CN114114903A - Variable-exponent power approach law-based sliding mode control method for integrating terminal of cricket system - Google Patents

Variable-exponent power approach law-based sliding mode control method for integrating terminal of cricket system Download PDF

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CN114114903A
CN114114903A CN202111214756.2A CN202111214756A CN114114903A CN 114114903 A CN114114903 A CN 114114903A CN 202111214756 A CN202111214756 A CN 202111214756A CN 114114903 A CN114114903 A CN 114114903A
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sliding mode
axis
cricket
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function
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毛剑琳
杨舒博
徐峰
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Kunming University of Science and Technology
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    • 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
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Abstract

The invention discloses a sliding mode control method of an integrating terminal of a cricket system based on a variable exponent power approach law, which can be known by analyzing a sliding mode controller of the cricket system, wherein the approach law in the controller consists of a variable exponent power term, a nonlinear function term and an index term; under the action of a nonsingular integral Terminal sliding mode control method, when the sliding mode function state is far away from a sliding mode, the system state is converged under the action of a Terminal attractor; when the sliding mode function state is close to the sliding mode, the convergence time is determined by a linear term and an integral term, and in addition, the sliding mode function introduces a nonlinear function integral term of tracking error, so that the steady-state error is reduced while the expected error is obtained.

Description

Variable-exponent power approach law-based sliding mode control method for integrating terminal of cricket system
Technical Field
The invention relates to a sliding mode control method of an integrating terminal of a cricket system based on a variable exponential power approach law, belonging to the technical field of cricket system control.
Background
The fixed height GPB2001 type plate-sphere system is a non-minimum phase and open loop unstable system, and is an important benchmark experimental object for verifying various control algorithms in the control field. The fixed-height GPB2001 cricket system controls a flat plate through a motor or other actuating mechanisms, and a freely rolling sphere realizes fixed-point motion control, fixed-track motion control and obstacle-detouring motion control on the flat plate. In the process of high-speed movement of the small ball, X, Y axes have strong coupling characteristics in two movement directions, the actual physical model of the cricket system is an under-actuated system, friction disturbance and unknown interference exist between the small ball and the flat plate, and the nonlinear characteristics make the traditional PID control difficult to achieve stable control requirements. And the sliding mode control can lead the system to change the directivity according to the current state in the dynamic process, so that the system moves up and down at high frequency according to the preset track. The sliding state design is irrelevant to a system model and interference, so that the sliding mode control has the advantages of strong robustness, rapidity, simple physical implementation and the like, and a better control effect can be obtained on a fixed-height GPB2001 type plate ball system. However, in the practical application process, high-frequency oscillation can be generated due to the fact that the system is in the sliding mode state, instability is caused, meanwhile, the system is also interfered when approaching the sliding mode state, and steady-state errors are large. Therefore, how to suppress the high-frequency buffeting and shorten the approaching sliding hyperplane time is the main direction of the sliding mode control research.
In addition, the cricket system also relates to the fields of design and construction of a mechanical experiment platform, dynamics mathematical analysis and simulation modeling of a nonlinear under-actuated system, a trajectory tracking motion control algorithm and the like. Therefore, the cricket system experiment platform not only can be regarded as an analysis and control platform of a nonlinear system, but also can be regarded as an experiment platform for testing the validity of a control algorithm, and a plurality of research results can be directly applied to actual engineering problems such as robot control and the like.
Disclosure of Invention
The invention provides a sliding mode control method of an integrating terminal of a cricket system based on a variable exponential power approach law, which is used for improving the tracking rate of small balls of the cricket system and effectively inhibiting buffeting in sliding mode control.
