CN112486191A - Balance car control method based on enhanced error model - Google Patents

Balance car control method based on enhanced error model Download PDF

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CN112486191A
CN112486191A CN202011177917.0A CN202011177917A CN112486191A CN 112486191 A CN112486191 A CN 112486191A CN 202011177917 A CN202011177917 A CN 202011177917A CN 112486191 A CN112486191 A CN 112486191A
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balance car
equation
inclination angle
control
follows
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CN112486191B (en
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余善恩
吴志涛
石玉超
杨文东
王坚
傅琳
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Hangzhou Dianzi University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0891—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for land vehicles
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
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    • Y02T10/40—Engine management systems

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Abstract

The invention relates to the field of balance car control algorithms, in particular to a balance car control method based on an enhanced error method, which comprises the following steps: step 1, establishing a simplified model for a balance car by utilizing a Newton second law; and 2, adopting an adaptive control method based on output feedback, and using an adaptive controller based on an enhanced error model to control the inclination angle of the balance car. The invention can estimate the change of each parameter of the vehicle body in real time and adjust and control in the running process of the balance vehicle, so that the state space equation of the system always tends to a reference model, when the balance vehicle has the problems of uneven road and the like, which causes the angular velocity data output by the gyroscope to be greatly interfered, the controller can quickly adjust the control parameters, thereby increasing the self-adaptability of the control method, and meanwhile, the method only adopts the inclination angle as the input feedback information, estimates the real-time angular velocity by adopting the estimator, avoids using the angular velocity information which is easy to be interfered and improves the anti-interference capability of the angular acceleration data.

Description

Balance car control method based on enhanced error model
Technical Field
The invention relates to the field of balance car control algorithms, in particular to a balance car control method based on an enhanced error model.
Background
The two-wheeled self-balancing electric vehicle is a complex nonlinear, under-actuated, strong-coupling and naturally unstable system. The longitudinal axis of the gravity center of the vehicle body is used as a reference line, and the motor is controlled to act according to the angular posture information of the vehicle body, so that the gravity center of the vehicle body and two wheels are kept on the same axis, and the self dynamic balance of the vehicle body is realized.
The current commonly used control algorithms of the two-wheel balance car include PID control, expert control, fuzzy control, neural network control and the like. When the effective load changes, the PID controller cannot automatically adjust the control parameters according to the load changes, so that effective control can be performed only for specific conditions, and the control performance is easy to deteriorate when the load changes. The expert control is composed of a knowledge base and an inference mechanism, and needs updating of the knowledge base and generation of rules, so that the expert control has great disadvantage in real-time performance. Fuzzy control has stronger robustness compared with the above control algorithm, but the control precision of the system is reduced due to the fuzzy characteristic of the fuzzy control. The neural network control also has strong robustness and adaptability, but needs a large amount of known engineering data samples for learning training, and has slow convergence rate. Aiming at the defects of the control algorithm, the self-adaptive algorithm based on the enhanced error model is adopted, so that the control system can estimate the parameter change of the control system in real time in the running process of the balance car and automatically adjust the control output quantity, and the controller has stronger robustness and adaptability.
Disclosure of Invention
In order to solve the problems and the technical requirements, the invention provides a balance car control method based on an enhanced error model, which can estimate the change of each parameter of a car body in real time in the running process of a balance car and adjust a control algorithm according to the estimated parameters so that a state space equation of a system always tends to a reference model, and the specific technical scheme is as follows.
A balance car control method based on an enhanced error model comprises the following steps:
step 1, establishing a simplified model for a balance car by utilizing a Newton second law;
and 2, adopting an adaptive control method based on output feedback, and controlling the inclination angle of the balance car by using an adaptive controller based on an enhanced error model.
