CN108828959A - A kind of novel bridge crane is anti-sway with position control method and device - Google Patents

Abstract

A new anti-swing and positioning control method for overhead cranes, including: establishing a T-S nonlinear fuzzy model; according to the T-S nonlinear fuzzy model, adopting a parallel distributed compensation control (PDC) structure to design a fuzzy controller, and obtaining PDC control law u(t); Feedback gain matrix F _i is calculated by using LMI with attenuation rate. The controller can guarantee the closed-loop asymptotic stability of the system, and has good positioning, anti-swing and anti-jamming performance. The maximum swing angle of the load during the operation of the trolley is small, and the swing of the payload is well suppressed. After the trolley reaches the target position, the swing angle disappears quickly, and the anti-interference performance does not change with the change of the expected position and load quality. Robustness strong.

Description

A new type of bridge crane anti-swing and positioning control method and device

technical field

The invention relates to the technical field of cranes, in particular to a novel anti-swing and positioning control method and device for bridge cranes.

Background technique

Bridge crane is an important tool and equipment to realize the mechanization and automation of production process in modern industrial production and lifting transportation. It is widely used in indoor and outdoor warehouses, workshops, port terminals, open-air storage and other places. The positioning and anti-sway control goal of the bridge crane is to drive the trolley, transport the load to the desired point quickly, accurately and safely, and minimize or eliminate the load swing angle during the transportation and the residual swing of the load after the trolley stops moving angle, so as to avoid the collision between the crane and the surrounding objects, causing huge economic losses.

There are many kinds of control strategies for the positioning and anti-swing of bridge cranes. According to the crane model when designing the controller, the control methods can be divided into two categories: one is based on linear mathematical models; the other is based on nonlinear models. control method.

The control method based on the linear mathematical model is to design the controller according to the linear control theory. This linear control strategy is simple and easy to implement. However, since the controller is designed based on the linear mathematical model, the system control performance depends heavily on the accuracy of the model. Model parameters (e.g. rope length, load magnitude) are sensitive to variations, and additionally, it requires lower trolley operating speeds and imposes unrealistic constraints on crane operation. In order to overcome this deficiency, many other linear control methods such as sliding mode control, model-free fuzzy control, neural network control, etc. are used in crane control. Although these control methods help to obtain better response characteristics, there are also some disadvantages, such as chattering in the control output in sliding mode control, and difficulty in system stability analysis when specific constraints are encountered in model-free fuzzy logic control, etc. .

With the development of nonlinear control technology, some researchers have proposed a series of nonlinear control strategies based on the analysis of the nonlinear mathematical model of the crane, such as energy/passivity-based control, nonlinear coupling control method, feedback linear optimization method, gain-scheduled nonlinear model predictive control, etc. However, the control effects of these control methods still have some shortcomings to varying degrees, such as: the maximum swing angle of the load during operation is large, the overshoot is large, and the anti-interference is not strong.

Contents of the invention

Aiming at the shortcomings of the prior art, the present application provides a new type of anti-swing and positioning control method and device for a bridge crane, which has better positioning, anti-swing and anti-interference performance.

According to the first aspect, the present application provides a novel anti-swing and positioning control method for bridge cranes, including:

Establish T-S nonlinear fuzzy model;

According to the T-S nonlinear fuzzy model, the T-S fuzzy controller is designed by using the parallel distributed compensation method, and the PDC is obtained

control law u(t);

The feedback gain matrix F _i is obtained by calculating the LMI with attenuation rate.

In some embodiments, the control law

In some embodiments, the control rules are:

IF z ₁ (t) is N _j and z ₂ (t) is R _k ;

THEN u _i (t)=-F _i x _m i=1, . . . r, r=8.

In some embodiments, including: using sector nonlinearity to establish a T-S nonlinear fuzzy model, the system state equation of the model is:

In some embodiments, the feedback gain matrix F _i =Q _i P ^-1 , P is a positive definite matrix, and P and Q _i satisfy:

PA _i ^T +A _i PQ _i ^T B _i ^T -B _i Q _i +2αP<0;α>0, i=1,2,...,r;

In some embodiments,

According to the second aspect, the present application provides a novel bridge crane anti-swing and positioning control device, including:

A module for establishing a T-S nonlinear fuzzy model;

According to the T-S nonlinear fuzzy model, the parallel distributed compensation method is used to design the T-S fuzzy controller, and the obtained

The module of the PDC control law u(t);

A module for calculating the feedback gain matrix F _i by using the LMI with attenuation rate.

