CN112748662A - Layered nonsingular terminal sliding mode-based control system and method for pendulum elimination and positioning of bridge crane - Google Patents
Layered nonsingular terminal sliding mode-based control system and method for pendulum elimination and positioning of bridge crane Download PDFInfo
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Abstract
The invention provides a control system and a method for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode, which comprises the following steps: step 1, determining a control target according to a two-dimensional fixed rope length bridge crane system; step 2, respectively constructing nonsingular terminal sliding mode surfaces of the two subsystems according to a state equation of the two-dimensional bridge crane system and by combining a layered sliding mode theory; step 3, constructing a nonsingular terminal sliding mode controller with saturated local control quantity according to the nonsingular terminal sliding mode surfaces of the two subsystems constructed in the step 2; step 4, controlling the two-dimensional fixed rope long bridge crane system through the nonsingular terminal sliding mode controller with saturated local control quantity constructed in the step 3; the method provided by the invention has good pertinence and practicability in the face of practical problems possibly occurring in bridge crane control.
Description
Technical Field
The invention belongs to the control technology of an under-actuated system, and mainly relates to a control system and a method for eliminating swing and positioning of a bridge crane based on a layered nonsingular terminal sliding mode.
Background
The bridge crane is typical under-actuated circulation and intermittent hoisting equipment for cargo transportation, and has a large number of applications in steel mills, repair factories, aerospace and other places. The bridge crane system adopts a steel wire rope to connect the crane and the load, and belongs to a strongly coupled and nonlinear under-actuated system. Due to the underactuated characteristic of the bridge crane system, the load can move along with the trolley to generate an undriven swing angle, and the uncertainty of parameters and the influence of various disturbance factors, so that a strong coupling and nonlinear relation is generated between the swing angle and the trolley position, the control of the bridge crane is difficult, the working efficiency is reduced, and the potential safety hazard is increased. In recent years, a great deal of intensive research is carried out on bridge crane system control methods by a plurality of domestic and foreign researchers, and certain progress is made in various control methods, including traditional control methods such as input shaping, linear control, optimal control and sliding mode control, and intelligent control methods such as neural network control, fuzzy control and self-adaptive control. The control target of the bridge crane is integrated, the load is quickly and accurately conveyed by controlling the trolley, and the load swing angle in the conveying process is kept as small as possible, so that the control difficulty is increased, and the control target is always a hotspot and difficult problem of research. Therefore, in the research of the sway suppression and positioning control method of the bridge crane system, researchers have intensively researched on the aspects of state convergence speed, robustness of the control method, positioning accuracy, state quantity or control quantity constraint and the like.
Disclosure of Invention
The invention aims to provide a layered nonsingular terminal sliding mode-based control system and method for pendulum elimination and positioning of a bridge crane, and overcomes the defects of low working efficiency and large potential safety hazard of the conventional bridge crane control method.
In order to achieve the purpose, the invention adopts the technical scheme that:
the invention provides a control method for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode, which comprises the following steps of:
step 3, constructing a nonsingular terminal sliding mode controller with saturated local control quantity according to the nonsingular terminal sliding mode surfaces of the two subsystems constructed in the step 2;
and 4, controlling the two-dimensional fixed rope long bridge crane system by the nonsingular terminal sliding mode controller with saturated local control quantity constructed in the step 3.
Preferably, in step 1, the control target refers to: under the action of the trolley, the bridge crane can quickly and accurately convey the load to a target position, and meanwhile, the quick convergence of the load swing is ensured;
the control target is represented by the following formula:
wherein epsilon represents the positioning error after the introduction of the anti-swing signal; gamma is a motion signal after the trolley displacement introduces an anti-swing signal; p is a radical ofdIs the desired trolley position.
