CN117105096B - Sliding mode control method suitable for rope-length-variable double-swing type ship crane - Google Patents
Sliding mode control method suitable for rope-length-variable double-swing type ship crane 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
- B66C13/18—Control systems or devices
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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
- B66C13/063—Auxiliary devices for controlling movements of suspended loads, or preventing cable slack for minimising or preventing longitudinal or transverse swinging of loads electrical
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Abstract
The invention provides a sliding mode control method suitable for a rope-length-variable double-swing type marine crane, and aims to solve the problems of poor swing reducing effect and low response speed of a nonlinear marine crane system caused by a traditional linear control method. Firstly, under the condition that the length of a cable can be changed and the quality of a lifting hook is not negligible, a dynamic model of a ship crane with double-pendulum characteristics is established, then, an interference observer is adopted to estimate and compensate the unknown wave bounded interference, finally, a sliding mode surface comprising a lifting hook swing angle derivative and a load swing angle derivative is constructed aiming at a system drivable state vector so as to ensure that the state output of the system is corrected early, and control laws based on the position of a trolley and the length of the cable are respectively designed so as to realize the stabilizing control of the variable-length double-pendulum ship crane. The invention improves the accuracy of the nonlinear model of the marine crane, ensures that the system state is quickly converged to the balance state in a limited time, and can effectively inhibit the swing angle of the load and the lifting hook.
Description
Technical Field
The invention belongs to the field of high-end ocean engineering equipment, in particular to a marine crane control technology, and particularly relates to a sliding mode control method suitable for a rope-length-variable double-swing marine crane.
Background
In the development background of global trade and marine transportation, the application field of the marine crane is also becoming wider and wider. A marine crane is a lifting device designed for use on a vessel for loading and unloading cargo at sea, maintaining a vessel and performing other lifting tasks. In the operation process of the marine crane, the complexity of the marine environment and the underactuated characteristic of the marine crane equipment are considered, and the load can generate complex nonlinear dynamic response, so that the overall efficiency can be reduced, the accurate placement operation of the load in the falling and hanging process is influenced, and the safety of the whole equipment and operators can be greatly threatened. For the reasons, designing a high-performance swing elimination positioning control method to improve the working efficiency and the safety performance of the system is a difficult problem to be solved in the field of marine cranes.
At present, domestic and foreign scholars conduct extensive research and exploration on a control method of the marine crane, wherein the control method comprises methods such as PID control, intelligent control, LQR control, sliding mode control and the like. PID control and LQR control belong to a typical linear system control method, and the under-actuated marine crane model needs to be linearized, and control design is performed on the basis of linearization. However, such linearization model-based controllers achieve better control only near the equilibrium point and may reduce robustness to external disturbances. In contrast, the sliding mode control has certain robustness on external interference and perturbation of system parameters, and is suitable for a nonlinear under-actuated crane system influenced by external environment.
However, in analyzing the under-actuated marine crane model, most researchers ignore the mass of the hook, or treat the hook and load as the same particle, i.e., reduce the dynamics of the system to a single-stage swing model. In fact, in many cases, when the trolley is running, the hook will oscillate around the trolley in a first stage, while the load will oscillate in a second stage with respect to the hook, i.e. the whole crane system will exhibit a significant two-stage oscillation.
In addition, there are few documents that study the effect of cable length changes on the tracking control of marine crane systems, i.e. taking into account the lifting movement of the load while the trolley moves horizontally. In fact, the cable length between trolley and hook in the under-actuated marine crane model is regarded as a constant rather than a state variable, on the one hand in order to reduce complex coupling relations between parameters and on the other hand in order to adapt its control method, but this results in a not perfect fit with the actual model, which, once in practical use, may result in the proposed control method failing to achieve the desired control effect due to the additional influence of the rope length variations.
When analyzing the under-actuated marine crane model in the patent 'a marine crane pendulum reduction control method based on a self-adaptive sliding mode variable structure', the dynamics of the system is simplified into a single-stage swing model, a large number of parameters are introduced in the design of the control law for estimating the unmodeled disturbance in the system, and although a satisfactory compensation effect is obtained, the design of the controller is too complex, and engineering realization is difficult.
