CN112947079B - Trajectory tracking control method of cricket system - Google Patents

Trajectory tracking control method of cricket system Download PDF

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CN112947079B
CN112947079B CN202110154835.2A CN202110154835A CN112947079B CN 112947079 B CN112947079 B CN 112947079B CN 202110154835 A CN202110154835 A CN 202110154835A CN 112947079 B CN112947079 B CN 112947079B
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cricket
follows
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axis
order
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CN112947079A (en
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平兆武
李坤鹏
宋阳
黄云志
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Hefei University of Technology
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    • 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
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Abstract

The invention discloses a track tracking control method of a plate ball system, which comprises the following steps: step 1, establishing a nonlinear mathematical model of a cricket system; step 2, describing the track tracking control problem of the cricket system into a nonlinear servo control problem; and 3, designing a 3-order state feedback controller by adopting a 3-order polynomial approximation regulator equation solution based on a nonlinear output regulation theory. The invention designs the feedback controller based on the nonlinear mathematical model of the cricket system, solves the problem of track tracking control of the system under a time-varying reference signal and has higher-precision tracking performance.

Description

Trajectory tracking control method of cricket system
Technical Field
The invention relates to the field of nonlinear system control, in particular to a trajectory tracking control method of a plate ball system.
Background
In recent decades, nonlinear output regulation theory has gained wide attention from the control world, and has the significant advantage of being able to achieve various control targets such as trajectory tracking, interference suppression, robustness, and the like. Common control strategies include feedback control and internal model control. The cricket system is a multi-input multi-output system with nonlinearity, strong coupling, and multiple variables, and is often used to test various advanced control algorithms. The trajectory tracking control problem of a cricket system can be described as a non-linear servo control problem. Although some documents adopt advanced control algorithms such as sliding mode control and the like to solve the problem of trajectory tracking of the cricket system, the track tracking is often based on a simplified cricket system model, and a satisfactory control effect is difficult to achieve. How to design a feedback controller based on an unreduced nonlinear mathematical model to realize the trajectory tracking control of the cricket system under a time-varying reference signal is to be further researched.
Disclosure of Invention
Based on the technical problems in the background art, the invention provides a track tracking control method of a cricket system. Aiming at the cricket system, a 3-order polynomial approximation regulator equation solution is adopted based on a nonlinear output regulation theory, and a 3-order state feedback controller is designed to realize a track tracking target of the cricket system under a time-varying reference signal.
The technical scheme adopted by the invention is as follows:
a trajectory tracking control method of a cricket system comprises the following steps:
step 1, establishing a nonlinear mathematical model of a cricket system;
step 2, describing the track tracking control problem of the cricket system into a nonlinear servo control problem;
and 3, designing a 3-order state feedback controller by adopting a 3-order polynomial approximation regulator equation solution based on a nonlinear output regulation theory.
