CN112181002B - Micro gyroscope dual-recursion disturbance fuzzy neural network fractional order sliding mode control method - Google Patents

Micro gyroscope dual-recursion disturbance fuzzy neural network fractional order sliding mode control method Download PDF

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CN112181002B
CN112181002B CN202010965764.XA CN202010965764A CN112181002B CN 112181002 B CN112181002 B CN 112181002B CN 202010965764 A CN202010965764 A CN 202010965764A CN 112181002 B CN112181002 B CN 112181002B
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陈放
费峻涛
陈云
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Changzhou Campus of Hohai University
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    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D13/00—Control of linear speed; Control of angular speed; Control of acceleration or deceleration, e.g. of a prime mover
    • G05D13/62—Control of linear speed; Control of angular speed; Control of acceleration or deceleration, e.g. of a prime mover characterised by the use of electric means, e.g. use of a tachometric dynamo, use of a transducer converting an electric value into a displacement
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    • G01C19/00—Gyroscopes; Turn-sensitive devices using vibrating masses; Turn-sensitive devices without moving masses; Measuring angular rate using gyroscopic effects
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Abstract

A micro gyroscope dual-recursion disturbance fuzzy neural network fractional order sliding mode control method comprises the following steps: s1: establishing a micro gyroscope mathematical model, and designing a fractional order sliding mode surface; s2: designing a fractional order sliding mode control law, and performing sliding mode control on the micro gyroscope by taking the fractional order sliding mode control law as control input; s3: and designing a self-adaptive control algorithm based on the double-recursive disturbance fuzzy neural network and the Lyapunov stability, updating unknown parameters of the neural network in real time, and ensuring that the track of the motion point of the system stably tracks the track of the dynamic model. The method utilizes the combination of the fuzzy system and the neural network to estimate the unknown part of the system on line in real time, and replaces the true value of the unknown part with the estimated value, so as to solve the problem that the unknown part containing the unknown parameters in the actual system can not be accurately obtained.

Description

Micro gyroscope dual-recursion disturbance fuzzy neural network fractional order sliding mode control method
Technical Field
The invention relates to a micro-gyroscope double-recursion disturbance fuzzy neural network fractional order sliding mode control method, and belongs to the technical field of micro-gyroscope control.
Background
The principle applied by the gyroscope is mainly the law of conservation of angular momentum, is a device with sensing, direction stability maintaining and angular motion detection functions, and has the tendency of resisting direction change. Compared with the traditional gyroscope, the micro gyroscope has the advantages of wide application range, can be used in the fields of aviation, aerospace, navigation, automobile safety, bioengineering, geodetic survey, environmental monitoring and the like, and particularly has remarkable advantages compared with the traditional gyroscope in the fields with strict requirements on size, weight and the like.
However, due to the limitations of the processing precision of the MEMS process and the design principle, the current technology has not yet made a qualitative leap, still stays at the rate level and is difficult to progress, and it is difficult to meet the requirements of tactical level and inertial level. The structure size is usually micron-sized, after the integrated packaging, the size is only millimeter-sized, so that the sensitivity, the precision and the like of the silicon micro gyroscope come in and go out with ideal conditions, and the micro gyroscope mainly solves the problems of compensating errors in the processing process and measuring the angular speed.
Disclosure of Invention
In order to solve the technical defects in the prior art, the invention provides a micro-gyroscope dual-recursion disturbance fuzzy neural network fractional order sliding mode control method, which is used for estimating an unknown part of a system on line in real time by combining a fuzzy system and a neural network, and replacing the true value of the unknown part with an estimated value, so as to solve the problem that the unknown part containing unknown parameters in the actual system cannot be accurately obtained.
A micro gyroscope double-recursion disturbance fuzzy neural network fractional order sliding mode control method comprises the following steps:
s1: establishing a micro-gyroscope mathematical model, and designing a fractional order sliding mode surface based on the micro-gyroscope mathematical model;
s2: designing a fractional order sliding mode control law based on the micro gyroscope mathematical model established in the step S1 and the designed fractional order sliding mode surface, and performing sliding mode control on the micro gyroscope by taking the fractional order sliding mode control law as control input, wherein the control law comprises an equivalent control law and a switching control law;
s3: and designing a self-adaptive control algorithm based on the double-recursive disturbance fuzzy neural network and the Lyapunov stability, updating unknown parameters of the neural network in real time, and ensuring that the track of the motion point of the system stably tracks the track of the dynamic model.
Preferably, the specific steps of establishing the micro gyroscope mathematical model in the step S1 are as follows:
s1-1: establish dynamics model's rotation coordinate system, rotation coordinate system includes the direction of little gyroscope drive vibration, the direction of detection vibration and the direction of input angular velocity, establish the basic dynamics model of little gyroscope drive mode and detection mode based on rotation coordinate system, wherein, set for the direction that the X axle is little gyroscope drive vibration, the Y axle is the direction that little gyroscope detects the vibration, the Z axle is the direction of input angular velocity, the basic dynamics model of little gyroscope drive mode and detection mode is shown as formula (1):
Figure BDA0002682253130000021
wherein m is the mass of the mass, x and y are the position vectors of the mass in the driving vibration direction and the detection vibration direction,
Figure BDA0002682253130000022
is the first derivative of x and is,
Figure BDA0002682253130000023
is the second derivative of x and is,
Figure BDA00026822531300000210
is the first derivative of y and is,
Figure BDA0002682253130000024
is the second derivative of y, d x Damping coefficient for driving vibration direction, d y Damping coefficient, k, for detecting direction of vibration x Stiffness coefficient, k, for driving the vibration direction y For detecting the stiffness coefficient in the direction of vibration, u x Control input for driving the direction of vibration u y Detecting a control input, Ω, of a direction of vibration z Is the angular velocity input on the z-axis,
Figure BDA0002682253130000025
is omega z The first derivative of (a);
s1-2: and (3) carrying out structural error correction on the basic dynamic model, as shown in formula (2):
Figure BDA0002682253130000026
