CN110716434A - Inverted pendulum system neural network tracking control method with self-adaptive friction compensation - Google Patents

Inverted pendulum system neural network tracking control method with self-adaptive friction compensation Download PDF

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CN110716434A
CN110716434A CN201911103644.2A CN201911103644A CN110716434A CN 110716434 A CN110716434 A CN 110716434A CN 201911103644 A CN201911103644 A CN 201911103644A CN 110716434 A CN110716434 A CN 110716434A
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friction
neural network
inverted pendulum
pendulum system
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平兆武
刘晨曦
张宏伟
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Hefei University of Technology
Hefei Polytechnic University
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Abstract

The invention discloses an inverted pendulum system neural network tracking control method with self-adaptive friction compensation, which is characterized by comprising the following steps of: step 1, establishing a discrete time model of an inverted pendulum system; step 2, describing the position tracking control problem of the inverted pendulum system as a discrete time nonlinear servo control problem; step 3, under the condition of not considering friction, designing a neural network controller based on a discrete time nonlinear output regulation theory; step 4, designing a self-adaptive friction compensator; and 5, combining the neural network controller with the self-adaptive friction compensator to obtain a final controller. The invention solves the problem that the friction force seriously influences the tracking performance in actual control, and has higher-precision position tracking performance. In addition, the self-adaptive friction compensator does not need to identify the friction force of the equipment off line, and is easy to realize.

Description

Inverted pendulum system neural network tracking control method with self-adaptive friction compensation
Technical Field
The invention relates to the field of nonlinear system control, in particular to an inverted pendulum system neural network tracking control method with adaptive friction compensation.
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 feed-forward control and internal model control. The inverted pendulum system is a nonlinear, strongly coupled, multivariable and naturally unstable system, which has become the benchmark system for testing various advanced control theories, and the position tracking control problem can be described as a discrete-time nonlinear servo control problem. Although some documents solve the position tracking control problem of the inverted pendulum system by using a polynomial approximation method and a neural network approximation method, the friction model only contains viscous friction and only provides a simulation result, which is greatly limited in practical application.
On the other hand, friction is prevalent in mechanical systems and is not negligible for control systems that require high precision tracking performance. Without proper compensation, friction may cause significant tracking errors. Friction compensation is therefore an important objective in motion control. A common approach is to use an additional control force to counteract the effect of friction. Friction estimation is a challenging problem due to the complexity of friction, often exhibiting unpredictable behavior.
Disclosure of Invention
Based on the technical problems in the background art, the invention provides an inverted pendulum system neural network tracking control method with adaptive friction compensation. Aiming at an inverted pendulum system, firstly, under the condition of not considering friction, a neural network controller is designed based on a discrete time nonlinear output regulation theory to realize position tracking; the adaptive friction compensator is then designed to overcome the effect of friction on tracking performance.
The technical scheme adopted by the invention is as follows:
the inverted pendulum system neural network tracking control method with the self-adaptive friction compensation is characterized by comprising the following steps of:
step 1, establishing a discrete time model of an inverted pendulum system;
step 2, describing the position tracking control problem of the inverted pendulum system as a discrete time nonlinear servo control problem;
step 3, under the condition of not considering friction, designing a neural network controller based on a discrete time nonlinear output regulation theory;
step 4, designing a self-adaptive friction compensator;
and 5, combining the neural network controller with the self-adaptive friction compensator to obtain a final controller.
Further, the inverted pendulum system neural network tracking control method with adaptive friction compensation is characterized in that in step 1, the discrete time model of the inverted pendulum system is described as follows:
x1(k+1)＝x1(k)+Tx2(k+1)
Figure BDA0002270591110000031
x3(k+1)＝x3(k)+Tx4(k)
Figure BDA0002270591110000032
y(k)＝x1(k)
(1)
wherein the content of the first and second substances,
Figure BDA0002270591110000033
indicating the position and speed of the vehicle, McThe mass of the trolley is represented and,
Figure BDA0002270591110000034
representing the angle between the pendulum and the vertical and its angular velocity, u representing the control force, lrRepresenting the length from the centre of mass to the bottom of the pendulum, MrIndicating the mass of the uniform pendulum bar, IrRepresenting the moment of inertia of the pendulum and y the output of the system.
