CN106647267B - With not knowing dynamic (dynamical) crane finite time contrail tracker and method - Google Patents

With not knowing dynamic (dynamical) crane finite time contrail tracker and method Download PDF

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CN106647267B
CN106647267B CN201611160077.0A CN201611160077A CN106647267B CN 106647267 B CN106647267 B CN 106647267B CN 201611160077 A CN201611160077 A CN 201611160077A CN 106647267 B CN106647267 B CN 106647267B
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pivot angle
crane
trolley
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estimated value
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马昕
张梦华
宋锐
荣学文
田新诚
田国会
李贻斌
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Shandong University
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

It is had the invention discloses one kind and does not know dynamic (dynamical) crane finite time contrail tracker and method, including：Two terminal sliding mode observers are designed, one of observer is used to estimate to load pivot angle, another observer is used to estimate uncertain dynamics.Then, by the information that these are estimated, finite time Trajectory Tracking Control method is proposed.The stability and convergence of closed-loop system are demonstrated using Lyapunov method and LaSalle principle of invariance.Simulation result shows the correctness and validity of proposed control method.

Description

With not knowing dynamic (dynamical) crane finite time contrail tracker and method
Technical field
The present invention relates to the control technology fields of drive lacking bridge type crane system, more particularly to a kind of have not to know power The overhead crane finite time contrail tracker and design method of and non-loaded pivot angle feedback.
Background technique
Crane, also known as crane are a kind of engineering haulage equipments of large size, be widely used in construction site, harbour, The numerous areas such as harbour.According to the difference of structure, crane can be broadly divided into overhead crane, tower crane, revolution cantilevered and hang Vehicle.Although crane is many kinds of, their Dou Youyige intrinsic propesties：Drive lacking characteristic.Under-actuated systems save part and hold Row device, therefore have many advantages, such as that hardware cost is low, electromechanical structure is simple, light-weight, energy consumption is small.It is all in view of under-actuated systems More advantages, the research of control method have become a big hot spot in recent years.In all kinds of cranes, overhead crane is most represented Property, using also the most extensive.Numerous scholars propose a series of significant control methods for drive lacking bridge type crane system.
Up to now, the control problem of drive lacking bridge type crane system is still an open project.On the one hand, due to Trolley quality, load quality, lifting rope length, frictional force, the uncertainty of external disturbance make the dynamics of bridge type crane system With indeterminate.Also, these indeterminates are difficult look-ahead, cause the control performance for having most digital control method big It gives a discount.Therefore, the design of overhead crane control method, which should fully consider, does not know dynamic (dynamical) influence.On the other hand, have most Digital control method is required to the feedback of load pivot angle.However, load pivot angle is unable to measure in many cases,.Therefore, if Counting out and not needing the high performance control method of load pivot angle feedback is the actual needs in crane scene.
In fact, input shaper method and PD control device do not need the feedback of load pivot angle.Input shaper method root According to lifting rope length information, basic command signal and a series of certain pulses signals for being referred to as input shaper are done into convolution fortune It calculates.The certifiable system of this method is without Residual oscillations, however its control performance but depends critically upon the levels of precision of model, works as model When parameter has uncertain, control effect can sharply decline.PD control device structure is simple, is easy to Project Realization.However, PD Controller is very sensitive to parameter uncertainty, limits the practicality.
Sliding-mode control and self-adaptation control method effectively processing system parameter can have uncertainty. For in detail, traditional single order sliding-mode control has been successfully applied in bridge type crane system, solves positioning and the pendulum that disappears Problem, and obtain control result well.But, traditional single order sliding-mode control is discontinuous, to driving device band Carry out potential danger and along with chattering.Also, the above control method only can guarantee the asymptotic stability of system, this is in height It is far from being enough in the transport task of required precision；In addition, the above control method assumes its uncertain dynamics and system Parameter is linear relationship, is required to the feedback of load pivot angle.
Summary of the invention
The purpose of the present invention is to solve the above-mentioned problems, proposes a kind of limited with dynamic (dynamical) crane is not known Time locus tracking control unit and method, the controller and method are based on two terminal sliding mode observers, one of observer For estimating to load pivot angle, another observer is used to estimate uncertain dynamics.