In the patent 'a sliding mode control method of a nonlinear sliding mode surface of a double-pendulum bridge crane', for the modeling process of the crane with double-pendulum characteristics, the lengths of ropes between a trolley and a lifting hook and between the lifting hook and a load are regarded as fixed lengths, and a nonlinear system function is introduced for adjusting the damping ratio of the system in the design of the sliding mode surface, but the artificially introduced nonlinear system function is too dependent on the output of the system, so that the method has weak adaptability.
The method provided in paper ASliding Mode Tracking Control Method for Double Pendulum Crane Systems With Variable Rope Length has the following problems:
(1) The under-actuated crane model established in the paper does not fully consider the friction force of the trolley in the horizontal movement direction, so that the nonlinear model established in the paper is not accurate enough;
(2) The failure to take the derivatives of hook and load pivot angles into account in the design of the slip form surface also makes it difficult to correct the state output of the system early.
Therefore, the control method provided by the document can not be directly applied to the variable-rope-length double-pendulum type ship crane system, and the method fully considers the nonlinear problem caused by the occurrence of two-stage swing characteristics and the influence of the change of the length of a cable on the control of the ship crane system.
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention provides a sliding mode control method suitable for a rope-length-variable double-pendulum type ship crane. An interference observer is introduced on the basis of a crane model for the variable-rope-length double-pendulum ship, so that the ship crane system can still keep good robustness under the condition of unknown bounded wave interference, and meanwhile, a sliding mode controller which is simple in structure and easy to engineering application is designed, the system state vector can be ensured to be converged to an expected value, and the swinging of a lifting hook and a load can be effectively restrained.
In order to achieve the purpose, the invention adopts the following specific technical scheme:
a sliding mode tracking control method suitable for a rope-length-variable double-swing type marine crane comprises the following steps:
s1: a Lagrangian mechanical equation is applied to deduce a crane dynamics model for the variable rope length double-pendulum ship under the condition that the length of a rope can be changed and the weight of a lifting hook is not negligible;
s2: when the double-pendulum type ship crane actually runs, the horizontal movement of the trolley, the lifting movement of the cable, the swing angle of the lifting hook and the swing angle of the load are mutually coupled, and the coupling degree between the trolley and the swing angle of the lifting hook is more serious under the condition of external wave interference, so that an interference observer based on a double-pendulum type ship crane model is adopted to estimate and compensate the external wave interference of the system;
s3: according to the crane dynamics equation for the double-pendulum ship, which is obtained in the step S1, a sliding mode surface is designed by utilizing an error function of a drivable state vector, and an underactuated state vector and a derivative term are added into the design of the position sliding mode surface, so that the swing angle of a system can be corrected in time, a control law is designed, and the stability of the system is proved;
the technical scheme of the method is characterized in that:
s1 specifically comprises the following steps:
s11: establishing a two-dimensional coordinate system, defining the horizontal position of the trolley as a zero potential energy point of the system, dynamically describing the crane system for the double-pendulum ship from an energy angle, and calculating the total kinetic energy K and the total potential energy P of the system by analyzing the motion states of all parts of the system:
P＝-(m h +m l )glcosθ 1 -m l grcosθ 2 (2)
wherein m is c For trolley mass, m h For the weight of the lifting hook, m l For the load mass, v c For trolley speed, v h For hook speed, v l Is the load speed, l is the cable length, r is the equivalent length between the hook centroid and the load centroid, θ 1 For the swing angle of the lifting hook, theta 2 G is the gravitational acceleration, g is the load pivot angle.
S12: motion displacement x of trolley, cable length l and hook swing angle theta 1 Load pivot angle θ 2 Generalized state vector as a systemDriving force f of taking trolley x And the lifting force f of the cable l Generalized input force as a system->The friction force between the trolley and the track in the horizontal movement process is assumed to bef, under the condition that the friction coefficient mu is known, the friction force f and the trolley displacement x are approximately considered to be in a linear relation, according to the motion state of each part of the system in S11, the displacement and speed relation of the trolley, the lifting hook and the load in different directions is obtained, and a Lagrange mechanical equation of the double-pendulum ship crane is established:
where L is a lagrangian, and l=k-P,for the first derivative of the generalized state vector, the dynamics of the system obtained by the arrangement are described as follows:
s2 specifically comprises the following steps:
s21: according to the dynamic equation of the generalized drivable state vector described in S12, the nonlinear model of the double-pendulum marine crane considering external wave interference can be expressed as follows:
in the method, in the process of the invention,is a nonlinear term, q [4×1] 、/>F is the state vector of the system [4×1] For the control input of the system d is unknown bounded wave disturbance and +.>Wherein->Represent the known upper bound, let xi 1 ＝q，The state equation is:
s22: the design of a disturbance observer based on a nonlinear model of a double pendulum marine crane according to equation (6) is represented as follows:
where z is the internal state of the disturbance observer, L is the gain matrix of the observer,is an observation value of external wave interference.
