CN111221250A - Nonlinear system with parameter uncertainty and multiple external disturbances and design method thereof - Google Patents

Nonlinear system with parameter uncertainty and multiple external disturbances and design method thereof Download PDF

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CN111221250A
CN111221250A CN202010038314.6A CN202010038314A CN111221250A CN 111221250 A CN111221250 A CN 111221250A CN 202010038314 A CN202010038314 A CN 202010038314A CN 111221250 A CN111221250 A CN 111221250A
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粟世玮
张思洋
尤熠然
熊炜
曹文康
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China Three Gorges University CTGU
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Abstract

A nonlinear system with parameter uncertainty and a plurality of external disturbances and a design method thereof are disclosed, wherein the nonlinear system introduces a virtual control function in control design, estimates unknown boundaries of the external disturbances by using an improved adaptive law, synthesizes a continuous adaptive robust state feedback controller u (t) for the uncertain nonlinear system by using updated values of the unknown boundaries, and ensures that the nonlinear system has progressive stability by using the Lyapunov stabilization theory, thereby ensuring that the complex nonlinear system still has stability when uncertain parameters and external disturbances exist.

Description

Nonlinear system with parameter uncertainty and multiple external disturbances and design method thereof
Technical Field
The invention belongs to the technical field of automatic control, and particularly relates to a nonlinear system with parameter uncertainty and multiple external disturbances and a design method thereof.
Background
In practical engineering applications, most controlled objects of many power systems, such as mechanical engineering, industrial control, computer networks, biological engineering, etc., have nonlinear characteristics. The system has uncertainty and random interference of different degrees, unstable signals can randomly appear, the system is difficult to process, the quality of the system is often reduced, and the stability of the system is even affected, so that the system has great significance for researching a nonlinear system. Especially since recent years, due to the continuous development of scientific technology, people continuously improve the accuracy of control, and some emerging fields are not suitable for using the traditional linear system control theory, so that many scholars have generated interest in nonlinear system research, received much attention from many people, and obtained great progress, most adaptive robust control laws can generally ensure the final limit of the adaptive closed-loop nonlinear system under the condition of existence of the uncertainty and external interference, but cannot ensure that the solution of the obtained adaptive closed-loop nonlinear system gradually converges to zero. In order to obtain accurate control results, it is necessary to propose some type of adaptive robust control scheme for uncertain nonlinear systems with unknown parameters and unknown bounds for external disturbances.
This system is not only designed with one uncertain system parameter but also with the problem of adaptive robust stability of the non-linear system with multiple external disturbances. Assuming that the upper bounds of these uncertainties are unknown, only the external disturbances are assumed to be any continuous bounded function and their time derivatives are not required to be bounded. For this type of uncertain non-linear system, an adaptation law adapted to this is used to estimate this unknown boundary. Then, by using the updated values of these unknown boundaries, a method is then proposed in which a class of continuous adaptive robust state feedback controllers can synthesize such an uncertain non-linear system. The end result of the design is the proposed adaptive robust state feedback control scheme that can guarantee a uniform asymptotic convergence of the state to zero in the presence of uncertainty and multiple external disturbances.
Disclosure of Invention
The invention provides a nonlinear system with parameter uncertainty and a plurality of external disturbances, and provides an adaptive control method based on backstepping, a virtual control function is introduced in control design, unknown boundaries of the external disturbances are estimated by using an improved adaptive law, a continuous adaptive robust state feedback controller u (t) is synthesized for the uncertain nonlinear system by using updated values of the unknown boundaries, and simultaneously the state feedback controller u (t) is ensured to have progressive stability by using the Lyapunov stabilization theory, so that the complex nonlinear system can still have stability when the complex nonlinear system has uncertainty parameters and external disturbances.
