Tower crane system anti-swing control method based on composite integral complementary sliding mode control

Abstract

The invention discloses a tower crane system anti-swing control method based on composite integral complementary sliding mode control, and belongs to the technical field of tower cranes, the method comprises the following steps: 1, defining a tower crane system error variable which is used for representing a difference between an actual operation state and an ideal state of a tower crane so as to realize a positioning anti-swing control target; 2, a composite integral complementary sliding mode controller is constructed, specifically, according to the tower crane system error variables defined in the step 1, under-actuated characteristics of the tower crane system are fully considered, the composite integral complementary sliding mode controller is constructed, and the controller inhibits load swing of the tower crane system and keeps stability and control precision of the system; and 3, the tower crane system is controlled to operate, specifically, the operation state of the tower crane system is controlled in real time through the composite integral complementary sliding mode controller constructed in the step 2, and the controller achieves stable and accurate carrying of the load of the tower crane according to the current load, position and speed state information of the tower crane and meanwhile effectively restrains swing of the load.

Description

Technical Field

[0001] The invention belongs to the technical field of tower cranes, and in particular proposes an anti-sway control method based on composite integral complementary sliding mode control to solve the problem of load swing control of tower cranes. Background Art

[0002] In recent years, the development momentum of intelligent manufacturing has been strong, not only has it achieved remarkable achievements in technological innovation, but also has shown great potential in industrial applications. As an important equipment in infrastructure construction, the development of intelligent control technology for tower cranes is particularly eye-catching.

[0003] With the acceleration of global infrastructure construction and urbanization, the demand for tower cranes has shown a significant growth, especially in emerging markets such as Asia, Africa and the Middle East. These regions have extremely high requirements for efficient and safe construction, which also poses new challenges and development opportunities for the intelligent control technology of tower cranes.

[0004] With the development trend of integration, automation and greening, the intelligent control technology of tower cranes is constantly upgrading. In order to improve construction efficiency and ensure operation safety, anti-sway control technology has become a hot spot for research and application. At present, research in this field mainly focuses on various control strategies such as PID control, adaptive control, fuzzy control, and sliding mode control. Among them, sliding mode control has attracted widespread attention in the industry for its simple structure, strong robustness, and easy engineering implementation.

[0005] However, the underactuated characteristics of tower cranes bring additional complexity to anti-sway control. Since the underactuated state cannot be controlled directly by the actuator, but must be achieved indirectly through a series of complex coupling relationships, this undoubtedly increases the difficulty of designing the control algorithm. In addition, the working environment of tower cranes is changeable, and they often need to face interference from external factors such as wind, temperature changes, and uneven ground. These uncertain factors will further aggravate the swing of the hook, making precise control more difficult. Summary of the invention

[0006] The present invention aims to provide an anti-sway control method based on compound integral complementary sliding mode control to solve the shortcomings of existing controllers in terms of control performance.

[0007] To achieve the above purpose, the following technical solution is adopted: The method of the present invention comprises the following steps:

[0008] Step 1, define the tower crane system error variable: used to characterize the difference between the actual operating state and the ideal state of the tower crane to achieve the positioning anti-sway control goal;

[0009] Step 2, construct a composite integral complementary sliding mode controller: Based on the tower crane system error variable defined in step 1, the under-actuated characteristics of the tower crane system are fully considered to construct a composite integral complementary sliding mode controller, which effectively suppresses the load swing of the tower crane system and maintains the stability and control accuracy of the system even in the face of complex and changeable loads and external environmental interference;

[0010] Step 3, control the operation of the tower crane system: Use the composite sliding mode controller constructed in step 2 to control the operation status of the tower crane system in real time. The controller realizes the smooth and accurate handling of the tower crane load based on the current load, position, speed and other status information of the tower crane, while effectively suppressing the swing of the load, thereby improving the safety and efficiency of the tower crane operation.

