Active-disturbance-rejection control method for underactuated crane sliding mode with distributed mass load

Abstract

The invention discloses an active-disturbance-rejection control method for an underactuated crane sliding mode with a distributed mass load, which is characterized by comprising the following steps of: firstly, designing a parallel-level linear expansion state observer capable of eliminating the mutual coupling influence of variables according to the characteristic that trolley movement, a lifting hook and load swinging of a crane system are mutually coupled; estimating total disturbance of the system caused by an unmodeled part of the system, a linearization error, a random initial swing angle and wind resistance, superposing an observation value of the total disturbance and an observation static error of the observation value, and then carrying out feedforward compensation so as to improve the control precision and the real-time performance; in order to solve the problem of load swing under input saturation constraint, an auxiliary system is designed to limit control force and is combined with sliding-mode active-disturbance-rejection control, and the swing control method for the under-actuated crane to hoist the distributed mass load is comprehensively formed. The crane can accurately reach a target position along a set track under the conditions of input saturation constraint, internal and external disturbance and random hoisting points, and swing of the distributed mass load is quickly eliminated.

Description

Technical Field

[0001] The invention relates to a sliding mode anti-disturbance control method for an underactuated crane with distributed mass load, and belongs to the field of automatic control. Background Art

[0002] As an indispensable lifting tool in the transportation industry, cranes undertake important loading and unloading, maintenance and rescue duties in transportation scenarios such as freight terminals, ports, railway stations and logistics centers. Their operational safety and efficiency have a significant impact on the social and economic benefits of the entire logistics system. The operating efficiency of cranes is affected by a variety of internal constraints such as load shape, center of gravity, driving force, and external constraints such as working space, wind speed and weather conditions. The key to improving the safety and efficiency of crane operations is to smoothly lift the load to the target location under various constraints and minimize the load swing.

[0003] Most of the existing studies regard the load as a point mass for anti-sway positioning control. In actual engineering, cranes are often used to transport aircraft wings, fan blades, rails, steel bars, profiles and other distributed mass loads whose volume cannot be ignored. Such loads can no longer be simplified as point masses for control, which means that the control method based on the point mass model is no longer practical for the positioning and anti-sway of distributed mass loads. The center of gravity of such loads is usually difficult to place directly under the hook, making the initial swing angle uncertain and highly variable during lifting; due to their large volume, such loads are easily disturbed by external factors such as wind disturbance and vibration, and are prone to strong double pendulum effects, making anti-sway positioning more difficult; and due to the constraints of the actual structure, the control force provided by the crane system is limited. When the required control force is too large, the actual control output cannot meet the requirements, and the stability of the system is difficult to guarantee. Summary of the invention

[0004] In view of the above problems and shortcomings, this paper proposes a sliding mode anti-disturbance control method for an underactuated crane with a distributed mass load. This method designs a parallel-level linear extended state observer that can eliminate the influence of mutual coupling of variables, and superimposes the observed value of the total disturbance and the observed static error, and performs feedforward compensation to improve the control accuracy and real-time performance; an auxiliary function is introduced to limit the control force, and combined with the sliding mode control, a sliding mode controller is designed for the crane to run according to the set trajectory under the saturation constraint, and a feedforward compensator is used to compensate for the factors that cause the crane disturbance, so that the crane can accurately reach the target position under the input saturation constraint and internal and external interference, and quickly eliminate the swing of the distributed mass load;

[0005] Specifically, the method is implemented according to the following steps:

[0006] Step A: Based on the dynamic model of crane lifting distributed mass load, a parallel linear extended state observer (PLESO) is designed to eliminate the influence of mutual coupling of variables, and a feedforward compensator is designed for the objective estimation error. At the same time, the total disturbance of the system caused by linearization error and time-varying air resistance is estimated to overcome the influence of the unmodeled part of the system, uncertain factors, internal and external interference, linearization error and random initial swing angle on the lifting process of distributed mass load;

[0007] Ignoring the damping caused by the mass and bending of the rope, the Lagrange theorem is used to obtain the dynamic model of the underactuated crane system lifting distributed mass loads with wind resistance. After linearization, the following model is obtained:

[0008]

[0009]