To achieve the goals above, the present invention adopts the following technical scheme that：
One kind has and does not know dynamic (dynamical) crane finite time contrail tracker, including：
Design first terminal sliding mode observer estimates load pivot angle θ；Second terminal sliding mode observer is designed to not Determine that dynamics h is estimated；The finite time track following control of non-loaded pivot angle feedback is designed according to obtained estimated value Device processed is as follows：
Wherein, k04∈R+The control gain being positive；For the evaluated error of machine speed,For crane displacement First derivative,The estimated value of first derivative is displaced for crane,The estimated value of second dervative is displaced for crane；p3,q3∈R+For Positive odd number, and have p3< q3；e3=xd- x is the tracking error of trolley, xdFor the target trajectory of trolley；M is trolley quality, mp For load quality；For γ2Estimated value, γ2=h；frxFor frictional force, l is lifting rope length；For load pivot angle estimated value,For load pivot angle first derivative estimated value,For the estimated value for loading pivot angle second dervative；
Further, the first terminal sliding mode observer is specially：
Wherein, auxiliary function is defined For the estimated value of p, peFor observation error,
k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1；peFor observation error, b1It is normal Number.
Further, the second terminal sliding mode observer is specially：
Wherein, defined variable：γ2=h；
Introducing state：
About time derivation：
Then：U is uncertain derivative of the dynamic h about the time；
AndRespectively γ1And γ2Estimation, k02∈R+The observation gain being positive, γv=l1sgn(e1), l1,l2,l3,l4,l5∈R+The observation gain being positive, q2And p2The odd number being positive, and q2> p2。
One kind has and does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, including：
(1) assume that not knowing dynamics f about the derivative of time is bounded, does not know derivative of the dynamics h about the time Bounded, load pivot angle θ andInitial estimation it is identical as actual value；
(2 definition about load pivot angle θ auxiliary function p andSolve the observation error p of auxiliary function pe, according to sight Survey error peFirst terminal sliding mode observer is designed, load pivot angle θ is estimated；So thatIn finite time ToInterior accurate receipts It holds back to p, andIn finite time ToIt is interior accurately to converge to load pivot angle θ；WhereinFor the estimated value of auxiliary function p,For load The estimated value of pivot angle θ；
(3) define about uncertain dynamics h auxiliary function Q andSecond terminal sliding mode observer is designed, is being had Uncertain dynamics h is accurately estimated in the time of limit；
(4) according to the estimated value of obtained load pivot angle θ and uncertain dynamics h, non-loaded pivot angle feedback is obtained Finite time contrail tracker；
(5) by actually detected trolley displacement x, machine speedBeing input to above-mentioned has with not knowing dynamic (dynamical) crane Between in limited time in contrail tracker, the torque F of output driving trolley movement, system trolley, load quality, lifting rope length, Frictional force Parameter uncertainties and there are external disturbance in the case where the accurate positioning of trolley can be realized in finite time And suspension hook swings, loads the effective inhibition and elimination swung around suspension hook.
Further, in the step (2), the auxiliary function p about load pivot angle θ is specially:
Wherein, g is acceleration of gravity, and l is lifting rope length,For the second dervative for loading pivot angle.
Further, in the step (2), first terminal sliding mode observer is specially：
Wherein,k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1；b1It is positive Constant.
Further, in the step (3), the auxiliary function Q about uncertain dynamics h is specially；
Q is obtained about time derivation：
Wherein, frxFor frictional force, l is lifting rope length；For load pivot angle estimated value,To load pivot angle first derivative Estimated value,For the estimated value for loading pivot angle second dervative；F is the resultant force being applied on trolley.
