A vibration suppression method and system for a flexible crane system based on a neural network

Abstract

The present invention discloses a vibration suppression method and control system for a flexible crane system, belonging to the technical field of position control and vibration suppression of flexible crane systems. The present invention considers the influence of unknown friction and output constraints in the actual system on the vibration control of the flexible crane system, uses the boundary state signals obtained by sensors, adopts neural networks to estimate the unknown friction of the system, and combines the barrier Lyapunov function method to design a boundary controller. This boundary controller solves the influence brought by the unknown friction to the system and confines the boundary curvature of the flexible rope within a given value, ensuring the safety of the system, thereby effectively suppressing the vibration of the flexible crane system and ensuring the stability of the crane system. Therefore, the vibration suppression method and system of the flexible crane system of the present invention can achieve the position control and vibration suppression control objectives under the conditions that the system contains unknown friction and output constraints.

Description

Technical Field

[0001] The present invention relates to a vibration suppression method and system for a flexible crane system based on a neural network, and belongs to the technical field of position control and vibration suppression of flexible crane systems. Background Art

[0002] With the rapid development of China's social economy, the application of crane systems in industries such as manufacturing, construction sites, ports, and marine industries is becoming increasingly widespread. The reason is that cranes can transport heavy or dangerous goods, which can not only greatly reduce labor but also effectively improve production efficiency.

[0003] A flexible crane system mainly consists of a top trolley, a flexible rope, and a bottom load. Compared with traditional rigid ropes, flexible rope materials have the advantages of light weight, low energy consumption, and strong shock absorption ability. However, this flexible characteristic of the rope easily causes vibrations in the system during transportation. Without effective measures to suppress such vibrations, the magnitude of the force on the rope will fluctuate to a certain extent, resulting in a series of problems such as the system being unable to complete precise position control, fatigue damage to the rope, and degradation of system performance. Therefore, suppressing the vibration problem of flexible crane systems is very important. In addition, a flexible crane system is a type of flexible mechanical system, and this system is usually described by a set of partial differential equations. In the control problem of partial differential equations, the infinite-dimensional characteristics of the system also bring certain difficulties to the design of the controller.

[0004] In the field of industrial control, boundary control only needs to set sensors and actuators at the boundaries of the system, and realizes the vibration suppression of the system through the action of the boundary controller. Therefore, the boundary control method is an easy-to-implement control method.

[0005] Since the top trolley of the flexible crane system moves left and right along the horizontal guide rail, there is friction between the trolley and the guide rail, and this friction is generally unknown and difficult to measure. In addition, at the connection point between the trolley end and the flexible rope, the flexible rope is prone to wear. Constraining the boundary curvature of the flexible rope within a given range can effectively avoid fatigue damage to the rope.

[0006] To solve the above problems, Entessari et al. proposed three boundary controllers acting on the trolley end, flexible cable, and bottom load respectively to suppress the vibration of the system (Entessari F., Ardekany A.N., Alasty A., Exponential stabilization of flexural sway vibration of gantry crane via boundary control method, Journal of Vibration and Control 26(1-2)(2020)36–55.doi:10.1177 / 1077546319876147.). However, this method using three boundary controllers is more complex and difficult to implement than the method using only one boundary controller. Therefore, d'AndreHa-Novel et al. studied a boundary controller acting only on the trolley end for the vibration control problem of the flexible crane system (d’Andréa Novel B., Coron J.M., Exponential stabilization of an over-head crane with flexible cable via a back-stepping approach, Automatica 36(4)(2000)587–593.doi:https: / / doi.org / 10.1016 / S0005-1098(99)00182-X.). However, the above schemes do not consider the influence of unknown friction forces in actual application scenarios and the constraint problem of the maximum value that the boundary curvature of the flexible cable in the flexible crane system can reach due to safety requirements, resulting in a great impact on the stability and position control accuracy of the flexible crane. Summary of the Invention

