CN113086844B - Variable-rope-length bridge crane anti-swing positioning control method based on second-order sliding mode disturbance observer - Google Patents
Variable-rope-length bridge crane anti-swing positioning control method based on second-order sliding mode disturbance observer Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B66—HOISTING; LIFTING; HAULING
- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
- B66C13/00—Other constructional features or details
- B66C13/04—Auxiliary devices for controlling movements of suspended loads, or preventing cable slack
- B66C13/06—Auxiliary devices for controlling movements of suspended loads, or preventing cable slack for minimising or preventing longitudinal or transverse swinging of loads
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B66—HOISTING; LIFTING; HAULING
- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
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- B66C13/22—Control systems or devices for electric drives
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Abstract
The invention provides a variable-rope-length crane anti-swing positioning control method based on a second-order sliding mode interference observer, aiming at the problem of bridge crane control for simultaneously carrying out lifting operation of a lifting rope in the horizontal transportation process, a system model of the variable-rope-length bridge crane is established by utilizing an Euler-Lagrangian method, and the second-order sliding mode interference observer is designed based on the model to observe composite interference consisting of internal uncertainty and external interference of the variable-rope-length bridge crane, so that the driving force calculated by a sliding mode controller is compensated, the positioning precision and the anti-swing effect of the system are improved, the problems of track friction force change and inconsistent load quality of the variable-rope-length bridge crane in actual operation are effectively solved, the robustness and the anti-interference performance of the control system are improved, and the application requirement of the bridge crane in a severe environment is met. Meanwhile, the horizontal movement and the lifting movement of the bridge crane are coupled, so that the conveying efficiency in actual operation is greatly improved.
Description
Technical Field
The invention particularly relates to a variable-rope-length crane anti-swing positioning control method based on a second-order sliding mode disturbance observer, and belongs to the technical field of crane anti-swing positioning control.
Background
The bridge crane has the advantages of simple structure, large load and the like, and is widely applied to transportation operation sites of ports, steel plants and the like. As a typical underactuated system, however, the underactuation of the overhead travelling crane makes it inevitable to cause jolts in the hoisted load during operation. The violent shaking of the load brings great potential safety hazards to an operation site on one hand, and also influences the positioning precision of the crane reaching a target position on the other hand. For the anti-swing positioning control problem of the bridge crane, domestic and foreign scholars study a series of control algorithms, such as open-loop control methods of input shaping, trajectory planning and the like and closed-loop control methods of sliding mode control, adaptive control, fuzzy control and the like.
The operation process of the bridge crane can be generally divided into: three stages of hoisting load, horizontally conveying and putting down load. In the actual crane operation, in order to improve the cargo conveying efficiency, the load lifting stage and the horizontal conveying stage are coupled, so that the control problem of the fixed rope length bridge crane is changed into the control problem of the variable rope length bridge crane. The introduction of a new control quantity, the driving force of the lifting rope lifting motor, brings new control problems: the length of a lifting rope of a bridge crane is changed from a constant value to a variable value which changes in real time, the swinging frequency of the load is influenced by the length of the rope, and if the variable value of the length of the rope cannot be processed, more severe load oscillation can be caused. On the other hand, the positioning control of the lifting rope also needs to compensate the gravity of the load so as to avoid the positioning steady-state error. Aiming at the anti-swing positioning control problem of the real-time variable rope length bridge crane, the existing control methods are few, and many of the methods need a more accurate crane mathematical model. Particularly in the process of lifting and lowering the load, the positioning accuracy in the vertical direction is greatly limited by the measurement accuracy of the load mass, and the accurate mass of each time of conveying the load is measured and calculated, which is difficult to realize in practical application. In addition, in the horizontal movement of the trolley, the rail friction force of the trolley is easily influenced by the ambient temperature, and the variation of the rail friction force parameters can be caused by the different temperatures in different seasons and different operation stages. Meanwhile, the bridge crane operating in the open air environment also has external disturbances such as wind power, ocean current and the like.
Therefore, in solving the problem of anti-swing positioning control of the variable-rope-length bridge crane, complex interference formed by internal uncertainty and external interference needs to be observed, the horizontal and vertical driving force of the crane is compensated, and the robustness and the anti-interference performance of a control system are improved.