Further, second terminal sliding mode observer is specially：
Wherein, defined variable：γ2=h；
Introducing state：
Then：U is uncertain derivative of the dynamic h about the time；
AndRespectively γ1And γ2Estimation, k02∈R+The observation gain being positive, γv=l1sgn(e1), l1,l2,l3,l4,l5∈R+The observation gain being positive, q2And p2The odd number being positive, and q2> p2。
Further, in the step (4), the finite time contrail tracker of non-loaded pivot angle feedback is specially：
Wherein, k04∈R+The control gain being positive；For the evaluated error of machine speed,For crane displacement First derivative,The estimated value of first derivative is displaced for crane,The estimated value of second dervative is displaced for crane；p3,q3∈R+For Positive odd number, and have p3< q3；e3=xd- x is the tracking error of trolley, xdFor the target trajectory of trolley；M is trolley quality, mp For load quality；For γ2Estimated value, γ2=h；frxFor frictional force, l is lifting rope length；For load pivot angle estimated value,For load pivot angle first derivative estimated value,For the estimated value for loading pivot angle second dervative；
Further, to inhibit and eliminating load pivot angle, desired trolley track is selected as：
Wherein,For target position；For trolley maximum permissible acceleration and speed；It indicates Adjust the parameter of initial acceleration；The control gain that κ > 1.0754 is positive.
The beneficial effects of the invention are as follows：
Compared with having most digital control method, the mentioned controller of the present invention does not need the feedback of load pivot angle, and solves Dynamic (dynamical) problem is not known existing for system.The mentioned control method of the present invention is directed to uncertain system parameter and external disturbance With very strong robustness；The feedback of load pivot angle is not needed, more actual motion is worth.Controller designed by the present invention can be real The convergence of existing finite time.The stabilization of closed-loop system is demonstrated using Lyapunov method and LaSalle principle of invariance Property and convergence.Simulation result shows the correctness and validity of proposed control method.
Detailed description of the invention
Fig. 1 is bridge type crane system schematic diagram；
Fig. 2 is control input, load pivot angle and the trolley track obtained under accurate model parameter using the method for the present invention Simulation result diagram；
Fig. 3 is that control input, load pivot angle and the trolley track obtained under accurate model parameter using LQR controller is imitated True result figure；
Fig. 4 is the control input obtained under accurate model parameter using enhancing coupling nonlinear controller, load pivot angle With trolley track simulation result diagram；
Fig. 5 is that control input, load pivot angle and the trolley track obtained under accurate model parameter using PD control device is imitated True result figure；
Fig. 6 is control input, load pivot angle and the trolley obtained under uncertain kinetics function using the method for the present invention Track emulation result figure；
Fig. 7 be the self-adaptation control method based on motion planning obtained under uncertain kinetics function control input, Load pivot angle and trolley track simulation result diagram.
Specific embodiment：
The present invention will be further described with example with reference to the accompanying drawing：
The invention proposes a kind of overhead crane finite times fed back with uncertain dynamics and non-loaded pivot angle Contrail tracker and design method.Specifically, two terminal sliding mode observers are based on, one of observer is used to estimate Meter load pivot angle, another observer are used to estimate uncertain dynamics.Then, it by the information that these are estimated, proposes limited Time locus tracking and controlling method.The steady of closed-loop system is demonstrated using Lyapunov method and LaSalle principle of invariance Qualitative and convergence.Simulation result shows the correctness and validity of proposed control method.
1. overhead crane kinetic model
Bridge type crane system model is as shown in Figure 1, its kinetic model can be described as：
Wherein, M is trolley quality, mpIndicate that load quality, l represent lifting rope length, h and f are uncertain dynamics, x (t) Crane displacement is represented, θ (t) indicates load pivot angle, frxFor frictional force, F is the resultant force being applied on trolley.
In fact, the uncertain dynamics h of (1) formula and the uncertain dynamics f of (2) formula are by uncertain trolley matter Measure Δ M, uncertain load quality Δ mp, uncertain lifting rope length Δ l, uncertain frictional force Δ frx, external disturbance d1 And d2It is caused.Not knowing dynamics h and f at this time can be written as：
Without loss of generality, to carry out following hypothesis：
Assuming that 1：Uncertain dynamics f is bounded, boundary lb about the derivative of time1, i.e.,：
Wherein, b1∈R+For known normal number.
Assuming that 2：For promote next analysis, it is assumed that θ andInitial estimation it is identical as actual value, i.e.,：
Assuming that 3：Uncertain dynamics h is expressed as u about the derivative of time.Although u is unknown, its amplitude is bounded, I.e. | | u | |≤π, wherein π ∈ R+ is known normal number.
2. the finite time contrail tracker design of non-loaded pivot angle feedback
2.1 load pivot angle estimations
To estimate the load pivot angle for being inconvenient to measure, a terminal sliding mode observer is devised.