The estimated error of the disturbance is defined as:assuming that in the actual process, the change of the disturbance is slow compared with the change of the system state, the time derivative of the disturbance is zero, so the dynamic property of the disturbance estimation error is expressed as:
since the unknown wave interference d is bounded, the interference estimation errorThe convergence can be ensured by only selecting the gain matrix L meeting the Hurwitz condition.
S3: according to the crane dynamics equation for the double-pendulum ship, which is obtained in the step S1, a sliding mode surface is designed by utilizing an error function of a drivable state vector, and an underactuated state vector and a derivative term are added into the design of the position sliding mode surface, so that the swing angle of a system can be corrected in time, a control law is designed, and the stability of the system is proved;
s31: considering that the crane system for double pendulum type ship is a typical under-actuated system, the swing angle of the lifting hook and the swing angle of the load belong to under-actuated amounts, and the dynamic equation in S12 is converted into a dynamic equation of a generalized drivable state vector by Gaussian elimination:
in the method, in the process of the invention,is the second derivative of the generalized state vector, +.>For controlling the input vector>Is an auxiliary item.
S32: two sliding die surfaces containing position variables and cable variables are respectively designed, due to the characteristics of an underactuated system, the hook swing angle and the load swing angle and the derivative of the hook swing angle and the derivative of the load swing angle are combined into the design of the sliding die surfaces, so that the state output of the system is corrected early, and an error vector is defined as follows:
e x ＝x-x d ,e l ＝l-l d (10)
the sliding mode surface of the system is designed as follows:
wherein k is x ，k l ，k 1 ，k 2 ，k 3 ，k 4 Are all positive integers.
An exponential approach law control method is adopted:
where sgn represents the standard sign function,is a positive definite diagonal matrix, the expression of which is as follows:
κ＝diag{κ 1 ,κ 2 },τ＝diag{τ 1 ,τ 2 } (13)
s33: in order to realize the stabilization control of the sliding mode tracking of the variable rope length shipborne crane, a control law is designed:
F a ＝M a [Ψ-κσ-τsgn(σ)]+D (14)
in the method, in the process of the invention,the expression for the constructed auxiliary term is as follows:
replacing discontinuous term sgn in the control law with tanh (5) to reduce buffeting phenomenon of sliding mode control, and designing a new control law as follows:
F a ＝M a [Ψ-κσ-τtanh(5σ)]+D (16)
s34: demonstrating system stability, defining Lyapunov function as:
the derivative is obtained by time, and the method comprises the following steps:
therefore, the constructed sliding mode surface is gradually stable, and the system converges.
The invention has the following beneficial effects:
(1) On the basis of a traditional double-pendulum type crane model for the ship, the influence of the heave motion of the load on other underactuated state vectors in the system is considered, the interference observer based on the nonlinear system compensates the external wave interference, and the designed rope-length-variable double-pendulum type crane control system for the ship has better performance, and the method is simple to operate, small in calculated amount and convenient to apply in practical engineering;
(2) According to the invention, on the premise of not linearizing the dynamic model, a sliding mode controller with two sliding mode surfaces is designed, the suspension hook and the load swing angle derivative are added in the design of the sliding mode surface of the system, so that the state vector can be ensured to be quickly converged to an expected value and simultaneously the swing angle of the load and the suspension hook is restrained, and the discontinuous items in the control law are smoothly approximated, so that the buffeting phenomenon in the sliding mode control is effectively weakened;
(3) Compared with a PID control method, the control performance of the variable-rope-length double-pendulum type ship crane is further improved, and the stability of the system is strictly proved by utilizing the Lyapunov stability theorem; according to the simulation experiment result, compared with the PID control method, the method has a certain improvement in the aspects of the pendulum reduction efficiency and the response time, and the pendulum reduction efficiency is improved by about 25 percent and the response time is improved by about 2 seconds under the sine wave interference condition that the fluctuation is 2.5 degrees and the frequency is about 1 Hz.
