follows: Introducing singular perturbation small parameters Defined as: ； in, This refers to the length of the hoisting rope. It is the acceleration due to gravity; Converting the dynamic model to the singular perturbation canonical form yields the dynamic model of the slowly varying subsystem, expressed as: ； in, For the quality of the cable tray, For the mass of the car, For load quality, , For slow-varying subsystems direction and The first-order time derivative of the orientation, position, and state is velocity. , For slow-varying subsystems direction and The second time derivative of the orientation, position, and state is acceleration. , This is the averaged term for the anti-pendulum reaction force. , The control force after anti-saturation compensation. , The damping coefficient; Introducing singular perturbation small parameters Subsequently, the dynamic model of the rapidly changing subsystem was derived as follows: ； in, The artificial damping ratio, , To quickly change control, This represents the small parameter of the singular perturbation. , The loads are respectively , The sway angle of the direction, , The loads are respectively , Angular velocity in the direction, , The loads are respectively , Angular acceleration in the direction; When the slow-varying forcing term is ignored, the dynamic model of the fast-varying subsystem is simplified to: 。 4. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 3, characterized in that: The specific method for step S3 is as follows: The tracking error is defined as follows: ， ， ， ； in, , For the desired position, for Orientation and position tracking error, for Orientation and position tracking error, for Directional load swing angle error, for Directional load swing angle error; For the design of a non-singular terminal sliding surface for a slowly varying subsystem, the expression is: ； ； in, for Directional velocity tracking error; for Directional velocity tracking error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain; For the design of a non-singular terminal sliding surface for a fast-changing subsystem, the expression is: ； ； in, for Directional angular velocity error; for Directional angular velocity error; , The sliding surface coefficient; , For power-order parameters, satisfying ,and ,in , It is a positive odd number; , This is the integral gain; The symbol preservation algorithm Defined as: ； in, For the defined sign-preserving algorithm, this algorithm ensures that when the base is... Negative and exponent When the value is a fraction, the calculation result remains a real number.

5. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 4, characterized in that: The specific method for step S4 is as follows: The design convergence law is expressed as follows: ； ； ； ； in, , , , This is a constant-rate approach gain; , , , For exponentially approaching gain; , , , Boundary layer thickness; , , , This is a system convergence law to ensure system convergence; , , , For the design of non-singular terminal sliding surfaces; It is a saturation function; The saturation function is defined as: ； in, , For sliding surface variables; Boundary layer thickness; It is a symbolic function; Calculate equivalent control force: Based on the principle that the derivative of the sliding surface equals the reaching law, the slowly varying equivalent control force can be calculated using the following formula: ； ； in, , for , Slowly changing direction equivalent control force; , Inertia matrix The middle corresponds to , Elements of direction; , for , Expected acceleration in the direction; , for , Directional acceleration tracking error; , The model compensation terms include Coriolis force, gravitational components, friction, and pendulum coupling terms, which counteract known nonlinear dynamics. The fast-changing equivalent control force is calculated using the following formula: ； ； in , , , For rapidly changing state variables, define as follows: , , , ; By weighting and combining the slow-changing equivalent control force with the fast-changing equivalent control force through coupling coefficients, the overall control law is obtained, expressed as: ； ； in, , for , The overall control law of direction, For singular perturbation coupling coefficients.

6. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 5, characterized in that: The specific method for designing the anti-saturation compensator in step S5 is as follows: The saturation error function is defined as follows: ； ； in, , for , Directional saturation error function; , The desired control force calculated for the controller; , The actual control force output by the actuator satisfies the amplitude limiting constraint. , , , for , The maximum permissible output force of the directional actuator; Establish an anti-saturation compensation dynamic system, with the following expression: ； ； in, , for , Directional anti-saturation compensation state; , for , The first derivative of the directional anti-saturation compensation state; For anti-saturation feedback gain; The expression for the compensated control quantity is: ； ； in, , for , Control quantity after directional compensation; Define a saturation flag, expressed as: ； in, A saturation flag is defined. When saturation occurs The integral term in the sliding surface decays exponentially, and its expression is: ； in, For the next moment The value of the integral term; For time; For the current moment The value of the integral term; It is an exponential decay factor; The attenuation coefficient controls how quickly the integral term is reset.

7. The bridge crane control method based on singular perturbation and non-singular terminal sliding mode according to claim 6, characterized in that: The specific method for step S6 is as follows: The following numerical stability enhancement measures are adopted: In progress When setting a protection threshold ,when season: ； in, To protect the threshold; The specific method for using singularity detection and pseudo-inverse processing is as follows: Real-time calculation of matrix condition number ,when The Moore-Penrose pseudo-inverse is used to solve the problem, where ; The calculated acceleration is limited, and the expression is: ； ； in, , , , .

Images