Further, the trajectory tracking control method of the cricket system is characterized in that in the step 1, assuming that the small balls are always in contact with the flat plate and the sliding friction force between the cricket is ignored, the mathematical model of the cricket system is described as follows:
Figure BDA0002932979900000021
wherein the content of the first and second substances,
Figure BDA0002932979900000022
respectively showing the displacement, speed and acceleration of the small ball on the flat plate in the directions of the x axis and the y axis,
Figure BDA0002932979900000023
respectively representing the deflection angle, deflection angular velocity and deflection angular acceleration of the flat plate in the directions of an x axis and a y axis, wherein m represents the mass of the small ball, (I)p,Ib) Respectively representing the moment of inertia of the plate and the ball (tau)x,τy) Respectively showing the moment applied to the flat plate in the x-axis direction and the moment applied to the flat plate in the y-axis direction, and g shows the gravity acceleration;
order to
Figure BDA0002932979900000024
The system (1) can be written as follows:
Figure BDA0002932979900000031
wherein
Figure BDA0002932979900000032
Figure BDA0002932979900000033
Figure BDA0002932979900000034
η3＝-mx1x5(-2mx4x5x6-mx2x4x5-mx1x4x6-mgx1cosx8)
η4＝-mx1x5(-2mx1x2x4-mx2x5x8-mx1x6x8-mgx1cosx3)
Figure BDA0002932979900000035
Further, in the trajectory tracking control method of the cricket system, in step 2, the trajectory tracking control problem of the cricket system is described as a non-linear servo control problem, and the process is as follows:
2.1, suppose cricketIn the system, the displacement reference tracks of the small ball in the directions of the x axis and the y axis are two sinusoidal signals yd1＝F1sin(ω1t+θ1),yd2＝F2sin(ω2t+θ2) This displacement reference trajectory may be generated by an external system of the form:
Figure BDA0002932979900000036
wherein v ═ v1,v2,v3,v4]TOutput is yd1＝v1,yd2＝v3；
Let lambdai(i ═ 1,2,3,4) is a matrix
Figure BDA0002932979900000037
Characteristic value of (a), easy to verifyiEach having zero real part and the equilibrium point v-0 of the external system (3) is lyapunov stable;
2.2, define the tracking error as
Figure BDA0002932979900000038
2.3, the cricket system (2), the external system (3) and the error function (4) can be written in a compact form as follows:
Figure BDA0002932979900000041
wherein F (x (t), u (t), v (t)) is described as follows:
Figure BDA0002932979900000042
at this time, the problem of tracking and controlling the small ball displacement trajectory of the cricket system has been described as a nonlinear servo control problem, and the control objective is to design the controller u so that the reference signal on the small ball displacement trajectory tracking and the steady-state tracking error are small enough on the premise of ensuring the stability of the closed-loop system.
Further, the trajectory tracking control method of the plate ball system is characterized in that in step 3, based on a nonlinear output regulation theory, a 3-order polynomial is adopted to approximate the solution of a regulator equation, and a 3-order state feedback controller is designed, wherein the process is as follows:
3.1, solving the Jacobian matrix of the cricket ball system as follows:
Figure BDA0002932979900000043
Figure BDA0002932979900000044
it can be proven by calculation that:
(1)
Figure BDA0002932979900000051
is calmable and there is a matrix K such that
Figure BDA0002932979900000052
All have negative real parts;
(2)
Figure BDA0002932979900000053
where n is the dimension 8 of the system state variable, p is the output dimension 2, λ ∈ { λ | λ ═ l1λ1+…+l4λ4,l1+…l4＝l,l1…l4＝0,1,2…l}，l＝1,2,3；
From 2.1 and 3.1 it is known that the 3 rd order nonlinear output tuning problem of the cricket system is solvable and that the solution of the tuner equation is unique because the input dimension is equal to the output dimension;
3.2, solving the regulator equation of the cricket system, wherein the process is as follows:
the cricket system regulator equation has the form:
Figure BDA0002932979900000054
wherein
Figure BDA0002932979900000055
Figure BDA0002932979900000056
η3(v)＝-mx1(v)x5(v)[-2mx4(v)x5(v)x6(v)-mx2(v)x4(v)x5(v)-mx1(v)x4(v)x6(v)-mgx1(v)cos x8(v)]
η4(v)＝-mx1(v)x5(v)[-2mx1(v)x2(v)x4(v)-mx2(v)x5(v)x8(v-mx1(v)x6(v)x8(v)-mgx1(v)cos x3(v)]
Figure BDA0002932979900000061
A partial solution of the regulator equation can be obtained by simple calculations as follows:
x1(v)＝v1
x2(v)＝ω1v2
x5(v)＝v3
x6(v)＝ω2v4
because the exact values of the other solutions to the regulator equation are not available, polynomial approximations are used to obtain their approximate solutions, and through mathematical calculations, the other 3 rd order approximate solutions can be obtained as follows:
Figure BDA0002932979900000062
Figure BDA0002932979900000063
Figure BDA0002932979900000064
Figure BDA0002932979900000065
Figure BDA0002932979900000066
Figure BDA0002932979900000067
here, the
Figure BDA0002932979900000068
Figure BDA0002932979900000069
Figure BDA00029329799000000610
Figure BDA00029329799000000611
Figure BDA00029329799000000612
Figure BDA00029329799000000613
Figure BDA00029329799000000614
Figure BDA00029329799000000615
d1000＝gm
Figure BDA00029329799000000616
Figure BDA00029329799000000617
Figure BDA00029329799000000618
Figure BDA00029329799000000619
Figure BDA0002932979900000071
Figure BDA0002932979900000072
3.3, designing a 3-stage state feedback controller as follows:
u＝y(3)(v)+Kx(x-x(3)(v)) (7)
wherein
Kx＝[K1,K2]T
Figure BDA0002932979900000073
Figure BDA0002932979900000074
The invention has the advantages that:
the invention designs the feedback controller based on the nonlinear mathematical model of the cricket system, solves the problem of track tracking control of the system under a time-varying reference signal and has higher-precision tracking performance.
