CN114890314A - Fault-tolerant control method for double-pendulum tower crane with online track correction - Google Patents
Fault-tolerant control method for double-pendulum tower crane with online track correction 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
- B66C13/22—Control systems or devices for electric drives
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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
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B66—HOISTING; LIFTING; HAULING
- B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
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Abstract
The invention discloses a fault-tolerant control method of a double-pendulum tower crane with online track correction, which comprises the following steps: establishing a double-pendulum tower crane model based on a Lagrange kinetic equation; linearly analyzing an under-actuated part of the double-pendulum tower crane model to establish an online correction track; coupling the original S-shaped track with the online correction track to restrain load swing; the invention can solve the problem of crane positioning anti-swing under complex working conditions; under-actuated parts are separated from a double-pendulum tower crane model in a normal state and analyzed, an online correction track with adjustable gain is designed through the internal coupling relation, and the online correction track is coupled to the original S-shaped track to inhibit swinging; a nonlinear observer is arranged in a fault state, possible disturbance and fault of the system are observed, and feedforward compensation is carried out in a fault-tolerant controller; and adding a self-adaptive sign function to be activated to inhibit observation errors and further improve the system stability.
Description
Technical Field
The invention relates to the technical field of motion control of double-pendulum tower cranes, in particular to a fault-tolerant control method of a double-pendulum tower crane with online track correction.
Background
Under-actuated systems, i.e., systems in which the system inputs fewer degrees of freedom than the system. The conventional under-actuated crane positioning and anti-swing control is mainly aimed at a bridge crane system, and even if the bridge crane moves in a three-dimensional space and has multiple degrees of freedom, the dynamic property of a driving mechanism still belongs to linear force, the dynamic characteristic is still simple, and the control is convenient. The crane system is a typical under-actuated system, takes cantilever rotating force and trolley running force as input, takes the angle of a load and a lifting hook as indirect control force, and has the advantages of simple structure, low power consumption, few actuating mechanisms, wide application range and the like.
The tower crane is a crane for transporting goods in space, and the transportation process of the tower crane is usually accompanied by two motions with different properties, wherein one direction is the translation force of a trolley, the other direction is the rotation force of a cantilever, and high coupling exists, so that the difficulty of designing a dynamic model and a controller is increased. Secondly, when the hook and the load mass are similar, or the lifting rope and the suspension rope are similar in length, the double-pendulum characteristic of the tower crane is obvious.
In addition, compared to a simple mass point double pendulum system, due to the different load volumes, the distributed mass load may rotate and translate during the swinging process, so that the controller designed for mass distributed loads with single or double pendulum may be disabled. For conventional controllers, on the one hand, they have poor coupling between the drivable and the non-drivable mechanisms, resulting in that usually only positioning can be achieved, but the wobble suppression effect is poor; on the other hand, most controllers use an adjustment control mode for a target position due to a complicated design process, but the adjustment control can generate a very large initial output value of the controller in practical application, so that inevitable initial fluctuation is caused, the service life of a driver is damaged, and the anti-oscillation effect is influenced.
Moreover, in long-term operation, the misoperation of workers can also cause the failure of the actuator and the inevitable external disturbance. The rapid suppression of the swinging of the hook and the load becomes a very challenging problem while achieving the accurate positioning of the cantilever and the trolley.
Disclosure of Invention
This section is for the purpose of summarizing some aspects of embodiments of the invention and to briefly introduce some preferred embodiments. In this section, as well as in the abstract and the title of the invention of this application, simplifications or omissions may be made to avoid obscuring the purpose of the section, the abstract and the title, and such simplifications or omissions are not intended to limit the scope of the invention.
In order to solve the above technical problems, the present invention provides the following technical solutions, including: establishing a double-pendulum tower crane model based on a Lagrange kinetic equation; linearly analyzing an under-actuated part of the double-pendulum tower crane model to establish an online correction track; and coupling the original S-shaped track with the online correction track to restrain the load swing.
