CN114326405A - Neural network backstepping control method based on error training - Google Patents

Neural network backstepping control method based on error training Download PDF

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CN114326405A
CN114326405A CN202111669359.4A CN202111669359A CN114326405A CN 114326405 A CN114326405 A CN 114326405A CN 202111669359 A CN202111669359 A CN 202111669359A CN 114326405 A CN114326405 A CN 114326405A
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CN114326405B (en
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高会军
郑晓龙
温克寒
李湛
杨学博
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Harbin Institute of Technology
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Abstract

A neural network backstepping control method based on error training solves the problems that the existing neural network backstepping control method is low in convergence speed, and the neural network cannot accurately estimate unmodeled dynamics, so that the tracking error of a system is large, and belongs to the field of neural network backstepping control methods of nonlinear systems. The invention comprises the following steps: s1, establishing a nonlinear n-order system state space model containing unmodeled dynamics, wherein the state variable is [ x ]1，...，xn]T(ii) a S2, determining an error variable z1And zi，z1＝x1‑yd，zi＝xi‑αi‑1Wherein α isi‑1Representing a virtual control function; s3, establishing an error ziThe input of the differential estimator is ziOutput is
Figure DDA0003449180450000011
Figure DDA0003449180450000016
Is composed of
Figure DDA0003449180450000012
(ii) an estimate of (d); s4, obtained in S3
Figure DDA0003449180450000013
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weight of the neural network based on the estimation error to obtain
Figure DDA0003449180450000014
S5, according to alphan、
Figure DDA0003449180450000015
A control input signal for the non-linear system is calculated.

Description

Neural network backstepping control method based on error training
Technical Field
The invention belongs to the field of a neural network backstepping control method of a nonlinear system.
Background
The basic principle of the method is that the neural network can approximate any unknown function with a certain error to estimate the unmodeled dynamics in the system, and the estimated value of the neural network is fed back to the nonlinear system through negative feedback in the design process of the backstepping method, so that the disturbance of the unmodeled dynamics to the system is reduced. Most of the current strategies employ net back-stepping control with final consistent bounding and sigma-adjustment-based neural network weight update strategies. The final consistency and the bounding are defined as stability in infinite time, and the system designed by the method has a low convergence speed and is not suitable for some control systems with strict convergence time of the system state. In addition, the method based on σ adjustment has the effect of reducing the lyapunov function, rather than directly reducing the approximation error of the neural network, and thus has a relatively limited ability to compensate for unmodeled dynamic quantities.
Disclosure of Invention
The invention provides a neural network backstepping control method based on error training, aiming at the problems that the existing neural network backstepping control method is low in convergence speed, and the neural network cannot accurately estimate unmodeled dynamics, so that the system tracking error is larger.
The invention discloses a neural network backstepping control method based on error training, which comprises the following steps:
s1, establishing a nonlinear n-order system state space model containing unmodeled dynamics, wherein the given target signal is ydThe state variable is [ x ]1,...,xn]T；
S2, determining an error variable z1And zi，z1＝x1-yd，zi＝xi-αi-1Wherein α isi-1Representing a virtual control function;
Figure BDA0003449180430000011
Figure BDA0003449180430000012
wherein, i is 1, …, n,
Figure BDA0003449180430000013
denotes the intermediate variable, giG, a non-linear n-th order system intrinsic parameter0＝z0＝0，μi1、μi2、εiAre all constant and are positive numbers, Wi TSi(Xi) Representing a radial basis neural network, Wi＝[wi1,...,wiN]TIs a weight matrix of the neural network, Si(Xi) Representing the basis function vector, S, of a neural networki(Xi) Each of the basis functions in (a) is a gaussian function having the same center and fixed width, Xi＝[x1,...,xi]The input vector of the neural network is defined, and N is the number of nodes of the neural network;
s3, establishing an error ziThe input of the differential estimator is ziOutput is
Figure BDA0003449180430000021
Is composed of
Figure BDA0003449180430000022
(ii) an estimate of (d);
s4, obtained in S3
Figure BDA0003449180430000023
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weight of the neural network based on the estimation error to obtain
Figure BDA0003449180430000024
S5, according to alphan、
Figure BDA0003449180430000025
A control input signal for the non-linear system is calculated.