Drawings
FIG. 1 is a diagram of a 3-level state feedback control method.
Fig. 2 is a diagram illustrating the tracking effect of the x-axis and y-axis displacement tracks of the small ball.
Fig. 3 is a diagram of the actual trajectory of the pellet.
Fig. 4 is a comparison graph of tracking errors of the displacement of the ball under the action of the 1 st order and 3 rd order controllers.
Detailed Description
The technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. It is to be understood that the described embodiments are merely exemplary of the invention, and not restrictive of the full scope of the invention.
A trajectory tracking control method of a cricket system comprises the following steps:
step 1, establishing a nonlinear mathematical model of a cricket system, wherein the process is as follows:
assuming that the ball is always in contact with the plate and the sliding friction between the cricket is negligible, the mathematical model of the cricket system is described as follows:
Figure BDA0002932979900000081
wherein the content of the first and second substances,
Figure BDA0002932979900000082
respectively showing the displacement, speed and acceleration of the small ball on the flat plate in the directions of the x axis and the y axis,
Figure BDA0002932979900000083
respectively representing the deflection angle, deflection angular velocity and deflection angular acceleration of the flat plate in the directions of an x axis and a y axis, wherein m represents the mass of the small ball, (I)p,Ib) Respectively representing the moment of inertia of the plate and the ball (tau)x,τy) Respectively showing the moment applied to the flat plate in the x-axis direction and the moment applied to the flat plate in the y-axis direction, and g shows the gravity acceleration;
order to
Figure BDA0002932979900000084
The system (1) can be written as follows:
Figure BDA0002932979900000085
wherein
Figure BDA0002932979900000086
Figure BDA0002932979900000087
Figure BDA0002932979900000088
η3＝-mx1x5(-2mx4x5x6-mx2x4x5-mx1x4x6-mgx1cos x8)
η4＝-mx1x5(-2mx1x2x4-mx2x5x8-mx1x6x8-mgx1cos x3)
Figure BDA0002932979900000091
Step 2, describing the track tracking control problem of the cricket system into a nonlinear servo control problem, wherein the process is as follows:
2.1, assuming that the displacement reference tracks of the small ball in the X-axis and Y-axis directions in the cricket system are two sinusoidal signals yd1＝F1sin(ω1t+θ1),yd2＝F2sin(ω2t+θ2) This displacement reference trajectory may be generated by an external system of the form:
Figure BDA0002932979900000092
wherein v ═ v1,v2,v3,v4]TOutput is yd1＝v1,yd2＝v3；
Let lambdai(i ═ 1,2,3,4) is a matrix
Figure BDA0002932979900000093
Characteristic value of (a), easy to verifyiEach having zero real part and the equilibrium point v-0 of the external system (3) is lyapunov stable;
2.2, define the tracking error as
Figure BDA0002932979900000094
2.3, the cricket system (2), the external system (3) and the error function (4) can be written in a compact form as follows:
Figure BDA0002932979900000095
wherein F (x (t), u (t), v (t)) is described as follows:
Figure BDA0002932979900000096
at this time, the problem of tracking and controlling the small ball displacement trajectory of the cricket system has been described as a nonlinear servo control problem, and the control objective is to design the controller u so that the reference signal on the small ball displacement trajectory tracking and the steady-state tracking error are small enough on the premise of ensuring the stability of the closed-loop system.
