Double-pendulum crane backstepping sliding mode control method based on generalized proportional-integral observer

Abstract

The invention discloses a double-pendulum crane backstepping sliding mode control method based on a generalized proportional-integral observer, and the method comprises the steps: constructing a dynamic model of a double-pendulum crane system, rewriting the dynamic model of the double-pendulum crane system, and designing the generalized proportional-integral observer; and effectively estimating the lumped interference and the state variable of the double-pendulum crane system by using a generalized proportional-integral observer to obtain an estimation signal, and designing the double-pendulum crane backstepping sliding mode control system according to the estimation signal. According to the double-pendulum crane system model, a generalized proportional-integral observer is designed, the observer can effectively estimate external interference and uncertainty in the model, and the anti-interference capability of the system is improved; and sliding mode surfaces are respectively designed for a drivable state variable and an under-actuated state variable of the system, so that the swinging of a load and a lifting hook is effectively inhibited.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent cranes, and in particular to a backstepping sliding mode control method for a double-pendulum crane based on a generalized proportional-integral observer. Background Art

[0002] As an important construction machinery, crane plays a very important role in completing large-scale engineering projects.

[0003] At present, most of the control methods for cranes are aimed at single pendulum crane systems. However, in some special occasions, such as when the volume of the load is too large or the mass of the hook cannot be ignored, the crane system will have the phenomenon that the hook rotates around the trolley and the load rotates around the hook, that is, the double pendulum effect occurs. In this case, some single pendulum control methods are often not applicable, and the control performance will also be reduced or even cannot be applied.

[0004] Since the working environment of the crane is often harsh and it is easily affected by external interference such as wind, it often causes the load to swing for a long time, which brings great safety hazards to the crane system and seriously affects the positioning accuracy of the trolley. The above reasons greatly affect the control performance of the crane and bring great difficulties to the design of the controller. Therefore, it is very necessary to design a controller with strong anti-interference ability. Summary of the invention

[0005] The purpose of the present application is to provide a backstepping sliding mode control method for a double-pendulum crane based on a generalized proportional-integral observer in order to address the technical defects in the prior art.

[0006] The technical solution adopted to achieve the purpose of this application is:

[0007] A backstepping sliding mode control method for a double pendulum crane based on a generalized proportional integral observer comprises the following steps:

[0008] Step 1, construct a dynamic model of the double-pendulum crane system, rewrite the dynamic model of the double-pendulum crane system, and design a generalized proportional integral observer;

[0009] The step 1 comprises the following steps:

[0010] S1.1: Rewrite the dynamic model of the double-pendulum crane system into a second-order standard form:

[0011]

[0012]

[0013] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, w i(i=1,2,3) represents the lumped interference of the double pendulum crane system, x1, x2, x3, x4, x5, x6 represent respectively, where x1=x, x3=θ1, x5=θ2,

[0014] S1.2: Redefine x1, x2, x3, x4, x5, x6, w1, w2, w3, and thus rewrite expressions (4)-(5):

[0015]

[0016] In the formula, z 1i 、z 2i 、z 3i 、z 4i denote the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance and its first-order derivative of the double-pendulum crane system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, δ i represents the second-order derivative of the aggregate interference residue; wherein, z 11 =x1,z 12 =x3,z 13 =x5,z 21 =x2,z 22 =x4,z 23 =x6,z 31 =w1,z 32 =w2,z 33 =w3,

[0017] S1.3: Define z 1i 、z 2i 、z 3i 、z 4i The estimated value of is obtained and the generalized proportional integral observer is designed according to expression (7):

[0018]

[0019] In the formula, Indicates z 1i , z 2i , z 3i , z 4i The estimated value of z 1i 、z 2i 、z 3i 、z 4i They represent the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance of the double-pendulum crane system and its first-order derivative, f represents the driving force, r1i , r 2i , r 3i , r 4i represents the gain of the generalized proportional-integral observer;

[0020] Step 2, using a generalized proportional integral observer to effectively estimate the lumped disturbance and state variables of the double-pendulum crane system to obtain an estimated signal, and designing a backstepping sliding mode controller for the double-pendulum crane according to the estimated signal;

[0021] The step 2 comprises the following steps:

[0022] S2.1: Define the state error vector of the double pendulum crane system:

[0023] e=[e1 e2 e3] T =[xx d θ1θ2] T .(9)

[0024] Where, e represents the state error vector of the double pendulum crane system, e1, e2, and e3 represent the errors of the state variables of the double pendulum crane system, x represents the displacement of the trolley, and x d represents the target displacement of the trolley, θ1 is the swing angle of the hook around the trolley, and θ2 is the swing angle of the load around the hook;

[0025] S2.2: Design the first sliding surface according to the state error vector of the double-pendulum crane system, and take the derivative of the first sliding surface with respect to time.