Preferably, in step 2, the nonsingular terminal sliding-mode surfaces of the two subsystems are respectively constructed, and the specific method is as follows:
s201, combining a system state equation and a layered sliding mode theory to respectively obtain nonsingular terminal sliding mode surfaces of two subsystems, wherein the two subsystems are respectively an independent position subsystem and a swing angle subsystem;
the nonsingular terminal sliding mode surfaces of the independent position subsystem and the swing angle subsystem are respectively represented by the following formulas:
wherein e is1Error in position of the trolley, e1＝x1-x1d；e2As error in the state of the pivot angle, e2＝x3-x3d；x1Is the trolley position; x is the number of3Is a swing angle; x is the number of1dAnd x3dAre all control targets; k is a radical of1And k2Are all positive real numbers; q. q.s1、q2、p1And p2Are positive and odd and satisfy
And S202, obtaining the total sliding mode surface of the sliding mode controllers of the two subsystems according to a layering theory.
Preferably, the total slip-form surface of the slip-form controller is represented by:
S＝α1s1+α2s2
wherein alpha is1,α2Are all positive real numbers.
Preferably, in step 3, a nonsingular terminal sliding mode controller with saturated local control quantity is constructed according to the nonsingular terminal sliding mode surfaces of the two subsystems constructed in step 2, and the specific method is as follows:
s301, setting a target positionx＝[x1d,0,0,0]TError vectore＝[x1-x1d,0,0,0]T；
S302, limiting local saturation overflow items which are easy to occur to the controller through a sat (t) function, and respectively obtaining non-singular terminal sliding mode controllers of the two subsystems;
s303, solving a master control law comprising equivalent control and switching control of the two subsystems by a Lyapunov feedback function method;
and S304, combining the saturation function sat (S) with the master control law to obtain a control law for buffeting compensation by adopting a nonlinear exponential approach law.
Preferably, in S302, the nonsingular terminal sliding mode controller of the two subsystems is represented by the following formula:
in the formula, K is a positive real number; omega is not less than (alpha)1+α2) D, D is the supremum limit of the external interference of the system; u. ofeqiThe equivalent control quantity of the ith subsystem; u. ofswiSwitching control quantity of the ith subsystem; u. ofiIs the control quantity of the ith subsystem.
A control system for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode comprises a controller and a computer program capable of running on a processor, wherein the processor executes the computer program to realize the steps of the method.
Preferably, the controller is connected with a motion control board card, and is used for transmitting the obtained control signal to the bridge crane system through the control board card.
Compared with the prior art, the invention has the beneficial effects that:
the invention provides a control method for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode, which is characterized in that the nonsingular terminal sliding mode control method with local saturation control constraint is applied to the rapid pendulum elimination and accurate positioning control of a bridge crane system, so that the rapid convergence of the state of the bridge crane system is realized, the dynamic response speed of the system is accelerated, and the phenomena of singularity and saturation of control quantity of the terminal sliding mode are avoided; the method has strong robustness to internal and external interference and uncertainty possibly existing in the system, and can still obtain good control effect under the conditions of initial swing angle, rope length change, outdoor wind load action and the like existing in the actual work of the bridge crane; the method provided by the invention has good pertinence and practicability in the face of practical problems possibly occurring in bridge crane control.
Drawings
FIG. 1 is a schematic view of a two-dimensional bridge crane system;
FIG. 2 is a disturbance dynamic response curve;
FIG. 3 is a schematic diagram illustrating the effect of different initial swing angles on the control effect;
FIG. 4 is a schematic view of the effect of different swing rope length variations;
FIG. 5 is a schematic view of a terminal sliding surface;
FIG. 6 is a control input comparison graph.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
The invention provides a bridge crane rapid pendulum elimination and positioning robust control method based on a nonsingular terminal sliding mode, which comprises the following steps:
The schematic diagram of the two-dimensional fixed rope length bridge crane system is shown in fig. 1, and the general form of the state equation of the two-dimensional fixed rope length bridge crane system is as follows:
wherein u (t) ═ FxRepresenting the control quantity of the running direction of the trolley;x＝[x1(t),x2(t),x3(t),x4(t)]Tis a system state vector, respectively representing the position, speed, load deflection angle and angular speed of the trolley,
the first derivative of the system state, representing the speed, angular velocity of the load and angular acceleration of the trolley, respectively; delta1,Δ2For uncertain variables, there is | Δ1,2|≤D； f1,f2,g1,g2The non-linear function of the state vector has the following specific meaning:
wherein M is the trolley mass; l is the length of the hoisting rope; m is the load mass; theta is an included angle between the load lifting rope and the vertical direction; f. ofrxIs frictional resistance to running of the trolley, dfThe wind resistance coefficient (such as indoor work can be not considered);
is the load movement speed.
