Bridge crane sliding mode anti-swing control method based on variable speed reaching law

Abstract

The invention discloses a bridge crane sliding mode anti-swing control method based on a variable speed reaching law. The method comprises the following steps: converting a state conversion system model according to a bridge crane dynamic model; a new variable speed reaching law is constructed by using an exponential function, so that the transient convergence speed of a sliding mode variable can be improved, and buffeting of a controller when the sliding mode variable enters a steady state can be reduced. And constructing a linear sliding mode surface, and designing a sliding mode controller based on the sliding mode surface and a variable speed reaching law. According to the method, the convergence speed of the sliding mode variable can be effectively improved, so that the rapid positioning problem of the bridge crane is solved, and the buffeting problem of the controller caused by a sign function is weakened.

Description

Technical Field

[0001] The invention relates to a sliding mode anti-sway control method for a bridge crane based on a variable speed reaching law, so as to meet the performance requirements of rapid positioning of a bridge crane system, effective sway elimination and weakened vibration. Background Art

[0002] An underactuated system is a type of nonlinear system in which the number of independent control variables is less than the number of system degrees of freedom. Compared with a fully driven system, an underactuated system has irreplaceable advantages in terms of energy saving, cost reduction, overall weight reduction, and system reliability enhancement. A bridge crane is a typical underactuated system, which is mainly used in workplaces such as construction sites and warehouses. It can deliver some heavy goods from a certain location to a target point. It is generally equipped with wire ropes and pulleys to raise, lower, or move the load horizontally. Cranes are also used in ports to load and unload goods and transport ships. In addition, there is a type of crane fixed on some movable platforms, such as ships or cars, so that the crane can quickly reach the work site, increasing the flexibility of crane application. However, bridge cranes have characteristics such as strong coupling and nonlinearity, which increase the difficulty of controller design. In addition, cranes are prone to swing when transporting goods, which not only reduces the efficiency of transporting goods, but also brings many safety hazards.

[0003] In order to achieve good positioning accuracy and good anti-sway effect, many researchers at home and abroad have proposed a variety of control methods. According to whether the crane control requires state information feedback, the crane control method can be divided into open-loop control method and closed-loop control method. The open-loop control method does not require feedback of system information, so it has the advantages of simple operation and low cost in actual operation. However, due to the lack of specific feedback information, the open-loop control is more sensitive to changes in system parameters and external disturbances. Common open-loop control methods include input shaping method, trajectory planning method, optimal control method, etc. The closed-loop control method uses the feedback characteristics of the system state to adjust the system state in real time according to the error between input and output, so that the closed-loop system has good robustness. Therefore, in order to achieve high-performance and high-precision control goals, relevant researchers at home and abroad have proposed many effective closed-loop control methods. Among them, the proposed sliding mode control has the advantages of fast response speed and strong robustness to system uncertainty and external interference. However, how to improve the rapidity while ensuring the smooth operation of the crane system is still a major problem. The sliding mode control based on the reaching law can realize discontinuous control, the sliding mode is designable, and it is not related to system parameters and disturbances. The sliding mode control based on the variable speed reaching law can not only reasonably design the speed to reach the sliding surface, reduce the time of the approaching stage, and improve the robustness of the system, but also effectively reduce the chattering problem in the sliding mode control. At present, in the field of bridge crane control, the sliding mode control based on the variable speed reaching law is rarely used, and the variable speed reaching law further accelerates the approaching speed of the system to the sliding surface on the basis of the traditional reaching law, while making the chattering smaller. Summary of the invention

[0004] In order to overcome the shortcomings of the prior art and to further improve the rapidity of the crane, that is, the trolley can quickly reach the target position, while reducing the adverse effects of jitter on the system. The present invention proposes a sliding mode anti-swing control method for a bridge crane based on a variable speed reaching law. The proposed method designs sliding mode variables through system state transformation, constructs auxiliary functions, and then designs a new sliding mode anti-swing controller based on a variable speed reaching law. This design can not only reduce the jitter of the sliding mode variable when entering a steady state, but also improve the transient convergence speed of the sliding mode variable, thereby ensuring that the crane system has a fast state convergence capability and a smooth control output effect.