As a preferred embodiment of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction, the model of the double-pendulum tower crane includes:
G(q)＝[0 0(m 1 +m 2 )gl 1 C 2 S 1 (m 1 +m 2 )gl 1 C 1 S 2 m 2 gl 2 C 4 S 3 m 2 gl 2 C 3 S 4 ] T
wherein ,
i＝1，…，4
wherein q is a system state variable of the double-pendulum tower crane,
is the first derivative of q and is,
is the second derivative of q, M (q) is the inertia matrix of the double pendulum tower crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, U is a control input vector, F s and Fa Mechanical friction and wind resistance, m, of double pendulum tower crane systems 1 And m 2 Mass of hook and load, respectively, /) 1 And l 2 The lengths from the suspension rope and the lifting hook to the center of mass of the load respectively, g is the gravity acceleration,
is cantilever rotation angle, xFor the trolley translation distance, theta i Is the swing angle between the lifting hook and the load,
for cantilever drive torque, F x Is the driving force of the trolley,
and
for the fault parameters of the potential fault signal in the boom and trolley directions,
k hx and μx In order to be a parameter of a fault,
and Fx Is used as the driving force of the system,
and
is a fault signal.
As a preferable scheme of the fault-tolerant control method for the double-pendulum tower crane with online trajectory correction, a friction feedforward compensation model is established to eliminate the friction generated by the driving mechanism of the double-pendulum tower crane model:
the mechanical and air friction of the cantilever is:
the mechanical friction and air friction of the trolley are as follows:
wherein ,
f x1 、f x2 、ε 1 and ε2 For the parameters of the friction feed-forward compensation model,
and fx1 The value of (c) corresponds to the maximum static friction force, ε 1 and ε2 In order to obtain a static coefficient of friction,
and dx Is an air friction parameter.
As a preferable aspect of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the under-actuated part of the model of the double-pendulum tower crane includes:
wherein ,xd Indicating the target position,/ b Indicating the length of the beam.
As a preferred embodiment of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the original S-shaped trajectory includes:
wherein ,
indicating the angle of rotation of the cantilever
Or the trolley is translated over a distance x,
and
respectively the target angle or position, the initial angle or position and the arrival time of the boom and the trolley,
to represent
Or x target position.
As a preferable scheme of the fault-tolerant control method for the double-pendulum tower crane with online track correction, the online track correction includes:
wherein ：
and xr Is the original S-shaped track, k 5 and k6 Is an adjustable parameter.
As a preferable solution of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the double-pendulum tower crane model further includes:
establishing a nonlinear observer to observe internal disturbance and faults of the double-pendulum tower crane model;
and establishing a fault-tolerant controller, and performing feedforward compensation on the disturbance and the fault observed by the nonlinear observer to the fault-tolerant controller so as to improve the stability of the double-pendulum tower crane model.
As a preferable solution of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the nonlinear observer includes:
wherein ,
e 1 ＝S 1_i -q u
e 2 ＝S 2_i -Γ
wherein ,S2_i For post-model-conversion disturbance and fault integration terms, q u The system state quantity after model conversion, upsilon, is the relevant item of model conversion,
for the output of the system after model transformation, the disturbance change related term after transformation of the gamma model, k 1 、k 2 、
Is an adjustable parameter.
As a preferable solution of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the fault-tolerant controller includes a linear sliding mode surface:
wherein ,
e x ＝x-x n
wherein ,
λ x 、
k px 、k dx and px In order to be able to adjust the gain,
and
in order to observe the parameters of the fault,
and kx0 In order to be able to adjust the gain,
and
is an adaptive gain.
As a preferable solution of the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the present invention, the fault-tolerant controller further includes:
the above-mentioned
And
for the adaptive sign function to be activated, observation errors can be suppressed.
