Anti-sway sliding mode control method for bridge crane with dynamic base based on Hurwitz stability

Abstract

The invention discloses a sliding mode control method for anti-sway of a moving base bridge crane based on Hurwitz stability, comprising the steps of S1. establishing a system dynamics model of the moving base bridge crane based on the Lagrange equation; S2. introducing the state variables of the system, and designing the control law system total input of the crane system of the controller according to the simplified model of the moving base bridge crane system; the method can effectively solve the problem of many parameters and difficult setting of the sliding mode controller by converting the sliding mode surface parameters into single parameter adjustment by utilizing the properties of the Hurwitz stability matrix, and has the characteristics of good trolley positioning, load anti-sway, and controller energy consumption and robustness.

Description

Technical Field

[0001] The invention relates to the technical field of bridge crane control, and in particular to an anti-sway sliding mode control method for a moving base bridge crane based on Hurwitz stability. Background Art

[0002] At present, bridge cranes, as transportation machinery, are widely used in various fields of national economic construction and play an extremely important role. In order to create greater value, the working efficiency of bridge cranes must be continuously improved, and the safety operation index, trolley positioning and load anti-sway performance largely determine the transportation efficiency of bridge cranes.

[0003] Marine cranes are easily affected by wind and waves during lifting operations and belong to a type of lifting equipment under the excitation of a dynamic base. Among them, dynamic base bridge cranes are mainly used for the transfer of containers between large container transport ships and floating platforms at sea. Considering the flexible traction drive mode of the wire rope, the swing of the hoisted objects will directly affect the safety and work efficiency of the crane. The strong coupling, nonlinearity and uncertainty of the hoisting system of the dynamic base bridge crane greatly increase the complexity of the system. The research on its anti-sway control is an important and challenging task, and is currently a hot and difficult issue in the research of marine cranes.

[0004] At present, there is no corresponding control method that can effectively solve the problems existing in the above-mentioned prior art. Therefore, it is urgent to design a method that can effectively control the swing of the hanging objects of the dynamic base bridge crane during use, and the problems of many parameters and difficult adjustment of the sliding mode controller. Summary of the invention

[0005] In view of the above-mentioned problems, the present invention aims to provide a sliding mode control method for anti-sway of a dynamic base bridge crane based on Hurwitz stability. The method converts the sliding mode surface parameters into single parameter adjustment by utilizing the properties of the Hurwitz stability matrix, which can effectively solve the problem of many parameters and difficult setting of the sliding mode controller, and has the characteristics of good trolley positioning, load anti-sway, and controller energy consumption and robustness.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A sliding mode control method for anti-swaying of a moving base bridge crane based on Hurwitz stability, comprising the steps of:

[0008] S1. Establishment of system dynamics model of moving base bridge crane based on Lagrange equation

[0009] S101. In a moving base bridge crane, three coordinate systems are introduced to establish a two-dimensional dynamic model of the moving base bridge crane;

[0010] S102. Linearize the two-dimensional dynamic model of the moving base bridge crane to obtain a simplified linear mathematical model of the moving base bridge crane system;

[0011] S2. Introduce the state variables of the system, and design the control law of the crane system based on the simplified linear mathematical model of the moving base bridge crane system.

[0012] Preferably, the process of establishing the two-dimensional dynamic model of the moving base bridge crane in step S101 includes:

[0013] (1) Construct a two-dimensional plane model of the moving base bridge crane system;

[0014] (2) Based on the two-dimensional plane model of the moving base bridge crane system, the plane reference coordinate system O is introduced. 0 X 0 Y 0 , the coordinate system O fixed at the center of mass of the base and moving with the base s X s Y s and the car moving coordinate system O t X t Y t , in the reference coordinate system O 0 X 0 Y 0 Under the geometric relationship between the load swing angle and the pendulum length, the displacement p of the trolley is M and the load displacement p m The horizontal and vertical components of are expressed as:

[0015]

[0016]

[0017] Among them, the plane reference coordinate system O 0 X 0 Y 0 The center of mass of the crane base at rest is taken as the origin, and the horizontal direction is taken as the X 0 Axis positive direction, with the direction perpendicular to the ground as Y 0 The axis is the coordinate axis established in the positive direction;

[0018] In equations (2) and (3), h is the height of the crane gantry, x is the position of the trolley at O s X s Y s The coordinate position in the reference coordinate system, l is the length from the center of mass of the trolley to the center of mass of the hanging weight, θ is the swing angle of the hanging weight along the direction of track operation, y is the heave displacement of the moving base, φ is the roll angle of the moving base caused by the wave motion, and (y, φ) is the motion state vector definition of the ship;

