CN112327623B - Double-pendulum crane sliding mode control method based on load swing state observation - Google Patents

Double-pendulum crane sliding mode control method based on load swing state observation Download PDF

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CN112327623B
CN112327623B CN202011216819.3A CN202011216819A CN112327623B CN 112327623 B CN112327623 B CN 112327623B CN 202011216819 A CN202011216819 A CN 202011216819A CN 112327623 B CN112327623 B CN 112327623B
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肖友刚
朱铖臻
童俊豪
李蔚
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Central South University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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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
    • B66C13/04—Auxiliary devices for controlling movements of suspended loads, or preventing cable slack
    • B66C13/06—Auxiliary devices for controlling movements of suspended loads, or preventing cable slack for minimising or preventing longitudinal or transverse swinging of loads
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Abstract

The invention discloses a double-pendulum type crane sliding mode control method based on load swing state observation, which is characterized by comprising the following steps of: firstly, the coupling relation between the swinging of a lifting hook and the swinging of a load is deduced according to a dynamic model of the double-pendulum type crane, then a Linear Extended State Observer (LESO) is designed to observe the swinging of the load by utilizing the limited model information and the coupling relation between the swinging of the load and the swinging of the lifting hook, so that the load swinging can be observed at high precision under the condition that the distance between the lifting hook and the gravity center of the load is unknown, the effect of replacing a load swinging sensor is achieved, the measurement problem of the swinging of the load is solved, the double-pendulum type crane anti-swinging positioning controller is obtained by feeding back an observed value of the swinging of the load to a sliding mode controller, and the positioning anti-swinging feedback control of the double-pendulum type crane is realized under the condition that the swinging of the load is not directly measured.

Description

Double-pendulum crane sliding mode control method based on load swing state observation
Technical Field
The invention relates to a double-pendulum type crane sliding mode control method based on load swing state observation, and belongs to the field of automatic control of engineering machinery.
Background
The under-actuated crane has strong transfer capacity and plays a very important role in logistics in places such as buildings, ports, warehouses, workshops and the like. However, external disturbance, crane start and stop, speed change and the like can cause the load to swing, so that the transportation efficiency of the crane is reduced, and huge potential safety hazards are brought. In order to solve this problem, researchers have simplified the crane into a simple pendulum, and have proposed many methods for suppressing the load swing. However, if the load transported by the crane is large in volume or the mass of the hook is not negligible, the hook swings around the trolley, and the load swings around the hook, thereby causing a complex double-swing phenomenon. The double pendulum effect increases the system motion dimension, the underactuation is stronger, and the pendulum elimination difficulty is increased.
At present, two control strategies are commonly used for a double-pendulum crane, the first strategy is to control the double-pendulum crane as a single-pendulum crane, and the influence of a double-pendulum effect is suppressed only by enhancing the robustness of an algorithm. The second strategy is to install an encoder between the lifting hook and the load, measure the load swing value in real time, and feed the swing value back to a control law containing load swing information to realize the control of the double-swing type crane. Although the encoder can be installed between the hook and the load by a special design under experimental conditions, the control device is complicated and the cost is increased. Moreover, in the actual hoisting operation, it is difficult to install a sensor on the hook, so that the method is greatly restricted in the practical application.
Disclosure of Invention
Aiming at the problems and the defects, the invention discloses a double-pendulum type crane sliding mode control method based on load swing state observation, which comprises the steps of firstly transforming a dynamic model of a double-pendulum type crane to obtain a coupling relation between the swing of a lifting hook and the swing of a load, further designing a Linear Extended State Observer (LESO), carrying out high-precision observation on the swing of the load through the LESO, feeding an observed value of the swing of the load back to a sliding mode controller to obtain a double-pendulum type crane anti-swing positioning sliding mode controller, and realizing good anti-swing control effect under the condition of not directly testing the swing state of the load.