According to a third aspect, the present application provides a computer-readable storage medium, which is characterized by including a program, and the program can be executed by a processor to implement the method according to any one of the first aspect.

According to the above embodiment, since the method of the present application establishes the TS nonlinear fuzzy model of the overhead crane, based on the TS nonlinear fuzzy model, the PDC fuzzy controller is designed using the sector nonlinear model, and the LMI with attenuation rate is used to calculate the obtained Feedback gain matrix F _i , the controller ensures the closed-loop asymptotic stability of the model, which makes the method have better control effect, better positioning, anti-swing and anti-jamming performance. The maximum swing angle of the load is small during operation, and the swing of the payload is well suppressed. After the trolley reaches the target position, the swing angle disappears quickly, and the anti-interference performance does not change with the change of the expected position, and the robustness is strong.

Description of drawings

Figure 1 is a schematic diagram of a simplified model of a two-dimensional bridge crane;

Fig. 2 is the flow chart of the novel bridge crane anti-swing and positioning control method of the present application;

Fig. 3 is the antecedent variable z ₁ (t) membership function diagram of the embodiment of the present application;

Fig. 4 is the antecedent variable z ₂ (t) attribute function diagram of the embodiment of the present application;

Fig. 5 is the antecedent variable z ₃ (t) membership function figure of the embodiment of the present application;

Fig. 6 is the simulation model diagram of the bridge crane of the embodiment of the present application;

Figure 7 shows the displacement of the trolley and the swing angle of the payload under different attenuation rates

Fig. 8 is a control effect simulation comparison diagram of the fuzzy PDC controller based on the local approximation model and the fuzzy PDC controller based on the sector nonlinear model;

Figure 9 is a simulation comparison diagram of the control effect of the traditional LMI based on the linear model and the LMI based on the sector nonlinear model with attenuation rate;

Fig. 10 is the simulation result under different expected target x _d conditions of the car;

Fig. 11 is the simulation result diagram of payload mass variation;

Fig. 12 is a simulation result diagram of the change of rope length.

Detailed ways

The present invention will be further described in detail below through specific embodiments in conjunction with the accompanying drawings. Wherein, similar elements in different implementations adopt associated similar element numbers. In the following implementation manners, many details are described for better understanding of the present application. However, those skilled in the art can readily recognize that some of the features can be omitted in different situations, or can be replaced by other elements, materials, and methods. In some cases, some operations related to the application are not shown or described in the description, this is to avoid the core part of the application being overwhelmed by too many descriptions, and for those skilled in the art, it is necessary to describe these operations in detail Relevant operations are not necessary, and they can fully understand the relevant operations according to the description in the specification and general technical knowledge in the field.

In addition, the characteristics, operations or characteristics described in the specification can be combined in any appropriate manner to form various embodiments. At the same time, the steps or actions in the method description can also be exchanged or adjusted in a manner obvious to those skilled in the art. Therefore, various sequences in the specification and drawings are only for clearly describing a certain embodiment, and do not mean a necessary sequence, unless otherwise stated that a certain sequence must be followed.

Most of the current control methods for intelligent cranes are designed for linear mathematical models. This linear control strategy based on linear models is simple and easy to implement, but the system control performance depends heavily on the accuracy of the model. Model parameters (such as rope length , load size) is more sensitive to changes. The T-S fuzzy model can approximate the nonlinear system with arbitrary precision, and the analysis and synthesis methods in the linear control theory can be applied to it, so that the effect of external disturbance and parameter changes on the control effect is greatly weakened, and the robustness of the system is enhanced. Therefore, the research on the fuzzy system based on T-S model has important practical significance.

Based on the analysis of the nonlinear mathematical model of the crane system, this application establishes a T-S nonlinear fuzzy model. Aiming at this nonlinear fuzzy model, a fuzzy controller is designed based on the PDC control scheme, and the feedback gain matrix is calculated by using the linear matrix inequality (LMI). Thereby realizing anti-swing and positioning of the bridge crane.