The technical scheme adopted by the invention is as follows:
a nonlinear system with parametric uncertainty and multiple external perturbations, represented by a parametric rigorous feedback system described by the following nonlinear differential equation:
Figure BDA0002366820180000021
wherein the content of the first and second substances,
Figure BDA0002366820180000022
for state variables, u e R is the system input, function f (-) Ri→RmiI-1, 2, …, n, is a known smooth vector field;
Figure BDA0002366820180000023
are constant parametric vectors, which are compact sets
Figure BDA00023668201800000210
And which defines all uncertainty ranges for a given system; unknown function deltai(t), i ═ 1,2, …, n denotes external disturbances and is assumed to be continuous and bounded, and the system described by the equation (1) state space equation is called a strict feedback nonlinear system. Strict feedback means that each xnNon-linear functions in the subsystem only with variable x1,x2,…,xnOf interest, only the feedback of these n state variables.
The state feedback controller u (t) is represented by a non-linear function:
Figure BDA0002366820180000024
u (t) is a state feedback controller, which can ensure the stability of the nonlinear power system (1) in the presence of system uncertainty and external disturbance, and how to synthesize the state feedback controller u (t) is described below.
Assume 1.1: non-linear function f (-) Ri→RmiI-1, 2, …, n is continuous and locally uniformly bounded, and for convenience is introduced by definition ①:
ρi(x1,…，xi):＝||fi(x1,…,xi)||
Figure BDA0002366820180000025
Figure BDA0002366820180000026
Figure BDA0002366820180000027
where ρ isi(x1,…，xi) Denotes fi(x1,…,xi) The boundary of (a) is determined,
Figure BDA0002366820180000028
are all unknown normal numbers in the control law,
Figure BDA0002366820180000029
to represent
Figure BDA0002366820180000031
The boundary of (a) is determined,
Figure BDA0002366820180000032
for external disturbance deltai(t) the maximum value of the absolute value,
Figure BDA0002366820180000033
is composed of
Figure BDA0002366820180000034
Is measured.
A design method of a nonlinear system with parameter uncertainty and multiple external disturbances is characterized in that an iterative design algorithm is used for designing a virtual controller and an adaptive law for each subsystem in the nonlinear system, and a Lyapunov function is constructed for each subsystem in the nonlinear system based on the Lyapunov stability theory to ensure the convergence of the subsystems.
A nonlinear system control method with parameter uncertainty and multiple external disturbances is used for ensuring the stability of the whole closed-loop system by designing an actual control law through reverse iteration.
A method of designing a non-linear system having a parameter uncertainty and a plurality of external perturbations, comprising the steps of:
step 1-first introduce the control function αi(·), i ═ 1,2, …, n-1, transformations of the form are performed on the state variables:
Figure BDA0002366820180000035
zias state variables, function αi(. 1,2, …, n-1 is the introduced virtual controller, and before designing an adaptive robust controller, a definition ② is introduced that for any i e {2,3,4, …, n-1},
Figure BDA0002366820180000036
Figure BDA0002366820180000037
wherein, mui-1(t)，ηi-1(t) for the convenience of controller design a newly introduced symbol whose value is defined to the right of the equation, where
Figure BDA0002366820180000038
For unknown parameters
Figure BDA0002366820180000039
Is determined by the estimated value of (c),
Figure BDA00023668201800000310
is an estimated value
Figure BDA00023668201800000311
Derivative of αiFor the introduced virtual control law, initial conditions
Figure BDA00023668201800000312
Step 2: the first equation of equation (3) is differentiated and equation (1) is substituted to obtain:
Figure BDA00023668201800000313
the virtual control function α is proposed according to equation (4)i(·)
Figure BDA00023668201800000314
Wherein k is any normal number, v11(. and v)12(. cndot.) is given by:
Figure BDA00023668201800000315
Figure BDA0002366820180000041
for any i e {2,3,4, …, n-1}, σi(t)∈R+Satisfy any positive uniform continuous bounded function, define
Figure BDA0002366820180000042
Wherein
Figure BDA0002366820180000043
Is any normal number;