[0010] Where m, m1, and m2 are the masses of the trolley, hook, and load, respectively; x is the displacement of the trolley; and u is x 、F rx are the driving force and friction force of the trolley, θ, are hook and load swing angles, respectively. R is the ratio of load mass to hook mass, R = m2 / m1. l1, l2, l p are the lengths of the wire rope, the inclined rope, and the distributed mass load, respectively, l h is the distance from the hook to the distributed mass load, They represent the wind resistance of the trolley, hook and load respectively. w x ,w θ , are the total disturbances affecting the trolley displacement, hook swing angle and load swing angle respectively, ξ1, ξ2, ξ3 are the errors generated in the linearization process;

[0011] The “total disturbance” of each loop in equations (1)-(3) is expanded into a new state variable in the original system and expressed as a parallel system containing the “total disturbance”:

[0012]

[0013] In the formula, in

[0014] Describe equation (4) using a block matrix:

[0015]

[0016] In the formula, O is an all-zero matrix, C = [EOO] ∈ R 3×9 is the output matrix;

[0017] In order to estimate the state of the crane system and the “total disturbance”, a parallel linear extended state observer (PLESO) is designed:

[0018]

[0019] In the formula, and They are and The observation state, the observation error is defined as is the observer gain matrix, where ⊙ represents the Hadamard product of two matrices,

[0020] The bandwidth method is used to configure the observation gain of the designed PLESO within the bandwidth On, and:

[0021]

[0022] Step B: In order to overcome the influence of disturbance observation static error and observation delay of PLESO on control accuracy and real-time performance, the observation value of the total disturbance and the observation static error are superimposed and feedforward compensation is performed to improve control accuracy and real-time performance;

[0023] In order to eliminate the influence of steady-state observation error on disturbance estimation, the estimation error of PLESO is obtained by subtracting equations (5) and (6):

[0024]

[0025] In the formula,

[0026] The PLESO observer gain L configured based on the bandwidth method makes A e It is Hurwitz stable, and the error equation can achieve asymptotic stability, that is:

[0027]

[0028] Substituting equation (9) into equation (8), we get the system observation static error:

[0029]

[0030] In the formula,

[0031] The total disturbance differential term observed using equation (6) Substitute H into equation (10) to obtain the observed static error D, and use it to compensate the observed value of PLESO to obtain the observed compensator PLESO+D:

[0032]

[0033] In the formula, D is the observed compensation, Z is the observed value, and Z' is the observed value after compensation. According to formula (11), the total disturbance w of the trolley displacement, hook swing angle and load swing angle loop can be obtained: x ,w θ , The compensated observation value z' x3 ,z' θ3 ,

[0034] Step C, introducing an auxiliary function to limit the control force, and combining it with the sliding mode control, designing a sliding mode controller for the crane to run according to the set trajectory under the saturation constraint, and using an observation compensator to compensate for the factors causing the crane disturbance, so that the crane can accurately reach the target position under the input saturation constraint, internal and external interference and random initial swing angle, and quickly eliminate the swing of the distributed mass load;

[0035] When the actuator power is limited, the control input of the trolley cannot exceed the maximum value u max ,u max > 0, let the actual control input of the trolley under input saturation be u x , the ideal control input is v x , then the control input saturation function sat(v x ) is expressed as:

[0036]

[0037] In order to make the system work normally when the control input is limited, an auxiliary control system is introduced to limit the control input extreme value. The designed auxiliary system is:

[0038]

[0039] In the formula, Δu x =u x -v x , χ=[χ1,χ2] T is the state variable of the auxiliary system, k1>0, k2>0, b is the control gain. Since K is Hurwitz stable, when t→∞, χ1→0, χ2→0;

[0040] Introduce the auxiliary variable χ1 into the actual displacement of the trolley and the target displacement trajectory x of the trolley r In the definition of error: e χ =xx r -χ1, then

[0041] Trolley target displacement trajectory x r Use the following formula:

[0042]

[0043] In the formula, n1 controls the running time of acceleration, acceleration and deceleration, deceleration and deceleration stages, n2 controls the running time of uniform acceleration stage, n3 controls the running time of uniform speed stage, n4 controls the running time of uniform deceleration stage, x d is the target position of the trolley, v m is the rated operating speed, a m is the rated operating acceleration, j m is the rated operating jerk, c∈R+ is the initial acceleration adjustment parameter;

[0044] In order to smoothly move the load along the target trajectory x r Lift to target position x d , the target values ​​of the hook and load swing angle and angular velocity are 0, so the sliding surface s is defined as:

[0045]

[0046] Taking the derivative of s with respect to time, we get:

[0047]

[0048] Substituting equations (1)-(3) into the above equation, we get:

[0049]

[0050] Let Δu=u x -v x Substitute into the above formula and change w x ,w θ , Use the compensated estimated value z' x3 ,z' θ3 , Instead, we get:

[0051]

[0052] make And add the switching function ξtanh(s) based on the sliding surface s (where ξ is a positive number), and the ideal control input is obtained as:

[0053]

[0054] Substituting the above equation into equation (12), we can obtain a controller for underactuated crane lifting distributed mass loads that takes into account input saturation and wind resistance. Under the control of this controller, the crane under input saturation constraints can accurately lift the distributed mass load to the target position under state variable coupling, external interference, parameter changes and random initial swing angles, and effectively suppress the load swing during the entire lifting process. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 System model for underactuated cranes lifting distributed mass loads;

[0056] Figure 2 Design flow chart for PLESO;

[0057] Figure 3 The figure shows a comparison of the control effects of the three algorithms on the distributed mass load when the target distance is 4m. The curves from top to bottom are the trolley displacement, hook swing angle, load swing angle and actuator control quantity.

[0058] Figure 4 The control effect of three algorithms on distributed mass load is compared when the target distance is 30m and the initial swing angle is 10°. The curves from top to bottom are the trolley displacement, hook swing angle, load swing angle and actuator control quantity.

[0059] Figure 5 The control effect of the proposed algorithm on the distributed mass load is compared when the target distance is 30m, the initial swing angle is 10°, and four different types of disturbances. The curves from top to bottom are load disturbance, trolley displacement, hook swing angle, load swing angle and actuator control quantity. DETAILED DESCRIPTION

[0060] In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention is described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0061] Step A: Based on the dynamic model of underactuated crane lifting distributed mass load, a parallel linear extended state observer (PLESO) is designed to eliminate the influence of mutual coupling of variables, and a feedforward compensator is designed for the objective estimation error. At the same time, the total disturbance of the system caused by linearization error and time-varying air resistance is estimated to overcome the influence of the unmodeled part of the system, uncertain factors, internal and external interference, linearization error and random initial swing angle on the lifting process of distributed mass load;

[0062] The system model of an underactuated crane lifting a distributed mass load is as follows: Figure 1As shown in the figure, m, m1, m2 are the masses of the trolley, hook, and load respectively, x is the displacement of the trolley, and u is x ,F rx are the trolley driving force and friction force, θ, are hook and load swing angles, l1, l2, l p are the lengths of the wire rope, the inclined rope, and the distributed mass load, respectively. Respectively represent the wind resistance of the trolley, hook, and load. Ignoring the damping caused by the mass and bending of the rope, using Lagrange's theorem, the dynamic model of the underactuated crane system lifting distributed mass load considering wind resistance can be obtained:

[0063]

[0064] Where R is the ratio of load mass to hook mass, R = m2 / m1; h is the distance from the hook to the distributed mass load,

[0065] During the load lifting process, the swing amplitude of the hook and the load usually does not exceed 10°, so sinθ≈θ, cosθ≈1, The original nonlinear model is linearized to obtain the following model:

[0066]

[0067]

[0068] In the formula, w x ,w θ , are the total disturbances affecting the trolley displacement, hook swing angle and load swing angle respectively, ξ1, ξ2, ξ3 are the errors generated in the linearization process;

[0069] It can be seen from equations (1)-(3) that the crane system needs to control the swing of the hook and the load by moving the trolley and lift the load to the target position. In this process, the displacement of the trolley, the swing of the hook and the load are coupled with each other, and the states of the trolley, the hook and the load are all affected by the unknown total disturbance. In order to eliminate the adverse effects of variable coupling on the control of the trolley, the hook and the load state, and suppress the total disturbance of the load lifting process, a parallel linear extended state observer (PLESO) is designed to eliminate the influence of the mutual coupling of variables and estimate the motion state and disturbance of the trolley, the hook and the load; the "total disturbance" of each loop in equations (1)-(3) is expanded into a new state variable in the original system and expressed as a parallel system form containing the "total disturbance":

[0070]

[0071] In the formula, in

[0072] Describe equation (4) using a block matrix:

[0073]

[0074] In the formula, O is an all-zero matrix, C = [EOO] ∈ R 3×9 is the output matrix;

