CN113093553A - Adaptive backstepping control method based on instruction filtering disturbance estimation - Google Patents

Adaptive backstepping control method based on instruction filtering disturbance estimation Download PDF

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CN113093553A
CN113093553A CN202110396089.8A CN202110396089A CN113093553A CN 113093553 A CN113093553 A CN 113093553A CN 202110396089 A CN202110396089 A CN 202110396089A CN 113093553 A CN113093553 A CN 113093553A
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CN113093553B (en
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郑晓龙
杨学博
李湛
高会军
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Harbin Institute of Technology
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Abstract

An adaptive backstepping control method based on instruction filtering disturbance estimation belongs to the field of nonlinear system adaptive control methods. The method solves the problems that the designed controller is too conservative and the energy consumption is large due to the fact that the disturbance upper bound is estimated by the current self-adaptive backstepping control technology. According to the state variable of a nonlinear system and an expected output signal of the system in practical application, a nonlinear second-order system state space model containing a disturbance term is established; establishing a dimensional-expanded nonlinear third-order system state space model according to a nonlinear second-order system state space model containing a disturbance term, setting an error variable, and designing a Lyapunov function by using the error variable; solving a first derivative of the Lyapunov function on time; designing a virtual control function and system control input by utilizing a backstepping method and an instruction filter; tracking of the desired output signal is achieved. The invention is suitable for the control of the nonlinear system.

Description

Adaptive backstepping control method based on instruction filtering disturbance estimation
Technical Field
The invention belongs to the field of a self-adaptive control method of a nonlinear system.
Background
The basic idea of the adaptive backstepping control technology is to estimate uncertain parameters in a system by using adaptive parameters and then to reduce the influence of the uncertainty of the parameters on the system performance by using the negative feedback of the estimated parameters. For the adaptive backstepping control technology, reference may be made to chinese patent nos. CN106329986A, CN106438593A, and CN 111679582A. The adaptive back-stepping control technology mainly aims at a nonlinear system with uncertain parameters, when the system contains unknown nonlinear disturbance items, the traditional adaptive back-stepping control technology is not applicable any more, the disturbance is generally assumed to be bounded at the moment, then the upper bound of the disturbance is estimated by using the adaptive parameters, and finally the influence of the disturbance on the system performance is reduced through negative feedback. Since this strategy directly estimates the upper disturbance bound, the designed controller is too conservative and consumes much energy.
Disclosure of Invention
The invention provides a self-adaptive backstepping control method based on instruction filtering disturbance estimation, which aims to solve the problems that a designed controller is too conservative and energy consumption is large due to the fact that the disturbance upper bound is estimated by the current self-adaptive backstepping control technology.
The invention relates to a self-adaptive backstepping control method based on instruction filtering disturbance estimation, which comprises the following steps:
step one, according to the state variable x of the nonlinear system applied in practice1、x2And the desired output signal ydEstablishing a nonlinear second-order system state space model containing a disturbance term;
step two, establishing a dimensional-expanded nonlinear third-order system state space model according to a nonlinear second-order system state space model containing a disturbance term, and simultaneously setting an error variable z1＝x1-yd，z2＝x2-α1And z3＝x3-α2Wherein α is1And alpha2Representing a virtual control function to be designed, a state variable x3U is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two1，z2And z3Designing a Lyapunov function V;
step four, step threeThe first derivative of the Lyapunov function V in the sequence is obtained by solving the time
Figure BDA0003018617620000011
Step five, according to the first derivative of the Lyapunov function
Figure BDA0003018617620000012
Design of virtual control function alpha using backstepping method and instruction filter1And alpha2And a system control input u; obtaining a self-adaptive backstepping controller based on instruction filtering disturbance estimation to realize the purpose of outputting an expected output signal ydThe tracking of (2).