The invention has the beneficial effects that: the invention can solve the problem of crane positioning anti-swing under complex working conditions such as actuator failure, system uncertainty, friction and external disturbance. Under a normal state, an under-actuated part is separated from a double-pendulum tower crane model and analyzed, a Lyapunov function is designed, an online correction track with adjustable gain is designed through an internal coupling relation, and the online correction track is coupled to an original S-shaped track, so that swinging can be effectively inhibited. Under the fault state, such as actuator fault, system uncertainty, friction and external disturbance state, a non-linear observer is designed for observing possible disturbance and fault of the system and performing feedforward compensation in a fault-tolerant controller. Further considering the possible error of the observation system, the adaptive sign function to be activated is designed to effectively restrain the observation error and improve the operation stability of the whole system.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive exercise. Wherein:
FIG. 1 is a schematic overall control flow diagram of a fault-tolerant control method for a double-pendulum tower crane with online trajectory modification according to a first embodiment of the present invention;
FIG. 2 is a schematic structural diagram of a mathematical model of a double-pendulum tower crane with an online trajectory correction fault-tolerant control method for the double-pendulum tower crane according to a first embodiment of the present invention;
fig. 3 is a schematic diagram of an experimental result of a controller LQR in a normal state of a fault-tolerant control method for a double-pendulum tower crane with online track correction according to a first embodiment of the present invention;
fig. 4 is a schematic diagram of an experimental result of a controller LQR in a fault state of a fault-tolerant control method for a double-pendulum tower crane with online track correction according to a first embodiment of the present invention;
FIG. 5 is a diagram illustrating experimental results of a controller in a normal state according to a fault-tolerant control method for a double-pendulum tower crane with online trajectory modification according to a first embodiment of the present invention;
FIG. 6 is a schematic diagram showing the experimental results of a controller in a fault state according to the fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to the first embodiment of the present invention;
FIG. 7 is a schematic diagram illustrating experimental results of observation error suppression of a controller in a fault state according to a fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to a first embodiment of the present invention;
fig. 8 is a schematic diagram of an experimental platform of a fault-tolerant control method for a double-pendulum tower crane with online trajectory correction according to a first embodiment of the present invention.
Detailed Description
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, specific embodiments accompanied with figures are described in detail below, and it is apparent that the described embodiments are a part of the embodiments of the present invention, not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making creative efforts based on the embodiments of the present invention, shall fall within the protection scope of the present invention.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways than those specifically described and will be readily apparent to those of ordinary skill in the art without departing from the spirit of the present invention, and therefore the present invention is not limited to the specific embodiments disclosed below.
Furthermore, reference herein to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one implementation of the invention. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments.
The present invention will be described in detail with reference to the drawings, wherein the cross-sectional views illustrating the structure of the device are not enlarged partially in general scale for convenience of illustration, and the drawings are only exemplary and should not be construed as limiting the scope of the present invention. In addition, the three-dimensional dimensions of length, width and depth should be included in the actual fabrication.
Meanwhile, in the description of the present invention, it should be noted that the terms "upper, lower, inner and outer" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, and are only for convenience of describing the present invention and simplifying the description, but do not indicate or imply that the referred device or element must have a specific orientation, be constructed in a specific orientation and operate, and thus, cannot be construed as limiting the present invention. Furthermore, the terms first, second, or third are used for descriptive purposes only and are not to be construed as indicating or implying relative importance.
The terms "mounted, connected and connected" in the present invention are to be understood broadly, unless otherwise explicitly specified or limited, for example: can be fixedly connected, detachably connected or integrally connected; they may be mechanically, electrically, or directly connected, or indirectly connected through intervening media, or may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
Example 1
Referring to fig. 1 to 2, a first embodiment of the present invention provides a fault-tolerant control method for a double-pendulum tower crane with online trajectory correction, including:
s1: and establishing a double-pendulum tower crane model based on a Lagrange kinetic equation.