[0075] When q3 is unknown and the interference model is uncertain, according to the design idea of ​​LESO, the parallel linear extended state observer (PLESO) shown in formula (6) is established:

[0076]

[0077] In the formula, and They are and The observation state, the observation error is defined as is the observer gain matrix, where ⊙ represents the Hadamard product of two matrices,

[0078] Using the bandwidth method, the observation gain of PLESO is configured in the bandwidth On, and:

[0079]

[0080] Step B: In order to overcome the influence of the disturbance observation error and observation delay of PLESO on the control accuracy and real-time performance, the observation value of the total disturbance and its observation static difference are superimposed and feedforward compensation is performed to improve the control accuracy and real-time performance;

[0081] It can be seen from equations (1) to (3) that due to the coupling relationship between different state quantities, the estimation errors of each link are superimposed on each other on the original basis, which affects the estimation accuracy. Although increasing the observer bandwidth can improve the observation accuracy and real-time performance, in practical applications, due to the influence of process noise and measurement noise, the observer bandwidth cannot be too high, and the efficiency of disturbance compensation is affected. To overcome this influence, the difference between equations (5) and (6) is made to obtain the estimation error of PLESO:

[0082]

[0083] In the formula,

[0084] The PLESO observer gain L configured based on the bandwidth method makes A e It is Hurwitz stable, and the error equation can achieve asymptotic stability, that is:

[0085]

[0086] Substituting equation (9) into equation (8), we get the system observation static error:

[0087]

[0088] In the formula,

[0089] The total disturbance differential term observed using equation (6) Substitute H into equation (10) to obtain the observed static error D, and use it to compensate the observed value of PLESO to obtain the observed compensator PLESO+D:

[0090]

[0091] In the formula, D is the observed compensation, Z is the observed value, and Z' is the observed value after compensation. Expanding formula (11) yields the total disturbance w of the trolley displacement, hook swing angle, and load swing angle loop: x ,w θ , The compensated observation value z' x3 ,z' θ3 ,

[0092] Step C, introducing the auxiliary function to limit the control force, and combining it with the sliding mode control, designing a sliding mode controller for the crane to run according to the set trajectory under the saturation constraint, and using an observation compensator to compensate for the factors causing the crane disturbance, so that the crane can accurately reach the target position under the input saturation constraint and internal and external interference, and quickly eliminate the swing of the distributed mass load;

[0093] In the case of total disturbance and excessive initial swing angle, the actuator may be saturated and uncontrollable even under rated load. Therefore, a sliding mode controller is designed to control input saturation to ensure that the crane can move accurately along the desired trajectory when the actuator input is limited. When the actuator power is limited, the control input of the trolley cannot exceed the maximum value u max ,u max > 0, let the actual control input of the trolley under input saturation be u x , the ideal control input is v x ,definition:

[0094]

[0095] In order to make the system work normally when the control input is limited, an auxiliary control system is introduced to limit the control input. The designed auxiliary system is:

[0096]

[0097] In the formula, Δu x =u x -v x , χ=[χ1,χ2] T is the state variable of the auxiliary system, k1>0, k2>0, b is the control gain. Since K is Hurwitz stable, when t→∞, χ1→0, χ2→0;

[0098] Introduce the auxiliary variable χ1 into the actual displacement of the trolley and the target displacement trajectory x of the trolley r In the definition of error: e χ =xx r -χ1, then

[0099] Trolley target displacement trajectory x r Use the following formula:

[0100]

[0101] In the formula, n1 controls the running time of acceleration, acceleration and deceleration, deceleration and deceleration stages, n2 controls the running time of uniform acceleration stage, n3 controls the running time of uniform speed stage, n4 controls the running time of uniform deceleration stage, x d is the target position of the trolley, v m is the rated operating speed, a m is the rated operating acceleration, j m is the rated operating jerk, c∈R+ is the initial acceleration adjustment parameter;

[0102] In order to smoothly move the load along the target trajectory x r Lift to target position x d , the target values ​​of the hook and load swing angle and angular velocity are 0, so the sliding surface s is defined as:

[0103]

[0104] The derivative with respect to time is:

[0105]

[0106] Substituting equations (1)-(3) into the above equation, we get:

[0107]

[0108] Let Δu=u x -v x Substitute into the above formula and change w x ,w θ , Use the compensated estimated value z' x3 ,z' θ3 , Instead, we get:

[0109]

[0110] make And add the switching function ξsgn(s) based on the sliding surface s (where ξ is a positive number), and the ideal control input is obtained as:

[0111]

[0112] Since the sign function sgn(s) is discontinuous, the above controller will cause system chattering. Therefore, the hyperbolic tangent function tanh(s) is used to replace the sign function sgn(s) to obtain the ideal control law for crane lifting distributed mass load:

[0113]

[0114] Step D: Experimentally illustrate that the proposed controller can enable the underactuated crane to accurately lift the distributed mass load to the target position under the conditions of input saturation constraints, external interference, parameter changes and random initial swing angles, and effectively suppress the load swing during the entire lifting process.