Further, in the invention, in the first step, establishing a nonlinear second-order system state space model containing a disturbance term is as follows:
Figure BDA0003018617620000021
wherein x is1，x2Represents the state variables of a non-linear second-order system,
Figure BDA0003018617620000022
denotes x2B is a constant, f (x)1,x2) Representing the non-linearity of the system for practically known non-linear functions, d (t) representing the disturbance term of a non-linear second-order system, u representing the control input signal of the non-linear second-order system, y representing the output of the non-linear second-order system, the control objective being to design the control input u such that the system output y tracks the desired output signal yd。
Preferably, in the present invention, the constant b is not 0.
Preferably, in the present invention, the non-linear function f (x)1,x2) Is a local Liphoz continuous function.
Further, in the present invention, in the second step, the establishment of the dimension-extended nonlinear third-order system state space model is:
Figure BDA0003018617620000023
wherein,
Figure BDA0003018617620000024
denotes x3The first derivative of (a) is,
Figure BDA0003018617620000025
the first derivative of u is indicated.
Furthermore, in the invention, the error variable z set in the step two is utilized in the step three1，z2And z3The designed Lyapunov function V is as follows:
Figure BDA0003018617620000026
further, in the fourth step, the first derivative of the lyapunov function V in the third step with respect to time is:
Figure BDA0003018617620000027
wherein,
Figure BDA0003018617620000028
representing the desired output signal ydThe first derivative of (a);
Figure BDA0003018617620000029
and
Figure BDA00030186176200000210
respectively representing virtual control functions alpha1And alpha2The first derivative of (a);
Figure BDA00030186176200000211
representing the first derivative of the control input u.
Further, in the present invention, in the fifth step, the obtained virtual isControl function alpha1And alpha2And the system control input u is:
Figure BDA0003018617620000031
Figure BDA0003018617620000032
Figure BDA0003018617620000033
wherein k is1,k2,k3Is a constant number of times, and is,
Figure BDA0003018617620000034
representing the desired output signal ydThe first derivative of (a);
Figure BDA0003018617620000035
is alpha1The first derivative of (a) is,
Figure BDA0003018617620000036
and
Figure BDA0003018617620000037
the outputs of the instruction filter are:
Figure BDA0003018617620000038
Figure BDA0003018617620000039
wherein λ is1，λ2Is a constant number of times, and is,
Figure BDA00030186176200000310
and
Figure BDA00030186176200000311
respectively, the state variables of the instruction filter.
Preferably, in the present invention, k is1，k2And k3Are all greater than 0.
Preferably, in the present invention, λ1And λ2Are all greater than 0.
The invention provides a self-adaptive backstepping control method based on instruction filtering disturbance estimation, which estimates system disturbance by using an instruction filter, and can directly and effectively estimate unknown disturbance of a system, consume less control energy and effectively realize that the system estimates an expected output signal y compared with the traditional self-adaptive backstepping control technology for estimating the disturbance upper bounddThe tracking of (2).