The technical scheme of the invention is as follows: a sliding mode control method of an integrating terminal of a plate-sphere system based on a variable exponential power approach law comprises the following steps:
s1, establishing a Lagrange mathematical model according to the plate-sphere system to obtain a system state equation, and carrying out linearization processing on the system state equation to obtain a linearization model; solving a state space expression based on the selected input quantity and the state variable according to the linearized model, and introducing external interference to respectively obtain final mathematical models of the x-axis subsystem and the y-axis subsystem of the cricket system;
s2, designing a variable exponent power approximation law of the sliding mode controller;
s3, designing a nonsingular integral terminal sliding mode function of the sliding mode controller;
s4, establishing an x-axis sliding mode controller and a y-axis sliding mode controller according to the subsystem mathematical model obtained in S1 and by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function;
and S5, inputting the set expected small ball position signal as a reference signal into the sliding mode controller to obtain the output quantity of the controller, driving the motor to change the disc inclination angle of the cricket system according to the output quantity of the controller to control the movement of the small ball, and feeding back the difference value between the reference signal of the cricket system and the actual small ball position signal to the input end of the controller to form a closed loop to complete the real-time control of the small ball in the cricket system.
The S1 specifically includes:
s11, constructing a specific mathematical model of the cricket system by adopting a Lagrange equation;
s12, obtaining a state equation of the plate-sphere system according to the Lagrange equation:
Figure BDA0003310334270000021
wherein m is the mass of the pellets, rbIs a radius of a sphere, IbMoment of inertia of the ball, g is acceleration of gravity, IpIs the rotational inertia of the disk, x and y are coordinates of the direction of the small sphere, alpha and beta are the inclination angles of the disk in the direction of the x axis and the direction of the y axis respectively,
Figure BDA0003310334270000022
and
Figure BDA0003310334270000023
is the first derivative of its corresponding variable,
Figure BDA0003310334270000024
and
Figure BDA0003310334270000025
for the second derivative of its corresponding variable, τx，τyThe torque in the x-axis direction and the y-axis direction of the disc;
s13, carrying out linearization treatment on the cricket system to obtain a linearization model of the cricket system:
Figure BDA0003310334270000026
s14, selecting the inclination angles alpha and beta of the disc as the input quantity u of the cricket system according to the linearized modelxAnd uyTaking the position and speed of the small ball in the x and y axis directions as the state variables of the system, and enabling the x to be1＝x,
Figure BDA0003310334270000027
x3＝y,
Figure BDA0003310334270000028
The state space expression of the cricket system is:
Figure BDA0003310334270000029
in the formula (I), the compound is shown in the specification,
Figure BDA0003310334270000031
is an equivalent control quantity;
s15 interference D introduced into x-axis subsystem and y-axis subsystemx(t) and Dy(t), the corrected mathematical model of the x-axis subsystem and the y-axis subsystem of the two subsystems is expressed as follows: an x-axis subsystem:
Figure BDA0003310334270000032
y-axis subsystem:
Figure BDA0003310334270000033
the S2 specifically includes:
s21, the variable exponential power approximation law is as follows:
Figure BDA0003310334270000034
in the formula (I), the compound is shown in the specification,
Figure BDA0003310334270000035
representing the approximation law of the first derivative of the sliding mode function, s represents the sliding mode function, and constant k1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure BDA0003310334270000036
The function of the symbol is represented by,
Figure BDA0003310334270000037
representing fal functions, constants
Figure BDA0003310334270000038
The constant δ is 1;
s22, using the tanh function instead of the sgn function, the modified variable exponential power approximation law is:
Figure BDA0003310334270000039
tanh(s) represents a hyperbolic tangent function.
The S3 specifically includes:
s31, introducing a nonlinear function g (e) with large saturation error:
Figure BDA00033103342700000310
in the formula, e is an error, and l is a design parameter;
s32, introducing a Terminal attractor, and forming a nonsingular integral Terminal sliding mode function as follows:
Figure BDA00033103342700000311
wherein the error e is x-xdX is the coordinate of the x direction of the small sphere, xdIs a target value, constant c1＞0,c2Is more than 0, eta is more than 0, n and m are positive odd numbers, n is more than m, and t represents time.