[0115] In order to verify the effectiveness, stability and robustness of the Anti-input-saturation and anti-wind-disturbance sliding mode controller for underacted crane with distributed mass (ASMC) considering input saturation and wind resistance, relevant simulation experiments were carried out. The system parameters of the crane in the experiment are as follows: m1=20kg, m2=1kg, m3=2kg, g=9.8m / s 2 , l1=2m, l2=0.5m, l p =0.6m, the initial position of the trolley is x0=0m, and the target position is x d , v m =1m / s, a m =0.2m / s2 , j m =0.2m / s 3 After parameter tuning, the parameters of the proposed controller are selected as follows: c1 = 0.455, c2 = -0.02, c3 = 7.8, c4 = 0.003, c5 = -3.2, k1 = 10, k2 = 10, ω x =8,ω θ =5,

[0116] In order to better demonstrate the effectiveness of the proposed ASMC, it is compared with the enhanced coupled sliding mode controller (PDSMC) and the time-varying sliding mode controller (VP-TVSMC). The control law expression of the PDSMC method is shown in Equation (24), and its control parameters are selected as: k p =15.89, k d =-2.6, H=25, λ=0.3.

[0117]

[0118] in,

[0119] The control law expression of VP-TVSMC method is shown in formula (25), and its control parameters are selected as: K = 19, ε = 0.15, k c =0.26, k α =0.23, α=-12.4, β=5.

[0120]

[0121] Among them, e ε =x d -x,

[0122] To intuitively represent the control effects of the three control methods, the following indicators are compared:

[0123] (1)x d , i.e. the target position of the trolley;

[0124] (2) Overshoot (OS), expressed as:

[0125] (3) Adjustment time T sx , T sθ , That is, the time it takes for the system to reach the target position, as well as the time it takes for the hook and load angles to clear. Since the uniform speed period is long in long-distance lifting operations, T sθ , It includes two indicators: acceleration section adjustment time and deceleration section adjustment time.

[0126] (4)θ max , θ re and That is, the maximum and residual swing angles of the hook and load during the entire operation;

[0127] (5)u x-max , which is the maximum driving force of the trolley.

[0128] In order to verify the good performance of the proposed control strategy in the long-distance lifting of distributed mass loads by underactuated double-pendulum crane, simulation experiments are conducted to compare the proposed ASMC method with the existing methods PDSMC and VP-TVSMC at the target distance x. d =4m. To increase the comparability of the three control strategies, in this group of experiments, the trolley trajectory is generated by equation (17) to avoid the influence of different ideal trajectories on the control effect. The simulation control effect of the proposed ASMC method and the existing methods PDSMC and VP-TVSMC is shown in Figure 3 As shown in Table 1, Figure 3 The curves from top to bottom are the trolley displacement, hook swing angle, load swing angle and actuator control quantity, x r Indicates the ideal running trajectory of the trolley.

[0129] Table 1x d =4mComparison of control performance of three controllers

[0130]

[0131] It can be analyzed that PDSMC performs the worst in trolley trajectory tracking and positioning, with an overshoot of about 1.38%, and takes the longest time to complete positioning. In contrast, ASMC performs the best in positioning time, reaching the target position in only 10.1 seconds. In terms of secondary swing angle control, the control time of ASMC is significantly less than the other two control strategies, 32.4 seconds faster than PDSMC and 6.4 seconds faster than VP-TVSMC. Although the maximum swing angle during the control process of VP-TVSMC is smaller than that of ASMC and PDSMC, the overall gap is not obvious. In terms of driving force, ASMC requires the smallest control force output by the actuator, showing a more efficient performance characteristic.