[0007] To solve the problem that the current flexible crane system is unstable or the position control is inaccurate, the present invention provides a vibration suppression method and system for a flexible crane system based on a neural network, and the solution is as follows:

[0008] The first object of the present invention is to provide a vibration suppression method for a flexible crane system based on a neural network, and the method includes:

[0009] Step 1: Model the flexible crane system containing unknown friction forces by using Hamilton's principle;

[0010] Step 2: Obtain the boundary state signal of the flexible crane system by using a boundary sensor;

[0011] Step 3: Estimate the unknown friction force in the flexible crane system using a neural network;

[0012] Step 4: Construct a boundary controller for the flexible crane system based on the boundary state signal obtained in Step 2 and the estimated value of the unknown friction force obtained in Step 3, in combination with a barrier Lyapunov function;

[0013] Step 5: Construct a Lyapunov function, prove the positive definiteness of the Lyapunov function, and perform a stability analysis of the control system under the boundary controller in Step 4;

[0014] Step 6: Verify whether the state variables of the control system in Step 5 are uniformly bounded. If they are, execute Step 7; if not, a new Lyapunov function and boundary controller need to be constructed;

[0015] Step 7: The boundary controller calculates a control signal based on the obtained boundary state signal and sends the control signal to the actuator;

[0016] Step 8: The actuator receives the control signal and applies the control signal to the flexible crane system. Finally, the crane system achieves the position control and vibration suppression control objectives.

[0017] Optionally, the main control equation of the flexible crane system modeled in Step 1 is:

[0018]

[0019] The boundary conditions of the flexible crane system are:

[0020]

[0021] where y(x,t) is the lateral position of the flexible rope at spatial position x and time t, ρ and l respectively represent the mass per unit length of the flexible rope and the length of the rope, M represents the mass of the trolley, y x (x,t) and y t (x,t) respectively represent the first-order partial derivatives of the lateral position y(x,t) of the flexible rope with respect to the spatial variable x and the time variable t, y tt (x,t) represents the second-order partial derivative of the lateral position y(x,t) of the flexible rope with respect to the time variable t, u(t) represents the boundary controller, F(t) is the unknown friction force between the trolley and the guide rail, T(x) = g[m + ρ(l - x)] is the tension of the rope at spatial position x, where m is the mass of the bottom load and g is the acceleration due to gravity.

[0022] Optionally, in step three, a radial basis function neural network (RBFNN) is used to estimate the unknown friction force in the flexible crane system. The output of the radial basis function neural network is:

[0023]

[0024] where χ = y t (0, t) is the input of the neural network, representing the speed of the moving trolley, is the activation function of the neural network, which is a Gaussian function, is the estimated value of the ideal weight W * ; is the estimated value of the unknown friction force F(t);

[0025] The weight update formula of the radial basis function neural network is:

[0026]

[0027] where γ, β, and τ are all constants greater than 0, and θ(t) is an auxiliary variable, expressed as θ(t) = k c (y(0, t) - y d ) - k d y x (0, t) + y t (0, t), y d is the desired position, k c and k d are both constants greater than 0, C > 0 is the constraint value of the boundary curvature y x (0, t) of the flexible rope, and the constraint condition is satisfied as |y x (0, t)| < C.

[0028] Optionally, the boundary controller in step four is:

[0029]

[0030] where k a and k b are both boundary controller gains greater than 0, y x (0, t) represents the boundary curvature of the flexible rope, and y xt (0, t) represents the boundary angular velocity of the flexible rope.