The double-pendulum tower crane model comprises:
G(q)＝[0 0(m 1 +m 2 )gl 1 C 2 S 1 (m 1 +m 2 )gl 1 C 1 S 2 m 2 gl 2 C 4 S 3 m 2 gl 2 C 3 S 4 ] T
wherein ,
i＝1,…,4
wherein q is a double-pendulum tower crane model state variable,
is the first derivative of q and is,
is the second derivative of q, M (q) is the inertia matrix of the double pendulum tower crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, U is a control input vector, F s and Fa Mechanical friction and wind resistance, m, of double pendulum tower crane systems 1 And m 2 Mass of hook and load, respectively, /) 1 And l 2 The lengths from the suspension rope and the lifting hook to the center of mass of the load respectively, g is the gravity acceleration,
is cantilever rotation angle, and x is trolley translation distanceAway, theta i Is the swing angle between the lifting hook and the load,
for cantilever drive torque, F x Is the driving force of the trolley,
and
for the fault parameters of the potential fault signal in the boom and trolley directions,
k hx and μx In order to be a parameter of a fault,
and Fx Is used as the driving force of the system,
and
is a fault signal.
Specifically, the inertia matrix and the partition matrix of the double-pendulum tower crane model are as follows:
m 11 ＝J+m 1 x 2 +m 2 x 2 +m t x 2 +l 1 2 m 1 +l 1 2 m 2 +l 2 2 m 2 -l 1 2 m 1 C 1 2 C 2 2 -l 1 2 m 2 C 1 2 C 2 2 -l 2 2 m 2 C 3 2 C 4 2 +2l 1 m 1 xC 2 S 1 +2l 1 m 2 xC 2 S 1 +2l 2 m 2 xC 4 S 3 +2l 1 l 2 m 2 S 2 S 4 +2l 1 l 2 m 2 C 2 C 4 S 1 S 3
m 12 ＝l 1 m 1 S 2 +l 1 m 2 S 2 +l 2 m 2 S 4
m 13 ＝l 1 2 m 1 C 1 C 2 S 2 +l 1 2 m 2 C 1 C 2 S 2 +l 1 l 2 m 2 C 1 C 2 S 4
m 14 ＝-l 1 2 m 1 S 1 -l 1 2 m 2 S 1 -l 1 m 1 xC 2 -l 1 m 2 xC 2 -l 1 l 2 m 2 C 2 C 4 S 3 -l 1 l 2 m 2 S 1 S 2 S 4
m 15 ＝l 2 2 m 2 C 3 C 4 S 4 +l 1 l 2 m 2 C 3 C 4 S 2
m 16 ＝-l 2 2 m 2 S 3 -l 2 m 2 xC 4 -l 1 l 2 m 2 S 2 S 3 S 4 -l 1 l 2 m 2 C 2 C 4 S 1
m 21 ＝l 1 m 1 S 2 +l 1 m 2 S 2 +l 2 m 2 S 4
m 22 ＝m 1 +m 2 +m t
m 23 ＝l 1 m 1 C 1 C 2 +l 1 m 2 C 1 C 2
m 24 ＝-l 1 m 1 S 1 S 2 -l 1 m 2 S 1 S 2
m 25 ＝l 2 m 2 C 3 C 4
m 26 ＝-l 2 m 2 S 3 S 4
m 31 ＝l 1 2 m 1 C 1 C 2 S 2 +l 1 2 m 2 C 1 C 2 S 2 +l 1 l 2 m 2 C 1 C 2 S 4
m 32 ＝l 1 m 1 C 1 C 2 +l 1 m 2 C 1 C 2
m 33 ＝l 1 2 m 1 C 2 2 +l 1 2 m 2 C 2 2
m 34 ＝0
m 35 ＝l 1 l 2 m 2 C 2 C 4 S 1 S 3 +l 1 l 2 m 2 C 1 C 2 C 3 C 4
m 36 ＝-l 1 l 2 m 2 C 1 C 2 S 3 S 4 +l 1 l 2 m 2 C 2 C 3 S 1 S 4
m 41 ＝-l 1 2 m 1 S 1 -l 1 2 m 2 S 1 -l 1 m 1 xC 2 -l 1 m 2 xC 2 -l 1 l 2 m 2 C 2 C 4 S 3 -l 1 l 2 m 2 S 1 S 2 S 4
m 42 ＝-l 1 m 1 S 1 S 2 -l 1 m 2 S 1 S 2
m 43 ＝0
m 44 ＝l 1 2 m 1 +l 1 2 m 2
