CN113219832A - Design method of adaptive iterative learning non-uniform target tracking controller - Google Patents

Design method of adaptive iterative learning non-uniform target tracking controller Download PDF

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CN113219832A
CN113219832A CN202110507272.0A CN202110507272A CN113219832A CN 113219832 A CN113219832 A CN 113219832A CN 202110507272 A CN202110507272 A CN 202110507272A CN 113219832 A CN113219832 A CN 113219832A
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张春丽
田旭
严雷
钱富才
谢国
王文卿
赵永红
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Xian University of Technology
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Abstract

The invention discloses a design method of a self-adaptive iterative learning non-uniform target tracking controller, which specifically comprises the following steps: the method comprises the following steps: step 1: performing system description aiming at a type of continuous time nonlinear system; step 2: describing a time-varying boundary layer and a radial basis function neural network; and step 3: designing a self-adaptive learning controller by respectively applying a Backstepping method and a radial basis function neural network in the conditions of asymmetric dead zone input and continuous nonlinear input; and 4, step 4: analyzing the stability and convergence of the designed adaptive learning controller; and 5: a simulation example is given to illustrate the feasibility and effectiveness of the method of the present invention. When the initial state error and the unknown input nonlinearity exist simultaneously, the problem of inconsistent target tracking control of self-adaptive iterative learning of a strict feedback nonlinear system is solved.

Description

Design method of adaptive iterative learning non-uniform target tracking controller
Technical Field
The invention belongs to the technical field of adaptive iterative learning control, and relates to a design method of an adaptive iterative learning non-uniform target tracking controller.
Background
The self-adaptive iterative learning control method is a powerful method for solving the problem of trajectory tracking in a finite time interval of a nonlinear system, and the self-adaptive iterative learning control is combined with iterative learning control. According to the design of the adaptive learning law, the AILC theory can be divided into two main categories: along the iteration domain direction and the time domain direction, are also called discrete AILC and continuous AILC, of course. Adaptive control is often used to deal with uncertainties that can be estimated by learning, and most adaptive controllers involve some type of function approximator, such as Neural Networks (NNs) and Fuzzy Logic Systems (FLSs), in their learning mechanism. The self-adaptive NN and FLS controllers can solve the problem of stability of closed-loop systems of various control systems, and the problem of tracking variable control tasks is solved by applying the self-adaptive fuzzy ILC to an unknown nonlinear system. The AILC is designed with a variable wavelet decomposition, and the wavelet decomposition terms are increased with the increase of the iteration number, so that the tracking error is converged uniformly in a limited time interval. The AILC based on a complex energy function or a Lyapunov function plays an important role in processing the learning estimation of time-varying parameters by connecting the time domain and the iterative domain. The AILC of a non-linear parameterized system remains a disclosed problem. No research efforts have been found for the aicc of non-linear parametric systems. Although traditional D-type and P-type ILCs have good research results on this problem, it remains a public problem for AILC. Due to the physical limitations of actuators, non-linearities are always present in control inputs, such as dead band inputs and continuous non-linear inputs. The presence of non-linearities in the control input tends to undermine system performance. At present, the performance of such control systems is improved by combining different methods for some nonlinear systems such as the tracking control problem of nonlinear uncertain systems with input saturation, very general nonlinear input uncertainty, class of time-lag systems with matched nonlinear time-lag disturbance and asymmetric dead zone saturation input, and uncertain nonlinear parameterized systems with unknown input nonlinearity (NLP-systems). However, there is currently no mature research effort for AILC for nonlinear systems with unknown input nonlinearities and initial state errors. When a possibly large initial state error exists, how to use the adaptive iterative learning control to solve the problem of tracking the non-uniform target track of a nonlinear strict feedback system on [0, T ] is a challenging problem.
Disclosure of Invention
The invention aims to provide a design method of a self-adaptive iterative learning non-uniform target tracking controller, which can solve the problem of self-adaptive iterative learning non-uniform target tracking control of a strict feedback non-linear system when an initial state error and unknown input non-linearity exist simultaneously.
The technical scheme adopted by the invention is that a design method of a self-adaptive iterative learning non-uniform target tracking controller specifically comprises the following steps:
step 1, the following formula (1) is adopted to represent a continuous time nonlinear system:
Figure BDA0003058917080000011
wherein the content of the first and second substances,
Figure BDA0003058917080000021
Γ(uk) E R represents the input-output characteristics of the actuator, ykE, R refers to system output;
Figure BDA0003058917080000022
is a smooth unknown non-linear function; k represents an iteration index;
step 2, the time-varying boundary layer network is expressed by the following formula (2):
Figure BDA0003058917080000023
wherein phi isi，k(t)＝εi，ke-ηt，zi,kAnd ziφ,kIs a function variable of time t, epsiloni,kIs a convergence series sequence, sat is a saturation function, defined as follows:
Figure BDA0003058917080000024
wherein phi isi,k(t) is a time-varying boundary layer,. phii,k(t) decreasing along the time axis, selecting an initial state error phi at the kth iterationi,k(0)＝εi,kThen 0 < epsiloni,ke-ηT≤φi,k(t)≤εi,k，
Figure BDA0003058917080000025
And 3, expressing the radial basis function neural network by adopting the following formula (4):
Figure BDA0003058917080000026
wherein the content of the first and second substances,
Figure BDA0003058917080000027
is a known smooth vector function, wherein the number l of NN nodes is more than 1; radial basis function
Figure BDA0003058917080000028
Is a gaussian function and is expressed by the following equation (5):
Figure BDA0003058917080000029
wherein, mujE.omega and eta > 0 are radial basis functions, respectively
Figure BDA00030589170800000210
Of (2) centerSum width, optimal weight vector W ═ ω1,...,ωl]TIs defined as:
Figure BDA00030589170800000211
wherein the content of the first and second substances,
Figure BDA00030589170800000212
is an approximation error inherent to the NN;
and 4, designing the self-adaptive learning controller by respectively applying a Backstepping method and a radial basis function neural network in the conditions of asymmetric dead zone input and continuous nonlinear input.
The invention is also characterized in that:
the specific process of the step 4 is as follows:
step 4.1, use function Γ (u)k) The actuator output with an asymmetric dead band is represented as shown in equation (7) below:
Figure BDA0003058917080000031
wherein the parameter mrAnd mlA right slope and a left slope representing the dead zone characteristic, respectively, parameter brAnd blA breakpoint representing a dead-zone input of the actuator;
and 4.2, modeling the asymmetric dead zone of the actuator into a form of the sum of a straight line and a disturbance-like term, wherein the form is shown in the following formula (8):
Γ(uk)＝m(t)uk+d(t) (8)；
wherein:
Figure BDA0003058917080000032
4.3, designing a controller for the uncertain nonlinear system with unknown asymmetric dead zone input and initial state error based on a self-adaptive iterative learning Backstepping frame;
and (4) performing a step (4.4),in the control input of a continuous-time nonlinear system of the type represented by equation (1), there is a nonlinearity which is assumed in the sector region s1 s2]A continuous non-linear function N (u)k) And N (0) ═ 0, where s is1And s2Represents two straight lines (t)1,t2) Of (2) i.e.
Figure BDA0003058917080000033
And 4.5, designing a controller for the condition of continuous nonlinear input.
The specific process of the step 4.3 is as follows:
step A) let z1,k＝x1,k-yr,k，z2,k＝x2,k-α1,kIn which α is1,kIs a virtual controller; because of the initial state error, according to the step 2 of representing the time-varying boundary layer, the following formula is adopted to respectively carry out the error function z1φ,kAnd z2φ,kTo show that:
Figure BDA0003058917080000034
wherein the content of the first and second substances,
Figure BDA0003058917080000035
Figure BDA0003058917080000036
wherein the content of the first and second substances,
Figure BDA0003058917080000041
ε1,kand ε2,kIs a positive series of convergent numbers, η1And η2Is a designed normal number;
because:
Figure BDA0003058917080000042
then z is1φ,kThe derivative with respect to time is shown in the following equation (13):
Figure BDA0003058917080000043
step B) according to the representation of the radial basis function neural network in the step 3, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure BDA0003058917080000044
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure BDA0003058917080000045
Figure BDA0003058917080000046
Wherein the content of the first and second substances,
Figure BDA0003058917080000047
is the approximation error, W1Is the optimal weight vector; for any real number a > 0 and positive integer l ≧ 2, take
Figure BDA0003058917080000048
Definition of
Figure BDA0003058917080000049
The virtual controller is taken as:
Figure BDA00030589170800000410
substituting equations (14) and (15) into (13) yields the following equation (16):
Figure BDA00030589170800000411
wherein the content of the first and second substances,
Figure BDA00030589170800000412
and
Figure BDA00030589170800000413
are respectively a parameter W1And N1Is estimated by the estimation of (a) a,
Figure BDA00030589170800000414
and
Figure BDA00030589170800000415
is the parameter estimation error, the last two terms to the right of the equal sign of equation (16) are rewritten as:
Figure BDA00030589170800000416
on the basis of the formula (17), the formula (16) is rewritten as the following formula (18):
Figure BDA0003058917080000051
order to
Figure BDA0003058917080000052
Then equation (18) is transformed into equation (19) as follows:
Figure BDA0003058917080000053
in step 4.3, define: z is a radical ofn,k＝xn,k-αn-1,kAccording to the step 2 representation of the time-varying boundary layer network, the following error function z is introducednφ,k：
Figure BDA0003058917080000054
Wherein the content of the first and second substances,
Figure BDA0003058917080000055
εn,kis a positive series of convergent numbers, ηnIs a designed normal number;
then z isnφ,kThe derivative of (c) is:
Figure BDA0003058917080000056
wherein the content of the first and second substances,
Figure BDA0003058917080000057
defining:
Figure BDA0003058917080000058
Figure BDA0003058917080000059
znφ,kthe derivative with respect to time is:
Figure BDA00030589170800000510
according to the representation of the radial basis function neural network in the step 3, the RBFNN unified approximation performance shows that the unknown nonlinear function is represented
Figure BDA00030589170800000511
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure BDA00030589170800000512
Figure BDA00030589170800000513
Wherein the content of the first and second substances,
Figure BDA0003058917080000061
is the approximation error, WnIs the optimal weight vector;
get
Figure BDA0003058917080000062
Selecting the actual controller as
uk＝u1,k+u2,k (26)；
Wherein u is2,kTo compensate for the unknown input gain m (t); substituting equations (25) and (26) into (24) yields the following equation (27):
Figure BDA0003058917080000063
wherein the content of the first and second substances,
Figure BDA0003058917080000064
and
Figure BDA0003058917080000065
are respectively a parameter WnAnd NnIs estimated by the estimation of (a) a,
Figure BDA0003058917080000066
and
Figure BDA0003058917080000067
is the parameter estimation error, the last two terms to the right of the equality sign of equation (27) are rewritten as:
Figure BDA0003058917080000068
Figure BDA0003058917080000069
get
Figure BDA00030589170800000610
Wherein
Figure BDA00030589170800000611
Is an uncertain parameter
Figure BDA00030589170800000612
(ii) an estimate of (d);
order to
Figure BDA00030589170800000613
Then, the formula (29) becomes the following formula (30):
Figure BDA00030589170800000614
the specific process of the step 4.5 is as follows: with N (u)k) As the output of the system instead of Γ (u)k) Definition of zn,k＝xn,k-αn-1,kAccording to the step 2 representation of the time-varying boundary layer network, the following error function z is introducednφ,k：
Figure BDA00030589170800000615
Wherein the content of the first and second substances,
Figure BDA00030589170800000616
εn,kis a positive series of convergent numbers, ηnIs a designed normal number; then z isnφ,kIs a derivative of
Figure BDA0003058917080000071
Figure BDA0003058917080000072
Defining:
Figure BDA0003058917080000073
Figure BDA0003058917080000074
then equation (32) is rewritten as equation (36) below:
Figure BDA0003058917080000075
according to the representation of the radial basis function neural network in the step 3, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure BDA0003058917080000076
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure BDA0003058917080000077
Figure BDA0003058917080000078
Wherein the content of the first and second substances,
Figure BDA0003058917080000079
is the approximation error, WnIs the optimal weight vector; according to the formulas (36) and (37), the following formula (38) is obtained:
Figure BDA00030589170800000710
the method for designing the self-adaptive iterative learning non-uniform target tracking controller has the advantages that the radial basis function neural network is introduced to learn the performance of unknown dynamics, and a typical series is used for effectively eliminating approximation errors. The problem of initial state errors is solved by applying a time-varying boundary layer. The problem of non-linearity of two types of input, namely asymmetric dead zone input and continuous input is effectively solved. It can be shown that all signals of the closed loop system are bounded over a given time interval 0, T and that the state tracking error will asymptotically converge to an adjustable set of residuals as the iteration goes to infinity.
Drawings
FIG. 1 shows the I/Z for asymmetric dead zone input in simulation research of the design method of adaptive iterative learning inconsistent target tracking controller1,kA graph of | | change with iteration index;
FIG. 2 is | | u for asymmetric dead zone input condition in simulation research of the design method of adaptive iterative learning inconsistent target tracking controller of the present inventionkA graph of | | change with iteration index;
FIG. 3 is a graph of asymmetric dead zone input conditions in simulation studies of a method for designing a non-uniform target tracking controller for adaptive iterative learning according to the present invention
Figure BDA00030589170800000711
A graph of variation with iteration index;
FIG. 4 shows asymmetric dead-zone input conditions in simulation studies of a method for designing a non-uniform target tracking controller according to the present invention
Figure BDA0003058917080000081
A graph of variation with iteration index;
FIG. 5 shows asymmetric dead-zone input conditions in simulation studies of a method for designing a non-uniform target tracking controller according to the present invention
Figure BDA0003058917080000082
A graph of variation with iteration index;
FIG. 6 is a graph of asymmetric dead zone input conditions in simulation studies of a method for designing a non-uniform target tracking controller for adaptive iterative learning according to the present invention
Figure BDA0003058917080000083
A graph of variation with iteration index;
FIG. 7 shows asymmetric dead-zone input conditions in simulation studies of a method for designing a non-uniform target tracking controller according to the present invention
Figure BDA0003058917080000084
A graph of variation with iteration index;
FIG. 8 is a plot of z for continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning according to the present invention1,kA graph of | | change with iteration index;
FIG. 9 is a graph of | | | u for continuous nonlinear input conditions in simulation studies of a method for designing a non-uniform target tracking controller for adaptive iterative learning according to the present inventionkA graph of | | change with iteration index;
FIG. 10 is a graph of continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning in accordance with the present invention
Figure BDA0003058917080000085
A graph of variation with iteration index;
FIG. 11 is a graph of continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning in accordance with the present invention
Figure BDA0003058917080000086
A graph of variation with iteration index;
FIG. 12 is a graph of continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning in accordance with the present invention
Figure BDA0003058917080000087
A graph of variation with iteration index;
FIG. 13 is a graph of continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning in accordance with the present invention
Figure BDA0003058917080000088
A graph of variation with iteration index;
FIG. 14 is a graph of continuous nonlinear input conditions in simulation studies of a method of designing a non-uniform target tracking controller for adaptive iterative learning in accordance with the present invention
Figure BDA0003058917080000089
Graph of variation with iteration index.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a design method of a self-adaptive iterative learning non-uniform target tracking controller, which specifically comprises the following steps: step 1, carrying out system description on a continuous time nonlinear system; consider a class of continuous-time nonlinear systems, as shown in equation (1) below:
Figure BDA00030589170800000810
wherein the content of the first and second substances,
Figure BDA0003058917080000091
is the state vector of the system that can be tested. Gamma (u)k) E R represents the input-output characteristics of the actuator, ykE R is the system output.
Figure BDA0003058917080000092
Is a smooth unknown non-linear function. k denotes an iteration index. The target is at [0, T]Upper design adaptive iterative learning control law uk(t) such that the system output y is repeatedk(t) tracking the upper target trajectory yr,k(t) and for some small positive error bound δ
Figure BDA0003058917080000093
And all signals of the closed loop system are guaranteed to be bounded. y isr,k(t) represents a sufficiently smooth target trajectory.
Step 2, a time-varying boundary layer and a radial basis function neural networkThe description is carried out; 1) a time-varying boundary layer; to overcome the uncertainty of the initial state error, a new function z is introduced as followsiφ,k：
Figure BDA0003058917080000094
Wherein z isi,kAnd ziφ,kIs a function variable of time t, epsiloni,kIs a convergence series sequence, sat is a saturation function, defined as follows:
Figure BDA0003058917080000095
wherein phi isi,k(t) is a time-varying boundary layer. Notice phii,k(t) decreasing along the time axis, selecting an initial state error phi at the kth iterationi,k(0)＝εi,kThen 0 < epsiloni,ke-ηT≤φi,k(t)≤εi,k，
Figure BDA0003058917080000096
Assuming 1, since the initial state error for each iteration of the initial state error is not necessarily zero, small and fixed, epsilon is constant for some known normality i,k1, m, assuming that the initial state error satisfies | zi,k(0)||≤εi,kWherein z isi,k(0) Is the initial state error of the state error at each iteration. Since z is easily obtained from equation (2) if it is assumed that 1 holdsiφ,k(0)＝0,
Figure BDA0003058917080000097
During the whole time interval [0, T]When η is increased properly, phii,k(t) will be as small as possible. If can prove that
Figure BDA0003058917080000098
Then the error function will satisfy that when k → ∞,
Figure BDA0003058917080000099
Figure BDA00030589170800000910
wherein
Figure BDA00030589170800000911
Is an arbitrarily small normal number, i.e., the control target is completed.
2) A radial basis function neural network; unknown smooth non-linear function
Figure BDA00030589170800000912
The approximation on the tight set Ω will be done by the following Radial Basis Function Neural Network (RBFNN):
Figure BDA0003058917080000101
wherein the content of the first and second substances,
Figure BDA0003058917080000102
is a known smooth vector function, where the number of NN nodes, l > 1. Radial basis function
Figure BDA0003058917080000103
Is selected as a commonly used gaussian function and
Figure BDA0003058917080000104
wherein mujE.omega and eta > 0 are radial basis functions, respectively
Figure BDA0003058917080000105
The center and the width of (c). The optimal weight vector W ═ ω1,...,ωl]TIs defined as:
Figure BDA0003058917080000106
wherein the content of the first and second substances,
Figure BDA0003058917080000107
is an approximation error inherent to the NN, which can be arbitrarily reduced by increasing the value of the NN network node number l. Following approximation error
Figure BDA0003058917080000108
The following assumptions are made.
Suppose 2, on tight set Ω, the error is approximated
Figure BDA0003058917080000109
Is assumed to be bounded and
Figure BDA00030589170800001010
wherein the unknown parameter thetai(1. ltoreq. i. ltoreq.n) represents
Figure BDA00030589170800001011
A minimum upper bound of where θi≥0。
And 3, respectively applying a Backstepping method and a radial basis function neural network to design the self-adaptive learning controller under the conditions of asymmetric dead zone input and continuous nonlinear input. Firstly, asymmetric dead zone input and continuous nonlinear input are described, and then corresponding controllers are respectively designed. 1) Asymmetric dead zone input; actuator dead band is common in mechanical connections, hydraulic servo valves, piezoelectric sensors, and electric servomotors. The presence of such non-linearities often undermines system performance. Step 3 will introduce an asymmetric dead zone input characteristic. Function Γ (u)k) Representing actuator output with asymmetric dead band, as expressed below
Figure BDA00030589170800001012
Wherein the parameter mrAnd mlRespectively representing the right and left slopes of the dead zone characteristic. Parameter brAnd blRepresenting a breakpoint of the actuator dead band input. The asymmetric dead zone of the actuator can be modeled as a straight line sumThe sum of one class perturbation term is in the form of:
Γ(uk)＝m(t)uk+d(t) (7)；
wherein:
Figure BDA00030589170800001013
let 3, parameter ml,mr,blAnd brIs an unknown normal number. There is an unknown parameter v small enough that 0 < v ≦ min { ml,mr}. Upper bound of disturbance d (t)
Figure BDA0003058917080000111
Are also unknown constants.
2) Continuous non-linear input; due to the physical limitations of the actuators, the output of the actuators acting on the system is not accurate, i.e., there is a non-linearity in the control input to the system (equation 1). These non-linearities are defined by the fact that the field of the device is a sector [ s ]1 s2]A continuous non-linear function N (u)k) And N (0) ═ 0, where s is1And s2Represents two straight lines (t)1,t2) Of (2) i.e.
Figure BDA0003058917080000112
Let 4, parameter s1And s2Is an unknown non-zero normal number.
3) Designing a controller under the condition of asymmetric dead zone input; and designing a controller for the uncertain nonlinear system with unknown asymmetric dead zone input and initial state error based on the adaptive iterative learning Backstepping framework.
Step 1, let z1,k＝x1,k-yr,k，z2,k＝x2,k-α1,kIn which α is1,kIs a virtual controller. Because of the initial state error, the following error function z is introduced according to step 2 for the description of the time-varying boundary layer1φ,kAnd z2φ,k：
Figure BDA0003058917080000113
Figure BDA0003058917080000114
Wherein epsilon1,kAnd ε2,kIs a positive series of convergent numbers, η1And η2Is a design normal number. Because:
Figure BDA0003058917080000115
z1φ,kthe derivative with respect to time is as follows:
Figure BDA0003058917080000116
according to the description of the radial basis function neural network in the step 2, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure BDA0003058917080000117
Can be approximated by RBFNN as the following form on a tight set, and has reconstruction error
Figure BDA0003058917080000118
Figure BDA0003058917080000119
Wherein the content of the first and second substances,
Figure BDA00030589170800001110
is an approximation error and W1Is the optimal weight vector. For any real number a > 0 and positive integer l ≧ 2, take
Figure BDA0003058917080000121
Definition of
Figure BDA0003058917080000122
The virtual controller is taken as:
Figure BDA0003058917080000123
substituting equations (13) and (14) into (12) can give:
Figure BDA0003058917080000124
wherein the content of the first and second substances,
Figure BDA0003058917080000125
and
Figure BDA0003058917080000126
are respectively a parameter W1And N1Is estimated.
Figure BDA0003058917080000127
And
Figure BDA0003058917080000128
is the parameter estimation error. The last two terms on the right of the equation of equation (15) can be rewritten as:
Figure BDA0003058917080000129
according to equation (16), equation (15) is rewritten as follows:
Figure BDA00030589170800001210
order to
Figure BDA00030589170800001211
Then equation (17) becomes equation (18) as follows:
Figure BDA00030589170800001212
suppose 5, the residue term ω1Is bounded and | ω1|≤ωM1Wherein ω isM1Is a normal number. Assuming 5 is reasonable, there are two reasons: 1) according to the assumption of 2, the method,
Figure BDA00030589170800001213
is bounded; 2) as long as eta2Is suitably large and is suitably large enough to be,
Figure BDA00030589170800001214
it can be small enough. Take the following non-negative function:
Figure BDA00030589170800001215
wherein, gamma is11And Γ21Is a symmetric positive definite matrix. V is given below1,kAlong the derivative of system equation (18) with time:
Figure BDA00030589170800001216
Figure BDA0003058917080000131
step i (i is more than or equal to 2 and less than or equal to n-1), define
Figure BDA0003058917080000132
As will be given later. Let zi+1,k＝xi+1,k-αi,kAnalogously to step 1, the following error function z is introduced according to the description of step 2 for the time-varying boundary layeriφ,kAnd z(i+1)φ,k：
Figure BDA0003058917080000133
Figure BDA0003058917080000134
Wherein epsiloni,kAnd ε(i+1),kIs a positive series of convergent numbers, ηiAnd ηi+1Is a design normal number. z is a radical ofiφ,kThe derivative with respect to time is as follows:
Figure BDA0003058917080000135
wherein the content of the first and second substances,
Figure BDA0003058917080000136
Figure BDA0003058917080000137
equation (23) can be rewritten as:
Figure BDA0003058917080000138
according to the description of the radial basis function neural network in the step 2, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure BDA0003058917080000139
Can be approximated by RBFNN as the following form on a tight set, and has reconstruction error
Figure BDA00030589170800001310
Figure BDA00030589170800001311
Wherein the content of the first and second substances,
Figure BDA00030589170800001312
is an approximation error and WiIs the optimal weight vector. The virtual controller is taken as:
Figure BDA00030589170800001313
substituting equation (25) and equation (26) into equation (24) yields the following equation (27):
Figure BDA0003058917080000141
Figure BDA0003058917080000142
wherein the content of the first and second substances,
Figure BDA0003058917080000143
and
Figure BDA0003058917080000144
are respectively a parameter WiAnd NiIs estimated.
Figure BDA0003058917080000145
And
Figure BDA0003058917080000146
is the parameter estimation error. The last two terms to the right of the equal sign of equation (27) can be rewritten as:
Figure BDA0003058917080000147
order to
Figure BDA0003058917080000148
Then, the formula (27) becomes the following formula (29):
Figure BDA0003058917080000149
suppose 6, the residue term ωiIs bounded and | ωi|≤ωMiWherein ω isMiIs an unknown normal number. The following non-negative function was chosen:
Figure BDA00030589170800001410
v is given belowi,kDerivative with respect to time along system equation (29):
Figure BDA00030589170800001411
step n, define zn,k＝xn,k-αn-1,kThe following error function z is introduced according to step 2 description of the time-varying boundary layernφ,k：
Figure BDA00030589170800001412
Wherein epsilonn,kIs a positive series of convergent numbers, ηnIs a design normal number. Then znφ,kThe derivative of (c) is:
Figure BDA00030589170800001413
Figure BDA00030589170800001414
defining:
Figure BDA0003058917080000151
Figure BDA0003058917080000152
then znφ,kThe derivative with respect to time is:
Figure BDA0003058917080000153
according to the description of the radial basis function neural network in the step 2, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure BDA0003058917080000154
Can be approximated by RBFNN as the following form on a tight set, and has reconstruction error
Figure BDA0003058917080000155
Figure BDA0003058917080000156
Wherein the content of the first and second substances,
Figure BDA0003058917080000157
is an approximation error and WnIs the optimal weight vector.
Get
Figure BDA0003058917080000158
Selecting the actual controller as
uk＝u1,k+u2,k (38)；
Wherein u is2,kTo compensate for the unknown input gain m (t). Substituting equation (37) and equation (38) into equation (36) yields the following equation (39):
Figure BDA0003058917080000159
wherein the content of the first and second substances,
Figure BDA00030589170800001510
and
Figure BDA00030589170800001511
are respectively a parameter WnAnd NnIs estimated.
Figure BDA00030589170800001512
And
Figure BDA00030589170800001513
is the parameter estimation error. The last two terms to the right of the equality sign of equation (39) can be rewritten as:
Figure BDA00030589170800001514
Figure BDA00030589170800001515
get
Figure BDA00030589170800001516
Wherein
Figure BDA00030589170800001517
Is an uncertain parameter
Figure BDA00030589170800001518
Is estimated.
Order to
Figure BDA00030589170800001519
Then the formula (41) becomes the following formula (42):
Figure BDA0003058917080000161
suppose 7, the residue term ωnIs bounded and | ωn|≤ωMn1Wherein ω isMn1Is an unknown normal number. The following adaptive iterative learning law is taken:
Figure BDA0003058917080000162
Figure BDA0003058917080000163
wherein, gamma is1i,Γ2i,Γ3Is a learning law that needs to be designed. The following non-negative function was chosen:
Figure BDA0003058917080000164
v is given belown,kAlong with the derivative of system equation (42) with respect to time, in conjunction with equations (43) through (45), the following equation (47) results:
Figure BDA0003058917080000165
for the setting of the initial estimate at each iteration, the following assumptions are given.
Assumption 8 for
Figure BDA0003058917080000166
When t is equal to 0, the first step is,
Figure BDA0003058917080000167
Figure BDA0003058917080000168
theorem 1, if controller equation (38) and parameter adaptation law equations (43) - (45) are applied to system equation (1) assuming 1-8 are satisfied, then all signals of the closed-loop system are bounded at [0, T ] and there are the following equations:
Figure BDA0003058917080000171
that is to say that the position of the first electrode,
Figure BDA0003058917080000172
then when k → ∞ is reached,
Figure BDA0003058917080000173
wherein
Figure BDA0003058917080000174
Is an arbitrarily small normal number. Here, the whole time interval [0, T]If let η1Suitably large then phi1,∞(t) may be arbitrarily small.
4) Designing a controller for the condition of continuous nonlinear input; in selecting a controller in an actual physical system, there is nonlinearity in the control input due to the limitations of the actuator. For controller design, the following arguments are used:
theorem 1, under the condition that assumption 4 is satisfied, for controller ukAnd error znφ,kThe following inequality holds:
znφ,kN(uk)≤sznφ,kuk,s∈{s1,s2} (49)；
and (3) proving that: if u is multiplied simultaneously on both sides of equation (9)kThen, the following formula (50) is obtained:
Figure BDA0003058917080000175
further, the inequality in the formula (50) is multiplied by each side
Figure BDA0003058917080000176
The following formula (51) is obtained:
Figure BDA0003058917080000177
the formula (51) is rewritten as the following formula (52):
s1(znφ,kuk)2≤znφ,k(znφ,kuk)N(uk)≤s2(znφ,kuk)2 (52)；
if the controller ukSatisfies znφ,kuk0 or less, the following inequality holds:
znφ,kN(uk)≤s1znφ,kuk (53)；
if the controller ukSatisfies znφ,kukGreater than or equal to 0, the following inequality holds:
znφ,kN(uk)≤s2znφ,kuk (54)；
obviously, the conclusion of lemma 1 holds. After the syndrome is confirmed. During the whole design process, use N (u)k) As the output of the system instead of Γ (u)k). Step 1, step i, i 2, n-1 is the same as in part 3) of step 3.
Step n, define zn,k＝xn,k-αn-1,kThe following error function z is introduced according to step 2 description of the time-varying boundary layernφ,k：
Figure BDA0003058917080000181
Wherein epsilonn,kIs a positive series of convergent numbers, ηnIs a design normal number. Then z isnφ,kThe derivative of (c) is:
Figure BDA0003058917080000182
Figure BDA0003058917080000183
defining:
Figure BDA0003058917080000184
Figure BDA0003058917080000185
equation (56) is rewritten as:
Figure BDA0003058917080000186
according to the description of the radial basis function neural network in the step 2, the uniform approximation performance of the RBFNN (radial basis function neural network) shows that the unknown nonlinear function is
Figure BDA0003058917080000187
Can be approximated by RBFNN as the following form on a tight set, and has reconstruction error
Figure BDA0003058917080000188
Figure BDA0003058917080000189
Wherein the content of the first and second substances,
Figure BDA00030589170800001810
is an approximation error and WnIs the optimal weight vector. From equation (57), equation (58) is rewritten as:
Figure BDA00030589170800001811
the following non-negative function was chosen:
Figure BDA00030589170800001812
wherein the content of the first and second substances,
Figure BDA00030589170800001813
and
Figure BDA00030589170800001814
are respectively a parameter Wn、NnAnd
Figure BDA00030589170800001815
is estimated.
Figure BDA00030589170800001816
Figure BDA00030589170800001817
And
Figure BDA00030589170800001818
is the parameter estimation error. Gamma-shaped1i,Γ2i,Γ4Is a learning gain that needs to be designed.
V is given belown,kAlong the derivative of system equation (59) with respect to time, and substituting equation (49), then there is:
Figure BDA0003058917080000191
get
Figure BDA0003058917080000192
Selecting an actual controller as follows:
uk＝u1,k+u2,k (62)；
wherein u is2,kTo compensate for the unknown input gain s. Order to
Figure BDA0003058917080000193
Then there are:
Figure BDA0003058917080000194
get
Figure BDA0003058917080000195
Equation (63) is changed to the following equation:
Figure BDA0003058917080000196
Figure BDA0003058917080000201
suppose 9, the residue term ωnIs bounded and | ωn|≤ωMn2Wherein ω isMn2Is an unknown normal number. The following adaptive iterative learning law is taken:
Figure BDA0003058917080000202
Figure BDA0003058917080000203
Figure BDA0003058917080000204
Figure BDA0003058917080000205
in order to set the initial estimation at each iteration, the following assumptions are specifically given: assuming 10, for any k, when t is 0,
Figure BDA0003058917080000211
theorem 2, if hypothesis 2, hypothesis 4-hypothesis 6, hypothesis 9, and hypothesis 10 are satisfied, and the controller equation (62) and the parametric adaptation rate equations (65) - (67) are applied to the system equation (1), then all signals of the closed-loop system are bounded at [0, T ] and have:
Figure BDA0003058917080000212
that is to say that the position of the first electrode,
Figure BDA0003058917080000213
then when k → ∞ is reached,
Figure BDA0003058917080000214
wherein
Figure BDA0003058917080000215
Is an arbitrarily small normal number. Here, the whole time interval [0, T]If let η1Suitably large then phi1,∞(t) may be arbitrarily small.
Step 4, analyzing the stability and convergence of the self-adaptive learning controller designed in the step 3; 1) proof theorem 1;
it is proved that according to the description of the time-varying boundary layer in step 2, there are
Figure BDA0003058917080000216
Wherein z isφ,k＝[z1φ,k,z2φ,k,...,znφ,k]T. From equation (46), the following holds:
Figure BDA0003058917080000217
wherein:
Figure BDA0003058917080000218
substituting equation (45) into equation (70) yields:
Figure BDA0003058917080000219
defining:
Figure BDA00030589170800002110
then equation (71) can be rewrittenComprises the following steps:
Figure BDA00030589170800002111
by
Figure BDA00030589170800002112
To obtain
Figure BDA00030589170800002113
Then V0,kIs bounded. And due to the fact that
Figure BDA00030589170800002114
By the formula (71) to
Figure BDA00030589170800002115
Is bounded, therefore
Figure BDA00030589170800002116
Is also bounded. And also
Figure BDA00030589170800002117
Therefore, the method comprises the following steps:
Figure BDA00030589170800002118
from equation (46), for any k,
Figure BDA0003058917080000221
substituting equation (47) results in the following equation (74):
Figure BDA0003058917080000222
from the formula (73),
Figure BDA0003058917080000223
is bounded. Due to DeltakIs bounded, and T ∈ [0, T ]]Thus, therefore, it is
Figure BDA0003058917080000224
Is also bounded. From the above discussion, for any k, Vn,k(t) is bounded, then there is xi,k，
Figure BDA0003058917080000225
Figure BDA0003058917080000226
And
Figure BDA0003058917080000227
is bounded. From equation (39), uk is bounded. As can be seen from the formula (29),
Figure BDA0003058917080000228
is bounded, therefore ziφ,kIs consistently continuous, then by the Barbalt theorem, we get when k → ∞, ziφ,k→ 0, that is, when k → ∞, z1φ,k→ 0, by definition of Limit, for
Figure BDA0003058917080000229
Such that:
when k is greater than N, the number of the transition metal,
Figure BDA00030589170800002210
and is also provided with
Figure BDA00030589170800002211
Figure BDA00030589170800002212
Then when k → ∞ is reached,
Figure BDA00030589170800002213
so theorem proves that the method is good. 2) Theorem 2 is the same as the certification process of theorem 1.
Example (b): a simulation example is given to illustrate the feasibility and effectiveness of the method of the present invention.
Considering a class of mass-spring mechanical systems to illustrate what is presented in this sectionThe effectiveness of the controller is learned adaptively and iteratively. A mass
Figure BDA00030589170800002214
Attached to the wall by a spring and a slide on a horizontal smooth surface, i.e. the resistance caused by friction is assumed to be zero. The mass being influenced by an external force ukCan be regarded as a control variable. Let ykIs a displacement from a reference position. In the presence of asymmetric dead-zone inputs, the dynamic equations for the system are given below
Figure BDA00030589170800002215
Wherein t is ∈ [0,1 ]]，Fms(. is) the restoring force of the spring. k denotes an iteration index. Definition of x1,k＝yk，
Figure BDA00030589170800002216
And is
Figure BDA00030589170800002217
Converting the system into a state space form:
Figure BDA00030589170800002218
the restoring force of the spring can be modeled as:
Figure BDA00030589170800002219
in the system, the selected parameter k is 1, a0＝0，a1＝a2＝a3＝a4＝1，q＝4。
1. The case of asymmetric dead-zone input; the asymmetric dead zone is described as follows:
Figure BDA00030589170800002220
the control target is such that the output of the system (formula 74) is at [0, π → ∞ when k → ∞]Upper trace upper reference track yr,k. Choose to haveThe same amplitude reference track yr,k＝gksin (2 π t), choosing g when k is even for the case of non-uniform tracesk-0.1, when k is an odd number, gk0.1. According to theorem 1, the adaptive iterative learning controller is selected as:
Figure BDA0003058917080000231
wherein the content of the first and second substances,
Figure BDA0003058917080000232
the parameter adaptive iterative learning law is given by equation (43), where η1＝20，η2＝50，
Figure BDA0003058917080000233
Γ21＝1，
Figure BDA0003058917080000234
Γ22＝1，Γ 31. The neural network consists of 31 neurons, with the centers of the basis functions uniformly covered [ -1,1 ]]The width of the basis function is chosen to be 1. The following parameters and initial values of the states and estimated parameters are selected:
Figure BDA0003058917080000235
Figure BDA0003058917080000236
the iteration number k is 30, and the simulation result is shown in fig. 1-fig. 7. It can be seen from fig. 1 that the tracking error can converge to zero. Further, fig. 2-7 show the control signal | | | uk||，
Figure BDA0003058917080000237
In the interval [0, pi]Upper is bounded. The simulation results shown in fig. 1 to fig. 7 further prove the effectiveness of the control method of the controller designed by the design method of the adaptive iterative learning non-uniform target tracking controller provided by the invention.
2. Continuous non-linear outputThe case of entry; here, a continuous non-linear input N (u) is usedk) Replacing asymmetric dead-zone input Γ (u)k). Function N (u)k) Is described as N (u)k)＝(0.5+0.1sin(uk))uk. Selecting reference tracks y having different amplitudesr,k＝gksin (2 π t), choosing g when k is even for the case of non-uniform tracesk-0.1, when k is an odd number, gk0.1. According to theorem 2, the adaptive iterative learning controller is selected as:
Figure BDA00030589170800002410
Figure BDA0003058917080000242
wherein the content of the first and second substances,
Figure BDA0003058917080000243
the parameter adaptive iterative learning law is given by equation (65), where η1＝20，η2＝50，
Figure BDA0003058917080000244
Γ21＝1，
Figure BDA0003058917080000245
Γ22＝1，Γ 31. The neural network consists of 31 neurons, with the centers of the basis functions uniformly covered [ -1,1 ]]The width of the basis function is chosen to be 1. The following parameters and initial values of the states and estimated parameters are selected:
Figure BDA0003058917080000246
Figure BDA0003058917080000247
the simulation result is shown in fig. 8-14 by taking the iteration number k as 30. It can be seen from fig. 8 that the tracking error can converge toAnd (4) zero. Further, fig. 9-14 show the control signal | | | uk||，
Figure BDA00030589170800002411
Figure BDA0003058917080000249
In the interval [0,1]Upper is bounded. The simulation results shown in fig. 8-fig. 14 further prove the effectiveness of the control method of the controller designed by the design method of the adaptive iterative learning non-uniform target tracking controller provided by the invention. In summary, the invention solves the problem of non-uniform target tracking control of adaptive iterative learning of a strict feedback nonlinear system when an initial state error and unknown input nonlinearity coexist. A backstepping method and a radial basis function neural network are utilized to solve the problem of trajectory tracking of a strict feedback nonlinear system. A radial basis function neural network is introduced to learn the performance of unknown dynamics, a typical series is utilized to effectively cancel approximation errors, and the problem of inconsistent target tracking is solved. The problem of initial state errors is solved by applying a time-varying boundary layer. The problem of non-linearity of two types of input, namely asymmetric dead zone input and continuous input is effectively solved. It can be shown that in a given time interval 0, T]All signals of the upper closed loop system are bounded and the state tracking error will asymptotically converge to an adjustable set of residuals as the iteration goes to infinity.

Claims (5)

1. A design method of a self-adaptive iterative learning non-uniform target tracking controller is characterized by comprising the following steps: the method specifically comprises the following steps:
step 1, the following formula (1) is adopted to represent a continuous time nonlinear system:
Figure FDA0003058917070000011
wherein the content of the first and second substances,
Figure FDA0003058917070000012
Γ(uk) E R represents the input-output characteristics of the actuator, ykE, R refers to system output;
Figure FDA0003058917070000013
is a smooth unknown non-linear function; k represents an iteration index;
step 2, the time-varying boundary layer network is expressed by the following formula (2):
Figure FDA0003058917070000014
wherein phi isi，k(t)＝εi，ke-ηt，zi,kAnd ziφ,kIs a function variable of time t, epsiloni,kIs a convergence series sequence, sat is a saturation function, defined as follows:
Figure FDA0003058917070000015
wherein phi isi,k(t) is a time-varying boundary layer,. phii,k(t) decreasing along the time axis, selecting an initial state error phi at the kth iterationi,k(0)＝εi,kThen 0 < epsiloni,ke-ηT≤φi,k(t)≤εi,k，
Figure FDA0003058917070000016
And 3, expressing the radial basis function neural network by adopting the following formula (4):
Figure FDA0003058917070000021
wherein the content of the first and second substances,
Figure FDA0003058917070000022
is known lightA sliding vector function, wherein the number l of NN nodes is more than 1; radial basis function
Figure FDA0003058917070000023
Is a gaussian function and is expressed by the following equation (5):
Figure FDA0003058917070000024
wherein, mujE.omega and eta > 0 are radial basis functions, respectively
Figure FDA0003058917070000025
The optimal weight vector W ═ ω1,...,ωl]TIs defined as:
Figure FDA0003058917070000026
wherein the content of the first and second substances,
Figure FDA0003058917070000027
is an approximation error inherent to the NN;
and 4, designing the self-adaptive learning controller by respectively applying a Backstepping method and a radial basis function neural network in the conditions of asymmetric dead zone input and continuous nonlinear input.
2. The design method of the adaptive iterative learning non-uniform target tracking controller according to claim 1, characterized in that: the specific process of the step 4 is as follows:
step 4.1, use function Γ (u)k) The actuator output with an asymmetric dead band is represented as shown in equation (7) below:
Figure FDA0003058917070000028
wherein the parameter mrAnd mlA right slope and a left slope representing the dead zone characteristic, respectively, parameter brAnd blA breakpoint representing a dead-zone input of the actuator;
and 4.2, modeling the asymmetric dead zone of the actuator into a form of the sum of a straight line and a disturbance-like term, wherein the form is shown in the following formula (8):
Γ(uk)＝m(t)uk+d(t) (8)；
wherein:
Figure FDA0003058917070000031
4.3, designing a controller for the uncertain nonlinear system with unknown asymmetric dead zone input and initial state error based on a self-adaptive iterative learning Backstepping frame;
step 4.4, there is a non-linearity in the control input of a class of continuous time non-linear systems represented by equation (1), which is taken to be in the sector region [ s ]1 s2]A continuous non-linear function N (u)k) And N (0) ═ 0, where s is1And s2Represents two straight lines (t)1,t2) Of (2) i.e.
Figure FDA0003058917070000032
And 4.5, designing a controller for the condition of continuous nonlinear input.
3. The design method of the adaptive iterative learning non-uniform target tracking controller according to claim 2, characterized in that: the specific process of the step 4.3 is as follows:
step A) let z1,k＝x1,k-yr,k，z2,k＝x2,k-α1,kIn which α is1,kIs a virtual controller; time ticks according to step 2 because of initial state errorsThe expression of the varying boundary layer is applied to the error function z by the following formula1φ,kAnd z2φ,kTo show that:
Figure FDA0003058917070000041
wherein the content of the first and second substances,
Figure FDA0003058917070000042
Figure FDA0003058917070000043
wherein the content of the first and second substances,
Figure FDA0003058917070000044
ε1,kand ε2,kIs a positive series of convergent numbers, η1And η2Is a designed normal number;
because:
Figure FDA0003058917070000045
then z is1φ,kThe derivative with respect to time is shown in the following equation (13):
Figure FDA0003058917070000046
step B) according to the representation of the radial basis function neural network in the step 3, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure FDA0003058917070000047
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure FDA0003058917070000048
Figure FDA0003058917070000049
Wherein the content of the first and second substances,
Figure FDA00030589170700000410
is the approximation error, W1Is the optimal weight vector;
for any real number a > 0 and positive integer l ≧ 2, take
Figure FDA00030589170700000411
Definition of
Figure FDA00030589170700000412
The virtual controller is taken as:
Figure FDA0003058917070000051
substituting equations (14) and (15) into (13) yields the following equation (16):
Figure FDA0003058917070000052
wherein the content of the first and second substances,
Figure FDA0003058917070000053
and
Figure FDA0003058917070000054
are respectively a parameter W1And N1Is estimated by the estimation of (a) a,
Figure FDA0003058917070000055
and
Figure FDA0003058917070000056
is the parameter estimation error, the last two terms to the right of the equal sign of equation (16) are rewritten as:
Figure FDA0003058917070000057
on the basis of the formula (17), the formula (16) is rewritten as the following formula (18):
Figure FDA0003058917070000058
order to
Figure FDA0003058917070000059
Then equation (18) is transformed into equation (19) as follows:
Figure FDA00030589170700000510
4. the design method of the adaptive iterative learning non-uniform target tracking controller according to claim 3, characterized in that: in the step 4.3:
defining: z is a radical ofn,k＝xn,k-αn-1,kAccording to the step 2 representation of the time-varying boundary layer network, the following error function z is introducednφ,k：
Figure FDA00030589170700000511
Wherein the content of the first and second substances,
Figure FDA0003058917070000061
εn,kis a positive series of convergent numbers, ηnIs a designed normal number;
then z isnφ,kThe derivative of (c) is:
Figure FDA0003058917070000062
wherein:
Figure FDA0003058917070000063
defining:
Figure FDA0003058917070000064
Figure FDA0003058917070000065
znφ,kthe derivative with respect to time is:
Figure FDA0003058917070000066
according to the representation of the radial basis function neural network in the step 3, the RBFNN unified approximation performance shows that the unknown nonlinear function is represented
Figure FDA0003058917070000067
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure FDA0003058917070000068
Figure FDA0003058917070000069
Wherein the content of the first and second substances,
Figure FDA00030589170700000610
is the approximation error, WnIs the optimal weight vector;
get
Figure FDA00030589170700000611
Selecting an actual controller as follows:
uk＝u1,k+u2,k (26)；
wherein u is2,kTo compensate for the unknown input gain m (t);
substituting equations (25) and (26) into (24) yields the following equation (27):
Figure FDA0003058917070000071
wherein the content of the first and second substances,
Figure FDA0003058917070000072
and
Figure FDA0003058917070000073
are respectively a parameter WnAnd NnIs estimated by the estimation of (a) a,
Figure FDA0003058917070000074
and
Figure FDA0003058917070000075
is the parameter estimation error, the last two terms to the right of the equality sign of equation (27) are rewritten as:
Figure FDA0003058917070000076
then:
Figure FDA0003058917070000077
get
Figure FDA0003058917070000078
Wherein
Figure FDA0003058917070000079
Is an uncertain parameter
Figure FDA00030589170700000710
(ii) an estimate of (d); order to
Figure FDA00030589170700000711
Then, the formula (29) becomes the following formula (30):
Figure FDA00030589170700000712
5. the design method of the adaptive iterative learning non-uniform target tracking controller according to claim 4, characterized in that: the specific process of the step 4.5 is as follows:
with N (u)k) As the output of the system instead of Γ (u)k) Definition of zn,k＝xn,k-αn-1,kAccording to the step 2 representation of the time-varying boundary layer network, the following error function z is introducednφ,k：
Figure FDA0003058917070000081
Wherein the content of the first and second substances,
Figure FDA0003058917070000082
εn,kis a positive series of convergent numbers, ηnIs a designed normal number; then z isnφ,kIs a derivative of
Figure FDA0003058917070000083
Wherein:
Figure FDA0003058917070000084
defining:
Figure FDA0003058917070000085
Figure FDA0003058917070000086
then equation (32) is rewritten as equation (36) below:
Figure FDA0003058917070000087
according to the representation of the radial basis function neural network in the step 3, the uniform approximation performance of the RBFNN shows that the unknown nonlinear function is
Figure FDA0003058917070000088
Approximated by RBFNN on a tight set to the following form, and with reconstruction error
Figure FDA0003058917070000089
Figure FDA00030589170700000810
Wherein the content of the first and second substances,
Figure FDA00030589170700000811
is the approximation error, WnIs the optimal weight vector;
according to the formulas (36) and (37), the following formula (38) is obtained:
Figure FDA00030589170700000812
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