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a graph comparing response curves output by a system using the method of the present invention and a conventional method;
FIG. 3 is a graph comparing tracking error response curves of a system using the method of the present invention and a conventional method;
FIG. 4 is a graph of unknown disturbance versus disturbance estimate using the method of the present invention;
FIG. 5 is a graph of an estimation of unknown disturbance and an upper disturbance bound using a conventional method;
FIG. 6 is a graph of a system control input signal profile using the method of the present invention;
FIG. 7 is a graph of a system control input signal using a conventional method;
FIG. 8 is a graph comparing energy consumption curves of a system employing the method of the present invention and a conventional method;
FIG. 9 is a graph comparing response curves of linear motor systems using the output of the method of the present invention and the conventional method;
FIG. 10 is a graph comparing tracking error response curves for a linear motor system using the method of the present invention and a conventional method;
FIG. 11 is a graph of unknown disturbance and disturbance estimation of a linear motor system using the method of the present invention;
FIG. 12 is a graph of an estimation of unknown disturbance and an upper disturbance bound of a linear motor system using a conventional method;
FIG. 13 is a graph of a control input signal for a linear motor system using the method of the present invention;
FIG. 14 is a graph of a control input signal for a linear motor system using a conventional method;
fig. 15 is a graph comparing power consumption curves of a linear motor system using the method of the present invention and a conventional method.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The first embodiment is as follows: the following describes the present embodiment with reference to fig. 1, where the present embodiment describes an adaptive backstepping control method based on instruction filtering disturbance estimation, the method includes: step one, according to the state variable x of the nonlinear system applied in practice1、x2And the desired output signal ydEstablishing a nonlinear second-order system state space model containing a disturbance term;
step two, establishing a dimensional-expanded nonlinear third-order system state space model according to a nonlinear second-order system state space model containing a disturbance term, and simultaneously setting an error variable z1＝x1-yd，z2＝x2-α1And z3＝x3-α2Wherein α is1And alpha2Representing a virtual control function to be designed, a state variable x3U is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two1，z2And z3Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Figure BDA0003018617620000041
Step five, according to the first derivative of the Lyapunov function
Figure BDA0003018617620000042
Design of virtual control function alpha using backstepping method and instruction filter1And alpha2And a system control input u; obtaining a self-adaptive backstepping controller based on instruction filtering disturbance estimation to realize the purpose of outputting an expected output signal ydThe tracking of (2).
The invention adopts a multi-medium nonlinear system suitable for a motor control system, an instrument control system and the like, and effectively realizes that the system outputs an expected output signal ydThe tracking of the system ensures the accuracy of the system output.
Further, in the invention, in the first step, establishing a nonlinear second-order system state space model containing a disturbance term is as follows:
Figure BDA0003018617620000051
wherein x is1，x2Represents the state variables of a non-linear second-order system,
Figure BDA0003018617620000052
denotes x2B is a constant, f (x)1,x2) Representing the non-linearity of the system for practically known non-linear functions, d (t) representing the disturbance term of a non-linear second-order system, u representing the control input signal of the non-linear second-order system, y representing the output of the non-linear second-order system, the control objective being to design the control input u such that the system output y tracks the desired output signal yd。
Further, in the present embodiment, the constant b is not 0.
Further, in the present embodiment, the nonlinear function f (x)1,x2) Is a local Liphoz continuous function.
Further, in the present embodiment, in the second step, the establishing of the dimension-extended nonlinear third-order system state space model is as follows:
Figure BDA0003018617620000053
wherein,
Figure BDA0003018617620000054
denotes x3The first derivative of (a) is,
Figure BDA0003018617620000055
the first derivative of u is indicated.
Further, in the present embodiment, the error variable z set in the second step is used in the third step1，z2And z3The designed Lyapunov function V is as follows:
Figure BDA0003018617620000056
further, in the fourth step of the present embodiment, the first derivative of the lyapunov function V in the third step with respect to time is calculated as:
Figure BDA0003018617620000057
wherein,
Figure BDA0003018617620000058
representing the desired output signal ydThe first derivative of (a);
Figure BDA0003018617620000059
and
Figure BDA00030186176200000510
respectively representing virtual control functions alpha1And alpha2The first derivative of (a);
Figure BDA00030186176200000511
representing the first derivative of the control input u.
Further, in this embodiment, in step five, the obtained virtual control function α is obtained1And alpha2And the system control input u is:
Figure BDA00030186176200000512
Figure BDA00030186176200000513
Figure BDA0003018617620000061
wherein k is1,k2,k3Is a constant number of times, and is,
Figure BDA0003018617620000062
representing the desired output signal ydThe first derivative of (a);
Figure BDA0003018617620000063
is alpha1The first derivative of (a) is,
Figure BDA0003018617620000064
and
Figure BDA0003018617620000065
the outputs of the instruction filter are:
Figure BDA0003018617620000066
Figure BDA0003018617620000067
wherein λ is1，λ2Is a constant number of times, and is,
Figure BDA0003018617620000068
and
Figure BDA0003018617620000069
respectively, the state variables of the instruction filter.