The S4 specifically includes:
the form of S4, the x-axis sliding mode controller and the y-axis sliding mode controller is the same, taking the x-axis sliding mode controller as an example:
obtaining an x-axis sliding mode controller U of the cricket system by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function according to an x-axis subsystem mathematical modelxComprises the following steps:
Figure BDA0003310334270000041
in the formula uxX-axis sliding mode controller U for input quantity of cricket systemxIs the cricket system target value xdThe controller output quantity obtained by calculation with the error e,
Figure BDA0003310334270000042
g is the gravity acceleration, and the error e is x-xd,xdIs a target value, constant c1＞0,c2Greater than 0, eta greater than 0, n and m are positive odd numbers, n is greater than m, Dx(t) interference introduced into the x-axis subsystem, s represents a sliding mode function, and k is a constant1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure BDA0003310334270000043
tanh(s) represents a hyperbolic tangent function,
Figure BDA0003310334270000044
representing fal functions, constants
Figure BDA0003310334270000045
The constant δ is 1.
The invention has the beneficial effects that:
1. compared with a PID controller, the sliding mode controller is insensitive to disturbance and has stronger robustness.
2. The invention provides a novel variable index power approximation law, which enables index parameters to be adaptively adjusted in different stages of a system, accelerates the approach process of the whole system, enables the approach process to converge to a balance point within limited time, and can effectively weaken buffeting in sliding mode control.
3. The invention provides a nonsingular integral terminal sliding mode control method. The system has the characteristics of small steady-state error and quick convergence when the terminal sliding mode control is far away from the sliding mode surface by combining the characteristics of integral sliding mode control, so that the system has better sliding mode and better control precision.
Drawings
FIG. 1 is a control structure diagram of a fixed-height GPB2001 cricket system;
FIG. 2 is a block diagram of the sliding mode control principle of the integrating terminal of the cricket system of the present invention;
FIG. 3 is a system phase trajectory diagram under the action of an integral terminal sliding mode surface;
FIG. 4 is a comparison simulation diagram of different approach law tracking curves of an x-axis subsystem of a solid-high GPB2001 type cricket system;
FIG. 5 is an enlarged view of a portion of FIG. 4;
FIG. 6 is a simulation diagram comparing different approach law errors of an x-axis subsystem of a fixed-height GPB2001 type cricket system;
FIG. 7 is a first enlarged view of a portion of FIG. 6;
FIG. 8 is a second enlarged view of a portion of FIG. 6;
FIG. 9 is a graph showing comparison simulation of outputs of different approach law controllers of an x-axis subsystem of a fixed-height GPB2001 type cricket system;
FIG. 10 is an enlarged view of a portion of FIG. 9;
FIG. 11 is a simulation diagram of the output of the x-axis subsystem index approach law controller of the Gu Gao GPB2001 cricket System;
FIG. 12 is a comparative simulation diagram of tracking curves of different sliding mode controllers of an x-axis subsystem of a fixed-height GPB2001 type cricket system;
FIG. 13 is an enlarged view of a portion of FIG. 12;
FIG. 14 is a simulation diagram comparing errors of different sliding mode controllers of an x-axis subsystem of a fixed-height GPB2001 type cricket system;
fig. 15 is a partially enlarged view of fig. 14.
Detailed Description
The invention will be further described with reference to the following figures and examples, without however restricting the scope of the invention thereto.
Example 1: as shown in fig. 1-15, as shown in fig. 1, the GPB2001 cricket system determines the position of a small ball by processing and analyzing image information collected by a camera, a PC sends a command to a motion controller to generate a control signal and input the control signal to an X-axis motor, a Y-axis motor performs angle control of a ball plate, so that the small ball completes a predetermined track, and a feedback signal obtained on the ball plate is output to an upper PC to form closed-loop control.
As shown in fig. 2, the control target amount is the actual position of the small ball, the controller is a sliding mode controller, the execution mechanism is an X-axis motor and a Y-axis motor, and the control object is a disc of the cricket system. When the small ball is far away from the preset track, the variable exponential power approach law part generates switching control to enable the small ball to follow the track, when the system tends to be stable, the integral terminal sliding mode function generates equivalent control to keep the small ball stable, and the sliding mode controller is formed by combined action of the switching control and the equivalent control.