in the formula, d xx Damping coefficient for the corrected driving vibration direction, d yy For a modified damping coefficient for detecting the direction of vibration, d xy To couple damping coefficients, k xx For the corrected stiffness coefficient in the driving vibration direction, k yy For the corrected stiffness coefficient, k, of the detected vibration direction xy For coupling stiffness systemsCounting;
s1-3: carrying out dimensionless treatment on the dynamic model subjected to structural error correction, dividing two sides of two equations in the formula (2) by the mass m of the mass block of the micro gyroscope respectively, and referring to the length q 0 And natural resonance frequency omega 0 And obtaining a dynamic model after dimensionless of the micro gyroscope, wherein the dynamic model is shown as a formula (3):
Figure BDA0002682253130000027
in the formula, the expression of each dimensionless quantity is:
Figure BDA0002682253130000028
Figure BDA0002682253130000029
ω x is k is xx Form after dimensionless, ω y Is k yy Form after dimensionless, ω xy Is k xy A non-dimensionalized form;
s1-4: rewriting the dynamic model after the dimensionless processing into a vector-form dynamic model, as shown in formula (4):
Figure BDA0002682253130000031
in the formula (I), the compound is shown in the specification,
Figure BDA0002682253130000032
q is the output trace of the micro-gyroscope system,
Figure BDA0002682253130000033
is the first derivative of q and is,
Figure BDA0002682253130000034
is the second derivative of q, and D is the damping coefficient composed of the corrected driving vibration direction, the corrected detection vibration direction and the coupling damping coefficientA matrix, wherein K is a matrix consisting of a dimensionless form of the corrected stiffness coefficient in the driving vibration direction, a dimensionless form of the corrected stiffness coefficient in the detection vibration direction and a dimensionless form of the coupling stiffness coefficient, omega is a matrix consisting of the angular velocity in the input direction and the inverse of the angular velocity in the input direction, and u is a system control law, namely a fractional order sliding mode control law;
s1-5: considering the uncertainty of parameters in the system and external interference, a plurality of variables are introduced into the dynamic model in the form of vectors, as shown in formula (5):
Figure BDA0002682253130000035
in the formula, Δ D is the uncertainty of an unknown parameter D +2 Ω, Δ K is the uncertainty of an unknown parameter K, and D is external interference;
defining psi (x) as the unknown part of the system, let
Figure BDA0002682253130000036
And define f m Is the lumped parameter uncertainty of the micro-gyroscope system
Figure BDA0002682253130000037
Assuming system lumped uncertainty f m Exists in the upper bound and satisfies | | | f m ||≤F d V. mixing psi (x) and f m Substituting into equation (5) and deriving to obtain equation (6):
Figure BDA0002682253130000038
preferably, the fractional order sliding mode surface design in step S1 is as follows:
Figure BDA0002682253130000039
in the formula, s is a fractional order sliding mode surface, c is a normal number, e is a tracking error,
Figure BDA00026822531300000310
is the first derivative of e, where:
e＝q-q r ＝[x-q r1 ,y-q r2 ] T (8)；
Figure BDA00026822531300000311
in the formula (I), the compound is shown in the specification,
Figure BDA0002682253130000041
is the output track of the micro-gyroscope system,
Figure BDA0002682253130000042
for the desired trajectory of the micro-gyroscope system,
Figure BDA0002682253130000043
is q r1 The first derivative of (a) is,
Figure BDA0002682253130000044
is q r2 First derivative of (q) r1 Desired trajectory for x-axis, q, of micro-gyroscope system r2 T represents the transpose of the vector for the desired trajectory of the y-axis of the micro-gyroscope system.
Preferably, in the step S2, a fractional order sliding mode control law u is designed based on a micro-gyroscope mathematical model and a fractional order sliding mode surface, which is specifically as follows:
s2-1: derivation is carried out on the fractional order sliding mode surface model, sliding mode control reaching conditions are led into the fractional order sliding mode surface model after derivation, and an equivalent control law u is obtained eq As shown in equation (10):
Figure BDA0002682253130000045
s2-2: the rate of the system motion point approaching the switching surface is represented by using external interference and uncertainty of system parameters, and a switching control law is obtained, as shown in a formula (11):
Figure BDA0002682253130000046
wherein a is the coefficient of the switching term, and a is more than F d And | s | | represents the norm of s;
s2-3: by adopting a method combining equivalent sliding mode control and switching control, designing a fractional order sliding mode control law u based on an equation (10) and an equation (11) as shown in an equation (12):
Figure BDA0002682253130000047
preferably, the double-recursive disturbance fuzzy neural network in the step S3 comprises a five-layer neural network of a closed-loop dynamic feedback and fuzzy system, which sequentially comprises an input layer, a membership function layer, a rule layer, a recursive layer and an output layer, and is set as [ e ] 1 e 2 ] T The output is an estimated value of an unknown part psi (x) of the micro-gyro system model for the input of the double-recursion disturbance fuzzy neural network
Figure BDA0002682253130000048
The specific design is as follows:
a first layer: the output of the input layer, the double recursive perturbation fuzzy neural network input layer, is shown as formula (13):
μ k ＝x k ·W rok ·exY,for k＝1,2 (13)；
in the formula, mu k Is the output signal of the first layer of the neural network, x k Is an input signal of a neural network, W rok The outer layer recursion weight value is used, and exY is a fifth layer feedback signal of the neural network;
a second layer: the membership function layer of the double recursive disturbance fuzzy neural network utilizes sine-cosine disturbance functions to process the uncertainty of the rule, and each membership function consists of a Gaussian function and a sine-cosine disturbance function, as shown in formula (14):
Figure BDA0002682253130000051
in the formula, σ kj As output signal of the second layer of the neural network, c kj As central vectors of membership functions of the neural network, b kj Is the basis width, h, of membership functions of the neural network kj Coefficient of perturbation, v, being membership functions of neural networks kj The frequency of the neural network membership function is shown, exp is an exponential function with a natural constant e as a base, and j is the number of nodes corresponding to each node output of the first layer of the neural network;
and a third layer: the rule layer, the output of the rule layer of the double recursive perturbation fuzzy neural network is shown as formula (15):
the output of each node of the layer is the product of all input signals of the node, namely:
Figure BDA0002682253130000052
in the formula, delta i The output signal of the third layer of the neural network is i, the number of nodes of the third layer of the neural network is i;
a fourth layer: the output of the recursion layer of the double recursion perturbation fuzzy neural network is shown as the formula (16):
Figure BDA0002682253130000053
in the formula, theta l Is the output signal of the fourth layer of the neural network, r i Is the inner layer recursion weight;
and a fifth layer: the output of the output layer and the fifth layer output layer of the double recursive disturbance fuzzy neural network is shown as the formula (17):
Figure BDA0002682253130000054
in the formula, Y is the output signal of the fifth layer of the neural network, namely the unknown part psi (x) of the system,w is the weight of the neural network, m 0 The number of nodes in the membership function layer.
Preferably, the specific design steps of the adaptive control algorithm in step S3 are as follows:
s3-1: obtaining estimated value of unknown part of system by using double-recursion disturbance fuzzy neural network
Figure BDA0002682253130000055
As shown in equation (18):
Figure BDA0002682253130000056
in the formula (I), the compound is shown in the specification,
Figure BDA0002682253130000057
is an estimate of the weights of the neural network,
Figure BDA0002682253130000058
with respect to the x-ray source,
Figure BDA0002682253130000059
is a function of (a) a function of (b),
Figure BDA00026822531300000510
is b, c, h, v, r, W ro The estimated parameter vector of (2) is,
Figure BDA00026822531300000511
an estimate of ψ (x);
s3-2: estimate of unknown part of system
Figure BDA0002682253130000061
Substituting the sliding mode control law into an estimated sliding mode control law u ', and obtaining an estimated sliding mode control law u' as shown in a formula (19):
Figure BDA0002682253130000062
s3-3: setting the estimated value and true value of unknown part in systemDifference of real value
Figure BDA0002682253130000063
Estimation error as an unknown part of the system;
wherein, through the approximation property of the Gaussian function of the double-recursion disturbance fuzzy neural network, the ideal neural network output Y exists * Then the real value of the unknown part ψ (x) of the system is:
Figure BDA0002682253130000064
where ε is the approximation error, b * ,c * ,h * ,v * ,r * ,
Figure BDA0002682253130000065
Are respectively b, c, h, v, r, W ro The optimal parameter vector of (2); the difference between the true value psi (x) and the estimated value of the unknown part in the system
Figure BDA0002682253130000066
Comprises the following steps:
Figure BDA0002682253130000067
wherein the content of the first and second substances,
Figure BDA0002682253130000068
Figure BDA00026822531300000616
with respect to the x-ray(s),
Figure BDA0002682253130000069
is a function of (a) a function of (b),
Figure BDA00026822531300000610
Figure BDA00026822531300000611
to pair
Figure BDA00026822531300000612
Taylor expansion is performed to obtain:
Figure BDA00026822531300000613
wherein, delta is a Taylor expansion remainder term,
Figure BDA00026822531300000614
Figure BDA00026822531300000615
Figure BDA0002682253130000071
Figure BDA0002682253130000072
substituting (22) into (21) yields:
Figure BDA0002682253130000073
wherein the content of the first and second substances,
Figure BDA0002682253130000074
for lumped approximation error, there is an upper bound |. Epsilon 0 E is less than or equal to E, and E is a normal number;
s3-4: simplifying the estimated dynamic model in a vector form, substituting the simplified dynamic model into a first derivative of a preset Lyapunov function with respect to time, and designing an adaptive control algorithm of unknown parameters of the system according to the Lyapunov stability principle, wherein the adaptive control algorithm specifically comprises the following steps:
selecting a Lyapunov function V as follows:
Figure BDA0002682253130000075
in the formula eta 1 ,η 2 ,η 3 ,η 4 ,η 5 ,η 6 ,η 7 The learning rates are normal numbers; tr {. Is } represents the trace-finding operation of the matrix.
The first derivative with respect to time is taken for the Lyapunov function:
Figure BDA0002682253130000076
in order to ensure the stability of the system, the order
Figure BDA0002682253130000081
Figure BDA0002682253130000082
Figure BDA0002682253130000083
Figure BDA0002682253130000084
The neural network parameter self-adaptive law is designed as follows:
Figure BDA0002682253130000085
Figure BDA0002682253130000086
Figure BDA0002682253130000087
Figure BDA0002682253130000088
Figure BDA0002682253130000089
Figure BDA00026822531300000810
Figure BDA00026822531300000811
in the formula (I), the compound is shown in the specification,
Figure BDA00026822531300000812
is that
Figure BDA00026822531300000813
The first derivative of (a) is,
Figure BDA00026822531300000814
is that
Figure BDA00026822531300000815
The first derivative of (a) is,
Figure BDA00026822531300000816
is that
Figure BDA00026822531300000817
The first derivative of (a) is,
Figure BDA00026822531300000818
is that
Figure BDA00026822531300000819
The first derivative of (a) is,
Figure BDA00026822531300000820
is that
Figure BDA00026822531300000821
The first derivative of (a) is,
Figure BDA00026822531300000822
is that
Figure BDA00026822531300000823
The first derivative of (a) is,
Figure BDA00026822531300000824
is that
Figure BDA00026822531300000825
The first derivative of (a);
Figure BDA00026822531300000826
for the estimation of the unknown parameter W,
Figure BDA00026822531300000827
is an estimate of the unknown parameter b,
Figure BDA00026822531300000828
is an estimate of the unknown parameter c and,
Figure BDA00026822531300000829
is an estimate of the unknown parameter h,
Figure BDA00026822531300000830
is an estimate of the unknown parameter v,
Figure BDA00026822531300000831
is an estimate of the unknown parameter r,
Figure BDA00026822531300000832
for an unknown parameter W ro Estimated value of, η 1 ,η 2 ,η 3 ,η 4 ,η 5 ,η 6 ,η 7 Learning rates for each unknown parameter.
The invention mainly adopts the technical scheme that:
has the advantages that: the invention provides a micro gyroscope double-recursion disturbance fuzzy neural network fractional order sliding mode control method, which has the following advantages:
(1) The fractional order sliding mode control law increases the order number which can adjust the fractional order, the control precision is improved, and the flexibility of the controller is improved;
(2) The self-adaptive approximation of the unknown part of the model is realized by using the double-recursion disturbance neural network, and the controller is not dependent on the accurate mathematical model of the controlled system;
(3) The Lyapunov stability theory is utilized to design an adaptive law with unknown parameters such as Gaussian function base width, central vector, disturbance coefficient and weight in a neural network, so that the system obtains good tracking performance under the condition of model uncertainty and external interference, and the robustness and anti-interference performance of the system are improved;
(4) The self-adaptive control algorithm can process the uncertainty of the system, realize the on-line automatic setting of the control system parameters and improve the stability and the robustness of the system.
Drawings
FIG. 1 is a block diagram of a micro-gyroscope system according to the invention;
FIG. 2 is a diagram of a dual recursive perturbation fuzzy neural network according to an embodiment of the present invention;
FIG. 3 is a trace of the x-axis trajectory of a micro gyroscope according to an embodiment of the present invention;
FIG. 4 is a trace plot of the y-axis trace of a micro gyroscope in an example of the present invention;
FIG. 5 is a graph illustrating adaptive identification of perturbation coefficient h of a micro-gyroscope according to an embodiment of the present invention;
FIG. 6 is a graph of the frequency v of the micro-gyroscope according to an embodiment of the present invention;
FIG. 7 is a graph of the x-axis direction of the unknown part estimated by the neural network in the example of the present invention;
FIG. 8 is a graph of the neural network estimating the unknown part in the y-axis direction in the example of the present invention.
Detailed Description
In order to make those skilled in the art better understand the technical solutions in the present application, the technical solutions in the embodiments of the present application are clearly and completely described below, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all 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 application.
A micro gyroscope self-adaptive double-recursive disturbance fuzzy neural network fractional order sliding mode control method comprises the following steps:
step one, establishing a mathematical model of the micro gyroscope.
The driving mode and the detection mode of the micro gyroscope are regarded as a second-order system of spring-mass-damping. Firstly, establishing a rotating coordinate system of a dynamic model; and then, establishing basic dynamic models of a driving mode and a detection mode of the micro gyroscope based on the rotating coordinate system.
In this embodiment, the x-axis is a driving vibration direction of the micro gyroscope, the y-axis is a detection vibration direction of the micro gyroscope, and the z-axis is an input angular velocity direction. The basic kinetic equation of the micro gyroscope is shown as formula (1) and formula (2):
Figure BDA0002682253130000091
wherein m is the mass of the mass, x and y are the position vectors of the mass in the driving direction and the detecting direction,
Figure BDA0002682253130000101
is the first derivative of x and is,
Figure BDA0002682253130000102
is the second derivative of x and is,
Figure BDA0002682253130000103
is the first derivative of y and is,
Figure BDA0002682253130000104
is the second derivative of y, d x Damping coefficient for driving direction, d y Damping coefficient, k, for detecting direction x Stiffness coefficient of driving direction, k y To measure the stiffness coefficient of the direction, u x For control input of drive direction, u y Control input for detecting direction, Ω z The angular velocity in the z-axis, i.e. the angular velocity in the input direction,
Figure BDA0002682253130000105
is omega z The first derivative of (a).
Considering the influence caused by the structural error of the micro gyroscope, in order to improve the control precision, the formula (1) is corrected as follows:
Figure BDA0002682253130000106
in the formula, d xx Damping coefficient for the corrected drive direction, d yy Damping coefficient for the corrected detection direction, d xy To couple damping coefficients, k xx For the corrected stiffness coefficient of the drive direction, k yy For the corrected stiffness coefficient of the detection direction, k xy Is the coupling stiffness coefficient.
In order to reduce the complexity of the controller design, the dynamic model is subjected to non-dimensionalization treatment, two sides of two equations in the formula (2) are divided by the mass m of the mass block of the micro-gyroscope respectively, and the reference length q is used 0 And natural resonance frequency omega 0 Obtaining a nondimensionalized dynamic model of the micro gyroscope, wherein the nondimensionalized dynamic model is shown as a formula (3):
Figure BDA0002682253130000107
in the formula, the expression of each dimensionless quantity is:
Figure BDA0002682253130000108
Figure BDA0002682253130000109
ω x is k is xx Form after dimensionless, ω y Is k yy Form after dimensionless, ω xy Is k xy Non-dimensionalized form.
Rewriting equation (3) to vector form, as shown in equation (4):
Figure BDA00026822531300001010
in the formula (I), the compound is shown in the specification,
Figure BDA00026822531300001011
q is the output trace of the micro-gyroscope system,
Figure BDA00026822531300001012
is the first derivative of q and is,
Figure BDA00026822531300001013
q is a second derivative of q, D is a matrix consisting of a corrected damping coefficient of the driving vibration direction, a corrected damping coefficient of the detecting vibration direction and a coupling damping coefficient, K is a matrix consisting of a dimensionless form of a corrected stiffness coefficient of the driving vibration direction, a dimensionless form of a corrected stiffness coefficient of the detecting vibration direction and a dimensionless form of a coupling stiffness coefficient, Ω is a matrix consisting of an angular velocity in the input direction and an opposite number of the angular velocity in the input direction, and u is a system control law, i.e., a fractional order sliding mode control law;
considering the uncertainty of parameters in the system and external interference, the vector form (4) of the dynamic model of the micro-gyroscope system is rewritten as follows:
Figure BDA0002682253130000111
in the formula, Δ D is the uncertainty of the unknown parameter D +2 Ω, Δ K is the uncertainty of the unknown parameter K, and D is the external interference.
Definition psi (x)) For the unknown part of the system, order
Figure BDA0002682253130000112
And define f m Is the lumped parameter uncertainty of the micro-gyroscope system
Figure BDA0002682253130000113
Assuming system lumped uncertainty f m Exists in the upper bound and satisfies | | f m ||≤F d Let ψ (x) and f m Substituting into equation (5) and deriving to obtain equation (6):
Figure BDA0002682253130000114
and step two, designing a micro gyroscope fractional order sliding mode control system.
FIG. 1 is a block diagram of a micro-gyroscope system according to an embodiment of the present invention;
firstly, designing a fractional order sliding mode surface of sliding mode control as follows:
Figure BDA0002682253130000115
in the formula, s is a fractional order sliding mode surface, c is a normal number, e is a tracking error,
Figure BDA0002682253130000116
is the first derivative of e, where:
e＝q-q r ＝[x-q r1 ,y-q r2 ] T (8)；
Figure BDA0002682253130000117
in the formula (I), the compound is shown in the specification,
Figure BDA0002682253130000118
is the output track of the micro-gyroscope system,
Figure BDA0002682253130000119
for the desired trajectory of the micro-gyroscope system,
Figure BDA00026822531300001110
is q r1 The first derivative of (a) is,
Figure BDA00026822531300001111
is q r2 First derivative of (q) r1 For the desired trajectory of the x-axis, q, of the micro-gyroscope system r2 T represents the transpose of the vector for the desired trajectory of the y-axis of the micro-gyroscope system.
Then, designing a control law u of fractional order sliding mode control, which is specifically as follows:
fractional order equivalent control law u capable of controlling arrival conditions by sliding mode eq ：
Figure BDA00026822531300001112
The method includes the steps of representing the speed of a system motion point approaching a switching surface by means of external interference and system parameter uncertainty, and obtaining a switching control law sw Comprises the following steps:
Figure BDA0002682253130000121
wherein a is the coefficient of the switching term, and a is more than F d And | s | | represents the norm of s.
By adopting a method combining equivalent sliding mode control and switching control, designing a control law u of fractional order sliding mode control based on an equation (10) and an equation (11) as follows:
Figure BDA0002682253130000122
and step three, designing a double-recursion disturbance fuzzy neural network.
FIG. 2 is a diagram showing a structure of a dual-recursive perturbation fuzzy neural network according to an embodiment of the present invention;
in this embodiment, the double-recursive disturbance fuzzy neural network includes a five-Layer neural network of a closed-loop dynamic feedback and fuzzy system, which mainly includes an Input Layer (Input Layer), a Membership Function Layer (Membership Function Layer), a Rule Layer (Rule Layer), a recursive Layer (recursive Layer), and an Output Layer (Output Layer), and is set to [ e ] e 1 e 2 ] T The output is an estimated value of an unknown part psi (x) of the micro-gyro system model for the input of the double-recursion disturbance fuzzy neural network
Figure BDA0002682253130000123
The specific design is as follows:
a first layer: the output of the input layer, the double recursive disturbance fuzzy neural network input layer, is shown as formula (13):
μ k ＝x k ·W rok ·exY,for k＝1,2 (13)；
in the formula, mu k Is the output signal of the first layer of the neural network, x k Is an input signal of a neural network, W rok The outer layer recursion weight value is used, and exY is a fifth layer feedback signal of the neural network;
a second layer: the membership function layer of the double recursive disturbance fuzzy neural network utilizes sine-cosine disturbance functions to process the uncertainty of the rule, and each membership function consists of a Gaussian function and a sine-cosine disturbance function, as shown in formula (14):
Figure BDA0002682253130000124
in the formula, σ kj As output signal of the second layer of the neural network, c kj As central vectors of membership functions of the neural network, b kj Is the basis width, h, of membership functions of the neural network kj Coefficient of perturbation, v, being membership functions of neural networks kj The frequency of the neural network membership function is shown, exp is an exponential function with a natural constant e as a base, and j is the number of nodes corresponding to each node output of the first layer of the neural network;
and a third layer: the rule layer, the output of the dual recursive perturbation fuzzy neural network rule layer is shown as formula (15):
the output of each node of the layer is the product of all the input signals of the node, namely:
Figure BDA0002682253130000131
in the formula, delta i The output signal of the third layer of the neural network i is the number of nodes of the third layer of the neural network;
a fourth layer: the output of the recursion layer of the double recursion perturbation fuzzy neural network is shown as the formula (16):
Figure BDA0002682253130000132
in the formula, theta l Is the output signal of the fourth layer of the neural network, r i Is the inner layer recursion weight;
and a fifth layer: the output layer and the output of the fifth layer of the double recursive perturbation fuzzy neural network are shown as the formula (17):
Figure BDA0002682253130000133
in the formula, Y is the output signal of the fifth layer of the neural network, namely the system unknown partial value psi (x), W is the weight of the neural network, and m is 0 The number of nodes in the membership function layer.
Estimating the unknown part of the system by using a double-recursion disturbance fuzzy neural network, wherein the formula (21) is as follows:
Figure BDA0002682253130000134
in the formula (I), the compound is shown in the specification,
Figure BDA0002682253130000135
weighted by neural networksThe value of the estimated value is,
Figure BDA0002682253130000136
with respect to the x-ray source,
Figure BDA0002682253130000137
as a function of (a) or (b),
Figure BDA0002682253130000138
is b, c, h, v, r, W ro The estimated parameter vector of (2) is,
Figure BDA0002682253130000139
an estimate of ψ (x);
therefore, the control law u of equation (12) can be adjusted to:
Figure BDA00026822531300001310
by the approximating nature of the function, there is an ideal recurrent neural network output Y * The unknown part of the system ψ (x) is then:
Figure BDA00026822531300001311
where ε is the approximation error, b * ,c * ,h * ,v * ,r * ,
Figure BDA00026822531300001312
Are respectively b, c, h, v, r, W ro The optimal parameter vector of (2). The difference between the true value psi (x) and the estimated value of the unknown part in the system
Figure BDA0002682253130000141
Comprises the following steps:
Figure BDA0002682253130000142
wherein the content of the first and second substances,
Figure BDA0002682253130000143
Figure BDA0002682253130000144
with respect to the x-ray(s),
Figure BDA0002682253130000145
is a function of (a) a function of (b),
Figure BDA0002682253130000146
Figure BDA0002682253130000147
to pair
Figure BDA0002682253130000148
Taylor expansion is carried out to obtain:
Figure BDA0002682253130000149
wherein, delta is a Taylor expansion remainder term,
Figure BDA00026822531300001410
Figure BDA00026822531300001411
Figure BDA00026822531300001412
Figure BDA00026822531300001413
substituting (22) into (21) yields:
Figure BDA00026822531300001414
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00026822531300001415
for lumped approximation error, there is an upper bound |. Epsilon 0 E is less than or equal to E, and E is a normal number;
and fourthly, identifying unknown parameters in the micro gyroscope system on line.
In this embodiment, the unknown parameters for designing the micro-gyroscope system are b, c, h, v, r, W ro And W;
the Lyapunov stability theory is used for replacing unknown actual values of the Lyapunov stability theory by utilizing estimated values of base width, central vector, weight and the like, and estimated values of base width of Gaussian function
Figure BDA0002682253130000151
Estimated value of central vector of Gaussian function
Figure BDA0002682253130000152
Estimation value of disturbance coefficient of membership function
Figure BDA0002682253130000153
Frequency of membership function
Figure BDA0002682253130000154
Estimation of inner layer recursive weights
Figure BDA0002682253130000155
And estimates of outer recursive weights
Figure BDA0002682253130000156
Adaptive law of (3), estimation of weight of neural network
Figure BDA0002682253130000157
The online real-time updating is realized, and the system stability is analyzed by using the theory, which specifically comprises the following steps:
to design for
Figure BDA0002682253130000158
And
Figure BDA0002682253130000159
the self-adaptive law of (1) selects a Lyapunov function V as follows:
Figure BDA00026822531300001510
in the formula eta 1 ,η 2 ,η 3 ,η 4 ,η 5 ,η 6 ,η 7 The learning rates are normal numbers; tr {. Cndot } represents the tracing operation of the matrix.
The first derivative with respect to time is taken for the designed Lyapunov function:
Figure BDA00026822531300001511
to ensure the stability of the system, let
Figure BDA00026822531300001512
Figure BDA00026822531300001513
Figure BDA0002682253130000161
Figure BDA0002682253130000162
The neural network parameter self-adaptive law is designed as follows:
Figure BDA0002682253130000163
Figure BDA0002682253130000164
Figure BDA0002682253130000165
Figure BDA0002682253130000166
Figure BDA0002682253130000167
Figure BDA0002682253130000168
Figure BDA0002682253130000169
equation (25) can be rewritten as:
Figure BDA00026822531300001610
due to epsilon 0 And f m Present in the upper bound E, F d So that when a.gtoreq.E + F is satisfied d Can ensure
Figure BDA00026822531300001611
The system stability is proved in a semi-negative mode, namely the system tracking track can reach the designed fractional order sliding mode surface and stay on the sliding mode surface. Inequality pair
Figure BDA00026822531300001612
Integral, can obtain
Figure BDA00026822531300001613
Since V ' (0) and V ' (t) are bounded and V ' (t) is not incremented, it can be seen that
Figure BDA00026822531300001614
Is bounded. According to the barbalt theorem and its deduction, it can be known
Figure BDA00026822531300001615
Therefore, the system is asymptotically stable, and the tracking error and the fractional sliding mode surface are asymptotically converged to zero.
To verify the feasibility and effectiveness of the invention, MATLAB/Simulink was used for simulation.
m＝1.8×10 -7 kg，d xx ＝1.8×10 -6 N·s/m，d xy ＝3.6×10 -7 N·s/m，
d yy ＝1.8×10 -6 N·s/m，k xx ＝63.955N/m，k xy ＝12.779N/m，k yy ＝95.92N/m，
Ω z ＝100rad/s，q 0 ＝1μm，ω 0 ＝1kHz。
The non-quantitative rigidization parameters of the micro gyroscope can be obtained as follows:
d xx ＝0.01，d xy ＝0.002，d yy ＝0.01，
Figure BDA0002682253130000171
ω xy ＝70.99，
Figure BDA0002682253130000172
Ω z ＝0.1。
in the simulation experiment, the simulation time is 60s, and the initial conditions of the system are as follows: q. q of 1 (0)＝0.001，
Figure BDA0002682253130000173
q 2 (0)＝0.001，
Figure BDA0002682253130000174
The two expected running tracks of the micro gyroscope are set as follows:
q r1 ＝sin(4.17t)，q r2 =1.2sin (5.11 t), the fractional sliding mode surface parameters are: m =3000, λ =10K =10, the order is set to 0.1,0.4,0.5,0.7,0.9, and the obtained root mean square errors are compared, as shown in the table, to determine that the order of the fractional order is α =0.9, and the initial values of the weight, the base width, the central vector, the disturbance coefficient, the frequency, the inner-layer recursion weight and the outer-layer recursion weight of the double-feedback fuzzy neural network are respectively taken
Figure BDA0002682253130000175
Figure BDA0002682253130000176
Figure BDA0002682253130000177
I.e. parameter m in the neural network structure 0 =5, self-adaptive law gain is respectively taken as eta 1 ＝80000，η 2 ＝10 7 ，η 3 ＝0.001，η 4 ＝0.01，η 5 ＝0.01，η 6 ＝10 -13 ，η 7 =10000. The simulation results are detailed in fig. 3 to 6.
As shown in fig. 3, which is a trace curve of the trace x-axis of the micro-gyroscope according to the embodiment of the present invention, the output signal can quickly trace the upper reference trace;
as shown in fig. 4, which is a trace curve of the trace y-axis of the micro-gyroscope in the embodiment of the present invention, the output signal can quickly trace the upper reference trace;
as shown in fig. 5, which is an adaptive identification curve of a disturbance coefficient h of a micro gyroscope in the embodiment of the present invention, by using adaptive fractional order sliding mode control, a parameter h can be converged within a limited time and tends to be stable;
as shown in fig. 6, which is a self-adaptive identification curve of the frequency v of the micro gyroscope in the embodiment of the present invention, by using the self-adaptive fractional order sliding mode control, the parameter v can be converged within a limited time and tends to be stable;
as shown in fig. 7, the curve of the micro gyroscope neural network in the x-axis direction of the estimated unknown part in the embodiment of the present invention is shown, and the value of the unknown part of the micro gyroscope system can be well estimated by using the double recursive disturbance fuzzy neural network;
as shown in fig. 8, the curve of the micro gyroscope neural network in the example of the present invention in the y-axis direction of the estimated unknown part can be well estimated by using the dual recursive disturbance fuzzy neural network;
the foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (1)

1. A micro gyroscope double-recursion disturbance fuzzy neural network fractional order sliding mode control method is characterized by comprising the following steps:
s1: establishing a micro-gyroscope mathematical model, and designing a fractional order sliding mode surface based on the micro-gyroscope mathematical model, wherein the micro-gyroscope mathematical model is specifically established by the following steps:
s1-1: establish dynamic model's rotation coordinate system, rotation coordinate system includes the direction of little gyroscope drive vibration, detects the direction of vibration and the direction of input angular velocity, establishes little gyroscope drive mode and detection mode's basic dynamic model based on rotation coordinate system, wherein, sets for the direction that the X axle is little gyroscope drive vibration, and the Y axle is little gyroscope detection vibration's direction, and the Z axle is the direction of input angular velocity, and little gyroscope drive mode and detection mode's basic dynamic model is shown as formula (1):
Figure FDA0004034800440000011
wherein m is the mass of the mass, x and y are the position vectors of the mass in the driving vibration direction and the detection vibration direction,
Figure FDA0004034800440000012
is a first order of xThe derivative(s) of the signal(s),
Figure FDA0004034800440000013
is the second derivative of x and is,
Figure FDA0004034800440000014
is the first derivative of y and is,
Figure FDA0004034800440000015
is the second derivative of y, d x Damping coefficient for driving vibration direction, d y Damping coefficient, k, for detecting direction of vibration x Coefficient of stiffness, k, for driving the direction of vibration y For detecting the stiffness coefficient in the direction of vibration, u x Control input for driving the direction of vibration u y Control input, omega, for detecting the direction of vibration z Is the angular velocity input on the z-axis,
Figure FDA0004034800440000016
is omega z The first derivative of (a);
s1-2: and (3) carrying out structural error correction on the basic dynamic model, as shown in an equation (2):
Figure FDA0004034800440000017
in the formula, d xx Damping coefficient for the corrected driving vibration direction, d yy For a modified damping coefficient for detecting the direction of vibration, d xy To couple damping coefficients, k xx For the corrected stiffness coefficient in the driving vibration direction, k yy For a corrected stiffness coefficient, k, for detecting the direction of vibration xy Is a coupling stiffness coefficient;
s1-3: carrying out dimensionless treatment on the dynamic model subjected to structural error correction, dividing two sides of two equations in the formula (2) by mass m of the mass block of the micro gyroscope respectively, and referring to length q 0 And natural resonance frequency omega 0 Obtaining a dynamic model after dimensionless of the micro gyroscopeAs shown in formula (3):
Figure FDA0004034800440000021
in the formula, the expression of each dimensionless quantity is:
Figure FDA0004034800440000022
Figure FDA0004034800440000023
ω x is k xx Form after dimensionless, omega y Is k yy Form after dimensionless, ω xy Is k xy (ii) a non-dimensionalized form;
s1-4: rewriting the dynamic model after the dimensionless processing into a vector-form dynamic model, as shown in formula (4):
Figure FDA0004034800440000024
in the formula (I), the compound is shown in the specification,
Figure FDA0004034800440000025
q is the output trace of the micro-gyroscope system,
Figure FDA0004034800440000026
is the first derivative of q and is,
Figure FDA0004034800440000027
is the second derivative of q, D is a matrix consisting of the corrected damping coefficient of the driving vibration direction, the corrected damping coefficient of the detected vibration direction and the coupling damping coefficient, K is a matrix consisting of the dimensionless form of the corrected stiffness coefficient of the driving vibration direction, the dimensionless form of the corrected stiffness coefficient of the detected vibration direction and the dimensionless form of the coupling stiffness coefficient, and Ω is a matrix consisting of the input direction stiffness coefficientU is a system control law, namely a fractional order sliding mode control law;
s1-5: considering the uncertainty of parameters in the system and external interference, a plurality of variables are introduced into the dynamic model in the form of vectors, as shown in equation (5):
Figure FDA0004034800440000031
in the formula, Δ D is the uncertainty of an unknown parameter D +2 Ω, Δ K is the uncertainty of an unknown parameter K, and D is external interference;
defining psi (x) as the unknown part of the system, let
Figure FDA0004034800440000032
And define f m For lumped parameter uncertainty of micro-gyroscope system
Figure FDA0004034800440000033
Assuming system lumped uncertainty f m Exists in the upper bound and satisfies | | f m ||≤F d Let ψ (x) and f m Substituting into equation (5) and deriving to obtain equation (6):
Figure FDA0004034800440000034
the fractional order sliding mode surface is designed as follows:
Figure FDA0004034800440000035
in the formula, s is a fractional order sliding mode surface, c is a normal number, e is a tracking error,
Figure FDA0004034800440000036
is the first derivative of e, where:
e＝q-q r ＝[x-q r1 ,y-q r2 ] T (8)；
Figure FDA0004034800440000037
in the formula (I), the compound is shown in the specification,
Figure FDA0004034800440000038
is the output track of the micro-gyroscope system,
Figure FDA0004034800440000039
for the desired trajectory of the micro-gyroscope system,
Figure FDA00040348004400000310
is q r1 The first derivative of (a) is,
Figure FDA00040348004400000311
is q r2 First derivative of (q) r1 For the desired trajectory of the x-axis, q, of the micro-gyroscope system r2 T represents the transposition of the vector for the y-axis expected track of the micro-gyroscope system;
s2: designing a fractional order sliding mode control law u based on the micro gyroscope mathematical model established in the step S1 and the designed fractional order sliding mode surface, and performing sliding mode control on the micro gyroscope by taking the fractional order sliding mode control law u as control input, wherein the control law comprises an equivalent control law and a switching control law;
the specific design method of the fractional order sliding mode control law u is as follows:
s2-1: derivation is carried out on the fractional order sliding mode surface model, sliding mode control reaching conditions are led into the fractional order sliding mode surface model after derivation, and an equivalent control law u is obtained eq As shown in equation (10):
Figure FDA0004034800440000041
s2-2: the rate of the system motion point approaching the switching surface is represented by using external interference and uncertainty of system parameters, and a switching control law is obtained, as shown in a formula (11):
Figure FDA0004034800440000042
wherein a is the coefficient of the switching term, and a is more than F d And | s | | represents the norm of s;
s2-3: by adopting a method combining equivalent sliding mode control and switching control, designing a fractional order sliding mode control law u based on an equation (10) and an equation (11) as shown in an equation (12):
Figure FDA0004034800440000043
s3: designing an adaptive control algorithm based on a double-recursion disturbance fuzzy neural network and Lyapunov stability, updating unknown parameters of the neural network in real time, and ensuring that the track of a system motion point stably tracks the track of a dynamic model;
the double-recursion disturbance fuzzy neural network comprises a five-layer neural network of a closed-loop dynamic feedback and fuzzy system, which sequentially comprises an input layer, a membership function layer, a rule layer, a recursion layer and an output layer, and is set by [ e ] 1 e 2 ] T The output is an estimated value of an unknown part psi (x) of the micro-gyro system model for the input of the double-recursion disturbance fuzzy neural network
Figure FDA0004034800440000044
The specific design is as follows:
a first layer: the output of the input layer, the double recursive disturbance fuzzy neural network input layer, is shown as formula (13):
μ k ＝x k ·W rok ·exY,for k＝1,2 (13)；
in the formula, mu k Is the output signal of the first layer of the neural network, x k Is an input signal of a neural network, W rok Is outer recursive weight, exY is neural netA fifth layer feedback signal;
a second layer: the membership function layer of the double recursive disturbance fuzzy neural network utilizes sine-cosine disturbance functions to process the uncertainty of the rule, and each membership function consists of a Gaussian function and a sine-cosine disturbance function, as shown in formula (14):
Figure FDA0004034800440000051
in the formula, σ kj As output signal of the second layer of the neural network, c kj As central vectors of membership functions of the neural network, b kj Is the basis width, h, of membership functions of the neural network kj Coefficient of perturbation, v, being membership functions of neural networks kj The frequency of the membership function of the neural network is shown, exp is an exponential function with a natural constant e as a base, and j is the number of nodes corresponding to each node output of the first layer of the neural network;
and a third layer: the rule layer, the output of the rule layer of the double recursive perturbation fuzzy neural network is shown as formula (15):
the output of each node of the layer is the product of all the input signals of the node, namely:
Figure FDA0004034800440000052
in the formula, delta i The output signal of the third layer of the neural network i is the number of nodes of the third layer of the neural network;
a fourth layer: the recursive layer and the output of the recursive layer of the double-recursive perturbation fuzzy neural network are shown as the formula (16):
Figure FDA0004034800440000061
in the formula, theta l Is the output signal of the fourth layer of the neural network, r i Is the inner layer recursion weight;
and a fifth layer: the output layer and the output of the fifth layer of the double recursive perturbation fuzzy neural network are shown as the formula (17):
Figure FDA0004034800440000062
in the formula, Y is the output signal of the fifth layer of the neural network, namely the unknown part psi (x) of the system, W is the weight of the neural network, and m is 0 The number of nodes of the membership function layer is;
the specific design steps of the self-adaptive control algorithm are as follows:
s3-1: obtaining estimated value of unknown part of system by using double-recursion disturbance fuzzy neural network
Figure FDA00040348004400000612
As shown in equation (18):
Figure FDA0004034800440000063
in the formula (I), the compound is shown in the specification,
Figure FDA0004034800440000064
is an estimate of the weights of the neural network,
Figure FDA0004034800440000065
with respect to the x-ray(s),
Figure FDA0004034800440000066
as a function of (a) or (b),
Figure FDA0004034800440000067
is b, c, h, v, r, W ro The estimated parameter vector of (2) is,
Figure FDA0004034800440000068
an estimate of ψ (x);
s3-2: estimate of unknown part of system
Figure FDA0004034800440000069
Substituting the sliding mode control law into the estimated sliding mode control law u ', and obtaining the estimated sliding mode control law u' as shown in a formula (19):
Figure FDA00040348004400000610
s3-3: setting the difference between the estimated value and the true value of the unknown part in the system
Figure FDA00040348004400000611
Estimation error as an unknown part of the system;
wherein, through the approximate property of the Gaussian function of the double-recursion disturbance fuzzy neural network, the ideal neural network output Y exists * Then the real value of the unknown part ψ (x) of the system is:
Figure FDA0004034800440000071
where ε is the approximation error, b * ,c * ,h * ,v * ,r * ,
Figure FDA0004034800440000072
Are respectively b, c, h, v, r, W ro The optimal parameter vector of (2); the difference between the true value psi (x) and the estimated value of the unknown part in the system
Figure FDA0004034800440000073
Comprises the following steps:
Figure FDA0004034800440000074
wherein the content of the first and second substances,
Figure FDA0004034800440000075
Figure FDA0004034800440000076
with respect to the x-ray source,
Figure FDA0004034800440000077
as a function of (a) or (b),
Figure FDA0004034800440000078
Figure FDA0004034800440000079
to pair
Figure FDA00040348004400000710
Taylor expansion is carried out to obtain:
Figure FDA00040348004400000711
wherein, delta is the Taylor expansion remainder term,
Figure FDA00040348004400000712
Figure FDA00040348004400000713
Figure FDA00040348004400000714
Figure FDA00040348004400000715
substituting (22) into (21) yields:
Figure FDA0004034800440000081
wherein the content of the first and second substances,
Figure FDA0004034800440000082
for lumped approximation error, there is an upper bound |. Epsilon 0 E is less than or equal to E, and E is a normal number;
s3-4: simplifying the estimated dynamic model in a vector form, substituting the simplified dynamic model into a first derivative of a preset Lyapunov function with respect to time, and designing an adaptive control algorithm of unknown parameters of the system according to the Lyapunov stability principle, wherein the adaptive control algorithm specifically comprises the following steps:
selecting a Lyapunov function V as follows:
Figure FDA0004034800440000083
in the formula eta 1 ,η 2 ,η 3 ,η 4 ,η 5 ,η 6 ,η 7 The learning rates are normal numbers; tr {. Is equal to } represents the trace calculation of the matrix;
the first derivative with respect to time is taken for the Lyapunov function:
Figure FDA0004034800440000091
in order to ensure the stability of the system, the order
Figure FDA0004034800440000092
Figure FDA0004034800440000093
Figure FDA0004034800440000094
The parameter self-adaptive law of the neural network is designed as follows:
Figure FDA0004034800440000095
Figure FDA0004034800440000096
Figure FDA0004034800440000097
Figure FDA0004034800440000098
Figure FDA0004034800440000099
Figure FDA0004034800440000101
Figure FDA0004034800440000102
in the formula (I), the compound is shown in the specification,
Figure FDA0004034800440000103
is that
Figure FDA0004034800440000104
The first derivative of (a) is,
Figure FDA0004034800440000105
is that
Figure FDA0004034800440000106
The first derivative of (a) is,
Figure FDA0004034800440000107
is that
Figure FDA0004034800440000108
The first derivative of (a) is,
Figure FDA0004034800440000109
is that
Figure FDA00040348004400001010
The first derivative of (a) is,
Figure FDA00040348004400001011
is that
Figure FDA00040348004400001012
The first derivative of (a) is,
Figure FDA00040348004400001013
is that
Figure FDA00040348004400001014
The first derivative of (a) is,
Figure FDA00040348004400001015
is that
Figure FDA00040348004400001016
The first derivative of (a);
Figure FDA00040348004400001017
for the estimation of the unknown parameter W,
Figure FDA00040348004400001018
is an estimate of the unknown parameter b,
Figure FDA00040348004400001019
is an estimate of the unknown parameter c,
Figure FDA00040348004400001020
is an estimate of the unknown parameter h and,
Figure FDA00040348004400001021
is an estimate of the unknown parameter v,
Figure FDA00040348004400001022
is an estimate of the unknown parameter r,
Figure FDA00040348004400001023
as a parameter W of unknown ro Estimated value of, η 1 ,η 2 ,η 3 ,η 4 ,η 5 ,η 6 ,η 7 Learning rates for each unknown parameter.
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