CN114572831A - Bridge crane sliding mode control method based on unknown input observer technology - Google Patents

Abstract

A bridge crane sliding mode control method based on an unknown input observer technology is characterized by establishing a dynamic model of a bridge crane system and initializing a system state and control parameters; designing an unknown input observer; designing a double-layer hyperbolic approaching law; and designing a sliding mode controller based on an unknown input observer and a double-layer hyperbolic approximation rule. The method can effectively solve the problem of rapid positioning of the bridge crane under different continuous non-matching disturbance conditions, effectively inhibit the residual swinging of the load of the bridge crane and ensure the stability of control input.

Description

A sliding mode control method of overhead crane based on unknown input observer technology

technical field

The invention belongs to the technical field of underactuated bridge crane control, and provides a bridge crane sliding mode control method based on unknown input observer technology, in particular to the bridge crane sliding mode control method containing different types of unmatched disturbances.

Background technique

Underactuated system is a special kind of system, its specific meaning is that the independent control input of the system is less than the degree of freedom of the system itself. Crane is a typical underactuated system, widely used in ports, construction sites, offshore drilling platforms and other places, and plays an irreplaceable role in modern industrial applications.

However, in actual production applications, cranes will be affected by external disturbances such as wind, and these uncertain factors can easily cause serious safety accidents to operators. In recent years, many scholars at home and abroad have devoted themselves to the research of crane sway elimination and positioning control, and have achieved many fruitful results. However, most of the existing research results are aimed at a series of control methods proposed for the crane in the undisturbed condition or when the crane has matching disturbance (including friction, etc.). When there are unmatched disturbances (including wind, air resistance, etc.) in the crane, there are few relevant research results that can well deal with the influence of such disturbances on the control performance of the crane. Therefore, it is very meaningful to study an effective method to suppress the unmatched disturbance in the crane system.

Unmatched disturbances exist in many control systems, and many scholars have proposed new control theories and methods to suppress such disturbances, including linear control methods based on generalized extended state observers, hierarchical sliding mode control methods, backstepping These methods can effectively suppress the influence of mismatched disturbances on control performance. At present, the research on the suppression of unmatched disturbances in underactuated cranes is still in its infancy, and many existing research works only target unmatched disturbances under certain types of disturbances or unmatched disturbances under non-continuous action. Therefore, it is very necessary to find a control method that can be quickly realized and has a good suppression effect for the unmatched disturbances under different types of continuous action in the crane system. .

SUMMARY OF THE INVENTION

In order to reduce the influence of unmatched disturbances under different types of continuous action on the anti-swing positioning of the crane, and at the same time, the crane can quickly reach a stable state and have good robustness, the present invention proposes a method based on the unknown input observer technology. The sliding mode control method for overhead cranes is proposed. The proposed method uses the unknown input observer technique to estimate the matching disturbance and non-matching disturbances in the crane system. Then, a double-layer hyperbolic reaching law is designed, based on the estimated matching disturbance. term and unmatched perturbation term and double-layer hyperbolic reaching law, design sliding mode controller. The design can effectively suppress the impact of matching disturbance and different types of unmatched disturbances under continuous action on the anti-sway positioning performance of the crane, and at the same time ensure that the crane system has a rapid state convergence ability and a stable control output effect.

The technical solutions proposed to solve the above technical problems are as follows:

A sliding mode control method for overhead cranes based on unknown input observer technology, comprising the following steps:

Step 1, establish a dynamic model of the overhead crane system, initialize the state and control parameters of the system, and the process is as follows:

1.1 The dynamics of the overhead crane is expressed as:

表示台车运动时的加速度；l表示吊绳的长度；

分别表示负载摆动的角度，角速度和角加速度；F表示控制输入；d_x表示匹配扰动集总项，包括摩擦力，未建模动力学等；d_θ表示非匹配扰动集总项，包括空气阻力，摩擦力等；g表示重力加速度；where M and m represent the mass of the trolley and the mass of the load, respectively;

Indicates the acceleration of the trolley when it moves; l represents the length of the sling;

represent the angle, angular velocity and angular acceleration of the load swing, respectively; F represents the control input; d _x represents the lumped term of the matched disturbance, including friction, unmodeled dynamics, etc.; d _θ represents the lumped term of the unmatched disturbance, including the air resistance , friction force, etc.; g represents the acceleration of gravity;