[0019] (3) Differentiate equations (2) and (3) with respect to time to obtain the velocity v of the car: M and the speed v of the load m The expression is as follows:

[0020] v M =[v Mx v My ] T (4)

[0021] v m =[v mx v my ] T (5)

[0022] Among them, in formula (4) and formula (5):

[0023]

[0024]

[0025]

[0026]

[0027] (4) The kinetic energy of the system is the kinetic energy of the car, T 1 , load kinetic energy T 2 The sum is:

[0028] T=T 1 +T 2 (6)

[0029] Where: In formula (6),

[0030] (5) Taking the center of mass of the car as the zero potential energy point, the potential energy of the system is:

[0031] U=Mg(y+hcosθ+xsinφ)-m p glcosθ+m p g(y+hcosφ+xsinφ) (7)

[0032] The kinetic potential energy of the above system does not include the kinetic energy and potential energy of the ship or platform carrying the crane, and the motion state (y, φ) generated by the movement of the base is treated as an interference term;

[0033] (6) During the horizontal swing of the load, the friction torque generated by wind resistance is f v , let μ be the wind resistance friction coefficient, then the generalized force in the direction of the state quantity θ is

[0034]

[0035] According to formula (8), the friction between the trolley and the horizontal track includes dynamic friction and static friction model F r for

[0036]

[0037] In formula (9), F r0 , ε f , k x is an adjustable parameter;

[0038] (7) According to formula (9), the generalized force in the x direction of the state quantity can be obtained:

[0039]

[0040] (8) The dynamic equation of the moving base bridge crane system with respect to the state variables x, θ is obtained from the Lagrange equation:

[0041]

[0042]

[0043] Among them, the mathematical models represented by equations (11) and (12) are nonlinear dynamic differential equations that take into account the swing damping of the moving base bridge crane.

[0044] Preferably, the simplification process of the two-dimensional dynamic model of the moving base bridge crane in step S102 includes:

[0045] Let sinθ≈θ, sinφ≈φ, cosθ≈1, cosφ≈1, θ 2 ≈0, simplifying equations (11) and (12), the linear mathematical model of the system is:

[0046]

[0047]

[0048] Preferably, the design process of the total input of the control law system of the crane system of the controller described in step S2 includes:

[0049] S201. Introduce the state variables that define the system

[0050]

[0051] According to the linear mathematical model of the dynamic base bridge crane system, the original system dynamics equation can be further transformed into:

[0052]

[0053] In equation (16), u(t) = F x -f rx , represents the input control quantity of the car operation, d 1 ,d 2 is the uncertain variable composed of external disturbance and system parameter perturbation, and |d 1,2 |≤D,g 1 and g 2 is the nonlinear dynamics of the model, b 1 and b 2 They represent the control gain of the displacement and swing angle loops respectively, and their values ​​are:

[0054]

[0055] In formula (17), Δ 1 ,Δ 2 The uncertain dynamics caused by the simplification of the system, M and m represent the masses of the trolley and the load, respectively, and g is the acceleration due to gravity;

[0056] S202. According to the underactuated system shown in equation (16), the system state variable X=[x 1 ,x 2 ,x 3 ,x 4 ] T Divided into the first group consisting of the position and speed state of the car (x 1 ,x 2 ), the second group (x 3 ,x 4 );

[0057] S203. According to the above (x 1 ,x 2 ) and (x 3 ,x 4 ), transforming the original system into a car subsystem Σ 1 and the swing angle subsystem Σ 2 ;

[0058] S204. Let x d ,θ d To control the target, the positioning error of the car during operation is e x Defined as e x =x 1 -x d , swing angle state error e θ Defined as e θ =x 3 -θ d, the sliding surface s including the positioning error of the trolley and the swing angle positioning error is:

[0059]

[0060] In formula (18), c 1 、c 2 、c 3 is the unknown coefficient, and the derivative of the sliding surface s can be obtained

[0061]

[0062] S205. Set the control target θ of the load swing angle d =0, then e θ =θ, Combined (16)

[0063] Available

[0064] S206. Order Design the equivalent control quantity u of the system eq for:

[0065]

[0066] S207. Add the switching control law of sliding surface to the control law sw , we can get the exponential reaching law

[0067]

[0068] S208. The control law of the crane system of the controller is obtained. The total input of the system is

[0069]

[0070] The beneficial effects of the present invention are as follows: the present invention discloses a sliding mode control method for anti-swaying of a moving base bridge crane based on Hurwitz stability. Compared with the prior art, the present invention has the following improvements:

[0071] (1) The present invention designs a sliding mode control method for anti-sway of a moving base bridge crane based on Hurwitz stability. The method introduces the sliding mode control method into a strongly nonlinear, coupled, and uncertain underactuated mechanical system such as a moving base bridge crane, which can effectively improve the robustness of the control system. Compared with traditional control methods, it has advantages in terms of system uncertain parameters and response speed, and can achieve better hoist positioning and anti-sway functions.

