CN115047768A - Flexible crane system vibration suppression method and system based on neural network - Google Patents
Flexible crane system vibration suppression method and system based on neural network Download PDFInfo
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- B66—HOISTING; LIFTING; HAULING
- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B66—HOISTING; LIFTING; HAULING
- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
- B66C13/00—Other constructional features or details
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- B66C13/46—Position indicators for suspended loads or for crane elements
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- B—PERFORMING OPERATIONS; TRANSPORTING
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- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
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Abstract
The invention discloses a vibration suppression method and a control system for a flexible crane system, and belongs to the technical field of position control and vibration suppression of the flexible crane system. The method considers the influence problem of unknown friction and output constraint in an actual system on the vibration control of the flexible crane system, utilizes boundary state signals acquired by a sensor, adopts the unknown friction of a neural network estimation system and combines a barrier Lyapunov function method, and designs a boundary controller. The boundary controller solves the problem of influence of unknown friction force on the system, restrains the boundary curvature of the flexible rope within a set value, and ensures the safety of the system, thereby effectively inhibiting the vibration of the flexible crane system and ensuring the stability of the crane system. Therefore, the flexible crane system vibration suppression method and system can achieve the position control and vibration suppression control targets under the condition that the system contains unknown friction and output constraints.
Description
Technical Field
The 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 the flexible crane system.
Background
With the rapid development of social economy in China, the application of a crane system in the industries such as manufacturing industry, construction sites, ports, marine industry and the like is more and more extensive. The reason for this is that the crane can transport heavy objects or dangerous goods, and this can not only reduce the labour by a wide margin but also can effectively improve production efficiency.
The flexible crane system mainly comprises a top end trolley, a flexible rope and a bottom load. Compared with the traditional rigid rope, the flexible rope material has the advantages of light weight, low energy consumption, strong shock absorption capacity and the like. However, the flexible nature of the ropes tends to cause the system to vibrate during transport. If no effective measures are taken to suppress such vibrations, the magnitude of the force applied to the rope fluctuates, which results in a system that cannot perform accurate position control and causes a series of problems such as fatigue failure of the rope and degradation of the system performance. Therefore, it is very important to suppress the vibration problem of the flexible crane system. In addition, a flexible crane system is a kind of flexible mechanical system, which is usually described by a set of partial differential equations, and in the control problem of the partial differential equations, the infinite dimensional characteristics of the system bring certain difficulties to the design of the controller.
In the field of industrial control, boundary control only needs to arrange a sensor and an actuator at the boundary of a system, and vibration suppression of the system is realized through the action of a controller at the boundary. Therefore, the boundary control method is a control method that is easy to implement.
As the top trolley of the flexible crane system moves left and right along the horizontal guide rail, there is a friction between the trolley and the guide rail which is generally unknown and difficult to measure. In addition, the flexible cord is prone to wear at the car end and flexible cord connection points. Constraining the boundary curvature of the flexible rope within a given range can effectively avoid fatigue damage of the rope.
To solve the above problems, Entessari et al propose three boundary controllers acting on the car end, flexible rope and bottom load, respectively, to dampen the vibrations of the system (Entessari F., ardekanny A.N., altitude A., exhibition stability of flexible Vibration of organic creature video boundary Control method, Journal of Vibration and Control 26(1-2) (2020) 36-55. doi: 10.1177/1077546319876147.). However, this approach using three boundary controllers is more complex and difficult to implement than the approach using only one boundary controller. Thus, d 'AndreHa-Novel et al have studied a boundary controller acting only on the trolley end for the vibration control problem of flexible crane systems (d' AndreHa Novel B., Coron J.M., Exponent stability of an over-head crane with a flexible cable a back-holding ap-proach, Automatica 36(4) (2000) 587-593. doi: https:// doi.org/10.1016/S0005-1098 (00182-X.). However, the above solutions do not consider the problem that the influence of unknown friction force and the constraint of the maximum value which can be reached by the boundary curvature of the flexible rope in the flexible crane system due to the safety requirement exist in the practical application, so that the stability and the position control accuracy of the flexible crane are greatly influenced.