For crane system, sin θ ≈ θ, cos θ ≈ 1 is to set up.Therefore, (2) formula can be written as：
Define auxiliary function p andRespectivelyWherein,For the estimated value of p, For the estimated value of θ.The observation error of p is：
Wherein, peFor observation error.
To estimate p, for the system (6) with uncertain dynamics f, the terminal sliding mode observer of following form is designed：
Wherein, k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1, b1∈R+For it is known just Constant.
Theorem 1：For the kinetics equation (6) containing uncertain dynamics f, sliding mode observer (8) be can guaranteeLimited Time ToIt is interior accurately to converge to p, andIn finite time ToIt is interior accurately to converge to θ, wherein：
So, as t >=ToWhen, pe≡ 0,
It proves：Choosing candidate's Lyapunov function is：
It can be obtained to (10) formula about time derivation and by (7), the substitution of (8) formula：
(5) formula is substituted into (11) Shi Ke get：
p1、q1The odd number being positive, then p1+q1For even number, thenAt the same time, (12) Shi Ke get is solved：
By (13) formula it is found that working as t >=T0When, VO1(t) 0 ≡, wherein：
By VO1(t) ≡ 0 can be obtained：
By (15) formula and p,Definition can obtain：
DefinitionIn conjunction with assuming 2, (16) formula can be written as：
Solve (17) Shi Ke get：
α=0 (18)
It is obtained by (18) formula：
By (15), (19) formula it is found that theorem 1 must be demonstrate,proved.
The estimation of 2.2 uncertain dynamics h
For the high-performance for guaranteeing controller, it should estimate and not determine dynamic h in crane system, and be effectively compensated for. For this purpose, designing a sliding mode observer to estimate uncertain dynamics h.
Define an auxiliary function：Q can be obtained about time derivation：
By (19) formula it is found that working as t >=T0When, An auxiliary function E is introduced, expression formula is：At this point, (20) formula can be written as：
For the design for promoting following sliding mode observer, a new state is introducedIts expression formula is：
Wherein, k02∈R+The observation gain being positive.
(22) formula can be obtained about time derivation：
So, the state estimation to linear augmented system (23) has been converted into the estimation problem of uncertain dynamic h.Its In in (23) formulaIt can measure/can find out.Assuming that uncertain dynamic h is u about the derivative of time, and introduce two new changes Measure γ1、γ2, expression formula is：γ2=h.At this point, (23) can be written as：
To estimate uncertain dynamic h (γ2), it is defined as follows the terminal sliding mode observer of form：
Wherein,AndRespectively γ1And γ2Estimation, γv=l1sgn(e1), l1,l2,l3,l4,l5∈R+The observation gain being positive,q2And p2The odd number being positive, and have q2> p2。
Defining observation error e is：E=[e1 e2]T.So, the dynamics side of observation error e can be obtained by (24)-(27) formula Cheng Wei：
Lemma 1：Under the action of terminal sliding mode observer (26) and (27), as t >=T0When, observation error system (28) and (29) the observation error e in is uniform ultimate bounded.In the process, it is assumed that γ1、γ2In t=T0When estimated value and reality Actual value is equal, i.e.,：
It proves：Consider the Lyapunov function of following form：
To (30) formula about time derivation, and (28), the substitution of (29) formula can be obtained：
Wherein：
Due to k02,l2,l3For positive number, then β is positive definite.Therefore, the minimal eigenvalue λ of βminIt is positive.
And then, (31) formula can be written as：
When | | e | | when ≠ 0, to guaranteeThe following conditions should meet：
In other words, when e is not gatheringWhen interior,It is negative.At this point, VO2Monotone decreasing.Obviously Ground, VO2Successively decrease finally will driving e enter in set D, then will be limited in set D.Due to I.e. | | e (T0) | |=0, by Lyapunov theorem and LaSalle principle of invariance it is found that working as t >=T0 When, observation error is limited in set D.This shows that e is uniform ultimate bounded.
Theorem 2：Consider that the error obtained by linear system (19), (20) and terminal sliding mode observer (21), (22) is seen Examining system (23), (24), selection observation gain l1,l2,l3,l4,l5So that：
l5- π > 0 (35)
Uncertain dynamics h can be accurately estimated within the limited time.
It proves：This process includes that following both sides proves.