[0072] (2) By utilizing the properties of the Hurwitz stability matrix, the sliding surface parameters are converted into single parameter adjustments, which can effectively solve the problems of multiple parameters and difficult tuning in the design process of the sliding mode controller, improve the feasibility of the control system, and have the advantages of good positioning of the trolley, load sway elimination, and controller energy consumption and robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 The present invention is a flow chart of the anti-sway sliding mode control method of the moving base bridge crane based on Hurwitz stability.

[0074] Figure 2 It is a schematic diagram of the two-dimensional moving base bridge crane system of the present invention.

[0075] Figure 3 This is a comparison diagram of the displacement simulation results of the three controller carts in Example 2 of the present invention.

[0076] Figure 4 This is a comparison diagram of load swing angle simulation results of three controllers in Example 2 of the present invention.

[0077] Figure 5 This is a simulation comparison result diagram of the control quantities of three controllers in Example 2 of the present invention.

[0078] Figure 6 This is a diagram of the simulation results of the trolley displacement under parameter changes in Example 2 of the present invention.

[0079] Figure 7 This is a diagram of the load swing angle simulation results under parameter changes in Example 2 of the present invention.

[0080] Figure 8 This is a diagram of the simulation results of the control quantity under parameter changes in Example 2 of the present invention. DETAILED DESCRIPTION

[0081] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0082] See attached Figure 1-8 The anti-sway sliding mode control method of a moving base bridge crane based on Hurwitz stability is shown, comprising the steps of

[0083] S1. Establishment of system dynamics model of moving base bridge crane based on Lagrange equation

[0084] The dynamic base bridge crane is a multivariable coupled nonlinear under-actuated system. Various factors in the actual operation process will affect the working performance, such as the change of the trolley's running speed, the change of the load's size and weight, the interference of wind force, the change of the wire rope length, the movement of the hull with the wave's heave and roll, the friction between the trolley and the track, etc. Therefore, it is necessary to establish a dynamic model of the anti-sway system of the dynamic base crane with high accuracy.

[0085] The motion of the hull of a gantry crane with a moving base under the action of waves is very complex. Because of the influence of waves, the hull has six degrees of freedom of base-excited motion. The following assumptions are made in the modeling:

[0086] (1) The wire rope has no mass, no elastic deformation, and is kept in a tensioned state during the lifting process, that is, the change in the length of the wire rope is ignored;

[0087] (2) Treat the load and the sling as point masses, ignoring their volume and moment of inertia;

[0088] (3) Ignore the air resistance of the sling during the swing process, and also ignore the friction between the wire rope and the trolley;

[0089] (4) Only the left and right swing of the spreader in the horizontal direction (x direction) is considered, and the influence of the front and back swing (y direction) is ignored;

[0090] (5) Ignore the torque of the wire rope;

[0091] (6) Ignore the nonlinear effects of the trolley motor and reducer and other transmission devices, and assume that the torque output by the inverter can directly control the input force of the trolley;

[0092] (7) Without considering extremely special conditions, assume that the swing of the hanging weight is always below the horizontal plane of the trolley, that is:

[0093]

[0094] S101. In the moving base bridge crane, three coordinate systems are introduced to establish the two-dimensional dynamic model of the moving base bridge crane

[0095] The dynamic base bridge crane system has the characteristics of multivariable, strong nonlinearity and under-actuation. It is difficult to establish the system model using Newtonian mechanics. The Lagrangian method describes the system dynamics from the perspective of energy, and is usually used to solve more complex particle system dynamics problems with ideal constraints. The equation does not need to analyze the unknown constraint force, but only the known active force, which can greatly simplify the motion equation analysis of the modeling process. Therefore, this method uses the second form of Lagrangian to establish the crane system dynamics model. The general form of the equation is:

[0096]