[0005] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0006] A sliding mode anti-sway control method for a bridge crane based on a variable speed reaching law comprises the following steps:

[0007] Step 1: Establish a dynamic model of the bridge crane system and initialize the system state and control parameters. The process is as follows:

[0008] 1.1 The dynamic model of the bridge crane is:

[0009]

[0010] Among them, x represents the position of the trolley during operation, Indicates the speed of the trolley when it is running. represents the acceleration of the trolley during movement, θ represents the amplitude of the load swing angle, represents the speed of the load swing angle, represents the acceleration of the load swing angle, M represents the mass of the trolley, m represents the mass of the load, l represents the length of the suspension rope, F represents the control input, g represents the acceleration of gravity, d represents the external disturbance, cos(·) represents the cosine function, sin(·) represents the sine function, and

[0011] 1.2 According to formula (1), the acceleration expression of the trolley during movement is:

[0012]

[0013] 1.3 Substituting equation (2) into equation (1), after mathematical transformation, the control input F is obtained as:

[0014]

[0015] in, tan(·) is the tangent function;

[0016] 1.4 In order to facilitate the design of subsequent controllers, an auxiliary control quantity v is defined:

[0017]

[0018] Substituting equation (4) into equation (3), the control input F can be rewritten as:

[0019]

[0020] 1.5 Define the following state variables:

[0021]

[0022] Among them, x d , are the reference trajectory and speed of the trolley motion, respectively, sec(·) represents the inverse of the cosine function;

[0023] 1.6 Combined with equation (6), the following state equation is constructed:

[0024]

[0025] in, Respectively expressed in formula (6) The first-order derivative of , λ represents the auxiliary variable defined, Represents state variables The constructor of ; λ and The specific expression is as follows:

[0026]

[0027] 1.7 For the actual application scenario of the crane, the load swing angle satisfies |θ(t)|≤π / 3, and the rope of the two-dimensional bridge crane is a rigid rope, that is, l is a constant;

[0028] Step 2: For the bridge crane system with unknown dynamics, the linear sliding surface and speed reaching law are designed through the defined state variables. The process is as follows:

[0029] 2.1 Design of linear sliding surface:

[0030]

[0031] Among them, a>0, b>0, c>0;

[0032] 2.2 Take the derivative of formula (9) and substitute formula (7) into it to obtain:

[0033]

[0034] 2.3 Design speed reaching law:

[0035]

[0036] in, represents the first-order derivative of the sliding surface to be designed, γ1>0,γ2>0,α1>1,0<α2<1; the table of E(s)

[0037] The expression is:

[0038]

[0039] Among them, 0<η<1, k>0;

[0040] Step 3, speed change nature description, the process is as follows:

[0041] When |s|>a, the value range of E(s) is between 1 and 2, then:

[0042]

[0043] Therefore, when |s|>a, the convergence time is shorter;

[0044] When |s|≤a, the value range of E(s) is between 0 and 1, the gain of the sign function decreases, and the jitter problem is weakened;

[0045] Step 4: Design of sliding mode control law based on variable speed reaching law.

[0046] Further, the process of step 4 is as follows:

[0047] 4.1 Combining equations (10)-(12), the sliding mode controller based on the speed reaching law is designed as:

[0048]

[0049] 4.2 Substituting the auxiliary control law designed by equation (14) into equation (5), the actual control law F in the crane system is obtained as:

[0050]

[0051] 4.3 Choose the following Lyapunov function:

[0052]

[0053] Taking the derivative of the above formula, we get That is, the system is judged to be stable.

[0054] The technical concept of the present invention is: for the bridge crane system that performs tasks with strict speed requirements, the present invention adopts the sliding mode control theory to construct an exponential auxiliary function and designs a new speed reaching law to adjust the convergence speed of the sliding mode variable, so that it has a faster convergence speed before reaching the deceleration point, and can effectively weaken the vibration after reaching the deceleration point. On this basis. According to the bridge crane model, the state transformation is carried out, and the sliding mode controller is constructed to ensure that the crane system can reach the target position quickly and stably, while ensuring that the load swing angle is as small as possible, ensuring the continuity of the control signal and reducing the control signal vibration.

[0055] The beneficial effects of the present invention are as follows: the sliding mode controller of the variable speed approaching law can make the system output have better transient performance, can ensure the rapid positioning of the bridge crane, ensure the stability of the control input, and reduce the system jitter, thereby further improving the steady-state performance of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is the state trajectory comparison diagram of x;

[0057] Figure 2 This is the state curve comparison diagram of θ;

[0058] Figure 3 It is the signal curve comparison diagram of F;

[0059] Figure 4 It is the state trajectory comparison diagram of sliding mode variables. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with the accompanying drawings.