The control targets of the bridge crane system are as follows: the method is characterized in that the load is quickly and accurately conveyed to a target position under the action of the trolley, meanwhile, the quick convergence of the load swing is ensured, and the control target is expressed by the following formula:
wherein epsilon represents the positioning error after the introduction of the anti-swing signal; gamma is a motion signal after the trolley displacement introduces an anti-swing signal; p is a radical ofdIs the desired trolley position.
Compared with the common sliding mode control, the terminal sliding mode control has better control performance, mainly shows that the robustness is stronger, the control precision is higher, the control error can be rapidly converged, and meanwhile, in order to eliminate the control singularity phenomenon of the terminal sliding mode, a non-singular terminal sliding mode surface is constructed.
Similar to the common sliding mode design method, the independent position subsystem or the swing angle subsystem contains less system information, and cannot ensure the integral stability of the bridge crane system, such as independent control quantity u1The state quantities such as the swing angle, the angular speed and the like of the system cannot be controlled; individual control quantity u2The state quantity such as the position, the speed and the like of the trolley can not be controlled; thus a hierarchical sliding mode control method is used.
Specifically, the method comprises the following steps:
s201, combining a system state equation and a layered sliding mode theory to respectively obtain nonsingular terminal sliding mode surfaces of two subsystems, wherein the two subsystems are respectively an independent position subsystem and a swing angle subsystem; expressions of nonsingular terminal sliding mode surfaces of the two subsystems are respectively as follows:
wherein e is1Error in position of the trolley, e1＝x1-x1d；e2As error in the state of the pivot angle, e2＝x3-x3d；x1Is a trolley positionPlacing; x is the number of3Is a swing angle; x is the number of1dAnd x3dAre all control targets; k is a radical of1And k2Are all positive real numbers; q. q.s1、q2、p1And p2Are positive and odd and satisfy
S202, obtaining the total sliding mode surfaces of the sliding mode controllers of the two subsystems according to a layering theory, and expressing the total sliding mode surfaces by the following formula:
S＝α1s1+α2s2 (5)
wherein alpha is1,α2Are all positive real numbers.
Step 3, designing a nonsingular terminal sliding mode controller with saturated local control quantity
S301, setting a target positionx＝[x1d,0,0,0]TError vectore＝[x1-x1d,0,0,0]T；
S302, in order to solve the contradiction between the system buffeting and the control efficiency and further weaken the system buffeting phenomenon caused by the discontinuous term, a nonlinear function is adopted to dynamically improve the approaching speed of the constant velocity approaching law in different approaching stages, and the nonlinear approaching rate is shown as the following formula (6):
wherein k is a positive real number; p is a positive real number between [0,1 ]; gamma is a positive real number; χ is a positive integer.
By system sliding modal reachability condition:
wherein Q (S) ═ p + (1-p) exp (-gamma | S-χ)＞0，
Representing the nonlinear approach rate.