[0026] The first sliding surface expression is as follows:

[0027]

[0028] Where, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first sliding surface, respectively; e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants; p, q are positive odd numbers, and p>q;

[0029] The time derivative expression of the first sliding surface is as follows:

[0030]

[0031] Where, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first sliding surface, respectively; e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants; p, q are positive odd numbers, and p>q;

[0032] S2.3: Substitute the dynamic model of the second-order standard double-pendulum crane system into the expression of the first-layer sliding surface derivative with respect to time and organize it:

[0033]

[0034] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double pendulum crane system, e1, e2, e3 represent the errors of the state variables of the double pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q;

[0035] S2.4: Perform coordinate transformation on the first sub-sliding surface of the first sliding surface:

[0036] v1=s1, (19)

[0037]

[0038] Where, s1 represents the first sub-sliding surface of the first sliding surface, u1 represents the virtual control amount of the first subsystem, χ1 represents the control parameter, where χ1>0; where u1=-χ1v1;

[0039] S2.5: Derivative the expression (19), and select the following Lyapunov candidate function to be differentiated;

[0040] Derivative of expression (19):

[0041]

[0042] Where s1 represents the first sub-sliding surface of the first sliding surface, and u1 represents the virtual control amount of the first subsystem;

[0043] The Lyapunov candidate function expression described in step 2.5 is as follows:

[0044]

[0045] Where V1(t) represents the Lyapunov candidate function described in step 2.5;

[0046] Derivative the Lyapunov candidate function described in step 2.5:

[0047]

[0048] In the formula, u1 represents the virtual control quantity of the first subsystem, and χ1 represents the control parameter;

[0049] S2.6: Let v2 = 0 and substitute into the derivative formula of the Lyapunov candidate function, we get Select the following Lyapunov candidate function, take its derivative, and substitute expressions (16) and (23) into it;

[0050] Substituting v2=0 into the derivative formula of the Lyapunov candidate function yields:

[0051]

[0052] Where V1(t) represents the Lyapunov candidate function described in step 2.5, and X1 represents the control parameter.

[0053] The Lyapunov candidate function expression described in step 2.6 is as follows:

[0054]

[0055] Where V2(t) represents the Lyapunov candidate function described in step 2.6, and V1(t) represents the Lyapunov candidate function described in step 2.5;

[0056] Derivate the Lyapunov candidate function described in step 2.6 and substitute expressions (16) and (23) into:

[0057]

[0058] Where, u1 represents the virtual control quantity of the first subsystem, X1 represents the control parameter, a1 represents the nonlinear function term of the dynamic model of the first subsystem, f represents the driving force, w1 represents the lumped disturbance of the first subsystem, e1 represents the error of the trolley displacement, α1 and β1 are both positive constants, p and q are positive odd numbers, and p>q;

[0059] S2.7 Design the equivalent control quantity of the first subsystem, and substitute the equivalent control quantity of the first subsystem into expression (26);

[0060] The equivalent control quantity expression of the first subsystem is as follows:

[0061]

[0062] In the formula, represents the equivalent control quantity of the first subsystem, a1 represents the nonlinear function term of the dynamic model of the first subsystem, e1 represents the error of the trolley displacement, u1 represents the virtual control quantity of the first subsystem, X2 represents the control parameter, α1, β1 are both positive constants, p, q are positive odd numbers, and p>q; among them, when χ2>0, the lumped disturbance w1 of the first subsystem is estimated by the generalized proportional integral observer z 31 to replace;

[0063] Substitute the equivalent control quantity of the first subsystem into the derivative of the Lyapunov candidate function (26) described in step 2.6:

[0064]

[0065] In the formula, χ1 and χ2 represent control parameters;

[0066] S2.8: Similar to steps 2.4-2.7, coordinate transformation is performed on the second and third sub-sliding surfaces of the first sliding surface, and a Lyapunov candidate function similar to step 2.5 is selected to design the equivalent control quantities of the second and third subsystems, and the equivalent control quantities of the second and third subsystems are respectively substituted into expressions (31) and (32);

[0067] Perform coordinate transformation on the second and third sub-sliding surfaces of the first sliding surface:

[0068]

[0069] Where s2 and s3 represent the second and third sub-sliding surfaces of the first sliding surface, and u2 and u3 represent the virtual control quantities of the second and third sub-systems, respectively. Where u2 = -χ3v3, u3 = -χ5v5, χ3χ5 represent control parameters, χ3>0, χ5>0;

[0070] The Lyapunov candidate function expression described in step 2.8 is as follows:

[0071]

[0072] Where V3(t), V4(t), V5(t), and V6(t) represent the Lyapunov candidate functions described in step 2.8;

[0073] The expressions of the equivalent control quantities of the second subsystem and the third subsystem are as follows:

[0074]

[0075]

[0076] In the formula, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the third subsystem, a2 represents the nonlinear function term of the dynamic model of the second subsystem, a3 represents the nonlinear function term of the dynamic model of the third subsystem, e2 and e3 represent the error of the hook around the swing angle of the trolley and the error of the load around the swing angle of the hook, respectively, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, χ4 and X6 represent control parameters, α2, α3, β2, β3 are all positive numbers, p, q are positive odd numbers, and p>q; among them, χ4>0, χ6>0; the lumped disturbances w2 and w3 of the second and third subsystems are estimated by the generalized proportional integral observer z 32 and z 33 to replace;