1.2 Combining equations (1)-(2), the acceleration expression of the trolley during motion is obtained as:

1.3 Substitute equation (3) into equation (2), and through mathematical transformation, the control input F is obtained as:

in,

1.4 In order to facilitate the design of subsequent controllers, define an auxiliary control variable v:

Substitute equation (5) into equation (4), and re-transform to get:

1.5 Define the following state variables:

分别为台车运动的参考轨迹和速度；sec(·)表示余弦函数的倒数；in,

are the reference trajectory and speed of the trolley motion, respectively; sec( ) represents the reciprocal of the cosine function;

1.6 Combine the formula (7) to construct the state equation of the following form:

分别表示式(7)中

的一阶导数；λ_i,i＝1,2,3表示定义的辅助变量；

表示状态变量

的构造函数；它们的具体表达式如下：in,

Respectively expressed in formula (7)

The first derivative of ; λ _i , i=1, 2, 3 represent the defined auxiliary variables;

Represents a state variable

constructors; their concrete expressions are as follows:

1.7 In order to facilitate the subsequent design of the unknown input observer, formula (8) is transformed into the following form:

d_m＝λ₂d_x-λ₃d_θ；in,

d _m =λ ₂ d _x -λ ₃ d _θ ;

因此式(9)定义的辅助变量λ₁,λ2₂,λ₃是有界的，同理构造函数

也是有界的；1.8 For the actual application scenario of the crane, the angle of the load swing meets the requirements

Therefore, the auxiliary variables λ ₁ , λ2 ₂ , λ ₃ defined by equation (9) are bounded, and the same is true for the constructor function

is also bounded;

Step 2: Design an unknown input observer to estimate the matching and non-matching disturbances in the crane system. The process is as follows:

2.1 It is assumed that matched perturbations and non-matched perturbations, as well as their first derivatives, are bounded, defined as follows:

分别表示匹配扰动与非匹配扰动的上界，

分别表示匹配扰动与非匹配扰动一阶导数的上界；in,

represent the upper bounds of matched perturbation and unmatched perturbation, respectively,

respectively represent the upper bound of the first derivative of matched disturbance and unmatched disturbance;

2.2 Perform first-order low-pass filter transformation on the transformed crane system model (10), and design the unknown input observer as follows:

Wherein, k>0 represents the filter coefficient; [·]/(ks+1)=(·) _f in the definition formula (12); formula (12) is transformed into:

表示原始状态变量

经过低通滤波变换得到的

的一阶导数，d_mmf,v_f,d_mf表示d_mm,v,d_m经过低通滤波后的结果；in,

represents the original state variable

After low-pass filtering, the

The first derivative of , d _mmf , v _f , d _mf represent the result of d _mm , v, d _m after low-pass filtering;

2.3 Each state variable in equation (13) has the following properties:

The expressions of d _mmf , d _mf can be obtained from equations (13)-(14):

Combined with equation (15), the disturbance observer is designed as follows:

表示扰动项d_mm,d_m的估计值；in,

represents the estimated value of the disturbance term d _mm , d _m ;

2.4 It is found from equation (16) that the disturbance observer has only one filter coefficient k that can be adjusted, so as to avoid the noise amplification problem caused by the system derivation;

Step 3, the design of the double-layer hyperbolic reaching law, the process is as follows:

Design a double-layer hyperbolic reaching law as follows:

表示待设计滑模面的一阶导数；γ₁＞0,γ₂＞0,k₁＞0,k₂＞0，p表示一个正奇数；tanh(k₁s)表示双曲正切函数，表达式如下：in,

represents the first derivative of the sliding mode surface to be designed; γ ₁ >0, γ ₂ >0, k ₁ >0, k ₂ >0, p represents a positive odd number; tanh(k ₁ s) represents the hyperbolic tangent function, which expresses The formula is as follows:

a sinh( ^k ₂ sp ) represents the inverse hyperbolic sine function, and the expression is as follows:

Step 4, the sliding mode control law design based on the unknown input observer technology, the process is as follows:

4.1 Combined with formula (7), the sliding mode surface is defined as follows:

Among them, a>0, b>0, c>0;

4.2 Derivation of formula (20) in combination with formula (10) can be obtained:

4.3 The sliding mode controller based on the double-layer hyperbolic reaching law combined with equations (16)-(17) to design equation (10) is:

4.4 Choose the following Lyapunov function:

即判定系统是稳定的；Taking the derivative of the above formula, and substituting the formula (22), we get

That is, it is determined that the system is stable;

4.5 Substitute the auxiliary control law designed in Equation (22) into Equation (6), and obtain the F expression as:

Equation (24) represents the actual control law in the crane system.

