CN110376902B - Design method of servo constraint tracking controller of under-actuated mechanical system - Google Patents

Design method of servo constraint tracking controller of under-actuated mechanical system Download PDF

Info

Publication number
CN110376902B
CN110376902B CN201910764399.3A CN201910764399A CN110376902B CN 110376902 B CN110376902 B CN 110376902B CN 201910764399 A CN201910764399 A CN 201910764399A CN 110376902 B CN110376902 B CN 110376902B
Authority
CN
China
Prior art keywords
under
constraint
matrix
actuated
servo
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201910764399.3A
Other languages
Chinese (zh)
Other versions
CN110376902A (en
Inventor
孙浩
杨路文
朱梓诚
王冕昊
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Hefei University of Technology
Original Assignee
Hefei University of Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Hefei University of Technology filed Critical Hefei University of Technology
Priority to CN201910764399.3A priority Critical patent/CN110376902B/en
Publication of CN110376902A publication Critical patent/CN110376902A/en
Application granted granted Critical
Publication of CN110376902B publication Critical patent/CN110376902B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Landscapes

  • Engineering & Computer Science (AREA)
  • Health & Medical Sciences (AREA)
  • Artificial Intelligence (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Evolutionary Computation (AREA)
  • Medical Informatics (AREA)
  • Software Systems (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention provides a design method of an under-actuated mechanical system servo constraint tracking controller, which comprises the following steps: constructing a dynamic model of the under-actuated mechanical system with parameter uncertainty, and effectively decomposing the uncertainty in the system; regarding the tracking performance requirement to be realized by the system as a servo constraint, and constructing a second-order form of the servo constraint; designing an adaptive robust controller based on the constructed dynamic model and second-order servo constraint; performing stability analysis on the constructed adaptive robust controller; and adjusting main parameters in the constructed adaptive robust controller, and analyzing the control effect. The design of the under-actuated mechanical system constraint tracking controller can effectively process the influence of uncertainty of system parameters and external interference, and simultaneously, the system can quickly and stably track the given tracking performance requirement accurately.