Drawings
In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a simplified model diagram of a variable rope length double pendulum marine crane in an embodiment of the present invention;
FIG. 2 is a block diagram of a sliding mode control in an embodiment of the present invention;
FIG. 3 is a schematic diagram of a disturbance observer observing a bounded wave in an embodiment of the present invention;
FIG. 4 is a schematic diagram showing the change of the horizontal position of the trolley according to the embodiment of the invention;
FIG. 5 is a diagram illustrating a load lifting position change in an embodiment of the present invention;
FIG. 6 is a diagram showing the change of the swing angle of the hook according to the embodiment of the invention;
fig. 7 is a diagram illustrating a change in load pivot angle according to an embodiment of the present invention.
Detailed Description
In order to make the above objects, features and advantages of the present invention more comprehensible, a sliding mode control method for a rope-length-variable double-pendulum type marine crane includes the steps of:
s1: by applying Lagrangian mechanical equation, deducing a double pendulum type ship crane dynamics model under the condition that the cable length can be changed and the lifting hook mass is not negligible, the specific steps are as follows:
s11: as shown in fig. 1, a two-dimensional coordinate system is established by taking an initial position of a trolley as an origin, a horizontal position of the trolley is defined as a zero potential energy point of the system, a crane system for a double-pendulum ship is dynamically described from an energy angle, and motion states of all parts of the system are analyzed:
carriage position and velocity components:
wherein x is c For displacement component of trolley in X-axis direction, y c For the displacement component of the carriage in the Y-axis direction,for the velocity component of the trolley in the X-axis direction, is->Is the velocity component of the trolley in the Y-axis direction.
Hook position and velocity component:
wherein x is h For the displacement component of the hook in the X-axis direction, y h As a displacement component of the hook in the Y-axis direction,for the velocity component of the hook in the X-axis direction, < >>Is the velocity component of the hook in the Y-axis direction.
Load position and velocity components:
wherein x is l For the displacement component of the load in the X-axis direction, y l For the displacement component of the load in the Y-axis direction,for the speed component of the load in the X-axis direction, < >>Is the velocity component of the load in the Y-axis direction.
The total kinetic energy of the double pendulum marine crane system is as follows:
the total potential energy of the double pendulum marine crane system is as follows:
P＝-(m h +m l )glcosθ 1 -m l grcosθ 2 (23)
wherein m is c For trolley mass, m h For the weight of the lifting hook, m l For the load mass, v c For trolley speed, v h For hook speed, v l For the load speed, l is the cable length, r is the equivalent constant length from the hook centroid to the load centroid, θ 1 For the swing angle of the lifting hook, theta 2 G is the gravitational acceleration, g is the load pivot angle.