The present invention designs a sliding mode control method for an overhead crane based on the unknown input observer technology. While solving the influence of matching disturbance and non-matching disturbance on the anti-swing positioning performance of the crane system, the proposed double-layer hyperbolic reaching law is effective Improve the convergence speed of the system state and achieve good control of the overhead crane system.

The technical concept of the present invention: for the bridge crane system with matching disturbance and different types of unmatched disturbances under continuous action, the present invention adopts the first-order low-pass filtering method in the unknown input observer technology to estimate the matching and matching in the crane system. According to the estimated value, a sliding mode controller based on the double-layer hyperbolic reaching law is designed. The controller can effectively suppress the influence of the matching disturbance and the unmatched disturbance on the crane system, while ensuring the state of the crane system. Can quickly calm down. The present invention proposes a sliding mode control method based on unknown input observer technology, which can effectively suppress matching disturbances and non-matching disturbances under different types of continuous action, and at the same time realizes rapid positioning and anti-swing of the crane, ensuring that the bridge crane system can achieve relatively high performance. good control effect.

The effective effects of the invention are: realize the rapid positioning of the overhead crane, suppress the residual swing of the load of the overhead crane, and ensure the stability of the control input.

Description of drawings

Fig. 1 is the control flow chart of the present invention;

Figure 2 shows the state trajectory of x when a nonlinear disturbance observer is added to the constant non-matching disturbance;

Fig. 3 is the state curve of θ when the nonlinear disturbance observer is added to the constant value unmatched disturbance;

Fig. 4 is the signal curve of F when the nonlinear disturbance observer is added to the constant value unmatched disturbance;

Figure 5 shows the state trajectory of x when a generalized extended state observer is added to the constant non-matching disturbance;

Fig. 6 is the state curve of θ when a generalized extended state observer is added to the constant-value unmatched disturbance;

Fig. 7 is the signal curve of F when the generalized extended state observer is added to the constant-value unmatched disturbance;

Fig. 8 is the state trajectory of x when adding the sliding mode controller of the present invention in the constant value unmatched disturbance;

Fig. 9 is the state curve of θ when adding the sliding mode controller of the present invention in the constant value unmatched disturbance;

Fig. 10 is the signal curve of F when adding the sliding mode controller of the present invention in the constant value unmatched disturbance;

Figure 11 shows the state trajectory of x when a nonlinear disturbance observer is added to the periodic unmatched disturbance;

Figure 12 is a state curve of θ when a nonlinear disturbance observer is added to the periodic unmatched disturbance;

Fig. 13 is the signal curve of F when the nonlinear disturbance observer is added to the periodic unmatched disturbance;

Figure 14 shows the state trajectory of x when a generalized extended state observer is added to the periodic unmatched disturbance;

Fig. 15 is the state curve of θ when a generalized extended state observer is added to the periodic unmatched disturbance;

Fig. 16 is the signal curve of F when the generalized extended state observer is added to the periodic unmatched disturbance;

Fig. 17 is the state trajectory of x when the sliding mode controller of the present invention is added to the periodic unmatched disturbance;

Fig. 18 is the state curve of θ when the sliding mode controller of the present invention is added to the periodic unmatched disturbance;

FIG. 19 is the signal curve of F when the sliding mode controller of the present invention is added to the periodic unmatched disturbance.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings.