The method is implemented according to the following steps:
step A, transforming a dynamic model of the double-pendulum type crane to obtain a dynamic coupling relation between the swinging of the lifting hook and the swinging of the load:
in the actual operation process, considering that the swing angles of the lifting hook and the load are small, the double-swing type crane dynamic model is subjected to linearization processing at a balance position, and a linearized double-swing type crane dynamic model is obtained:
Figure BDA0002760687790000011
wherein, M, m1、m2Respectively including trolley, hook and load mass, beta is the vertical swing angle of the hook, measured by an angle sensor,
Figure BDA0002760687790000013
the vertical swing angle of the load is difficult to measure in practice; l1For the length of the lifting rope, /)2Is the distance between the hook and the center of gravity of the load, g is the acceleration of gravity, and F is the driving force of the trolley.
Processing the dynamic model of the double-pendulum type crane shown in the formula (1) to obtain the dynamic coupling relation between the swinging of the lifting hook and the swinging of the load:
Figure BDA0002760687790000012
b, designing an extended state observer according to the dynamic coupling relation between the swinging of the lifting hook and the swinging of the load, and accurately observing the swinging state of the load:
let beta equal theta1By expanding the formula (2), it is possible to obtain:
Figure BDA0002760687790000021
in the formula (I), the compound is shown in the specification,
Figure BDA0002760687790000022
equation (3) is transformed into the following matrix:
Figure BDA0002760687790000023
wherein the content of the first and second substances,
Figure BDA0002760687790000024
in the matrix a, the matrix b is,
Figure BDA0002760687790000025
designing a Linear Extended State Observer (LESO) according to equation (4):
Figure BDA0002760687790000026
wherein z is [ z ]1 z2 z3]TIs the state estimator for the vector theta,
Figure BDA0002760687790000027
is the state estimator of y, L is the observation gain vector, L ═ 3 ω0 3ω0 2 ω0 3]T，ω0Is the bandwidth of the LESO.
Step C, designing a double-pendulum type crane sliding mode controller according to the trolley state, the actual measurement value of the swing state of the lifting hook and the estimated value of the load swing state;
transforming equation (1) to obtain:
Figure BDA0002760687790000028
defining the slip form surface as:
Figure BDA0002760687790000029
wherein, c1、c2、c3、c4、c5Is the coefficient to be set.
And (3) obtaining a time derivative of the sliding mode surface s:
Figure BDA00027606877900000210
substituting formula (6) for formula (8) to obtain:
Figure BDA0002760687790000031
order to
Figure BDA0002760687790000034
Obtaining a driving force:
Figure BDA0002760687790000032
adding a switching function xi sgn(s) based on a sliding mode surface s (wherein xi is a positive number) into the formula (11), and replacing the switching function xi sgn(s) with xi tanh(s) to reduce the buffeting phenomenon of sliding mode control, thereby obtaining the sliding mode controller of the double-pendulum crane:
Figure BDA0002760687790000033
the invention has the beneficial effects that: a load swing Linear Extended State Observer (LESO) is designed by utilizing limited model information and the coupling relation between load swing and lifting hook swing, the load swing can be observed with high precision by replacing a sensor under the condition that the distance between the gravity center of the lifting hook and the gravity center of a load is unknown, the measurement difficulty of the load swing is solved, the observed value of the load swing is fed back to a sliding mode controller, a double-swing type crane sway eliminating and positioning controller is obtained, and the positioning sway eliminating feedback control of the double-swing type crane under the condition that the load swing angle is not directly measured is realized.
Drawings
FIG. 1 is a schematic view of a model of a double pendulum type crane system;
FIG. 2 is a control result of the present invention under different lifting loads, wherein the control result comprises a trolley position, a hook swing angle, a load swing angle and a trolley driving force in sequence from top to bottom;
FIG. 3 shows the control results of the present invention for different rope lengths, wherein the control results include the position of the trolley, the swing angle of the hook, the swing angle of the load, and the driving force of the trolley from top to bottom;
FIG. 4 is a control result of the present invention at different target positions, in which a trolley position, a hook swing angle, a load swing angle, and a trolley driving force are sequentially set from top to bottom;
fig. 5 is a control result of the present invention under the disturbance, in which the estimated values of the carriage position, the hook swing angle, the carriage driving force, and the load swing angle are compared with the actual values in the order from top to bottom.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in detail with reference to the accompanying drawings and detailed description.