[0011] Furthermore, in step 1, a 4-DOF tower crane system with model uncertainty and external disturbance is considered. According to the Euler-Lagrange principle, the dynamic equation of the system is:

[0012]

[0013] in θ2(t)] T ∈R 4 is a 4-dimensional column vector, is the rotation angle of the cantilever of the tower crane system, x(t) is the displacement of the trolley, θ1(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal front and rear direction, θ2(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal plane left and right direction; M(q)∈R 4×4 is a 4-dimensional positive definite inertia matrix; is the matrix of Coriolis force and centrifugal force; F(t)=[u1(t),u2(t),0,0] T ∈R 4 is the system control input; in addition, the uncertainty and external disturbance of the system are expressed as d(t) = [d1(t), d2(t), d3(t), d4(t)] T ∈R 4 .

[0014] The tower crane system has the following properties:

[0015] Property 1: The inertia matrix M(q) is a symmetric positive definite matrix, and there are two positive constants δ m and δ M Make

[0016]

[0017] Where I is the 4-dimensional identity matrix.

[0018] Property 2: For a tower crane system, there exists a positive constant c, ρ such that

[0019]

[0020] (1) By simplifying the derivation, we can get:

[0021]

[0022] in represents the known nonlinear term, represents the aggregate uncertainty, u(t) = [u1(t),u2(t)] T ∈R 2 represents the system control input, h=[h1,h2,h3,h4] T =M -1 B∈R 4×2 is a known nonlinear term, and

[0023]

[0024] Defining the error variable

[0025]

[0026] Where i = 1, 2, 3, 4, q id is the expected position of the tower crane. Without loss of generality, we hope that the load swing angle is as small as possible during the operation of the tower crane. Therefore, q 3d =q 4d =0.

[0027] Furthermore, in step 2, the integral complementary sliding surface is defined as:

[0028]

[0029] Among them, λ is the positive parameter of the sliding surface.

[0030] The composite sliding surface is defined as:

[0031]

[0032] Where S = [S1, S2] T , S c =[S c1 ,S c2 ] T is a composite complementary sliding surface, and α1 and α2 are adjustable parameters.

[0033] Setting the compound integral complementary sliding mode controller:

[0034]

[0035] Where, e′=[e1+α1e4,e2+α2e3] T ∈R 2 , is the nonlinear control function, f′=[f1+α1f4,f2+α2f3] T ∈R 2 and h′ -1 =[(h1+α1h4) -1 ,(h2+α2h3) -1 ] T ∈R 2×2 is a known nonlinear term in the system, sat(·) is the saturation function, K is the controller parameter, and Φ is the boundary layer parameter.

[0036] Taking the derivative of the composite sliding surface (8), we get:

[0037]

[0038] Combining (4) and (7), we can get

[0039]

[0040] Furthermore, the complementary sliding modes S and S c The derivatives of

[0041]

[0042] It is deduced that there is the following relationship between the coupled sliding surfaces

[0043]

[0044] From (4), (6)

[0045]

[0046] in, is the lumped uncertainty and external disturbance, assuming There is an upper bound, namely D is a positive constant.

[0047] Substituting (9) and (14) into (12), we obtain

[0048]

[0049] The Lyapunov candidate function is selected as:

[0050]

[0051] Derivation of the Lyapunov candidate function V along the system motion trajectory yields:

[0052]

[0053] If K>D, then according to the definition of Lyapunov equation (16) and equation (17), all signals in the tower crane system are bounded, and the composite integral complementary sliding mode surfaces S and S c It will converge to 0 in a finite time T.

[0054] Therefore, when t>T, S=0, From (8) we get

[0055]

[0056] make x5=e1, but

[0057]

[0058] can be rewritten in compact form in

[0059] X=[x1, x2, x3, x4, x5] T (20)

[0060]

[0061] According to (6) and (19), it is not difficult to find that

[0062]

[0063] In addition, when t ≥ T, the composite integral complementary sliding surfaces S and S c Converges to 0, the controller (9) is simplified to

[0064] u(t)=-h′ -1 H(e′,t) (24)

[0065] Choose appropriate controller parameters α1 and λ so that the matrix A is Hurwitz. Therefore, there exists a unique positive definite matrix P that satisfies PA+A T P = -Q, where Q is a positive definite symmetric matrix. For further analysis, the Lyapunov candidate function is selected

[0066] V′=X T PX (25)