Referring to Figure 1, it is a simplified model diagram of a two-dimensional bridge crane. In the figure, M and m represent the mass of the trolley and the mass of the load, respectively, l is the length of the suspension rope, x is the displacement of the trolley, and θ is the swing angle of the load in the vertical direction , F represents the external force acting on the crane, F _x and F _l are the external forces acting on the horizontal direction and the rope length direction respectively, and g is the acceleration of gravity.

With reference to Fig. 2, the application provides a kind of bridge crane anti-swing and positioning non-linear control method based on LMI, the method comprises:

Step 100: Establish a T-S nonlinear fuzzy model;

Step 200: According to the T-S nonlinear fuzzy model, adopt the parallel distributed compensation method to design the T-S fuzzy controller, and obtain the PDC control law u(t);

Step 300: Calculate and obtain the feedback gain matrix F _i by using the LMI with attenuation rate.

For, step 100, utilize Euler-Lagrange method to establish the dynamic model of crane, Lagrange equation is as follows:

Where Q _i is the force applied to the system, x _i is the generalized coordinate or state variable, L is the Lagrangian operator L=TU, T and U are the total kinetic energy and total potential energy of the system, respectively.

Considering the movement of the bridge crane on a two-dimensional plane, the total kinetic energy of the system can be expressed as

The total potential energy is expressed as

U=mg(h-l cosθ) (3)

Among them, ignoring the mass and stiffness of the wire rope, the load is considered as a point mass. If the rope length remains constant, the nonlinear dynamic model of the bridge crane is as follows:

Take the state vector Then the state equation is as follows:

It can be seen from the above formula that the bridge crane is a nonlinear system. The state equation of any nonlinear system can be expressed as follows:

In the formula, x(t) represents the state vector, u(t) represents the control vector, A represents the state matrix, and B represents the control matrix.

The T-S fuzzy model is approximately expressed as

In the formula,

z(t) refers to the system input and is the antecedent variable; ω _i represents the weight of the i-th rule, i=1, 2,... r, h _i represent the corresponding normalized weight, where,

In some embodiments, step 100 of this embodiment utilizes sector nonlinearity to establish a TS nonlinear fuzzy model. There are five non-linear terms in formula (6), namely sinx ₃ , cosx ₃ , x ₄ ² , sinx ₃ cosx ₃ , 1/((M+m)-mcos ² x ₃ ), using TS fuzzy model requires 2 ⁵ rules. In order to simplify the mathematical model and reduce the nonlinear terms in the model, let

F＝(M+msin ² x ₃ )u-mg sin x ₃ cos x ₃ -mlx ₄ ² sin x ₃ (10)

where u is the new control input.

The mathematical model of formula (6) is simplified as

It can be seen from the above formula that there are three nonlinear terms in the model as antecedent variables, which are respectively defined as z ₁ =sinx ₃ , z ₂ =cos x ₃ and under the constraints In the case of sector nonlinear theory, the membership function corresponding to the antecedent variable can be determined by the following formula.

In the formula

again

From formula (12) - formula (15), the membership function can be calculated as follows:

Figures 3 to 5 show the membership function diagrams of the antecedent variables z ₁ (t), z ₂ (t) and z ₃ (t).

According to equations (16)-(21), the dynamic model (6) of the bridge crane is and Under the constraints, it can be described as the following formula:

In the formula, M _i , N _j and P _k are the membership functions of the antecedent variables z ₁ (t), z ₂ (t) and z ₃ (t) respectively, and u(t) is the input quantity.

Assuming that all state variables of the crane system can be measured, a state feedback controller can be designed for each fuzzy subsystem. Step 200 adopts the Parallel Distributed Compensation (PDC) method to design the TS fuzzy controller, and calculates the feedback gain F _i using LMI. In the PDC design, each control rule is designed according to the corresponding rules of the TS fuzzy model, that is, the antecedent part of the designed fuzzy controller has the same fuzzy set as the fuzzy model.