Disclosure of Invention
In order to overcome the defects and shortcomings of the prior art, the invention provides a variable-rope-length crane anti-swing positioning control method based on a second-order sliding mode disturbance observer, which specifically comprises the following steps:
step 1) taking external interference and internal uncertainty of a bridge crane as composite interference, and establishing a mathematical model of a variable rope length bridge crane system by an Euler-Lagrange method;
wherein M and M are respectively the trolley mass and the load mass; x (t) is the sum of the values of,
representing the displacement and second derivative, i.e. acceleration, of the trolley; l (t) of,
representing the rope length, rope speed and rope acceleration of the lifting rope; the sum of θ (t),
representing the angle, angular velocity and angular acceleration of the swing of the hoisted load; f. of x (t),f l (t) motor driving forces of the crane in the horizontal direction and the vertical direction respectively; phi is a x (t),φ l (t) composite disturbances for the bridge crane system combined with external disturbances and internal uncertainties, respectively.
And transforming the dynamic model equation to obtain:
wherein d is x ,d l ,d θ Another manifestation of the complex interference experienced by the system in three directions.
Step 2) introducing trolley displacement, rope length and load swinging angle into state variables, and converting a mathematical model equation of the state variables into a nonlinear dynamics state equation with a compound interference term, wherein the method specifically comprises the following steps:
introducing a state variable χ 1 (t)＝[x(t),l(t),θ(t)] T ，
Transforming a mathematical model equation of the nonlinear dynamical state equation, wherein the nonlinear dynamical state equation is as follows:
wherein
D＝[d x ,d l ,d θ ] T ，U c ＝[f x ,f l ] T
Step 3) tracking the track x by using the target of the trolley and the lifting rope r ,l r Error e from actual displacement x ,e l Respectively forming a slip form surface sigma x ,σ l ；
Step 4) sliding mode surface sigma of trolley x Adding a suspended load swing angle term theta to realize the suppression of load swing, and specifically performing the following steps 3) and 4):
defining a target position tracking error:
e x ＝x-x r ,e l ＝l-l r
wherein e is x ,e l Track following errors, x, of trolley and lifting rope, respectively r ,l r Respectively the tracking tracks of the trolley and the lifting rope.
Constructing a sliding mode surface according to a target tracking track:
the slip form of which is dynamically
Wherein
Step 5) designing a second-order sliding mode disturbance observer according to the coupling relation between the sliding mode surface sliding mode dynamics and the composite disturbance constructed in the step 3), and utilizing a high-order sliding mode disturbance observerCharacteristics of slip form
Estimating the compound interference of the bridge crane with variable rope length,
the method comprises the following specific steps:
designing a second-order sliding mode disturbance observer based on super-twisting:
wherein z, V = [ V = x ,v l ] T And the injection control items are respectively an auxiliary sliding mode surface and an auxiliary sliding mode surface. To slip form surface s And (5) obtaining a derivative:
when the system state converges to the second-order sliding mode surface s When is at time
The injection control term V can be considered as a prediction of the composite disturbance D.
Input control item v i And i belongs to { x, l } and is designed based on the super-twisting control rate as follows:
wherein alpha is i ,λ i Are all positive constants and satisfy the inequality
α＞f +
Wherein f is + Which is a threshold value, can be regarded as twice the maximum second derivative of the system state variable, p satisfying the inequality 0 < p < 1.
Step 6) designing a sliding mode controller according to the composite interference observation value of the bridge crane obtained in the step 5), and obtaining the driving force of a trolley motor of the crane in the horizontal direction and the driving force of a lifting rope motor of the crane in the vertical direction, wherein the concrete steps are as follows:
firstly, a sliding mode surface sigma designed according to a tracking track and a composite interference item D in sliding mode dynamic can be replaced by V, namely
Selecting an index sliding mode approach rate
Wherein rho and epsilon are both positive constants, and beta is epsilon (0,1).
In conclusion, the anti-swing positioning control rate of the variable-rope-length bridge crane is obtained as
Wherein x is related to the system state 1 Matrix B (χ) 1 ) Is constantly reversible.