S12: motion displacement x of trolley, cable length l and hook swing angle theta 1 Load pivot angle θ 2 Generalized state vector as a systemDriving force f of taking trolley x And the lifting force f of the cable l Generalized input force as a system->Assuming that the friction force f between the trolley and the track in the horizontal movement process is in a linear relation with the displacement x of the trolley under the condition of a known friction force coefficient mu, according to the movement states of all parts of the system in the step S11, the displacement and speed relations of the trolley, the lifting hook and the load in different directions are obtained, and a Lagrange mechanical equation of the double-pendulum marine crane is established:
where L is a lagrangian, and l=k-P,for the first derivative of the generalized state vector, the dynamics of the system obtained by the arrangement are described as follows:
according to the under-actuated system model, the reduction formula (26) is in a standard form:
wherein M (q),g (q) is an inertia matrix, a coriolis matrix, and a gravity matrix, respectively, which are defined as follows:
wherein m is 23 ＝0，m 11 ＝m c +m h +m l ，m 12 ＝(m h +m l )sinθ 1 ，m 13 ＝(m h +m l )lcosθ 1 ，m 14 ＝m l rcosθ 2 ，m 21 ＝(m h +m l )sinθ 1 ，m 22 ＝m h +m l ，m 24 ＝m l rsin(θ 1 -θ 2 )，m 31 ＝(m h +m l )lcosθ 1 ，m 32 ＝0，m 33 ＝(m h +m l )l 2 ，m 34 ＝m l lrcos(θ 1 -θ 2 )，m 41 ＝m l rcosθ 2 ，m 42 ＝m l rsin(θ 1 -θ 2 )，m 43 ＝m l lrcos(θ 1 -θ 2 )，m 44 ＝m l r。
Wherein c 11 ＝μ，c 21 ＝0，/>c 22 ＝0，c 31 ＝0，c 33 ＝0，c 41 ＝0，c 44 ＝0。
G(q)＝[0 -(m h +m l )cosθ 1 (m h +m l )glsinθ 1 m l grsinθ 2 ] T (29)
In order to facilitate the design of disturbance observers, the mathematical model of the rope-length-variable double-pendulum marine crane can be rewritten into the following form:
s2: when the double-pendulum type ship crane actually runs, the horizontal movement of the trolley, the lifting movement of the cable, the swing angle of the lifting hook and the swing angle of the load are mutually coupled, and the coupling degree between the trolley and the swing angle of the lifting hook is more serious under the condition of external wave interference, so that an interference observer based on a double-pendulum type ship crane model is adopted to estimate and compensate the external wave disturbance of the system, and the specific steps are as follows:
s21: according to the dynamic equation of the generalized drivable state vector in S12, the variable rope length double pendulum type marine crane nonlinear model considering external wave interference can be expressed as follows:
in the method, in the process of the invention,g(q)＝M -1 (q) is a nonlinear term of the system, q [4×1] 、F is the state vector of the system [4×1] D is unknown bounded wave interference and is the control input of the systemWherein->Represent the known upper bound, let xi 1 ＝q，/>The state equation is:
s22: the design of a disturbance observer based on a nonlinear model of a double pendulum marine crane according to equation (32) is represented as follows:
where z is the internal state of the disturbance observer, L is the gain matrix of the observer,is an observation value of external wave interference.
The estimated error of the disturbance is defined as:
assuming that in the actual process, the change of the disturbance is slow compared with the change of the system state, the time derivative of the disturbance is zero, so the dynamic property of the disturbance estimation error is expressed as:
since the unknown wave interference d is bounded, the interference estimation errorThe convergence can be ensured by only selecting the gain matrix L meeting the Hurwitz condition.
S3: according to the crane dynamics equation for the double-pendulum ship obtained in the step S1, a sliding mode surface is designed by utilizing an error function of a drivable state vector, and an underactuated state vector and a derivative term are added into the design of the position sliding mode surface, so that the swing angle of a system can be corrected in time, a control law is designed, and the stability of the system is proved, and the method comprises the following specific steps:
s31: considering that the crane system for double pendulum type ship is a typical under-actuated system, the swing angle of the lifting hook and the swing angle of the load belong to under-actuated amounts, and the dynamic equation in S12 is converted into a dynamic equation of a generalized drivable state vector by Gaussian elimination:
in the method, in the process of the invention,is the second derivative of the generalized state vector, +.>For controlling the input vector>As an auxiliary term, the expression is as follows:
s32: two sliding die surfaces containing position variables and cable variables are respectively designed, and due to the characteristics of an underactuated system, the swing angle of a lifting hook and a load swing angle and the derivative of the swing angle of the lifting hook and the derivative of the swing angle of the load are combined into the design of the sliding die surfaces, so that the state output of the system is corrected early, the final aim of a designed controller is to track the positions of a trolley and a cable to the expected positions, and meanwhile, the swing angle of the lifting hook and the load is inhibited to a certain extent, and therefore, the definition error vector is as follows:
e x ＝x-x d ,e l ＝l-l d (43)
the sliding mode surface of the system is designed as follows:
wherein k is x ，k l ，k 1 ，k 2 ，k 3 ，k 4 Are all positive integers.