[0077] Substitute the equivalent control quantity of the second subsystem and the equivalent control quantity of the third subsystem into expressions (31) and (32) respectively:

[0078]

[0079] Where, χ3, χ5, χ4, and χ6 represent control parameters;

[0080] S2.9: Design the total control quantity of the backstepping sliding mode controller of the double pendulum crane, and construct the second sliding mode surface (total sliding mode surface) of the backstepping sliding mode controller of the double pendulum crane based on the first sliding mode surface, and derive the second sliding mode surface;

[0081] The total control quantity calculation formula of the double-pendulum crane backstepping sliding mode controller is as follows:

[0082]

[0083] Where f represents the total control quantity of the backstepping sliding mode controller for the double pendulum crane, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, f usw represents the switching control quantity of the backstepping sliding mode controller of the double pendulum crane;

[0084] The expression of the second sliding surface is as follows:

[0085] S=λ1s1+λ2s2+λ3s3, (38)

[0086] Where S represents the second sliding surface, s1 represents the first sub-sliding surface of the first sliding surface, s2 represents the second sub-sliding surface of the first sliding surface, s3 represents the third sub-sliding surface of the first sliding surface, λ1, λ2, λ3 represent the control parameters of the system;

[0087] The expression for the derivation of the second layer sliding surface is as follows:

[0088]

[0089] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double pendulum crane system, e1, e2, e3 represent the error of the trolley displacement, the error of the hook swing angle around the trolley, and the error of the load swing angle around the hook, respectively, λ1, λ2, λ3 represent the control parameters of the system, α2, α3, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q;

[0090] S2.10: Design an exponential reaching law, and substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem, the total control quantity of the double-pendulum crane backstepping sliding mode controller, and the exponential reaching law into the derivative of the second-layer sliding surface to obtain the switching control quantity of the double-pendulum crane backstepping sliding mode controller;

[0091] The expression of the exponential reaching law is as follows:

[0092]

[0093] In the formula, S represents the second sliding surface, ε and k represent control parameters;

[0094] The expression of the switching control quantity of the double-pendulum crane backstepping sliding mode controller is as follows:

[0095]

[0096] In the formula, f usw represents the switching control quantity of the backstepping sliding mode controller of the double pendulum crane, χ2, χ4, χ6 represent the control parameters, λ1, λ2, λ3 represent the control parameters of the system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, u1 represents the virtual control quantity of the first subsystem, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, γ, η, λ, ε, k all represent control parameters, and S represents the second-layer sliding surface;

[0097] S2.11: Substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem and the switching control quantity of the double-pendulum crane backstepping sliding mode controller into the total control quantity expression of the double-pendulum crane backstepping sliding mode controller and arrange the terms to obtain the total control quantity of the double-pendulum crane backstepping sliding mode controller;

[0098] The final expression of the total control quantity of the double-pendulum crane backstepping sliding mode controller is as follows:

[0099]

[0100] Where f represents the total control quantity of the backstepping sliding mode controller for the double pendulum crane, λ1, λ2, λ3 represent the control parameters of the system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, z 31 、z 32 、z 33 They respectively represent the estimated values ​​of the generalized proportional-integral observer, e1, e2, and e3 respectively represent the error of the trolley displacement, the error of the hook swing angle around the trolley, and the error of the load swing angle around the hook, α2, α3, β2, and β3 are all positive constants, p, q are positive odd numbers, and p>q, S represents the second-layer sliding surface, and ε and k represent control parameters.

[0101] The beneficial effects of the present invention are as follows:

[0102] 1. The present invention designs a generalized proportional integral observer for the double-pendulum crane system model, which can effectively estimate external interference and uncertainty in the model, thereby improving the anti-interference ability of the system.

[0103] 2. The present invention designs sliding surfaces for the system's drivable state variables and under-actuated state variables, respectively, to effectively suppress the swing of the load and the hook.

[0104] 3. The present invention utilizes the control concept of backstepping to make the structure of the system clearer and more readable. BRIEF DESCRIPTION OF THE DRAWINGS

[0105] In order to more clearly illustrate the technical solutions in the present application or the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0106] Figure 1 This is a block diagram of the backstepping sliding mode control of a double-pendulum crane based on a generalized proportional-integral observer of the present invention.

[0107] Figure 2This is a diagram of the backstepping sliding mode control results of the double-pendulum crane based on the generalized proportional-integral observer of the present invention. DETAILED DESCRIPTION

[0108] 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 specific embodiments.

[0109] A backstepping sliding mode control method for a double pendulum crane based on a generalized proportional-integral observer, see Figure 1 , including the following steps:

[0110] Step 1, construct the dynamic model of the double pendulum crane system, rewrite the dynamic model of the double pendulum crane system, and design a generalized proportional integral observer. Rewriting the dynamic model of the double pendulum crane system can facilitate the design of the generalized proportional integral observer and the double pendulum crane backstepping sliding mode controller.