A sliding mode control method of an integrating terminal of a plate-sphere system based on a variable exponential power approach law comprises the following steps:
s1, establishing a Lagrange mathematical model according to the plate-sphere system to obtain a system state equation, and carrying out linearization processing on the system state equation to obtain a linearization model; solving a state space expression based on the selected input quantity and the state variable according to the linearized model, and introducing external interference to respectively obtain final mathematical models of the x-axis subsystem and the y-axis subsystem of the cricket system;
s2, designing a variable exponent power approximation law of the sliding mode controller;
s3, designing a nonsingular integral terminal sliding mode function of the sliding mode controller;
s4, establishing an x-axis sliding mode controller and a y-axis sliding mode controller according to the subsystem mathematical model obtained in S1 and by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function;
s5, setting an expected small ball position signal as a reference signal through a PC, inputting the signal into a sliding mode controller to obtain the output quantity of the controller, driving a motor to change the disc inclination angle of a cricket system according to the output quantity of the controller to control the movement of the small ball, and feeding back the difference value between the reference signal of the cricket system and the actual small ball position signal (acquired through a camera) to the input end of the controller to form a closed loop to complete the real-time control of the small ball in the cricket system.
By the control method, the small balls on the GPB2001 type cricket system can complete the tracking of a target track (a circular track is made on a ball disc) under the action of the proposed controller.
Optionally, the S1 specifically is:
s11, constructing a specific mathematical model of the cricket system by adopting a Lagrange equation, and constructing the model by adopting the Lagrange equation can better embody that the cricket system is a strongly coupled under-actuated system:
Figure BDA0003310334270000061
where T is the kinetic energy of the cricket system, qiIn the form of a generalized coordinate system,
Figure BDA0003310334270000062
is the derivative of the generalized coordinate, L is the difference between the potential energy V and the kinetic energy T of the cricket system, QiN, i ═ 1,2,3,. n;
s12, obtaining a state equation of the solid high GPB2001 type plate sphere system according to the Lagrange equation:
Figure BDA0003310334270000063
wherein m is the mass of the pellets, rbIs a radius of a sphere, IbMoment of inertia of the ball, g is acceleration of gravity, IpIs the rotational inertia of the disk, x and y are coordinates of the direction of the small sphere, alpha and beta are the inclination angles of the disk in the direction of the x axis and the direction of the y axis respectively,
Figure BDA0003310334270000064
and
Figure BDA0003310334270000065
is the first derivative of its corresponding variable,
Figure BDA0003310334270000066
and
Figure BDA0003310334270000067
for the second derivative of its corresponding variable, τx，τyThe torque in the x-axis direction and the y-axis direction of the disc;
s13, carrying out linearization treatment on the board ball system, when the system is stable, the disc is in a horizontal position, the inclination angle of the disc in the x and y axis directions is regarded as zero, and the inclination angle range of the disc rotation is not large, so that the sine function of the inclination angle can be approximated to the independent variable thereof, namely sin alpha is approximately equal to a, sin beta is approximately equal to beta, and the angular speed of the disc can also be approximated to zero, namely sin alpha is approximately equal to a, sin beta is approximately equal to beta
Figure BDA0003310334270000068
Obtaining a linear model of a solid high GPB2001 type plate ball system:
Figure BDA0003310334270000069
s14, selecting the inclination angles alpha and beta of the discs as input quantities u of the solid-high GPB2001 type cricket system according to the linearization modelxAnd uy(input quantity of cricket system, namely output quantity of controller), and taking position and speed of the small ball in x and y axis directions as the systemState variable of (2), let x1＝x,
Figure BDA00033103342700000610
x3＝y,
Figure BDA00033103342700000611
The state space expression of the cricket system is:
Figure BDA0003310334270000071
in the formula (I), the compound is shown in the specification,
Figure BDA0003310334270000072
is an equivalent control quantity;
s15 interference D introduced into x-axis subsystem and y-axis subsystemx(t) and Dy(t), the corrected mathematical model of the x-axis subsystem and the y-axis subsystem of the two subsystems is expressed as follows: an x-axis subsystem:
Figure BDA0003310334270000073
y-axis subsystem:
Figure BDA0003310334270000074
the S2 specifically includes:
s21, the variable exponential power approximation law is as follows:
Figure BDA0003310334270000075
in the formula (I), the compound is shown in the specification,