Further, the step 1 specifically includes:
when the steering problem is not considered temporarily, the balance car system is equivalent to a primary inverted pendulum, the mass of a pendulum rod of the inverted pendulum is m, the total length of the pendulum rod is 2l, the gravity center of the pendulum rod is located in the middle of the rod, namely, the pendulum rod is away from the position with the length of l at the two ends, the inclination angle of the pendulum rod is theta, the rotational inertia of the pendulum rod around a rotating shaft below is J, the transverse acting force of the pendulum rod on the pendulum rod is equal to the driving force of a trolley on the pendulum rod is u, and the vertical acting force of the pendulum rod on the; the mass of the trolley with the bottom capable of moving left and right is M, and the transverse movement distance of the trolley is z;
the horizontal position of the center of gravity of the swing link can be expressed as:
x＝z+lsinθ (1)
the formula (1) is derived, and the acceleration of the gravity center of the swing rod in the horizontal direction can be obtained as follows:
Figure BDA0002747029260000021
similarly, the vertical position of the center of the swing rod is only expressed as:
y＝lcosθ (3)
the formula (3) is derived, and the acceleration in the direction perpendicular to the gravity center of the swing rod can be obtained as follows:
Figure BDA0002747029260000022
because the stress of the swing rod in the horizontal direction is H, according to the Newton second law, the equation of the horizontal direction of the swing rod can be obtained as follows:
Figure BDA0002747029260000023
similarly, because the swing rod is subjected to the action of the gravity mg and the supporting force v in the vertical direction, the vertical direction equation is as follows:
Figure BDA0002747029260000024
meanwhile, under the action of horizontal thrust H and vertical thrust v, the swing rod rotates around the gravity center according to the following equation:
Figure BDA0002747029260000025
carrying out stress analysis on the bottom trolley, wherein the trolley is subjected to driving force u and reaction force H in the horizontal direction, and the motion equation is as follows:
Figure BDA0002747029260000026
the equations of motion of the balance car can be simplified for the formula (5) and the formula (8) as follows:
Figure BDA0002747029260000027
the second motion equation of the balance car can be simplified by the equations (5), (6), (7) and (8):
Figure BDA0002747029260000028
because the inclination angle theta is always near zero degrees when the balance car is in a normal running state, the trigonometric function can be simplified as follows:
Figure BDA0002747029260000031
the approximation relation (11) is brought into the equations of motion (9), (10), and because
Figure BDA0002747029260000032
The term is smaller than other terms, and if the term is abandoned, the motion equation of the balance car can be simplified as follows:
Figure BDA0002747029260000033
Figure BDA0002747029260000034
equivalently converting differential equations (12), (13):
Figure BDA0002747029260000035
Figure BDA0002747029260000036
controlling the inclination angle theta through the driving force variable u, so that a transfer function from the variable u to the variable theta is stable, setting the set value of the inclination angle theta to be zero, and when a user standing on the balance car controls the gravity center of the body of the user to incline forwards, the adjustment action u on the control of the inclination angle theta can lead the balance car to accelerate forwards; the balance car can accelerate backwards when the body gravity center of the user inclines backwards;
compared with the first-stage inverted pendulum model shown in the equations (14) and (15), the variable M of the balance vehicle is equal to zero, so the equation (15) can be simplified as follows:
Figure BDA0002747029260000037
simultaneously, the barycenter position with the balance car is located the intermediate position of inverted pendulum pole approximately, then inertia is:
J＝ml2 (17)
substituting the moment of inertia into an equation (16), simplifying the balance car tilt angle control equation into:
Figure BDA0002747029260000038
let variable x1＝θ，
Figure BDA0002747029260000039
And (3) converting the balance car inclination angle control equation into a state space equation form:
Figure BDA00027470292600000310
Figure BDA00027470292600000311
further, the step 2 specifically includes:
the inclination angle of the balance car is used as an input variable, the estimator is used for estimating the real-time angular velocity, the direct adoption of the gyroscope angular velocity data which is easy to interfere is avoided, and the self-adaptive controller is used for automatically estimating the unknown quantity in the motion process of the balance car
Figure BDA0002747029260000041
And simultaneously adjusting the parameters of the controller, knowing that the differential equation of the controlled object is equation (18), the variable y represents the inclination angle of the balance car, the relationship between the inclination angle y and the input driving force variable u is expressed by the differential equation:
Figure BDA0002747029260000042
the differential equation of the reference model represents the control performance of the desired controlled object, and the differential equation of the reference model is:
Figure BDA0002747029260000043
wherein a ism1，am0，bm0Is a transfer function coefficient of the reference model;
the error in the tilt angle y can be defined as:
ε＝y-ym (22)
converting differential equations of the controlled object and the reference model into an error model through a filter, and obtaining the error model of the inclination angle as follows:
ε＝Wm(s)(bm0g-θTω-b0u) (23)
Figure BDA0002747029260000044
Figure BDA0002747029260000045
wherein the content of the first and second substances,
Figure BDA0002747029260000046
is a filter equation, and k is a filter transfer function coefficient;
to convert the error model into a static error model form, the control terms are selected as:
Figure BDA0002747029260000047
and (3) bringing the control item into an error model to obtain:
Figure BDA0002747029260000048
the enhanced error model is defined as:
Figure BDA0002747029260000049
and (3) bringing the error epsilon into an enhanced error model, and converting the error into a static error model form:
Figure BDA00027470292600000410
the adaptive formula of the static error model is expressed as:
Figure BDA0002747029260000051
the normalized gain can be expressed as:
Figure BDA0002747029260000052
the beneficial technical effects of the invention are as follows:
the method can estimate parameters in a balance car model in real time, and then adjust the parameters of the controller according to the estimated parameters, so that the characteristics of the balance car tend to a reference model, when the effective load on the balance car changes, the controller can quickly adjust the control parameters, the adaptability of the control method is improved, meanwhile, the method only adopts the inclination angle as input feedback information, adopts the estimator to estimate the real-time angular velocity, avoids using angular velocity information which is easy to be interfered, and improves the anti-interference capability of the angular acceleration data.