Meanwhile, in order to avoid the saturation phenomenon of the control quantity, the controller is limited to easily generate a local saturation overflow item, specifically, the controller is limited to easily generate a local saturation overflow item through a sat (t) function, and non-singular terminal sliding mode controllers of the two subsystems are respectively obtained, and are expressed by the following formulas (8) and (9):
in the formula, K is a positive real number; omega is not less than (alpha)1+α2) D, D is the supremum limit of the external interference of the system; u. ofeqiThe equivalent control quantity of the ith subsystem; u. ofswiSwitching control quantity of the ith subsystem; u. ofiIs the control quantity of the ith subsystem.
Setting saturation functions separately
Threshold parameter u insAnd
wherein, set up us＞0；
Setting as shown in formula (10):
s303, solving a master control law including equivalent control and switching control of the two subsystems through a Lyapunov feedback function method, and further ensuring that the system state is quickly converged through the subsystem sliding mode surface, and ensuring that the sub sliding mode surface sliding mode reaches the condition through the master sliding mode surface; specifically, the method comprises the following steps:
firstly, setting the control input quantity of the sliding mould surface of the second layer as follows:
u′＝ueq1+ueq2+u′sw (11)
wherein u'swAnd the sliding mode surface switching control quantity of the second layer of the system is shown.
Then, the formula (5) is derived with respect to time, a method of a nonlinear exponential approximation law (6) is adopted, and the formula (6), the formula (8) and the formula (9) are substituted into the formula to obtain a system second-layer sliding mode surface switching control quantity:
finally, a saturation function sat(s) is used instead of the usual sign function sgn(s) to reduce buffeting, the saturation function being given by:
setting the boundary layer thickness usAnd delta, obtaining the total control law of the system for buffeting compensation by adopting a nonlinear index approach law, wherein the total control law comprises the following steps:
wherein u' represents the final overall control law of the system.
Theorem 1: for a two-dimensional bridge crane system, if a subsystem sliding mode surface shown in a formula (4), a total sliding mode surface shown in a formula (5) and a total sliding mode control quantity shown in a formula (12) are adopted, the bridge crane control system is stable.
And (3) proving that: take the following Lyapunov function:
after derivation of (13) have
Substituting the nonlinear approximation law shown in formula (10) into:
similarly, the approach velocity is known to satisfy Ω < Ω/q(s) < Ω/p, integrated over the above equation:
and V (t) ═ S2[ 2 ] 0, it is possible to obtain:
here Ψ < 0 is a bounded negative real number.
From the above formula, one can obtain:
namely have
S∈L∝It is known that
In the same way
By
The available S is equal to L2According to the barbalt theorem, the method can be obtained
Therefore, it is
I.e. the total slip-form surface of the system is stable.
Step 5, simulation analysis
And carrying out simulation analysis on the control effect of the proposed terminal sliding mode control method through a Matlab/Simulink numerical simulation environment, specifically, carrying out dynamic test on uncertain factors and parameter perturbation by the control method, and checking the pendulum elimination and positioning effects, the state convergence speed, the robust performance and the like of the control method. The simulation parameters of the bridge crane system are that the trolley mass M is 9.5kg, the load mass M is 5kg, the rope swing length l is 1M, and the control target is also set to be Xd＝[1,0,0,0]T. The control law of the nonsingular terminal sliding mode control method is as follows:
the relevant parameters of the nonsingular terminal sliding mode controller are shown in table 1.
TABLE 1 nonsingular terminal sliding mode controller parameters
1) External disturbance experiment
A robustness test is carried out, in order to simulate the influence of external interference such as sudden impact or wind load, a Gaussian signal is used for interference in the running process of a system, and the interference signal is as follows:
wherein, parameter A is 2, c1＝5,b1＝0.1。
From the simulation result, fig. 2 shows that the position in the graph represents the trolley displacement, and the angle represents the swing angle, which is the same as the following. The load swing caused by the external interference can be eliminated within a swing period, the trolley position cannot move greatly, and the control method has strong robustness to the external interference.
2) Internal uncertain factor disturbance experiment
Considering the fact that in actual operation of the bridge crane, the conditions of initial swing angle caused by inclined pulling and inclined hoisting, rope length time variation caused by synchronous operation of a large trolley and a lifting and the like cause uncertainty of a system, and inspecting the anti-interference performance test of the control method on the conditions.