The invention discloses a double-pendulum type crane sliding mode control method based on load swing state observation, which has the following basic ideas: firstly, deriving a coupling relation between swing angles according to a dynamic model of the double-swing crane, then designing a Linear Extended State Observer (LESO) to observe the swing state of a load, and feeding the swing state to a sliding mode controller to obtain the double-swing crane sway elimination positioning controller. The method is implemented by the following steps:
step A, transforming a dynamic model of the double-pendulum type crane to obtain a dynamic coupling relation between the swinging of the lifting hook and the swinging of the load:
according to the double pendulum type crane model shown in fig. 1, a lagrangian method is applied to establish the following dynamic model:
Figure BDA0002760687790000041
Figure BDA0002760687790000042
Figure BDA0002760687790000043
wherein, M, m1、m2The mass of the trolley, the lifting hook and the load respectively, beta is the vertical swing angle of the lifting hook, and is easy to measure by an encoder,
Figure BDA0002760687790000044
the vertical swing angle of the load is difficult to measure in practice; l1Indicating the length of the lifting rope, /)2Represents the distance between the hook and the center of gravity of the load, g is the acceleration of gravity, F (t) is the driving force of the trolley, and x (t) is the displacement of the trolley.
Considering that the primary and secondary swing angles are usually within 10 ° in the actual operation process, therefore:
Figure BDA0002760687790000045
in the case of the formula (1), (2) or (3), theta is 0,
Figure BDA0002760687790000046
Carrying out linearization treatment to obtain an approximate linear model of the double-pendulum crane:
Figure BDA0002760687790000047
Figure BDA0002760687790000048
Figure BDA0002760687790000049
the general formula (4) × l1- (5) obtaining:
Figure BDA00027606877900000410
the general formula (6) × l1-(5)×l2The following can be obtained:
Figure BDA00027606877900000411
substituting (7) into (8) and simplifying to obtain:
Figure BDA00027606877900000412
b, designing an extended state observer according to the dynamic coupling relation between the swinging of the lifting hook and the swinging of the load, and accurately observing the swinging state of the load:
let beta equal theta1Expanding equation (9) to obtain:
Figure BDA00027606877900000413
in the formula (I), the compound is shown in the specification,
Figure BDA00027606877900000414
equation (10) is expressed in matrix form:
Figure BDA00027606877900000415
wherein the content of the first and second substances,
Figure BDA00027606877900000416
in the matrix a, the matrix b is,
Figure BDA0002760687790000051
a Linear Extended State Observer (LESO) is designed according to equation (11):
Figure BDA0002760687790000052
wherein z is [ z ]1 z2 z3]TIs an estimate of the state of the vector theta,
Figure BDA0002760687790000053
is the state estimate of y, L is the observation gain vector, L ═ 3 ω0 3ω0 2 ω0 3]TAnd ω 0 is the designed bandwidth of the LESO.
The observed error of LESO is defined as: e.g. of the type1＝θ1-z1，e2＝θ2-z2，e3＝θ3-z3. And (3) obtaining an error equation of the LESO according to the formula (12) to the formula (11):
Figure BDA0002760687790000054
let Y1＝e1，Y2＝-3ω0e1+e2，Y3＝-3ω0(-3ω0e1+e2)+(-3ω0 2+m)e1+e3Then the error equation is updated as:
Figure BDA0002760687790000055
let a equal to 3 omega0，b＝3ω0 2，c＝ω0 3I.e. by
Figure BDA0002760687790000056
The characteristic equation of the formula (14) is thus
λ3+aλ2+bλ+c＝0 (15)
From the Hurwitz condition, the essential condition that all characteristic roots have negative real parts is:
a＞0，c＞0，ab-c＞0
order to
Figure BDA0002760687790000057
Obtaining a Lyapunov function V of an error system according to a Barbarsin formula:
Figure BDA0002760687790000058
from ab-c > 0
Figure BDA0002760687790000059
Then:
Figure BDA00027606877900000510
after derivation of equation (17), equation (14) is substituted to obtain:
Figure BDA00027606877900000511
thus, V is positive, when θ3When the average molecular weight is 0, the average molecular weight,
Figure BDA00027606877900000512
the designed LESO is in e1＝0,e2＝0,e3A wide range with 0 as the equilibrium point is progressively stabilized. When theta is3When not equal to 0, | θ is assumed3Eta (eta is a normal number) is less than or equal to | eta, and the following can be obtained:
Figure BDA00027606877900000513
the error range obtained according to equation (13) is:
Figure BDA0002760687790000061
therefore, only need to make ω0>>Eta, then e1≈e2≈e30, i.e.:
Figure BDA0002760687790000062