Description

Design method of servo constraint tracking controller of under-actuated mechanical system
Technical Field
The invention relates to the field of mechanical system dynamics control, in particular to a design method of an under-actuated mechanical system servo constraint tracking controller.
Background
The under-actuated system is a nonlinear system with the number of independent control variables of the system smaller than the number of degrees of freedom of the system, and is superior to a completely-actuated system in the aspects of saving energy, reducing manufacturing cost, lightening weight, enhancing system flexibility and the like. The under-actuated system has a simple structure and is convenient for integral dynamics analysis and test. Meanwhile, due to the reasons of high nonlinearity of the system, parameter perturbation, multi-target control requirements, limited control quantity and the like, the under-actuated system is complex enough, and is convenient for researching and verifying the effectiveness of various algorithms. Bridge cranes, inverted pendulum systems, vertical take-off and landing aircraft, flexible manipulators, and the like are typical under-actuated systems.
In contrast, the difficulty of controlling an under-actuated system is much higher than that of a fully actuated system, and theoretically, the system is expected to be fully actuated, but in practice, in many cases, due to physical and economic reasons, many systems cannot achieve full actuation, and meanwhile, due to the fact that the number of control variables of the under-actuated system is small, the under-actuated system has the advantages of low cost and manufacturing cost, high flexibility, weight reduction, energy consumption reduction and the like in control, so that the under-actuated system is required to be used in many cases. In addition, some fully-actuated systems automatically switch to under-actuated systems when they fail. In a certain sense, the under-actuated system can be regarded as the fault-tolerant situation of the full-actuated system, and has application value in practical application, both from the aspect of cost and the aspect of result. Therefore, research into under-actuated systems is necessary.
For the control problem of the under-actuated mechanical system, researchers at home and abroad successively develop various control methods, and obtain certain effects, which are typically PID control, linear feedback control, LQR (linear quadratic regulator) control, robust control and the like. The system is easy to vibrate under nonlinear disturbance by PID control and linear feedback control, the robustness of LQR control is not strong, and overshoot and unnecessary control consumption are easy to generate by robust control.
Therefore, it is highly desirable to provide a control scheme with high stability and robustness.
Disclosure of Invention
Aiming at the technical problems, the invention provides a design method of an under-actuated mechanical system servo constraint tracking controller, which can effectively process the influence of uncertainty of system parameters and external interference, and simultaneously enables the system to quickly, stably and accurately control torque output.
The technical scheme adopted by the invention is as follows:
a design method of an under-actuated mechanical system servo constraint tracking controller comprises the following steps:
constructing an under-actuated mechanical system dynamics model containing parameter uncertainty, and effectively decomposing the uncertainty in the system;
the tracking performance requirement to be realized by the under-actuated mechanical system is regarded as servo constraint, and the second-order expression form of the servo constraint is obtained by derivation of the constraint;
a certain assumption requirement is put forward aiming at the characteristics of an under-actuated mechanical system and the servo constraint characteristics;
designing a self-adaptive robust constraint tracking controller based on the established dynamic model of the under-actuated system, second-order servo constraint and the provided hypothesis requirement, wherein the self-adaptive law can be adjusted in real time according to the tracking error;
performing stability analysis on the constructed adaptive robust controller;
and adjusting main parameters in the constructed adaptive robust controller, and analyzing the control effect.
Optionally, the constructing a dynamic model of the under-actuated system with parameter uncertainty includes:
constructing a dynamic model of the under-actuated mechanical system shown in the following equation (1):
Figure GDA0003506467940000021
wherein t is time, q is the generalized coordinates of the system,
Figure GDA0003506467940000022
in order to be a broad-sense speed of the system,
Figure GDA0003506467940000023
the method comprises the following steps of taking generalized acceleration of a system, delta as an uncertainty parameter of the system, M as an inertia matrix of the system, C as a centrifugal force/Coriolis force matrix of the system, G as a gravity term matrix of the system, tau as control input of the system, and B as a control input matrix of the system;
decomposing an uncertainty matrix in the constructed dynamics model of the under-actuated system according to the following equations (2) to (5):
Figure GDA0003506467940000024
Figure GDA0003506467940000025
Figure GDA0003506467940000031
Figure GDA0003506467940000032
wherein the content of the first and second substances,
Figure GDA0003506467940000033
the deterministic portions of the inertia matrix, the Coriolis force/centrifugal force matrix, the gravity matrix and the control input matrix of the under-actuated system are Delta M (q, delta, t),
Figure GDA0003506467940000034
Δ G (q, δ, t), Δ B (q, δ, t) are the uncertainty parts of the inertia matrix, coriolis force/centrifugal force matrix, gravity matrix, control input matrix of the under-actuated system.
Order to
Figure GDA0003506467940000035
Figure GDA0003506467940000036
Then
Δχ(q，δ，t)＝χ(q，t)Φ(q，δ，t) (6)
Based on matrix
Figure GDA0003506467940000037
We decompose Φ into two parts:
Figure GDA0003506467940000038
wherein
Figure GDA0003506467940000039
And
Figure GDA00035064679400000310
the matrices can be selected as:
Figure GDA00035064679400000311
Figure GDA00035064679400000312
for the same reason, based on matrices
Figure GDA00035064679400000313
We decompose the matrices Δ C, Δ G, Δ B into two parts:
Figure GDA00035064679400000314
Figure GDA00035064679400000315
Figure GDA00035064679400000316
wherein
Figure GDA00035064679400000317
And
Figure GDA00035064679400000318
and
Figure GDA00035064679400000319
and
Figure GDA00035064679400000320
the matrices can be selected as:
Figure GDA00035064679400000321
Figure GDA00035064679400000322
Figure GDA00035064679400000323
Figure GDA00035064679400000324
Figure GDA00035064679400000325
Figure GDA00035064679400000326