[0031] Optionally, the process of performing the stability analysis of the control system in step five includes:

[0032] Step 5-1: Construct a Lyapunov function, and its expression is:

[0033] E(t) = E a (t) + Eb (t) + E c (t) + E d (t)

[0034] Among them,

[0035]

[0036]

[0037]

[0038]

[0039] Among them, α is a positive constant, is the weight error term, E a (t) is the energy term of the flexible crane system, including kinetic energy and potential energy, E b (t) is the barrier Lyapunov term related to the auxiliary variable θ(t), E c (t) is the energy term related to friction, E d (t) is the signal y x (x, t)y t (x, t) and the cross - term of the cross - multiplication of the signal (y(x, t) - y d )y t (x, t);

[0040] Step 5 - 2: Verify the positive definiteness of the constructed Lyapunov function, that is, E(t) > 0, and obtain:

[0041] 0 ≤ (1 - η)[E a (t) + E b (t) + E c (t)] ≤ E(t) ≤ (1 + η)[E a (t) + E b (t) + E c (t)]

[0042] Among them,

[0043] Step 5 - 3: Conduct the stability analysis of the said control system, take the first - order derivative of the said Lyapunov function with respect to time t, substitute the said boundary conditions and the said boundary controller, and obtain:

[0044]

[0045] Among them,

[0046] Step 5-4: Further derivation shows that the lateral offset ω(x,t) = y(x,t) - y(0,t) of the flexible crane system rope satisfies:

[0047]

[0048] The above equation indicates that the lateral offset ω(x,t) of the flexible crane system rope is uniformly bounded;

[0049] In addition, since y(x,t) = ω(x,t) + y(0,t), the state y(0,t) satisfies:

[0050]

[0051] That is, when t → ∞, y(x,t) → y d , the above equation indicates that position control is finally achieved.

[0052] Optionally, the boundary state signals of the flexible crane system include: the position y(0,t) of the moving trolley, the speed y t (0,t) of the moving trolley, the boundary curvature y x (0,t) of the flexible rope, and the boundary angular velocity y xt (0,t) of the flexible rope. Among them, the position y(0,t) of the moving trolley and the speed y t (0,t) of the moving trolley are measured by the motor encoder at the trolley end, and the boundary curvature y x (0,t) of the flexible rope is measured by an inclinometer, and the boundary angular velocity y xt (0,t) of the flexible rope is calculated by using the backward difference method based on the measured y x (0,t).

[0053] The second object of the present invention is to provide a vibration suppression system for a flexible crane system based on a neural network. The system includes:

[0054] A boundary sensor for acquiring the boundary state signals of the flexible crane system;

[0055] A boundary controller that calculates the flexible crane control signal by using a vibration suppression method for a flexible crane system based on a neural network as described in any one of the above;

[0056] An actuator for applying the received flexible crane control signal to the controlled flexible crane system to control the operation of the flexible crane.

[0057] Optionally, the boundary sensor includes: a motor encoder and an inclinometer.

[0058] The beneficial effects of the present invention are:

[0059] A vibration suppression method and system for a flexible crane system based on a neural network provided by the present invention consider the influence of unknown friction and output constraints in an actual system on the vibration control of the flexible crane system. The present invention uses the boundary state signals obtained by sensors, estimates the unknown friction of the system by using a neural network, and combines the barrier Lyapunov function method to design a boundary controller. This boundary controller solves the influence brought by the unknown friction to the system and confines the boundary curvature of the flexible rope within a given value, ensuring the safety of the system, thereby effectively suppressing the vibration of the flexible crane system and ensuring the stability of the crane system. Therefore, the vibration suppression method and system of the flexible crane system of the present invention can achieve the position control and vibration suppression control objectives under the conditions that the system contains unknown friction and output constraints, and are applicable to actual engineering applications. Description of the Drawings

[0060] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0061] Figure 1 It is a flowchart of the vibration suppression method for a flexible crane system under unknown friction and output constraints based on a neural network provided by the present invention.

[0062] Figure 2 It is a schematic structural diagram of the flexible crane system provided by the present invention.

[0063] Figure 3 It is a three-dimensional diagram of the lateral position of the flexible rope under the designed boundary controller provided by the present invention.

[0064] Figure 4 It is a three-dimensional diagram of the lateral offset of the flexible rope under the designed boundary controller provided by the present invention.