step C, designing a double-pendulum type crane sliding mode controller according to the trolley state, the actual measurement value of the lifting hook swinging state and the estimated value of the load swinging state, and proving the stability of the double-pendulum type crane sliding mode controller;
the following formulas (4), (5) and (6) are arranged to obtain:
Figure BDA0002760687790000063
Figure BDA0002760687790000064
Figure BDA0002760687790000065
defining the slip form surface as:
Figure BDA0002760687790000066
wherein, c1、c2、c3、c4、c5For the real number to be set, the derivation of the sliding mode surface s can be obtained:
Figure BDA0002760687790000067
by substituting formulae (22), (23), and (24) for formula (26):
Figure BDA0002760687790000068
order to
Figure BDA0002760687790000069
The driving force can be found as:
Figure BDA00027606877900000610
to equation (28), a switching function ξ sgn(s) (where ξ is a positive number) based on the sliding mode surface s is added, so that the control law can be updated as:
Figure BDA0002760687790000071
the controller shown in equation (29) causes system chattering due to discontinuity of the sign function, and therefore, the sign function is replaced by a hyperbolic tangent function:
Figure BDA0002760687790000072
controlling the Lyapunov function V of the system1Is defined as:
Figure BDA0002760687790000073
is apparent from V1The function is positive and is derived over time from equation (31):
Figure BDA0002760687790000074
the control law (30) can be substituted for the formula (27):
Figure BDA0002760687790000075
the formula (33) can be substituted for the formula (32):
Figure BDA0002760687790000076
thus, the designed control system is progressively stable.
Step D, designing three groups of simulation experiments, verifying that the double-pendulum sliding mode crane controller can effectively observe the swinging state of the load, and can perform anti-pendulum positioning control on the crane;
in order to verify the effectiveness of the LESO designed by the invention and the control effect after the LESO is fed back to the controller, four groups of simulation experiments are designed, the first three groups of experiments respectively change the load mass, the length of the lifting rope and the positioning distance, the set control parameters and other model parameters are kept unchanged, and all the parameters of the fourth group of experiments are kept unchanged and only disturbance is applied. The basic parameters in the simulation are as follows: m20 kg, M1＝1kg，m2＝5kg，l1＝2m，l2＝0.4m，g2＝9.8m/s2，frox＝8，εx＝0.01，kr-1.2 desired target position p of the trolley d2 m. After full parameter setting, the control parameter is c1＝0.6、c2＝0.45、c3＝5.12、c4＝-0.04、c5＝-1.78。
At the load mass m2When the load is 1kg, 3kg and 5kg respectively, the control result of the invention is shown in fig. 2, and it can be seen from the figure that the trolley reaches the target position within 8s without any overshoot, which shows that the load quality change has almost no influence on the positioning and adjusting time of the trolley; the amplitude of the swing of the lifting hook and the load is slightly reduced with the increase of the load mass, but can be basically ignored, the time of the swing of the lifting hook and the load is almost unchanged, the amplitude of the change of the driving force of the trolley is also not large, and the controller is not sensitive to the change of the mass when the mass is changed.
At the length l of the lifting rope1When the length of the rope is 1m, 2m and 4m respectively, the control result of the invention is shown in figure 3, and it can be seen from the figure that when the length of the rope takes different values, the time for reaching the target position is almost unchanged and is stable within 8 s; the swing amplitude is slightly increased along with the increase of the swing length, but the change amplitude is within 0.1 degrees and can be ignored, the swing stabilization time is about 8s, the driving force is not changed greatly in the whole process, the maximum value is not more than 14N and is within a reasonable range, and the robustness is better when the swing length is changed.
At a positioning distance of pdWhen the power of the actuator is limited, the control result of the invention is as shown in fig. 4, and it can be seen from the figure that the trolley can reach the designated position without any overshoot and the swing amplitude increases with the increase of the target position for different designated positions, but the swing amplitude of the lifting hook and the load in the whole process does not exceed 2 °, and the trolley does not swing, and the driving force increases with the increase of the positioning distance, but the increase amplitude is small.