Drawings
FIG. 1 is a schematic view of a one-stage inverted pendulum of the present invention;
FIG. 2 is a diagram of the force analysis of a part of a pendulum rod of the primary inverted pendulum of the present invention;
FIG. 3 is a force analysis diagram of a portion of the primary inverted pendulum cart of the present invention;
FIG. 4 is a block diagram of an enhanced error model based adaptive control method of the present invention;
FIG. 5 is a simulation of the enhanced error model method of the present invention in Matlab/Simulink;
FIG. 6 is a portion of a regressor in an enhanced error controller subsystem of a simulation diagram of the present invention;
FIG. 7 is an enhanced error model portion of the enhanced error controller subsystem of the simulation diagram of the present invention;
FIG. 8 is a normalized gain section in the enhanced error controller subsystem of the simulation diagram of the present invention;
FIG. 9 is a portion of the adaptive algorithm in the enhanced error controller subsystem of the simulation diagram of the present invention;
FIG. 10 is a control item portion of the enhanced error controller subsystem of the simulation diagram of the present invention;
FIG. 11 shows the difference ε ═ x between the controlled object state variable and the reference model state variable in accordance with the present inventionM-simulation results of x;
FIG. 12 is a graph of the difference between the unknown parameter value and the true value obtained by the estimator of the present invention
Figure BDA0002747029260000053
And (4) obtaining a simulation result.
Detailed Description
In order to make the objects, technical solutions and technical effects of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings.
The invention discloses a balance car control method based on an enhanced error model, which can estimate the change of each parameter of a car body in real time in the running process of a balance car and adjust a control algorithm according to the estimated parameters, so that a state space equation of a system always tends to a reference model. When the balance car runs, the problems of uneven road and the like are met, so that the angular speed data output by the gyroscope is greatly interfered. Therefore, the algorithm adopts the inclination angle of the balance car as output feedback, and simultaneously adopts the estimator to estimate the real-time angular velocity, thereby improving the anti-interference capability of the angular acceleration data.
A balance car control method based on an enhanced error method comprises the following steps:
step 1, establishing a simplified model for a balance car by utilizing a Newton second law;
and 2, adopting an adaptive control method based on output feedback, and controlling the inclination angle of the balance car by using an adaptive controller based on an enhanced error model.