function in the formula (6) or (7)
Figure BDA0002366820180000044
And
Figure BDA0002366820180000045
are respectively unknown parameters
Figure BDA0002366820180000046
And
Figure BDA0002366820180000047
are updated by the adaptation law equations (8), (9):
Figure BDA0002366820180000048
Figure BDA0002366820180000049
Figure BDA00023668201800000410
is composed of
Figure BDA00023668201800000411
And
Figure BDA00023668201800000412
error therebetween, is recorded as
Figure BDA00023668201800000413
In the same way
Figure BDA00023668201800000414
Is composed of
Figure BDA00023668201800000415
And
Figure BDA00023668201800000416
error therebetween, is recorded as
Figure BDA00023668201800000417
γ11,γ12Rewriting the adaptive law equations (8) and (9) as system parameters are the following error systems:
Figure BDA00023668201800000418
Figure BDA00023668201800000419
according to the uncertain nonlinear system described by the formula (4) and the formula (5), a Lyapunov function is introduced:
Figure BDA00023668201800000420
V1(t) is the introduced Lyapunov function, and V is obtained by following the trajectories of the formula (4) and the formula (5)1(t) derivative of t ≧ t for any t ≧ t0，
Figure BDA00023668201800000421
From the above derivation, any t ≧ t can be obtained0When the temperature of the water is higher than the set temperature,
Figure BDA00023668201800000422
according to the inequality
Figure BDA00023668201800000423
Being positive, it can be obtained from equation (14):
Figure BDA0002366820180000051
wherein the content of the first and second substances,
Figure BDA0002366820180000052
and step 3: first, the second equation of equation (3) is differentiated, and equation (1) is substituted to obtain:
Figure BDA0002366820180000053
from the uncertain nonlinear system described by equation (3), the following virtual control function α is proposed2(·)
Figure BDA0002366820180000054
Wherein v is21(·),v22(·),p11(·),p12(. cndot.) is a relationship function relating the virtual control law to the unknown parameter, the relationship being given by:
Figure BDA0002366820180000055
function(s)
Figure BDA0002366820180000056
Are respectively unknown parameters
Figure BDA0002366820180000057
Is updated by the following adaptation law:
Figure BDA0002366820180000058
wherein, γ21，γ22，m1Is a system parameter, being any normal number.
Figure BDA0002366820180000059
Are respectively as
Figure BDA00023668201800000510
And
Figure BDA00023668201800000511
Figure BDA00023668201800000512
and
Figure BDA00023668201800000513
Figure BDA00023668201800000514
and
Figure BDA00023668201800000515
is recorded as the error value of
Figure BDA00023668201800000516
Figure BDA00023668201800000517
Is finite, (19) can be rewritten as:
Figure BDA0002366820180000061
for the uncertain nonlinear systems described by equation (16), equation (17) and equation (20), the lyapunov function is introduced in the form:
Figure BDA0002366820180000062
V2(t) is the introduced Lyapunov function, gamma21，γ22，m1Is a system parameter, being any normal number.
Along the trajectories of the equations (16) and (17), V is obtained2(t) derivative, one can obtain for any t ≧ t0，
Figure BDA0002366820180000063
This can be obtained from equation (22):
Figure BDA0002366820180000064
as can be seen from equation (23):
Figure BDA0002366820180000065
substituting formula (24) for formula (22) to obtain:
Figure BDA0002366820180000066
ρ2(x1,x2) Is f2(x1,x2) Can be obtained from the formula (17), the formula (18), the formula (20) and the formula (25) in a similar manner to the first step:
Figure BDA0002366820180000067
when in use
Figure BDA0002366820180000071
When the formula (26) is satisfied.
Step i: starting from the i-th equation differential of equation (3) and replacing equation (1) one can obtain:
Figure BDA0002366820180000072
for the uncertain nonlinear system described by equation (27), the following finite control function α is proposedi(·)
Figure BDA0002366820180000073
Wherein v isi1(·),vi2(·),ρ(i-1)1(·),ρ(i-2)2(. cndot.) is obtained by the following equation:
Figure BDA0002366820180000074
function(s)
Figure BDA0002366820180000075
Are respectively unknown parameters
Figure BDA0002366820180000076
And updating the law by adapting equation (30)
Figure BDA0002366820180000077
Wherein, γi1,γi2,mi-1Is any normal number of the components to be tested,
Figure BDA0002366820180000078
is limited.