Compared with the prior art, the invention has the following advantages: the invention designs a novel second-order sliding mode interference observer based on super-twisting to observe composite interference formed by internal uncertainty and external interference of a variable-rope-length bridge crane, further compensates driving force calculated by a sliding mode controller, improves positioning precision and anti-shaking effect of the system, effectively solves the problems of track friction force change and load quality inconsistency of the variable-rope-length bridge crane in actual operation, improves robustness and anti-interference performance of a control system, and meets application requirements of a bridge crane in severe environment. Meanwhile, the horizontal movement and the lifting movement of the bridge crane are coupled, so that the conveying efficiency in actual operation is greatly improved.
Drawings
FIG. 1 is a block diagram of the system of the present invention;
FIG. 2 is a schematic model of a hoist for a long bridge of a hoist rope according to an embodiment of the present invention;
FIG. 3 is a graph of the change of state in an example of the present invention;
FIG. 4 illustrates compensation of vertical driving force by a disturbance observer in an embodiment of the present invention;
fig. 5 is a sliding mode dynamics of the sliding mode surfaces in an example of the present invention.
Detailed Description
The technical solution of the present invention is further explained with reference to the accompanying drawings and specific embodiments.
Example (b): referring to fig. 1-5, the invention provides a variable-rope-length crane anti-swing positioning control method based on a second-order sliding mode disturbance observer, which comprises the following specific implementation steps:
1) Taking external interference and internal uncertainty of the bridge crane as composite interference, and establishing a mathematical model of the variable rope length bridge crane system by an Euler-Lagrange method;
wherein M and M are respectively the trolley mass and the load mass; x (t) is the sum of the values of,
representing the displacement and second derivative, i.e. acceleration, of the trolley; l (t) of,
representing the rope length, rope speed and rope acceleration of the lifting rope; the sum of the values of theta (t),
representing the angle, angular velocity and angular acceleration of the swing of the hoisted load; f. of x (t),f l (t) motor driving forces of the crane in the horizontal direction and the vertical direction respectively; phi is a x (t),φ l (t) composite disturbances for the bridge crane system combining external disturbances and internal uncertainties, respectively. In the present example, M =6.5kg, M =0.5kg for the bogie mass and the load mass, respectively.
And transforming the dynamic model equation to obtain:
wherein d is x ,d l ,d θ Another manifestation of the complex interference experienced by the system in three directions.
2) The method is characterized in that the displacement, the rope length and the load swinging angle of the trolley are introduced into state variables, and a mathematical model equation of the state variables is converted into a nonlinear dynamical state equation with a compound interference term, and the method specifically comprises the following steps:
introducing a state variable χ 1 (t)＝[x(t),l(t),θ(t)] T ，
Transforming a mathematical model equation of the nonlinear dynamical state equation, wherein the nonlinear dynamical state equation is as follows:
wherein
D＝[d x ,d l ,d θ ] T ，U c ＝[f x ,f l ] T
In this example, the initial state variable of the system is set to χ 1 (0)＝[0,0.4,0] T ，χ 2 (0)＝[0,0,0] T 。
3) Target tracking trajectory x using trolley and lifting rope r ,l r Error e from actual displacement x ,e l Respectively constructing slip form surface sigma x ,σ l ；
4) At trolley slip form surface sigma x Adding a suspended load swing angle term theta to realize the suppression of load swing, and specifically performing the following steps 3) and 4):
defining target position tracking error:
e x ＝x-x r ,e l ＝l-l r
wherein e is x ,e l Trajectory tracking error, x, of trolley and lifting rope, respectively r ,l r Respectively the tracking tracks of the trolley and the lifting rope. Note the book
Respectively, the final target position of the tracking track, in this example, is selected as
And in order to control the system to start smoothly, the following tracking track is set
Constructing a sliding mode surface according to a target tracking track:
the slip form of which is dynamically
Wherein
In this example, the parameter for constructing the sliding surface is specifically selected as c after debugging x ＝0.65,c l ＝0.68,k x ＝1.2。
5) Designing a second-order sliding mode disturbance observer according to the coupling relation between the sliding mode surface sliding mode dynamics and the composite disturbance constructed in the step 3), and utilizing the characteristics of a high-order sliding mode
Estimating the composite interference of a bridge crane with a variable rope length, comprising the following specific steps:
designing a super-twisting-based second-order sliding mode disturbance observer:
wherein z, V = [ V = x ,v l ] T And the injection control items are respectively an auxiliary sliding mode surface and an auxiliary sliding mode surface. To slip form surface s And (5) obtaining a derivative:
when the system stateConverging on a second-order slip form surface s When is at time
The injection control term V can be considered as a prediction of the composite disturbance D.