Fig. 3 shows the influence of the initial swing angle on the control effect, and it can be seen that, regardless of the forward and reverse swing angles, under the condition of the initial swing angle of 17.2 °, the swing angle can be suppressed to about 1.8 ° within about 3.8s, the robustness to the initial swing angle is good, and the swing elimination and the positioning are realized within a limited time.
Fig. 4 shows the influence of the swing rope length variation, and the rope lengths are 0.8m and 1.2m respectively, so that the method has a good control effect on the condition of uncertain rope length. The method has the advantages that the method shortens the trolley positioning and load sway elimination time by using the local control quantity saturation nonsingular terminal sliding mode control method, obtains a better robust control effect, and is suitable for the conditions of internal and external interference, initial swing angle, rope length change and the like of a system.
3) State convergence time
The state convergence time directly determines the anti-sway rapidity and the positioning time of the bridge crane control system. The system sliding mode surface function curve is shown in fig. 5, the convergence speed of the system sliding mode surface is about 4.2s, and the system state of nonsingular terminal sliding mode control or the rapidity of system error convergence are reflected.
The anti-shake control quantity obtained through the control quantity simulation curve diagram 6 has the advantages that the improved approach rate is introduced, the slip form buffeting phenomenon is greatly weakened after the quasi-slip mode is improved, and the saturation phenomenon of the control quantity is effectively limited.
Implementation method
In actual bridge crane pendulum elimination and positioning control, an algorithm programming environment is built by a computer, appropriate system parameters and control parameters are selected according to a bridge crane system in actual application, corresponding control signals are obtained according to a formula (18), and the control signals are sent to the bridge crane system by a motion control panel card, so that real-time control over the crane system is realized, and further, a control target of the bridge crane system is realized.
Innovation and effect
The nonsingular terminal sliding mode control method with local saturation control constraint is applied to the rapid pendulum elimination and accurate positioning control of a bridge crane system, so that the rapid convergence of the state of the bridge crane system is realized, the dynamic response speed of the system is accelerated, and the singularity and control quantity saturation phenomena of the terminal sliding mode are avoided; the method has strong robustness to internal and external interference and uncertainty possibly existing in the system, and can still obtain good control effect under the conditions of initial swing angle, rope length change, outdoor wind load action and the like existing in the actual work of the bridge crane. The method provided by the invention has good pertinence and practicability in the face of practical problems possibly occurring in bridge crane control.
Claims (8)
1. A control method for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode is characterized by comprising the following steps:
step 1, determining a control target according to a two-dimensional fixed rope length bridge crane system;
step 2, respectively constructing nonsingular terminal sliding mode surfaces of the two subsystems according to a state equation of the two-dimensional bridge crane system and by combining a layered sliding mode theory;
step 3, constructing a nonsingular terminal sliding mode controller with saturated local control quantity according to the nonsingular terminal sliding mode surfaces of the two subsystems constructed in the step 2;
and 4, controlling the two-dimensional fixed rope length bridge crane system by the nonsingular terminal sliding mode controller with saturated local control quantity constructed in the step 3.
2. The method for controlling the pendulum elimination and the positioning of the bridge crane based on the layered nonsingular terminal sliding mode according to claim 1, wherein in the step 1, the control target refers to: under the action of the trolley, the bridge crane can quickly and accurately convey the load to a target position, and meanwhile, the quick convergence of the load swing is ensured;
the control target is represented by the following formula:
wherein epsilon represents the positioning error after the introduction of the anti-swing signal; gamma is a motion signal after the trolley displacement introduces an anti-swing signal; p is a radical ofdIs the desired trolley position.