[0065] Figure 5 It is a load position diagram under the designed boundary controller provided by the present invention.

[0066] Figure 6 It is a diagram of unknown friction and its estimation provided by the present invention.

[0067] Figure 7 It is a boundary curvature diagram of the flexible rope under the designed boundary controller provided by the present invention.

[0068] Figure 8 It is a structural block diagram of the vibration suppression control system for a flexible crane under unknown friction and output constraints based on a neural network provided by the present invention. Detailed implementation manners

[0069] To make the objectives, technical solutions and advantages of the present invention clearer, the following will further describe in detail the implementation manners of the present invention with reference to the accompanying drawings.

[0070] First, the basic knowledge related to the present invention is introduced as follows:

[0071] I. Boundary and boundary state: The boundary refers to the physical boundary of the system. In the flexible crane system of the present invention, it refers to the top end or the trolley end (or called the 0 end) and the bottom end (or called the l end) of the flexible rope. Figure 2 The boundary state refers to the set of motion information of the 0 end or the l end. For example, in the flexible crane system of the present invention, the position of the trolley y(0,t) moved by the 0 end, the speed of the trolley y t (0,t) moved by the 0 end, the lateral position y(l,t) of the rope at the l end, etc. In the present invention, only the boundary state information that is easy to measure at the 0 end is utilized, which is easy to implement measurement or calculation in actual engineering.

[0072] II. Boundary control: Boundary control is a control method for partial differential equation control. This control method only applies the control force at the boundary of the system. In the flexible crane system of the present invention, the control only acts on the 0 end, that is, the trolley end.

[0073] III. Lyapunov function and barrier Lyapunov function: The Lyapunov function V(t) is a quadratic function of the system state. The constructed Lyapunov function needs to satisfy the condition V(t)>0 (i.e., the positive definiteness of Lyapunov). There are various forms of the barrier Lyapunov function. Here, the logarithmic barrier Lyapunov function is selected.

[0074] IV. Lyapunov stability analysis: This is a commonly used stability analysis method, that is, a Lyapunov function V(t) is selected and satisfies V(t)>0, and the properties of the derivative of V(t) along the solution of the system are analyzed to judge the stability.

[0075] V. Uniform boundedness: A commonly used professional term in the control discipline. Briefly speaking, in the flexible crane system of the present invention, the lateral offset ω(x,t) of the flexible rope satisfies |ω(x,t)|≤A, A∈R + .

[0076] The above are all common professional terms in partial differential equation control. For reference, please refer to the following article:

[0077] [1] He Xiuyu. Design and research on boundary control of flexible systems [D]. University of Science and Technology Beijing, 2020.

[0078] [2] Fu Yun. Research on Vibration Control of Flexible Spacecraft Boundaries [D]. South China University of Technology, 2019.

[0079] [3] Guo Fang, Liu Yu, Zhao Zhijia, et al. Vibration Control of Flexible Marine Risers Coupled with Internal Flow Dynamics [J]. Journal of Vibration and Shock, 2017, 36(21): 157-162.

[0080] [4] Guo Fang. Research on Vibration Control of Flexible Marine Riser Systems [D]. South China University of Technology, 2019.

[0081] Example 1:

[0082] This example provides a vibration suppression method for a flexible crane system based on a neural network. The method includes:

[0083] Step 1: Use Hamilton's principle to model the flexible crane system with unknown friction forces.

[0084] Step 2: Use boundary sensors to obtain the boundary state signals of the flexible crane system.

[0085] Step 3: Use a neural network to estimate the unknown friction forces in the flexible crane system.

[0086] Step 4: According to the boundary state signals obtained in Step 2 and the estimated values of the unknown friction forces obtained in Step 3, construct a boundary controller for the flexible crane system in combination with a barrier Lyapunov function.

[0087] Step 5: Construct a Lyapunov function, prove the positive definiteness of the Lyapunov function, and perform a stability analysis of the control system under the boundary controller in Step 4.