Further, in the present invention, k is1，k2And k3Are all greater than 0.
Further, in the present embodiment, λ1And λ2Are all greater than 0.
Proving that the control input (7) designed based on the instruction filter can ensure that the tracking error of the system is converged near the origin; the demonstration process is as follows:
defining error variables
Figure BDA00030186176200000610
Selecting the Lyapunov function as
Figure BDA00030186176200000611
To V1The first derivative can be found:
Figure BDA00030186176200000612
wherein delta1，δ2Is a normal number, and is,
Figure BDA00030186176200000613
Figure BDA00030186176200000614
from formula (10):
Figure BDA00030186176200000615
wherein, constant c1And c2Satisfy V1(0)≥c2/c1。
From formula (11):
Figure BDA00030186176200000616
Figure BDA00030186176200000617
the following formulae (5) to (7) and formulae (12) to (13) may be substituted for formula (4):
Figure BDA0003018617620000071
in which ξ1，ξ2Is a normal number, and is,
Figure BDA0003018617620000072
from formula (14):
Figure BDA0003018617620000073
from formula (15):
Figure BDA0003018617620000074
the formula (16) shows that the designed control input (7) can ensure that the tracking error of the system is converged near the origin, and the verification of the method is realized.
Detailed description of the preferred embodiment
Taking the initial value of a nonlinear second-order system state space model (1) as x1(0)＝0.8，x2(0)＝-0.3, and the constant b is 2. To demonstrate the simulation, assume that the known system nonlinear function is
Figure BDA0003018617620000075
The system perturbation term d (t) is 0.5cos0.04t, which is unknown to the designer and cannot be used directly to design the system control input u. The desired output signal of the system is set to yd(t)＝1.2sin(0.5t)。
The parameters in the virtual control functions (5) and (6) and the control input (7) are taken to be k1＝2，k2＝2，k 32; the parameter in the command filters (8) and (9) is lambda1＝20，λ 260; the initial value of the state variable of the instruction filters (8) and (9) is
Figure BDA0003018617620000076
For the purpose of comparing results, the adaptive backstepping control method based on instruction filtering disturbance estimation (hereinafter referred to as the method of the invention) is adopted to compare with the traditional adaptive backstepping control method based on disturbance upper bound estimation (hereinafter referred to as the traditional method).
FIG. 2 is a graph showing a comparison of the output response curves of a system under the action of the method of the present invention and a conventional method, wherein the solid line represents the desired output signal ydThe dashed line is the system output y in the method of the invention, and the dotted line is the system output y in the traditional method; FIG. 3 is a graph comparing the tracking error response curves of the system according to the method of the present invention and the conventional method, wherein the dashed line represents the difference between the output signal of the system according to the method of the present invention and the expected output signal, and the dashed line represents the difference between the output signal of the system according to the conventional method and the expected output signal; FIG. 4 is a graph of the command filtering disturbance estimation and system unknown disturbance under the method of the present invention, in which the dotted line is the system unknown disturbance and the dash-dot line is the disturbance estimation under the method of the present invention; FIG. 5 is a graph illustrating an upper bound estimation of disturbance and an unknown disturbance of a system according to a conventional method, where a dotted line represents an unknown disturbance of the system and a dashed line represents an upper bound estimation of disturbance according to a conventional method; FIG. 6 is a graph of a system control input curve, shown in dashed lines, according to the method of the present invention; FIG. 7 is a graph of system control input curves in a conventional manner, using a dash-dot lineRepresents; FIG. 8 is a graph comparing the control input energy consumption curves of the system according to the present invention and the conventional method, wherein the dashed line represents the control input energy consumption according to the present invention, and the dotted line represents the control input energy consumption according to the conventional method.