[0132] In actual transportation, the distributed mass load is prone to a certain initial swing angle due to uneven mass distribution. If it is not properly controlled, it will swing during lifting, seriously affecting safety and work efficiency. d =30m, when the initial swing angles of the first and second stages are both 10°, and the controller parameters remain unchanged, the simulation results under the three control strategies are as follows Figure 4As shown in Table 2. It can be seen that when there is an initial swing angle, PDSMC can no longer meet the control requirements, and both swing angles are divergent. In contrast, ASMC and VP-TVSMC have better control performance, both of which can enable the system to overcome the impact of the initial swing angle and make the system stable. However, whether in the acceleration stage or the deceleration stage, the time and control force required to achieve complete swing elimination using the ASMC method are much lower than those of VP-TVSMC. This shows that in the presence of an initial swing angle, the ASMC control strategy can accurately, smoothly, quickly and efficiently deliver the distributed mass load to the target position with lower energy consumption.

[0133] Table 2 Comparison of control performance of three controllers with initial swing angle of 10°

[0134]

[0135] In order to test the anti-interference ability of ASMC under actuator input saturation, the upper limit of the actuator control force is set to 24N, the initial swing angle of the load is 10°, and the load is subjected to four different types of interference as shown in formula (26): pulse disturbance at 30-31s, linear disturbance at 35-45s, random disturbance at 45-50s, and sinusoidal disturbance with an intensity of 5° at 50-55s. The load disturbance and simulation results are shown in Figure 26. Figure 5 As shown in the figure, it can be seen that under the action of this algorithm, the double-swing crane system can quickly suppress and control external disturbances when the actuator input is limited, and can effectively eliminate the swing, has strong anti-interference ability, and has good control effect.

[0136]