[0088] Step 6: Verify whether the state variables of the control system in Step 5 are uniformly bounded. If they are, execute Step 7; if not, a new Lyapunov function and boundary controller need to be constructed.

[0089] Step 7: The boundary controller calculates a control signal based on the obtained boundary state signals and sends the control signal to the actuator.

[0090] Step 8: The actuator receives the control signal and applies the control signal to the flexible crane system. Finally, the crane system achieves the position control and vibration suppression control objectives.

[0091] Example 2:

[0092] This example provides a vibration suppression method for a flexible crane system based on a neural network. The method flow is as follows Figure 1As shown in the figure, it specifically includes the following steps:

[0093] Step 1: Use Hamilton's principle to model the flexible crane system with unknown friction force:

[0094]

[0095] The boundary conditions of the flexible crane system are:

[0096]

[0097] Among them, y(x, t) is the lateral position of the flexible rope at the spatial position x and time t, ρ and l respectively represent the mass per unit length of the flexible rope and the length of the rope, M represents the mass of the trolley, y x (x, t) and y t (x, t) respectively represent the first-order partial derivatives of the lateral position y(x, t) of the flexible rope with respect to the spatial variable x and the time variable t, y tt (x, t) represents the second-order partial derivative of the lateral position y(x, t) of the flexible rope with respect to the time variable t, u(t) represents the boundary controller, F(t) is the unknown friction force between the trolley and the guide rail, T(x) = g[m + ρ(l - x)] is the tension of the rope at the spatial position x, where m is the mass of the bottom load and g is the acceleration due to gravity.

[0098] Step 2: Use boundary sensors to obtain the boundary state signals of the flexible crane system:

[0099] The boundary state signals of the flexible crane system include: the position y(0, t) of the moving trolley, the speed y t (0, t) of the moving trolley, the boundary curvature y x (0, t) of the flexible rope, and the boundary angular velocity y xt (0, t) of the flexible rope. Among them, the position y(0, t) of the moving trolley and the speed y t (0, t) of the moving trolley are measured by the motor encoder at the trolley end. The boundary curvature y x (0, t) of the flexible rope is measured by an inclinometer, and the boundary angular velocity y xt (0, t) of the flexible rope is calculated by using the measured y x (0, t) with the backward difference method.

[0100] Step 3: Use a neural network to estimate the unknown friction force in the flexible crane system. The output of the radial basis function neural network is:

[0101]

[0102] Among them, χ = yt (0, t) is the input of the neural network, representing the speed of the moving trolley. is the activation function of the neural network, which is a Gaussian function. is the estimated value of the ideal weight W * ; is the estimated value of the unknown friction force F(t);

[0103] The weight update formula of the radial basis function neural network is:

[0104]

[0105] where γ, β, and τ are all constants greater than 0, and θ(t) is an auxiliary variable, expressed as θ(t) = k c (y(0, t) - y d ) - k d y x (0, t) + y t (0, t), y d is the desired position, k c and k d are both constants greater than 0, C > 0 is the constraint value of the boundary curvature y x (0, t) of the flexible cable, and the constraint condition is satisfied as |y x (0, t)| < C.

[0106] Step Four: Based on the boundary state signal obtained in Step Two and the estimated value of the unknown friction force obtained in Step Three, construct the boundary controller of the flexible crane system by combining with the obstacle Lyapunov function:

[0107] The boundary controller is:

[0108]

[0109] where k a and k b are both boundary controller gains greater than 0, y x (0, t) represents the boundary curvature of the flexible cable, and y xt (0, t) represents the boundary angular velocity of the flexible cable.