Disclosure of Invention
In order to solve the problems of instability or inaccurate position control of the conventional flexible crane system, the invention provides a flexible crane system vibration suppression method and system based on a neural network, and the scheme is as follows:
a first object of the present invention is to provide a flexible crane system vibration suppression method based on a neural network, the method comprising:
the method comprises the following steps: modeling the flexible crane system with unknown friction by using Hamilton principle;
step two: acquiring a boundary state signal of the flexible crane system by using a boundary sensor;
step three: estimating an unknown friction force in the flexible crane system using a neural network;
step four: constructing a boundary controller of the flexible crane system by combining a barrier Lyapunov function according to the boundary state signal obtained in the second step and the unknown friction estimated value obtained in the third step;
step five: constructing a Lyapunov function, proving the positive nature of the Lyapunov function, and performing stability analysis on a control system under the boundary controller in the fourth step;
step six: verifying whether the state quantity of the control system in the fifth step meets the consistency bounded condition, and if so, executing a seventh step; if not, the Lyapunov function and the boundary controller need to be reconstructed;
step seven: the boundary controller calculates according to the acquired boundary state signal to obtain a control signal and sends the control signal to the actuator;
step eight: and the actuator receives the control signal and acts the control signal on the flexible crane system, and finally the crane system achieves the position control and vibration suppression control targets.
Optionally, the main control equation of the flexible crane system obtained by modeling in the step one is as follows:
the boundary conditions of the flexible crane system are as follows:
wherein y (x, t) is the transverse position of the flexible rope at the time when the space position is x and the time is t, rho and l respectively represent the mass of the flexible rope per unit length and the length of the rope, M represents the mass of the trolley, y x (x, t) and y t (x, t) represents the first partial derivatives of the transverse position y (x, t) of the flexible rope with respect to the space variable x and the time variable t, respectively, y tt (x, t) represents the second partial derivative of the transverse position y (x, t) of the flexible line with respect to the time variable t, u (t) represents the boundary controller, f (t) is the unknown friction between the trolley and the rail, t (x) g [ m + ρ (l-x)]Is the tension of the rope at a spatial position x, where m is the bottom load mass and g is the acceleration of gravity.
Optionally, in the third step, the 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:
wherein x ═ y t (0, t) is the input of the neural network, which represents the speed of the moving trolley,
is an activation function of the neural network, is a gaussian function,
is an ideal weight W * Is determined by the estimated value of (c),
an estimate of the unknown friction force f (t);
the weight updating formula of the radial basis function neural network is as follows:
where γ, β, and τ are 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 To a desired position, k c And k d Are all constants greater than 0, C > 0 is the flexible cord boundary curvature y x Constraint value of (0, t), constraint condition satisfying y x (0,t)|＜C。
Optionally, the boundary controller in step four includes:
wherein k is a And k b Boundary controller gain, y, both greater than 0 x (0, t) denotes the boundary curvature of the flexible rope, y xt (0, t) represents the boundary angular velocity of the flexible cord.
Optionally, the process of analyzing the stability of the control system in the fifth step includes:
step 5-1: constructing a Lyapunov function, wherein the expression of the Lyapunov function is as follows:
E(t)＝E a (t)+E b (t)+E c (t)+E d (t)
wherein the content of the first and second substances,
wherein, alpha is a normal number,
to weight the error term, E a (t) is the energy term of the flexible crane system, including kinetic and potential energy, E b (t) is the term Lyapunov for the barrier associated with the auxiliary variable θ (t), E c (t) is an energy term related to friction, E d (t) is the signal y x (x,t)y t (x, t) and signal (y (x, t) -y d )y t (x, t) cross-multiplied cross terms;
step 5-2: verifying the positive nature of the constructed Lyapunov function, i.e., E (t) > 0, yields:
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)]
wherein the content of the first and second substances,
step 5-3: and performing stability analysis on the control system, solving a first derivative of the Lyapunov function to time t, and substituting the first derivative into the boundary condition and the boundary controller to obtain:
wherein the content of the first and second substances,
step 5-4: further derivation, we find that the lateral offset ω (x, t) of the flexible crane system rope satisfies y (x, t) -y (0, t):
the above equation indicates that the lateral offset ω (x, t) of the flexible crane system rope is consistently bounded;
further, since y (x, t) ═ ω (x, t) + y (0, t), the state y (0, t) satisfies:
i.e., when t → ∞ y (x, t) → y d The above equation indicates that position control is finally achieved.