[0111] The dynamic model expression of the double pendulum crane system is as follows:

[0112]

[0113] Where x is the displacement of the trolley; θ1 is the swing angle of the hook around the trolley; θ2 is the swing angle of the load around the hook; m0 is the mass of the trolley; m1 is the mass of the hook; m2 is the mass of the load; l1 is the length of the rope; l2 is the length from the center of mass of the hook to the center of mass of the load; f is the driving force; f d is the interference term of the double-pendulum crane system, such as external disturbance and model uncertainty.

[0114] The step 1 comprises the following steps:

[0115] S1.1: Rewrite the dynamic model (1)-(3) of the double-pendulum crane system into a second-order standard form.

[0116] The dynamic model expression of the second-order standard double-pendulum crane system is as follows:

[0117]

[0118] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, w i (i=1,2,3) represents the lumped disturbance of the double pendulum crane system, where x1=x, x3=θ1, x5=θ2, The lumped disturbance includes the nonlinear function term of the double-pendulum crane system dynamics model and the collection term of factors such as external unknown disturbance; the expression of the collection term is as follows:

[0119]

[0120] Where ω1, ω2, and ω3 represent the lumped interference of the double-pendulum crane system; a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system; f d is the interference term of the double-pendulum crane system, such as external disturbance and model uncertainty.

[0121] S1.2: Redefine x1, x2, x3, x4, x5, x6, w1, w2, and w3, and rewrite expressions (4)-(5) to obtain the following expressions:

[0122]

[0123] In the formula, z 1i 、z 2i 、z 3i 、z 4i denote the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance and its first-order derivative of the double-pendulum crane system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, δ i represents the second-order derivative of the aggregate interference residual. 11 =x1,z 12 =x3,z 13 =x5,z 21 =x2,z 22 =x4,z 23 =x6,z 31 =w1,z 32 =w2,z 33 =w3,

[0124] S1.3: Define z 1i 、z 2i 、z 3i 、z 4i The estimated value of is obtained and the generalized proportional integral observer (GPIO) is designed according to expression (7).

[0125] The expression of the generalized proportional integral observer is as follows:

[0126]

[0127] In the formula, Indicates z 1i , z 2i , z 3i , z 4i The estimated value of z1i 、z 2i 、z 3i 、z 4i They represent the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance of the double-pendulum crane system and its first-order derivative, f represents the driving force, r 1i , r 2i , r 3i , r 4i represents the gain of the generalized proportional-integral observer.

[0128] Step 2: Use a generalized proportional integral observer (GPIO) to effectively estimate the lumped disturbance and state variables of the double-pendulum crane system to obtain an estimated signal, and design a backstepping sliding mode controller for the double-pendulum crane based on the estimated signal.

[0129] The step 2 comprises the following steps:

[0130] S2.1: Define the state error vector of the double pendulum crane system.

[0131] The state error vector expression of the double pendulum crane system is as follows:

[0132] e=[e1 e2 e3] T =[xx d θ1θ2] T .(9)

[0133] Where e represents the state error vector of the double-pendulum crane system, e1, e2, and e3 represent the errors of the state variables of the double-pendulum crane system (the state variables of the double-pendulum crane system are x (trolley displacement), θ1 (hook swing angle around the trolley), and θ2 (load swing angle around the hook)), x represents the trolley displacement, and x d represents the target displacement of the trolley, θ1 is the swing angle of the hook around the trolley, and θ2 is the swing angle of the load around the hook.

[0134] S2.2: Design the first sliding surface according to the state error vector of the double-pendulum crane system, and take the derivative of the first sliding surface with respect to time.

[0135] The first sliding surface expression is as follows:

[0136]

[0137] In the formula, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first layer of sliding surface respectively (that is, the sliding surfaces of state variables x, θ1, θ2), e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system respectively, α1, α2, α3, β1, β2, β3 are all positive numbers, p, q are positive odd numbers, and p>q.

[0138] The time derivative expression of the first sliding surface (first sub-sliding surface, second sub-sliding surface, third sub-sliding surface) is as follows:

[0139]

[0140] In the formula, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first layer of sliding surface respectively (that is, the sliding surfaces of state variables x, θ1, θ2), e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system respectively, α1, α2, α3, β1, β2, β3 are all positive numbers, p, q are positive odd numbers, and p>q.

[0141] S2.3: Substitute the dynamic model of the second-order standard double-pendulum crane system into the expression of the first-layer sliding surface derivative with respect to time and organize it:

[0142]

[0143] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double-pendulum crane system, e1, e2, and e3 represent the errors of the state variables of the double-pendulum crane system respectively, α1, α2, α3, β1, β2, and β3 are all positive constants, p, q are positive odd numbers, and p>q.

[0144] S2.4: Perform coordinate transformation on the first sub-sliding surface of the first sliding surface:

[0145] v1=s1, (19)

[0146]

[0147] In the formula, s1 represents the first sub-sliding surface of the first sliding surface, u1 represents the virtual control amount of the first subsystem, χ1 represents the control parameter, where χ1>0, v1 and v2 have no special meanings and are set for the convenience of subsequent formula deduction. In the formula, u1=-χ1v1.