[0061] Reference Figure 1-Figure 4 A sliding mode anti-sway control method for a bridge crane based on a variable speed reaching law comprises the following steps:

[0062] Step 1: Establish a dynamic model of the bridge crane system and initialize the system state and control parameters. The process is as follows:

[0063] 1.1 The dynamic model of the bridge crane is:

[0064]

[0065] Among them, x represents the position of the trolley during operation, Indicates the speed of the trolley when it is running. represents the acceleration of the trolley during movement, θ represents the amplitude of the load swing angle, represents the speed of the load swing angle, represents the acceleration of the load swing angle, M represents the mass of the trolley, m represents the mass of the load, l represents the length of the suspension rope, F represents the control input, g represents the acceleration of gravity, d represents the external disturbance, cos(·) represents the cosine function, sin(·) represents the sine function, and

[0066] 1.2 According to formula (1), the acceleration expression of the trolley during movement is:

[0067]

[0068] 1.3 Substituting equation (2) into equation (1), after mathematical transformation, the control input F is obtained as:

[0069]

[0070] in, tan(·) is the tangent function;

[0071] 1.4 In order to facilitate the design of subsequent controllers, an auxiliary control quantity v is defined:

[0072]

[0073] Substituting equation (4) into equation (3), the control input F can be rewritten as:

[0074]

[0075] 1.5 Define the following state variables:

[0076]

[0077] Among them, x d , are the reference trajectory and speed of the trolley motion, respectively, sec(·) represents the inverse of the cosine function;

[0078] 1.6 Combined with equation (6), the following state equation is constructed:

[0079]

[0080] in, Respectively expressed in formula (6) The first-order derivative of , λ represents the auxiliary variable defined, Represents state variables The constructor of ; λ and The specific expression is as follows:

[0081]

[0082] 1.7 For the actual application scenario of the crane, the load swing angle satisfies |θ(t)|≤π / 3, and the rope of the two-dimensional bridge crane is a rigid rope, that is, l is a constant;

[0083] Step 2: For the bridge crane system with unknown dynamics, the linear sliding surface and speed reaching law are designed through the defined state variables. The process is as follows:

[0084] 2.1 Design of linear sliding surface:

[0085]

[0086] Among them, a>0, b>0, c>0;

[0087] 2.2 Take the derivative of formula (9) and substitute formula (7) into it to obtain:

[0088]

[0089] 2.3 Design speed reaching law:

[0090]

[0091] in, represents the first-order derivative of the sliding surface to be designed, γ1>0,γ2>0,α1>1,0<α2<1; the expression of E(s) is:

[0092]

[0093] Among them, 0<η<1, k>0;

[0094] Step 3, speed change nature description, the process is as follows:

[0095] When |s|>a, the value range of E(s) is between 1 and 2, then:

[0096]

[0097] Therefore, when |s|>a, the convergence time is shorter;

[0098] When |s|≤a, the value range of E(s) is between 0 and 1, the gain of the sign function decreases, and the jitter problem is weakened;

[0099] Step 4: Design of sliding mode control law based on variable speed reaching law. The process is as follows:

[0100] 4.1 Combining equations (10)-(12), the sliding mode controller based on the speed reaching law is designed as:

[0101]

[0102] 4.2 Substituting the auxiliary control law designed by equation (14) into equation (5), the actual control law F in the crane system is obtained as:

[0103]

[0104] 4.3 Choose the following Lyapunov function:

[0105]

[0106] Taking the derivative of the above formula, we get That is, the system is judged to be stable.

[0107] In order to verify the effectiveness of the proposed method, the present invention provides the proposed method, namely, method 1: sliding mode controller of bridge crane based on variable speed reaching law, method 2: fast power reaching law control method and method 3: comparative simulation experiment of enhanced power reaching law control method:

[0108] Among them, method 2: the reaching law of the fast power reaching law control method is

[0109]

[0110] The controller is

[0111]

[0112] Method 3: The reaching law in the double power reaching law control method is

[0113]

[0114] The controller is

[0115]

[0116] In order to make a more effective comparison, the initial conditions and control parameters in the experiment are set as follows: the sampling step is 0.001s, all system parameters are set the same, and the expected positioning target is x d =10m, system parameters are M=5kg, m=10kg, l=6m, g=9.8m / s 2 , a=0.512, b=1.92, c=2.4, the reaching law parameters are k1=0.5, k2=0.5, α1=0.3, α2=3, the speed change gain is k1=1000, a=0.001, the interference is d=0.5sin(t), and the initial state of the system is