Optionally, the boundary state signal of the flexible crane system includes: position y (0, t) of the carriage, velocity y of the carriage t (0, t), boundary curvature y of the flexible rope x (0, t) and boundary angular velocity y of the flexible rope xt (0, t), wherein the position y (0, t) of the traveling carriage and the velocity y of the traveling carriage t (0, t) is measured by a motor encoder at the trolley end, and the boundary curvature y of the flexible rope x (0, t) is measured by an inclinometer, and the boundary angular velocity y of the flexible rope xt (0, t) by measured y x (0, t) by the reverseAnd calculating by a direction difference method.
It is a second object of the present invention to provide a neural network based flexible crane system vibration suppression system, the system comprising:
a boundary sensor for acquiring boundary status signals of the flexible crane system;
the boundary controller is used for calculating a flexible crane control signal by adopting any one of the flexible crane system vibration suppression method based on the neural network;
and the actuator is used for applying the received flexible crane control signal to a controlled flexible crane system to control the operation of the flexible crane.
Optionally, the boundary sensor includes: motor encoder, inclinometer.
The invention has the beneficial effects that:
the invention provides a vibration suppression method and system for a flexible crane system based on a neural network, and the method and system provided by the invention consider the influence problem of unknown friction force and output constraint in an actual system on the vibration control of the flexible crane system. According to the invention, a boundary controller is designed by utilizing a boundary state signal acquired by a sensor, adopting unknown friction of a neural network estimation system and combining a barrier Lyapunov function method. The boundary controller solves the problem of influence of unknown friction force on the system, restrains the boundary curvature of the flexible rope within a given value, and ensures the safety of the system, thereby effectively inhibiting the vibration of the flexible crane system and ensuring the stability of the crane system. Therefore, the method and the system for suppressing the vibration of the flexible crane system can realize the position control and the vibration suppression control under the condition that the system contains unknown friction force and output constraint, and are suitable for practical engineering application.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a flow chart of a method for suppressing vibration of a flexible crane system under unknown friction and output constraints based on a neural network.
Fig. 2 is a schematic structural diagram of a flexible crane system provided by the invention.
Fig. 3 is a three-dimensional view of the lateral position of a flexible cord under a boundary controller of a design provided by the present invention.
Fig. 4 is a three-dimensional view of the lateral offset of a flexible cord under a boundary controller using a design provided by the present invention.
Fig. 5 is a diagram of the lower load position of the boundary controller with the design provided by the present invention.
FIG. 6 is a graph of unknown friction and its estimation provided by the present invention.
Fig. 7 is a graph of the boundary curvature of a flexible cord under a boundary controller of the present invention using a design.
FIG. 8 is a structural block diagram of a flexible crane vibration suppression control system under unknown friction and output constraints based on a neural network provided by the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be described in detail with reference to the accompanying drawings.
The basic knowledge involved in the present invention is first described as follows:
first, boundary and boundary state: the boundary refers to the physical boundary of the system, and what is meant in the flexible crane system of the present invention is the top or trolley end (or called 0 end) and the bottom end (or called l end) of the flexible rope see fig. 2; and the boundary state refers to the set of motion information of the 0-terminal or the l-terminal, such as the y (0, t) position of the 0-terminal moving vehicle and the y (0, t) speed of the 0-terminal moving vehicle in the flexible crane system of the present invention t (0, t), the lateral position y (l, t) of the end of the rope l, etc. The invention only utilizes the boundary state information which is easy to measure at the 0 end, and is easy to realize measurement or calculation in the actual engineering.
