CN113885314A - Nonlinear system tracking control method with unknown gain and interference - Google Patents

Nonlinear system tracking control method with unknown gain and interference Download PDF

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CN113885314A
CN113885314A CN202111230756.1A CN202111230756A CN113885314A CN 113885314 A CN113885314 A CN 113885314A CN 202111230756 A CN202111230756 A CN 202111230756A CN 113885314 A CN113885314 A CN 113885314A
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李猛
苗朕海
陈勇
刘越智
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a non-linear system tracking control method with unknown gain and interference, and relates to the design of an interference observer, the design of a gain compensation algorithm and the design of a tracking controller which comprise a non-linear system, wherein the design of the interference observer, the design of the gain compensation algorithm and the design of the tracking controller are included. Aiming at the interference problem in a nonlinear system, the invention designs an interference observer based on sliding mode control; aiming at the problem of unknown gain of a nonlinear system, a gain compensation algorithm based on a Nussbaum (Nussbaum) technology is designed; in order to realize tracking control, a tracking controller based on a backstepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under unknown gain and interference.

Description

Nonlinear system tracking control method with unknown gain and interference
Technical Field
The invention belongs to the technical field of nonlinear system tracking control, and particularly relates to a nonlinear system tracking control method with unknown gain and interference.
Background
In recent years, nonlinear systems have been the focus of research because they can better describe real systems. Typically, fuzzy or neural network techniques are used to estimate the non-linear functions in the system and design the controller using a back-stepping control method. Although many excellent research results have been reported, there are still many unsolved problems such as disturbance and unknown gain function. "Full-order observer for a class of systems with infinite induced errors and sliding modes" (B.S 'anchez, C.Cuvas, P.Ordaz, O.Santos-S' anchez, and A.Poznyak, IEEE Transactions on Industrial Electronics, vol.67, No.7, pp.5677-5686,2020.) "for affine nonlinear systems with uncertainty and perturbation, a Full order observer combining extreme uniformly bounded stability and sliding modes is proposed. "Output feedback adaptive control of a class of non-linear control systems with unknown control gains" (C.Yang, S.Ge, T.Lee, Automatica, vol.45, pp.270-276,2009.) ] proposes an adaptive control based on Output feedback. In order to overcome the unknown control direction, a discrete Nussbaum gain method is adopted. However, to date, the tracking control problem of nonlinear systems with unknown gain and disturbance has not been fully studied, as solving for the unknown gain while suppressing the disturbance is more challenging.
Disclosure of Invention
The present invention is directed to overcome the deficiencies of the prior art and provide a tracking control method for a nonlinear system with unknown gain and interference, so as to effectively solve the problems of unknown gain compensation, interference suppression and tracking control in the nonlinear system.
In order to achieve the purpose, the invention provides a nonlinear system tracking control method with unknown gain and interference, which designs a compensation algorithm based on Nussbaum technology aiming at the problem of unknown gain in a nonlinear system; aiming at the interference problem of a nonlinear system, a base sliding mode controlled interference observer is designed; in order to realize tracking control, a tracking controller adopting a backstepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under unknown gain and interference.
The disturbance observer is designed, and an optimal weight parameter is defined as wi *Designing a sliding mode function as follows:
Figure BDA0003315831390000021
wherein
Figure BDA0003315831390000022
ei＝zi-xi，ziIs an auxiliary variable, and
Figure BDA0003315831390000023
the following observer is then designed:
Figure BDA0003315831390000024
wherein δiIs an intermediate variable, ηi＞0,kiAnd the observer adjustment parameter is more than 0.
The design of the tracking controller based on the backstepping method comprises the following design
Figure BDA0003315831390000025
And:
Figure BDA0003315831390000026
Figure BDA0003315831390000027
wherein εn,σ＞0,εn,f＞0,εn,ω＞0,fn0Is an adjustment parameter. Parameter(s)
Figure BDA0003315831390000028
And fn0The calculation of (a) will be given in the specification.
The object of the invention is thus achieved.
The invention discloses a non-linear system tracking control method with unknown gain and interference, and relates to the design of an interference observer, the design of a gain compensation algorithm and the design of a tracking controller which comprise a non-linear system, wherein the design of the interference observer, the design of the gain compensation algorithm and the design of the tracking controller are included. Aiming at the interference problem in a nonlinear system, the invention designs an interference observer based on sliding mode control; aiming at the problem of unknown gain of a nonlinear system, a gain compensation algorithm based on a Nussbaum (Nussbaum) technology is designed; in order to realize tracking control, a tracking controller based on a backstepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under unknown gain and interference.
Drawings
Fig. 1 is a schematic structural diagram of an embodiment of a tracking control method of a nonlinear system with unknown gain and interference according to the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.
Fig. 1 is a schematic structural diagram of an embodiment of a tracking control method of a nonlinear system with unknown gain and interference according to the present invention.
As shown in fig. 1, the present invention relates to a disturbance observer design with a nonlinear system, a gain compensation algorithm design based on the Nussbaum (Nussbaum) technology, and a tracking controller design based on the back-stepping method.