Further, the method for neural network tracking control of the inverted pendulum system with adaptive friction compensation is characterized in that in step 2, the problem of position tracking control of the inverted pendulum system is described as a discrete-time nonlinear servo control problem, and the process is as follows:
2.1, assuming that the position reference track of the inverted pendulum system is a sinusoidal signal yd(k) This position reference trajectory may be generated by an external system of the form:
Figure BDA0002270591110000035
its output is yd(k)＝v1(k)；
2.2, define the tracking error as
e(k)＝x1(k)-v1(k) (3)
2.3, the inverted pendulum system (1) and the external system (2) can be written in a compact form as the system (4). The position tracking control problem of the inverted pendulum system has been described as a discrete time nonlinear servo control problem, and the control target is to make the steady-state tracking error small enough on the premise of ensuring the stability of the closed-loop system;
Figure BDA0002270591110000041
wherein x (k) ═ x1(k),x2(k),x3(k),x4(k))·，H(x(k),u(k),v(k))＝x1(k)-v1(k)，
Figure BDA0002270591110000042
Further, the method for tracking and controlling the neural network of the inverted pendulum system with the adaptive friction compensation is characterized in that in step 3, under the condition of not considering the friction, a neural network controller is designed based on a discrete-time nonlinear output regulation theory, and the process is as follows:
3.1, solving the Jacobian matrix of the system as follows:
Figure BDA0002270591110000043
can prove that
Figure BDA0002270591110000044
Is calmable and there is a matrix K such thatAll within the unit circle;
3.2, solving the discrete regulator equation, wherein the process is as follows:
the discrete regulator equation has the form:
Figure BDA0002270591110000051
the partial solution to the discrete regulator equation can be found by calculation as follows:
Figure BDA0002270591110000052
where E is the identity matrix. Let χ (v) col (x)3(v),x4(v) ). Then χ (v) satisfies the following equation:
Figure BDA0002270591110000053
wherein
χ＝col(x3,x4)
Figure BDA0002270591110000054
Figure BDA0002270591110000055
Figure BDA0002270591110000056
3.3, solving the approximate solution of the chi (v) by using the feedforward neural network function, wherein the process is as follows:
since the exact solution for χ (v) is not available, the approximate solution for χ (v) can be found by the following feed forward neural network function
Figure BDA0002270591110000061
Wherein ψ (λ) ═ 1-e-λ)/(1+e-λ) The weight vector W is formed by real numbersWherein N is 1, …, N, m is 1, 2;
let χ (W, v) col (x)3(W,v),x4(W, v)). According to the general approximation theorem, given>0 and a tight set containing the origin
Figure BDA0002270591110000063
Since χ (v) is sufficiently smooth, N is present and W satisfies
Figure BDA0002270591110000064
Searching a group of proper weight vectors W by a parameter optimization method; order to
J(W，v)＝0.5F1 2(W，v)+0.5F2 2(W，v) (11)
To make it possible to
Figure BDA0002270591110000067
Small enough, a new objective function is constructed:
Figure BDA0002270591110000068
wherein
Figure BDA0002270591110000069
Is a dense finite set of Γ; appropriate weight vector
Figure BDA00022705911100000610
Can be obtained by the following weight update law
Figure BDA00022705911100000611
Where ρ isjFor training step length, the initial weight W0Is a set of randomly generated vectors; according to the weight value obtained by off-line training, x can be calculated3(v),x4(v) An approximation of (d);
3.4, designing a neural network controller as follows:
Figure BDA00022705911100000612
wherein
Figure BDA0002270591110000071
Figure BDA0002270591110000072
Further, the inverted pendulum system neural network tracking control method with adaptive friction compensation is characterized in that in step 4, an adaptive friction compensator is designed, specifically:
4.1, the above design considerations are only for the ideal case, i.e. no friction effect,
in practice, small cars are often affected by friction, typically including static friction, coulomb friction, and viscous friction, and the models are as follows:
Ff(k)＝fc(k)sign(vc(k))+bvc(k) (15)
wherein f isc(k) B is a viscous friction parameter, v is a time-varying friction parameter, b is a viscous friction parameter, and v is a coefficient of frictionc＝x2Is the speed;
4.2, to eliminate the effect of friction, the form of the compensator is defined as follows:
Figure BDA0002270591110000073
wherein
Figure BDA0002270591110000074
Is fc(k) The estimated amount of (a) is,
Figure BDA0002270591110000075
is an estimate of b;
4.3, designing an updating law: to determine
Figure BDA0002270591110000076
And
Figure BDA0002270591110000077
the following update law is adopted:
Figure BDA0002270591110000079
wherein P is1,P2,λ∈R+Adjustable parameters defined for the user.