[0097] In formula (1), T and U represent the kinetic energy and potential energy of the system, respectively, and Q k is the generalized force of the system, q k Represents the generalized state quantity of the system; in order to establish the system dynamics equation using the Lagrange equation, it is necessary to analyze the kinetic potential energy and generalized force of the crane system;

[0098] (1) The relative motion between the moving base gantry crane and the large container ship or fixed platform is more critical. Since the container ship is usually large in size, its rotational maneuverability is not affected by the wave action. To simplify the analysis, it is assumed that the mother ship is stationary. Since the moving base crane is moored to the mother ship, its movement can be suppressed to a certain extent by the mother ship. Therefore, the simplified model can ignore the sway, pitch and pitch of the base, and only discuss the heave and roll of the base. At this time, the external angle of the load swing surface is regarded as zero, so the two-dimensional plane model of the system can be constructed as follows: Figure 2 As shown;

[0099] (2) Based on the two-dimensional plane model of the moving base bridge crane system, the plane reference coordinate system O is introduced. 0 X 0 Y 0 , the coordinate system O fixed at the center of mass of the base and moving with the base s X s Y s and the car moving coordinate system O t X t Y t , in the reference coordinate system O 0 X 0 Y 0 Under the geometric relationship between the load swing angle and the pendulum length, the displacement p of the trolley is M and the load displacement p m The horizontal and vertical components of are expressed as:

[0100]

[0101]

[0102] Among them, the plane reference coordinate system O 0 X 0 Y 0 The center of mass of the crane base at rest is taken as the origin, and the horizontal direction is taken as the X 0 Axis positive direction, with the direction perpendicular to the ground as Y 0 The axis is the coordinate axis established in the positive direction;

[0103] In equations (2) and (3), h is the height of the crane gantry, g is the acceleration of gravity, and x is the velocity of the trolley at O.s X s Y s The coordinate position in the reference coordinate system, l is the length from the center of mass of the trolley to the center of mass of the load, and θ is the swing angle of the load in the direction of the track; F x To represent the driving force applied to the trolley, the friction force during the system movement is considered, and the friction damping between the trolley and the track is expressed as F r , its direction is opposite to the direction of motion; y is the heave displacement of the dynamic base, which reciprocates in the reference coordinate system; φ is the roll angle of the dynamic base caused by the wave motion, so the motion state vector of the ship can be defined as (y, φ);

[0104] (3) Differentiate equations (2) and (3) with respect to time to obtain the velocity v of the car: M and the speed v of the load m The expression is as follows:

[0105] v M =[v Mx v My ] T (4)

[0106] v m =[v mx v my ] T (5)

[0107] Among them, in formula (4) and formula (5):

[0108]

[0109]

[0110]

[0111]

[0112] (4) The kinetic energy of the system is the kinetic energy of the car, T 1 , load kinetic energy T 2 The sum is:

[0113] T=T 1 +T 2 (6)

[0114] Where: In formula (6),

[0115] (5) Taking the center of mass of the car as the zero potential energy point, the potential energy of the system is:

[0116] U=Mg(y+hcosθ+xsinφ)-m pglcosθ+m p g(y+hcosφ+xsinφ) (7)

[0117] The kinetic potential energy of the above system also does not include the kinetic energy and potential energy of the ship or platform carrying the crane, and the motion state (y, φ) generated by the movement of the base is treated as an interference term;

[0118] (6) In actual operation, the crane is affected by friction from many aspects. During the horizontal swing of the load, it will be affected by wind resistance, generating a friction torque f v The direction of the wind resistance is opposite to the swing speed. Let μ be the wind resistance friction coefficient, then the generalized force in the direction of the state quantity θ is:

[0119]

[0120] According to formula (8), the friction between the trolley and the horizontal track includes dynamic friction and static friction model F r for

[0121]

[0122] In formula (9), F r0 , ε f , k x is an adjustable parameter;

[0123] (7) According to formula (9), the generalized force in the x direction of the state quantity can be obtained:

[0124]

[0125] (8) The dynamic equation of the moving base bridge crane system with respect to the state variables x, θ is obtained from the Lagrange equation:

[0126]

[0127]

[0128] Among them, the mathematical models expressed by equations (11) and (12) are nonlinear dynamic differential equations that take into account the swing damping of the moving base bridge crane;

[0129] S102. Simplify the system model of the moving base bridge crane

[0130] (1) It can be seen from the above system dynamics model that there is a complex nonlinear coupling relationship between the various state quantities of the dynamic base bridge crane system, which makes the design of the controller complicated. In order to facilitate the analysis of the system dynamics characteristics under different working conditions and design a suitable anti-sway device, the two-dimensional crane mathematical model is reasonably simplified and some high-order terms in the equation are processed. In actual situations, the roll angle of the base movement and the maximum swing angle of the load are generally small, so it is assumed that sinθ≈θ, sinφ≈φ, cosθ≈1, cosφ≈1, θ 2 ≈0; at this time, the system model can be linearized to obtain the linear mathematical model of the system as follows:

[0131]

[0132]

[0133] S2. Introduce the state variables of the system, design the control law of the crane system based on the simplified model of the moving base bridge crane system.