Consider the following nonlinear system
Figure BDA0003315831390000031
Where y ∈ R and u (t) ∈ R denote the output and input of the system respectively,
Figure BDA0003315831390000032
and
Figure BDA0003315831390000033
which is indicative of the state of the system,
Figure BDA0003315831390000034
the non-linear function is represented by a linear function,
Figure BDA0003315831390000035
representing unknown gain, ξi(t), i ═ 1,2.., n denotes external interference.
The nonlinear system (1) satisfies the assumption: (1) for any i e {1,. eta., n }, the function
Figure BDA0003315831390000036
Is known, and
Figure BDA0003315831390000037
is bounded, satisfies
Figure BDA0003315831390000038
wherein f iAnd
Figure BDA0003315831390000039
is a positive normal quantity. Without loss of generality, we assume
Figure BDA00033158313900000310
(2) The disturbance and its first derivative are bounded, i.e. the disturbance and its first derivative are
Figure BDA00033158313900000311
And
Figure BDA00033158313900000312
wherein the upper bound
Figure BDA00033158313900000313
Are available, but the upper bound
Figure BDA00033158313900000314
Is unknown.
Generally, a Nussbaum function N (κ) is used to handle unknown gains
Figure BDA00033158313900000315
Neural network estimators are used to estimate nonlinear functions
Figure BDA00033158313900000316
Namely, it is
Figure BDA00033158313900000317
wherein wiThe weight is represented by a weight that is,
Figure BDA00033158313900000318
the function of the excitation is represented by,
Figure BDA00033158313900000319
represents an estimation error, and
Figure BDA00033158313900000320
Figure BDA00033158313900000321
representing an upper bound for error.
Disturbance observer design based on sliding mode control
In the system (1), let
Figure BDA00033158313900000322
Then
Figure BDA00033158313900000323
The estimation can be done by a neural network estimator:
Figure BDA00033158313900000324
wherein wFiN represents a weight, which satisfies the following conditionThe following adaptation law:
Figure BDA0003315831390000041
wherein ρ i1,2, n denotes a normal quantity, and a matrix QiSatisfy the following requirements
Figure BDA0003315831390000042
Optimal weight
Figure BDA0003315831390000043
Is defined as:
Figure BDA0003315831390000044
wherein
Figure BDA0003315831390000045
And
Figure BDA0003315831390000046
represent two compact sets, and
Figure BDA0003315831390000047
Figure BDA0003315831390000048
is a constant. Further, auxiliary variables are defined
ei＝zi-xi,i＝1,2,...,n(i＝1,...,n) (4)
Wherein the variable ziHas the following dynamics:
Figure BDA0003315831390000049
wherein ciRepresents a constant, satisfies
Figure BDA00033158313900000410
Variable deltai(i 1,2.., n) will be designed so as to estimateError counting
Figure BDA00033158313900000411
Can converge to 0 within a limited time, wherein
Figure BDA00033158313900000412
Representing the disturbance xii(t) an estimated value.
The following sliding-mode function is defined:
Figure BDA00033158313900000413
wherein ki and ηiExpressing the adjustment parameter to satisfy ηi＞0 and
Figure BDA00033158313900000414
sgn (·) represents a sign function. The estimated value of the interference can be calculated by the following equation:
Figure BDA00033158313900000415
then the error is estimated
Figure BDA00033158313900000416
Will converge to 0 within a finite time.
Tracking controller design based on backstepping method
Defining an error variable: tau isi＝xi-αi-11,2, n, wherein α isi-1Represents a virtual control signal, and0＝yr，yrrepresenting the desired signal. According to the backstepping method, the virtual control input and the actual control input are designed as follows:
step 1: for error tau1Differentiating to obtain
Figure BDA00033158313900000417
Order to
Figure BDA00033158313900000418
Function(s)
Figure BDA00033158313900000419
The estimation can be done with a neural network:
Figure BDA00033158313900000420
wherein
Figure BDA00033158313900000421
w1 *Representing the ideal weights. Then
Figure BDA00033158313900000422
wherein
Figure BDA00033158313900000423
Designing a virtual control input and parameter adaptation law as follows:
Figure BDA0003315831390000051
and is
N(κ1)＝κ1 2cos(κ1 2) (11)
Figure BDA0003315831390000052
Figure BDA0003315831390000053
wherein θiN is a normal quantity, and the matrix P is a normal quantityiSatisfy Pi＝Pi TN, parameter epsilon > 0, i ═ 1,21,σ＞0。
Step i (i ═ 2.., n-1): for variable tauiDifferentiating to obtain
Figure BDA0003315831390000054
In the above formula, due to existence
Figure BDA0003315831390000055
The complexity of calculation is increased, so the supercoiled estimator is adopted to estimate the supercoiled estimator, which comprises the following steps:
Figure BDA0003315831390000056
wherein λil(l ═ 0,1) and fi0Indicating the state of the supercoiled system, μil(l is 0,1) is a constant number satisfying μil＞0。
Then the parameter
Figure BDA0003315831390000057
Can be obtained by the following formula:
Figure BDA0003315831390000058
wherein ωi-1Represents an estimation error with an upper bound of
Figure BDA0003315831390000059
The nonlinear function is estimated as:
Figure BDA00033158313900000510
the virtual control inputs and parameter adaptation laws are then designed as follows:
Figure BDA00033158313900000511
and is
N(κi)＝κi 2cos(κi 2) (18)
Figure BDA00033158313900000512
Figure BDA00033158313900000513
Wherein the parameter epsiloni,σ＞0,εi,ω＞0。
Step i ═ n, for error τnDifferentiating to obtain
Figure BDA0003315831390000061
Non-linear function
Figure BDA0003315831390000062
Can be estimated as
Figure BDA0003315831390000063
wherein wnRepresent weights and
Figure BDA0003315831390000064
parameter(s)
Figure BDA0003315831390000065
Can be calculated as
Figure BDA0003315831390000066
wherein ωn-1Represents an estimation error with an upper bound of
Figure BDA0003315831390000067
The control inputs u (t) and the parameter adaptation law are designed as follows:
Figure BDA0003315831390000068
and is
Figure BDA0003315831390000069
Figure BDA00033158313900000610
Wherein the parameter epsilonn,σ＞0,εn,f＞0,εn,ω＞0，fn0To adjust the parameters.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (5)