Further, the inverted pendulum system neural network tracking control method with adaptive friction compensation is characterized in that the final controller in step 5 is:
Figure BDA0002270591110000081
the invention has the advantages that:
the neural network tracking control method with the self-adaptive friction compensation solves the problem that the tracking performance is seriously influenced by the friction force in the actual control, and has higher-precision position tracking performance. In addition, the self-adaptive friction compensator provided by the invention does not need to carry out off-line identification on the friction force of the equipment, and is easy to realize.
Drawings
FIG. 1 is a block diagram of a neural network control method with adaptive friction compensation.
FIG. 2 is a schematic diagram of experimental apparatus control.
Fig. 3 is a diagram illustrating the tracking effect of the position signal.
Fig. 4 is a position signal tracking error map.
Fig. 5 is a diagram of the controller output.
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.
Examples are given.
The inverted pendulum system neural network tracking control method with the self-adaptive friction compensation comprises the following steps:
step 1, establishing a discrete time model of an inverted pendulum system, wherein the discrete time model of the inverted pendulum system is described as follows:
x1(k+1)＝x1(k)+Tx2(k+1)
Figure BDA0002270591110000091
x3(k+1)＝x3(k)+Tx4(k)
Figure BDA0002270591110000092
y(k)＝x1(k)
(1)
wherein the content of the first and second substances,
Figure BDA0002270591110000093
indicating the position and speed of the vehicle, McThe mass of the trolley is represented and,representing the angle between the pendulum and the vertical and its angular velocity, u representing the control force, lrRepresenting the length from the centre of mass to the bottom of the pendulum, MrIndicating the mass of the uniform pendulum bar, IrRepresenting the moment of inertia of the pendulum and y the output of the system.
Step 2, describing the position tracking control problem of the inverted pendulum system as a discrete time nonlinear servo control problem, and the process is as follows:
2.1, assuming that the position reference track of the inverted pendulum system is a sinusoidal signal yd(k) This position reference trajectory may be generated by an external system of the form:
Figure BDA0002270591110000095
its output is yd(k)＝v1(k)。
2.2, define the tracking error as
e(k)＝x1(k)-v1(k) (3)
2.3, the inverted pendulum system (1) and the external system (2) can be written in a compact form as the system (4). The position tracking control problem of the inverted pendulum system has been described as a discrete time nonlinear servo control problem, and the control target is to make the steady-state tracking error small enough on the premise of ensuring the stability of the closed-loop system;
Figure BDA0002270591110000101
wherein x (k) ═ x1(k),x2(k),x3(k),x4(k))·，H(x(k),u(k),v(k))＝x1(k)-v1(k)，
Figure BDA0002270591110000102
And 3, under the condition of not considering friction, designing a neural network controller based on a discrete time nonlinear output regulation theory, wherein the process is as follows:
3.1, solving the Jacobian matrix of the system as follows:
can prove that
Figure BDA0002270591110000104
Is calmable and there is a matrix K such that
Figure BDA0002270591110000105
Are all within the unit circle.
3.2, solving the discrete regulator equation, wherein the process is as follows:
the discrete regulator equation has the form:
Figure BDA0002270591110000111
the partial solution to the discrete regulator equation can be found by calculation as follows:
Figure BDA0002270591110000112
where E is the identity matrix. Let χ (v) col (x)3(v),x4(v) ). Then χ (v) satisfies the following equation:
Figure BDA0002270591110000113
wherein
χ＝col(x3,x4)
Figure BDA0002270591110000114
Figure BDA0002270591110000115
Figure BDA0002270591110000116
3.3, solving the approximate solution of the chi (v) by using the feedforward neural network function, wherein the process is as follows:
since the exact solution for χ (v) is not available, the approximate solution for χ (v) can be found by the following feed forward neural network function
Figure BDA0002270591110000121
Wherein ψ (λ) ═ 1-e-λ)/(1+e-λ) The weight vector W is formed by real numbers
Figure BDA0002270591110000122
The composition is shown in the specification, wherein N is 1, …, and N, m is 1, 2.
Let χ (W, v) col (x)3(W,v),x4(W, v)). According to the general approximation theorem, given>0 and a tight set containing the origin
Figure BDA0002270591110000123
Since χ (v) is sufficiently smooth, N is present and W satisfies
Figure BDA0002270591110000124
Next, a set of suitable weight vectors W is found by a parameter optimization method. Order to
Figure BDA0002270591110000125
J(W，v)＝0.5F1 2(W，v)+0.5F2 2(W，v) (11)
To make it possible to
Figure BDA0002270591110000127
Small enough, a new objective function is constructed:
wherein
Figure BDA0002270591110000129
Is a dense finite set of Γ. Appropriate weight vectorCan be obtained by the following weight update law
Where ρ isjFor training step length, the initial weight W0Is a set of randomly generated vectors. According to the weight value obtained by off-line training, x can be calculated3(v),x4(v) An approximation of (d).