Secondly, boundary control: boundary control is a control mode of partial differential equation control that applies control force only at the boundary of the system, and in the flexible crane system of the present invention, control is applied only at the 0 end, i.e., the trolley end.
Three, Lyapunov function and barrier Lyapunov function: the Lyapunov function V (t) is a quadratic function of the system state, and the constructed Lyapunov function needs to satisfy the condition V (t) > 0 (namely the positive nature of Lyapunov). The barrier Lyapunov function has various forms, and is selected from logarithmic barrier Lyapunov function
Fourthly, analysis of Lyapunov stability: the method is a common stability analysis method, namely a Lyapunov function V (t) is selected, V (t) is satisfied, and the property of the derivative of the function along the system solution is analyzed to judge the stability.
And fifthly, uniformly bounding: the common professional names in the control subject, in brief, in the flexible crane system of the invention, the transverse offset omega (x, t) of the flexible rope meets the condition that omega (x, t) is less than or equal to A, and A belongs to R + 。
The above are all common terms used in partial differential equation control, and reference may be made to the following articles:
[1] the border control design and study of the flexible system [ D ]. university of beijing technology, 2020.
[2] Flexible spacecraft boundary vibration control study [ D ]. university of south china, 2019.
[3] Guo Fang, Liu, Zhao Shi, et al.
[4] Guo fang flexible marine riser system vibration control study [ D ]. university of south china, 2019.
The first embodiment is as follows:
the embodiment provides a flexible crane system vibration suppression method based on a neural network, which comprises the following steps:
the method comprises the following steps: modeling the flexible crane system with unknown friction by using Hamilton principle;
step two: acquiring a boundary state signal of the flexible crane system by using a boundary sensor;
step three: estimating an unknown friction force in the flexible crane system using a neural network;
step four: constructing a boundary controller of the flexible crane system by combining a barrier Lyapunov function according to the boundary state signal obtained in the second step and the unknown friction estimated value obtained in the third step;
step five: constructing a Lyapunov function, proving the positive nature of the Lyapunov function, and performing stability analysis on a control system under the boundary controller in the fourth step;
step six: verifying whether the state quantity of the control system in the fifth step meets the consistency bounded condition, and if so, executing a seventh step; if not, the Lyapunov function and the boundary controller need to be reconstructed;
step seven: the boundary controller calculates according to the acquired boundary state signal to obtain a control signal and sends the control signal to the actuator;
step eight: and the actuator receives the control signal and acts the control signal on the flexible crane system, and finally the crane system achieves the position control and vibration suppression control targets.
Example two:
the embodiment provides a vibration suppression method for a flexible crane system based on a neural network, and the flow of the method is shown in fig. 1, and the method specifically comprises the following steps:
the method comprises the following steps: modeling the flexible crane system with unknown friction using Hamilton principles:
the boundary conditions of the flexible crane system are as follows:
wherein y (x, t) is the transverse position of the flexible rope at the time when the space position is x and the time is t, rho and l respectively represent the mass of the flexible rope per unit length and the length of the rope, M represents the mass of the trolley, y x (x, t) and y t (x, t) represents the first partial derivatives of the transverse position y (x, t) of the flexible rope with respect to the space variable x and the time variable t, respectively, y tt (x, t) represents the second partial derivative of the transverse position y (x, t) of the flexible line with respect to the time variable t, u (t) represents the boundary controller, f (t) is the unknown friction between the trolley and the rail, t (x) g [ m + ρ (l-x)]Is the tension of the rope at a spatial position x, where m is the bottom load mass and g is the acceleration of gravity.
Step two: acquiring boundary state signals of the flexible crane system by using a boundary sensor:
the boundary state signal of the flexible crane system comprises: position y (0, t) of the carriage, velocity y of the carriage t (0, t), boundary curvature y of the flexible rope x (0, t) and boundary angular velocity y of the flexible rope xt (0, t), wherein the position y (0, t) of the traveling carriage and the velocity y of the traveling carriage t (0, t) is measured by a motor encoder at the trolley end, and the boundary curvature y of the flexible rope x (0, t) is measured by an inclinometer, and the boundary angular velocity y of the flexible rope xt (0, t) by measured y x And (0, t) is obtained by calculating by using an inverse difference method.