And 3, designing a 3-order state feedback controller by adopting a 3-order polynomial to approximate the solution of a regulator equation based on a nonlinear output regulation theory, wherein the process is as follows:
3.1, solving the Jacobian matrix of the cricket ball system as follows:
Figure BDA0002932979900000101
Figure BDA0002932979900000102
it can be proven by calculation that:
(1)
Figure BDA0002932979900000103
is calmable and there is a matrix K such that
Figure BDA0002932979900000104
All have negative real parts;
(2)
Figure BDA0002932979900000105
where n is the dimension 8 of the system state variable, p is the output dimension 2, λ ∈ { λ | λ ═ l1λ1+…+l4λ4,l1+…l4＝l,l1…l4＝0,1,2…l}，l＝1,2,3；
From 2.1 and 3.1 it is known that the 3 rd order nonlinear output tuning problem of the cricket system is solvable and that the solution of the tuner equation is unique because the input dimension is equal to the output dimension;
3.2, solving the regulator equation of the cricket system, wherein the process is as follows:
the cricket system regulator equation has the form:
Figure BDA0002932979900000111
wherein
Figure BDA0002932979900000112
Figure BDA0002932979900000113
η3(v)＝-mx1(v)x5(v)[-2mx4(v)x5(v)x6(v)-mx2(v)x4(v)x5(v-mx1(v)x4(v)x6(v)-mgx1(v)cos x8(v)]
η4(v)＝-mx1(v)x5(v)[-2mx1(v)x2(v)x4(v)-mx2(v)x5(v)x8(v-mx1(v)x6(v)x8(v)-mgx1(v)cos x3(v)]
Figure BDA0002932979900000114
A partial solution of the regulator equation can be obtained by simple calculations as follows:
x1(v)＝v1
x2(v)＝ω1v2
x5(v)＝v3
x6(v)＝ω2v4
because the exact values of the other solutions to the regulator equation are not available, a multi-top approximation is used to obtain their approximate solutions, and through mathematical calculations, the other 3 rd order approximate solutions can be obtained as follows:
Figure BDA0002932979900000121
Figure BDA0002932979900000122
Figure BDA0002932979900000123
Figure BDA0002932979900000124
Figure BDA0002932979900000125
Figure BDA0002932979900000126
here, the
Figure BDA0002932979900000127
Figure BDA0002932979900000128
Figure BDA0002932979900000129
Figure BDA00029329799000001210
Figure BDA00029329799000001211
Figure BDA00029329799000001212
Figure BDA00029329799000001213
Figure BDA00029329799000001214
d1000＝gm
Figure BDA00029329799000001215
Figure BDA00029329799000001216
Figure BDA00029329799000001217
Figure BDA00029329799000001218
Figure BDA00029329799000001219
Figure BDA00029329799000001220
3.4, designing a 3-stage state feedback controller as follows:
u＝u(3)(v)+Kx(x-x(3)(v)) (7)
wherein
Kx＝[K1,K2]T
Figure BDA00029329799000001221
Figure BDA00029329799000001222
In order to verify the validity of the proposed method, the invention provides a 3-stage state feedback controller (7) and a feedback control method of the type u-u(1)(v)+Kx(x-x(1)(v) The control effect of the 1 st order state feedback controller is verified by simulation, wherein
u(1)(v)＝[c1000v1,d1000v1+d0010v3]T
x(1)(v)＝[x1(v),x2(v),a1000v1,a1000ω1v2,x5(v),x6(v),v0010v3,v0010ω2v4]T
The specific parameters of the simulation verification are as follows:
acceleration of gravity g-9.8 m/s2Mass m of pellet is 0.038kg, and inertia moment of pellet Ib＝4.2×10- 6kg·m2Radius of the sphere rb0.015m, moment of inertia of plate Ip＝1.1kg·m2。
The state feedback control gain matrix is set to:
Figure BDA0002932979900000131
the present embodiment tracks the amplitude F1＝F2＝AmOf the sinusoidal reference signal yd1＝Amsin2t and
Figure BDA0002932979900000132
in FIGS. 2 to 4, Am0.5. Fig. 2 and 3 are a tracking effect diagram of x-axis and y-axis of the ball and an actual trajectory diagram of the ball under the action of a 3-order controller, respectively, and fig. 4 is a comparison diagram of tracking error of displacement of the ball under the action of a 1-order controller and a 3-order controller, respectively.