optionally, regarding the tracking performance requirement to be achieved by the under-actuated mechanical system as a servo constraint, deriving the constraint to obtain a second-order expression form of the servo constraint:
the target performance requirement of the under-actuated system is written in the form shown in equation (19) below:
Figure GDA0003506467940000041
the following equations (20) and (21) are obtained by suitably sorting and deriving equation (19):
Figure GDA0003506467940000042
Figure GDA0003506467940000043
wherein A is a constraint matrix; c is a first order constraint vector; b is a second order constraint vector.
Optionally, a certain assumption requirement is proposed for the under-actuated mechanical system characteristic and the servo constraint characteristic, and specifically includes:
1) the servo constraint equation is solvable: equation of
Figure GDA0003506467940000044
Are consistent.
2) Controllability of the under-actuated system: equation of
Figure GDA0003506467940000045
Is in agreement, wherein
Figure GDA0003506467940000046
Figure GDA0003506467940000047
3) Matrix array
Figure GDA0003506467940000048
Is reversible.
4) Order to
Figure GDA0003506467940000049
Then there is a constant pΨ>-1 such that:
Figure GDA00035064679400000410
5) for a given constant positive definite matrix Q, there are constantsλ>0 is such that
Figure GDA00035064679400000411
6) There is a vector η sum function
Figure GDA00035064679400000415
So that
Figure GDA00035064679400000412
Simultaneous letterNumber of
Figure GDA00035064679400000413
Can be linearly decomposed into:
Figure GDA00035064679400000414
optionally, designing an adaptive robust constraint tracking controller based on the constructed under-actuated system dynamic model and second-order servo constraint and the proposed hypothesis requirement, wherein adaptive parameters can be adjusted in real time according to tracking errors;
constructing a controller shown in the following equation (26) based on the constructed dynamic model and the second-order constraint form:
Figure GDA0003506467940000051
wherein the content of the first and second substances,
Figure GDA0003506467940000052
Figure GDA0003506467940000053
Figure GDA0003506467940000054
wherein the content of the first and second substances,
Figure GDA0003506467940000055
Figure GDA0003506467940000056
Figure GDA0003506467940000057
p3is used for solving the problem that the system has uncertainty, the theta function is the upper bound of the uncertainty of the system,
Figure GDA0003506467940000058
and e is a control precision adjusting parameter for the adaptive parameter.
The adaptive parameter
Figure GDA0003506467940000059
The adaptation law is determined as shown by equation (30) below:
Figure GDA00035064679400000510
therein, ζ0,ζ1,ζ2Parameters are adjusted for the adaptive law.
Optionally, the performing the stability analysis on the constructed adaptive robust controller includes:
the final stable bound of the constructed adaptive robust controller is analyzed using the lyapunov function as shown in equation (31) below:
Figure GDA00035064679400000511
where Q is a positive definite matrix, ρΨIs an arbitrary constant greater than-1, and η is an upper bound parameter of uncertainty for an under-actuated mechanical system.
Optionally, the analyzing the final stable boundary of the constructed adaptive law robust controller by using the lyapunov function shown in equation (31) specifically includes:
calculation of equation (31) yields the following formula (32):
Figure GDA0003506467940000061
wherein the content of the first and second substances,
Figure GDA0003506467940000062
ζ 1＝min{2κλ(1+ρΨ)，2ζ0 -1ζ2(1+ρΨ)}，ζ 2＝2ζ0 -1(1+ρΨ)(ζ1+ζ2)η，ζ 3＝2(1+ρΨ)∈；
and obtaining a balance parameter R of the final consistent and stable limit of the under-actuated mechanical system based on the formula (32), as shown in the following formula (33):
Figure GDA0003506467940000063
and (3) obtaining the final consistent stable limit of the active anti-roll system based on the formula (32), as shown in the following formula (34):
Figure GDA0003506467940000064
wherein the content of the first and second substances,da lower limit value representing the size of the final stable limit of the under-actuated mechanical system,
Figure GDA0003506467940000065
Figure GDA0003506467940000071
λmin(Q) represents the minimum eigenvalue, λ, of the positive definite matrix Qmax(Q) represents the maximum eigenvalue of the positive definite matrix Q;
according to the lyapunov stability theory, the time for the active anti-roll system to reach the final stable limit can be obtained, as shown in the following formula (35):
Figure GDA0003506467940000072
wherein T represents the time for the under-actuated mechanical system to reach the final consistent stable limit, r represents the initial state of the system,
Figure GDA0003506467940000073
is arbitrarily greater thandPositive number of (c).
Optionally, the adjusting main parameters in the constructed adaptive robust controller, and analyzing the control effect includes:
adjusting initial condition incompatible compensation parameters, adaptive law adjustment parameters and control precision adjustment parameters in the constructed adaptive law robust controller;
and analyzing whether the constraint tracking error of the system meets the preset error requirement or not based on the adjusted parameters.
The invention has the beneficial effects that:
the design method of the servo constraint tracking controller of the under-actuated mechanical system comprises the following steps of firstly constructing a dynamic model of the under-actuated mechanical system with parameter uncertainty, and effectively decomposing the uncertainty in the system; secondly, regarding the tracking performance requirement to be realized by the system as servo constraint, and constructing a second-order form of the servo constraint; then, designing an adaptive robust controller based on the constructed dynamic model and second-order servo constraint; finally, performing stability analysis on the constructed adaptive robust controller; and adjusting main parameters in the constructed adaptive robust controller, and analyzing the control effect. The design of the under-actuated mechanical system constraint tracking controller can effectively process the influence of uncertainty of system parameters and external interference, and simultaneously, the system can quickly and stably track the given tracking performance requirement accurately.
Drawings
Fig. 1 is a schematic flow chart of a design method of an under-actuated mechanical system servo constraint tracking controller according to an embodiment of the present invention;
fig. 2 is a schematic overall structure diagram of a controller according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the overall structure of a trolley-swing link system provided in the embodiment of the present invention;
