CN113885314B - 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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CN113885314B
CN113885314B CN202111230756.1A CN202111230756A CN113885314B CN 113885314 B CN113885314 B CN 113885314B CN 202111230756 A CN202111230756 A CN 202111230756A CN 113885314 B CN113885314 B CN 113885314B
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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 nonlinear system tracking control method with unknown gain and interference, which relates to an interference observer design, a gain compensation algorithm design and a tracking controller design of a nonlinear system, including the interference observer design, the gain compensation algorithm design and the tracking controller design. Aiming at the interference problem in a nonlinear system, the invention designs an interference observer based on sliding mode control; aiming at the unknown gain problem of a nonlinear system, a gain compensation algorithm based on a Nussbaum technology is designed; in order to realize tracking control, a tracking controller based on a back-stepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under the 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 become a research hotspot because of their ability to better describe practical systems. Generally, fuzzy or neural network techniques are used to estimate the nonlinear function in the system and to design the controller using a backstepping control method. Although many excellent research results have been reported, there are many unresolved problems such as disturbance and unknown gain functions. [ "Full-order observer for a class of nonlinear systems with unmatched uncertainties: joint attractive ellipsoid and sliding mode concepts" (B.S 'ankez, C.Cuvas, P.Ordaz, O.Santos-S' ankez, and A.Poznylak, 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 limited uniformly bounded stability and sliding mode was proposed. For a type of nonlinear system with unknown control gain, an adaptive control based on output feedback is proposed by [ (Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions "(C.Yang, S.Ge, T.Lee, automation, vol.45, pp.270-276,2009) ]. To overcome the unknown control direction, a discrete Nussbaum gain method is employed. However, to date, the tracking control problem of nonlinear systems with unknown gain and disturbances has not been fully studied, as solving the unknown gain while suppressing the disturbances is more challenging.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a nonlinear system tracking control method with unknown gain and interference so as to effectively solve the problems of unknown gain compensation, interference suppression and tracking control in a nonlinear system.
In order to achieve the aim of the invention, the nonlinear system tracking control method with unknown gain and interference is used for solving the problem of the unknown gain in a nonlinear system, and a compensation algorithm based on a Nussbaum technology is designed; aiming at the interference problem of a nonlinear system, an interference observer controlled by a basic sliding mode is designed; in order to realize tracking control, a tracking controller of a back-stepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under the unknown gain and interference.
The design of the disturbance observer defines the optimal weight parameter as w i * The sliding mode function is designed as follows:
Figure BDA0003315831390000021
wherein
Figure BDA0003315831390000022
e i ＝z i -x i ，z i Is an auxiliary variable, and->
Figure BDA0003315831390000023
The following observers were then designed:
Figure BDA0003315831390000024
wherein δi As intermediate variable, eta i ＞0,k i > 0 is the observer tuning parameter.
The tracking controller based on the back-stepping method is designed as follows
Figure BDA0003315831390000025
And:
Figure BDA0003315831390000026
Figure BDA0003315831390000027
wherein εn,σ ＞0,ε n,f ＞0,ε n,ω ＞0,f n0 Is an adjustment parameter. Parameters (parameters)
Figure BDA0003315831390000028
F n0 The calculation of (2) will be given in the description.
The object of the present invention is thus achieved.
The invention discloses a nonlinear system tracking control method with unknown gain and interference, which relates to an interference observer design, a gain compensation algorithm design and a tracking controller design of a nonlinear system, including the interference observer design, the gain compensation algorithm design and the tracking controller design. Aiming at the interference problem in a nonlinear system, the invention designs an interference observer based on sliding mode control; aiming at the unknown gain problem of a nonlinear system, a gain compensation algorithm based on a Nussbaum technology is designed; in order to realize tracking control, a tracking controller based on a back-stepping method is designed. The invention can effectively solve the tracking control problem of the nonlinear system under the unknown gain and interference.
Drawings
FIG. 1 is a schematic diagram of an embodiment of a nonlinear system tracking control method with unknown gain and disturbance of the present invention.
Detailed Description
The following description of the embodiments of the invention is presented in conjunction with the accompanying drawings to provide a better understanding of the invention to those skilled in the art. It is to be expressly noted that in the description below, detailed descriptions of known functions and designs are omitted here as perhaps obscuring the present invention.
FIG. 1 is a schematic diagram of an embodiment of a nonlinear system tracking control method with unknown gain and disturbance of the present invention.