In order to verify the anti-interference capability of the invention, pulse interference is applied at 9s and sine interference is applied between 15-16 s for the load, and the amplitude is 2 degrees. It can be seen from the control result under the interference effect shown in fig. 5 that after the interference is applied, the disturbance can be quickly eliminated, and the maximum amplitude of the pivot angle is within 2 °, which indicates that the anti-interference capability of the invention is strong. In the context of figure 5, it is shown,
Figure BDA0002760687790000081
representing an estimate of the load swing angle.
Similar technical solutions can be derived from the solutions given in the figures and the description, as described above. However, any simple modification, equivalent change and modification made according to the technical spirit of the present invention still fall within the scope of the technical solution of the present invention.

Claims (1)

1. A double-pendulum type crane sliding mode control method based on load swing state observation is characterized by comprising the following steps:
step A, transforming a dynamic model of the double-pendulum type crane to obtain a dynamic coupling relation between the swinging of the lifting hook and the swinging of the load:
in the actual operation process, considering that the swing angles of the lifting hook and the load are small, the double-swing type crane dynamic model is subjected to linearization processing at a balance position, and a linearized double-swing type crane dynamic model is obtained:
Figure FDA0003467814650000011
wherein, M, m1、m2Respectively including trolley, hook and load mass, beta is the vertical swing angle of the hook, measured by an angle sensor,
Figure FDA0003467814650000012
vertical pivot angle of the load, difficult to measure in practice,. l1For the length of the lifting rope, /)2The distance between the hook and the center of gravity of the load,g is the gravity acceleration, and F is the driving force of the trolley;
processing the dynamic model of the double-pendulum type crane shown in the formula (1) to obtain the dynamic coupling relation between the swinging of the lifting hook and the swinging of the load:
Figure FDA0003467814650000013
b, designing an extended state observer according to the dynamic coupling relation between the swinging of the lifting hook and the swinging of the load, and accurately observing the swinging state of the load:
let beta equal theta1When formula (2) is expanded, it can be obtained:
Figure FDA0003467814650000014
in the formula (I), the compound is shown in the specification,
Figure FDA0003467814650000015
equation (3) is transformed into the following matrix:
Figure FDA0003467814650000016
wherein the content of the first and second substances,
Figure FDA0003467814650000017
in the matrix a, the matrix b is,
Figure FDA0003467814650000018
designing a Linear Extended State Observer (LESO) according to equation (4):
Figure FDA0003467814650000019
wherein z is [ z ]1 z2 z3]TIs the state estimator for the vector theta,
Figure FDA00034678146500000110
is the state estimator of y, L is the observation gain vector, L ═ 3 ω0 3ω0 2 ω0 3]T，ω0Bandwidth for LESO;
step C, designing a double-pendulum type crane sliding mode controller according to the trolley state, the actual measurement value of the swing state of the lifting hook and the estimated value of the load swing state;
by transforming equation (1), we can obtain:
Figure FDA0003467814650000021
defining the slip form surface as:
Figure FDA0003467814650000022
wherein, c1、c2、c3、c4、c5Is a coefficient to be set;
derivation of the slip form surface s can be obtained:
Figure FDA0003467814650000023
the formula (6) may be substituted for the formula (7):
Figure FDA0003467814650000024
order to
Figure FDA0003467814650000025
The driving force can be found as:
Figure FDA0003467814650000026
adding a switching function xi sgn(s) based on a sliding mode surface s into the formula (11), wherein xi is a positive number, and replacing the switching function xi sgn(s) with xi tanh(s) to reduce the buffeting phenomenon of sliding mode control, thereby obtaining the sliding mode controller of the double-pendulum crane:
Figure FDA0003467814650000027
d, utilizing the controller (12) to perform anti-swing positioning simulation on the double-swing crane according to the system parameters of the specific double-swing crane, and performing control parameter setting according to the simulation result to obtain an optimal control parameter c1、c2、c3、c4、c5Then the control signal is input into a controller (12) to form a double-pendulum type crane sliding mode controller of a specific model, so that the pendulum eliminating positioning control of the double-pendulum type crane sliding mode controller is realized.
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