[0134] S201. Introduce the state variables that define the system

[0135]

[0136] According to the linear mathematical model of the dynamic base bridge crane system, the original system dynamics equation can be further transformed into:

[0137]

[0138] In equation (16), u(t) = F x -f rx , represents the input control quantity of the car operation, d 1 ,d 2 is the uncertain variable composed of external disturbance and system parameter perturbation, and |d 1,2 |≤D,g 1 and g 2 is the nonlinear dynamics of the model, b 1 and b 2 They represent the control gain of the displacement and swing angle loops respectively, and their values ​​are:

[0139]

[0140] In formula (17), Δ 1 ,Δ 2 The uncertain dynamics caused by the simplification of the system, M and m represent the masses of the trolley and the load, respectively, and g is the acceleration due to gravity;

[0141] S202. For the mobile base bridge crane system, the purpose of designing the controller is to overcome the influence of the harsh water environment and transport the load to the target position accurately and quickly. d , and can it fully suppress the swing of the load in the entire process? Therefore, the design of the controller needs to consider the dual goals of swing angle suppression and car positioning; According to the physical nature of the under-actuated system shown in formula (16), the system state variable X = [x 1 ,x 2 ,x 3 ,x 4 ] T It is divided into two groups, including the first group consisting of the position and speed state of the car (x 1 ,x 2 ), the second group (x 3 ,x 4 );

[0142] S203. According to the above (x 1 ,x 2 ) and (x 3 ,x 4 ), transforming the original system into a car subsystem Σ 1 and the swing angle subsystem Σ 2 ;

[0143] S204. Let x d ,θ d To control the target, the positioning error of the car during operation is e x Defined as e x =x 1 -x d , swing angle state error e θ Defined as e θ =x 3 -θ d In order to solve the problem that one control variable of the underactuated crane system can achieve the dual control objectives of positioning and anti-sway at the same time, a sliding surface including the positioning error of the trolley and the swing angle positioning error is constructed to realize the joint control of the two sets of objectives:

[0144]

[0145] In formula (18), c 1 、c 2 、c 3 is the unknown coefficient, and the derivative of the sliding surface s can be obtained

[0146]

[0147] S205. Set the control target θ of the load swing angle d =0, then e θ=θ, Combined (16)

[0148] Available

[0149] S206. Order Design the equivalent control quantity u of the system eq for:

[0150]

[0151] S207. In order to ensure the rapid convergence of the sliding surface, that is, s = 0, and thus ensure the stability of the crane system, the selection of the control quantity u must meet the sliding mode arrival condition Therefore, the switching control law of the sliding surface is added to the control law sw , after an in-depth study of the sliding mode control law, the following exponential reaching law is selected:

[0152]

[0153] S208. Exponential terms included in the exponential reaching law: It can ensure that the system state has a faster convergence speed when it deviates greatly from the sliding mode, but as the system state point approaches the switching surface, the approach speed gradually decreases; at this time, the general constant speed approach law is introduced: The sliding surface switching function improves the approach speed of the system state to the sliding surface, improves its control quality, and further improves the stability of the system. The total input of the design system is:

[0154]

[0155] That is, through the above steps S1 and S2, the anti-sway sliding mode control method for the moving base bridge crane based on Hurwitz stability is established.

[0156] Example 1: In order to verify the feasibility of the above-mentioned anti-sway sliding mode control method for a moving base bridge crane based on Hurwitz stability, the stability of the above-mentioned method is verified:

[0157] Theorem 1: Let ε>L, L=(1+c 2 )D represents the upper bound of the disturbance, then the control law (23) of the crane system can make the sliding surface s shown in equation (17) converge to zero in a finite time;

[0158] Proof: Define the Lyapunov function as:

[0159]

[0160] Obviously, the V function is positive definite, and its derivative yields:

[0161]

[0162] When ε>L, substitute the control law into In, can be obtained

[0163]

[0164] From equation (26) and equation (24), we can get

[0165]

[0166] From equation (27), we can see that the Lyapunov function V(t) and the sliding surface s(t) of the system are both bounded, that is, s,V∈L ∞ ;

[0167] Therefore, according to the Lyapunov stability theory in equation (16), the sliding surface s of the system is asymptotically stable, that is, there exists a finite time t s , so that t ≥ t s When , the sliding surface converges to zero, and the proof is complete.

