CN114326405B - 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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CN114326405B
CN114326405B CN202111669359.4A CN202111669359A CN114326405B CN 114326405 B CN114326405 B CN 114326405B CN 202111669359 A CN202111669359 A CN 202111669359A CN 114326405 B CN114326405 B CN 114326405B
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CN114326405A (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 can not accurately estimate unmodeled dynamics, so that system tracking errors are causedThe method has a large problem, 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 a state variable is [ x ] 1 ，...，x n ] T (ii) a S2, determining an error variable z 1 And z i ，z 1 ＝x 1 ‑y d ，z i ＝x i ‑α i‑1 Wherein α is i‑1 Representing a virtual control function; s3, establishing an error z i The input of the differential estimator is z i Output is as
Figure DDA0003449180450000011
Figure DDA0003449180450000016
Is composed of
Figure DDA0003449180450000012
(ii) is estimated; s4, obtained by 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 alpha n 、
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 a given target signal is y d The state variable is [ x ] 1 ,...,x n ] T ；
S2, determining an error variable z 1 And z i ，z 1 ＝x 1 -y d ，z i ＝x i -α i-1 Wherein α is i-1 Representing a virtual control function;
Figure GDA0003936571700000011
Figure GDA0003936571700000012
wherein i =1, \8230;, n,
Figure GDA0003936571700000013
denotes the intermediate variable, g i Is a non-linear n-th order system intrinsic parameter, g 0 ＝z 0 ＝0，μ i1 、μ i2 、ε i Are all constant and are positive numbers, W i T S i (X i ) Representing a radial basis neural network, W i ＝[w i1 ,...,w iN ] T Is a weight matrix of the neural network, S i (X i ) Representing the basis function vector, S, of a neural network i (X i ) Each of the basis functions in (1) is a gaussian function having the same center and fixed width, X i ＝[x 1 ,...,x i ]The input vector is an input vector of the neural network, and N is the number of nodes of the neural network;
s3, establishing an error z i The input of the differential estimator is z i Output is
Figure GDA0003936571700000021
Is->
Figure GDA0003936571700000022
(ii) an estimate of (d);
s4, obtained by S3
Figure GDA0003936571700000023
Calculating the estimation error of the current radial basis function neural network, performing gradient descent training on the weight of the neural network based on the estimation error, and obtaining->
Figure GDA0003936571700000024
S5, according to alpha n 、
Figure GDA0003936571700000025
A control input signal for the non-linear system is calculated. />
Preferably, in S3, the differential estimator is:
Figure GDA0003936571700000026
where Ψ (y) = (1-e) -by )/(1+e -by )，
Figure GDA0003936571700000027
λ, η, b are constants and positive numbers, ω i Ψ (y) is used to estimate the sign function sgn (y), - @, for the state quantity of the differential estimator>
Figure GDA0003936571700000028
Is z i In the first derivative of (D), in conjunction with a signal from a signal pickup device>
Figure GDA0003936571700000029
Is omega i ξ denotes the estimation error of the differential estimator, b @>
Figure GDA00039365717000000210
Is the upper limit of ξ.
Preferably, in S5, the control input signal u of the nonlinear system:
Figure GDA00039365717000000211
Figure GDA00039365717000000212
preferably, in S1, the nonlinear n-order system state space model including unmodeled dynamics is:
Figure GDA00039365717000000213
wherein f is i (. 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 nonlinear lineAn output of the sexual system;
preferably, in S2, the method for acquiring the virtual control function includes:
using the obtained error z i Designing a Lyapunov function V:
Figure GDA00039365717000000214
the derivative values are:
Figure GDA00039365717000000310
and selecting a virtual control function according to the derivative of the Lyapunov function.
Preferably, in S1, S is i (X i ) The kth element s of k (X i ) Represents:
Figure GDA0003936571700000031
where exp represents an exponential function, b k Representing the k-th basis function width, C, of the neural network k Is the k-th basis function center vector, C, of the neural network k ＝[c 1,k ,...,c n,k ] T Wherein c is i,k Is a constant.
Preferably, in S4, the compound obtained in S3 is used
Figure GDA0003936571700000032
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 is i The update rate is specifically expressed as:
Figure GDA0003936571700000033
W n the update rate is specifically expressed as:
Figure GDA0003936571700000034
/>
wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0003936571700000035
represents a weight W i First derivative of (a), r is the learning rate, r>0；/>
Figure GDA0003936571700000036
And/or>
Figure GDA0003936571700000037
Are respectively in>
Figure GDA0003936571700000038
And/or>
Figure GDA0003936571700000039
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 diagram of a dynamic curve of the neural network estimation and system unmodeled 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 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 a motor input torque signal plotted on the ordinate versus time on the abscissa, in accordance with the method of the present invention.
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 comprises the following steps:
Figure GDA0003936571700000041
wherein f is i () 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 y d The state variable is [ x ] 1 ,...,x n ] T ；
Step two, determining an error variable z 1 And z i ，z 1 ＝x 1 -y d ，z i ＝x i -α i-1 Wherein α is i-1 Representing 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 GDA0003936571700000042
Figure GDA0003936571700000043
wherein i =1, \8230;, n,
Figure GDA0003936571700000044
denotes the intermediate variable, g i G, a non-linear n-th order system intrinsic parameter 0 ＝z 0 ＝0，μ i1 、μ i2 、ε i Are all constant and positive, are>
Figure GDA0003936571700000045
Representing a radial basis neural network, W i ＝[w i1 ,...,w iN ] T Is a weight matrix of the neural network, S i (X i ) Representing the basis function vector, S, of a neural network i (X i ) Each of the basis functions in (a) is a gaussian function having the same center and fixed width, X i ＝[x 1 ,...,x i ]The input vector of the neural network is defined, and N is the number of nodes of the neural network;
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 GDA0003936571700000051
And selecting a virtual control function according to the first derivative of the Lyapunov function.
In this step, the lyapunov function V:
Figure GDA0003936571700000052
having a derivative value of：
Figure GDA0003936571700000053
Step three, establishing error z i The input of the differential estimator is z i Output is
Figure GDA0003936571700000054
Is->
Figure GDA0003936571700000055
(ii) an estimate of (d);
step four, utilizing the product obtained in step three
Figure GDA0003936571700000056
Calculating the estimation error of the current radial basis function neural network, performing gradient descent training on the weight of the neural network based on the estimation error, and obtaining->
Figure GDA0003936571700000057
Step five, according to alpha n 、
Figure GDA0003936571700000058
And determining a fixed time neural network backstepping controller to calculate a control input signal of the nonlinear system.
The fixed time neural network backstepping controller of the embodiment is as follows:
Figure GDA0003936571700000059
wherein u is a control input signal of the nonlinear system; mu.s n1 ，μ n2 ，ε n Is constant and is a positive number, W n ＝[w n1 ,...,w nN ] T Is a weight matrix of the neural network, S n (X n ) Representing the basis function vector, S, of a neural network n (X n ) Each of which is a gaussian function having the same center and fixed width，X n ＝[x 1 ,...,x n ]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 from Gaussian function S i (X i ) The kth element s of k (X i ) Represents:
Figure GDA00039365717000000510
wherein exp represents an exponential function, b k Representing the k-th basis function width, C, of the neural network k Is the k-th basis function center vector, C, of the neural network k ＝[c 1,k ,...,c n,k ] T Wherein c is i,k I = 1.. And n is a constant.
In the step of the present embodiment, the compound obtained in the third step is used
Figure GDA0003936571700000061
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, wherein the weight W is i The update rate is specifically expressed as:
Figure GDA0003936571700000062
W n the update rate is specifically expressed as:
Figure GDA0003936571700000063
wherein the content of the first and second substances,
Figure GDA0003936571700000064
represents a weight W i First derivative of (a), r is the learning rate, r>0；/>
Figure GDA0003936571700000065
And/or>
Figure GDA0003936571700000066
Are respectively in>
Figure GDA0003936571700000067
And/or>
Figure GDA0003936571700000068
Is estimated.
The differential estimator in step three of the present embodiment is a differential estimator with fixed time convergence, and specifically includes:
Figure GDA0003936571700000069
where Ψ (y) = (1-e) -by )/(1+e -by )，
Figure GDA00039365717000000610
λ, η, b are constants and are positive numbers, ω i Ψ (y) is used to estimate the sign function sgn (y), - @, for the state quantity of the differential estimator>
Figure GDA00039365717000000611
Is z i Is first derivative of->
Figure GDA00039365717000000612
Is omega i ξ denotes the estimation error of the differential estimator, b @>
Figure GDA00039365717000000613
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 GDA00039365717000000614
Figure GDA00039365717000000615
wherein
Figure GDA00039365717000000616
Is 0. From (8) (9), the estimation error of the neural network can be written as
Figure GDA00039365717000000617
Figure GDA0003936571700000071
For any continuous function f (X), there is an ideal network weight W * So that
Figure GDA0003936571700000072
Let f be the function to be evaluated of the neural network if the weight of the neural network is based on a gradient descent algorithm>
Figure GDA0003936571700000073
Update then has->
Figure GDA0003936571700000074
Therefore, when we update the network weights according to (5) and (6), we can get:
Figure GDA0003936571700000075
Figure GDA0003936571700000076
substituting (2), (3), (12) and (13) into
Figure GDA0003936571700000077
To obtain
Figure GDA0003936571700000078
For any constant x, there is a constant ε > 0, such that
Figure GDA0003936571700000079
Therefore, there are:
Figure GDA00039365717000000710
from the inequality of mean, have
Figure GDA00039365717000000711
For x i ∈R,i＝1,...,n，ι∈[0，1]Of (| x) 1 |+...+|x n |) ι ≤(|x 1 |) ι +...+(|x n |) ι And (| x) 1 |+...+|x n |) 2 ≤[(|x 1 |) 2 +...+(|x n |) 2 ]N according to (14), (15) and (16) are
Figure GDA0003936571700000081
Wherein mu 1 ＝min{μ 11 ,...,μ n1 }，μ 2 ＝min{μ 12 ,...,μ n2 }/n，
Figure GDA0003936571700000082
According to the fixed time bounded stability theory, the system meets the condition
Figure GDA0003936571700000083
Wherein
Figure GDA0003936571700000084
Is a constant, the tracking error of the system will be at a fixed time T s Converge into a small neighborhood around the origin, T s The values of (A) are:
Figure GDA0003936571700000085
after the syndrome is confirmed.
Example 1:
consider the following motor rotation angle system:
Figure GDA0003936571700000086
wherein x is 1 (unit rad) is the output angle, x, of the motor angle system 2 (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 x 1 (0)＝0,x 2 (0) =0, constant j =20. To demonstrate the simulation, assume that the system unmodeled dynamics are
Figure GDA0003936571700000087
This function cannot be used directly to design the system control input u. System target signal set to y d (t)＝0.1。
The parameters in the virtual control function (4) and the system control input function (5) are taken as mu 11 ＝15，μ 12 ＝20，ε 1 ＝0.4，μ 21 ＝150，μ 22 ＝3，ε 2 And =0.01. (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. The parameters in the differential estimators (7), (8) are λ =10, η =10, b =1. The parameters in the error-training based update strategy (11) are 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 an estimate for 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.
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 (5)

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 a given target signal is y d The state variable is [ x ] 1 ,...,x n ] T ；
The nonlinear n-order system state space model is a motor corner system model, and the motor corner system model is as follows:
Figure FDA0004088359350000011
x 1 is the output angle, x, of the motor angle system 2 The angular velocity of the motor is shown, j is the rotational inertia of the system, and u is the input torque; f (x) 1 ，x 2 ) For unmodeled dynamics of the system, y represents the output of the motor rotation angle system;
s2, determining an error variable z 1 And z i ，z 1 ＝x 1 -y d ，z i ＝x i -α i-1 Wherein α is i-1 Representing a virtual control function;
Figure FDA0004088359350000012
Figure FDA0004088359350000013
wherein, i =1, \8230, n,
Figure FDA0004088359350000014
denotes the intermediate variable, g i G, a non-linear n-th order system intrinsic parameter 0 ＝z 0 ＝0，μ i1 、μ i2 、ε i Are all constant and are positive numbers, W i T S i (X i ) Representing a radial basis neural network, W i ＝[w i1 ,...,w iN ] T Is a weight matrix of the neural network, S i (X i ) Representing the basis function vector, S, of a neural network i (X i ) Each of the basis functions in (1) is a gaussian function having the same center and fixed width, X i ＝[x 1 ,...,x i ]The input vector of the neural network is defined, and N is the number of nodes of the neural network;
s3, establishing an error z i The input of the differential estimator is z i Output is
Figure FDA00040883593500000111
Is->
Figure FDA00040883593500000112
(ii) is estimated;
the differential estimator is:
Figure FDA0004088359350000015
where Ψ (y) = (1-e) -by )/(1+e -by )，
Figure FDA0004088359350000016
λ, η, b are constants and are positive numbers, ω i Ψ (y) is used to estimate the sign function sgn (y), or->
Figure FDA0004088359350000017
Is z i Is first derivative of->
Figure FDA0004088359350000018
Is omega i ξ denotes the estimation error of the differential estimator, b @>
Figure FDA0004088359350000019
An upper limit of ξ;
s4, obtained by S3
Figure FDA00040883593500000110
Calculating estimation error of the current radial basis function neural network, performing gradient descent training on the weight of the neural network based on the estimation error to obtain->
Figure FDA00040883593500000214
Weight W i The update rate is specifically expressed as:
Figure FDA0004088359350000021
W n the update rate is specifically expressed as:
Figure FDA0004088359350000022
wherein it is present>
Figure FDA0004088359350000023
Represents a weight W i First derivative of r i To the learning rate, r i >0；/>
Figure FDA0004088359350000024
And &>
Figure FDA0004088359350000025
Are respectively based on>
Figure FDA0004088359350000026
And/or>
Figure FDA0004088359350000027
(ii) an estimate of (d);
s5, according to alpha n 、
Figure FDA0004088359350000028
Calculating a control input signal of the nonlinear system; />
Control input signal u of nonlinear system:
Figure FDA0004088359350000029
Figure FDA00040883593500000210
2. 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 comprises:
using the obtained error z i Designing a Lyapunov function V:
Figure FDA00040883593500000211
the derivative values are:
Figure FDA00040883593500000212
and selecting a virtual control function according to the derivative of the Lyapunov function.
3. The neural network backstepping control method based on error training as claimed in claim 1, wherein in S1, the S i (X i ) The kth element s of k (X i ) Represents:
Figure FDA00040883593500000213
wherein exp represents an exponential function, b k Representing the k-th basis function width, C, of the neural network k Is the k-th basis function center vector, C, of the neural network k ＝[c 1,k ,...,c n,k ] T Wherein c is i,k Is a constant.
4. A computer-readable storage device, in which a computer program is stored, which computer program, when being executed, carries out the method according to any one of claims 1 to 3.
5. 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 3.
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