Input control item v i And i belongs to { x, l } and is designed based on the super-twisting control rate as follows:
wherein alpha is i ,λ i Are all positive constants and satisfy the inequality
α＞f +
Wherein f is + For the limit, it can be considered as twice the maximum second derivative of the system state variable, p satisfying the inequality 0 < p < 1.
In the present example, the parameter of the second-order sliding mode disturbance observer is specifically selected to be alpha after debugging i ＝0.86,λ i ＝0.56i∈{x,l}。
6) Designing a sliding mode controller according to the composite interference observation value of the bridge crane obtained in the step 5), and obtaining the driving force of a trolley motor of the crane in the horizontal direction and the driving force of a lifting rope motor of the crane in the vertical direction, wherein the method comprises the following specific steps:
firstly, a sliding mode surface sigma designed according to a tracking track and a composite interference item D in sliding mode dynamic can be replaced by V, namely
Selecting an index sliding mode approach rate
Wherein rho and epsilon are both positive constants, and beta is epsilon (0,1).
In conclusion, the anti-swing positioning control rate of the variable-rope-length bridge crane is obtained as
Wherein x is related to the system state 1 Matrix B (χ) 1 ) Is constantly reversible.
In this example, the parameter of the sliding mode controller is specifically selected to be ρ after debugging x ＝ρ l ＝0.5，ε x ＝0.24，ε l ＝0.15。
In order to prove the observation effect of the second-order sliding-mode observer designed by the invention for composite interference, 10% of uncertainty is added to the crane model parameters M and M in simulation, and the load swing angle dynamic state is carried out at 2-3 seconds
External interference 2.3sin (t/8) is added.
As can be seen from FIG. 3, the trolley and the lifting rope reach the target position gradually within 8 seconds, overshoot and steady-state errors do not occur under the conditions that the track friction parameters are unknown and the load quality is inaccurate, and the positioning accuracy is high. As can be seen from the angle change curve of fig. 3, the external disturbance added at 2 to 3 seconds obviously increases the swing angle, but then gradually converges to zero, and the swing suppression effect is obvious. Figure 4 shows that the observer compensates the driving force on the hoisting rope by observing the error between the actual weight of the load and the given parameter. As shown in fig. 5, slip form surface s x ，s l It is possible to converge to zero quickly. Simulation experiment results show that the anti-swing positioning control method for the variable-rope-length bridge crane based on the second-order sliding-mode disturbance observer can effectively solve the problems of uncertain system parameters and existence of external partsThe anti-shaking positioning problem of the interfered bridge crane.
The above is only a preferred implementation of the present invention, and it should be noted that: it will be apparent to those skilled in the art that modifications and variations can be made in the present invention without departing from the spirit or essential attributes thereof, and it is intended to cover all modifications and equivalents included within the scope of the invention.