[0168] Theorem 2: When s = 0, by adjusting c 1 ,c 2 ,c 3 The car can reach the target position within a limited time and quickly eliminate the swing. The state variables of the system converge asymptotically to the equilibrium point, that is:

[0169]

[0170] Proof: Substituting the control law u into equation (16) yields:

[0171]

[0172] Substituting formula (17) into the above formula, we can get

[0173]

[0174] Define y 1 =θ, y 3 =e x , from equations (16), (18), and (30), we can get:

[0175]

[0176] Then formula (31) can be written as:

[0177]

[0178] in

[0179] No interference is considered 2 When the eigenvalue of matrix A is located in the negative half plane far away from the imaginary axis, A can be considered a Hurwitz stable matrix. Therefore, from |A-λE|=0, we can get:

[0180]

[0181] The expected eigenvalue of A is taken as At the same point, solving the above equation yields:

[0182]

[0183] Since A is a Hurwitz stable matrix, there exists a positive definite matrix P such that A T P+PA=-Q, where Q is a positive definite matrix, and the following Lyapunov function is constructed:

[0184] V 1 =y T Py (35)

[0185] When the disturbance d 2 When V is bounded, 1 Taking the derivative and combining it with the matrix properties, we can get:

[0186]

[0187] Among them, λ min (Q) represents the minimum eigenvalue of Q, so when the condition of formula (37) is satisfied, it can be guaranteed that Established.

[0188] λ min (Q)||y|| 2 -2||y||||PB||D>0 (37)

[0189] It can be seen that formula (32) is asymptotically convergent, that is, y 1 →0,y 2 →0,y 3 →0;

[0190] Therefore θ→0, e x →0 holds, and since e x =x→x d , so x→x d , So far, Theorem 2 has been proved.

[0191] In this way, the eigenvalues ​​are arranged in the same position through the Hurwitz stability matrix, and the c of the sliding mode controller of the crane system is 1 ,c2 ,c 3 Converting three parameters to be adjusted into one parameter adjustment not only ensures the convergence of the system tracking state, but also greatly reduces the complexity of parameter adjustment. It can be seen that the anti-sway sliding mode control method for the moving base bridge crane based on Hurwitz stability described in the present invention is theoretically feasible.

[0192] Example 2: Simulation results and analysis

[0193] 1. Comparative analysis of controllers

[0194] In order to verify the positioning and anti-sway effect of the proposed sliding mode control, a Matlab / Simulink simulation experiment is designed with a sampling time of 0.05s; the control effect of the proposed sliding mode controller is compared and analyzed with the traditional PID and LQR; the S-type function based on the sigmoid function is used to plan the trajectory of the car motion; the maximum acceleration A of the car is set max =0.2m / s 2 , rated speed of the trolley V max =1m / s, tracking distance is 10m; in the simulation, the system parameters are set as: M=20kg, m=4kg, g=9.81m / s 2 , l = 0.7m, h = 2.5m; its initial state is set to x(0) = 0, θ(0)=0, The roll angle of the dynamic base in severe sea conditions is φ = 0.01sin (0.8t) rad;

[0195] The sliding mode controller is shown in formula (23). Since the switching function sgn(s) is prone to frequent switching when the state quantity is close to the sliding mode surface, the sliding mode controller has chattering phenomenon. In order to reduce the chattering, the continuous saturation function sat(s) is used to approximate the discontinuous sign function sgn(s). The function sat(s) is defined as:

[0196]

[0197] The sliding mode controller parameters are selected as:

[0198] ε=3, k=2.5, φ=0.2, where ζ is set to ζ=2.2 through parameter optimization;

[0199] The mathematical description of the PID controller is:

[0200]

[0201] The position loop PID controller parameters are selected as: K px =5,K Ix =0.12,Kdx =12, the parameters of the swing angle loop PID controller are selected as: K pθ =12,K Iθ =0.15,K dθ =10;

[0202] The mathematical expression of the LQR controller is:

[0203]

[0204] The corresponding parameter setting is: [k 1 ,k 2 ,k 3 ,k 4 ] = [100.00 68.40 -23.53 -0.13];