Preferably, in S3, the differential estimator is:
Figure BDA0003449180430000026
wherein Ψ (y) ═ 1-e-by)/(1+e-by)，
Figure BDA0003449180430000027
λ, η, b are constants and are positive numbers, ωiFor the state quantities of the differential estimator, Ψ (y) is used to estimate the sign function sgn (y),
Figure BDA0003449180430000028
is ziThe first derivative of (a) is,
Figure BDA0003449180430000029
is omegaiThe first derivative of (c), ξ represents the estimation error of the differential estimator,
Figure BDA00034491804300000210
xi isAn upper limit.
Preferably, in S5, the control input signal u of the nonlinear system:
Figure BDA00034491804300000211
Figure BDA00034491804300000212
preferably, in S1, the nonlinear n-order system state space model including unmodeled dynamics is:
Figure BDA00034491804300000213
wherein f isi() is an unknown nonlinear continuous function representing unmodeled dynamics of the system, u represents the control input signal of the nonlinear system, and y represents the output of the nonlinear system;
preferably, in S2, the method for acquiring the virtual control function includes:
using the obtained error ziDesigning a Lyapunov function V:
Figure BDA00034491804300000214
the derivative values are:
Figure BDA0003449180430000031
and selecting a virtual control function according to the derivative of the Lyapunov function.
Preferably, in S1, S isi(Xi) The kth element s ofk(Xi) Represents:
Figure BDA0003449180430000032
where exp represents an exponential function, bkRepresenting the k-th basis function width, C, of the neural networkkIs the k-th basis function center vector, C, of the neural networkk＝[c1,k,...,cn,k]TWherein c isi,kIs a constant.
Preferably, in S4, the one obtained in S3 is used
Figure BDA0003449180430000033
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weights of the neural network based on the estimation error, wherein the weight W isiThe update rate is specifically expressed as:
Figure BDA0003449180430000034
Wnthe update rate is specifically expressed as:
Figure BDA0003449180430000035
wherein the content of the first and second substances,
Figure BDA0003449180430000036
represents a weight WiFirst derivative of (a), r is the learning rate, r>0；
Figure BDA0003449180430000037
And
Figure BDA0003449180430000038
are respectively as
Figure BDA0003449180430000039
And
Figure BDA00034491804300000310
is estimated.
Preferably, the nonlinear n-order system is a motor rotation angle system.
The method has the advantages that by utilizing a fixed time bounded stability theory and an updating strategy based on error training neural network weight, compared with the traditional adaptive neural network control method, the method has the advantages that the system has shorter convergence time and smaller tracking error.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a graph of the motor turn angle response with time on the abscissa, under the method of the present invention;
FIG. 3 is a graph of the unmodeled dynamic curve of the neural network estimation and system under the method of the present invention, with the ordinate being the estimation value of the neural network and the abscissa being time;
FIG. 4 is a graph of neural network weights plotted on the ordinate versus time for the neural network weights under the method of the present invention;
FIG. 5 is a graph of the neural network estimation error in the method of the present invention, with the ordinate being the neural network estimation error and the abscissa being time;
fig. 6 is a graph of the input torque signal of the motor according to the method of the present invention, wherein the ordinate is the input torque of the motor and the abscissa is time.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The invention is further described with reference to the following drawings and specific examples, which are not intended to be limiting.