Further, the step 1 specifically includes:
when the steering problem is not considered temporarily, the balance car system is equivalent to a primary inverted pendulum, the mass of a pendulum rod of the inverted pendulum is m, the total length of the pendulum rod is 2l, the gravity center of the pendulum rod is located in the middle of the rod, namely, the pendulum rod is away from the position with the length of l at the two ends, the inclination angle of the pendulum rod is theta, the rotational inertia of the pendulum rod around a rotating shaft below is J, the transverse acting force of the pendulum rod on the pendulum rod is equal to the driving force of a trolley on the pendulum rod is u, and the vertical acting force of the pendulum rod on the; the mass of the trolley with the bottom capable of moving left and right is M, and the transverse movement distance of the trolley is z;
the horizontal position of the center of gravity of the swing link can be expressed as:
x＝z+lsinθ (1)
the formula (1) is derived, and the acceleration of the gravity center of the swing rod in the horizontal direction can be obtained as follows:
Figure BDA0002747029260000061
similarly, the vertical position of the center of the swing rod is only expressed as:
y＝lcosθ (3)
the formula (3) is derived, and the acceleration in the direction perpendicular to the gravity center of the swing rod can be obtained as follows:
Figure BDA0002747029260000062
because the stress of the swing rod in the horizontal direction is H, according to the Newton second law, the equation of the horizontal direction of the swing rod can be obtained as follows:
Figure BDA0002747029260000063
similarly, because the swing rod is subjected to the action of the gravity mg and the supporting force v in the vertical direction, the vertical direction equation is as follows:
Figure BDA0002747029260000071
meanwhile, under the action of horizontal thrust H and vertical thrust v, the swing rod rotates around the gravity center according to the following equation:
Figure BDA0002747029260000072
carrying out stress analysis on the bottom trolley, wherein the trolley is subjected to driving force u and reaction force H in the horizontal direction, and the motion equation is as follows:
Figure BDA0002747029260000073
the equations of motion of the balance car can be simplified for the formula (5) and the formula (8) as follows:
Figure BDA0002747029260000074
the second motion equation of the balance car can be simplified by the equations (5), (6), (7) and (8):
Figure BDA0002747029260000075
because the inclination angle theta is always near zero degrees when the balance car is in a normal running state, the trigonometric function can be simplified as follows:
sinθ≈θ，cosθ≈1 (11)
the approximation relation (11) is brought into the equations of motion (9), (10), and because
Figure BDA0002747029260000076
The term is smaller than other terms, and if the term is abandoned, the motion equation of the balance car can be simplified as follows:
Figure BDA0002747029260000077
Figure BDA0002747029260000078
equivalently converting differential equations (12), (13):
Figure BDA0002747029260000079
Figure BDA00027470292600000710
controlling the inclination angle theta through the driving force variable u, so that a transfer function from the variable u to the variable theta is stable, setting the set value of the inclination angle theta to be zero, and when a user standing on the balance car controls the gravity center of the body of the user to incline forwards, the adjustment action u on the control of the inclination angle theta can lead the balance car to accelerate forwards; the balance car can accelerate backwards when the body gravity center of the user inclines backwards;
compared with the first-stage inverted pendulum model shown in the equations (14) and (15), the variable M of the balance vehicle is equal to zero, so the equation (15) can be simplified as follows:
Figure BDA0002747029260000081
simultaneously, the barycenter position with the balance car is located the intermediate position of inverted pendulum pole approximately, then inertia is:
J＝ml2 (17)
substituting the moment of inertia into an equation (16), simplifying the balance car tilt angle control equation into:
Figure BDA0002747029260000082
let variable x1＝θ，
Figure BDA0002747029260000083
And (3) converting the balance car inclination angle control equation into a state space equation form:
Figure BDA0002747029260000084
Figure BDA0002747029260000085
further, the step 2 specifically includes:
the inclination angle of the balance car is used as an input variable, the estimator is used for estimating the real-time angular velocity, the direct adoption of the gyroscope angular velocity data which is easy to interfere is avoided, and the self-adaptive controller is used for automatically estimating the unknown quantity in the motion process of the balance car
Figure BDA0002747029260000086
And simultaneously adjusting the parameters of the controller, knowing that the differential equation of the controlled object is equation (18), the variable y represents the inclination angle of the balance car, the relationship between the inclination angle y and the input driving force variable u is expressed by the differential equation:
Figure BDA0002747029260000087
the differential equation of the reference model represents the control performance of the desired controlled object, and the differential equation of the reference model is:
Figure BDA0002747029260000088