Figure BDA0003310334270000076
representing the approximation law, s represents a sliding mode function, and k is a constant1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure BDA0003310334270000077
The function of the symbol is represented by,
Figure BDA0003310334270000078
representing fal functions, constants
Figure BDA0003310334270000079
The constant δ is 1;
s22, using the tanh function instead of the sgn function, to reduce the characteristic influence of the sgn function and further suppress the buffeting, the modified variable exponent power approximation law is:
Figure BDA00033103342700000710
the design principle is as follows: the approach law is formed by a variable exponential power term-k1|s|γtan h(s), nonlinear function term-k2fal (s, a, delta) and exponential term-k3And s. When the | s | is more than 1 and the | s | is less than or equal to 1, because the index parameters in the approximation law are automatically adjusted due to different values of the gamma, the fast approximation time can be obtained in different stages through the interaction of the power approximation term and the index approximation term, and the buffeting is effectively weakened.
Optionally, the S3 is specifically as follows:
s31, introducing a nonlinear function g (e) with large saturation error:
Figure BDA0003310334270000081
in the formula, e is an error, and l is a design parameter;
s32, taking the nonsingular requirement of Terminal sliding mode control into consideration, introducing a Terminal attractor, and forming a nonsingular integral Terminal sliding mode function as follows:
Figure BDA0003310334270000082
wherein e ═ x-xd,xdIs a target value, constant c1＞0,c2Is more than 0, eta is more than 0, n and m are positive odd numbers, n is more than m, and t represents time.
The form of S4, the x-axis sliding mode controller and the y-axis sliding mode controller is the same, taking the x-axis sliding mode controller as an example:
obtaining an x-axis sliding mode controller U of the cricket system by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function according to an x-axis subsystem mathematical modelxComprises the following steps:
Figure BDA0003310334270000083
in the formula uxX-axis sliding mode controller U for input quantity of cricket systemxIs the cricket system target value xdThe controller output quantity obtained by calculation with the error e,
Figure BDA0003310334270000084
g is the gravity acceleration, and the error e is x-xd,xdIs a target value, constant c1＞0,c2Greater than 0, eta greater than 0, n and m are positive odd numbers, n is greater than m, Dx(t) interference introduced into the x-axis subsystem, s represents a sliding mode function, and k is a constant1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure BDA0003310334270000085
tanh(s) represents a hyperbolic tangent function,
Figure BDA0003310334270000086
representing fal functions, constants
Figure BDA0003310334270000087
The constant δ is 1.
The design principle is as follows: when the state of the GPB2001 type cricket system is far away from the sliding mode function, the state is converged under the action of a Terminal attractor; when the state of the plate-sphere system is close to a sliding mode function, the convergence time and the steady-state error are determined by a linear term and an integral term. Therefore, the rapidity of the track tracking of the cricket system is ensured under the action of the integral terminal sliding mode surface, and the steady-state error is reduced while the expected error is obtained. Sliding mode controlThe phase trace of the system is shown in FIG. 3, when s > 0, x2Decreases rapidly, when s < 0, x2Increase rapidly, therefore, when x2When s is 0, the controller may realize s is 0 for a limited time.
By analyzing the sliding mode controller of the solid-high GPB2001 type plate-ball system, the approximation law in the controller is changed from the exponential power term k1|s|γtan h(s), nonlinear function term k2fal (s, a, delta) and exponential term k3s, when the controller is more than 1 in the absolute value of s and less than or equal to 1 in the absolute value of s, because the index parameters in the approaching law are automatically adjusted according to different values of gamma, the controller can obtain faster tracking rate in different stages through the interaction of the power approaching term and the index approaching term, and effectively weaken buffeting; under the action of a nonsingular integral Terminal sliding mode control method, when the sliding mode function state is far away from a sliding mode, the system state is converged under the action of a Terminal attractor; when the sliding mode function state is close to the sliding mode, the convergence time is determined by a linear term and an integral term, and in addition, the sliding mode function introduces a nonlinear function integral term of tracking error, so that the steady-state error is reduced while the expected error is obtained. Therefore, under the action of the controller, the fixed-height GPB2001 type cricket system can be converged in a limited time, and the system is ensured to have smaller steady-state error under the convergence condition.