3. The method for controlling the pendulum elimination and the positioning of the bridge crane based on the layered nonsingular terminal sliding mode according to claim 1, wherein in the step 2, nonsingular terminal sliding mode surfaces of the two subsystems are respectively constructed, and the specific method is as follows:
s201, combining a system state equation and a layered sliding mode theory to respectively obtain nonsingular terminal sliding mode surfaces of two subsystems, wherein the two subsystems are respectively an independent position subsystem and a swing angle subsystem;
the nonsingular terminal sliding mode surfaces of the independent position subsystem and the swing angle subsystem are respectively represented by the following formulas:
wherein e is1Error in position of the trolley, e1＝x1-x1d；e2As error in the state of the pivot angle, e2＝x3-x3d；x1Is the trolley position; x is the number of3Is a swing angle; x is the number of1dAnd x3dAre all control targets; k is a radical of1And k2Are all positive real numbers; q. q.s1、q2、p1And p2Are all positive odd numbers and satisfy q1＜p1＜2q1，q2＜p2＜2q2；
And S202, obtaining the total sliding mode surface of the sliding mode controllers of the two subsystems according to a layering theory.
4. The method for controlling the pendulum elimination and the positioning of the bridge crane based on the layered nonsingular terminal sliding mode according to claim 3, wherein the total sliding mode surface of the sliding mode controller is represented by the following formula:
S＝α1s1+α2s2
wherein alpha is1，α2Are all positive real numbers.
5. The method for controlling the pendulum elimination and the positioning of the bridge crane based on the layered nonsingular terminal sliding mode according to claim 1, wherein in step 3, a nonsingular terminal sliding mode controller with saturated local control quantity is constructed according to the nonsingular terminal sliding mode surfaces of the two subsystems constructed in step 2, and the specific method is as follows:
s301, setting a target positionx＝[x1d，0，0，0]TError vectore＝[x1-x1d，0，0，0]T；
S302, limiting local saturation overflow items which are easy to occur to the controller through a sat (t) function, and respectively obtaining nonsingular terminal sliding mode controllers of the two subsystems;
s303, solving a master control law comprising equivalent control and switching control of the two subsystems by a Lyapunov feedback function method;
and S304, combining the saturation function sat (S) with the master control law to obtain a control law for buffeting compensation by adopting a nonlinear exponential approach law.
6. The method for controlling the pendulum elimination and the positioning of the bridge crane based on the layered nonsingular terminal sliding mode according to claim 5, wherein in S302, the nonsingular terminal sliding mode controllers of the two subsystems are represented by the following formula:
in the formula, K is a positive real number; omega is not less than (alpha)1+α2) D, D is the supremum limit of the external interference of the system; u. ofeqiThe equivalent control quantity of the ith subsystem; u. ofswiSwitching control quantity of the ith subsystem; u. ofiIs the control quantity of the ith subsystem.
7. A control system for pendulum cancellation and positioning of a bridge crane based on a layered nonsingular terminal sliding mode, comprising a controller and a computer program capable of running on a processor, the processor implementing the steps of the method according to any one of claims 1 to 6 when executing the computer program.
8. The control system for pendulum elimination and positioning of a bridge crane based on a layered nonsingular terminal sliding mode according to claim 7, wherein the controller is connected with a motion control board card for transmitting an obtained control signal to the bridge crane system through the control board card.
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| CN113321122B (en) * | 2021-05-07 | 2023-08-29 | 武汉理工大学 | Layered nonsingular terminal sliding mode control method for crane double-pendulum system |
| CN114572828A (en) * | 2022-02-18 | 2022-06-03 | 武汉理工大学 | Nonsingular terminal sliding mode anti-sway control method for slender load vertical hoisting process |
| CN115453870A (en) * | 2022-08-31 | 2022-12-09 | 南京工业大学 | Bridge crane global robust disturbance rejection control method based on sliding mode theory |
| CN115453870B (en) * | 2022-08-31 | 2023-06-30 | 南京工业大学 | Bridge crane global robust disturbance rejection control method based on sliding mode theory |
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