[0148] S2.5: Derivative expression (19) and select the following Lyapunov candidate function to be differentiated.

[0149] Derivative of expression (19):

[0150]

[0151] Where s1 represents the first sub-sliding surface of the first sliding surface, and u1 represents the virtual control amount of the first subsystem.

[0152] The Lyapunov candidate function expression described in step 2.5 is as follows:

[0153]

[0154] Where V1(t) represents the Lyapunov candidate function described in step 2.5.

[0155] Derivative the Lyapunov candidate function described in step 2.5:

[0156]

[0157] Where u1 represents the virtual control quantity of the first subsystem, and χ1 represents the control parameter.

[0158] S2.6: Let v2 = 0 and substitute into the derivative formula of the Lyapunov candidate function, we get Select the following Lyapunov candidate function, take its derivative, and substitute expressions (16) and (23) into it.

[0159] Substituting v2=0 into the derivative formula of the Lyapunov candidate function yields:

[0160]

[0161] Where V1(t) represents the Lyapunov candidate function described in step 2.5, and X1 represents the control parameter.

[0162] The Lyapunov candidate function expression described in step 2.6 is as follows:

[0163]

[0164] Where V2(t) represents the Lyapunov candidate function described in step 2.6, and V1(t) represents the Lyapunov candidate function described in step 2.5.

[0165] Derivate the Lyapunov candidate function described in step 2.6 and substitute expressions (16) and (23) into:

[0166]

[0167] Wherein, u1 represents the virtual control quantity of the first subsystem, χ1 represents the control parameter, a1 represents the nonlinear function term of the dynamic model of the first subsystem, f represents the driving force, w1 represents the lumped disturbance of the first subsystem, e1 represents the error of the trolley displacement, α1 and β1 are both positive constants, p and q are positive odd numbers, and p>q.

[0168] S2.7 Design the equivalent control quantity of the first subsystem and substitute the equivalent control quantity of the first subsystem into expression (26).

[0169] The equivalent control quantity expression of the first subsystem is as follows:

[0170]

[0171] In the formula, f ueq1 represents the equivalent control quantity of the first subsystem, a1 represents the nonlinear function term of the dynamic model of the first subsystem, e1 represents the error of the trolley displacement, u1 represents the virtual control quantity of the first subsystem, χ2 represents the control parameter, α1, β1 are both positive constants, p, q are positive odd numbers, and p>q. Among them, when χ2>0, the lumped disturbance w1 of the first subsystem is estimated by the generalized proportional integral observer z 31 to replace.

[0172] Substitute the equivalent control quantity of the first subsystem into the derivative of the Lyapunov candidate function (26) described in step 2.6:

[0173]

[0174] Where χ1 and χ2 represent control parameters. Since χ1>0 and χ2>0, the first subsystem is stable in the sense of Lyapunov.

[0175] S2.8: Similar to steps 2.4-2.7, coordinate transformation is performed on the second sub-sliding surface and the third sub-sliding surface of the first sliding surface, and a Lyapunov candidate function similar to step 2.5 is selected to design the equivalent control quantities of the second subsystem and the third subsystem, and the equivalent control quantities of the second subsystem and the equivalent control quantities of the third subsystem are substituted into expressions (31) and (32) respectively.

[0176] Perform coordinate transformation on the second and third sub-sliding surfaces of the first sliding surface:

[0177]

[0178] In the formula, v3, v4, v5, and v6 have no special meanings and are set for the convenience of subsequent formula deduction; s2 and s3 represent the second and third sub-sliding surfaces of the first sliding surface, and u2 and u3 represent the virtual control quantities of the second and third subsystems, respectively. Among them, u2 = -χ3v3, u3 = -χ5v5, χ3X5 represents the control parameters, X3>0, χ5>0.

[0179] The Lyapunov candidate function expression described in step 2.8 is as follows:

[0180]

[0181] Where V3(t), V4(t), V5(t), and V6(t) represent the Lyapunov candidate functions described in step 2.8.

[0182] The expressions of the equivalent control quantities of the second subsystem and the third subsystem are as follows:

[0183]

[0184]

[0185] In the formula, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the third subsystem, a2 represents the nonlinear function term of the dynamic model of the second subsystem, a3 represents the nonlinear function term of the dynamic model of the third subsystem, e2 and e3 represent the error of the hook around the swing angle of the trolley and the error of the load around the swing angle of the hook, respectively, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, χ4 and χ6 represent control parameters, α2, α3, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q. Among them, χ4>0, χ6>0. The lumped disturbances w2 and w3 of the second and third subsystems are estimated by the generalized proportional integral observer z 32 and z 33 to replace.

[0186] Substitute the equivalent control quantity of the second subsystem and the equivalent control quantity of the third subsystem into expressions (31) and (32) respectively:

[0187]

[0188] Wherein, χ3, χ5, χ4, and χ6 represent control parameters.