wherein a ism1，am0，bm0Is a transfer function coefficient of the reference model;
the error in the tilt angle y can be defined as:
ε＝y-ym (22)
converting differential equations of the controlled object and the reference model into an error model through a filter, and obtaining the error model of the inclination angle as follows:
ε＝Wm(s)(bm0g-θTω-b0u) (23)
Figure BDA0002747029260000089
Figure BDA0002747029260000091
Figure BDA0002747029260000092
wherein the content of the first and second substances,
Figure BDA0002747029260000093
is a filter equation, and k is a filter transfer function coefficient;
to convert the error model into a static error model form, the control terms are selected as:
Figure BDA0002747029260000094
and (3) bringing the control item into an error model to obtain:
Figure BDA0002747029260000095
the enhanced error model is defined as:
Figure BDA0002747029260000096
and (3) bringing the error epsilon into an enhanced error model, and converting the error into a static error model form:
Figure BDA0002747029260000097
the adaptive formula of the static error model is expressed as:
Figure BDA0002747029260000098
the normalized gain can be expressed as:
Figure BDA0002747029260000099
the control method of the invention is simulated by Matlab/Simulink software:
Figure BDA00027470292600000910
from the above balance car model, the differential equation between the inclination angle and the output u is:
Figure BDA0002747029260000101
Figure BDA0002747029260000102
after the differential equation of the reference model is substituted into the simulation value, it can be expressed as:
Figure BDA0002747029260000103
am1＝5，am0＝6，bm0＝6
the differential equations of the controlled object and the reference model can be converted into an error model using a filter, the filter equation being selected as:
Figure BDA0002747029260000104
the resulting error model can be expressed as:
ε＝Wm(s)(bm0g-θTω-b0u) (33)
Figure BDA0002747029260000105
Figure BDA0002747029260000106
after the control quantity is substituted into the simulation value, the control quantity can be expressed as:
Figure BDA0002747029260000107
the enhanced error model is:
Figure BDA0002747029260000108
the adaptive method can be expressed as:
Figure BDA0002747029260000109
the normalized gain can be expressed as:
Figure BDA00027470292600001010
fig. 5 is a simulation diagram of the whole system in Matlab/Simulink, fig. 6 is a regressor part in an enhanced error controller subsystem in the simulation diagram, fig. 7 is an enhanced error model part in the enhanced error controller subsystem in the simulation diagram, fig. 8 is a normalized gain part in the enhanced error controller subsystem in the simulation diagram, fig. 9 is an adaptive algorithm part in the enhanced error controller subsystem in the simulation diagram, and fig. 10 is a control item part in the enhanced error controller subsystem in the simulation diagram.
FIG. 11 shows the difference ε -y between the controlled object state variable and the reference model state variablem-y simulation results; as can be seen from the figure, the state variable error gradually converges to 0, which means that the output of the controlled object gradually tends to the output of the reference model, so that the control characteristic of the controlled object is the same as the characteristic of the reference model.
FIG. 12 is the difference between the estimated balance car parameter and the actual parameter
Figure BDA0002747029260000111
The simulation result of (2); it can be seen from the figure that the error of the estimated parameters gradually converges to 0, which shows that the estimator can estimate the unknown parameters in the balance car model in real time, especially when the effective load on the balance car changes, the estimator can estimate the true value of the load and adjust the control parameters in a targeted manner.

Claims (3)

1. A balance car control method based on an enhanced error model is characterized by comprising the following steps:
step 1, establishing a simplified model for a balance car by utilizing a Newton second law;
and 2, adopting an adaptive control method based on output feedback, and controlling the inclination angle of the balance car by using an adaptive controller based on an enhanced error model.
2. The method as claimed in claim 1, wherein the step 1 specifically comprises:
when the steering problem is not considered temporarily, the balance car system is equivalent to a primary inverted pendulum, the mass of a pendulum rod of the inverted pendulum is m, the total length of the pendulum rod is 2l, the gravity center of the pendulum rod is located in the middle of the rod, namely, the pendulum rod is away from the position with the length of l at the two ends, the inclination angle of the pendulum rod is theta, the rotational inertia of the pendulum rod around a rotating shaft below is J, the transverse acting force of the pendulum rod on the pendulum rod is equal to the driving force of a trolley on the pendulum rod is u, and the vertical acting force of the pendulum rod on the; the mass of the trolley with the bottom capable of moving left and right is M, and the transverse movement distance of the trolley is z;
the horizontal position of the center of gravity of the swing link can be expressed as:
x＝z+lsinθ (1)
the formula (1) is derived, and the acceleration of the gravity center of the swing rod in the horizontal direction can be obtained as follows:
Figure FDA0002747029250000011