As shown in fig. 1, the present invention relates to an interference observer design with a nonlinear system, a gain compensation algorithm design based on the nussemer (Nussbaum) technique, and a tracking controller design based on a back-stepping method.
Consider the following nonlinear system
Figure BDA0003315831390000031
Where y e R and u (t) e R represent the output and input of the system respectively,
Figure BDA0003315831390000032
and />
Figure BDA0003315831390000033
Representing the state of the system->
Figure BDA0003315831390000034
Representing a nonlinear function>
Figure BDA0003315831390000035
Representing unknown gain, ζ i (t), i=1, 2,..n represents external interference.
The nonlinear system (1) satisfies the assumption that: (1) For any i e { 1..n }, function
Figure BDA0003315831390000036
The sign of (2) is known, and
Figure BDA0003315831390000037
is bounded, satisfy->
Figure BDA0003315831390000038
wherein f i and />
Figure BDA0003315831390000039
Is a determined normal quantity. Without loss of generality, we assume that
Figure BDA00033158313900000310
(2) The disturbance and its first derivative are bounded, i.e.>
Figure BDA00033158313900000311
and />
Figure BDA00033158313900000312
Wherein the upper bound->
Figure BDA00033158313900000313
Is available, but upper bound +.>
Figure BDA00033158313900000314
Is unknown.
Typically, a knowledgeable baum (Nussbaum) function N (κ) is used to handle the unknown gain
Figure BDA00033158313900000315
The neural network estimator is used to estimate the nonlinear function +.>
Figure BDA00033158313900000316
I.e. < ->
Figure BDA00033158313900000317
wherein wi Representing weights +.>
Figure BDA00033158313900000318
Representing the excitation function +.>
Figure BDA00033158313900000319
Represents an estimation error, and->
Figure BDA00033158313900000320
Figure BDA00033158313900000321
Representing the upper bound of the error.
Interference observer design based on sliding mode control
In the system (1), let
Figure BDA00033158313900000322
Then->
Figure BDA00033158313900000323
The estimation may be performed by a neural network estimator:
Figure BDA00033158313900000324
wherein wFi I=1,..n represents a weight that satisfies the following adaptive law:
Figure BDA0003315831390000041
wherein ρi I=1, 2,..n represents a normal quantity, matrix Q i Satisfies the following conditions
Figure BDA0003315831390000042
Optimal weight
Figure BDA0003315831390000043
The definition is as follows: />
Figure BDA0003315831390000044
wherein />
Figure BDA0003315831390000045
and />
Figure BDA0003315831390000046
Represents two compact sets, and +.>
Figure BDA0003315831390000047
Figure BDA0003315831390000048
Is constant. Further, define auxiliary variables
e i ＝z i -x i ,i＝1,2,...,n(i＝1,...,n) (4)
Wherein the variable z i Has the following dynamic states:
Figure BDA0003315831390000049
wherein ci Express constant, satisfy
Figure BDA00033158313900000410
Variable delta i (i=1, 2,., n) will be designed so as to estimate the error
Figure BDA00033158313900000411
Can converge to 0 in a limited time, wherein +.>
Figure BDA00033158313900000412
Indicating interference xi i An estimate of (t).
The following sliding mode function is defined:
Figure BDA00033158313900000413
wherein ki and ηi Represents the adjustment parameters to satisfy eta i＞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 for a finite time.