[0067] Then the derivative of V′ along the system motion trajectory can be expressed as

[0068]

[0069] Among them, Q has a minimum eigenvalue σ m Therefore, based on properties 1 and 2 of the tower crane system, as well as (23) and (24), it can be deduced that:

[0070]

[0071] in, is a known positive parameter. In addition, y=Ω1||X|| 2 +Ω2||X||+D is a quadratic function with respect to ||X|| that opens upward, so when the initial value of ||X|| is in the neighborhood

[0072]

[0073] Internal time, And ||X|| will decay as time goes by until ||X||=0, that is Therefore, it can be concluded that

[0074]

[0075] Right now Therefore, from (18),

[0076] Similarly, through similar analysis of the composite sliding surface S2, it can be concluded that That is to say, the proof that all error states of the tower crane system are asymptotically stable is completed.

[0077] In step 3, appropriate parameters of the composite integral complementary sliding mode controller are selected so that the composite integral complementary sliding mode controller of the invention can control the smooth and precise handling of the tower crane load and effectively suppress the swing of the load.

[0078] Among them, the parameters of the compound integral complementary sliding mode controller include:

[0079] (1) λ in formula (7) is used to adjust the convergence speed of the sliding surface. The larger λ is, the faster the sliding surface converges.

[0080] (2) α1 and α2 in formula (8) are used to adjust the weights of the driven sliding surface and the under-actuated sliding surface in the controller to control the load swing suppression effect. The larger the α1 and α2, the more obvious the load swing suppression is. However, too large a value will cause the trolley and cantilever to shake, so a trade-off needs to be made.

[0081] (3) K and Φ in formula (9) are used to suppress the influence of external disturbances on the system. K should be slightly larger than the upper limit of the external disturbance of the system. Φ is the boundary layer thickness of the composite integral complementary sliding surface and is a small positive parameter. The thicker the boundary layer, the less sensitive it is to disturbances, but the positioning accuracy deteriorates. There is a trade-off between the two.

[0082] There is coupling between all parameters, which need to be adjusted according to actual conditions and operating results.

[0083] Due to the adoption of the above technical solution, the technical effects achieved by the present invention are as follows:

[0084] The invention proposes a tower crane system anti-sway control method based on a composite integral complementary sliding mode, without any model simplification and conversion, thereby improving the accuracy and stability of the controller.

[0085] The composite sliding surface includes both the driven and under-actuated states, which improves the coupling relationship between the driven and under-actuated states and enhances the controller's control over the under-actuated state.

[0086] The integral complementary sliding surface not only avoids the arrival stage of the sliding surface, but also constructs the complementary sliding surface by introducing the complementary function, making the movement of the system on the sliding surface smoother, thereby reducing the chattering phenomenon and improving the control accuracy of the system.

[0087] The inventive method is applicable to a class of under-actuated systems, such as various crane systems, inverted pendulum systems, and flexible robotic arm systems, and can be easily extended to other systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 is a flow chart of the control method of the present invention;

[0089] Figure 2 It is a structural schematic diagram of a tower crane system;

[0090] Figure 3 is a control block diagram of the present invention;

[0091] Figure 4 This is a graph showing comparative experimental results of the present invention and other inventive solutions;

[0092] Figure 5 This is a diagram of the robustness test experimental results of the present invention. DETAILED DESCRIPTION

[0093] The present invention will be further described below in conjunction with the accompanying drawings:

[0094] like Figure 1-5 As shown, the method of the present invention comprises the following steps:

[0095] Step 1, define the tower crane system error variable: used to characterize the difference between the actual operating state and the ideal state of the tower crane to achieve the positioning anti-sway control goal;

[0096] Considering a 4-DOF tower crane system with model uncertainty and external disturbance, according to the Euler-Lagrange principle, the dynamic equation of the system is:

[0097]

[0098] in is the rotation angle of the cantilever of the tower crane system, x(t) is the displacement of the trolley, θ1(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal front and rear direction, θ2(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal plane left and right direction; M(q)∈R 4×4 is the positive definite inertia matrix of the system; is the vector of Coriolis force and centrifugal force; F(t)=[u1(t),u2(t),0,0] T ∈R 4 is the system control input; in addition, the uncertainty and external disturbance of the system are expressed as d(t) = [d1(t), d2(t), d3(t), d4(t)] T ∈R 4 .