[0189] S2.9: Design the total control quantity of the double-pendulum crane backstepping sliding mode controller, and construct the second-layer sliding surface (total sliding surface) of the double-pendulum crane backstepping sliding mode controller based on the first-layer sliding surface, and derive the second-layer sliding surface.

[0190] The total control quantity calculation formula of the double-pendulum crane backstepping sliding mode controller is as follows:

[0191]

[0192] Where f represents the total control quantity (driving force) of the double-pendulum crane backstepping sliding mode controller, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, fusw Represents the switching control quantity of the backstepping sliding mode controller for the double-pendulum crane.

[0193] The expression of the second sliding surface is as follows:

[0194] S=λ1s1+λ2s2+λ3s3, (38)

[0195] Where S represents the second sliding surface, s1 represents the first sub-sliding surface of the first sliding surface, s2 represents the second sub-sliding surface of the first sliding surface, s3 represents the third sub-sliding surface of the first sliding surface, and λ1, λ2, and λ3 represent the control parameters of the system.

[0196] The expression for the derivation of the second layer sliding surface is as follows:

[0197]

[0198] In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double-pendulum crane system, e1, e2, e3 represent the error of the trolley displacement, the error of the hook around the trolley swing angle, and the error of the load around the hook swing angle, respectively, λ1, λ2, λ3 represent the control parameters of the system, α2, α3, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q.

[0199] S2.10: Design an exponential reaching law, and substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem, the total control quantity of the double-pendulum crane backstepping sliding mode controller, and the exponential reaching law into the derivative of the second-layer sliding surface to obtain the switching control quantity of the double-pendulum crane backstepping sliding mode controller.

[0200] The expression of the exponential reaching law is as follows:

[0201]

[0202] Where S represents the second sliding surface, and ε and k represent control parameters.

[0203] The expression of the switching control quantity of the double-pendulum crane backstepping sliding mode controller is as follows:

[0204]

[0205] In the formula, f usw represents the switching control quantity of the backstepping sliding mode controller of the double pendulum crane, χ2, χ4, χ6 represent the control parameters, λ1, λ2, λ3 represent the control parameters of the system, and a i(i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, u1 represents the virtual control quantity of the first subsystem, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, γ, η, λ, ε, k all represent control parameters, and S represents the second-layer sliding surface.

[0206] S2.11: Substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem and the switching control quantity of the double-pendulum crane backstepping sliding mode controller into the total control quantity expression of the double-pendulum crane backstepping sliding mode controller and sort them out to obtain the total control quantity of the double-pendulum crane backstepping sliding mode controller.

[0207] The final expression of the total control quantity of the double-pendulum crane backstepping sliding mode controller is as follows:

[0208]

[0209] In the formula, f represents the total control quantity (driving force) of the backstepping sliding mode controller of the double pendulum crane, λ1, λ2, λ3 represent the control parameters of the system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, z 31 、z 32 、z 33 They respectively represent the estimated values ​​of the generalized proportional-integral observer, e1, e2, and e3 respectively represent the error of the trolley displacement, the error of the hook swing angle around the trolley, and the error of the load swing angle around the hook, α2, α3, β2, and β3 are all positive constants, p, q are positive odd numbers, and p>q, S represents the second-layer sliding surface, and ε and k represent control parameters.

[0210] See also Figure 2 From the first sub-graph, it can be found that the trolley can reach the target position within 8 seconds, and the double-pendulum crane backstepping sliding mode control method of this embodiment can suppress the influence of external disturbances by changing the displacement of the trolley, and there is no trolley positioning error; from the second and third sub-graphs, it can be found that the swing angles of the load and the hook are limited to 4.5 degrees, and the load and hook swings will be quickly eliminated with the change of the trolley displacement after adding external interference, and the influence of external swing angle disturbances on the crane system will be eliminated after 3 seconds and 4 seconds respectively, and there is basically no residual swing. From the simulation results, it can be concluded that the proposed control method can achieve the corresponding control objectives and has good anti-interference ability to external interference, which has very important application significance in the practical application of cranes.