[0110] Step Five: Construct a Lyapunov function, prove the positive definiteness of the Lyapunov function, and conduct a stability analysis of the control system under the boundary controller in Step Four. The steps include:

[0111] Step 5-1: Construct a Lyapunov function, and its expression is:

[0112] E(t) = E a (t) + Eb (t) + E c (t) + E d (t)

[0113] Among them,

[0114]

[0115]

[0116]

[0117]

[0118] Among them, α is a positive constant, is the weight error term, E a (t) is the energy term of the flexible crane system, including kinetic energy and potential energy, E b (t) is the barrier Lyapunov term related to the auxiliary variable θ(t), E c (t) is the energy term related to friction, E d (t) is the signal y x (x, t)y t (x, t) and the cross term of the signal (y(x, t) - y d )y t (x, t) cross - multiplied;

[0119] Step 5 - 2: Verify the positive definiteness of the constructed Lyapunov function, that is, E(t) > 0, and obtain:

[0120] 0 ≤ (1 - η)[E a (t) + E b (t) + E c (t)] ≤ E(t) ≤ (1 + η)[E a (t) + E b (t) + E c (t)]

[0121] Among them,

[0122] Step 5 - 3: Conduct the stability analysis of the control system. Take the first - order derivative of the Lyapunov function with respect to time t, substitute the boundary conditions and the boundary controller, and obtain:

[0123]

[0124] Among them, The above formula shows that the Lyapunov function E(t) is bounded.

[0125] Step 6: Verify whether the state variables of the control system in Step 5 are uniformly bounded. If so, execute Step 7; if not, reconstruct the Lyapunov function and the boundary controller.

[0126] Further derivation shows that the lateral offset ω(x,t) = y(x,t) - y(0,t) of the cable of the flexible crane system satisfies:

[0127]

[0128] The above equation indicates that the lateral offset ω(x,t) of the cable of the flexible crane system is uniformly bounded.

[0129] In addition, since y(x,t) = ω(x,t) + y(0,t), the state y(0,t) satisfies:

[0130]

[0131] That is, when t → ∞, y(x,t) → y d , the above equation indicates that position control is finally achieved.

[0132] Step 7: The boundary controller calculates the control signal based on the acquired boundary state signal and sends the control signal to the actuator.

[0133] Step 8: The actuator receives the control signal and applies the control signal to the flexible crane system. Finally, the crane system achieves the position control and vibration suppression control objectives.

[0134] Next, the effectiveness of the proposed method will be illustrated with specific parameters and figures.

[0135] First, select the parameters of the flexible crane system as follows:

[0136] l = 1m, g = 9.8N / kg, M = 2.1kg, m = 10kg, ρ = 0.2kg / m

[0137] The frictional force is:

[0138] F(t) = F co tanh(k fr y t (0,t)) + F vi y t (0,t)

[0139] where, F co = 17.5, F vi = 0.5, k fr = 100.

[0140] The initial values of the system are set as follows: the initial position of the cart y(x,0) = 0.1m, the initial velocity of the cart y t (x,0) = 0m / s, and the boundary curvature of the flexible rope y x (0,0) = 0rad. The boundary output constraint C = 0.009rad, and the desired position y d = 0.5m.

[0141] Secondly, the described boundary controller is adopted, and its controller gains are set as follows: k a = 65, k b = 20, k c = 0.6, k d = 5. The parameters of the neural network update law: γ = 10, β = 50, τ = 0.001. The parameters of the neural network are set as follows: c j ∈ R, b j ∈ R, j = 1, 2,..., 9, b j = 1, [c 1 , c 2 ,..., c 9 = [-0.04, -0.03, -0.02, -0.01, 0.01, 0.01, 0.02, 0.03, 0.04],

[0142] Figure 3 This is the three-dimensional diagram of the lateral position of the flexible rope under the designed boundary controller provided by the present invention. It can be seen from Figure 3 that y(x,t) → y d , and the position control goal is achieved.

[0143] Figure 4 This is the three-dimensional diagram of the lateral offset of the flexible rope under the designed boundary controller provided by the present invention. It can be seen from Figure 4 that the lateral offset ω(x,t) is restricted within a small area near 0, and the vibration suppression goal is achieved.