Claims

1. A sliding mode auto-disturbance rejection control method for an underactuated crane with distributed mass load, It is characterized in that Follow these steps to implement: Step A: Based on the dynamic model of underactuated crane lifting distributed mass load, a parallel linear extended state observer (PLESO) is designed to eliminate the influence of mutual coupling of variables, and a feedforward compensator is designed for the objective estimation error. At the same time, the total disturbance of the system caused by linearization error and time-varying air resistance is estimated to overcome the influence of the unmodeled part of the system, uncertain factors, internal and external interference, linearization error and random initial swing angle on the lifting process of distributed mass load; Ignoring the damping caused by the mass and bending of the rope, and using Lagrange's theorem, the dynamic model of the underactuated crane system lifting distributed mass loads with wind resistance can be obtained: In the formula, m,m 1 ,m 2 are the masses of the trolley, hook and load respectively, x is the displacement of the trolley, u x ,F rx are the driving force and friction force of the trolley, θ, are hook and load swing angles respectively, R is the ratio of load mass to hook mass, R=m 2 / m 1 ; l 1 ,l 2 ,l p are the lengths of the wire rope, the inclined rope, and the distributed mass load, respectively, l h is the distance from the hook to the distributed mass load, They represent the wind resistance of the trolley, hook and load respectively. During the load lifting process, the swing amplitude of the hook and the load usually does not exceed 10°, so sinθ≈θ, cosθ≈1, The original nonlinear model is linearized to obtain the following model: In the formula, w x ,w θ , are the total disturbances affecting the trolley displacement, hook swing angle and load swing angle, ξ 1 ,ξ 2 ,ξ 3 is the error generated during the linearization process; It can be seen from equations (4)-(6) that the crane system needs to control the swing of the hook and the load by moving the trolley and lift the load to the target position. In this process, the displacement of the trolley and the swing of the hook and the load are coupled with each other, and the states of the trolley, hook and load are all affected by the unknown total disturbance. In order to eliminate the adverse effects of variable coupling on the control of the trolley, hook and load states and suppress the total disturbance of the load lifting process, a parallel linear extended state observer (PLESO) is designed to eliminate the influence of variable coupling and estimate the motion states and disturbances of the trolley, hook and load; the "total disturbance" of each loop in equations (4)-(6) is expanded into a new state variable in the original system and expressed as a parallel system form containing the "total disturbance": In the formula, in Describe equation (7) using a block matrix: In the formula, O is an all-zero matrix, C = [EOO] ∈ R 3×9 is the output matrix; In q 3 In the case of unknown and uncertain interference models, according to the design idea of ​​LESO, the parallel linear extended state observer (PLESO) shown in formula (9) is established: In the formula, and They are and The observation state, the observation error is defined as is the observer gain matrix, where ⊙ represents the Hadamard product of two matrices, Using the bandwidth method, the observation gain of PLESO is configured in the bandwidth On, and: Step B: In order to overcome the influence of the disturbance observation error and observation delay of PLESO on the control accuracy and real-time performance, the observation value of the total disturbance and its observation static difference are superimposed and feedforward compensation is performed to improve the control accuracy and real-time performance; It can be seen from equations (4) to (6) that due to the coupling relationship between different state quantities, the estimation errors of each link are superimposed on each other on the original basis, which affects the estimation accuracy. Although increasing the observer bandwidth can improve the observation accuracy and real-time performance, in practical applications, due to the influence of process noise and measurement noise, the observer bandwidth cannot be too high, and the efficiency of disturbance compensation is affected. To overcome this influence, the difference between equations (8) and (9) is made to obtain the estimation error of PLESO: In the formula, The PLESO observer gain L configured based on the bandwidth method makes A e It is Hurwitz stable, and the error equation can achieve asymptotic stability, that is: Substituting equation (12) into equation (11), we get the system observation static error: In the formula, The total disturbance differential term observed using equation (9) Substitute H into equation (13) to obtain the observed static error D, and use it to compensate the observed value of PLESO to obtain the observed compensator PLESO+D: In the formula, D is the observed compensation, Z is the observed value, and Z' is the observed value after compensation. According to formula (14), the total disturbance w of the trolley displacement, hook swing angle and load swing angle loop can be obtained: x ,w θ , The compensated observation value z' x3 ,z' θ3 , Step C, introducing the auxiliary function to limit the control force, and combining it with the sliding mode control, designing a sliding mode controller for the crane to run according to the set trajectory under the saturation constraint, and using an observation compensator to compensate for the factors causing the crane disturbance, so that the crane can accurately reach the target position under the input saturation constraint and internal and external interference, and quickly eliminate the swing of the distributed mass load; In the case of total disturbance and excessive initial swing angle, the actuator may be saturated and uncontrollable even under rated load. Therefore, a sliding mode controller is designed to control input saturation to ensure that the crane can move accurately along the desired trajectory when the actuator input is limited. When the actuator power is limited, the control input of the trolley cannot exceed the maximum value u max ,u max > 0, let the actual control input of the trolley under input saturation be u x , the ideal control input is v x ,definition: In order to make the system work normally when the control input is limited, an auxiliary control system is introduced to limit the control input. The designed auxiliary system is: In the formula, Δu x =u x -v x , χ=[χ 1 ,x 2 ] T is the state variable of the auxiliary system, k 1 >0,k 2 > 0, b is the control gain, since K is Hurwitz stable, when t→∞, χ 1 →0,χ 2 →0; The auxiliary variable χ 1 Introduced into the actual displacement of the trolley and the target displacement trajectory of the trolley x r In the definition of error: e χ =xx r -x 1 ,but Trolley target displacement trajectory x r Use the following formula: In the formula, n 1 Control the running time of acceleration, deceleration, acceleration and deceleration stages, n 2 Control the running time of the uniform acceleration stage, n 3 Control the running time of the uniform speed stage, n 4 Control the running time of the uniform deceleration stage, x d is the target position of the trolley, v m is the rated operating speed, a m is the rated operating acceleration, j m is the rated operating jerk, c∈R+ is the initial acceleration adjustment parameter; In order to smoothly move the load along the target trajectory x r Lift to target position x d , the target values ​​of the hook and load swing angle and angular velocity are 0, so the sliding surface s is defined as: The derivative with respect to time is: Substituting equations (4)-(6) into the above equation, we get: Let Δu=u x -v x Substitute into the above formula and change w x ,w θ , Use the compensated estimated value z' x3 ,z' θ3 , Instead, we get: make And add the switching function ξsgn(s) based on the sliding surface s (where ξ is a positive number), and the ideal control input is obtained as: Since the sign function sgn(s) is discontinuous, the above controller will cause system chattering. Therefore, the hyperbolic tangent function tanh(s) is used to replace the sign function sgn(s) to obtain the ideal control law for crane lifting distributed mass load: Substituting the above equation into equation (15), we can obtain the actual control law of the trolley under the input saturation constraint. Under the control of this control law, the crane under the input saturation constraint can accurately lift the distributed mass load to the target position under the conditions of state variable coupling, external interference, parameter changes and random initial swing angle, and effectively suppress the load swing during the entire lifting process.

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