In order to verify the stability of the sliding mode controller, Lyapunov is selected as follows:
Figure BDA0003310334270000091
only when
Figure BDA0003310334270000092
When satisfied, the system is stable. Substituting a sliding mode controller formula to obtain:
Figure BDA0003310334270000093
substituting the nonsingular integral terminal sliding mode function into the Lyapunov to obtain:
Figure BDA0003310334270000094
in the formula, k1＞0,k2＞0,k3Is > 0, and therefore,
Figure BDA0003310334270000095
the system is stable.
Simulation experiment verification:
since the external disturbance term in the experiment cannot be directly obtained by the sensor, it cannot be estimated whether the verified disturbance is accurate. For this purpose, simulation verification was carried out using the Simulink platform in MATLAB.
Experiment one: compared with the approximation Law on the basis of the linear Sliding Mode Function, the approximation laws respectively include a novel Variable exponent Power approximation Law (NVEPRL), a Double Power Combination Function approximation Law (DPRL), a Power approximation Law (PRL) and an Exponential approximation Law (ERL). Wherein, DPRL, PRL and ERL are as follows:
Figure BDA0003310334270000096
Figure BDA0003310334270000097
Figure BDA0003310334270000098
experiment two: on the basis of the novel variable exponent power approach law, the Control methods of a nonsingular Integral Terminal Sliding Mode Control (I _ Terminal), a Terminal Sliding Mode Control (Terminal Sliding Mode Control) and a Linear Sliding Mode Control (Linear) are respectively adopted for comparison, and the Terminal Sliding Mode Control method and the Linear Sliding Mode Control method are as follows:
Figure BDA0003310334270000099
Figure BDA0003310334270000101
disturbance D given an x-axis subsystem and a y-axis subsystemxAnd DyIn the form of: 0.1sint +0.1 cost. Experiment one, in the given formula
Figure BDA0003310334270000102
And (4) representing different approximation laws, wherein s represents a sliding mode function in the second experiment, the sliding mode function corresponds to different sliding mode control methods, and e represents a system error.
Relevant parameters of the solid high GPB2001 type cricket system are shown in the table 1.
TABLE 1 simulation-related parameter values
Variable names Parameter value Variable names Parameter value
m 0.264kg Ib 4.2×10-5kg·m2
r 0.02m g 9.8m/s2
c1 12 c2 0.1
η 1.5 k1 18
k2 12 k3 30
a 1.5 b 0.5
δ 1 l 0.002
n 11 m 9
p 2 q 1
The initial position coordinates of the small ball are (0.5 ), the initial position coordinates of the circular curve are (0,0), the control target of the simulation tracking track is selected to be the circular curve, and the expression of the curve is as follows:
Figure BDA0003310334270000103
in the formula, xxdAnd xydRepresenting the expected values of the x-axis and y-axis of the circular tracking curve.
The thick black solid line in fig. 4 represents the desired tracking track, the thin gray solid line represents the track-tracking effect of the NVEPRL approximation law, and the remaining different types of broken lines represent the track-tracking effects of the DPRL, PRL, and ERL approximation laws, respectively. In the enlarged partial view of fig. 5, about 0.33s, the NVEPRL approximation law and the DPRL approximation law proposed by the present invention complete trajectory tracking. The NVEPRL approach law has a faster response speed, and the PRL approach law and the ERL approach law complete track tracking after 0.4 s. The black solid line in fig. 6 represents the track-following error of NVEPRL approach law, and the remaining different types of broken lines represent the track-following errors of DPRL, PRL and ERL approach laws, respectively. From the partial enlarged views of fig. 7 and fig. 8, at 1.56s, the NVEPR approximation law tracking error not only converges to zero in error approximation but also has no error fluctuation, and thus has better control accuracy, compared with other approximation laws. In fig. 9 and 10, the black solid line represents the controller output of the NVEPRL approach law, the remaining two types of broken lines represent the controller outputs of the DPRL and PRL approach laws, respectively, and the black solid line in fig. 11 represents the controller output of the ERL approach law. The results show that although the DPRL approach law, the PRL approach law and the ERL approach law can complete track tracking, the problem of buffeting exists to a certain extent, and the novel approach law provided by the invention can effectively weaken buffeting.