Step three: estimating an unknown friction force in the flexible crane system using a neural network, the output of the radial basis function neural network being:
wherein x ═ y t (0, t) is the input of the neural network, which represents the speed of the moving trolley,
as an activation function of a neural network, as a Gaussian function，
Is an ideal weight W * Is determined by the estimated value of (c),
an estimate of the unknown friction force f (t);
the weight updating formula of the radial basis function neural network is as follows:
where γ, β, and τ are 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 To a desired position, k c And k d Are all constants greater than 0, C > 0 is the flexible cord boundary curvature y x Constraint value of (0, t), constraint condition satisfying y x (0,t)|＜C。
Step four: constructing a boundary controller of the flexible crane system by combining a barrier Lyapunov function according to the boundary state signal obtained in the second step and the unknown friction estimated value obtained in the third step:
the boundary controller is as follows:
wherein k is a And k b Boundary controller gain, y, both greater than 0 x (0, t) denotes the boundary curvature of the flexible rope, y xt (0, t) represents the boundary angular velocity of the flexible cord.
Step five: constructing a Lyapunov function, proving the positive nature of the Lyapunov function and carrying out stability analysis on a control system under the boundary controller in the fourth step, wherein the steps comprise:
step 5-1: constructing a Lyapunov function, wherein the expression of the Lyapunov function is as follows:
E(t)＝E a (t)+E b (t)+E c (t)+E d (t)
wherein the content of the first and second substances,
wherein, alpha is a normal number,
to weight the error term, E a (t) is the energy term of the flexible crane system, including kinetic and potential energy, E b (t) is the term Lyapunov for the barrier associated with the auxiliary variable θ (t), E c (t) is an energy term related to friction, E d (t) is the signal y x (x,t)y t (x, t) and signal (y (x, t) -y d )y t (x, t) cross-multiplied cross terms;
step 5-2: verifying the positive nature of the constructed Lyapunov function, i.e., E (t) > 0, yields:
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)]
wherein the content of the first and second substances,
step 5-3: and performing stability analysis on the control system, solving a first derivative of the Lyapunov function to time t, and substituting the first derivative into the boundary condition and the boundary controller to obtain:
wherein the content of the first and second substances,
the above equation indicates that the Lyapunov function E (t) is bounded.
Step six: verifying whether the state quantity of the control system in the step five meets the consistency and the boundary, and if so, executing a step seven; if not, the Lyapunov function and the boundary controller need to be reconstructed;
further derivation, we find that the lateral offset ω (x, t) of the flexible crane system rope satisfies y (x, t) -y (0, t):
the above equation indicates that the lateral offset ω (x, t) of the flexible crane system rope is consistently bounded;
further, since y (x, t) ═ ω (x, t) + y (0, t), the state y (0, t) satisfies:
i.e., when t → ∞ y (x, t) → y d The above equation indicates that position control is finally achieved.
Step seven: the boundary controller calculates according to the acquired boundary state signal to obtain a control signal and sends the control signal to the actuator;
step eight: and the actuator receives the control signal and acts the control signal on the flexible crane system, and finally the crane system achieves the position control and vibration suppression control targets.
The effectiveness of the proposed method is explained below in connection with specific parameters and the attached figures.
Firstly, the parameters of the flexible crane system are selected as follows:
l＝1m,g＝9.8N/kg,M＝2.1kg,m＝10kg,ρ＝0.2kg/m
the friction force is as follows:
F(t)＝F co tanh(k fr y t (0,t))+F vi y t (0,t)
wherein, F co ＝17.5,F vi ＝0.5,k fr ＝100。
The initial values of the system are set as follows: initial position y (x,0) of the carriage is 0.1m, and initial velocity y of the carriage t (x,0) 0m/s, boundary curvature y of flexible cord x (0,0) ═ 0 rad. Boundary output constraint C0.009 rad, desired position y d ＝0.5m。
Secondly, with the border controller, the controller gain is set as follows: k is a radical of a ＝65,k b ＝20,k c ＝0.6,k d 5. Neural network update law parameters: γ is 10, β is 50, and τ is 0.001. The parameters of the neural network are set as follows: c. 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],
FIG. 3 is a three-dimensional view of the lateral position of a flexible cord under a boundary controller of the present invention using a design, as seen in FIG. 3, y (x, t) → y d And the position control target is realized.