Figure BDA0002366820180000079
Are respectively as
Figure BDA00023668201800000710
And
Figure BDA00023668201800000711
Figure BDA00023668201800000712
and
Figure BDA00023668201800000713
Figure BDA00023668201800000714
and
Figure BDA00023668201800000715
is recorded as the error value of
Figure BDA00023668201800000716
The rewrite formula (30) is the following adaptive system.
Figure BDA00023668201800000717
For the uncertain nonlinear system described by (27), (28) and (31), the following Lyapunov function was introduced
Figure BDA0002366820180000081
Vi(t) is the introduced Lyapunov function, and by using a method similar to the first and second steps, it can be concluded that for any t ≧ t0All have:
Figure BDA0002366820180000082
when in use
Figure BDA0002366820180000083
When the formula (33) is satisfied.
And a last step: and synthesizing an actual control law.
The last equation of equation (3) is first differentiated and substituted from equation (1):
Figure BDA0002366820180000084
for the uncertain system equation (34), a practical controller u (t) is proposed as follows:
Figure BDA0002366820180000085
wherein v isn1(·),vn2(·),p(n-1)1(·),p(n-1)2(. cndot.) is obtained by the following equation:
Figure BDA0002366820180000086
wherein the function
Figure BDA0002366820180000087
Are respectively unknown parameters
Figure BDA0002366820180000088
Is updated by the following adaptation rules:
Figure BDA0002366820180000089
wherein, γn1,γn2,mn-1Is any normal number of the components to be tested,
Figure BDA00023668201800000810
is limited.
Figure BDA00023668201800000811
Are respectively as
Figure BDA00023668201800000812
And
Figure BDA0002366820180000091
Figure BDA0002366820180000092
and
Figure BDA0002366820180000093
Figure BDA0002366820180000094
and
Figure BDA0002366820180000095
is recorded as the error value of
Figure BDA0002366820180000096
Figure BDA0002366820180000097
Rewrite equation (37) is the following system:
Figure BDA0002366820180000098
for the uncertainty systems described by equation (34), equation (35) and equation (38), the following forms of the Lyapunov function are introduced
Figure BDA0002366820180000099
V is obtained by following the trajectories of (48) and (49)n(t) derivative, one can obtain for any t ≧ t0，
Figure BDA00023668201800000910
Wherein
Figure BDA00023668201800000911
Introduction definition ③:
Figure BDA00023668201800000912
ε:＝max{εj:j＝1,2,…,n}
wherein:
z(t):＝[z1z2…zn]T
Figure BDA00023668201800000913
Figure BDA00023668201800000914
Figure BDA00023668201800000915
from the above procedure, the following theorem 1 can be obtained: theorem: the uncertain nonlinear system described by equation (1) is in existence under the given adaptive robust control scheme.
In the case of fixed parameters and external disturbances, a uniform asymptotic convergence to zero is possible.
And (3) proving that: the formula (40) can be rewritten as t ≧ t0All have:
Figure BDA00023668201800000916
according to the definition of the Lyapunov function given by the formula (39), it can be obtained that t is larger than t0，
Figure BDA00023668201800000917
Wherein:
Figure BDA0002366820180000101
Figure BDA0002366820180000102
δmax，δmintwo normal numbers.
It can be seen from equations (41) and (42) that the solutions of the adaptive nonlinear system are uniformly bounded,
Figure BDA0002366820180000103
progressively converging to zero.
The invention relates to a nonlinear system with parameter uncertainty and a plurality of external disturbances and a design method thereof, aiming at a nonlinear system with uncertain system parameters and a plurality of external disturbances, introducing a virtual control function in the control design by a backstepping method, estimating unknown boundaries of the external disturbances by using an improved adaptive law, synthesizing a continuous adaptive robust state feedback controller by using updated values of the unknown boundaries, and finally showing that the adaptive robust state feedback control scheme can ensure that the state of the uncertain nonlinear system is uniformly asymptotically converged to zero under the conditions of uncertainty and external disturbance, and the synthesized adaptive robust controller can stabilize the uncertain nonlinear system.