Tracking controller design based on back-stepping method
Defining an error variable: τ i ＝x i -α i-1 I=1, 2,..n, where α i-1 Representing a virtual control signal, and alpha 0 ＝y r ，y r Representing the desired signal. According to the back-stepping method, the virtual control input and the actual control input are designed as follows:
step 1: for error τ 1 Differentiation to obtain
Figure BDA00033158313900000417
Order the
Figure BDA00033158313900000418
Function->
Figure BDA00033158313900000419
The estimation can be done by a neural network: />
Figure BDA00033158313900000420
wherein
Figure BDA00033158313900000421
w 1 * Representing the ideal weights. Then
Figure BDA00033158313900000422
wherein
Figure BDA00033158313900000423
Virtual control inputs and parameter adaptation laws are designed as follows:
Figure BDA0003315831390000051
and is also provided with
N(κ 1 )＝κ 1 2 cos(κ 1 2 ) (11)
Figure BDA0003315831390000052
Figure BDA0003315831390000053
wherein θi I=1, 2,..n is the normal amount, matrix P i Satisfy P i ＝P i T > 0, i=1, 2..n, parameter epsilon 1,σ ＞0。
Step i (i=2,., n-1): for variable τ i Differentiation to obtain
Figure BDA0003315831390000054
In the above formula, due to the existence of
Figure BDA0003315831390000055
The complexity of calculation is increased, so that the patent adopts the supercoiled estimator to estimate the supercoiled estimator, and the method is as follows:
Figure BDA0003315831390000056
wherein λil (l=0, 1) and f i0 Represents the state of the supercoiled system, mu il (l=0, 1) is constant, satisfying μ il ＞0。
Parameters of
Figure BDA0003315831390000057
Can be obtained by the following formula:
Figure BDA0003315831390000058
wherein ωi-1 Representing the estimated 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 also provided with
N(κ i )＝κ i 2 cos(κ i 2 ) (18)
Figure BDA00033158313900000512
/>
Figure BDA00033158313900000513
Wherein the parameter epsilon i,σ ＞0,ε i,ω ＞0。
Step i=n, error τ n Differentiation is carried out to obtain
Figure BDA0003315831390000061
Nonlinear characteristicsFunction of
Figure BDA0003315831390000062
Can be estimated as +.>
Figure BDA0003315831390000063
wherein wn Represent weights and
Figure BDA0003315831390000064
parameter->
Figure BDA0003315831390000065
Can be calculated as +.>
Figure BDA0003315831390000066
wherein ωn-1 Represents an estimation error, whose upper bound is +.>
Figure BDA0003315831390000067
The control input u (t) and the parameter adaptation law are designed as follows:
Figure BDA0003315831390000068
and is also provided with
Figure BDA0003315831390000069
Figure BDA00033158313900000610
Wherein the parameter epsilon n,σ ＞0,ε n,f ＞0,ε n,ω ＞0，f n0 To adjust the parameters.
While the foregoing describes illustrative embodiments of the present invention to facilitate an 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, but is to be construed as protected by the accompanying claims insofar as various changes are within the spirit and scope of the present invention as defined and defined by the appended claims.

Claims (1)

1. A nonlinear system tracking control method with unknown gain and interference is characterized by comprising an interference observer design, a gain compensation algorithm design and a tracking controller design;
the disturbance observer design includes 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 designed to be a control gain compensation algorithm based on a Knoop Bowm technology; the tracking controller design comprises differential estimation based on virtual control input of the supercoiled estimator and tracking controller design based on a back-stepping method;
the nonlinear system is as follows:
Figure FDA0004162209860000011
wherein y epsilon R and u (t) epsilon R respectively represent the output and input of the system,
Figure FDA0004162209860000012
and />
Figure FDA0004162209860000013
Representing the state of the system x i For the i-th state component,/->
Figure FDA0004162209860000014
Representing a nonlinear function>
Figure FDA0004162209860000015
Representing unknown gain, ζ i (t), i=1, 2,..n represents external interference;
to any oneIntended nonlinear continuous function
Figure FDA0004162209860000016
There is a neural network such that:
Figure FDA0004162209860000017
wherein ,wi The weight vector is represented by a weight vector,
Figure FDA0004162209860000018
representing the excitation function of the neural network, T representing the transpose of the vector or matrix,
Figure FDA0004162209860000019
representing the estimation error, defining the optimal weight parameter as w i * The sliding mode function is designed as follows:
Figure FDA00041622098600000110
wherein ,
Figure FDA00041622098600000111
e i ＝z i -x i ，z i is an auxiliary variable, and->
Figure FDA00041622098600000112
Figure FDA00041622098600000117
Transpose of representing optimal weights, c i Being constant, the following observer is then designed:
Figure FDA00041622098600000113
Figure FDA00041622098600000114
wherein ,δi As intermediate variable, eta i＞0 and ki The values of > 0 are the observer tuning parameters,
Figure FDA00041622098600000115
representing an interference estimate;
the design is as follows
Figure FDA00041622098600000116
And is also provided with
Figure FDA0004162209860000021
Figure FDA0004162209860000022
wherein ,εn,σ ＞0,ε n,f ＞0,ε n,ω ＞0,f n0 Is the controller regulating parameter, τ n Is the error, w n The weight is represented by a weight that,
Figure FDA0004162209860000023
representing the excitation function +.>
Figure FDA0004162209860000024
Representing the upper bound of the estimation error,/->
Figure FDA0004162209860000025
Representing the upper bound of the nonlinear function, +.>
Figure FDA0004162209860000026
For the upper bound of the estimation error, +.>
Figure FDA0004162209860000027
Representing the interference estimate. />
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