Meanwhile, in order to illustrate that the higher the order, the higher the controller precision and the better the tracking effect, this embodiment lists amThe maximum steady state tracking error results of the ball under the action of the 1 st and 3 rd order controllers when different values are taken are shown in table 1:
Figure BDA0002932979900000133
TABLE maximum Steady State tracking error under order 11 and 3 controllers
As can be seen from the simulation results of fig. 2-4: under the action of a 3-stage state feedback controller, the displacement of the small ball can quickly track the upper reference signal, and the steady-state tracking error is very small; it can be seen from fig. 4 and table 1 that the maximum steady-state tracking error of the 3-step state feedback controller is less than 1 step, and the tracking effect is better.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims (1)

1. A trajectory tracking control method of a cricket system is characterized by comprising the following steps:
step 1, establishing a nonlinear mathematical model of a cricket system;
step 2, describing the track tracking control problem of the cricket system into a nonlinear servo control problem;
3, designing a 3-order state feedback controller by adopting a 3-order polynomial to approximate the solution of a regulator equation based on a nonlinear output regulation theory;
in step 1, assuming that the small ball is always in contact with the flat plate and the sliding friction force between the cricket is ignored, the nonlinear mathematical model of the cricket system is described as follows:
Figure FDA0003510586770000011
wherein, the ratio of (x, y),
Figure FDA0003510586770000012
respectively showing the displacement, speed and acceleration (alpha, beta) of the small ball on the flat plate in the directions of the x axis and the y axis,
Figure FDA0003510586770000013
respectively representing the deflection angle, deflection angular velocity and deflection angular acceleration of the flat plate in the directions of an x axis and a y axis, wherein m represents the mass of the small ball, (I)p,Ib) Respectively representing the moment of inertia of the plate and the ball (tau)x,τy) Respectively showing the moment applied to the flat plate in the x-axis direction and the moment applied to the flat plate in the y-axis direction, and g shows the gravity acceleration;
order to
Figure FDA0003510586770000014
u＝[u1,u2]T＝[τx,τy]T
The system (1) can be written as follows
Figure FDA0003510586770000015
Figure FDA0003510586770000016
Figure FDA0003510586770000021
Figure FDA0003510586770000022
Figure FDA0003510586770000023
Figure FDA0003510586770000024
Figure FDA0003510586770000025
Figure FDA0003510586770000026
Wherein
Figure FDA0003510586770000027
Figure FDA0003510586770000028
Figure FDA0003510586770000029
η3＝-mx1x5(-2mx4x5x6-mx2x4x5-mx1x4x6-mgx1cosx8)
η4＝-mx1x5(-2mx1x2x4-mx2x5x8-mx1x6x8-mgx1cosx3)
Figure FDA00035105867700000210
In step 2, the trajectory tracking control problem of the cricket system is described as a nonlinear servo control problem, and the process is as follows:
2.1, assuming that the displacement reference tracks of the small ball in the X-axis and Y-axis directions in the cricket system are two sinusoidal signals yd1＝F1sin(ω1t+θ1),yd2＝F2sin(ω2t+θ2) This displacement reference trajectory may be generated by an external system of the form:
Figure FDA00035105867700000211
wherein v ═ v1,v2,v3,v4]TOutput is yd1＝v1,yd2＝v3；
Let lambdai(i ═ 1,2,3,4) is a matrix