fig. 4 is a schematic diagram illustrating a simulation of the stability of the trolley-swing link system according to the embodiment of the present invention.
Detailed Description
In order to make the technical problems, technical solutions and advantages of the present invention more apparent, the following detailed description is given with reference to the accompanying drawings and specific embodiments.
Fig. 1 is a schematic flow chart of a design method of an under-actuated mechanical system servo constraint tracking controller according to an embodiment of the present invention. As shown in fig. 1, a method for designing an under-actuated mechanical system servo constraint tracking controller according to an embodiment of the present invention includes the following steps:
s101, constructing a dynamic model of the under-actuated mechanical system with parameter uncertainty;
s102, analyzing uncertainty parameters in the system and performing effective decomposition;
s103, providing a target performance constraint requirement of the under-actuated system, and constructing a second-order constraint form of the servo constraint;
s104, aiming at the system characteristics and the servo constraint characteristics, providing certain assumption requirements;
s105, designing a self-adaptive robust controller based on the constructed dynamic model and the second-order constraint form, wherein self-adaptive parameters can be adjusted in real time according to tracking errors;
s106, performing stability analysis on the constructed adaptive robust controller;
and S107, adjusting main parameters in the constructed adaptive robust controller, and analyzing the control effect.
In the present invention, the dynamic model constructed in step S101 is the basis for the model-based adaptive robust controller design in step S105; step S102, analyzing uncertainty parameters in the system and performing effective decomposition is to design a model-based adaptive robust controller to process a reference of uncertainty in step S105; the second-order form of the performance constraint constructed in step S103 is the final control target for the model-based adaptive robust controller design in step S105; step S104 is to make a certain request for the mechanical system characteristics and the servo constraint characteristics involved in step S101, step S102 and step S103; s106, analyzing the stability of the designed adaptive robust controller to ensure that the controller can realize the final stability of the system; step S107 is to analyze the influence of the main parameters in the adaptive robust controller designed in step S105 on the control effect.
Specifically, in step S101, a dynamic model of the under-actuated mechanical system shown in the following equation (1) is constructed:
Figure GDA0003506467940000091
wherein t is time, q is the generalized coordinates of the system,
Figure GDA0003506467940000092
is the generalized speed of the system and,
Figure GDA0003506467940000093
the generalized acceleration of the system is shown, delta is an uncertainty parameter of the system, M is an inertia matrix of the system, C is a centrifugal force/Coriolis force matrix of the system, G is a gravity term matrix of the system, tau is a control input of the system, and B is a control input matrix of the system.
Further, step S102 decomposes the uncertainty matrix in the constructed dynamics model of the under-actuated system according to the following equations (2) to (5):
Figure GDA0003506467940000094
Figure GDA0003506467940000095
Figure GDA0003506467940000096
Figure GDA0003506467940000097
wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0003506467940000098
the deterministic portions of the inertia matrix, the Coriolis force/centrifugal force matrix, the gravity matrix and the control input matrix of the under-actuated system are Delta M (q, delta, t),
Figure GDA0003506467940000099
Δ G (q, δ, t), Δ B (q, δ, t) are the uncertainty parts of the inertia matrix, coriolis force/centrifugal force matrix, gravity matrix, control input matrix of the under-actuated system.
Order to
Figure GDA00035064679400000910
Figure GDA00035064679400000911
Then
Δχ(q，δ，t)＝χ(q，t)Φ(q，δ，t) (6)
Based on matrix
Figure GDA00035064679400000912
We decompose Φ into two parts:
Figure GDA00035064679400000913
wherein
Figure GDA00035064679400000914
And
Figure GDA00035064679400000915
the matrices can be selected as:
Figure GDA00035064679400000916
Figure GDA00035064679400000917
for the same reason, based on matrices
Figure GDA00035064679400000918
We decompose the matrices Δ C, Δ G, Δ B into two parts:
Figure GDA0003506467940000101
Figure GDA0003506467940000102
Figure GDA0003506467940000103
wherein
Figure GDA0003506467940000104
And
Figure GDA0003506467940000105
and
Figure GDA0003506467940000106
and
Figure GDA0003506467940000107
the matrices can be selected as:
Figure GDA0003506467940000108
Figure GDA0003506467940000109
Figure GDA00035064679400001010
Figure GDA00035064679400001011
Figure GDA00035064679400001012
Figure GDA00035064679400001013
further, in step S103, the tracking performance requirement to be achieved by the under-actuated mechanical system is regarded as a servo constraint, and the constraint is derived to obtain a second-order expression form of the servo constraint:
the target performance requirement of the under-actuated system is written in the form shown in equation (19) below:
Figure GDA00035064679400001014
the following equations (20) and (21) are obtained by suitably sorting and deriving equation (19):
Figure GDA00035064679400001015
Figure GDA00035064679400001016
wherein A is a constraint matrix; c is a first order constraint vector; b is a second order constraint vector.
Further, in step S104, a certain assumption requirement is proposed for the under-actuated mechanical system characteristic and the servo constraint characteristic, specifically including:
1) the servo constraint equation is solvable: equation of
Figure GDA00035064679400001017
Are consistent.
2) Controllability of the under-actuated system: equation of
Figure GDA00035064679400001018
Is in agreement, wherein
Figure GDA00035064679400001019
Figure GDA00035064679400001020
3) Matrix of
Figure GDA00035064679400001021
Is reversible.
4) Order to
Figure GDA0003506467940000111
Then there is a constant pΨ>-1 such that:
Figure GDA0003506467940000112
5) for a given constant positive definite matrix Q, there are constantsλ>0 is such that
Figure GDA0003506467940000113
6) There is a vector η sum function
Figure GDA0003506467940000114
So that
Figure GDA0003506467940000115
Simultaneous function
Figure GDA0003506467940000116
Can be linearly decomposed into:
Figure GDA0003506467940000117
further, in step S105, based on the constructed under-actuated system dynamic model and second-order servo constraint and the proposed assumption requirement, a self-adaptive robust constraint tracking controller is designed, wherein self-adaptive parameters can be adjusted in real time according to tracking errors;