The comparison experiment of the sliding mode function is carried out on the premise of also using the novel variable exponential power approach law provided by the invention. The black solid line in fig. 12 represents the expected tracking trajectory, the dashed line I _ Terminal represents the trajectory tracking effect of the nonsingular integral Terminal sliding mode function, and the remaining two different dashed lines represent the trajectory tracking effects of the Terminal sliding mode function and the linear sliding mode function, respectively. In the partial enlarged view of fig. 13, the nonsingular integral terminal sliding mode control proposed by the present invention completes the track tracking after about 0.25s, and the nonsingular terminal sliding mode control and the linear sliding mode control follow the predetermined track after 0.33s, so the dynamic response performance of the system is improved by 24.24%. In fig. 14, a black solid line represents a trajectory tracking error of the nonsingular integral terminal sliding mode function, and the remaining two different dotted lines represent trajectory tracking errors of the terminal sliding mode function and the linear sliding mode function, respectively. In the enlarged partial view of fig. 15, at 1.27s, the error of the nonsingular integral terminal sliding mode control method is already approximately converged to zero, and the smaller steady-state error is achieved.
According to the analysis of the two experimental results, on the premise of the same sliding mode function, the novel variable exponential power approach law can weaken the buffeting phenomenon, and has high convergence speed and high control precision. After the nonsingular integral terminal sliding mode function is introduced, the controller has a faster convergence rate and smaller steady-state error.
While the present invention has been described in detail with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

Claims (5)

1. A sliding mode control method of an integrating terminal of a plate-sphere system based on a variable exponential power approach law is characterized by comprising the following steps: the method comprises the following steps:
s1, establishing a Lagrange mathematical model according to the plate-sphere system to obtain a system state equation, and carrying out linearization processing on the system state equation to obtain a linearization model; solving a state space expression based on the selected input quantity and the state variable according to the linearized model, and introducing external interference to respectively obtain final mathematical models of the x-axis subsystem and the y-axis subsystem of the cricket system;
s2, designing a variable exponent power approximation law of the sliding mode controller;
s3, designing a nonsingular integral terminal sliding mode function of the sliding mode controller;
s4, establishing an x-axis sliding mode controller and a y-axis sliding mode controller according to the subsystem mathematical model obtained in S1 and by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function;
and S5, inputting the set expected small ball position signal as a reference signal into the sliding mode controller to obtain the output quantity of the controller, driving the motor to change the disc inclination angle of the cricket system according to the output quantity of the controller to control the movement of the small ball, and feeding back the difference value between the reference signal of the cricket system and the actual small ball position signal to the input end of the controller to form a closed loop to complete the real-time control of the small ball in the cricket system.