The neural network backstepping control method based on error training of the embodiment comprises the following steps:
step one, establishing a nonlinear n-order system state space model containing unmodeled dynamics as follows:
Figure BDA0003449180430000041
wherein f isi() is an unknown nonlinear continuous function representing unmodeled dynamics of the system, u represents the control input signal of the nonlinear system, and y represents the output of the nonlinear system; given a target signal of ydThe state variable is [ x ]1,...,xn]T；
Step two, determining an error variable z1And zi，z1＝x1-yd，zi＝xi-αi-1Wherein α isi-1Representing a virtual control function;
the virtual control function of the present embodiment is designed according to a fixed-time bounded stability theory, and the virtual control function designed in the present embodiment is:
Figure BDA0003449180430000042
wherein, i is 1, …, n,
Figure BDA0003449180430000043
denotes the intermediate variable, giG, a non-linear n-th order system intrinsic parameter0＝z0＝0，μi1、μi2、εiAre all constant and are positive numbers, Wi TSi(Xi) Representing a radial basis neural network, Wi＝[wi1,...,wiN]TIs a weight matrix of the neural network, Si(Xi) Representing the basis function vector, S, of a neural networki(Xi) Each of the basis functions in (a) is a gaussian function having the same center and fixed width, Xi＝[x1,...,xi]Is a neural netInputting vectors of the network, wherein N is the number of the neural network nodes;
in this embodiment, the process of designing the virtual control function is as follows: designing a Lyapunov function V by using an error variable; first derivative of time of Lyapunov function
Figure BDA0003449180430000051
And selecting a virtual control function according to the first derivative of the Lyapunov function.
In this step, the lyapunov function V:
Figure BDA0003449180430000052
the derivative values are:
Figure BDA0003449180430000053
step three, establishing error ziThe input of the differential estimator is ziOutput is
Figure BDA0003449180430000054
Is composed of
Figure BDA0003449180430000055
(ii) an estimate of (d);
step four, utilizing the product obtained in step three
Figure BDA0003449180430000056
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weight of the neural network based on the estimation error to obtain
Figure BDA0003449180430000057
Step five, according to alphan、
Figure BDA0003449180430000058
A fixed time neural network back-stepping controller is determined to calculate the control input signal of the nonlinear system.
The fixed time neural network backstepping controller of the embodiment is as follows:
Figure BDA0003449180430000059
wherein u is a control input signal of the nonlinear system; mu.sn1，μn2，εnIs constant and is a positive number, Wn＝[wn1,...,wnN]TIs a weight matrix of the neural network, Sn(Xn) Representing the basis function vector, S, of a neural networkn(Xn) Each of the basis functions in (a) is a gaussian function having the same center and fixed width, Xn＝[x1,...,xn]Is the input vector of the neural network, and N is the number of the nodes of the neural network.
In this embodiment, the vector of the basis function of the neural network is selected as the Gaussian function Si(Xi) The kth element s ofk(Xi) Represents:
Figure BDA00034491804300000510
where exp represents an exponential function, bkRepresenting the k-th basis function width, C, of the neural networkkIs the k-th basis function center vector, C, of the neural networkk＝[c1,k,...,cn,k]TWherein c isi,kN is a constant.
In the step I of the embodiment, the method obtained in the step III is utilized
Figure BDA0003449180430000061
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weights of the neural network based on the estimation error, wherein the weight W isiThe update rate is specifically expressed as:
Figure BDA0003449180430000062
Wnthe update rate is specifically expressed as:
Figure BDA0003449180430000063
wherein the content of the first and second substances,
Figure BDA0003449180430000064
represents a weight WiFirst derivative of (a), r is the learning rate, r>0；
Figure BDA0003449180430000065
And
Figure BDA0003449180430000066
are respectively as
Figure BDA0003449180430000067
And
Figure BDA0003449180430000068
is estimated.
The differential estimator in step three of the present embodiment is a differential estimator with fixed time convergence, and specifically includes:
Figure BDA0003449180430000069
wherein Ψ (y) ═ 1-e-by)/(1+e-by)，
Figure BDA00034491804300000610
λ, η, b are constants and are positive numbers, ωiFor the state quantities of the differential estimator, Ψ (y) is used to estimate the sign function sgn (y),
Figure BDA00034491804300000611
is ziThe first derivative of (a) is,
Figure BDA00034491804300000612
is omegaiThe first derivative of (c), ξ represents the estimation error of the differential estimator,
Figure BDA00034491804300000613
is the upper limit of ξ.