[0211] The above is only a preferred embodiment of the present invention. It should be pointed out that, for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A backstepping sliding mode control method for a double pendulum crane based on a generalized proportional integral observer, characterized in that: The following steps are involved: Step 1, construct a dynamic model of the double-pendulum crane system, rewrite the dynamic model of the double-pendulum crane system, and design a generalized proportional integral observer; The step 1 comprises the following steps: S1.1: Rewrite the dynamic model of the double-pendulum crane system into a second-order standard form: In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, w i (i=1,2,3) represents the lumped interference of the double pendulum crane system, x1, x2, x3, x4, x5, x6 represent respectively, where x1=x, x3=θ1, x5=θ2, S1.2: Redefine x1, x2, x3, x4, x5, x6, w1, w2, w3, and thus rewrite expressions (4)-(5): In the formula, z 1i 、z 2i 、z 3i 、z 4i denote the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance and its first-order derivative of the double-pendulum crane system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, δ i represents the second-order derivative of the aggregate interference residue; wherein, z 11 =x1,z 12 =x3,z 13 =x5,z 21 =x2,z 22 =x4,z 23 =x6,z 31 =w1,z 32 =w2,z 33 =w3, S1.3: Define z 1i 、z 2i 、z 3i 、z 4i The estimated value of is obtained and the generalized proportional integral observer is designed according to expression (7): In the formula, Indicates z 1i , z 2i , z 3i , z 4i The estimated value of z 1i 、z 2i 、z 3i 、z 4i They represent the state variables and their first-order derivatives of the double-pendulum crane system, the lumped disturbance of the double-pendulum crane system and its first-order derivative, f represents the driving force, r 1i , r 2i , r 3i , r 4i represents the gain of the generalized proportional-integral observer; Step 2, using a generalized proportional integral observer to effectively estimate the lumped disturbance and state variables of the double-pendulum crane system to obtain an estimated signal, and designing a backstepping sliding mode controller for the double-pendulum crane according to the estimated signal; The step 2 comprises the following steps: S2.1: Define the state error vector of the double pendulum crane system: e=[e1 e2 e3] T =[x-x d θ1 θ2] T . (9) Where, e represents the state error vector of the double pendulum crane system, e1, e2, and e3 represent the errors of the state variables of the double pendulum crane system, x represents the displacement of the trolley, and x d represents the target displacement of the trolley, θ1 is the swing angle of the hook around the trolley, and θ2 is the swing angle of the load around the hook; S2.2: Design the first sliding surface according to the state error vector of the double-pendulum crane system, and take the derivative of the first sliding surface with respect to time. The first sliding surface expression is as follows: Where, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first sliding surface, respectively; e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants; p, q are positive odd numbers, and p>q; The time derivative expression of the first sliding surface is as follows: Where, s1, s2, s3 represent the first sub-sliding surface, the second sub-sliding surface, and the third sub-sliding surface of the first sliding surface, respectively; e1, e2, e3 represent the errors of the state variables of the double-pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants; p, q are positive odd numbers, and p>q; S2.3: Substitute the dynamic model of the second-order standard double-pendulum crane system into the expression of the first-layer sliding surface derivative with respect to time and organize it: In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double pendulum crane system, e1, e2, e3 represent the errors of the state variables of the double pendulum crane system, α1, α2, α3, β1, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q; S2.4: Perform coordinate transformation on the first sub-sliding surface of the first sliding surface: v1=s1,(19) Where, s1 represents the first sub-sliding surface of the first sliding surface, u1 represents the virtual control amount of the first subsystem, χ1 represents the control parameter, where χ1>0; where u1=-X1v1; S2.5: Derivative the expression (19), and select the following Lyapunov candidate function to be differentiated; Derivative of expression (19): Where s1 represents the first sub-sliding surface of the first sliding surface, and u1 represents the virtual control amount of the first subsystem; The Lyapunov candidate function expression described in step 2.5 is as follows: Where V1(t) represents the Lyapunov candidate function described in step 2.5; Derivative the Lyapunov candidate function described in step 2.5: In the formula, u1 represents the virtual control quantity of the first subsystem, and X1 represents the control parameter; S2.6: Let v2 = 0 and substitute into the derivative formula of the Lyapunov candidate function, we get Select the following Lyapunov candidate function, take its derivative, and substitute expressions (16) and (23) into it; Substituting v2=0 into the derivative formula of the Lyapunov candidate function yields: Where V1(t) represents the Lyapunov candidate function described in step 2.5, and X1 represents the control parameter. The Lyapunov candidate function expression described in step 2.6 is as follows: Where V2(t) represents the Lyapunov candidate function described in step 2.6, and V1(t) represents the Lyapunov candidate function described in step 2.5; Derivate the Lyapunov candidate function described in step 2.6 and substitute expressions (16) and (23) into: Where, u1 represents the virtual control quantity of the first subsystem, X1 represents the control parameter, a1 represents the nonlinear function term of the dynamic model of the first subsystem, f represents the driving force, w1 represents the lumped disturbance of the first subsystem, e1 represents the error of the trolley displacement, α1 and β1 are both positive constants, p and q are positive odd numbers, and p>q; S2.7 Design the equivalent control quantity of the first subsystem, and substitute the equivalent control quantity of the first subsystem into expression (26); The equivalent control quantity expression of the first subsystem is as follows: In the formula, represents the equivalent control quantity of the first subsystem, a1 represents the nonlinear function term of the dynamic model of the first subsystem, e1 represents the error of the trolley displacement, u1 represents the virtual control quantity of the first subsystem, χ2 represents the control parameter, α1, β1 are both positive constants, p, q are positive odd numbers, and p>q; among them, when χ2>0, the lumped disturbance w1 of the first subsystem is estimated by the generalized