Detailed description of the invention
The system (1) can describe the dynamics of a linear motor position control system, and the state space model of the system is as follows:
Figure BDA0003018617620000081
wherein x1(unit m) represents the motor coil position, x2The unit m/s is the motor coil speed, m (unit kg) represents the motor coil mass, u (unit V) is the motor control voltage, sigma is the viscous friction coefficient, and d (t) is the system disturbance term.
The initial value of the system is x1(0)＝0.8m，x2(0) The mass m of the motor coil is 1.2kg and the viscous friction coefficient sigma is 0.011. To demonstrate the simulation, assuming that the system disturbance term is d (t) ═ 0.2sin0.25t, the function is unknown to the designer and cannot be used directly to design the motor control voltage u. The desired output signal of the system is set to yd＝0.6m。
The parameters in the virtual control functions (5) and (6) and the control input (7) are taken to be b-0.83, k1＝2，k2＝2，k 32; the parameter in the command filters (8) and (9) is lambda1＝20，λ 260; the initial value of the state variable of the instruction filters (8) and (9) is
Figure BDA0003018617620000082
For the purpose of comparing results, the adaptive backstepping control method based on instruction filtering disturbance estimation (hereinafter referred to as the method of the invention) is adopted to compare with the traditional adaptive backstepping control method based on disturbance upper bound estimation (hereinafter referred to as the traditional method).
FIG. 9 is a graph comparing the output response curves of the system under the action of the method of the present invention and the conventional method, wherein the solid line is the expected outputSignal ydThe dashed line is the system output y in the method of the invention, and the dotted line is the system output y in the traditional method; FIG. 10 is a graph of a system tracking error response for the method of the present invention and the conventional method, where the dashed line represents the difference between the system output and the expected output signal for the method of the present invention and the dashed line represents the difference between the system output and the expected output signal for the conventional method; FIG. 11 is a graph of the command filtering disturbance estimation and system unknown disturbance according to the method of the present invention, in which the dotted line is the system unknown disturbance and the dash-dot line is the disturbance estimation value according to the method of the present invention; FIG. 12 is a graph illustrating an upper bound estimation of disturbance and an unknown disturbance of a system according to a conventional method, where a dotted line represents an unknown disturbance of the system and a dashed line represents an upper bound estimation of disturbance according to a conventional method; FIG. 13 is a graph of system control input curves, shown in dashed lines, according to the method of the present invention; FIG. 14 is a graph of system control inputs for a conventional method, shown in phantom; FIG. 15 is a graph comparing the control input energy consumption curves of the system according to the present invention and the conventional method, wherein the dashed line represents the control input energy consumption according to the present invention, and the dotted line represents the control input energy consumption according to the conventional method.
And conclusion one: as can be seen from fig. 4, 5, 11 and 12, the conventional method can only obtain the estimation value of the upper bound of the disturbance, whereas the method of the present invention can directly obtain the estimation value of the disturbance by using the instruction filter.
And a second conclusion: as can be seen from fig. 8 and 15, the system control input energy consumption under the method of the present invention is smaller than that of the conventional method.
The above-described calculation examples of the present invention are merely to explain the calculation model and the calculation flow of the present invention in detail, and are not intended to limit the embodiments of the present invention. It will be apparent to those skilled in the art that other variations and modifications of the present invention can be made based on the above description, and it is not intended to be exhaustive or to limit the invention to the precise form disclosed, and all such modifications and variations are possible and contemplated as falling within the scope of the invention.