[0144] Figure 5 This is the load position diagram under the designed boundary controller provided by the present invention. It can be seen from Figure 5 that the final load is transported to the specified position y d .

[0145] Figure 6 This is the diagram of the unknown friction force F(t) and its estimation provided by the present invention. It can be known from Figure 6 that the neural network can estimate the unknown friction force F(t) well.

[0146] Figure 7 This is the boundary curvature diagram of the flexible rope under the designed boundary controller provided by the present invention. It can be seen from Figure 7It can be seen that the boundary curvature y of the rope x (0,t) does not violate the set output constraint C.

[0147] A vibration suppression method and system for a flexible crane system based on a neural network provided in this embodiment consider the influence of unknown friction and output constraints in an actual system on the vibration control of the flexible crane system. Using the boundary state signals acquired by sensors, the unknown friction of the system is estimated by a neural network and combined with the barrier Lyapunov function method to design a boundary controller. This boundary controller solves the influence brought by unknown friction to the system and confines the boundary curvature of the flexible rope within a given value, ensuring the safety of the system. It not only effectively suppresses the vibration of the flexible crane system but also ensures the stability of the crane system. Therefore, the vibration suppression method and system for the flexible crane system of the present invention can achieve the position control and vibration suppression control objectives under the condition that the system contains unknown friction and output constraints, and is applicable to actual engineering applications.

[0148] Embodiment Three:

[0149] This embodiment provides a vibration suppression control system for a flexible crane system based on a neural network. The system includes:

[0150] Boundary sensors, including: a motor encoder and an inclinometer, for acquiring the boundary state signals of the flexible crane system;

[0151] A boundary controller that calculates the flexible crane control signal by using a vibration suppression method for a flexible crane system based on a neural network described in Embodiment Two;

[0152] An actuator for applying the received flexible crane control signal to the controlled flexible crane system to control the operation of the flexible crane.

[0153] Some steps in the embodiments of the present invention can be implemented by software, and the corresponding software program can be stored in a readable storage medium, such as an optical disc or a hard disk, etc.