constructing a controller shown in the following equation (26) based on the constructed dynamic model and the second-order constraint form:
Figure GDA0003506467940000118
wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0003506467940000119
Figure GDA00035064679400001110
Figure GDA00035064679400001111
wherein the content of the first and second substances,
Figure GDA0003506467940000121
Figure GDA0003506467940000122
Figure GDA0003506467940000123
p3is used for solving the problem that the system has uncertainty, the theta function is the upper bound of the uncertainty of the system,
Figure GDA0003506467940000124
and e is a control precision adjusting parameter for the adaptive parameter.
The adaptive parameter
Figure GDA0003506467940000125
The adaptation law is determined as shown by equation (30) below:
Figure GDA0003506467940000126
therein, ζ0，ζ1，ζ2Parameters are adjusted for the adaptive law.
Further, the performing the stability analysis on the constructed adaptive robust controller in step S106 includes:
the final stable bound of the constructed adaptive robust controller is analyzed using the lyapunov function as shown in equation (31) below:
Figure GDA0003506467940000127
where Q is a positive definite matrix, ρΨIs an arbitrary constant greater than-1, and η is an upper bound parameter of uncertainty for an under-actuated mechanical system.
Calculation of equation (31) yields the following formula (32):
Figure GDA0003506467940000128
wherein the content of the first and second substances,
Figure GDA0003506467940000129
ζ 1＝min{2κλ(1+ρΨ)，2ζ0 -1ζ2(1+ρΨ)}，ζ2＝2ζ0 -1(1+ρΨ)(ζ1+ζ2)η，ζ 3＝2(1+ρΨ)∈；
and obtaining a balance parameter R of the final consistent and stable limit of the under-actuated mechanical system based on the formula (32), as shown in the following formula (33):
Figure GDA0003506467940000131
and (3) obtaining the final consistent stable limit of the active anti-roll system based on the formula (32), as shown in the following formula (34):
Figure GDA0003506467940000132
wherein the content of the first and second substances,da lower limit value representing the size of the final stable limit of the under-actuated mechanical system,
Figure GDA0003506467940000133
Figure GDA0003506467940000134
λmin(Q) represents the minimum eigenvalue, λ, of the positive definite matrix Qmax(Q) represents the maximum eigenvalue of the positive definite matrix Q;
according to the lyapunov stability theory, the time for the active anti-roll system to reach the final stable limit can be obtained, as shown in the following formula (35):
Figure GDA0003506467940000135
wherein T represents an under-actuated mechanical system toFor a final consistent and stable bound, r represents the initial state of the system,
Figure GDA0003506467940000136
is arbitrarily greater thandPositive number of (c).
Further, the step S107 adjusts main parameters in the constructed adaptive robust controller, and analyzing the control effect includes:
an initial condition incompatibility compensation parameter kappa and an adaptive law adjustment parameter zeta in the constructed adaptive law robust controller0、ζ1、ζ2Carrying out adjustment; and analyzing whether the constraint tracking error of the system meets the preset error requirement or not based on the adjusted parameters. Specifically, Matlab software can be used for performing performance simulation of the control system, whether the constrained tracking error of the system meets the preset error requirement is analyzed, if the constrained tracking error of the system meets the preset error requirement, the process is ended, and if the constrained tracking error of the system does not meet the preset error requirement, the parameters are continuously adjusted until the preset error requirement is met.
In the embodiment of the invention, the parameter kappa is related to the incompatibility problem of the compensation initial conditions, and the larger the value of kappa is, the better the effect is and the larger the corresponding control cost is; parameter ζ0、ζ1、ζ2In relation to the compensation uncertainty, the influence of the adjustment of these three parameters on the control performance of the system and the influence of the control cost are a comprehensive adjustment process. Namely, the larger the value of the initial condition incompatible compensation parameter is, the better the control effect is, and the larger the corresponding control cost is; adaptive control parameter ζ0、ζ1、ζ2The value of (A) is a comprehensive consideration, and needs to be comprehensively adjusted by combining the control effect and the control cost. The specific values of the parameters can be determined by a designer according to the actual control precision of the system. In one exemplary embodiment (trolley-swing-lever system), the better control parameter values may be: k 5, ζ0＝1,ζ1＝1,ζ2＝1。
Fig. 2 shows the overall structure of the controller of the present invention. As shown in FIG. 2, first, the nominal control of the system is written from the kinetic equations, target constraints, and assumptionsSystem p1Then, a controller p is proposed to compensate the initial condition incompatibility problem according to the error of the system2And then a controller p for compensating the uncertainty of the system is provided according to a self-adaptive law designed3。
Fig. 3 shows a trolley swing link system provided by the embodiment of the invention. As shown in fig. 3, the swing link with length l is mounted on the trolley, and the trolley moves along the direction of the horizontal plane x under the action of the external force F.
FIG. 4 shows a graph at p1、p2And p3Under the cooperative control of the three-dimensional simulation model, the position of the swing rod follows the simulation structure schematic diagram of the expected required track. The figure shows that the force output by the controller can make the position of the swing link quickly and stably follow the expected track under the condition of uncertainty of the system. Through simulation results, the method of the invention can be found to be accurate and smooth, and the effectiveness and superiority of the design method of the invention are proved.
The above-mentioned embodiments are only specific embodiments of the present invention, which are used for illustrating the technical solutions of the present invention and not for limiting the same, and the protection scope of the present invention is not limited thereto, although the present invention is described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: any person skilled in the art can modify or easily conceive the technical solutions described in the foregoing embodiments or equivalent substitutes for some technical features within the technical scope of the present disclosure; such modifications, changes or substitutions do not depart from the spirit and scope of the embodiments of the present invention, and they should be construed as being included therein. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (4)