1)e1Finite time convergence
Consider the Lyapunov function of following form：
To (36) formula about time derivation, and the substitution of (28) formula can be obtained：
For | | e1| | ≠ 0, to guaranteeSelection：
For the establishment of guarantee (38) formula, selection：
Solve (39) Shi Ke get：
So,
It sets up.
To (41) formula about time integral, can obtain：
By (42) formula it is found that working as t=T1When：
By (28) Shi Ke get：
2)e2Finite time convergence：As t >=T1When,And have：
For the proof for completing theorem 2, the positive definite scalar function of following form is considered：
To (45) formula about time derivation, and (35), the substitution of (44) formula can be obtained:
Solve (46) Shi Ke get:
By (47) Shi Ke get, work as t=T2When：
||e2| |=0.In other words, Uncertain dynamic can be in finite time T2It is interior, by accurately estimating uncertain dynamics h.
The finite time contrail tracker of 2.3 non-loaded feedbacks
For the design for completing contrail tracker, it is defined as follows the trolley Displacement Estimation expression formula of form：
Wherein, k03,δ0∈R+The gain being positive, p3,q3∈R+The odd number being positive, and have p3< q3, e3=xd- x is trolley Tracking error, xdFor the target trajectory of trolley,
Therefore, the finite time contrail tracker of non-loaded pivot angle feedback is designed as：
Wherein,For the evaluated error of machine speed, k04The control gain that ∈ R+ is positive.
Theorem 3：Mentioned tracking control unit (50) and terminal sliding mode observer (8), (26), (27) certifiable trolley track In Finite-time convergence to desired trajectory.
It proves：By e3, e4Definition can obtain：
From (49), (51) formula：
On the other hand, by (1) Shi Ke get:
To prove theorem 3, following Lyapunov candidate functions are selected：
To (54) formula about time derivation, and (50), (51), the substitution of (53) formula can be obtained:
By theorem 1 it is found that working as t >=T0When,By theorem 2 it is found that working as t >=T2 When,Wherein：T2≥T0.So, (55) formula can be written as:
By finite time T2Afterwards, (49) formula can be reduced to:
Wherein:
Solve (50) Shi Ke get:
Therefore, there is (58) formula it is found that as t >=T3When, e3≡ 0, e4≡ 0, wherein:
This shows the tracking error e of trolley3In limited time T3Inside converge to 0.
Remarks 1：To inhibit and eliminating load pivot angle, desired trolley track xdIt is selected as
Wherein,For target position；For trolley maximum permissible acceleration and speed；It indicates Adjust the parameter of initial acceleration；The control gain that κ > 1.0754 is positive.
The desired target trajectory of trolley (being indicated by (60) formula) consists of two parts：
(i) reference locus x is positionedd1：Drive trolley to target position；
(ii) disappear and put part xd2：Quickly eliminate the positioning performance that hunting of load has no effect on trolley.
3. numerical simulation
For the correctness and validity for verifying proposed control method, following two groups of emulation experiments are carried out.For in detail, In first group of emulation experiment, by comparing LQR controller, enhance coupling nonlinear controller, PD control device, verifying proposes control The superiority of method control performance.In first group of group experiment, due to LQR controller, enhance coupling nonlinear controller, PD control What device processed proposed in the case where being all based on precise kinetic, so h and f are set as 0.Verifying is proposed control by second group of emulation experiment Method processed is directed to and does not know dynamic (dynamical) robustness, and compares with the adaptive Gaussian filtering device based on motion planning.
LQR controller, enhancing coupling nonlinear controller, PD control device and the adaptive tracing control based on motion planning The expression formula of device processed is as follows：
1) LQR controller
Wherein,To control gain.
2) enhance coupling nonlinear controller
Wherein,The control gain being positive, ξxFor following auxiliary function：
3) PD control device
Wherein, kp,kd∈R+The control gain being positive.
4) the adaptive Gaussian filtering device based on motion planning
Wherein,The control gain being positive, r=x-xd1For trolley tracking error,For the online of parameter vector Estimation, is generated by following turnover rate：
Wherein, Γ is that positive definite symmetrically diagonally updates gain matrix.