The fixed-time bounded stability of the present embodiment is demonstrated:
substituting (2) and (3) into coordinate transformation comprises:
Figure BDA00034491804300000614
Figure BDA00034491804300000615
wherein
Figure BDA00034491804300000616
Is 0. From (8) (9), the estimation error of the neural network can be written as
Figure BDA00034491804300000617
Figure BDA0003449180430000071
For any continuous function f (X), there is an ideal network weight W*So that
Figure BDA0003449180430000072
Assuming that f is the function to be estimated of the neural network, if the weight of the neural network is decreased according to the gradient algorithm
Figure BDA0003449180430000073
The update is then
Figure BDA0003449180430000074
Therefore, when we follow (5),(6) When the network weight is updated, the following results can be obtained:
Figure BDA0003449180430000075
Figure BDA0003449180430000076
substituting (2), (3), (12) and (13) into
Figure BDA0003449180430000077
To obtain
Figure BDA0003449180430000078
For any constant x, there is a constant ε > 0, such that
Figure BDA0003449180430000079
Therefore, there are:
Figure BDA00034491804300000710
from the inequality of mean, have
Figure BDA00034491804300000711
For xi∈R,i＝1,...,n，ι∈[0，1]Is (| x)1|+...+|xn|)i≤(|x1|)i+...+(|xn|)iAnd (| x)1|+...+|xn|)2≤[(|x1|)2+...+(|xn|)2]N according to (14), (15) and (16) are
Figure BDA0003449180430000081
Wherein mu1＝min{μ11,...,μn1}，μ2＝min{μ12,...,μn2}/n，
Figure BDA0003449180430000082
According to the fixed-time bounded stability theory, the system satisfies
Figure BDA0003449180430000083
Wherein
Figure BDA0003449180430000084
Is a constant, the tracking error of the system will be at a fixed time TsConverge into a small neighborhood around the origin, TsThe values of (A) are:
Figure BDA0003449180430000085
after the syndrome is confirmed.
Example 1:
consider the following motor rotation angle system:
Figure BDA0003449180430000086
wherein x is1(unit rad) is the output angle, x, of the motor angle system2(unit rad/s) is the angular velocity of the motor. j (unit N.m)2) The system input u (in N · m) is the input torque for the system moment of inertia.
Taking the initial value of the system as x1(0)＝0,x2(0) 0 and 20. To demonstrate the simulation, assume that the system unmodeled dynamics are
Figure BDA0003449180430000087
This function cannot be used directly to design the system control input u. System target signal set to yd(t)＝0.1。
The parameters in the virtual control function (4) and the system control input function (5) are taken as mu11＝15，μ12＝20，ε1＝0.4，μ21＝150，μ22＝3，ε20.01. (6) The medium neural network comprises 12 nodes, the elements of the central vector of the basis function are all 0, the widths are all 0.5, and the initial values of the weights are all 0. In the differential estimators (7) and (8), the parameters λ and η are 10, and b and 1, respectively. The parameter in the update strategy (11) based on error training is taken as r-0.0002. The system sampling interval time is 0.001 seconds.
The control is carried out in the manner of fig. 1, and fig. 2 shows a motor rotation angle response curve under the action of the method of the invention; FIG. 3 is a graph of estimated values of a neural network under the method of the present invention; FIG. 4 is a graph illustrating neural network weight update under the method of the present invention; FIG. 5 is a graph of the neural network estimated error under the method of the present invention; FIG. 6 is a graph of motor input torque in accordance with the method of the present invention.
And conclusion one: from fig. 2, it can be obtained that the convergence rate of the system is faster and the tracking error of the system is smaller under the method of the present invention.
And a second conclusion: 3-5, the neural network can adjust the network weight according to the error better under the method of the present invention, thereby estimating the unmodeled dynamics in the system more accurately, and leading the system to have smaller tracking error.
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that features described in different dependent claims and herein may be combined in ways different from those described in the original claims. It is also to be understood that features described in connection with individual embodiments may be used in other described embodiments.