Fig. 4 is a three-dimensional diagram of the lateral offset of the flexible rope under the boundary controller adopting the design provided by the invention, and it can be seen from fig. 4 that the lateral offset ω (x, t) is limited to a small area near 0, and the vibration suppression target is realized.
FIG. 5 is a diagram of the position of the load under the boundary controller with design according to the present invention, and it can be seen from FIG. 5 that the final load is transported to the designated position y d 。
Fig. 6 is a diagram of the unknown frictional force f (t) and an estimation diagram thereof according to the present invention, and it can be seen from fig. 6 that the neural network can well estimate the unknown frictional force f (t).
FIG. 7 is a graph of the boundary curvature of a flexible rope under a boundary controller of the present invention using a design, as can be seen in FIG. 7, the boundary curvature y of the rope x (0, t) does not violate the set output constraint C.
According to the flexible crane system vibration suppression method and system based on the neural network, the problem of influence of unknown friction force and output constraint on vibration control of the flexible crane system in an actual system is considered, boundary state signals obtained by a sensor are utilized, the unknown friction force of the neural network estimation system is adopted, and a boundary controller is designed by combining a barrier Lyapunov function method. The boundary controller solves the problem of influence of unknown friction force on the system, restrains the boundary curvature of the flexible rope within a set value, ensures the safety of the system, effectively inhibits the vibration of the flexible crane system, and ensures the stability of the crane system. Therefore, the method and the system for suppressing the vibration of the flexible crane system can realize the position control and the vibration suppression control under the condition that the system contains unknown friction force and output constraint, and are suitable for practical engineering application.
Example three:
the embodiment provides a flexible crane system vibration suppression control system based on a neural network, which comprises:
a boundary sensor, comprising: the motor encoder and the inclinometer are used for acquiring boundary state signals of the flexible crane system;
the boundary controller is used for calculating a flexible crane control signal by adopting the flexible crane system vibration suppression method based on the neural network described in the second embodiment;
and the actuator is used for applying the received flexible crane control signal to a controlled flexible crane system to control the operation of the flexible crane.
Some steps in the embodiments of the present invention may be implemented by software, and the corresponding software program may be stored in a readable storage medium, such as an optical disc or a hard disk.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (8)
1. A flexible crane system vibration suppression method based on a neural network is characterized by comprising the following steps:
the method comprises the following steps: modeling the flexible crane system with unknown friction by using Hamilton principle;
step two: acquiring a boundary state signal of the flexible crane system by using a boundary sensor;
step three: estimating an unknown friction force in the flexible crane system using a neural network;
step four: constructing a boundary controller of the flexible crane system by combining an obstacle Lyapunov function according to the boundary state signal obtained in the second step and the unknown friction estimated value obtained in the third step;
step five: constructing a Lyapunov function, proving the positive nature of the Lyapunov function, and performing stability analysis on a control system under the boundary controller in the fourth step;
step six: verifying whether the state quantity of the control system in the fifth step meets the consistency bounded condition, and if so, executing a seventh step; if not, the Lyapunov function and the boundary controller need to be reconstructed;
step seven: the boundary controller calculates according to the acquired boundary state signal to obtain a control signal and sends the control signal to the actuator;
step eight: and the actuator receives the control signal and acts the control signal on the flexible crane system, and finally the crane system achieves the position control and vibration suppression control targets.