3.4, designing a neural network controller as follows:
Figure BDA00022705911100001212
wherein
Figure BDA0002270591110000131
Step 4, designing a self-adaptive friction compensator:
4.1, the above design considerations are only without frictional effects in the ideal case.
In practice, small cars are often affected by friction, typically including static friction, coulomb friction, and viscous friction, and the models are as follows:
Ff(k)＝fc(k)sign(vc(k))+bvc(k) (15)
wherein f isc(k) Is time-varying at > 0The friction parameter of (a) represents the sum of static friction and coulomb friction, b is a viscous friction parameter, v isc＝x2Is the velocity.
4.2, to eliminate the effect of friction, the form of the compensator is defined as follows:
wherein
Figure BDA0002270591110000134
Is fc(k) The estimated amount of (a) is,
Figure BDA0002270591110000135
is an estimate of b.
And 4.3, designing an updating law. To determineAnd
Figure BDA0002270591110000137
the following update law is adopted:
Figure BDA0002270591110000139
wherein P is1,P2,λ∈R+Adjustable parameters defined for the user.
And 5, combining the neural network controller with the self-adaptive friction compensator to obtain a final controller:
FIG. 1 is a schematic block diagram of a neural network control method with adaptive friction compensation. In order to verify the effectiveness of the method, the invention carries out experimental verification on the control effect of the neural network controller (19) with the adaptive friction compensation. The hardware schematic of the experimental setup is shown in fig. 2. The specific parameters of the experimental equipment were as follows:
trolley mass Mc1.29Kg, mass M of pendulum rodr0.075Kg pendulum bar length l of 1/2r0.175Kg, moment of inertia
Figure BDA0002270591110000142
The sampling time T is 0.005 s.
The controller parameters are set as:
K＝[38.9217 26.1885 87.2114 13.3724]，λ＝10，P1＝5，P2＝10，
and
Figure BDA0002270591110000145
the parameters of training are set as follows:
N＝25，Γ＝{v∈R2|||v||≤0.12}，ω＝2.5π，
Figure BDA0002270591110000146
Figure BDA0002270591110000147
this example tracks a sinusoidal reference signal with a trace of 0.06sin (2.5 π kT) m. Fig. 3 is a diagram showing effects of a position tracking experiment in the inverted pendulum system, fig. 4 is a diagram showing a position tracking error, and fig. 5 is a diagram showing an output of the controller (19). As can be seen from the experimental results of fig. 3-4: under the action of the neural network controller with the self-adaptive friction compensation, the interference of friction force can be effectively inhibited, and the position reference signal can be quickly tracked.
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 (6)

1. The inverted pendulum system neural network tracking control method with the self-adaptive friction compensation is characterized by comprising the following steps of:
step 1, establishing a discrete time model of an inverted pendulum system;
step 2, describing the position tracking control problem of the inverted pendulum system as a discrete time nonlinear servo control problem;
step 3, under the condition of not considering friction, designing a neural network controller based on a discrete time nonlinear output regulation theory;
step 4, designing a self-adaptive friction compensator;
and 5, combining the neural network controller with the self-adaptive friction compensator to obtain a final controller.
2. The neural network tracking control method for an inverted pendulum system with adaptive friction compensation as claimed in claim 1, wherein in step 1, the discrete time model of the inverted pendulum system is described as follows:
x1(k+1)＝x1(k)+Tx2(k+1)
Figure FDA0002270591100000011
x3(k+1)＝x3(k)+Tx4(k)
Figure FDA0002270591100000012
y(k)＝x1(k) (1)
wherein the content of the first and second substances,
Figure FDA0002270591100000013
indicating the position and speed of the vehicle, McRepresenting the mass of the car, x3＝β∈(-π,π),
Figure FDA0002270591100000014
Representing the angle between the pendulum and the vertical and its angular velocity, u representing the control force, lrRepresenting the length from the centre of mass to the bottom of the pendulum, MrIndicating the mass of the uniform pendulum bar, IrRepresenting the moment of inertia of the pendulum and y the output of the system.