Let x _d be the expected position of the car, x _x -x _d is the position deviation, let x _m =[x ₁ -x _d x ₂ x ₃ x ₄ ]′, in order to make the car reach the desired position, the control rules are as follows:

IF z ₁ (t) is N _j and z ₂ (t) is R _k ,

THEN u _i (t)=-F _i x _m i=1, . . . r, r=8. (twenty three)

The PDC control law is obtained by the weighted combination of the above rules:

In some specific embodiments, the 8 rules that can establish the T-S fuzzy model are as follows:

Rule 1:

Rule 2:

Rule 3:

Rule4:

Rule 5:

Rule 6:

Rule 7:

Rule 8:

In this rule, every linear continuity equation is called a "subsystem", and the subsystem coefficient matrix can be expressed as:

Substituting Equation (24) into Equation (8), the closed-loop control system equation is obtained as follows:

In the formula, G _ij ＝A _i -B _i F _j

The controllability and observability of each "subsystem" are as follows:

(1) The rank of the controllable matrix

rank[B _i A _i *B _i A _i ² *B _i A _i ³ *B _i ]＝4

(2) The rank of observable matrix

rank[C*B _i C*A _i *B _i C*A _i ² *B _i C*A _i ³ *B _i ]＝2

It can be seen that the system is controllable and observable, so a feedback control device can be added to the system to make the system closed-loop stable.

In step 300, when using the linear matrix inequality (LMI) to determine the feedback gain matrix, in order to ensure the stability of the system, according to the existing literature: Fuzzy control systems design and analysis edited by K.Tanaka and H.O.Wang published in 2001 for the continuous control system to give derived the following theorem.

Theorem 1: If there exists a positive definite matrix P that satisfies the following inequality

Then the continuous fuzzy control system described by formula (25) is globally asymptotically stable.

The above inequality is not a Linear Matrix Inequality (LMI) due to the presence of a product of unknown matrix/vector variables. In order to use the stability condition of the theorem to determine the feedback gain Fi of the control system, the left and right sides of equation (26) are multiplied by P ^-1 respectively, and the variable P=P ^-1 and a new variable Q _i =F _i P are redefined, which can be The following linear matrix inequality is obtained:

Matrix P and vector Q _i can be obtained by solving linear matrix inequality (27). If a symmetric positive definite matrix P can be obtained, the stability of the system can be guaranteed.

The feedback gain matrix can be calculated by the following formula:

F _i =Q _i P ^-1 (28)

The feedback gain matrix can be easily obtained by using the LMIs in Equation (27) and Equation (28), but the obtained feedback gain matrix is unique, which may not necessarily enable the system to obtain better control performance. In order to meet the actual requirements of the system, this application adopts LMIs with attenuation rate α. The response speed of the T-S fuzzy control system is related to the decay rate. By changing the decay rate parameters, the system can be better controlled.

Definition: For the system (25), the selected Lyapunov function is V(x(t))=x ^T (t)Px(t), P>0, if there is a real number α>0, satisfy Then the system is said to be globally asymptotically stable with decay rate α.

Theorem 2: For the system (25), if the real number α>0, for all i, satisfy

and for all i<j, st h _i ∩h _j ≠φ, satisfy

Then the system is said to be globally asymptotically stable with decay rate α.

Multiply the left and right sides of inequalities (29) and (30) by P ^-1 respectively, redefine the variable P=P ^-1 and a new variable Q _i =F _i P, the following linear matrix inequality can be obtained:

PA _i ^T +A _i PQ _i ^T B _i ^T -B _i Q _i +2αP<0 α>0, i=1,2,...,r; (31)

Matrix P and vector Q _i can be obtained by solving linear matrix inequalities (31) and (32).

Correspondingly, the present application also provides a novel anti-swing and positioning control device for bridge cranes, which includes:

A module for establishing a T-S nonlinear fuzzy model;

According to the T-S nonlinear fuzzy model, adopt the parallel distributed compensation method to design the T-S fuzzy controller, and obtain the module of the PDC control law u(t);

A module for calculating the feedback gain matrix F _i by using the LMI with attenuation rate.

Correspondingly, the present application also provides a computer-readable storage medium, including a program that can be executed by a processor to implement the LMI-based non-linear control method for anti-sway and positioning of an overhead crane.

In order to prove the effectiveness of the method of the present application, the following is compared with the existing control method based on local approximate model and traditional LMI based on linear model.