1. A non-linear system tracking control method with unknown gain and interference is characterized by comprising interference observer design, gain compensation algorithm design and tracking controller design.
2. The disturbance observer design of claim 1, comprising a nonlinear system description with unknown gain and disturbance, a neural estimation of a nonlinear function, and a sliding mode based observer design; the gain compensation algorithm is specifically a control gain compensation algorithm based on a Nussbaum (Nussbaum) technology; the tracking controller design comprises differential estimation based on virtual control input of the supercoiled estimator and a tracking controller design based on a backstepping method.
3. The non-linear system description with unknown gain and interference according to claim 2, characterized by: for the following non-linear system
Figure FDA0003315831380000011
Where y ∈ R and u (t) ∈ R denote the output and input of the system respectively,
Figure FDA0003315831380000012
and
Figure FDA0003315831380000013
which is indicative of the state of the system,
Figure FDA0003315831380000014
the non-linear function is represented by a linear function,
Figure FDA0003315831380000015
representing unknown gain, ξi(t), i ═ 1,2.., n denotes external interference.
4. The neural estimation and disturbance observer design of a nonlinear function as claimed in claim 2, characterized in that: for arbitrary non-linear continuous function
Figure FDA0003315831380000016
A neural network exists such that
Figure FDA0003315831380000017
wherein wiA vector of weights is represented by a vector of weights,
Figure FDA0003315831380000018
represents the excitation function of the neural network, T represents the transpose of the solution vector or matrix,
Figure FDA00033158313800000115
indicating the estimation error. Defining the optimal weight parameter as
Figure FDA0003315831380000019
Designing a sliding mode function as follows:
Figure FDA00033158313800000110
wherein
Figure FDA00033158313800000111
ei＝zi-xi，ziIs an auxiliary variable, and
Figure FDA00033158313800000112
the following observer is then designed:
Figure FDA00033158313800000113
Figure FDA00033158313800000114
wherein δiIs an intermediate variable, ηi＞0 and kiAnd the observer adjustment parameter is more than 0.
5. The tracking controller design according to claim 2, characterized in that: designed as follows
Figure FDA0003315831380000021
And is
Figure FDA0003315831380000022
Figure FDA0003315831380000023
wherein εn，σ＞0,εn，f＞0,εn，ω＞0,fn0Is the controller adjustment parameter.
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