An exponential approach law control method is adopted:
where sgn represents the standard sign function,is a positive definite diagonal matrix, the expression of which is as follows:
κ＝diag{κ 1 ,κ 2 },τ＝diag{τ 1 ,τ 2 } (46)
s33: in order to realize the calm control of the crane for the variable rope length double-pendulum ship, a control law is designed:
F a ＝M a [Ψ-κσ-τsgn(σ)]+D (47)
in the method, in the process of the invention,the expression for the constructed auxiliary term is as follows:
replacing discontinuous term sgn in the control law with tanh (5) to reduce buffeting phenomenon of sliding mode control, and designing a new control law as follows:
F a ＝M a [Ψ-κσ-τtanh(5σ)]+D (49)
s34: demonstrating system stability, defining Lyapunov function as:
the derivative is obtained by time, and the method comprises the following steps:
therefore, according to Lyapunov stability criteria, the constructed sliding mode surface is gradually stable, and the system converges.
The effectiveness of the control method of the invention is verified by experiments, and the specific steps are as follows:
at the beginning of simulation, adding the bounded wave interference into a mathematical model of the double-pendulum type marine crane, wherein the bounded wave interference is equivalently approximated by adopting a sinusoidal signal, and a function expression of the sinusoidal signal is selected in a numerical simulation experiment as follows: d (t) =2.5 sin (pi/3.16×t).
In the simulation experiment, the parameters of the controller are as follows: k (k) x ＝4，k l ＝2，k 1 ＝1，k 2 ＝1，k 3 ＝3，k 4 ＝3，τ 1 ＝10，τ 2 ＝10，κ 1 ＝5，κ 2 ＝5。
In the simulation experiment, the parameters of the PID controller are as follows: k (k) xp ＝6.8，k xi ＝9.4，k xd ＝0.8，k lp ＝4.2，k li ＝8.3，k ld ＝2.9。
Based on Matlab/Simulink, a crane model for the variable-rope-length double-pendulum ship shown in fig. 2 is established for simulation, the change condition of the swinging angle motions of the trolley, the cable, the lifting hook and the load under the condition of bounded wave interference is analyzed, the effects of the control method and the PID control method are compared, and the feasibility of the proposed control method is verified. The selected system parameters are shown in table 1.
Table 1 simulation parameter settings
/>
As can be seen from fig. 3, the estimated value of the interference observer to the bounded wave interference based on the crane model for the variable rope length double-pendulum ship basically realizes effective tracking of the actual interference situation in the simulation time.
As can be seen from fig. 4, the trolley controlled by the method of the present invention reaches the target position for about 5s, and the trolley controlled by the PID method reaches the target position for about 7s, and the response time of the method of the present invention is improved by about 2s compared with the PID method; as can be seen from FIG. 5, the cable under the control of the method reaches the target position at about 8s, and the cable under the control of the PID method reaches the target position at about 9s, but the cable length also shows obvious overshoot, and as can be seen from FIGS. 6 and 7, the maximum swing angle of the lower lifting hook and the load under the control of the method is about 1.5 DEG, and the maximum swing angle of the lower lifting hook and the load under the control of the PID method is about 2 DEG, and the swing reduction efficiency of the method is improved by about 25% compared with that of the PID method. Therefore, the method can realize quick positioning of the trolley, reduce swinging of the lifting hook and the load, and effectively improve the working efficiency and the safety performance of the marine crane during operation.
The above embodiments further illustrate the objects, technical solutions and advantageous effects of the present invention, and the above examples are only for illustrating the technical solutions of the present invention, but not for limiting the scope of protection of the present invention, and it should be understood by those skilled in the art that modifications, equivalents and alternatives to the technical solutions of the present invention are included in the scope of protection of the present invention.