[0099] The tower crane system has the following properties:

[0100] Property 1: The inertia matrix M(q) is a symmetric positive definite matrix, and there are two positive constants δ m and δ M Make

[0101]

[0102] Where I is the 4th-order identity matrix.

[0103] Property 2: For a tower crane system, there exists a positive constant c, ρ such that

[0104]

[0105] (1) By simplifying the derivation, we can get:

[0106]

[0107] in represents the known nonlinear term, represents the aggregate uncertainty, u(t) = [u1(t),u2(t)] T ∈R 2represents the system control input, h=[h1,h2,h3,h4] T =M -1 B∈R 4×2 is a known nonlinear term, and

[0108]

[0109] Defining the error variable

[0110]

[0111] Where i = 1, 2, 3, 4, q id is the expected position of the tower crane. Without loss of generality, we hope that the load swing angle is as small as possible during the operation of the tower crane. Therefore, q 3d =q 4d =0,q 1d is the desired angle of the cantilever angular operation; q 2d is the desired angle of the cantilever angular operation; q 3d and q 4d It is the expected offset angle of the load in the horizontal direction in the front-back and left-right directions relative to the center line of the tower crane boom.

[0112] Step 2, construct a composite integral complementary sliding mode controller: Based on the tower crane system error variable defined in step 1, the under-actuated characteristics of the tower crane system are fully considered to construct a composite integral complementary sliding mode controller, which effectively suppresses the load swing of the tower crane system and maintains the stability and control accuracy of the system even in the face of complex and changeable loads and external environmental interference;

[0113] The integral complementary sliding surface is defined as:

[0114]

[0115] Among them, λ is the positive parameter of the sliding surface;

[0116] The composite sliding surface is defined as:

[0117]

[0118] Where S = [S1, S2] T , S c =[S c1 ,S c2 ] T is a composite complementary sliding surface, and α1 and α2 are adjustable parameters.

[0119] Setting the compound integral complementary sliding mode controller:

[0120]

[0121] Where, e′=[e1+α1e4,e2+α2e3] T ∈R 2 , H(e′,t)=f′+2λe′+λ 2 e′ is the nonlinear control function, f′=[f1+α1f4,f2+α2f3] T ∈R 2 and h′ -1 =[(h1+α1h4) -1 ,(h2+α2h3) -1 ] T ∈R 2×2 is a known nonlinear term in the system, sat(·) is the saturation function, K is the controller parameter, and Φ is the boundary layer parameter.

[0122] Taking the derivative of the composite sliding surface (8), we get:

[0123]

[0124] Combining (4) and (7), we can get

[0125]

[0126] Furthermore, the complementary sliding modes S and S c The derivatives of

[0127]

[0128] It is deduced that there is the following relationship between the coupled sliding surfaces

[0129]

[0130] From (4), (6)

[0131]

[0132] in, is the lumped uncertainty and external disturbance, assuming There is an upper bound, namely D is a positive constant.

[0133] Substituting (9) and (14) into (12), we obtain

[0134]

[0135] The Lyapunov candidate function is selected as:

[0136]

[0137] Derivation of the Lyapunov candidate function V along the system motion trajectory yields:

[0138]

[0139] If K>D, then according to the definition of Lyapunov equation (16) and equation (17), all signals in the tower crane system are bounded, and the composite integral complementary sliding mode surfaces S and S c It will converge to 0 in a finite time T.