[0117] Figure 1-Figure 4 The simulation results of the sliding mode control method based on the variable speed reaching law on the system (1) and (2) are shown in Fig. Figure 1 The comparison diagram of crane position tracking under different reaching law methods is shown in Figure 2. Figure 1 It can be seen that compared with method 2 and method 3, method 1 has better transient performance. The position tracking accuracy of method 1 is 10.092, while the position tracking accuracy of method 2 is 10.096 and the position tracking accuracy of method 3 is 10.097. Therefore, the position tracking accuracy of method 1 is higher. Figure 2 This is a comparison diagram of the load swing angle convergence process. It can be seen from the figure that the load swing angle has converged to 0 before 15s in method 1. Compared with methods 2 and 3, the convergence law proposed in this paper can make the load swing angle converge faster. Figure 3 This is a controller comparison chart. Figure 4 It can be seen that the control signal of method 1 is relatively smooth, while methods 2 and 3 have certain jitter after 2s. Therefore, E(s) has a certain weakening effect on jitter. Figure 4 The figure shows the comparison of the sliding mode variable convergence process. It can be seen from the figure that the sliding mode variable converges to 0 in about 1.5s for method 1, in about 2s for method 2, and in about 4s for method 3. Compared with methods 2 and 3, the method proposed in this paper can ensure a faster convergence speed. In summary, the sliding mode control method of the bridge crane based on the variable speed reaching law can effectively increase the speed of the trolley to reach the target position, improve the position tracking accuracy, and reduce the jitter to a certain extent, making the system more stable.

[0118] The above describes the excellent optimization effect shown by an embodiment of the present invention. Obviously, the present invention is not limited to the above embodiment, and various modifications can be made to it without departing from the basic spirit of the present invention and without exceeding the scope of the essential content of the present invention.

Claims

1. A sliding mode anti-sway control method for a bridge crane based on a variable speed reaching law, characterized in that the method comprises the following steps: Step 1: Establish a dynamic model of the bridge crane system and initialize the system state and control parameters. The process is as follows: 1.1 The dynamic model of the bridge crane is: Among them, x represents the position of the trolley during operation, Indicates the speed of the trolley when it is running. represents the acceleration of the trolley during movement, θ represents the amplitude of the load swing angle, represents the speed of the load swing angle, represents the acceleration of the load swing angle, M represents the mass of the trolley, m represents the mass of the load, l represents the length of the suspension rope, F represents the control input, g represents the acceleration of gravity, d represents the external disturbance, cos(·) represents the cosine function, sin(·) represents the sine function, and 1.2 According to formula (1), the acceleration expression of the trolley during movement is: 1.3 Substitute equation (2) into equation (1) and after mathematical transformation, the control input F is obtained as: in, tan(·) is the tangent function; 1.4 Define an auxiliary control quantity v: Substituting equation (4) into equation (3), the control input F can be rewritten as: 1.5 Define the following state variables: Among them, x d , are the reference trajectory and speed of the trolley motion, respectively, sec(·) represents the inverse of the cosine function; 1.6 Combined with equation (6), the following state equation is constructed: in, Respectively expressed in formula (6) The first-order derivative of , λ represents the auxiliary variable defined, Represents state variables The constructor of λ and The specific expression is as follows: 1.7 For the actual application scenario of the crane, the load swing angle satisfies |θ(t)|≤π / 3, and the rope of the two-dimensional bridge crane is a rigid rope, that is, l is a constant; Step 2: For the bridge crane system with unknown dynamics, the linear sliding surface and speed reaching law are designed through the defined state variables. The process is as follows: 2.1 Design of linear sliding surface: in, a>0, b>0, c>0; 2.2 Take the derivative of formula (9) and substitute formula (7) into it to obtain: 2.3 Design speed reaching law: in, represents the first-order derivative of the sliding surface to be designed, γ1>0,γ2>0,α1>1,0<α2<1; the expression of E(s) is: Among them, 0<η<1, k>0; Step 3, speed change nature description, the process is as follows: When |s|>a, the value range of E(s) is between 1 and 2, then: Therefore, when |s|>a, the convergence time is shorter; When |s|≤a, the value range of E(s) is between 0 and 1, the gain of the sign function decreases, and the jitter problem is weakened; Step 4: Design of sliding mode control law based on variable speed reaching law.

2. The sliding mode anti-sway control method for a bridge crane based on a variable speed reaching law as claimed in claim 1, characterized in that: The process of step 4 is as follows: 4.1 Combining equations (10)-(12), the sliding mode controller based on the speed reaching law is designed as: 4.2 Substituting the auxiliary control law designed by equation (14) into equation (5), the actual control law F in the crane system is obtained as: 4.3 Choose the following Lyapunov function: Taking the derivative of the above formula, we get That is, the system is judged to be stable.

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