1-19, a sliding mode control method for overhead cranes based on unknown input observer technology includes the following steps:

Step 1, establish a dynamic model of the overhead crane system, initialize the state and control parameters of the system, and the process is as follows:

1.1 The dynamics of the overhead crane is expressed as:

表示台车运动时的加速度；l表示吊绳的长度；

分别表示负载摆动的角度，角速度和角加速度；F表示控制输入；d_x表示匹配扰动集总项，包括摩擦力，未建模动力学等；d_θ表示非匹配扰动集总项，包括空气阻力，摩擦力等；g表示重力加速度；where M and m represent the mass of the trolley and the mass of the load, respectively;

Indicates the acceleration of the trolley when it moves; l represents the length of the sling;

represent the angle, angular velocity and angular acceleration of the load swing, respectively; F represents the control input; d _x represents the lumped term of the matched disturbance, including friction, unmodeled dynamics, etc.; d _θ represents the lumped term of the unmatched disturbance, including the air resistance , friction force, etc.; g represents the acceleration of gravity;

1.2 Combining equations (1)-(2), the acceleration expression of the trolley during motion is obtained as:

1.3 Substitute equation (3) into equation (2), and through mathematical transformation, the control input F is obtained as:

in,

1.4 In order to facilitate the design of subsequent controllers, define an auxiliary control variable v:

Substitute equation (5) into equation (4), and re-transform to get:

1.5 Define the following state variables:

分别为台车运动的参考轨迹和速度；sec(·)表示余弦函数的倒数；in,

are the reference trajectory and speed of the trolley motion, respectively; sec( ) represents the reciprocal of the cosine function;

1.6 Combine the formula (7) to construct the state equation of the following form:

分别表示式(7)中

的一阶导数；λ_i,i＝1,2,3表示定义的辅助变量；

表示状态变量

的构造函数；它们的具体表达式如下：in,

Respectively expressed in formula (7)

The first derivative of ; λ _i , i=1, 2, 3 represent the defined auxiliary variables;

Represents a state variable

constructors; their concrete expressions are as follows:

1.7 In order to facilitate the subsequent design of the unknown input observer, formula (8) is transformed into the following form:

d_m＝λ₂d_x-λ₃d_θ；in,

d _m =λ ₂ d _x -λ ₃ d _θ ;

因此式(9)定义的辅助变量λ₁,λ₂,λ₃是有界的，同理构造函数

也是有界的；1.8 For the actual application scenario of the crane, the angle of the load swing meets the requirements

Therefore, the auxiliary variables λ ₁ , λ ₂ , λ ₃ defined by equation (9) are bounded, and the same is true for the constructor function

is also bounded;

Step 2: Design an unknown input estimator to estimate the matching and non-matching disturbances in the crane system. The process is as follows:

2.1 It is assumed that matched perturbations and non-matched perturbations, as well as their first derivatives, are bounded, defined as follows:

分别表示匹配扰动与非匹配扰动的上界，

分别表示匹配扰动与非匹配扰动一阶导数的上界；in,

represent the upper bounds of matched perturbation and unmatched perturbation, respectively,

respectively represent the upper bound of the first derivative of matched disturbance and unmatched disturbance;

2.2 Perform first-order low-pass filter transformation on the transformed crane system model (10), and design the unknown input observer as follows:

Wherein, k>0 represents the filter coefficient; [·]/(ks+1)=(·) _f in the definition formula (12); formula (12) is transformed into:

表示原始状态变量

经过低通滤波变换得到的

的一阶导数，d_mmf,v_f,d_mf表示d_mm,v,d_m经过低通滤波后的结果；in,

represents the original state variable

After low-pass filtering, the

The first derivative of , d _mmf , v _f , d _mf represent the result of d _mm , v, d _m after low-pass filtering;

2.3 Each state variable in equation (13) has the following properties:

The expressions of d _mmf , d _mf can be obtained from equations (13)-(14):

Combined with equation (15), the disturbance observer is designed as follows:

表示扰动项d_mm,d_m的估计值；in,

represents the estimated value of the disturbance term d _mm , d _m ;

2.4 It is found from equation (16) that the disturbance observer has only one filter coefficient k that can be adjusted, so as to avoid the noise amplification problem caused by the system derivation;

Step 3, the design of the double-layer hyperbolic reaching law, the process is as follows:

Design a double-layer hyperbolic reaching law as follows:

表示待设计滑模面的一阶导数；γ₁＞0,γ₂＞0,k₁＞0,k₂＞0，p表示一个正奇数；tanh(k₁s)表示双曲正切函数，表达式如下：in,

represents the first derivative of the sliding mode surface to be designed; γ ₁ >0, γ ₂ >0, k ₁ >0, k ₂ >0, p represents a positive odd number; tanh(k ₁ s) represents the hyperbolic tangent function, which expresses The formula is as follows:

asinh( ^k ₂ sp ) represents the inverse hyperbolic sine function, and the expression is as follows:

Step 4, the sliding mode control law design based on the unknown input observer technology, the process is as follows:

4.1 Combined with formula (7), the sliding mode surface is defined as follows:

Among them, a>0, b>0, c>0;

4.2 Derivation of formula (20) in combination with formula (10) can be obtained:

4.3 The sliding mode controller based on the double-layer hyperbolic reaching law combined with equations (16)-(17) to design equation (10) is:

4.4 Choose the following Lyapunov function:

即判定系统是稳定的；Taking the derivative of the above formula, and substituting the formula (22), we get

That is, it is determined that the system is stable;

4.5 Substitute the auxiliary control law designed by Equation (22) into Equation (6), and obtain the F expression as:

Equation (24) represents the actual control law in the crane system.

图2-图19是对含有不同类型持续非匹配扰动的桥式吊车系统的仿真结果对比图。图2-图10是系统(8)中加入持续性常值非匹配扰动时的仿真对比结果。其中图2-图4是系统(8)中加入基于非线性扰动观测器的传统滑模控制器的状态曲线图x,θ,u，由图可以看出x在18秒左右进入稳定状态，θ在24秒进入稳定状态且存在一定的抖动问题，F存在明显的输入抖动问题；图5-图7是在系统(8)中加入基于广义扩张状态观测器的反馈控制器的状态曲线图x,θ,u，从图中可以看出x在17秒左右进入稳定状态，θ在21秒左右进入稳定状态，F在20秒左右进入稳定范围，但存在较为明显的上下波动，不够平稳；图8-图10是在系统(8)中加入本发明的基于未知输入观测器技术的滑模控制器状态曲线图x,θ,u，由图可发现x在13秒左右进入稳定状态，θ在15秒左右进入稳定状态，且负载不存在残余摆动现象，F在2秒左右进入稳定状态，且不存在明显的颤振或者波动现象。图11-图19是系统(8)中加入持续性周期非匹配扰动时的仿真对比结果。图11-图13是系统(8)中加入基于非线性扰动观测器的传统滑模控制器的状态曲线图x,θ,u，由图可以看出x在18秒左右进入稳定状态，θ在22秒进入稳定范围但负载存在明显残余摆动现象，F存在剧烈的抖振现象；图14-图16是在系统(8)中加入基于广义扩张状态观测器的反馈控制器的状态曲线图x,θ,u，从图中可以看出x在18秒左右进入稳定状态，θ在21秒左右进入稳定范围但负载也存在明显的残余摆动现象，F在22秒左右进入稳定范围，但存在较为明显的上下波动，不够平稳；图17-图19是在系统(8)中加入本发明的基于未知输入观测器技术的滑模控制器状态曲线图x,θ,u，由图可发现x在15秒左右进入稳定状态，θ在16秒左右进入稳定状态，且负载不存在明显的残余摆动现象，F在5秒左右进入稳定状态，且不存在明显的颤振或者波动现象，输出平稳。由三种方法的仿真对比结果可得，与基于非线性扰动观测器的传统滑模控制器和基于广义扩张状态观测器的反馈控制器相比较，基于本发明的未知输入观测器技术滑模控制器，可以保证桥式吊车系统状态快速到达稳定范围，且对不同的持续性非匹配扰动类型有着良好的控制效果。综上，基于未知输入观测器技术的桥式吊车滑模控制方法可以有效解决桥式吊车面对持续性非匹配扰动时的抑制问题，同时减少系统状态到达稳态的过渡时间，使桥式吊车具有良好的定位抗摆性能。In order to verify the effectiveness of the proposed method, the present invention conducts a simulation experiment on the control effect of the sliding mode controller of the bridge crane based on the unknown input observer technology shown in Eq. The effects of traditional sliding mode controller and state feedback controller based on generalized extended state observer are compared. Set the initial conditions and control parameters in the experiment as follows: the sampling step is 0.001s; the selected system parameters are M=5kg, m=10kg, l=6m, g=9.8m/s ² ; the controller parameters are selected as a=0.44 , b=1.74, c=2.28, γ ₁ =28, k ₁ =15, γ ₂ =15, k ₂ =10, p=3; the desired positioning target is x _d =10m; the filter coefficients in the unknown input observer k=0.01; the matching disturbance is d _x =0.5sin(t); the constant non-matching disturbance is d _θ =2, and the periodic unmatching disturbance is d _θ =sin(t); the initial state of the system is