[0205] The simulation results of the sliding mode controller SMC are compared with those of the PID and LQR controllers. The simulation comparison results of the real-time position curve of the car, the swing angle curve of the load and the control force curve are shown in the following figure. Figure 3-5 As shown;

[0206] Depend on Figure 3-5 It can be seen that the three control methods can achieve the dynamic base bridge crane trolley to accurately reach the target position x to a certain extent. d That is, the displacement is from 0m to 10m, and it can suppress the swing of the load swing angle to varying degrees; specifically, in view of the requirements of trolley position tracking, although the PID controller has a faster overall movement speed, there is an obvious overshoot, which makes the trolley reach the expected position after 22s of PID control; compared with PID control, the trolley of both SMC and LQR controllers takes about 17s to reach the target position, and can reach it with almost no overshoot; therefore, compared with PID control, LQR and the proposed SMC controller can greatly reduce the running time, improve the operation efficiency of the crane, and can better realize the accurate positioning of the trolley;

[0207] From the swing angle simulation comparison curve, it can be seen that during the operation, the maximum swing angle of the PID controller exceeds 2°, the maximum swing angle of the LQR is about 2°, and the maximum swing angle of the proposed SMC algorithm is only about 1.5°, which is the smallest of the three. After reaching the target position, the continuous swing angle oscillation caused by the SMC controller overcoming the interference of the moving base is significantly smaller than the swing amplitude of the PID controller, and slightly smaller than the swing amplitude of the LQR controller. Therefore, the proposed SMC controller performs best in both the maximum swing angle and the residual swing angle of the whole process.

[0208] In terms of control force, the maximum control force of the SMC controller during the operation of the crane is about 10N, which is less than the maximum control force of the LQR and PID controllers (about 14N). Moreover, it can be observed from the figure that the total energy consumed by the three controllers, that is, the ∫udt index SMC is also the smallest, indicating that the SMC controller is the most energy-efficient;

[0209] In summary, the proposed SMC controller is good in terms of trolley positioning, load sway elimination and controller energy consumption. Overall, the control effect of using SMC is better.

[0210] 2. Robustness Analysis

[0211] Considering that in actual operation, the weight of the goods transported by the crane each time and the height of each lifting may be different, the load mass and rope length are parameters that are easy to change; in order to verify the robustness of the proposed SMC controller to the change of model parameters, different load mass and rope length parameters are set. Parameter 1: m = 2kg, l = 0.4m; Parameter 2: m = 4kg, l = 0.7m; Parameter 3: m = 8kg, l = 1.5m; the tracking target, controller gain, other parameters of the crane and the initial state remain unchanged, and the obtained simulation curve is as follows Figure 6-8 As shown;

[0212] It can be seen that when the model parameters change greatly by changing the load mass and the length of the suspension rope, the anti-sway control effect is not much different; the displacement and swing angle curves under the three sets of parameters are basically coincident. SMC makes the controller insensitive to changes in load mass and rope length by changing the control quantity, which effectively illustrates the good robustness of the control method proposed in the present invention.