3. The neural network tracking control method for the inverted pendulum system with adaptive friction compensation as claimed in claim 2, wherein in step 2, the position tracking control problem of the inverted pendulum system is described as a discrete time nonlinear servo control problem by the following process:
2.1, assuming that the position reference track of the inverted pendulum system is a sinusoidal signal yd(k) This position reference trajectory may be generated by an external system of the form:
Figure FDA0002270591100000021
its output is yd(k)＝v1(k)；
2.2, define the tracking error as
e(k)＝x1(k)-v1(k) (3)
2.3, the inverted pendulum system and the external system can be written as a compact form as the system (4), and the position tracking control problem of the inverted pendulum system has been described as a discrete-time nonlinear servo control problem, and the control target is to make the steady-state tracking error small enough on the premise of ensuring the stability of the closed-loop system;
Figure FDA0002270591100000022
wherein x (k) ═ x1(k),x2(k),x3(k),x4(k))·，H(x(k),u(k),v(k))＝x1(k)-v1(k)，
4. The inverted pendulum system neural network tracking control method with adaptive friction compensation as claimed in claim 3, wherein step 3, under the condition of not considering friction, based on discrete time nonlinear output regulation theory, designing the neural network controller, and the process is as follows:
3.1, solving the Jacobian matrix of the system as follows:
Figure FDA0002270591100000032
can prove that
Figure FDA0002270591100000033
Is calmable and there is a matrix K such that
Figure FDA0002270591100000034
All within the unit circle;
3.2, solving the discrete regulator equation, wherein the process is as follows:
the discrete regulator equation has the form:
Figure FDA0002270591100000041
the partial solution to the discrete regulator equation can be found by calculation as follows:
Figure FDA0002270591100000042
wherein E is a monoA bit matrix; let χ (v) col (x)3(v),x4(v) χ (v) satisfies the following equation:
Figure FDA0002270591100000043
wherein
χ＝col(x3,x4)
Figure FDA0002270591100000044
Figure FDA0002270591100000045
Figure FDA0002270591100000046
3.3, solving the approximate solution of the chi (v) by using the feedforward neural network function, wherein the process is as follows:
since the exact solution for χ (v) is not available, the approximate solution for χ (v) can be found by the following feed forward neural network function
Figure FDA0002270591100000051
Wherein ψ (λ) ═ 1-e-λ)/(1+e-λ) The weight vector W is formed by real numbers
Figure FDA0002270591100000052
Wherein N is 1, …, N, m is 1, 2;
let χ (W, v) col (x)3(W,v),x4(W, v)); according to the general approximation theorem, given>0 and a tight set containing the origin
Figure FDA0002270591100000053
Since χ (v) is sufficiently smooth, N is present and W satisfies
Figure FDA0002270591100000054
Searching a group of proper weight vectors W by a parameter optimization method; order to
Figure FDA0002270591100000055
J(W,v)＝0.5F1 2(W,v)+0.5F2 2(W,v) (11)
To make it possible to
Figure FDA0002270591100000056
Small enough, a new objective function is constructed:
Figure FDA0002270591100000057
wherein
Figure FDA0002270591100000058
Is a dense finite set of Γ; appropriate weight vector
Figure FDA0002270591100000059
Can be obtained by the following weight update law
Figure FDA00022705911000000510
Where ρ isjFor training step length, the initial weight W0Is a set of randomly generated vectors; according to the weight value obtained by off-line training, x can be calculated3(v),x4(v) An approximation of (d);
3.4, designing a neural network controller as follows:
Figure FDA00022705911000000511
wherein
Figure FDA0002270591100000061
Figure FDA0002270591100000062
5. The inverted pendulum system neural network tracking control method with adaptive friction compensation as claimed in claim 4, wherein in step 4, an adaptive friction compensator is designed, specifically:
4.1, the above design consideration is only the ideal case, i.e. no friction effect, and in practice, small and medium vehicles are often influenced by friction, which usually includes static friction, coulomb friction and viscous friction, and the model is as follows:
Ff(k)＝fc(k)sign(vc(k))+bvc(k) (15)
wherein f isc(k) B is a viscous friction parameter, v is a time-varying friction parameter, b is a viscous friction parameter, and v is a coefficient of frictionc＝x2Is the speed;
4.2, to eliminate the effect of friction, the form of the compensator is defined as follows:
Figure FDA0002270591100000063
wherein
Figure FDA0002270591100000064
Is fc(k) The estimated amount of (a) is,is an estimate of b;
4.3, designing an updating law: to determine
Figure FDA0002270591100000066
And
Figure FDA0002270591100000067
the following update law is adopted:
Figure FDA0002270591100000068
wherein P is1,P2,λ∈R+Adjustable parameters defined for the user.
6. The inverted pendulum system neural network tracking control method with adaptive friction compensation of claim 5, wherein the final controller of step 5 is:
Figure FDA0002270591100000071
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Application publication date: 20200121