similarly, the vertical position of the center of the swing rod is only expressed as:
y＝lcosθ (3)
the formula (3) is derived, and the acceleration in the direction perpendicular to the gravity center of the swing rod can be obtained as follows:
Figure FDA0002747029250000012
because the stress of the swing rod in the horizontal direction is H, according to the Newton second law, the equation of the horizontal direction of the swing rod can be obtained as follows:
Figure FDA0002747029250000013
similarly, because the swing rod is subjected to the action of the gravity mg and the supporting force v in the vertical direction, the vertical direction equation is as follows:
Figure FDA0002747029250000014
meanwhile, under the action of horizontal thrust H and vertical thrust v, the swing rod rotates around the gravity center according to the following equation:
Figure FDA0002747029250000021
carrying out stress analysis on the bottom trolley, wherein the trolley is subjected to driving force u and reaction force H in the horizontal direction, and the motion equation is as follows:
Figure FDA0002747029250000022
the equations of motion of the balance car can be simplified for the formula (5) and the formula (8) as follows:
Figure FDA0002747029250000023
the second motion equation of the balance car can be simplified by the equations (5), (6), (7) and (8):
Figure FDA0002747029250000024
because the inclination angle theta is always near zero degrees when the balance car is in a normal running state, the trigonometric function can be simplified as follows:
sinθ≈θ，cosθ≈1 (11)
the approximation relation (11) is brought into the equations of motion (9), (10), and because
Figure FDA0002747029250000025
The term is smaller than other terms, and if the term is abandoned, the motion equation of the balance car can be simplified as follows:
Figure FDA0002747029250000026
Figure FDA0002747029250000027
equivalently converting differential equations (12), (13):
Figure FDA0002747029250000028
Figure FDA0002747029250000029
controlling the inclination angle theta through the driving force variable u, so that a transfer function from the variable u to the variable theta is stable, setting the set value of the inclination angle theta to be zero, and when a user standing on the balance car controls the gravity center of the body of the user to incline forwards, the adjustment action u on the control of the inclination angle theta can lead the balance car to accelerate forwards; the balance car can accelerate backwards when the body gravity center of the user inclines backwards;
compared with the first-stage inverted pendulum model shown in the equations (14) and (15), the variable M of the balance vehicle is equal to zero, so the equation (15) can be simplified as follows:
Figure FDA00027470292500000210
simultaneously, the barycenter position with the balance car is located the intermediate position of inverted pendulum pole approximately, then inertia is:
J＝ml2 (17)
substituting the moment of inertia into an equation (16), simplifying the balance car tilt angle control equation into:
Figure FDA0002747029250000031
let variable x1＝θ，
Figure FDA0002747029250000032
And (3) converting the balance car inclination angle control equation into a state space equation form:
Figure FDA0002747029250000033
Figure FDA0002747029250000034
。
3. the method as claimed in claim 2, wherein the step 2 specifically includes:
the inclination angle of the balance car is used as an input variable, the estimator is used for estimating the real-time angular velocity, the direct adoption of the gyroscope angular velocity data which is easy to interfere is avoided, and the self-adaptive controller is used for automatically estimating the unknown quantity in the motion process of the balance car
Figure FDA0002747029250000035
And simultaneously adjusting the parameters of the controller, knowing that the differential equation of the controlled object is equation (18), the variable y represents the inclination angle of the balance car, the relationship between the inclination angle y and the input driving force variable u is expressed by the differential equation:
Figure FDA0002747029250000036
the differential equation of the reference model represents the control performance of the desired controlled object, and the differential equation of the reference model is:
Figure FDA0002747029250000037
wherein a ism1，am0，bm0Is a transfer function coefficient of the reference model;
the error in the tilt angle y can be defined as:
ε＝y-ym (22)
the error model for the tilt angle can be obtained as follows:
ε＝Wm(s)(bm0g-θTω-b0u) (23)
Figure FDA0002747029250000038
Figure FDA0002747029250000039
Figure FDA00027470292500000310
wherein the content of the first and second substances,
Figure FDA00027470292500000311
is a filter equation, and k is a filter transfer function coefficient;
to convert the error model into a static error model form, the control terms are selected as:
Figure FDA0002747029250000041
and (3) bringing the control item into an error model to obtain:
Figure FDA0002747029250000042
the enhanced error model is defined as:
Figure FDA0002747029250000043
and (3) bringing the error epsilon into an enhanced error model, and converting the error into a static error model form:
Figure FDA0002747029250000044
the adaptive formula of the static error model is expressed as:
Figure FDA0002747029250000045
the normalized gain can be expressed as:
Figure FDA0002747029250000046
。
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