1. A design method of an under-actuated mechanical system servo constraint tracking controller is characterized by comprising the following steps:
constructing an under-actuated mechanical system dynamics model containing parameter uncertainty, and effectively decomposing the uncertainty in the system;
the tracking performance requirement to be realized by the under-actuated mechanical system is regarded as servo constraint, and the second-order expression form of the servo constraint is obtained by derivation of the constraint;
a certain assumption requirement is put forward aiming at the characteristics of an under-actuated mechanical system and the servo constraint characteristics;
designing a self-adaptive robust constraint tracking controller based on the established dynamic model and second-order servo constraint of the under-actuated system and the provided hypothesis requirement, wherein self-adaptive parameters can be adjusted in real time according to tracking errors;
performing stability analysis on the constructed adaptive robust controller;
adjusting main parameters in the constructed adaptive robust controller, and analyzing a control effect;
the method for constructing the dynamic model of the under-actuated system with parameter uncertainty comprises the following steps:
constructing a dynamic model of the under-actuated mechanical system shown in the following equation (1):
Figure FDA0003506467930000011
wherein t is time, q is the generalized coordinates of the system,
Figure FDA0003506467930000012
in order to be a broad-sense speed of the system,
Figure FDA0003506467930000013
the method comprises the following steps of taking generalized acceleration of a system, delta as an uncertainty parameter of the system, M as an inertia matrix of the system, C as a centrifugal force/Coriolis force matrix of the system, G as a gravity term matrix of the system, tau as control input of the system, and B as a control input matrix of the system;
decomposing an uncertainty matrix in the constructed dynamics model of the under-actuated system according to the following equations (2) to (5):
Figure FDA0003506467930000014
Figure FDA0003506467930000015
Figure FDA0003506467930000016
Figure FDA0003506467930000017
wherein the content of the first and second substances,
Figure FDA0003506467930000018
the deterministic portions of the inertia matrix, the Coriolis force/centrifugal force matrix, the gravity matrix and the control input matrix of the under-actuated system are Delta M (q, delta, t),
Figure FDA0003506467930000021
Δ G (q, δ, t), Δ B (q, δ, t) are the uncertainty parts of the inertia matrix, coriolis force/centrifugal force matrix, gravity matrix, control input matrix of the under-actuated system;
order to
Figure FDA0003506467930000022
Figure FDA0003506467930000023
Then
Δχ(q，δ，t)＝χ(q，t)Φ(q，δ，t) (6)
Based on matrix
Figure FDA0003506467930000024
We decompose Φ into two parts:
Figure FDA0003506467930000025
wherein
Figure FDA0003506467930000026
And
Figure FDA0003506467930000027
the matrices can be selected as:
Figure FDA0003506467930000028
Figure FDA0003506467930000029
for the same reason, based on matrices
Figure FDA00035064679300000210
We decompose the matrices Δ C, Δ G, Δ B into two parts:
Figure FDA00035064679300000211
Figure FDA00035064679300000212
Figure FDA00035064679300000213
wherein
Figure FDA00035064679300000214
And
Figure FDA00035064679300000215
and
Figure FDA00035064679300000216
and
Figure FDA00035064679300000217
the matrices can be selected as:
Figure FDA00035064679300000218
Figure FDA00035064679300000219
Figure FDA00035064679300000220
Figure FDA00035064679300000221
Figure FDA00035064679300000222
Figure FDA00035064679300000223
the tracking performance requirement to be realized by the under-actuated mechanical system is regarded as servo constraint, and the second-order expression form of the servo constraint is obtained by derivation of the constraint:
the target performance requirement of the under-actuated system is written in the form shown in equation (19) below:
Figure FDA0003506467930000031
the following equations (20) and (21) are obtained by suitably sorting and deriving equation (19):
Figure FDA0003506467930000032
Figure FDA0003506467930000033
wherein A is a constraint matrix; c is a first order constraint vector; b is a second order constraint vector;
the method specifically comprises the following steps of aiming at the characteristics of an under-actuated mechanical system and the servo constraint characteristics, providing certain assumed requirements, specifically comprising the following steps:
1) the servo constraint equation is solvable: equation of
Figure FDA0003506467930000034
Are consistent;
2) controllability of the under-actuated system: equation of
Figure FDA0003506467930000035
Is in agreement, wherein
Figure FDA0003506467930000036
Figure FDA0003506467930000037
3) Matrix array
Figure FDA0003506467930000038
Is reversible;
4) order to
Figure FDA0003506467930000039
Then there is a constant pΨ>-1 such that:
Figure FDA00035064679300000310
5) for a given constant positive definite matrix Q, there are constantsλ>0 is such that
Figure FDA00035064679300000311
6) There is a vector η sum function
Figure FDA00035064679300000312
So that
Figure FDA00035064679300000313
Simultaneous function
Figure FDA00035064679300000314
Can be linearly decomposed into:
Figure FDA00035064679300000315
designing a self-adaptive robust constraint tracking controller based on the established dynamic model of the under-actuated system, second-order servo constraint and the provided hypothesis requirement, wherein the self-adaptive law can be adjusted in real time according to the tracking error;
constructing a controller shown in the following equation (26) based on the constructed dynamic model and the second order constraint form:
Figure FDA0003506467930000041
wherein the content of the first and second substances,
Figure FDA0003506467930000042
Figure FDA0003506467930000043
Figure FDA0003506467930000044
wherein the content of the first and second substances,
Figure FDA0003506467930000045
Figure FDA0003506467930000046
Figure FDA0003506467930000047
p3is used for solving the problem that the system has uncertainty, the theta function is the upper bound of the uncertainty of the system,
Figure FDA0003506467930000048
is a self-adaptive parameter, and belongs to a control precision adjusting parameter;
the adaptive parameter
Figure FDA0003506467930000049
The adaptation law is determined as shown by equation (30) below:
Figure FDA00035064679300000410
therein, ζ0,ζ1,ζ2Parameters are adjusted for the adaptive law.
2. The design method of the servo constraint tracking controller of the under-actuated mechanical system according to claim 1, wherein the performing the stability analysis on the constructed adaptive law robust controller comprises:
analyzing the final stable limit of the constructed adaptive law-robust controller by using a Lyapunov function shown in the following equation (31):
Figure FDA00035064679300000411
where Q is a positive definite matrix, ρΨIs any constant greater than-1, and η is an upper bound parameter of uncertainty for an under-actuated mechanical system.
3. The design method of the servo constraint tracking controller of the under-actuated mechanical system according to claim 2, wherein the analyzing the final stable bound of the constructed adaptive robust controller by using the lyapunov function shown in equation (31) specifically comprises:
calculation of equation (31) yields the following formula (32):
Figure FDA0003506467930000051
wherein the content of the first and second substances,
Figure FDA0003506467930000052
Figure FDA0003506467930000053
and obtaining a balance parameter R of the final consistent and stable limit of the under-actuated mechanical system based on the formula (32), as shown in the following formula (33):
Figure FDA0003506467930000054
and (3) obtaining the final consistent stable limit of the active anti-roll system based on the formula (32), as shown in the following formula (34):
Figure FDA0003506467930000055
wherein the content of the first and second substances,da lower limit value representing the size of the final stable limit of the under-actuated mechanical system,
Figure FDA0003506467930000056
Figure FDA0003506467930000057
λmin(Q) represents the minimum eigenvalue, λ, of the positive definite matrix Qmax(Q) represents the maximum eigenvalue of the positive definite matrix Q;
according to the lyapunov stability theory, the time for the active anti-roll system to reach the final stable limit can be obtained, as shown in the following formula (35):
Figure FDA0003506467930000061
wherein T represents the time for the under-actuated mechanical system to reach the final consistent stable limit, r represents the initial state of the system,
Figure FDA0003506467930000062
is arbitrarily greater thandA positive number of (c).
4. The design method of the servo constraint tracking controller of the under-actuated mechanical system according to claim 1, wherein the adjusting the main parameters of the constructed adaptive robust controller and the analyzing the control effect comprise:
adjusting initial condition incompatible compensation parameters, adaptive law adjustment parameters and control precision adjustment parameters in the constructed adaptive law robust controller;
and analyzing whether the constraint tracking error of the system meets the preset error requirement or not based on the adjusted parameters.
CN201910764399.3A 2019-08-19 2019-08-19 Design method of servo constraint tracking controller of under-actuated mechanical system Active CN110376902B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201910764399.3A CN110376902B (en) 2019-08-19 2019-08-19 Design method of servo constraint tracking controller of under-actuated mechanical system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201910764399.3A CN110376902B (en) 2019-08-19 2019-08-19 Design method of servo constraint tracking controller of under-actuated mechanical system