Emulation 1：The verifying of control performance under accurate model parameter：In the experiment of this group, the actual value of crane system parameter Be with nominal value it is identical, be set as：
M=7kg, mp=1kg, l=0.6m, h=f=0
Frictional force has following form：
Trolley target position is：
pd=1m
The parameters of desired trolley track (60) are set as：
ka=0.5, kv=0.5, ε=2, κ=4
Controller designed by the present invention, LQR controller, the control gain for enhancing coupling nonlinear controller, PD control device It is shown in Table 1.
1 control gain of the emulation of table 1.
Controller designed by the present invention, LQR controller, the emulation knot for enhancing coupling nonlinear controller and PD control device Fruit is as shown in Figure 2-5.By comparison diagram 2 and Fig. 3-5 it is found that under similar haulage time (in 5s), mentioned control method Maximum load pivot angle and driving force are the smallest in these four control methods.These results show that mentioned control method control The superiority of performance.
Emulation 2：The verifying of control performance under uncertain kinetics function：In the experiment of this group, the name of crane system parameter Adopted value is set as：
M=12kg, mp=9kg, l=0.7m
The actual value of trolley quality, load quality and lifting rope length is respectively：14kg,10kg,1.0m.So, following formula It can obtain：
Δ M=2kg, Δ mp=1kg, Δ l=0.3m
The nominal value of frictional force is identical with emulation 1, and actual value is：
To simulate external disturbance, by sinusoidal perturbation d1And random perturbation d2It is applied in crane system, amplitude is 10。
The target position of trolley is set as：
pd=1m
It is identical in the parameter and emulation 1 of expectation target track (60).
The control gain of controller designed by the present invention and the adaptive controller based on motion planning is shown in Table 2.
2 control gain of the emulation of table 2
Fig. 6-7 show mentioned control method and the self-adaptation control method based on motion planning and there is uncertain power Simulation result.By Fig. 6-7 it is found that uncertain dynamics influences less the tracing control performance of proposed control method.So And when there is uncertain dynamics, the control performance of the self-adaptation control method based on motion planning is had a greatly reduced quality.By Fig. 6 It is found that the curve of the load pivot angle of estimation is almost identical as the load actual curve of pivot angle, this shows for load pivot angle design Terminal sliding mode observer correctness.These advantages bring convenience for the practical application of the mentioned control method of the present invention.
Above-mentioned, although the foregoing specific embodiments of the present invention is described with reference to the accompanying drawings, not protects model to the present invention The limitation enclosed, those skilled in the art should understand that, based on the technical solutions of the present invention, those skilled in the art are not Need to make the creative labor the various modifications or changes that can be made still within protection scope of the present invention.

Claims (10)

1. one kind has and does not know dynamic (dynamical) crane finite time contrail tracker, characterized in that including：
Design first terminal sliding mode observer estimates load pivot angle θ；Second terminal sliding mode observer is designed to uncertain Dynamics h is estimated；The finite time contrail tracker of non-loaded pivot angle feedback is designed according to obtained estimated value It is as follows：
Wherein, F is the resultant force being applied on trolley, k04∈R+The control gain being positive；It is missed for the estimation of machine speed Difference,For crane displacement first derivative,The estimated value of first derivative is displaced for crane,Estimating for second dervative is displaced for crane Evaluation；p3,q3∈R+The odd number being positive, and have p3< q3；e3=xd- x is the tracking error of trolley, xdFor the target trajectory of trolley； M is trolley quality, mpFor load quality；For γ2Estimated value, γ2=h；frxFor frictional force, l is lifting rope length；It is negative The estimated value of pivot angle is carried,For load pivot angle first derivative estimated value,For the estimated value for loading pivot angle second dervative.
2. a kind of have as described in claim 1 does not know dynamic (dynamical) crane finite time contrail tracker, special Sign is that the first terminal sliding mode observer is specially：
Wherein, auxiliary function is defined For the estimated value of p, peFor observation error,
k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1；peFor observation error, b1For normal number.
3. a kind of have as described in claim 1 does not know dynamic (dynamical) crane finite time contrail tracker, special Sign is that the second terminal sliding mode observer is specially：
Wherein, defined variable：γ2=h；
Introducing state：
About time derivation：
Then：U is uncertain derivative of the dynamic h about the time；
AndRespectively γ1And γ2Estimation, k02∈R+The observation gain being positive, γv=l1sgn(e1), l1,l2,l3,l4,l5∈R+The observation gain being positive, q2And p2The odd number being positive, and q2> p2；
Wherein, k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1, peIt (0) is auxiliary function p negative Carry observation error when pivot angle is zero.