m 45 ＝l 1 l 2 m 2 C 1 C 4 S 2 S 3 -l 1 l 2 m 2 C 3 C 4 S 1 S 2
m 46 ＝l 1 l 2 m 2 C 2 C 4 +l 1 l 2 m 2 C 1 C 3 S 2 S 4 +l 1 l 2 m 2 S 1 S 2 S 3 S 4
m 51 ＝l 2 2 m 2 C 3 C 4 S 4 +l 1 l 2 m 2 C 3 C 4 S 2
m 52 ＝l 2 m 2 C 3 C 4
m 53 ＝l 1 l 2 m 2 C 1 C 2 C 3 C 4 +l 1 l 2 m 2 C 2 C 4 S 1 S 3
m 54 ＝l 1 l 2 m 2 C 1 C 4 S 2 S 3 -l 1 l 2 m 2 C 3 C 4 S 1 S 2
m 55 ＝l b 2 m 2 /12+l 2 2 m 2 C 4 2
m 56 ＝0
m 61 ＝-l 2 m 2 xC 4 -l 2 2 m 2 S 3 -l 1 l 2 m 2 C 2 C 4 S 1 -l 1 l 2 m 2 S 2 S 3 S 4
m 62 ＝-l 2 m 2 S 3 S 4
m 63 ＝-l 1 l 2 m 2 C 1 C 2 S 3 S 4 +l 1 l 2 m 2 C 2 C 3 S 1 S 4
m 64 ＝l 1 l 2 m 2 C 2 C 4 +l 1 l 2 m 2 C 1 C 3 S 2 S 4 +l 1 l 2 m 2 S 1 S 2 S 3 S 4
m 65 ＝0
m 66 ＝m 2 l 2 2
wherein ,mij Indicating matrix coordinates, i 1, 2 … 6, j 1, 2 … 6.
Centripetal coriolis matrix
The following were used:
c 22 ＝0
c 32 ＝0
c 44 ＝0
c 52 ＝0
c 66 ＝0
wherein cij Indicating matrix coordinates, i 1, 2 … 6, j 1, 2 … 6.
Establishing a friction feedforward compensation model to eliminate the friction generated by a driving mechanism of the double-pendulum tower crane model:
the mechanical and air friction of the cantilever is:
the mechanical friction and air friction of the trolley are as follows:
wherein ,
f x1 、f x2 、ε 1 and ε2 For the parameters of the friction feed-forward compensation model,
and fx1 The value of (c) corresponds to the maximum static friction force, ε 1 and ε2 In order to obtain a static coefficient of friction,
and dx Is an air friction parameter.
It should be noted that the swing characteristic of the double-pendulum system and the rod translation characteristic of the load of the double-pendulum tower crane model are analyzed to find the underactuated portion of the double-pendulum tower crane model.
The under-actuated portion of the double-pendulum tower crane model includes:
wherein ,xd Indicating the target position,/ b Indicating the length of the beam.
It should be noted that the original system is divided into a driving part and an under-driving part as follows:
wherein the original M (q) is,
G(q)、U、F s 、F a And q is divided in the form as above, namely: m 11 ∈R 2 ×2 ，M 12 ∈R 2×4 ，M 21 ∈R 4×2 ，M 22 ∈R 4×4 ，C 11 ∈R 2×2 ，C 12 ∈R 2×4 ，C 21 ∈R 4×2 ，C 22 ∈R 4×4 ，G 1 ∈R 2 ，G 2 ∈R 4 ，U 1 ∈R 2 ，U 2 ∈R 4 ，F s1 ∈R 2 ，F s2 ∈R 4 ，F 1 ∈R 2 ，F 2 ∈R 4 ，
q b ＝[θ 1 θ 2 θ 3 θ 4 ] T 。
Then consider the under-driven part | M 22 | ≠ 0, which can be rewritten as:
after bringing it into the drive section:
wherein ,
and because of
Reissue to order
It can be found that:
and establishing a non-linear observer to observe internal disturbance and faults of the double-pendulum tower crane model.