Drawings
FIG. 1 is a schematic view of a suspension system for a vehicle according to an embodiment of the present invention.
FIG. 2 is a graph illustrating the variation of the control function according to an embodiment of the present invention.
FIG. 3(a) is a diagram of a state variable x according to an embodiment of the present invention1、x2A variation graph;
FIG. 3(b) is a diagram of a state variable x according to an embodiment of the present invention3、x4The graph is varied.
FIG. 4(a) is a diagram of a state variable x according to an embodiment of the present invention1、x2Error variation curve diagram;
FIG. 4(b) is a diagram of a state variable x according to an embodiment of the present invention3、x4Error variation graph.
FIG. 5(a) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000104
A graph of variation of (d);
FIG. 5(b) is a diagram illustrating unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000105
A graph of variation of (d);
FIG. 5(c) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000106
A graph of variation of (d);
FIG. 5(d) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000107
Graph of the variation of (c).
FIG. 6(a) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000108
A graph of variation of (d);
FIG. 6(b) is a diagram illustrating unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000109
A graph of variation of (d);
FIG. 6(c) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000111
A graph of variation of (d);
FIG. 6(d) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000112
Graph of the variation of (c).
FIG. 7(a) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000113
A graph of variation of (d);
FIG. 7(b) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000114
A graph of variation of (d);
FIG. 7(c) is a diagram of unknown parameters in an embodiment of the present invention
Figure BDA0002366820180000115
Graph of the variation of (c).
Detailed Description
The following is one embodiment of the present invention:
and (3) selecting an automobile independent suspension system for modeling and simulation. Its main function is to make the wheel have good adhesion with the ground, and make the dynamic load of the wheel change little, thus lighten the uneven impact of road surface, make the vehicle control more stable. Variations in vehicle load, suspension stiffness, and non-linearity of the shock absorber, among others, can lead to uncertainties in system parameters. The influence of uneven road conditions on the vehicle during driving can be regarded as an uncertain disturbance of the system. In consideration of uncertainty of vehicle dynamics model parameters and external disturbances, it is necessary to adopt robust control. The automobile independent suspension system has external energy input, can make active response according to the driving state and road conditions of an automobile, and generates corresponding power to be balanced with external excitation. The suspension system of a vehicle is shown in fig. 1.
The single wheel vehicle suspension system equation of state of FIG. 1 is:
Figure BDA0002366820180000116
wherein m is1,m2To mass, k1,k2Is the spring coefficient, b is the damping coefficient, u is the external input, y and x are the vertical displacements of the two springs, respectively, assuming that both y and x vary sinusoidally.
Let y be x1，
Figure BDA0002366820180000119
x＝x3，
Figure BDA0002366820180000117
After considering the uncertainty factor, the nonlinear dynamical system is given by the following differential equation:
Figure BDA0002366820180000118
wherein:
f1(x1)＝0
f2(x1,x2)＝-x1-0.5(x2-x4)|x2-x4|
f3(x1,x2,x3)＝0
f4(x1,x2,x3,x4)＝x1+0.5(x2-x4)|x2-x4|
where u is the control input, θi(t), i is 1,2,3,4 to limit the range of uncertainty factors, and d (t) is an external perturbation. The main problem is to determine the system form as formula (3), formula (35) and adaptive robust control law as formula (37), so that the stability of formula (44) can be ensured in the presence of uncertainty and external disturbance. For the proposed adaptive robust control scheme, the following parameters are selected:
Figure BDA0002366820180000121
σi(t)＝50e-0.05t,i＝1,2,3,4；
adapting the robust control scheme, the following parameters are selected:
Figure BDA0002366820180000122
σi(t)＝50e-0.05t,i＝1,2,3,4；
for system equation (43), a continuous adaptive robust controller can be obtained from equations (3), (35), and (37), and the controller can ensure uniform and bounded closed-loop dynamic system, and the output of system equation (43) can be stable in the presence of uncertainty and external disturbance. The uncertain time-varying parameter θ (t) and initial condition values are as follows:
θi(t)＝0.1sin(0.01t),i＝1,2,3,4
x(t)＝[0.3；-0.3；0.3；-0.3]T
Figure BDA0002366820180000123
Figure BDA0002366820180000124
Figure BDA0002366820180000125
Figure BDA0002366820180000126
with the selected parameter settings, the simulation results are as shown in fig. 2, fig. 3(a), fig. 3(b), fig. 4(a), fig. 4(b), fig. 5(a), fig. 5(b), fig. 5(c), fig. 5(d), fig. 6(a), fig. 6(b), fig. 6(c), fig. 6(d), fig. 7(a), fig. 7(b), and fig. 7(c), and the ordinate in the figures has no unit because of pure simulation.