2. The method of claim 1, wherein the flexible crane system master control equation modeled in step one is:
the boundary conditions of the flexible crane system are as follows:
wherein y (x, t) is the transverse position of the flexible rope at the time when the space position is x and the time is t, rho and l respectively represent the mass of the flexible rope per unit length and the length of the rope, M represents the mass of the trolley, y x (x, t) and y t (x, t) represents the first partial derivatives of the transverse position y (x, t) of the flexible rope with respect to the space variable x and the time variable t, respectively, y tt (x, t) represents the second partial derivative of the transverse position y (x, t) of the flexible line with respect to the time variable t, u (t) represents the boundary controller, f (t) is the unknown friction between the trolley and the rail, t (x) g [ m + ρ (l-x)]Is the tension of the rope at a spatial position x, where m is the bottom load mass and g is the acceleration of gravity.
3. The method of claim 2, wherein step three estimates the unknown friction force in the flexible crane system using a radial basis function neural network RBFNN whose outputs are:
wherein x ═ y t (0, t) is the input of the neural network, which represents the speed of the moving trolley,
is an activation function of the neural network, is a gaussian function,
is an ideal weight W * Is determined by the estimated value of (c),
an estimate of the unknown friction force f (t);
the weight updating formula of the radial basis function neural network is as follows:
where γ, β, and τ are 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 To a desired position, k c And k d Are all constants greater than 0, C > 0 is the flexible cord boundary curvature y x Constraint value of (0, t), constraint condition satisfying y x (0,t)|＜C。
4. The method of claim 3, wherein the step four middle boundary controller is:
wherein k is a And k b Boundary controller gain, y, both greater than 0 x (0, t) denotes the boundary curvature of the flexible cord, y xt (0, t) represents the boundary angular velocity of the flexible cord.
5. The method of claim 4, wherein the step five process of performing the stability analysis of the control system comprises:
step 5-1: constructing a Lyapunov function, wherein the expression of the Lyapunov function is as follows:
E(t)＝E a (t)+E b (t)+E c (t)+E d (t)
wherein the content of the first and second substances,
wherein, alpha is a normal number,
to weight the error term, E a (t) is the energy term of the flexible crane system, including kinetic and potential energy, E b (t) is the Lyapunov term of the barrier associated with the auxiliary variable θ (t), E c (t) is an energy term related to friction, E d (t) is the signal y x (x,t)y t (x, t) and signal (y (x, t) -y d )y t (x, t) cross-multiplied cross terms;
step 5-2: verifying the positive nature of the constructed Lyapunov function, i.e., E (t) > 0, yields:
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)]
wherein the content of the first and second substances,
step 5-3: and performing stability analysis on the control system, solving a first derivative of the Lyapunov function to time t, and substituting the first derivative into the boundary condition and the boundary controller to obtain:
wherein the content of the first and second substances,
step 5-4: further derivation, we find that the lateral offset ω (x, t) of the flexible crane system rope satisfies y (x, t) -y (0, t):
the above equation indicates that the lateral offset ω (x, t) of the flexible crane system rope is consistently bounded;
further, since y (x, t) ═ ω (x, t) + y (0, t), the state y (0, t) satisfies:
i.e., when t → ∞ y (x, t) → y d The above equation indicates that position control is finally achieved.
6. The method of claim 1, wherein the boundary condition signals of the flexible crane system comprise: position y (0, t) of the carriage, velocity y of the carriage t (0, t), boundary curvature y of the flexible rope x (0, t) and boundary angular velocity y of the flexible rope xt (0, t), wherein the position y (0, t) of the traveling carriage and the velocity y of the traveling carriage t (0, t) is measured by a motor encoder at the trolley end, and the boundary curvature y of the flexible rope x (0, t) is measured by an inclinometer, and the boundary angular velocity y of the flexible rope xt (0, t) by measured y x And (0, t) is obtained by calculating by using an inverse difference method.
7. A neural network based flexible crane system vibration suppression system, the system comprising:
a boundary sensor for acquiring boundary status signals of the flexible crane system;
a boundary controller for calculating a limp home control signal using a neural network based limp home system vibration suppression method according to any one of claims 1 to 6;
and the actuator is used for applying the received flexible crane control signal to a controlled flexible crane system to control the operation of the flexible crane.
8. The system of claim 7, wherein the boundary sensor comprises: motor encoder, inclinometer.
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