The non-linear observer includes:
wherein ,
e 1 ＝S 1_i -q u
e 2 ＝S 2_i -Γ
wherein ,S2_i For post-model-conversion disturbance and fault integration terms, q u The system state quantity after model conversion, upsilon, is the relevant item of model conversion,
for the output of the system after model transformation, the disturbance change related term after transformation of the gamma model, k 1 、k 2 、
Is an adjustable parameter.
And establishing a fault-tolerant controller, and performing feedforward compensation on the disturbance and the fault observed by the nonlinear observer to the fault-tolerant controller so as to improve the stability of the double-pendulum tower crane model.
It should be noted that the fault tolerant controller is designed to linearize the model driven part for the observed errors as follows:
wherein ,J1 Is moment of inertia, m 0 As trolley mass, x n Target position and x after track correction d And (5) the consistency is achieved.
The above equation is transformed:
wherein ,
the error system is derived as follows:
wherein ,
|H x |≤P x
the fault tolerant controller comprises a linear sliding mode surface:
wherein ,
e x ＝x-x n
wherein ,
λ x 、
k px 、k dx and px In order to be able to adjust the gain,
and
in order to observe the parameters of the fault,
and kx0 In order to be able to adjust the gain,
and
is an adaptive gain.
It should be noted that in the case of actuator faults and external disturbances, a non-linear observer is first designed to observe the internal disturbances of the system and faults, and then a fault-tolerant controller is used for suppression. The nonlinear observer only needs to observe the fault and disturbance of the original system, and the fault-tolerant controller is conveniently and pertinently designed. In addition, considering the possible observation error of the observation system, the adaptive sign function to be activated is designed to inhibit the observation error, so that the stability of the system is further improved, and the system can still ensure stable operation even if the observer has an error.
S2: and linearly analyzing an under-actuated part of the double-pendulum tower crane model to establish an online correction track.
The original S-shaped trajectory includes:
wherein ,
indicating the angle of rotation of the cantilever
Or the trolley is translated over a distance x,
and
respectively the target angle or position, the initial angle or position and the arrival time of the boom and the trolley,
to represent
Or x target position.
It should be noted that the original S-type track is derived from the existing paper Aneficient online track generating method for indirect create systems.
The on-line track correction comprises the following steps:
wherein ：
and xr Is the original S-shaped track, k 5 and k6 Is an adjustable parameter.
Specifically, the following lyapunov equation is designed based on the under-actuated part function of the double-pendulum tower crane model following the dynamics rule of the tower crane model:
wherein ,k3 and k4 Is a positive parameter.
Using the basic inequality, one can derive:
wherein ,
can obtain V 1 And (4) positive determination.
P-Lyapunov equation V 1 The derivation is carried out to obtain:
wherein ,
and then scaling to obtain:
multiplying the formulas in the under-actuated part by the formulas respectively
And
substituting the formula obtained by scaling to obtain:
can obtain the product
And
and embedding the second integral into the original S-shaped track to obtain an online correction track.
It should be noted that, the invention designs an online correction track aiming at the normal running condition, and restrains the swing of the crane in the positioning completion process; the method is easy to realize, and the load swing item with adjustable gain is coupled on the original S-shaped track.
S3: the original S-shaped track is coupled with the online correction track to restrain the load swing.
Example 2
Referring to fig. 3 to 7, in order to verify and explain the technical effects adopted in the method, the present embodiment performs a simulation experiment on the method to verify the real effects of the method.