[0140] Therefore, when t>T, S=0, From (8) we get

[0141]

[0142] make x2=e4, x5=e1, but

[0143]

[0144] can be rewritten in compact form in

[0145] X=[x1, x2, x3, x4, x5] T (47)

[0146]

[0147] According to (6) and (19), it is not difficult to find that

[0148]

[0149] In addition, when t ≥ T, the composite integral complementary sliding surfaces S and S c Converges to 0, the controller (9) is simplified to

[0150] u(t)=-h′ -1 H(e′,t) (51)

[0151] Choose appropriate controller parameters α1 and λ so that the matrix A is Hurwitz. Therefore, there exists a unique positive definite matrix P that satisfies PA+A T P = -Q, where Q is a positive definite symmetric matrix. For further analysis, the Lyapunov candidate function is selected

[0152] V′=X T PX (52)

[0153] Then the derivative of V′ along the system motion trajectory can be expressed as

[0154]

[0155] Among them, Q has a minimum eigenvalue σ m Therefore, based on properties 1 and 2 of the tower crane system, as well as (23) and (24), it can be deduced that:

[0156]

[0157] in, is a known positive parameter. In addition, y=Ω1||X|| 2 +Ω2||X||+D is a quadratic function with respect to ||X|| that opens upward, so when the initial value of ||X|| is in the neighborhood

[0158]

[0159] Internal time, And ||X|| will decay as time goes by until ||X||=0, that is Therefore, it can be concluded that

[0160]

[0161] Right now Therefore, from (18),

[0162] Similarly, through similar analysis of the composite sliding surface S2, it can be concluded that That is to say, the proof that all error states of the tower crane system are asymptotically stable is completed.

[0163] Step 3, control the operation of the tower crane system: Use the composite sliding mode controller constructed in step 2 to control the operation status of the tower crane system in real time. The controller realizes the smooth and accurate handling of the tower crane load based on the current load, position, speed and other status information of the tower crane, while effectively suppressing the swing of the load, thereby improving the safety and efficiency of the tower crane operation.

[0164] Among them, the parameters of the compound integral complementary sliding mode controller include:

[0165] (1) λ in formula (7) is used to adjust the convergence speed of the sliding surface. The larger λ is, the faster the sliding surface converges.

[0166] (2) α1 and α2 in formula (8) are used to adjust the weights of the driven sliding surface and the under-actuated sliding surface in the controller to control the load swing suppression effect. The larger the α1 and α2, the more obvious the load swing suppression is. However, too large a value will cause the trolley and cantilever to shake, so a trade-off needs to be made.

[0167] (3) K and Φ in formula (9) are used to suppress the influence of external disturbances on the system. K should be slightly larger than the upper limit of the external disturbance of the system. Φ is the boundary layer thickness of the composite integral complementary sliding surface and is a small positive parameter. The thicker the boundary layer, the less sensitive it is to disturbances, but the positioning accuracy deteriorates. There is a trade-off between the two.

[0168] There is coupling between all parameters, which need to be adjusted according to actual conditions and operating results.

[0169] The present invention does not require model simplification or conversion, and improves the accuracy and stability of the controller. The composite sliding surface integrates the driven and under-actuated states, strengthens the coupling between the two, and enhances the controller's control over the under-actuated state. The integral complementary sliding surface design avoids the arrival stage, and the complementary function makes the system motion smoother, reduces jitter, and improves control accuracy. This method is applicable to under-actuated systems such as cranes, inverted pendulums, and flexible robotic arms, and can be easily extended to other systems.

[0170] The embodiments described above are merely descriptions of preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A tower crane system anti-sway control method based on composite integral complementary sliding mode control, characterized in that The following steps are involved: Step 1, define the tower crane system error variable: used to characterize the difference between the actual operating state and the ideal state of the tower crane to achieve the positioning anti-sway control goal; Step 2, constructing a composite integral complementary sliding mode controller: According to the tower crane system error variable defined in step 1, the under-actuated characteristics of the tower crane system are fully considered, and a composite integral complementary sliding mode controller is constructed, which suppresses the load swing of the tower crane system and maintains the stability and control accuracy of the system; Step 3, control the operation of the tower crane system: use the composite integral complementary sliding mode controller constructed in step 2 to control the operation status of the tower crane system in real time. The controller realizes smooth and precise handling of the tower crane load according to the current load, position, and speed status information of the tower crane, while effectively suppressing the swing of the load.

2. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 1, characterized in that: In step 1, the position error variable of the 4-DOF tower crane system is defined as e i (t)=q i (t)-q id (6) Among them, e i (t) is the position error variable, t is the system operation time, i=1,2,3,4 represents four degrees of freedom, represents a 4-dimensional column vector, is the rotation angle of the cantilever of the tower crane system, x(t) is the displacement of the trolley, θ1(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal front and rear direction, θ2(t) is the offset angle of the load relative to the center line of the tower crane boom in the horizontal plane left and right direction; q 1d is the desired angle of the cantilever angular operation; q 2d is the desired angle of the cantilever angular operation; q 3d and q 4d is the expected offset angle of the load in the horizontal direction in the front, back, left and right directions relative to the center line of the tower crane boom, which is set as: 3d =q 4d =0.

3. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 1, characterized in that: The integral complementary sliding surface in step 2 is defined as: Among them, λ is the positive parameter of the sliding surface.

4. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 1, characterized in that: The composite sliding surface in step 2 is defined as: Where S = [S1, S2] T , S c =[S c1 ,S c2 ] T is a composite complementary sliding surface, α1 and α2 are adjustable parameters; Setting the compound integral complementary sliding mode controller: Where, e′=[e1+α1e4,e2+α2e3] T ∈R 2 , is the nonlinear control function, f′=[f1+α1f4,f2+α2f3] T ∈R 2 and h′ -1 =[(h1+α1h4) -1 ,(h2+α2h3) -1 ] T ∈R 2×2 is a known nonlinear term in the system, sat(·) is the saturation function, K is the controller parameter, and Φ is the boundary layer parameter.

5. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 3, characterized in that: In step 2, select the Lyapunov candidate function: Derivation of the Lyapunov candidate function V along the system motion trajectory yields: Select K>D, where D>0 is the upper bound of the unknown disturbance. According to the definition of Lyapunov equation (16) and equation (17), all signals in the tower crane system are bounded, and the composite integral complementary sliding mode surfaces S and S c It will converge to 0 in a finite time T.

6. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 5, characterized in that: The composite integral complementary sliding surfaces S and S in step 2 c After converging to 0, select the Lyapunov candidate function V′=X T PX (25) Then the derivative of V′ along the system motion trajectory can be expressed as Where X is the system state, A is the system state matrix, and appropriate parameters α1 and λ are selected so that there exists a unique positive definite matrix P that satisfies PA+A T P = -Q, so when the initial value of ||X|| is in the neighborhood Internal time, And ||X|| will decay as time goes by until ||X||=0, that is Therefore, it can be concluded that Therefore, all position error states of the system are asymptotically stable.

7. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 1, characterized in that: In step 3, appropriate parameters of the composite integral complementary sliding mode controller are selected so that the composite integral complementary sliding mode controller can control the smooth and precise handling of the tower crane load and suppress the swing of the load; the parameters of the composite integral complementary sliding mode controller include: the positive parameter λ of the sliding surface, the adjustable parameters α1 and α2, the controller parameter K and the boundary layer parameter Φ. There is a coupling characteristic between all parameters, and they should be adjusted according to the actual situation and operation effect.

8. The anti-sway control method for a tower crane system based on composite integral complementary sliding mode control according to claim 7, characterized in that: λ is used to adjust the convergence speed of the sliding surface. The larger the λ is, the faster the sliding surface converges. α1 and α2 are used to adjust the weights of the driven sliding surface and the underdriven sliding surface in the controller to control the suppression effect of load swing. The larger the α1 and α2 are, the more obvious the suppression of the load swing angle is. However, too large a value will cause the jitter of the trolley and cantilever, so a trade-off needs to be made. K and Φ are used to suppress the influence of external disturbances on the system. K should be slightly larger than the upper bound of the external disturbance of the system. Φ is the boundary layer thickness of the composite integral complementary sliding surface, which is a small positive parameter. The thicker the boundary layer, the less sensitive it is to disturbances, but the positioning accuracy becomes worse. There is a trade-off between the two.

9. A computer device, characterized in that: The computer device includes a memory and a processor, the memory stores a computer program, and when the computer program is executed by the processor, the processor executes the anti-sway control method for a tower crane system based on a composite integral complementary sliding mode as described in any one of claims 1-8.

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