Among them, the local approximate model-based control method is to linearize the nonlinear model of the crane at three operating points of 0° and ±45° to obtain the corresponding linearized state-space model. For formula (7),

The state-space model at 0° is in

State-space model of an overhead crane at 45°:

Using the triangular membership function, the local approximate model of the bridge crane is as follows:

This application uses MATLAB for simulation to verify the effectiveness of the given method. Figure 6 shows the dynamic model of the bridge crane represented by formula (6) and the simulation model of the fuzzy PDC controller represented by formula (24). In the simulation, the target position of the trolley is set as x _d =0.6m. In order to verify the anti-interference performance of the system, a pulse interference is added to the system after the car stabilizes. The simulation results of the four cases are discussed below.

Case 1: Discuss the case of different decay rate α values.

The response speed is related to the attenuation rate α. Figure 7(a) and Figure 7(b) respectively show the displacement of the car and the swing angle of the payload under different α values. It can be seen from Figure 7 that the larger the value of α, the faster the displacement response speed and the stronger the anti-interference ability, but the larger the payload swing angle. Therefore, this application chooses α=0.53.

Case 2: Comparative study.

First, the control effect of the fuzzy PDC controller with 2 rules designed based on the local approximation model is compared with the fuzzy PDC controller with 8 rules designed based on the sector nonlinear model, both cases use the decay rate The feedback gain matrix of the LMI is calculated, and the feedback gains of the eight rules are calculated according to formulas (31) and (32) as

F ₁ =[17.7510 32.9452 -175.0808 -1.7266]

F ₂ =[17.7554 32.9467 -175.1205 -1.7257]

F ₃ =[18.5326 34.7355 -190.4480 -4.2406]

F ₄ =[18.5275 34.7206 -190.4016 -4.2411] (34)

F ₅ =[19.1524 35.4767 -188.3394 -1.3522]

F ₆ =[19.1548 35.4745 -188.3598 -1.3512]

F ₇ =[20.9061 39.0106. -212.2236 -3.5317]

F ₈ ＝[20.8961 38.9868 -212.1307 -3.5330]

The gain matrix of the two rules is

F ₁ =[24.4625 30.4433 -63.5695 -14.4929]

F ₂ =[37.9780 47.7883 -183.1461 -18.1274] (35)

The simulation results for both cases are shown in Fig. 8. Figure 8 shows the displacement, speed and load swing angle of the trolley, as well as the swing angle velocity response curve, and Table 1 gives the specific comparison results.

Table 1 Comparison of performance indicators

It can be seen from the chart that compared with the fuzzy PDC controller with 2 rules designed based on the local approximate model, when the fuzzy PDC controller with 8 rules designed based on the sector nonlinear model is used, the running time of the car is the same, The maximum swing angle of the load during operation is small, and there is no overshoot.

Secondly, the feedback gains obtained by using the traditional LMI based on the linear model and the LMI based on the sector nonlinear model with attenuation rate are respectively used in the control of the nonlinear system of the bridge crane for comparative research. The simulation results are shown in Figure 9 . Table 2 presents the detailed quantification results when using the two methods.

Table 2 Comparison of system response in two cases

The simulation results show that both LMI control methods can ensure that there is no residual swing of the load at the target position. However, compared with the traditional linear LMI, the controller designed in this application has better positioning, anti-swing and anti-jamming performance.

Case 3: Different transport distances.

In order to verify the control performance of the method of the present application under different transportation distances, x _d =0.4m, x _d =0.6m and x _d =0.8m, x _d =1.0m were respectively selected. Figure 11 shows the simulation results for the four cases.

The simulation results show that during the transportation process, the trolley can accurately reach the predetermined position, the swing of the payload is well suppressed, the swing angle is between [-5°, 5°], and the swing angle disappears quickly after the trolley reaches the target position , and the anti-interference performance does not change with the change of the expected position.

Case 4: Robustness study when load mass changes and rope length changes.

Load mass and rope length are two important parameters affecting system performance. In practical industrial applications, different transportation tasks require changing the load mass or rope length. Therefore, it is necessary to consider the robustness of the controller when these two parameters change. Figure 11 and Figure 12 of the present application respectively give the simulation results of payload mass changing from 2kg to 8kg, and rope length changing from 0.5m to 0.85m.

It can be seen from Figure 11 that when the load mass changes, neither the rapidity of the trolley nor the swing angle of the payload changes. It can be seen from Figure 12 that when the rope length decreases, the rapidity of the trolley does not change much, and the swing angle of the payload increases, but the angle is within the allowable range of the control performance. But when the two parameters are changed, the interference suppression performance does not change. The results show that the method is robust to variations in payload mass and rope length, which is of great significance in practical applications.

In summary, the method of this application establishes the TS nonlinear fuzzy model of the bridge crane, based on the TS nonlinear fuzzy model, uses the sector nonlinear model to design the PDC fuzzy controller, and uses the LMI calculation with decay rate to obtain the feedback Gain matrix F _i , the controller ensures the closed-loop asymptotic stability of the model, and has a good control effect. The maximum swing angle of the load during operation is small, and it has better positioning, anti-swing and anti-interference performance, and is effective The load swing is well suppressed, and the swing angle disappears quickly after the trolley reaches the target position, and the anti-interference performance does not change with the change of the expected position, and the robustness is strong. Simulation results verify the effectiveness of the method.

Those skilled in the art can understand that all or part of the functions of the various methods in the foregoing implementation manners can be realized by means of hardware, or by means of computer programs. When all or part of the functions in the above embodiments are implemented by means of a computer program, the program can be stored in a computer-readable storage medium, and the storage medium can include: read-only memory, random access memory, magnetic disk, optical disk, hard disk, etc., through The computer executes the program to realize the above-mentioned functions. For example, the program is stored in the memory of the device, and when the processor executes the program in the memory, all or part of the above-mentioned functions can be realized. In addition, when all or part of the functions in the above embodiments are realized by means of a computer program, the program can also be stored in a storage medium such as a server, another computer, a magnetic disk, an optical disk, a flash disk, or a mobile hard disk, and saved by downloading or copying. To the memory of the local device, or to update the version of the system of the local device, when the processor executes the program in the memory, all or part of the functions in the above embodiments can be realized.

The above uses specific examples to illustrate the present invention, which is only used to help understand the present invention, and is not intended to limit the present invention. For those skilled in the technical field to which the present invention belongs, some simple deduction, deformation or replacement can also be made according to the idea of the present invention.

Claims (8)

1. A novel bridge crane anti-swing and positioning control method, characterized in that it comprises: Establish T-S nonlinear fuzzy model; According to the T-S nonlinear fuzzy model, the T-S fuzzy controller is designed by using the parallel distributed compensation method, and the PDC control law u(t) is obtained; The feedback gain matrix F _i is obtained by calculating the LMI with the attenuation rate. 2. The method of claim 1, wherein the control law 3. The method according to claim 1, wherein the control rule is: IF z ₁ (t) is N _j and z ₂ (t) is R _k ; THEN u _i (t)=-F _i x _m i=1, . . . r, r=8. 4. method as claimed in claim 1, is characterized in that, comprises: utilize sector non-linear establishment T-S nonlinear fuzzy model, the system state equation of model is: 5. method as claimed in claim 1, is characterized in that, feedback gain matrix F _i =Q _i P ^-1 , P is positive definite matrix, P, Q _i satisfy: PA _i ^T +A _i PQ _i ^T B _i ^T -B _i Q _i +2αP<0;α>0,i=1,2,...,r; PA _i ^T +A _i P+PA _j ^T +A _j PQ _i ^T B _j ^T -B _i Q _j -Q _j ^T B _i ^T -B _j Q _i +4αP≤0; sth _i ∩ _{h j} ≠ φ. 6. Apparatus/method according to claim 6, characterized in that, P is a positive definite symmetric matrix. 7. A new type of bridge crane anti-swing and positioning control device, characterized in that it includes: A module for establishing a T-S nonlinear fuzzy model; According to the T-S nonlinear fuzzy model, adopt the parallel distributed compensation method to design the T-S fuzzy controller, and obtain the module of the PDC control law u(t); A module for calculating the feedback gain matrix F _i by using the LMI with attenuation rate. 8. A computer-readable storage medium, comprising a program, the program can be executed by a processor to implement the method according to any one of claims 1-6.