Figures 2-19 are comparison diagrams of simulation results for overhead crane systems with different types of persistent unmatched disturbances. Fig. 2-Fig. 10 are the simulation comparison results when a persistent constant non-matching disturbance is added to the system (8). Among them, Fig. 2-Fig. 4 are the state curves x, θ, u of the traditional sliding mode controller based on the nonlinear disturbance observer added to the system (8). It can be seen from the figure that x enters a stable state in about 18 seconds, and θ Entering a stable state in 24 seconds and there is a certain jitter problem, F has an obvious input jitter problem; Figures 5-7 are the state curves of the feedback controller based on the generalized extended state observer added to the system (8) x, θ, u, it can be seen from the figure that x enters a stable state in about 17 seconds, θ enters a stable state in about 21 seconds, and F enters a stable state in about 20 seconds, but there are obvious up and down fluctuations, which are not stable enough; Figure 8 - Figure 10 is a state curve graph x, θ, u of the sliding mode controller based on the unknown input observer technology of the present invention added to the system (8). It can be found from the graph that x enters a stable state in about 13 seconds, and θ is 15 It enters a stable state in about 2 seconds, and the load has no residual swing phenomenon, and F enters a stable state in about 2 seconds, and there is no obvious flutter or fluctuation phenomenon. Fig. 11-Fig. 19 are the simulation comparison results when continuous periodic unmatched disturbance is added to the system (8). Figures 11-13 are the state curves x, θ, u of the traditional sliding mode controller based on the nonlinear disturbance observer added to the system (8). It can be seen from the figure that x enters a stable state in about 18 seconds, and θ is in the stable state. 22 seconds into the stable range, but the load has obvious residual swing phenomenon, and F has a severe chattering phenomenon; Figure 14-Figure 16 is the state curve diagram x of adding a feedback controller based on a generalized expansion state observer in system (8), θ, u, it can be seen from the figure that x enters the stable state at about 18 seconds, θ enters the stable range at about 21 seconds, but the load also has obvious residual swing phenomenon, and F enters the stable range at about 22 seconds, but there are obvious residual swings. The up and down fluctuations are not stable enough; Figures 17-19 are the state curves x, θ, u of the sliding mode controller based on the unknown input observer technology of the present invention added to the system (8). It can be found from the figure that x is at 15 It enters a stable state in about 5 seconds, θ enters a stable state in about 16 seconds, and the load has no obvious residual swing phenomenon, F enters a stable state in about 5 seconds, and there is no obvious flutter or fluctuation phenomenon, and the output is stable. Compared with the traditional sliding mode controller based on the nonlinear disturbance observer and the feedback controller based on the generalized extended state observer, the sliding mode control based on the unknown input observer technology of the present invention can be obtained from the simulation comparison results of the three methods. The controller can ensure that the state of the overhead crane system quickly reaches the stable range, and has a good control effect on different types of persistent non-matching disturbances. In conclusion, the sliding mode control method of bridge crane based on unknown input observer technology can effectively solve the problem of suppression when the bridge crane faces continuous unmatched disturbances, and at the same time reduce the transition time for the system state to reach a steady state, so that the bridge crane can be effectively controlled. Has good positioning and anti-swing performance.

What is described above is that the simulation comparison experiment given by the present invention shows the superiority of the designed method. Obviously, the present invention is not only limited to the above-mentioned examples, but does not deviate from the basic spirit of the present invention and does not exceed the scope of the essential content of the present invention. Various modifications can be made to it. The control method designed by the invention has good positioning and anti-swing control effect on bridge cranes with different continuous non-matching disturbances, and can effectively improve the working efficiency of the crane.