proportional integral observer z 31 to replace; Substitute the equivalent control quantity of the first subsystem into the derivative of the Lyapunov candidate function (26) described in step 2.6: In the formula, χ1 and χ2 represent control parameters; S2.8: Similar to steps 2.4-2.7, coordinate transformation is performed on the second and third sub-sliding surfaces of the first sliding surface, and a Lyapunov candidate function similar to step 2.5 is selected to design the equivalent control quantities of the second and third subsystems, and the equivalent control quantities of the second and third subsystems are respectively substituted into expressions (31) and (32); Perform coordinate transformation on the second and third sub-sliding surfaces of the first sliding surface: Where s2 and s3 represent the second and third sub-sliding surfaces of the first sliding surface, and u2 and u3 represent the virtual control quantities of the second and third sub-systems, respectively. Where u2 = -χ3v3, u3 = -χ5v5, χ3χ5 represent control parameters, χ3>0, χ5>0; The Lyapunov candidate function expression described in step 2.8 is as follows: Where V3(t), V4(t), V5(t), and V6(t) represent the Lyapunov candidate functions described in step 2.8; The expressions of the equivalent control quantities of the second subsystem and the third subsystem are as follows: In the formula, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the third subsystem, a2 represents the nonlinear function term of the dynamic model of the second subsystem, a3 represents the nonlinear function term of the dynamic model of the third subsystem, e2 and e3 represent the error of the hook around the swing angle of the trolley and the error of the load around the swing angle of the hook, respectively, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, χ4 and χ6 represent control parameters, α2, α3, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q; among them, χ4>0, χ6>0; the lumped disturbances w2 and w3 of the second and third subsystems are estimated by the generalized proportional integral observer z 32 and z 33 to replace; Substitute the equivalent control quantity of the second subsystem and the equivalent control quantity of the third subsystem into expressions (31) and (32) respectively: Where, χ3, X5, χ4, and χ6 represent control parameters; S2.9: Design the total control quantity of the backstepping sliding mode controller of the double pendulum crane, and construct the second sliding mode surface (total sliding mode surface) of the backstepping sliding mode controller of the double pendulum crane based on the first sliding mode surface, and derive the second sliding mode surface; The total control quantity calculation formula of the double-pendulum crane backstepping sliding mode controller is as follows: Where f represents the total control quantity of the backstepping sliding mode controller for the double pendulum crane, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, f usw represents the switching control quantity of the backstepping sliding mode controller of the double pendulum crane; The expression of the second sliding surface is as follows: S=λ1s1+λ2s2+λ3s3, (38) Where S represents the second sliding surface, s1 represents the first sub-sliding surface of the first sliding surface, s2 represents the second sub-sliding surface of the first sliding surface, s3 represents the third sub-sliding surface of the first sliding surface, λ1, λ2, λ3 represent the control parameters of the system; The expression for the derivation of the second layer sliding surface is as follows: In the formula, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, f represents the driving force, w i (i=1,2,3) represents the lumped disturbance of the double pendulum crane system, e1, e2, e3 represent the error of the trolley displacement, the error of the hook swing angle around the trolley, and the error of the load swing angle around the hook, respectively, λ1, λ2, λ3 represent the control parameters of the system, α2, α3, β2, β3 are all positive constants, p, q are positive odd numbers, and p>q; S2.10: Design an exponential reaching law, and substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem, the total control quantity of the double-pendulum crane backstepping sliding mode controller, and the exponential reaching law into the derivative of the second-layer sliding surface to obtain the switching control quantity of the double-pendulum crane backstepping sliding mode controller; The expression of the exponential reaching law is as follows: In the formula, S represents the second sliding surface, ε and k represent control parameters; The expression of the switching control quantity of the double-pendulum crane backstepping sliding mode controller is as follows: In the formula, f usw represents the switching control quantity of the backstepping sliding mode controller of the double pendulum crane, X2, X4, X6 represent the control parameters, λ1, λ2, λ3 represent the control parameters of the system, a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, represents the equivalent control quantity of the first subsystem, represents the equivalent control quantity of the second subsystem, represents the equivalent control quantity of the second subsystem, u1 represents the virtual control quantity of the first subsystem, u2 represents the virtual control quantity of the second subsystem, u3 represents the virtual control quantity of the third subsystem, γ, η, λ, ε, k all represent control parameters, and S represents the second-layer sliding surface; S2.11: Substitute the equivalent control quantities of the first subsystem, the second subsystem, and the third subsystem and the switching control quantity of the double-pendulum crane backstepping sliding mode controller into the total control quantity expression of the double-pendulum crane backstepping sliding mode controller and arrange the terms to obtain the total control quantity of the double-pendulum crane backstepping sliding mode controller; The final expression of the total control quantity of the double-pendulum crane backstepping sliding mode controller is as follows: Where f represents the total control quantity of the backstepping sliding mode controller for the double pendulum crane, λ1, λ2, λ3 represent the control parameters of the system, and a i (i=1,2,3) represents the nonlinear function term of the dynamic model of the double pendulum crane system, z 31 、z 32 、z 33 They respectively represent the estimated values ​​of the generalized proportional-integral observer, e1, e2, and e3 respectively represent the error of the trolley displacement, the error of the hook swing angle around the trolley, and the error of the load swing angle around the hook, α2, α3, β2, and β3 are all positive constants, p, q are positive odd numbers, and p>q, S represents the second-layer sliding surface, and ε and k represent control parameters.

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