Figure FDA00035105867700000212
Characteristic value of (a), easy to verifyiEach have zero real part, and the balance point v of the external system (3) is 0 is lieThe compound is stable;
2.2, define the tracking error as
Figure FDA00035105867700000213
2.3, the cricket system (2), the external system (3) and the error function (4) can be written in a compact form as follows:
Figure FDA0003510586770000031
wherein F (x (t), u (t), v (t)) is described as follows:
Figure FDA0003510586770000032
at the moment, the trajectory tracking control problem of the cricket system is described as a nonlinear servo control problem, and the control target is to design a controller u to enable a reference signal on the displacement tracking of a small ball and a steady-state tracking error to be sufficiently small on the premise of ensuring the stability of a closed-loop system;
in step 3, based on the nonlinear output regulation theory, a 3-order polynomial approximation regulator equation solution is adopted to design a 3-order state feedback controller, and the process is as follows:
3.1, solving the Jacobian matrix of the cricket ball system as follows:
Figure FDA0003510586770000033
Figure FDA0003510586770000034
it can be proven by calculation that:
(1)
Figure FDA0003510586770000035
is calmable and there is a matrix K such that
Figure FDA0003510586770000036
All have negative real parts;
(2)
Figure FDA0003510586770000037
where n is the dimension 8 of the system state variable, p is the output dimension 2, λ ∈ { λ | λ ═ l1λ1+…+l4λ4,l1+…l4＝l,l1…l4＝0,1,2…l}，l＝1,2,3；
From 2.1 and 3.1 it is known that the 3 rd order nonlinear output tuning problem of the cricket system is solvable and that the solution of the tuner equation is unique because the input dimension is equal to the output dimension;
3.2, solving the regulator equation of the cricket system, wherein the process is as follows:
the cricket system regulator equation has the form:
Figure FDA0003510586770000041
wherein
Figure FDA0003510586770000042
Figure FDA0003510586770000043
η3(v)＝-mx1(v)x5(v)[-2mx4(v)x5(v)x6(v)-mx2(v)x4(v)x5(v)-mx1(v)x4(v)x6(v)-mgx1(v)cosx8(v)]
η4(v)＝-mx1(v)x5(v)[-2mx1(v)x2(v)x4(v)-mx2(v)x5(v)x8(v)-mx1(v)x6(v)x8(v)-mgx1(v)cosx3(v)]
Figure FDA0003510586770000044
Figure FDA0003510586770000051
A partial solution of the regulator equation can be obtained by simple calculations as follows:
x1(v)＝v1
x2(v)＝ω1v2
x5(v)＝v3
x6(v)＝ω2v4
because the exact values of the other solutions to the regulator equation are not available, polynomial approximations are used to obtain their approximate solutions, and through mathematical calculations, the other 3 rd order approximate solutions can be obtained as follows:
Figure FDA0003510586770000052
Figure FDA0003510586770000053
Figure FDA0003510586770000054
Figure FDA0003510586770000055
Figure FDA0003510586770000056
Figure FDA0003510586770000057
here, the
Figure FDA0003510586770000058
Figure FDA0003510586770000059
Figure FDA00035105867700000510
Figure FDA00035105867700000511
Figure FDA00035105867700000512
Figure FDA00035105867700000513
Figure FDA00035105867700000514
Figure FDA00035105867700000515
d1000＝gm
Figure FDA00035105867700000516
Figure FDA00035105867700000517
Figure FDA00035105867700000518
Figure FDA00035105867700000519
Figure FDA00035105867700000520
Figure FDA00035105867700000521
3.3, designing a 3-stage state feedback controller as follows:
u＝u(3)(v)+Kx(x-x(3)(v)) (7)
wherein
Kx＝[K1,K2]T
Figure FDA0003510586770000061
Figure FDA0003510586770000062
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