Claims (3)
1. A variable-rope-length crane anti-swing positioning control method based on a second-order sliding mode disturbance observer is characterized by comprising the following steps:
step 1) taking external interference and internal uncertainty of a bridge crane as composite interference, and establishing a mathematical model of a variable rope length bridge crane system;
step 2) introducing trolley displacement, rope length and load swinging angle into state variables, and converting a mathematical model equation of the state variables into a nonlinear dynamical state equation with a compound interference term;
step 3) tracking track x by using trolley and lifting rope r ,l r Error e from actual displacement x ,e l Respectively forming a slip form surface sigma x ,σ l ；
Step 4) sliding mode surface sigma of trolley x Adding a suspended load swing angle term theta to realize the suppression of load swing;
step 5) designing a second-order sliding mode disturbance observer according to the coupling relation between the sliding mode surface sliding mode dynamics and the composite disturbance constructed in the step 3), and utilizing the characteristic of a high-order sliding mode
Estimating the compound interference of a bridge crane with a variable rope length;
step 6) designing a sliding mode controller according to the composite interference observation value of the bridge crane obtained in the step 5), and obtaining the driving force of a trolley motor of the crane in the horizontal direction and the driving force of a lifting rope motor of the crane in the vertical direction;
step 1) establishing a mathematical model, wherein the specific method comprises the following steps:
wherein M and M are respectively the trolley mass and the load mass;
representing the displacement and second derivative, i.e. acceleration, of the trolley; l, the ratio of the total amount of the catalyst,
representing the rope length, rope speed and rope acceleration of the lifting rope; the number of the theta's is,
representing the angle, angular velocity and angular acceleration of the swing of the hoisted load; f. of x ,f l The motor driving forces of the crane in the horizontal direction and the vertical direction are respectively; phi is a unit of x ,φ l Respectively combining external disturbance and internal uncertainty compound interference for a bridge crane system;
transforming the mathematical model to obtain:
wherein d is x ,d l ,d θ Another manifestation of the complex interference experienced by the system in three directions;
step 2) introducing trolley displacement, rope length and load swinging angle into state variables, and converting a mathematical model equation into a nonlinear dynamics state equation with a compound interference term, wherein the specific steps are as follows:
introducing a state variable χ 1 ＝[x,l,θ] T ，
Transforming a mathematical model equation of the nonlinear dynamical state equation, wherein the nonlinear dynamical state equation is as follows:
wherein
D＝[d x ,d l ,d θ ] T ，U c ＝[f x ,f l ] T
Wherein F (x) 1 ,χ 2 ),B(χ 1 ) Dependent on the state and parameters of the system, U c ＝[f x ,f l ] T D is a composite interference item;
defining target position tracking error:
e x ＝x-x r ,e l ＝l-l r
wherein e is x ,e l Track following errors, x, of trolley and lifting rope, respectively r ,l r Respectively a trolley andtracking the target of the lifting rope;
the design of the sliding mode surface in the step 3) comprises the following steps:
constructing a sliding mode surface according to a target tracking track:
the slip form of which is dynamically
Wherein C (x) 1 ,χ 2 ) Parameters of the system, state variables and sliding mode surface sigma structural parameters c x ,c l ,k x (ii) related;
in the step 5), the design of the second-order sliding mode disturbance observer comprises the following steps:
designing a super-twisting-based second-order sliding mode disturbance observer:
wherein z, V = [ V = x ,v l ] T Respectively obtaining injection control items of the auxiliary sliding mode surface and the auxiliary sliding mode surface by derivation of the sliding mode surface s:
when the system state converges to the second-order slip form surface s, i.e.
The injected control term V can be considered as a prediction of the composite disturbance D;
input control item v i And i belongs to { x, l } and is designed based on the super-twisting control rate as follows:
where sign is a sign function, α i ,λ i Are all positive constants and satisfy the inequality
Wherein f is + Which is a limit value, can be regarded as twice the maximum second derivative of the system state variable, the constant p satisfies the inequality 0 < p < 1,
in step 6), the design of the sliding mode controller comprises the following steps:
designing a sliding mode controller according to an observation V of a designed second-order sliding mode disturbance observer based on super-twisting to a composite disturbance D of a bridge crane:
firstly, a sliding mode surface sigma designed according to a tracking track and a composite interference item D in sliding mode dynamic can be replaced by V, namely
Selecting an index sliding mode approach rate
Where ρ is i ，σ i Are all positive constants, are beta e (0,1),
in conclusion, the anti-swing positioning control rate of the variable-rope-length bridge crane is obtained as
Wherein x is related to the system state 1 Matrix B (χ) 1 ) Is constantly reversible.
2. The method for controlling the anti-swing positioning of the variable-rope-length crane according to claim 1, wherein the observation V of the second-order sliding mode disturbance observer on the complex disturbance D comprises an integral of a sign function, i.e., V is uninterrupted, and a continuous control input U can be obtained c 。
3. The method for controlling the anti-shaking positioning of the variable-rope-length crane based on the second-order sliding-mode disturbance observer according to claim 2, wherein the bridge crane is a variable-rope-length bridge crane, namely, the bridge crane comprises a lifting mechanism to lift and lower a load during horizontal transportation.
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