Claims (1)
1. The sliding mode control method suitable for the rope-length-variable double-swing marine crane is characterized by comprising the following steps of:
s1: applying Lagrangian mechanical equation to deduce crane dynamics model for double pendulum ship in case of changeable cable length and non-negligible hook quality;
s11: establishing a two-dimensional coordinate system, defining the horizontal position of the trolley as a zero potential energy point of the system, dynamically describing the crane system for the double-pendulum ship from an energy angle, and calculating the total kinetic energy K and the total potential energy P of the system by analyzing the motion states of all parts of the system:
P＝-(m h +m l )glcosθ 1 -m l grcosθ 2 (2)
wherein m is c For trolley mass, m h For the weight of the lifting hook, m l For the load mass, v c For the speed of the trolley it is possible,v h for hook speed, v l Is the load speed, l is the cable length, r is the equivalent length between the hook centroid and the load centroid, θ 1 For the swing angle of the lifting hook, theta 2 G is the gravity acceleration, and g is the load swing angle;
s12: motion displacement x of trolley, cable length l and hook swing angle theta 1 Load pivot angle θ 2 Generalized state vector as a systemDriving force f of taking trolley x And the lifting force f of the cable l Generalized input force as a systemAssuming that the friction force between the trolley and the track is f in the horizontal movement process, under the condition of a known friction force coefficient mu, approximately considering that the friction force f and the trolley displacement x are in a linear relation, obtaining the displacement and speed relation of the trolley, the lifting hook and the load in different directions according to the movement states of each part of the system in S11, and then establishing a Lagrange mechanical equation of the double-pendulum marine crane:
where L is a lagrangian, and l=k-P,is the first derivative of the generalized state vector;
s2: when the double-pendulum type ship crane actually runs, the horizontal movement of the trolley, the lifting movement of the cable, the swing angle of the lifting hook and the swing angle of the load are mutually coupled, and the coupling degree between the trolley and the swing angle of the lifting hook is more serious under the condition of external wave interference, so that an interference observer based on a double-pendulum type ship crane model is adopted to estimate and compensate the external wave interference of the system;
s21: according to the dynamics equation described in S12, the nonlinear model of the double pendulum type marine crane considering the external wave interference can be expressed as follows:
in the method, in the process of the invention,g (q) is a nonlinear term, q [4×1] 、/>F is the state vector of the system [4×1] For the control input of the system d is unknown bounded wave disturbance and +.>Wherein->Represent the known upper bound, let xi 1 ＝q，/>The state equation is:
s22: the design of a disturbance observer based on a nonlinear model of a double pendulum marine crane according to equation (5) is represented as follows:
where z is the internal state of the disturbance observer, L is the gain matrix of the observer,is an observed value of external wave interference;
because the unknown wave interference d is bounded, the convergence of a designed interference observer can be ensured only by selecting a gain matrix L meeting the Hurwitz condition;
s3: according to the crane dynamics equation for the double-pendulum ship, which is obtained in the step S1, a sliding mode surface is designed by utilizing an error function of a drivable state vector, and an underactuated state vector and a derivative term are added into the design of the position sliding mode surface, so that the swing angle of a system can be corrected in time, a control law is designed, and the stability of the system is proved;
s31: considering that the crane system for double pendulum type ship is a typical under-actuated system, the swing angle of the lifting hook and the swing angle of the load belong to under-actuated amounts, and the dynamic equation in S12 is converted into a dynamic equation of a generalized drivable state vector by Gaussian elimination:
in the method, in the process of the invention,is the second derivative of the generalized state vector, +.>For controlling the input vector> Is an auxiliary item;
s32: two sliding die surfaces containing position variables and cable variables are respectively designed, due to the characteristics of an underactuated system, the hook swing angle and the load swing angle and the derivative of the hook swing angle and the derivative of the load swing angle are combined into the design of the sliding die surfaces, so that the state output of the system is corrected early, and an error vector is defined as follows:
e x ＝x-x d ,e l ＝l-l d (8)
the sliding mode surface of the system is designed as follows:
wherein k is x ，k l ，k 1 ，k 2 ，k 3 ，k 4 Are all positive integers;
an exponential approach law control method is adopted:
where sgn represents the standard sign function,is a positive definite diagonal matrix, the expression of which is as follows:
κ＝diag{κ 1 ,κ 2 },τ＝diag{τ 1 ,τ 2 } (11)
s33: in order to realize the calm control of the variable rope length double-pendulum ship crane, a control law is designed:
F a ＝M a [Ψ-κσ-τsgn(σ)]+D (12)
in the method, in the process of the invention,the expression for the constructed auxiliary term is as follows:
replacing discontinuous term sgn in the control law with tanh (5) to reduce buffeting phenomenon of sliding mode control, and designing a new control law as follows:
F a ＝M a [Ψ-κσ-τtanh(5σ)]+D (14)
s34: demonstrating system stability, defining Lyapunov function as:
the derivative is obtained by time, and the method comprises the following steps:
therefore, the constructed sliding mode surface is gradually stable, and the system converges.
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