Publications (2)

Publication Number Publication Date
CN110376902A CN110376902A (en) 2019-10-25
CN110376902B true CN110376902B (en) 2022-05-06

Family

ID=68259718

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201910764399.3A Active CN110376902B (en) 2019-08-19 2019-08-19 Design method of servo constraint tracking controller of under-actuated mechanical system

Country Status (1)

Country Link
CN (1) CN110376902B (en)

Families Citing this family (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113219823B (en) * 2021-01-22 2022-11-04 南京理工大学 Mechanical system tracking-escape-arrival point control method based on constraint tracking
CN113419433B (en) * 2021-07-23 2022-07-05 合肥中科深谷科技发展有限公司 Design method of tracking controller of under-actuated system of self-balancing electric wheelchair
CN115047769B (en) * 2022-06-29 2023-10-31 南京理工大学 Unmanned combat platform obstacle avoidance-arrival control method based on constraint following
CN115237164B (en) * 2022-08-12 2024-01-23 南京理工大学 Constraint following-based two-degree-of-freedom cradle head stability control method and system
CN116009594B (en) * 2023-03-27 2023-06-02 合肥工业大学 Hypersonic aircraft state constraint fault tolerance control method based on zero and game

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103645638A (en) * 2013-12-09 2014-03-19 天津工业大学 Design method for robustness controller of under-actuated vehicle
CN108942928A (en) * 2018-07-06 2018-12-07 合肥工业大学 One kind being based on the servo-controlled drive lacking flexible mechanical arm system of restraining force robust
CN108983606A (en) * 2018-07-09 2018-12-11 南京理工大学 A kind of Sliding mode self-adaptation control method of mechanical arm system
CN109129464A (en) * 2018-07-06 2019-01-04 合肥工业大学 The restraining force robust method of servo-controlling and device of drive lacking flexible mechanical arm system
CN109732614A (en) * 2019-03-19 2019-05-10 合肥工业大学 The control design case method and device thereof and controller of SCARA robot based on restraining force

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103645638A (en) * 2013-12-09 2014-03-19 天津工业大学 Design method for robustness controller of under-actuated vehicle
CN108942928A (en) * 2018-07-06 2018-12-07 合肥工业大学 One kind being based on the servo-controlled drive lacking flexible mechanical arm system of restraining force robust
CN109129464A (en) * 2018-07-06 2019-01-04 合肥工业大学 The restraining force robust method of servo-controlling and device of drive lacking flexible mechanical arm system
CN108983606A (en) * 2018-07-09 2018-12-11 南京理工大学 A kind of Sliding mode self-adaptation control method of mechanical arm system
CN109732614A (en) * 2019-03-19 2019-05-10 合肥工业大学 The control design case method and device thereof and controller of SCARA robot based on restraining force

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
双臂空间机器人的自适应鲁棒性联合控制;郭益深 等;《系统仿真学报》;20090228;第21卷(第3期);第625-633页 *

Also Published As

Publication number Publication date
CN110376902A (en) 2019-10-25

Similar Documents

Publication Publication Date Title
CN110376902B (en) Design method of servo constraint tracking controller of under-actuated mechanical system
CN106873624B (en) Four-rotor unmanned aerial vehicle suspension flight control method based on partial feedback linearization
Seifried Two approaches for feedforward control and optimal design of underactuated multibody systems
Lee et al. Multi-input noncertainty-equivalent adaptive control of an aeroelastic system
CN108052117A (en) Flight control method is hung based on Partial feedback linearization quadrotor unmanned plane
Seifried Integrated mechanical and control design of underactuated multibody systems
Khatri et al. Aircraft maneuver design using bifurcation analysis and sliding mode control techniques
Deshmukh et al. Bridging the gap between open-loop and closed-loop control in co-design: A framework for complete optimal plant and control architecture design
Lee et al. Robust translational force control of multi-rotor UAV for precise acceleration tracking
CN111007724A (en) Hypersonic aircraft designated performance quantitative tracking control method based on interval II type fuzzy neural network
Lee et al. Linear parameter-varying control of variable span-sweep morphing aircraft
Zhao et al. A differential-flatness-based approach for autonomous helicopter shipboard landing
Wang et al. Intelligent control of air-breathing hypersonic vehicles subject to path and angle-of-attack constraints
Cao et al. Tracking control of transition window for aerospace vehicles based on robust preview control
Zhao et al. Manipulator trajectory tracking based on adaptive sliding mode control
Yuan et al. Sliding mode observer controller design for a two dimensional aeroelastic system with gust load
Akmeliawati et al. Nonlinear energy-based control method for aircraft automatic landing systems
Schlagenhauf et al. Cascaded nonlinear mpc for realtime quadrotor position tracking
Bouzid et al. 3D trajectory tracking control of quadrotor UAV with on-line disturbance compensation
CN110109357B (en) Semi-global self-adaptive control method for non-standard type non-linear aircraft
Di et al. Low-level control with actuator dynamics for multirotor UAVs
He et al. Disturbance compensation based piecewise linear control design for perching maneuvers
Mavrommati et al. Real-time dynamic-mode scheduling using single-integration hybrid optimization
Ahmadian et al. A novel L1 adaptive-hybrid control with guaranteed stability for a class of uncertain nonlinear systems: A case study on SA330 Puma
Zhang et al. Co-regulation of computational and physical effectors in a quadrotor unmanned aircraft system

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant