CN110362110B - Fixed self-adaptive neural network unmanned aerial vehicle track angle control method - Google Patents

Fixed self-adaptive neural network unmanned aerial vehicle track angle control method Download PDF

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CN110362110B
CN110362110B CN201910628565.7A CN201910628565A CN110362110B CN 110362110 B CN110362110 B CN 110362110B CN 201910628565 A CN201910628565 A CN 201910628565A CN 110362110 B CN110362110 B CN 110362110B
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倪骏康
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Abstract

The invention relates to a fixed self-adaptive neural network unmanned aerial vehicle track angle control method, which comprises the following steps: establishing a track angle dynamic mathematical model of an unmanned aerial vehicle longitudinal system, and establishing an actuator model with an unknown nonlinear dead zone; determining an ideal output value and an output limit; designing a fixed-time self-adaptive neural network controller, a self-adaptive parameter updating law and a fixed-time differentiator, so that the output of a system can track an upper reference output track in fixed time, and meanwhile, all state variables are guaranteed to be bounded; and carrying out stability analysis on the control system, and determining the parameters of the controller according to the stability analysis result. The method provided by the invention fully considers the limiting factors such as dead zones, system uncertainty, output limitation and the like existing in the actual system, is suitable for a more general nonlinear system such as a non-strict feedback system, and can be better applied to the actual system to ensure that the track angle of the unmanned aerial vehicle tracks an ideal track in a fixed time.

Description

Fixed self-adaptive neural network unmanned aerial vehicle track angle control method
Technical Field
The invention relates to the field of industrial control, in particular to a track angle control method of a fixed self-adaptive neural network unmanned aerial vehicle.
Background
Drones exhibit advantages over conventional aircraft in many respects and have been used to perform many complex tasks. The automatic flight control system can guarantee the performance of the unmanned aerial vehicle when the unmanned aerial vehicle executes a special task. The complexity and specificity of the task executed by the unmanned aerial vehicle puts high requirements on the control time and control precision of the unmanned aerial vehicle and the transient and steady-state performance of the system. Due to the complexity and the variability of the flight environment, the unmanned aerial vehicle system is an uncertain nonlinear system which has a non-strict feedback structure and is influenced by input dead zones and output limits, and the design of a controller is difficult.
Because the neural network has good unknown nonlinear function approximation capability, the neural network control is a good control method for an uncertain nonlinear system. In recent years, many research results have been achieved in the control of neural networks. However, these research efforts are only directed to non-linear systems with a strict form of feedback. A non-critical feedback system is a more general form of system and a critical feedback system may be considered to be a special form thereof. Because the non-linear function in the non-strict feedback system contains the whole state variable, the algebraic loop problem can occur when the existing neural network controller designed aiming at the strict feedback system is used for controlling the non-strict feedback system. Therefore, there is a need to extend existing neural network control methods to non-rigid feedback nonlinear systems.
The convergence rate is an important performance index of a control system, and the existing neural network control method can only realize asymptotic stability or stability in limited time. Asymptotic stabilization cannot ensure that the system is stable within a limited time and cannot be used in application occasions with strict requirements on convergence time. The finite stable convergence time depends on the initial value of the system, however, for many practical systems, the initial state is difficult to obtain. Furthermore, the convergence time of the finite time stability increases endlessly with the increase of the initial value, which limits the application of the finite time stability control to the actual system control whose initial value is very large. To overcome the above-mentioned deficiencies, researchers have proposed stability upon fixation. The main feature of the stability at fixed time is that the stability time boundary is a constant independent of the initial value, which helps the convergence time estimation and controller design to meet the convergence time requirement. However, no fixed-time adaptive neural network control method is reported in the literature at present.
For input dead zones, existing literature adopts neural networks and fuzzy logic to estimate and compensate for dead zone nonlinearity. However, due to the non-smooth nature of the dead zone function, more nodes, training times, and fuzzy rules need to be used to approximate the dead zone non-linearity, which increases the computational burden. An adaptive dead-zone inverse method is used to solve the dead-zone problem. However, the adaptive law of unknown dead-zone parameters contains the actuator input u, which can only be obtained after determining the dead-zone parameters to be estimated, which makes the method difficult to implement practically. Another approach to deal with dead zones is to model the dead zones as a combination of linear and interference terms, using adaptive or robust methods to estimate and compensate for the interference. For output limitation, the existing literature adopts a convex optimization method, but the method depends on an algorithm with large calculation amount. The systematic transformation method has proven to be very good for avoiding output limitations. Recently, a design method based on the barrier lyapunov function is used to deal with the output limitation problem. However, there is currently no literature reporting system fixed time control with dead band and output limitations.
Disclosure of Invention
Technical problem to be solved
In order to avoid the defects of the prior art, the invention provides a fixed time self-adaptive neural network unmanned aerial vehicle track angle control method, which is used for meeting the high requirements of an unmanned aerial vehicle system on tracking time, tracking precision and system transient and steady tracking performance, and considering the ubiquitous control dead zone and output limitation in an actual system, so that the unmanned aerial vehicle track angle can track an ideal track within fixed time.
Technical scheme
A fixed time self-adaptive neural network unmanned aerial vehicle track angle control method is characterized by comprising the following steps:
step 1: establishing a dynamic mathematical model of a track angle of a longitudinal system of the unmanned aerial vehicle:
Figure BDA0002127980450000031
wherein γ represents a track angle, α represents an attack angle, q B Representing pitch angle velocity, V t Denotes space velocity, F T Representing engine thrust, δ e Indicating elevator yaw angle, M and I y Denotes mass and inertia, L (α, q) B ) And
Figure BDA0002127980450000032
representing aerodynamic lift and pitching moment, and having the following expressions:
Figure BDA0002127980450000033
where S is the reference airfoil, ρ represents the air density,
Figure BDA0002127980450000034
denotes the mean chord length, C L And C m Representing lift and pitching moment coefficients, can be written as:
Figure BDA0002127980450000035
Figure BDA0002127980450000036
wherein the content of the first and second substances,
Figure BDA0002127980450000037
is the aerodynamic coefficient contributed by pitch angle velocity to lift and pitch moment;
Figure BDA0002127980450000038
aerodynamic coefficients contributing to lift and pitching moment by angle of attack;
Figure BDA0002127980450000039
is the aerodynamic coefficient contributed by the elevator yaw angle to the pitching moment;
Figure BDA00021279804500000310
the lift coefficient is zero attack angle and lift coefficient under pitch angle speed;
deflecting the elevator by an angle delta e As actuator output, u has the following expression:
Figure BDA00021279804500000311
where v denotes the actuator input to be designed, m r And m l Representing dead band input slope, b r And b l Representing dead zone right and left breakpoints; it is assumed here that there are normal numbers
Figure BDA00021279804500000312
So that
Figure BDA00021279804500000313
Figure BDA00021279804500000314
Let x be 1 ＝γ，x 2 ＝α，x 3 ＝q B Then the system (1) can be represented as:
Figure BDA0002127980450000041
wherein y is the system output, and y is the system output,
Figure BDA0002127980450000042
g 1 (x 1 )＝1，
Figure BDA0002127980450000043
Figure BDA0002127980450000044
f due to uncertain parameters in the actual system i (x) And
Figure BDA0002127980450000045
is an unknown function, where i ═ 1,2, 3; it is assumed here that
Figure BDA0002127980450000046
Is known if
Figure BDA0002127980450000047
Is positive, a constant can be found
Figure BDA0002127980450000048
So that
Figure BDA0002127980450000049
If it is not
Figure BDA00021279804500000410
Is negative, a constant can be found
Figure BDA00021279804500000411
So that
Figure BDA00021279804500000412
Assuming a non-linear function f i (x) Satisfying the Lipschitz condition, i.e., the presence of an arbitrary real number X 1 ,X 2 ∈R l So that | f i (X 2 )-f i (X 1 )|≤L i ||X 2 -X 1 I satisfies, wherein L i Is the Lipschitz constant;
step 2: determining the ideal output value y d (10+2sin (0.5 pi t)) ° with an output limit of y ≦ k d Assuming ideal output y d And its derivatives are bounded, i.e. a normal B can be found 0 ，B 1 Make | y d |≤B 0 ，
Figure BDA00021279804500000413
And step 3: designing a fixed-time adaptive neural network controller, an adaptive parameter updating law and a fixed-time differentiator to enable the system output to track an upper reference output track in a fixed time and ensure that all state variables are bounded, wherein the specific steps are as follows:
the design actual control inputs are:
Figure BDA0002127980450000051
in the formula
Figure BDA0002127980450000052
In the formula, chi 3 Is a normal number, an auxiliary variable
Figure BDA0002127980450000053
Is defined as:
Figure BDA0002127980450000054
in the formula of 3 ，θ 3 Is a normal number, r is the number of cryptic neurons, m, n, p, q satisfy m>n，p<Positive odd number of q, E 3 Has the following form:
Figure BDA0002127980450000055
adaptive parameters
Figure BDA0002127980450000056
The dynamics of (A) are:
Figure BDA0002127980450000057
Λ 3 is a normal number, ζ 22 The state of the differentiator is fixed as follows:
Figure BDA0002127980450000058
in the formula of i I μ ∈ (i-1), μ ∈ (1,1+ iota), and iota are sufficiently small positive numbers, and differentiator gains L, M>0，k 1 ，k 2 ，σ 1 ，σ 2 The selection is such that the following matrices are Hurwitz matrices:
Figure BDA0002127980450000059
error e 3 ＝x 3 -α 2 Virtual control of alpha 2 The expression of (a) is:
Figure BDA0002127980450000061
in the formula, x 2 Is a normal number, an auxiliary variable
Figure BDA0002127980450000062
Is defined as follows:
Figure BDA0002127980450000063
in the formula (I), the compound is shown in the specification,
Figure BDA0002127980450000064
k c ＝k d -B 0 ，λ 2 ,θ 2 is a normal number, E 2 Is a normal number, and satisfies:
Figure BDA0002127980450000065
where psi is a positive number, adaptive parameter
Figure BDA0002127980450000066
The dynamics of (A) are:
Figure BDA0002127980450000067
in the formula, Λ 2 Is a normal number, ζ 21 The state of the differentiator is fixed as follows:
Figure BDA0002127980450000068
ζ 11 a fixed time differentiator;
error e 2 ＝x 2 -α 1 Virtual control of alpha 1 Is defined as
Figure BDA0002127980450000069
In the formula, the auxiliary variable
Figure BDA00021279804500000610
Is defined as follows:
Figure BDA00021279804500000611
λ 1 and theta 1 Is a normal number, B 1 And E 1 Is defined as
Figure BDA0002127980450000071
Figure BDA0002127980450000072
Wherein psi is a normal number;
adaptive parameters
Figure BDA0002127980450000073
The dynamics of (A) are:
Figure BDA0002127980450000074
in the formula, Λ 1 Is a normal number;
and 4, step 4: and (3) controlling the unmanned aerial vehicle by adopting the control parameters determined in the step (3) so that the track angle can track the upper reference track angle within a fixed time.
Advantageous effects
Compared with the prior art, the invention provides a fixed self-adaptive neural network unmanned aerial vehicle track angle control method, which is innovatively realized in the following four aspects:
(a) the invention extends fixed-time control to non-rigid feedback nonlinear systems. To our knowledge, no fixed-time stable control method for non-rigid feedback nonlinear systems has been reported in the literature.
(b) The invention provides a simple and effective method for overcoming the algebraic ring problem.
(c) The invention combines the reverse-deducing design, the neural network control and the fixed time control, solves the problem of complexity explosion, reduces the number of self-adaptive parameters and realizes the convergence of the tracking error to the small neighborhood of the origin in fixed time.
(d) Dead zones and output limits are considered in the control design during fixing, so that the designed control scheme has more universal applicability to actual engineering systems.
Compared with the prior art, the invention has the following beneficial effects:
(a) the fixed time self-adaptive neural network control method provided by the invention fully considers the limiting factors such as dead zones, system uncertainty, output limitation and the like in an actual system, is suitable for a more general nonlinear system such as a non-strict feedback system, and can be better applied to the actual system.
(b) The control scheme reduces the calculation amount and is easy to implement. The derivative of the virtual control is estimated by using a fixed-time differentiator, so that the problem of 'computation complexity explosion' of a reverse-thrust design is avoided; compared with the traditional neural network control method, the number of the self-adaptive parameters needing to be updated is greatly reduced, the self-adaptive updating law is simpler in design, and the calculated amount of a control algorithm is reduced; in actual controller design, the designer does not need to compromise on control accuracy, computational burden and real-time performance, and the proposed control scheme reduces the requirements on the computational performance of the processor, thus reducing implementation complexity and difficulty.
(c) The proposed control scheme can ensure that the unmanned aerial vehicle tracks along the ideal track within a fixed time, and is suitable for the task occasion with strict requirement on convergence time.
Drawings
FIG. 1 is a control flow chart of a fixed-time adaptive neural network control method provided by the invention
FIG. 2 is a time response plot of track angle and its reference trajectory in an embodiment of the present invention
FIG. 3 is a time response plot of angle of attack in an embodiment of the present invention
FIG. 4 is a time response plot of pitch rate in an embodiment of the present invention
FIG. 5 is a dead band control input time plot for an embodiment of the present invention
Detailed Description
The invention will now be further described with reference to the following examples and drawings:
drones exhibit advantages over conventional aircraft in many respects and have been used to perform many complex tasks. The automatic flight control system can guarantee the performance of the unmanned aerial vehicle when the unmanned aerial vehicle executes a special task. The complexity and specificity of the task executed by the unmanned aerial vehicle puts high requirements on the control time and control precision of the unmanned aerial vehicle and the transient and steady-state performance of the system. Due to the complexity and the variability of the flight environment, the unmanned aerial vehicle system is an uncertain nonlinear system which has a non-strict feedback structure and is influenced by input dead zones and output limits, and the design of a controller is difficult.
Referring to fig. 1 to 5, the present invention provides a fixed time adaptive neural network control method, including the following steps:
the mathematical model of the flight path angle dynamic of the longitudinal system of the unmanned aerial vehicle in the step (1) is as follows:
Figure BDA0002127980450000091
wherein γ represents a track angle, q B Representing pitch angle velocity, V t Denotes the space velocity, F T Representing engine thrust, δ e Indicating elevator yaw angle, M and I y Denotes mass and inertia, L (α, q) B ) And
Figure BDA0002127980450000092
representing aerodynamic lift and pitching moment, and having the following expressions:
Figure BDA0002127980450000093
where S is the reference airfoil, ρ represents the air density,
Figure BDA0002127980450000094
denotes the mean chord length, C L And C m Representing lift and pitch moment coefficients, can be written as:
Figure BDA0002127980450000095
Figure BDA0002127980450000096
wherein the content of the first and second substances,
Figure BDA0002127980450000097
is the aerodynamic coefficient contributed by pitch angle velocity to lift and pitch moment;
Figure BDA0002127980450000098
Figure BDA0002127980450000099
aerodynamic coefficients that are contributions to lift and pitching moment from angle of attack;
Figure BDA00021279804500000910
is the aerodynamic coefficient contributed by the elevator yaw angle to the pitching moment;
Figure BDA00021279804500000911
the lift coefficient at zero angle of attack and pitch angle rates.
Elevator drift angle u ═ δ e Often referred to as actuator output, has the following expression:
Figure BDA00021279804500000912
where v denotes the actuator input to be designed, m r And m l Representing dead zone input slope, b r And b l Representing dead zone right and left breakpoints.
The reference output in the step (2) is y d (10+2sin (0.5 pi t)) °, and the output is limited to | y | ≦ k d 。
And (3) designing a fixed self-adaptive neural network control law to realize a control target. First, the system (1) is written as a standard form of control system. Let x be 1 ＝γ，x 2 ＝α，x 3 ＝q B Then the system (1) can be represented as:
Figure BDA0002127980450000101
wherein y is the output of the system, and y is the output of the system,
Figure BDA0002127980450000102
g 1 (x 1 )＝1，
Figure BDA0002127980450000103
Figure BDA0002127980450000104
f due to uncertain parameters in the real system i (x) And
Figure BDA0002127980450000105
(i ═ 1,2,3) is an unknown function.
Next, a fixed time adaptive neural network control law is designed for the control system (6). Prior to controller design, the following assumptions were made for the control parameters, control gain and reference output signal:
assume that 1: the parameters in the dead zone (5) are unknown, but its breakpoint b l ，b r And slope m r ，m l Is bounded, i.e. there are normal numbers
Figure BDA0002127980450000106
So that
Figure BDA0002127980450000107
Assume 2:
Figure BDA0002127980450000108
the symbol of (b) is known. If it is not
Figure BDA0002127980450000109
Is positive, a constant can be found
Figure BDA00021279804500001010
So that
Figure BDA00021279804500001011
If it is not
Figure BDA00021279804500001012
Is negative, a constant can be found
Figure BDA00021279804500001013
So that
Figure BDA00021279804500001014
Assume that 3: reference output y d And its derivatives are bounded, i.e. a normal B can be found 0 ,B 1 Make | y d |≤B 0 ，
Figure BDA00021279804500001015
Assume 4: non-linear function f i (x) Satisfying the Lipschitz condition, i.e., the presence of an arbitrary real number X 1 ,X 2 ∈R l So that | f i (X 2 )-f i (X 1 )|≤L i ||X 2 -X 1 I satisfies, wherein L i Is a Lipschitz constant.
The first step is as follows: defining the tracking error as e 1 ＝x 1 -y d The time derivative thereof can be expressed as:
Figure BDA0002127980450000111
radial basis function neural networks are used to approximate unknown nonlinear functions:
f 1 (x)＝W 1 *T S 1 (x)+ε 1 (8)
in the formula W 1 * ,S 1 (x) And ε 1 Respectively representing the optimal weight of the radial basis function neural network, the Gaussian basis function and the approximation error. Presence of normal number
Figure BDA0002127980450000112
So that
Figure BDA0002127980450000113
Defining auxiliary variables:
Figure BDA0002127980450000114
in the formula
Figure BDA0002127980450000115
λ 1 And theta 1 Is a normal number, r is the number of cryptic neurons, m, n, p, q satisfy m>n,p<Odd number of q, B 1 And E 1 Is defined as follows:
Figure BDA0002127980450000116
Figure BDA0002127980450000117
where ψ is a normal number.
Figure BDA0002127980450000118
For adaptive parameters, the update law is:
Figure BDA0002127980450000119
in the formula Λ 1 Is a positive number.
The virtual control laws can be designed as follows:
Figure BDA0002127980450000121
because the system output and the reference output are limited to | y | ≦ k d And | y d |≤B 0 Then there is | e 1 |≤k c In the formula, k c +B 0 ＝k d 。
The second step is that: approximating an unknown non-linear function f using a radial basis function neural network 2 (x),e 2 The time derivative of (a) is:
Figure BDA0002127980450000122
in the formula
Figure BDA00021279804500001211
S 2 (x) And epsilon 2 Respectively representing the optimal weight of the neural network of the radial basis function, the Gaussian basis function and the approximation error, and having normal constants
Figure BDA0002127980450000123
So that
Figure BDA0002127980450000124
Fixed time differentiator for estimating virtual control alpha 1 Derivative of (a):
Figure BDA0002127980450000125
zeta in the formula 11 ,ζ 21 Is a fixed time differentiator state variable, L, M>0, select the proper k 1 ,k 2 ,σ 1 ,σ 2 So that the matrix defined by (16) and (17) is a Hurwitz matrix, mu i I μ ∈ (i-1) and μ ∈ (1,1+ iota), iota being a sufficiently small positive number, sig (·) α ＝|·| α sign(·)。
Figure BDA0002127980450000126
Figure BDA0002127980450000127
The design auxiliary variables are:
Figure BDA0002127980450000128
in the formula of 2 ,θ 2 ,χ 2 Is a normal number, E 2 Has the following form:
Figure BDA0002127980450000129
Figure BDA00021279804500001210
the adaptive parameters are adaptive parameters, and the adaptive law is as follows:
Figure BDA0002127980450000131
the virtual control law is designed as follows:
Figure BDA0002127980450000132
the third step: approximating an unknown non-linear function f using a radial basis function neural network 3 (x) Then e is 3 The derivative with respect to time can be expressed as:
Figure BDA0002127980450000133
in the formula, W 3 * ,S 3 (x) And ε 3 The normal number can be found for the optimal weight, Gaussian radial basis function and approximation error of the radial basis function neural network
Figure BDA0002127980450000134
So that
Figure BDA0002127980450000135
Fixed-time differentiators for obtaining the virtual control alpha 2 The time derivative of (c):
Figure BDA0002127980450000136
the following auxiliary variables are defined:
Figure BDA0002127980450000137
in the formula of lambda 3 ,θ 3 ,χ 3 Is a normal number, E 3 Has the following expression:
Figure BDA0002127980450000138
Figure BDA0002127980450000139
for adaptive parameters, the update law can be described as:
Figure BDA00021279804500001310
the auxiliary control inputs may be designed as:
Figure BDA0002127980450000141
the actuator inputs are:
Figure BDA0002127980450000142
further, stability analysis is carried out on the control system, and stability of the control system during fixing is proved. First, the following arguments are introduced:
lemma 1 for the following system:
Figure BDA0002127980450000143
wherein α >0, β >0, m, n, p, q are normal numbers satisfying m > n, q > p. The system (29) will arrive at the origin within a finite time bounded by:
Figure BDA0002127980450000144
2, leading: for z 1 E.g. R and x 1 >0, then there are:
Figure BDA0002127980450000145
and 3, introduction: for any c >0, a ≧ 0, b >0, then:
Figure BDA0002127980450000146
and (4) introduction: for any c >1, a is more than or equal to 0, b is less than or equal to a, then:
(a-b) c ≥b c -a c (33)
and (5) introduction: for any 0<c is less than or equal to 1 and x i If the ratio is more than or equal to 0, the following components are adopted:
Figure BDA0002127980450000147
and (4) introduction 6: for any c>1 and x i If the ratio is more than or equal to 0, the following components are adopted:
Figure BDA0002127980450000151
and (4) introduction 7: for the differentiator (15), the alpha is virtually controlled 1 The time derivative of (a) can be obtained within a limited time bounded by the following equation:
Figure BDA0002127980450000152
in the formula
Figure BDA0002127980450000153
Symmetric positive definite matrix P 1 And Q 1 Satisfies the following conditions:
Figure BDA0002127980450000154
matrix A 1 As shown in formula (16).
The symmetric positive definite matrices P and Q satisfy:
PA+A T P＝-Q (38)
the matrix a is represented by the formula (17).
Next, in a first step, consider the following lyapunov function:
Figure BDA0002127980450000155
in the formula
Figure BDA0002127980450000156
V 1 The time derivative of (a) is:
Figure BDA0002127980450000157
in the formula e 2 ＝x 2 -α 1 . Order to
Figure BDA0002127980450000158
Then there are:
Figure BDA0002127980450000161
Figure BDA0002127980450000162
in the formula
Figure BDA0002127980450000163
Using theorem 2, there are:
Figure BDA0002127980450000164
then, (40) becomes:
Figure BDA0002127980450000165
using the lemma 3-4, one can obtain:
Figure BDA0002127980450000166
similarly, it can be derived:
Figure BDA0002127980450000167
substituting (45) and (46) into (44) yields:
Figure BDA0002127980450000168
in a second step, the following Lyapunov function is constructed:
Figure BDA0002127980450000169
in the formula
Figure BDA0002127980450000171
Ask for V 2 Derivative of (a):
Figure BDA0002127980450000172
order to
Figure BDA0002127980450000173
Then there are:
Figure BDA0002127980450000174
Figure BDA0002127980450000175
in the formula
Figure BDA0002127980450000176
The use lemma 2 has:
Figure BDA0002127980450000177
substituting (50) - (52) into (49) then has:
Figure BDA0002127980450000178
when T is more than or equal to T d1 The differentiator may give an accurate estimate of the virtual control derivative, i.e.,
Figure BDA0002127980450000179
then it is possible to obtain:
Figure BDA0002127980450000181
applying a method similar to the first step, one can obtain:
Figure BDA0002127980450000182
Figure BDA0002127980450000183
from lemma 2 we can get:
Figure BDA0002127980450000184
substituting (55) - (57) into (54) can result in:
Figure BDA0002127980450000185
third, the control input error may be expressed as:
Figure BDA0002127980450000186
when in use
Figure BDA0002127980450000187
Since the sign of u' is represented by e 3 Determined when e 3 >0, then there are u'<0, and when e 3 <0, then there are u'>0. Therefore, we have
Figure BDA0002127980450000188
Similarly, when
Figure BDA0002127980450000189
We also have
Figure BDA00021279804500001810
Consider the following Lyapunov function:
Figure BDA0002127980450000191
in the formula
Figure BDA0002127980450000192
V 3 The time derivative of (a) is:
Figure BDA0002127980450000193
order to
Figure BDA0002127980450000194
Then there are:
Figure BDA0002127980450000195
Figure BDA0002127980450000196
in the formula
Figure BDA0002127980450000197
Using lemma 2, there are:
Figure BDA0002127980450000198
substituting (62) - (64) into (61) yields:
Figure BDA0002127980450000199
similar to the first step, we have:
Figure BDA0002127980450000201
Figure BDA0002127980450000202
using lemma 2, we have:
Figure BDA0002127980450000203
when T is more than or equal to T d(l-1) The differentiator may estimate the derivative of the virtual control, i.e.
Figure BDA0002127980450000204
Substituting (66) - (68) into (65), (65) becomes:
Figure BDA0002127980450000205
using the theorem 5-6, there are:
Figure BDA0002127980450000211
in the formula
Figure BDA0002127980450000212
Figure BDA0002127980450000213
Figure BDA0002127980450000214
From (70), we can calculate the final boundary of the closed-loop system:
Figure BDA0002127980450000215
outside the boundary
Figure BDA0002127980450000216
However, it is difficult to give an analytical solution of equation (71). The final boundary of the closed loop system can be estimated as:
Figure BDA0002127980450000217
V 3 means e i And
Figure BDA0002127980450000218
is bounded. Since theta i Is a constant, we have
Figure BDA0002127980450000219
Is bounded. Due to e 1 And y d We have system output y bounded. Considering e 1 ,
Figure BDA00021279804500002110
Is that the material is bounded by the surface,
Figure BDA00021279804500002111
χ 1 ,λ 1 ,θ 1 r is a constant, we have
Figure BDA00021279804500002112
And alpha 1 Is bounded. Due to e 2 And alpha 1 Is bounded, we have a system state x 2 Is bounded. Due to the fact that
Figure BDA00021279804500002113
And
Figure BDA00021279804500002114
is that the continuous function has a bounded domain of definition, we have
Figure BDA00021279804500002115
And ζ 12 Is bounded. Zeta is shown in (15) 11 Is bounded. Due to zeta 12 ,e 1 ,e 2 ，
Figure BDA00021279804500002116
Is bounded, and
Figure BDA00021279804500002117
χ 2 ，λ 2 ，θ 2 and r is constant, then
Figure BDA0002127980450000221
And alpha 2 Is bounded. Due to e 3 And alpha 2 Is bounded, then the system state x 3 Is bounded. Similarly, x can be demonstrated i ,α i ,ζ 1i ,ζ 2i And v is bounded. Thus, all closed loop signals are bounded。
Assuming that the system output y crosses the output limit at time T ═ T, we have a Lyapunov function V 3 Will become infinite, this is in accordance with the Lyapunov function V 3 Bounded runs counter to each other, so the system output will not exceed the output limit | y ≦ k d 。
We can find a constant
Figure BDA0002127980450000222
So that
Figure BDA0002127980450000223
Then (70) becomes:
Figure BDA0002127980450000224
order to
Figure BDA0002127980450000225
Then (73) becomes:
Figure BDA0002127980450000226
using introduction 1, we have V 3 Will be at the same time
Figure BDA0002127980450000227
Convergence to the region within a limited time for an upper bound
Figure BDA0002127980450000228
If V 3 Reach area
Figure BDA0002127980450000229
Then there is
Figure BDA00021279804500002210
Therefore, the tracking error will be
Figure BDA00021279804500002211
Reach a compact set for a limited time of the upper bound
Figure BDA00021279804500002212
(4) And (4) dynamically controlling the unmanned aerial vehicle track angle by adopting the control law determined in the step (3), so that the unmanned aerial vehicle track angle can track an ideal motion track, and the system output is ensured not to violate the limit.
The embodiment is as follows: unmanned aerial vehicle track angle developments
The effectiveness of the fixed time self-adaptive neural network control method in tracking the unmanned aerial vehicle track angle on an ideal track is illustrated by taking the unmanned aerial vehicle track angle dynamic as an example. The drone flight path angle dynamics may be expressed as:
Figure BDA0002127980450000231
wherein
Figure BDA0002127980450000232
And is provided with
Figure BDA0002127980450000233
Figure BDA0002127980450000234
The system parameter is selected as V t ＝100m/s，F T ＝8000N，M＝9295.44kg，S＝27.87m 2 ，I y ＝75673.6kg·m 2 ，
Figure BDA0002127980450000235
ρ＝1.7g/L，
Figure BDA0002127980450000236
Figure BDA0002127980450000237
The parameter of the dead zone is selected as m r ＝1，b r ＝0.6°，m l ＝1.05，b l ＝-0.8°。
The self-adaptive neural network control method for the unmanned aerial vehicle during the dynamic fixation of the track angle comprises the following steps:
(1) determining a control target: the reference output signal is chosen to be y d The output is limited to | y | ≦ 22 ° (10+2sin (0.5 π t)) °. The control target is determined such that the system output can track the reference output of the upper system in a fixed time while the system output does not exceed the limit.
(2) To achieve the control objective, the design control inputs are:
Figure BDA0002127980450000238
wherein u' has the expression:
Figure BDA0002127980450000241
in the formula
Figure BDA0002127980450000242
Has the following expression form:
Figure BDA0002127980450000243
(3) based on the Lyapunov function stability analysis, the controller and differentiator parameters are selected as lambda i ＝θ i ＝10，r＝5，χ i ＝0.1，p＝5，q＝9,m＝9，n＝5，ψ＝0.05，Λ i ＝5，k 1 ＝5，k 2 ＝10，σ 1 ＝5，σ 2 ＝10，L＝10，μ 1 ＝1.2，μ 2 ＝1.4，
Figure BDA0002127980450000244
It can be demonstrated that this set of control parameters satisfies lyapunov stability.
(4) And (4) dynamically controlling the unmanned aerial vehicle track angle by adopting the control parameters determined in the step (3), so that the unmanned aerial vehicle track angle can track an ideal motion track, and the system output meets the output limit that y is less than or equal to 22 degrees.
A flow chart of a method of fixed time adaptive neural network control is provided and shown in fig. 1. The time response of the flight path angle and its reference trajectory is shown in fig. 2. The time response of the angle of attack is shown in figure 3. The time response of the pitch angle rate is shown in fig. 4. The dead band control input time profile is shown in fig. 5. It can be seen from these figures that the system output tracks the upper reference trajectory in a fixed time, the system output does not exceed the limit, and the other state variables and dead band control inputs are bounded.

Claims (1)

1. A fixed time self-adaptive neural network unmanned aerial vehicle track angle control method is characterized by comprising the following steps:
step 1: establishing a dynamic mathematical model of a track angle of a longitudinal system of the unmanned aerial vehicle:
Figure FDA0002127980440000011
wherein γ represents a track angle, α represents an attack angle, q B Representing pitch angular velocity, V t Denotes the space velocity, F T Representing engine thrust, δ e Indicating elevator yaw angle, M and I y Denotes mass and inertia, L (α, q) B ) And
Figure FDA0002127980440000012
representing aerodynamic lift and pitching moment, and having the following expressions:
Figure FDA0002127980440000013
where S is a reference airfoil and ρ represents the air density，
Figure FDA0002127980440000014
Denotes the mean chord length, C L And C m Representing lift and pitch moment coefficients, can be written as:
Figure FDA0002127980440000015
Figure FDA0002127980440000016
wherein the content of the first and second substances,
Figure FDA0002127980440000017
is the aerodynamic coefficient contributed by pitch angle velocity to lift and pitch moment;
Figure FDA0002127980440000018
aerodynamic coefficients contributing to lift and pitching moment by angle of attack;
Figure FDA0002127980440000019
is the aerodynamic coefficient contributed by the elevator yaw angle to the pitching moment;
Figure FDA00021279804400000110
the lift coefficient is zero attack angle and lift coefficient under pitch angle speed;
deflecting the elevator by an angle delta e As an actuator output, u has the following expression:
Figure FDA00021279804400000111
where v denotes the actuator input to be designed, m r And m l Representing dead zone input slope, b r And b l Representing dead zone right and left breakagesPoint; it is assumed here that there are normal numbers
Figure FDA00021279804400000115
Figure FDA00021279804400000114
So that
Figure FDA00021279804400000116
Figure FDA00021279804400000113
Figure FDA00021279804400000214
Let x 1 ＝γ，x 2 ＝α，x 3 ＝q B Then the system (1) can be represented as:
Figure FDA0002127980440000021
wherein y is the system output, and y is the system output,
Figure FDA0002127980440000022
g 1 (x 1 )＝1，
Figure FDA0002127980440000023
Figure FDA0002127980440000024
f due to uncertain parameters in the actual system i (x) And
Figure FDA0002127980440000025
is an unknown function, where i ═ 1,2, 3; it is assumed here that
Figure FDA0002127980440000026
Is known if
Figure FDA0002127980440000027
Is positive, a constant can be found
Figure FDA00021279804400000215
Figure FDA0002127980440000028
So that
Figure FDA0002127980440000029
If it is not
Figure FDA00021279804400000210
Is negative, a constant can be found
Figure FDA00021279804400000211
So that
Figure FDA00021279804400000212
Assuming a non-linear function f i (x) Satisfying the Lipschitz condition, i.e., the presence of an arbitrary real number X 1 ,X 2 ∈R l So that | f i (X 2 )-f i (X 1 )|≤L i ||X 2 -X 1 I satisfies, wherein L i Is Lipschitz constant;
step 2: determining the ideal output value y d (10+2sin (0.5 pi t)) ° with an output limit of y ≦ k d Assuming ideal output y d And its derivatives are bounded, i.e. a normal B can be found 0 ，B 1 Make y d |≤B 0 ，
Figure FDA00021279804400000213
And step 3: designing a fixed-time adaptive neural network controller, an adaptive parameter updating law and a fixed-time differentiator to enable the system output to track an upper reference output track in a fixed time and ensure that all state variables are bounded, wherein the specific steps are as follows:
the design actual control inputs are:
Figure FDA0002127980440000031
in the formula
Figure FDA0002127980440000032
In the formula, chi 3 Is a normal number, an auxiliary variable
Figure FDA0002127980440000033
Is defined as:
Figure FDA0002127980440000034
in the formula of 3 ，θ 3 Is a normal number, r is the number of cryptic neurons, m, n, p, q are numbers satisfying m>n，p<Positive odd number of q, E 3 Has the following form:
Figure FDA0002127980440000035
adaptive parameters
Figure FDA0002127980440000036
The dynamics of (A) are:
Figure FDA0002127980440000037
Λ 3 is a normal number, ζ 22 The state of the differentiator is fixed as follows:
Figure FDA0002127980440000038
in the formula mu i I μ ∈ (1,1+ ι), and ι is a sufficiently small positive number, and differentiator gains L, M>0，k 1 ，k 2 ，σ 1 ，σ 2 The selection is such that the following matrices are Hurwitz matrices:
Figure FDA0002127980440000039
error e 3 ＝x 3 -α 2 Virtual control of alpha 2 The expression of (a) is:
Figure FDA0002127980440000041
in the formula, x 2 Is a normal number, an auxiliary variable
Figure FDA0002127980440000042
Is defined as:
Figure FDA0002127980440000043
in the formula (I), the compound is shown in the specification,
Figure FDA0002127980440000044
k c ＝k d -B 0 ，λ 2 ,θ 2 is a normal number, E 2 Is a normal number, and satisfies:
Figure FDA0002127980440000045
where psi is positive number, adaptive parameter
Figure FDA0002127980440000046
The dynamics of (A) are:
Figure FDA0002127980440000047
in the formula, Λ 2 Is a normal number, ζ 21 The state of the differentiator is fixed as follows:
Figure FDA0002127980440000048
ζ 11 a fixed time differentiator;
error e 2 ＝x 2 -α 1 Virtual control of alpha 1 Is defined as
Figure FDA0002127980440000049
In the formula, the auxiliary variable
Figure FDA00021279804400000410
Is defined as:
Figure FDA00021279804400000411
λ 1 and theta 1 Being a normal number, B 1 And E 1 Is defined as
Figure FDA0002127980440000051
Figure FDA0002127980440000052
Wherein psi is a normal number;
adaptive parameters
Figure FDA0002127980440000053
The dynamics of (A) are:
Figure FDA0002127980440000054
in the formula, Λ 1 Is a normal number;
and 4, step 4: and (3) controlling the unmanned aerial vehicle by adopting the control parameters determined in the step (3) so that the track angle can track the upper reference track angle within a fixed time.
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