As can be seen from fig. 2, the control function is continuous and converges to zero.
As can be seen from fig. 3(a), 3(b), the adaptive robust controller can ensure that the vehicle suspension system (43) achieves gradual stabilization in the presence of time-varying uncertainties and external disturbances.
From fig. 5(a), 5(b), 5(c), and 5(d), the adaptive law can be derived to make the unknown parameters
Figure BDA0002366820180000131
The estimated value of (a) is reduced, and the adaptive function of the controller is embodied.
FIG. 6(a), FIG. 6(b), FIG. 6(c), FIG. 6(d) can be derived from the adaptation law to make the unknown parameters
Figure BDA0002366820180000132
The estimated value of (a) is reduced, and the adaptive function of the controller is embodied.
FIG. 7(a), FIG. 7(b), FIG. 7(c) can derive the adaptive law to make the unknown parameters
Figure BDA0002366820180000133
The estimated value of (a) is reduced, and the adaptive function of the controller is embodied.
The controller has the self-adaptive function and can change according to the change of the external disturbance.

Claims (3)

1. A nonlinear system with parameter uncertainty and a plurality of external disturbances, characterized in that the nonlinear system has the expression:
Figure FDA0002366820170000011
wherein the content of the first and second substances,
Figure FDA0002366820170000012
for state variables, u e R is the system input, function f (-) Ri→Rmi,i＝1,2, …, n, is a known smooth vector field;
Figure FDA0002366820170000013
are constant parametric vectors, which are compact sets
Figure FDA0002366820170000014
And which defines all uncertainty ranges for a given system; unknown function deltai(t), i ═ 1,2, …, n denotes external interference;
the state feedback controller u (t) is represented by a non-linear function:
Figure FDA0002366820170000015
setting: non-linear function f (-) Ri→RmiI-1, 2, …, n is continuous and locally uniformly bounded, the following definitions are introduced:
pi(x1,…，xi):＝||fi(x1,…,xi)｜｜
Figure FDA0002366820170000016
Figure FDA0002366820170000017
Figure FDA0002366820170000018
where ρ isi(x1,…，xi) Denotes fi(x1,…,xi) The boundary of (a) is determined,
Figure FDA0002366820170000019
are all unknown normal numbers in the control law,
Figure FDA00023668201700000110
to represent
Figure FDA00023668201700000111
The boundary of (a) is determined,
Figure FDA00023668201700000112
for external disturbance deltai(t) the maximum value of the absolute value,
Figure FDA00023668201700000113
is composed of
Figure FDA00023668201700000114
Is measured.
2. A method of designing a nonlinear system with parameter uncertainty and multiple external perturbations in accordance with claim 1, wherein: and designing a virtual controller and an adaptive law for each subsystem in the nonlinear system by using an iterative design algorithm, and constructing a Lyapunov function for each subsystem in the nonlinear system based on a Lyapunov stability theory to ensure the convergence of the subsystems.
3. A method of designing a nonlinear system with parameter uncertainty and multiple external perturbations in accordance with claim 1, wherein: and the stability of the whole closed-loop system is ensured by designing an actual control law through reverse iteration.
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