Claims (10)

1. An adaptive backstepping control method based on instruction filtering disturbance estimation is characterized by comprising the following steps:
step one, according to the state variable x of the nonlinear system applied in practice1、x2And the desired output signal ydEstablishing a nonlinear second-order system state space model containing a disturbance term;
step two, establishing a dimensional-expanded nonlinear third-order system state space model according to a nonlinear second-order system state space model containing a disturbance term, and simultaneously setting an error variable z1＝x1-yd，z2＝x2-α1And z3＝x3-α2Wherein α is1And alpha2Representing a virtual control function to be designed, a state variable x3U is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two1，z2And z3Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Figure FDA0003018617610000011
Step five, according to the first derivative of the Lyapunov function
Figure FDA0003018617610000012
Design of virtual control function alpha using backstepping method and instruction filter1And alpha2And a system control input u; obtaining a self-adaptive backstepping controller based on instruction filtering disturbance estimation to realize the purpose of outputting an expected output signal ydThe tracking of (2).
2. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 1, wherein in the first step, the establishment of the nonlinear second-order system state space model containing the disturbance term comprises:
Figure FDA0003018617610000013
wherein x is1，x2Represents the state variables of a non-linear second-order system,
Figure FDA0003018617610000014
denotes x2B is a constant, f (x)1,x2) Representing the non-linearity of the system for practically known non-linear functions, d (t) representing the disturbance term of a non-linear second-order system, u representing the control input signal of the non-linear second-order system, y representing the output of the non-linear second-order system, the control objective being to design the control input u such that the system output y tracks the desired output signal yd。
3. The adaptive backstepping control method based on command filter disturbance estimation according to claim 2, wherein the constant b is not 0.
4. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 3, characterized in that the nonlinear function f (x)1,x2) Is a local Liphoz continuous function.
5. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 1, wherein in the second step, the establishment of the dimensional-expanded nonlinear third-order system state space model is as follows:
Figure FDA0003018617610000021
wherein,
Figure FDA0003018617610000022
denotes x3The first derivative of (a) is,
Figure FDA0003018617610000023
the first derivative of u is indicated.
6. The adaptive backstepping control method based on command filter disturbance estimation as claimed in claim 1, wherein the error variable z set in step two is utilized in step three1，z2And z3The designed Lyapunov function V is as follows:
Figure FDA0003018617610000024
7. the adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 6, wherein in step four, the first derivative of time on the Lyapunov function V in step three is:
Figure FDA0003018617610000025
wherein,
Figure FDA0003018617610000026
representing the desired output signal ydThe first derivative of (a);
Figure FDA0003018617610000027
and
Figure FDA0003018617610000028
respectively representing virtual control functions alpha1And alpha2The first derivative of (a);
Figure FDA0003018617610000029
representing the first derivative of the control input u.
8. Instruction-based filtering disturbance estimation according to claim 7The self-adaptive backstepping control method of the meter is characterized in that in the fifth step, the obtained virtual control function alpha1And alpha2And the system control input u is:
Figure FDA00030186176100000210
Figure FDA00030186176100000211
Figure FDA00030186176100000212
wherein k is1,k2,k3Is a constant number of times, and is,
Figure FDA00030186176100000213
representing the desired output signal ydThe first derivative of (a);
Figure FDA00030186176100000214
is alpha1The first derivative of (a) is,
Figure FDA00030186176100000215
and
Figure FDA00030186176100000216
the outputs of the instruction filter are:
Figure FDA00030186176100000217
Figure FDA00030186176100000218
wherein λ is1，λ2Is a constant number of times, and is,
Figure FDA00030186176100000219
and
Figure FDA00030186176100000220
respectively, the state variables of the instruction filter.
9. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 8, characterized in that k is1，k2And k3Are all greater than 0.
10. The adaptive backstepping control method based on command filter disturbance estimation according to claim 8, wherein λ1And λ2Are all greater than 0.
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CN114779628B (en) * 2022-03-16 2024-05-24 哈尔滨工业大学 Active disturbance rejection motion control method based on RBF and multi-mode switching mechanism
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CN114937993A (en) * 2022-06-22 2022-08-23 福州大学 State estimation method of remote terminal equipment of power system based on enhanced filter algorithm

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