Claims (1)

1. an overhead crane sliding mode control method based on unknown input observer technology, is characterized in that: described method comprises the following steps: Step 1, establish the dynamic model of the overhead crane system, initialize the state and control parameters of the system, and the process is as follows: 1.1 The dynamics of the overhead crane is expressed as:

where M and m represent the mass of the trolley and the mass of the load, respectively;

Represents the acceleration of the trolley when it moves; l represents the length of the sling; θ,

represent the angle, angular velocity and angular acceleration of the load swing, respectively; F represents the control input; d _x represents the lumped term of the matched disturbance, including friction, unmodeled dynamics, etc.; d _θ represents the lumped term of the unmatched disturbance, including the air resistance , friction force, etc.; g represents the acceleration of gravity; 1.2 Combining equations (1)-(2), the acceleration expression of the trolley during motion is obtained as:

1.3 Substitute equation (3) into equation (2), and through mathematical transformation, the control input F is obtained as:

in,

1.4 Define an auxiliary control quantity v:

Substitute equation (5) into equation (4), and re-transform to get:

1.5 Define the following state variables:

where x _d ,

are the reference trajectory and speed of the trolley motion, respectively; sec( ) represents the reciprocal of the cosine function; 1.6 Combine the formula (7) to construct the state equation of the following form:

in,

Respectively expressed in formula (7)

The first derivative of ; λ _i , i=1, 2, 3 represent the defined auxiliary variables;

Represents a state variable

constructors; their concrete expressions are as follows:

1.7 Transform equation (8) into the following form:

in,

d _m =λ ₂ d _x -λ ₃ d _θ ; 1.8 For the actual application scenario of the crane, the angle of the load swing meets the requirements

Therefore, the auxiliary variables λ ₁ , λ ₂ , λ ₃ defined by equation (9) are bounded, and the same is true for the constructor function

is also bounded; Step 2: Design an unknown input estimator to estimate the matching and non-matching disturbances in the crane system. The process is as follows: 2.1 It is assumed that matched perturbations and non-matched perturbations, as well as their first derivatives, are bounded, defined as follows:

in,

represent the upper bounds of matched perturbation and unmatched perturbation, respectively,

respectively represent the upper bound of the first derivative of matched disturbance and unmatched disturbance; 2.2 Perform first-order low-pass filter transformation on the transformed crane system model (10), and design the unknown input observer as follows:

Wherein, k>0 represents the filter coefficient; [·]/(ks+1)=(·) _f in the definition formula (12); formula (12) is transformed into:

in,

represents the original state variable

After low-pass filtering, the

The first derivative of , d _mmf , v _f , d _mf represent the result of d _mm , v, d _m after low-pass filtering; 2.3 Each state variable in equation (13) has the following properties:

The expressions of d _mmf , d _mf can be obtained from equations (13)-(14):

Combined with equation (15), the disturbance observer is designed as follows:

in,

represents the estimated value of the disturbance term d _mm , d _m ; 2.4 It is found from equation (16) that the disturbance observer has only one filter coefficient k that can be adjusted, so as to avoid the noise amplification problem caused by the system derivation; Step 3, the design of the double-layer hyperbolic reaching law, the process is as follows: Design a double-layer hyperbolic reaching law as follows:

in,

represents the first derivative of the sliding mode surface to be designed; γ ₁ >0, γ ₂ >0, k ₁ >0, k ₂ >0, p represents a positive odd number; tanh(k ₁ s) represents the hyperbolic tangent function, which expresses The formula is as follows:

asinh( ^k ₂ sp ) represents the inverse hyperbolic sine function, and the expression is as follows:

Step 4, the sliding mode control law design based on the unknown input observer technology, the process is as follows: 4.1 Combined with formula (7), the sliding mode surface is defined as follows:

Among them, a>0, b>0, c>0; 4.2 Derivation of formula (20) in combination with formula (10) can be obtained:

4.3 The sliding mode controller based on the double-layer hyperbolic reaching law combined with equations (16)-(17) to design equation (10) is:

4.4 Choose the following Lyapunov function:

Taking the derivative of the above formula, and substituting the formula (22), we get

That is, it is determined that the system is stable; 4.5 Substitute the auxiliary control law designed by Equation (22) into Equation (6), and obtain the expression of F as:

Equation (24) represents the actual control law in the crane system.

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