[0154] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A vibration suppression method for a flexible crane system based on a neural network, characterized in that, the method includes: Step 1: Model the flexible crane system with unknown friction using Hamilton's principle; Step 2: Use boundary sensors to obtain the boundary state signals of the flexible crane system; Step 3: Use a neural network to estimate the unknown friction in the flexible crane system; Step 4: According to the boundary state signals obtained in Step 2 and the estimated values of the unknown friction obtained in Step 3, construct a boundary controller for the flexible crane system in combination with a barrier Lyapunov function; Step 5: Construct a Lyapunov function, prove the positive definiteness of the Lyapunov function, and perform stability analysis of the control system under the boundary controller in Step 4; Step 6: Verify whether the state variables of the control system in Step 5 are uniformly bounded. If so, execute Step 7; if not, it is necessary to reconstruct the Lyapunov function and the boundary controller; Step 7: The boundary controller calculates a control signal based on the obtained boundary state signals and sends the control signal to the actuator; Step 8: The actuator receives the control signal and applies the control signal to the flexible crane system. Finally, the crane system achieves the position control and vibration suppression control objectives; The boundary controller in Step 4 is: where k a and k b are both boundary controller gains greater than 0, y x (0, t) represents the boundary curvature of the flexible cable, and y xt (0, t) represents the boundary angular velocity of the flexible cable; C > 0 is the constraint value of the boundary curvature y x (0, t) of the flexible cable, and the constraint condition is satisfied as |y x (0, t)| < C; θ(t) is an auxiliary variable, expressed as θ(t) = k c (y(0, t) - y d ) - k d y x (0, t) + y t (0, t), where y d is the desired position, and k c and k d are both constants greater than 0, y(x, t) is the lateral position of the flexible cable at the spatial position x and time t; M represents the mass of the cart; T(x) = g[m + ρ(l - x)] is the tension of the cable at the spatial position x, where m is the mass of the bottom load, g is the acceleration due to gravity, and ρ and l respectively represent the mass per unit length of the flexible cable and the length of the cable; y x (x, t) and y t (x, t) respectively represent the first-order partial derivatives of the lateral position y(x, t) of the flexible cable with respect to the spatial variable x and the time variable t; is the estimated value of the unknown friction force F(t) between the cart and the guide rail; The process of performing the stability analysis of the control system in Step 5 includes: Step 5-1: Construct a Lyapunov function, and its expression is: E(t) = E a (t) + E b (t) + E c (t) + E d (t) where, where γ and β are both constants greater than 0, and α is a positive constant. is the neural network weight error term. is the estimated value of the ideal neural network weight W * ; E a (t) is the energy term of the flexible crane system, including kinetic energy and potential energy, and E b (t) is the barrier Lyapunov term related to the auxiliary variable θ(t), and E c (t) is the energy term related to friction, and E d (t) is the signal y x (x, t)y t (x, t) and the cross-term obtained by cross-multiplying the signal (y(x, t) - y d )y t (x, t). Step 5-2: Verify the positive definiteness of the constructed Lyapunov function, that is, E(t)>0, and obtain: 0 ≤ (1 - η)[E a (t) + E b (t) + E c (t)] ≤ E(t) ≤ (1 + η)[E a (t) + E b (t) + E c (t)] Among them, Step 5-3: Perform the stability analysis of the control system, take the first derivative of the Lyapunov function with respect to time t, substitute the boundary conditions and the boundary controller, and obtain: wherein, τ is a constant greater than 0; Step 5-4: Further derivation, and obtain that the lateral offset ω(x,t)=y(x,t)-y(0,t) of the flexible crane system rope satisfies: The above formula shows that the lateral offset ω(x,t) of the flexible crane system rope is uniformly bounded; In addition, since y(x,t)=ω(x,t)+y(0,t), the state y(0,t) satisfies: That is, when t → ∞, y(x, t) → y d , which indicates that position control is finally achieved.

2. The method according to claim 1, characterized in that, The main control equation of the flexible crane system obtained by modeling in Step 1 is: The boundary conditions of the flexible crane system are: where y tt (x, t) represents the second-order partial derivative of the lateral position y(x, t) of the flexible cable with respect to the time variable t, and u(t) represents the boundary controller.

3. The method according to claim 2, characterized in that, In Step 3, a radial basis function neural network (RBFNN) is used to estimate the unknown friction in the flexible crane system, and the output of the radial basis function neural network is: where χ = y t (0, t) is the input of the neural network, representing the speed of the moving trolley, is the activation function of the neural network and is a Gaussian function; The weight update formula of the radial basis function neural network is:

4. The method according to claim 1, characterized in that, The boundary state signals of the flexible crane system include: the position y(0,t) of the moving trolley, the speed y t (0,t) of the moving trolley, the boundary curvature y x (0,t) of the flexible rope, and the boundary angular velocity y xt (0,t) of the flexible rope. Among them, the position y(0,t) and the speed y t (0,t) of the moving trolley are measured by the motor encoder at the trolley end. The boundary curvature y x (0,t) of the flexible rope is measured by an inclinometer. The boundary angular velocity y xt (0,t) of the flexible rope is calculated by using the backward difference method based on the measured y x (0,t).

5. A vibration suppression system for a flexible crane system based on a neural network, characterized in that, the system includes: Boundary sensors, used to obtain the boundary state signals of the flexible crane system; A boundary controller that calculates a flexible crane control signal by using the vibration suppression method of a flexible crane system based on a neural network according to any one of claims 1-4; An actuator for applying the received flexible crane control signal to the controlled flexible crane system to control the operation of the flexible crane.

6. The system according to claim 5, wherein, the boundary sensor includes: a motor encoder and an inclinometer.

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