An experiment platform is built in the embodiment and comprises a PC (personal computer), a control board card, a servo motor driver, a trolley, a cantilever, a swing angle measuring mechanism and the like. On the aspect of an upper computer, codes are generated through MATLAB/Simulink compiling, then a control board (DSP) is used, the sampling period is 0.005s, experimental data on the control board are monitored and recorded in real time through serial port communication, and position signals of a driving part come from counting of the encoder; the load/hook swing angle information comes from the contact type potentiometer sensor mechanism, the voltage signal of the contact type potentiometer sensor mechanism is transmitted to the control panel through the A/D converter, and for the output of the controller, the control panel generates a designed voltage signal to the motor driver through the D/A converter to drive the servo motor to operate.
Setting parameters:
k px ＝150，k dx ＝18，k x0 ＝4，p x ＝0.6，
λ x ＝1，k 1 ＝diag[1 1]，k 2 ＝diag[200 200]，k 3 ＝0.1，k 4 ＝0.1，k 5 ＝-0.6，k 6 ＝0.7，
f x1 ＝0.5，f x2 ＝0.5，ε 1 ＝0.001，ε 2 ＝0.001，
d x ＝0.6，
k hx ＝0.6，
μ x ＝0.1，
the controller LQR and the controller using the control method are used for carrying out experiments, and the control formula of the controller LQR is as follows:
for LQR controllers, the state vector
And Q matrix and R matrix are set to Q ═ diag {200,100,20,20,20,20,5,5,5,5, 5}, and R ═ 1,1] T Finally, the gain of the controller is k 11 ＝56.6，k 12 ＝14.5，k 13 ＝8.9，k 14 ＝3.4，k 15 ＝-2.5，k 16 ＝-1.1，k 21 ＝40.0，k 22 ＝9.2，k 23 ＝-19.1，k 24 ＝-0.9，k 25 ＝13.3，k 26 ＝0.80。
The amplitude of the method used by the method and the method used by the LQR controller are calculated by utilizing the experimental platform constructed as described above, and as can be seen by referring to FIGS. 3 and 5, under the condition that the positioning time is basically the same, the controller can completely track the target track and realize the positioning function, and the LQR controller is in the position
The direction can not realize the positioning, the tracking and positioning process of the method is smooth, the positioning task can be completed within 3 seconds, and the amplitude of a lifting hook and a load caused by a controller of the method is not large and can not exceed 1.15[ deg ] in the aspect of swing inhibition]And the amplitude of the hook and the load caused by the traditional LQR method controller is too large, and is not lower than 1[ deg ]]And up to approximately 1.7[ deg. ]]And the swing of the method can be completely eliminated within 2-4 seconds after the positioning of the driving mechanism is finished, but the traditional method has a poor suppression effect, and the swing still remains after the driving mechanism is subjected to repeated violent oscillation until 15 seconds, so that the swing suppression efficiency of the method is extremely high, the positioning is accurate, and no overshoot and no steady-state error exist. Referring to fig. 4 and 6, when a crane system fails, it can be seen that the LQR controller cannot overcome the system failure, which results in a large swing and fails to implement a positioning function. Finally, fig. 7 shows that when an error occurs in the observation system, the adaptive term is activated to suppress the disturbance and improve the overall stability of the system.
It should be noted that the above-mentioned embodiments are only for illustrating the technical solutions of the present invention and not for limiting, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, which should be covered by the claims of the present invention.
Claims (10)
1. A fault-tolerant control method for a double-pendulum tower crane with online track correction is characterized by comprising the following steps:
establishing a double-pendulum tower crane model based on a Lagrange kinetic equation;
linearly analyzing an under-actuated part of the double-pendulum tower crane model to establish an online correction track;
and coupling the original S-shaped track with the online correction track to restrain the load swing.