[0213] The above shows and describes the basic principles, main features and advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. An anti-sway sliding mode control method for a moving base bridge crane based on Hurwitz stability. Features: Included steps S1. Establishment of system dynamics model of moving base bridge crane based on Lagrange equation S101. In a moving base bridge crane, three coordinate systems are introduced to establish a two-dimensional dynamic model of the moving base bridge crane; S102. Linearize the two-dimensional dynamic model of the moving base bridge crane to obtain a simplified linear mathematical model of the moving base bridge crane system; S2. Introduce the state variables of the system, and design the control law of the crane system of the controller according to the simplified linear mathematical model of the moving base bridge crane system. The design process of the total input of the control law system of the crane system of the controller described in step S2 includes: S201. Introduce the state variables that define the system According to the linear mathematical model of the dynamic base bridge crane system, the original system dynamics equation can be further transformed into: In equation (16), u(t) = F x -f rx , represents the input control quantity of the car operation, d 1 ,d 2 is the uncertain variable composed of external disturbance and system parameter perturbation, and |d 1,2 |≤D,g 1 and g 2 is the nonlinear dynamics of the model, b 1 and b 2 They represent the control gain of the displacement and swing angle loops respectively, and their values ​​are: In formula (17), Δ 1 ,Δ 2 The uncertain dynamics caused by the simplification of the system, M and m represent the masses of the trolley and the load, respectively, and g is the acceleration due to gravity; S202. According to the underactuated system shown in equation (16), the system state variable X=[x 1 ,x 2 ,x 3 ,x 4 ] T Divided into the first group consisting of the position and speed state of the car (x 1 ,x 2 ), the second group (x 3 ,x 4 ); S203. According to the above (x 1 ,x 2 ) and (x 3 ,x 4 ), transforming the original system into a car subsystem Σ 1 and the swing angle subsystem Σ 2 ; S204. Let x d ,θ d To control the target, the positioning error of the car during operation is e x Defined as e x =x 1 -x d , swing angle state error e θ Defined as e θ =x 3 -θ d , the sliding surface s including the positioning error of the trolley and the swing angle positioning error is: In formula (18), c 1 、c 2 、c 3 is the unknown coefficient, and the derivative of the sliding surface s can be obtained S205. Set the control target θ of the load swing angle d =0, then e θ =θ, Combining formula (16), we can get S206. Order Design the equivalent control quantity u of the system eq for: S207. Add the switching control law of sliding surface to the control law sw , we can get the exponential reaching law S208. The control law of the crane system of the controller is obtained. The total input of the system is 2. According to claim 1, a sliding mode control method for anti-sway of a moving base bridge crane based on Hurwitz stability, Features: The process of establishing the two-dimensional dynamic model of the moving base bridge crane described in step S101 includes: (1) Construct a two-dimensional plane model of the moving base bridge crane system; (2) Based on the two-dimensional plane model of the moving base bridge crane system, the plane reference coordinate system O is introduced. 0 X 0 Y 0 , the coordinate system O fixed at the center of mass of the base and moving with the base s X s Y s and the car moving coordinate system O t X t Y t , in the reference coordinate system O 0 X 0 Y 0 Under the geometric relationship between the load swing angle and the pendulum length, the displacement p of the trolley is M and the load displacement p m The horizontal and vertical components of are expressed as: Among them, the plane reference coordinate system O 0 X 0 Y 0 The center of mass of the crane base at rest is taken as the origin, and the horizontal direction is taken as the X 0 Axis positive direction, with the direction perpendicular to the ground as Y 0 The axis is the coordinate axis established in the positive direction; In equations (2) and (3), h is the height of the crane gantry, x is the position of the trolley at O s X s Y s The coordinate position in the reference coordinate system, l is the length from the center of mass of the trolley to the center of mass of the hanging weight, θ is the swing angle of the hanging weight along the direction of track operation, y is the heave displacement of the moving base, φ is the roll angle of the moving base caused by the wave motion, and (y, φ) is the motion state vector definition of the ship; (3) Differentiate equations (2) and (3) with respect to time to obtain the velocity v of the car: M and the speed v of the load m The expression is as follows: v M =[v Mx v My ] T (4) v m =[v mx v my ] T (5) Among them, in formula (4) and formula (5): (4) The kinetic energy of the system is the kinetic energy of the car, T 1 , load kinetic energy T 2 The sum is: T= T 1 +T 2 (6) Where: In formula (6), (5) Taking the center of mass of the car as the zero potential energy point, the potential energy of the system is: U=Mg(y+h cosθ+x sinφ)-m p gcosθ+m p g(y+h cosφ+x sinφ) (7) The kinetic potential energy of the above system does not include the kinetic energy and potential energy of the ship or platform carrying the crane, and the motion state (y, φ) generated by the movement of the base is treated as an interference term; (6) During the horizontal swing of the load, the friction torque generated by wind resistance is f v , let μ be the wind resistance friction coefficient, then the generalized force in the direction of the state quantity θ is According to formula (8), the friction between the trolley and the horizontal track includes dynamic friction and static friction model F r for In formula (9), F r0 , ε f , k x is an adjustable parameter; (7) According to formula (9), the generalized force in the x direction of the state quantity can be obtained: (8) The dynamic equation of the moving base bridge crane system with respect to the state quantities x, θ is obtained from the Lagrange equation: Among them, the mathematical models represented by equations (11) and (12) are nonlinear dynamic differential equations that take into account the swing damping of the moving base bridge crane.

3. According to claim 2, a sliding mode control method for anti-swaying bridge crane based on Hurwitz stability, Features: The simplification process of the two-dimensional dynamic model of the moving base bridge crane described in step S102 includes: Let sinθ≈θ, sinφ≈φ, cosθ≈1, cosφ≈1, θ 2 ≈0, simplifying equations (11) and (12), the linear mathematical model of the system is:

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