4. one kind has and does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, characterized in that including：
(1) the first uncertain dynamics f of hypothesis is bounded about the derivative of time, and the second uncertain dynamics h is about the time Derivative bounded, load pivot angle θ andInitial estimation it is identical as actual value；
(2) define about load pivot angle θ auxiliary function p andSolve the observation error p of auxiliary function pe, missed according to observation Poor peFirst terminal sliding mode observer is designed, load pivot angle θ is estimated；So thatIn finite time ToIt is interior accurately to converge to P, andIn finite time ToIt is interior accurately to converge to load pivot angle θ；WhereinFor the estimated value of auxiliary function p,To load pivot angle θ Estimated value；
(3) define about the second uncertain dynamics h auxiliary function Q andSecond terminal sliding mode observer is designed, is being had Uncertain dynamics h is accurately estimated in the time of limit；
(4) according to the estimated value of the obtained load uncertain dynamics h of pivot angle θ and first, non-loaded pivot angle feedback is obtained Finite time contrail tracker；
(5) by actually detected trolley displacement x, machine speedBe input to it is above-mentioned with do not know dynamic (dynamical) crane it is limited when Between in contrail tracker, the resultant force F being applied on trolley is exported, in system trolley, load quality, lifting rope length, friction Force parameter it is uncertain and there are external disturbance in the case where can be realized in finite time trolley accurate positioning and Suspension hook swings, loads the effective inhibition and elimination swung around suspension hook.
5. a kind of have as claimed in claim 4 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, in the step (2), the auxiliary function p about load pivot angle θ is specially:
Wherein, g is acceleration of gravity, and l is lifting rope length,For the second dervative for loading pivot angle.
6. a kind of have as claimed in claim 5 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, in the step (2), first terminal sliding mode observer is specially：
Wherein,k01∈R+The observation gain being positive, p1,q1∈R+For positive odd number, and there is p1< q1；b1For normal number.
7. a kind of have as claimed in claim 4 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, in the step (3), the auxiliary function Q about uncertain dynamics h is specially；
Q is obtained about time derivation：
Wherein, frxFor frictional force, l is lifting rope length；For load pivot angle estimated value,For estimating for load pivot angle first derivative Evaluation,For the estimated value for loading pivot angle second dervative；F is the resultant force being applied on trolley, and M is trolley quality, mpBe negative charge material Amount.
8. a kind of have as claimed in claim 7 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, second terminal sliding mode observer is specially：
Wherein, defined variable：γ2=h；
Introducing state：
Then：U is uncertain derivative of the dynamic h about the time；
AndRespectively γ1And γ2Estimation, k02∈R+The observation gain being positive, γv=l1sgn(e1), l1,l2,l3,l4,l5∈R+The observation gain being positive, q2And p2The odd number being positive, and q2> p2。
9. a kind of have as claimed in claim 4 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, in the step (4), the finite time contrail tracker of non-loaded pivot angle feedback is specially：
Wherein, F is the resultant force being applied on trolley, k04∈R+The control gain being positive；It is missed for the estimation of machine speed Difference,For crane displacement first derivative,The estimated value of first derivative is displaced for crane,Estimating for second dervative is displaced for crane Evaluation；p3,q3∈R+The odd number being positive, and have p3< q3；e3=xd- x is the tracking error of trolley, xdFor the target trajectory of trolley； M is trolley quality, mpFor load quality；For γ2Estimated value, γ2=h；frxFor frictional force, l is lifting rope length；It is negative The estimated value of pivot angle is carried,For load pivot angle first derivative estimated value,For the estimated value for loading pivot angle second dervative.
10. a kind of have as claimed in claim 4 does not know dynamic (dynamical) crane finite time Trajectory Tracking Control method, It is characterized in, to inhibit and eliminating load pivot angle, desired trolley track is selected as：
Wherein,For target position；ka,For trolley maximum permissible acceleration and speed；It indicates to adjust The parameter of initial acceleration；The control gain that κ > 1.0754 is positive.
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