Claims (10)

1. A neural network backstepping control method based on error training is characterized by comprising the following steps:
s1, establishing a nonlinear n-order system state space model containing unmodeled dynamics, wherein the given target signal is ydThe state variable is [ x ]1，...，xn]T；
S2, determining an error variable z1And zi，z1＝x1-yd，zi＝xi-αi-1Wherein α isi-1Representing a virtual control function;
Figure FDA0003449180420000011
Figure FDA0003449180420000012
wherein, i is 1, …, n,
Figure FDA0003449180420000013
denotes the intermediate variable, giG, a non-linear n-th order system intrinsic parameter0＝z0＝0，μi1、μi2、εiAre all constant and are positive numbers, Wi TSi(Xi) Representing a radial basis neural network, Wi＝[wi1，...，wiN]TIs a weight matrix of the neural network, Si(Xi) Representing the basis function vector, S, of a neural networki(Xi) Each of the basis functions in (a) is a gaussian function having the same center and fixed width, Xi＝[x1，...，xi]The input vector of the neural network is defined, and N is the number of nodes of the neural network;
s3, establishing an error ziThe input of the differential estimator is ziOutput is
Figure FDA0003449180420000014
Figure FDA0003449180420000015
Is composed of
Figure FDA0003449180420000016
(ii) an estimate of (d);
s4, obtained in S3
Figure FDA0003449180420000017
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weight of the neural network based on the estimation error to obtain
Figure FDA0003449180420000018
S5, according to
Figure FDA0003449180420000019
The control input signal of the nonlinear system is calculated.
2. The error-training-based neural network backstepping control method of claim 1, wherein in S3, the differential estimator is:
Figure FDA00034491804200000110
wherein Ψ (y) ═ 1-e-by)/(1+e-by)，
Figure FDA00034491804200000111
λ, η, b are constants and are positive numbers, ωiFor the state quantities of the differential estimator, Ψ (y) is used to estimate the sign function sgn (y),
Figure FDA00034491804200000112
is ziThe first derivative of (a) is,
Figure FDA00034491804200000113
is omegaiThe first derivative of (c), ξ represents the estimation error of the differential estimator,
Figure FDA00034491804200000114
is the upper limit of ξ.
3. The neural network backstepping control method based on error training as claimed in claim 1, wherein in S5, the control input signal u of the nonlinear system:
Figure FDA0003449180420000021
Figure FDA0003449180420000022
4. the neural network backstepping control method based on error training as claimed in claim 1, wherein in S1, the non-linear n-th order system state space model containing unmodeled dynamics is:
Figure FDA0003449180420000023
wherein f isi(. cndot.) is an unknown nonlinear continuous function representing unmodeled dynamics of the system, u represents the control input signal of the nonlinear system, and y represents the output of the nonlinear system.
5. The neural network backstepping control method based on error training as claimed in claim 1, wherein in S2, the method for obtaining the virtual control function includes:
using the obtained error ziDesign ofLyapunov function V:
Figure FDA0003449180420000024
the derivative values are:
Figure FDA0003449180420000025
and selecting a virtual control function according to the derivative of the Lyapunov function.
6. The neural network backstepping control method based on error training as claimed in claim 1, wherein in S1, the S isi(Xi) The kth element s ofk(Xi) Represents:
Figure FDA0003449180420000026
where exp represents an exponential function, bkRepresenting the k-th basis function width of the neural network,
Figure FDA0003449180420000027
is the k-th basis function center vector, C, of the neural networkk＝[c1，k，…，cn，k]TWherein c isi，kIs a constant.
7. The neural network backstepping control method based on error training as claimed in claim 1, wherein in S4, the data obtained in S3 is utilized
Figure FDA0003449180420000028
Calculating the estimation error of the current radial basis function neural network, and performing gradient descent training on the weights of the neural network based on the estimation error, wherein the weight W isiThe update rate is specifically expressed as:
Figure FDA0003449180420000031
Wnthe update rate is specifically expressed as:
Figure FDA0003449180420000032
wherein the content of the first and second substances,
Figure FDA0003449180420000033
represents a weight WiR is the learning rate, r > 0;
Figure FDA0003449180420000034
and
Figure FDA0003449180420000035
are respectively as
Figure FDA0003449180420000036
And
Figure FDA0003449180420000037
is estimated.
8. The error-training-based neural network back-stepping control method of claim 1, wherein the nonlinear n-th order system is a motor rotation angle system.
9. A storage device readable by a computer, the storage device storing a computer program, wherein the computer program when executed implements the method of any of claims 1 to 8.
10. An error-training-based neural network back-stepping control system, comprising a storage device, a processor, and a computer program stored in the storage device and executable on the processor, wherein the processor executes the computer program to implement the method of any one of claims 1 to 8.
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