2. The sliding mode control method of the integrating terminal of the cricket system based on the variable exponential power approach law according to claim 1, characterized in that: the S1 specifically includes:
s11, constructing a specific mathematical model of the cricket system by adopting a Lagrange equation;
s12, obtaining a state equation of the plate-sphere system according to the Lagrange equation:
Figure FDA0003310334260000011
wherein m is the mass of the pellets, rbIs a radius of a sphere, IbMoment of inertia of the ball, g is acceleration of gravity, IpIs the rotational inertia of the disk, x and y are coordinates of the direction of the small sphere, alpha and beta are the inclination angles of the disk in the direction of the x axis and the direction of the y axis respectively,
Figure FDA0003310334260000012
and
Figure FDA0003310334260000013
is the first derivative of its corresponding variable,
Figure FDA0003310334260000014
and
Figure FDA0003310334260000015
for the second derivative of its corresponding variable, τx，τyThe torque in the x-axis direction and the y-axis direction of the disc;
s13, carrying out linearization treatment on the cricket system to obtain a linearization model of the cricket system:
Figure FDA0003310334260000021
s14, selecting the inclination angles alpha and beta of the disc as the input quantity u of the cricket system according to the linearized modelxAnd uyTaking the position and speed of the small ball in the x and y axis directions as the state variables of the system, and enabling the x to be1＝x,
Figure FDA0003310334260000022
x3＝y,
Figure FDA0003310334260000023
The state space expression of the cricket system is:
Figure FDA0003310334260000024
in the formula (I), the compound is shown in the specification,
Figure FDA0003310334260000025
is an equivalent control quantity;
s15 interference D introduced into x-axis subsystem and y-axis subsystemx(t) and Dy(t), the corrected mathematical model of the x-axis subsystem and the y-axis subsystem of the two subsystems is expressed as follows: an x-axis subsystem:
Figure FDA0003310334260000026
y-axis subsystem:
Figure FDA0003310334260000027
3. the sliding mode control method of the integrating terminal of the cricket system based on the variable exponential power approach law according to claim 1, characterized in that: the S2 specifically includes:
s21, the variable exponential power approximation law is as follows:
Figure FDA00033103342600000214
in the formula (I), the compound is shown in the specification,
Figure FDA0003310334260000028
representing the approximation law of the first derivative of the sliding mode function, s represents the sliding mode function, and constant k1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure FDA0003310334260000029
Figure FDA00033103342600000210
The function of the symbol is represented by,
Figure FDA00033103342600000211
representing fal functions, constants
Figure FDA00033103342600000212
The constant δ is 1;
s22, using the tanh function instead of the sgn function, the modified variable exponential power approximation law is:
Figure FDA00033103342600000213
tanh(s) represents a hyperbolic tangent function.
4. The sliding mode control method of the integrating terminal of the cricket system based on the variable exponential power approach law according to claim 1, characterized in that: the S3 specifically includes:
s31, introducing a nonlinear function g (e) with large saturation error:
Figure FDA0003310334260000031
in the formula, e is an error, and l is a design parameter;
s32, introducing a Terminal attractor, and forming a nonsingular integral Terminal sliding mode function as follows:
Figure FDA0003310334260000032
wherein the error e is x-xdX is the coordinate of the x direction of the small sphere, xdIs a target value, constant c1＞0,c2Is more than 0, eta is more than 0, n and m are positive odd numbers, n is more than m, and t represents time.
5. The sliding mode control method of the integrating terminal of the cricket system based on the variable exponential power approach law according to claim 1, characterized in that: the S4 specifically includes:
the form of S4, the x-axis sliding mode controller and the y-axis sliding mode controller is the same, taking the x-axis sliding mode controller as an example:
obtaining an x-axis sliding mode controller U of the cricket system by combining a variable exponent power approach law and a nonsingular integral terminal sliding mode function according to an x-axis subsystem mathematical modelxComprises the following steps:
Figure FDA0003310334260000033
in the formula uxX-axis sliding mode controller U for input quantity of cricket systemxIs a cricket systemTarget value xdThe controller output quantity obtained by calculation with the error e,
Figure FDA0003310334260000034
g is the gravity acceleration, and the error e is x-xd,xdIs a target value, constant c1＞0,c2Greater than 0, eta greater than 0, n and m are positive odd numbers, n is greater than m, Dx(t) interference introduced into the x-axis subsystem, s represents a sliding mode function, and k is a constant1＞0,k2＞0,k3Is greater than 0; coefficient of performance
Figure FDA0003310334260000035
tanh(s) represents a hyperbolic tangent function,
Figure FDA0003310334260000036
representing fal functions, constants
Figure FDA0003310334260000037
The constant δ is 1.
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