2. The fault-tolerant control method for the double-pendulum tower crane with the online track correction as claimed in claim 1, wherein the double-pendulum tower crane model comprises:
G(q)＝[0 0(m 1 +m 2 )gl 1 C 2 S 1 (m 1 +m 2 )gl 1 C 1 S 2 m 2 gl 2 C 4 S 3 m 2 gl 2 C 3 S 4 ] T
wherein ,
i＝1,…,4
F x a ＝k hx F x +μ x F x a
wherein q is a system state variable of the double-pendulum tower crane,
is the first derivative of q and is,
is the second derivative of q, M (q) is the inertia matrix of the double pendulum tower crane system,
is a centripetal-Coriolis matrix, G (q) is a gravity vector, U is a control input vector, F s and Fa Mechanical friction and wind resistance, m, of double pendulum tower crane systems 1 And m 2 Mass of hook and load, respectively, /) 1 And l 2 The lengths from the suspension rope and the lifting hook to the center of mass of the load respectively, g is the gravity acceleration,
is the cantilever rotation angle, x is the trolley translation distance, theta i Is the swing angle between the lifting hook and the load,
for cantilever drive torque, F x Is the driving force of the trolley,
and
for the fault parameters of the potential fault signal in the boom and trolley directions,
k hx and μx In order to be a parameter of a fault,
and Fx Is used as the driving force of the system,
and
is a fault signal.
3. The fault-tolerant control method for the double-pendulum tower crane with the online trajectory correction as claimed in claim 2, characterized in that a friction feedforward compensation model is established to eliminate the friction generated by the driving mechanism of the double-pendulum tower crane model:
the mechanical and air friction of the cantilever is:
the mechanical friction and air friction of the trolley are as follows:
wherein ,
f x1 、f x2 、ε 1 and ε2 For the parameters of the friction feed-forward compensation model,
and
the value of (c) corresponds to the maximum static friction force, ε 1 and ε2 In order to obtain a static coefficient of friction,
and dx Is an air friction parameter.
4. The fault-tolerant control method for the double-pendulum tower crane with the online trajectory correction as claimed in claim 2, wherein the under-actuated part of the model of the double-pendulum tower crane comprises:
wherein ,xd Indicating the target position,/ b Indicating the length of the beam.
5. The fault-tolerant control method for the double-pendulum tower crane with the online track correction as claimed in claim 1, wherein the original S-shaped track comprises:
wherein l represents the cantilever rotation angle
Or trolley translation distance x, l d 、l 0 and td Target angle or position, initial angle or position and arrival time, l, of the boom and trolley, respectively r To represent
Or x target position.
6. The fault-tolerant control method for the double-pendulum tower crane with the online track correction as claimed in claim 1, wherein the online track correction comprises:
wherein ：
and xr Is the original S-shaped track, k 5 and k6 Is an adjustable parameter.
7. The fault-tolerant control method for the double-pendulum tower crane with the online trajectory correction as claimed in claim 2, wherein the double-pendulum tower crane model further comprises:
establishing a nonlinear observer to observe internal disturbance and faults of the double-pendulum tower crane model;
and establishing a fault-tolerant controller, and performing feedforward compensation on the disturbance and the fault observed by the nonlinear observer to the fault-tolerant controller so as to improve the stability of the double-pendulum tower crane model.
8. The fault-tolerant control method for the double-pendulum tower crane with the online trajectory correction as claimed in claim 7, wherein the non-linear observer comprises:
wherein ,
e 1 ＝S 1_i -q u
e 2 ＝S 2_i -Γ
wherein ,S2_i For post-model-conversion disturbance and fault integration terms, q u The system state quantity after model conversion, upsilon, is the relevant item of model conversion,
for the output of the system after model transformation, the disturbance change related term after transformation of the gamma model, k 1 、k 2 、
Is an adjustable parameter.
9. The method for fault-tolerant control of a double-pendulum tower crane with online trajectory modification according to claim 7, wherein the fault-tolerant controller comprises a linear sliding mode surface:
wherein ,
e x ＝x-x x
wherein ,
λ x 、
k px 、k dx and px In order to be able to adjust the gain,
and
in order to observe the parameters of the fault,
and kx0 In order to be able to adjust the gain,
and
is an adaptive gain.
10. The method of claim 9, wherein the fault-tolerant controller further comprises:
the above-mentioned
And
for the adaptive sign function to be activated, observation errors can be suppressed.
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