CN113093553B - Adaptive backstepping control method based on instruction filtering disturbance estimation - Google Patents
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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
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 CN111679582A. 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 the strategy directly estimates the upper bound of the disturbance, the designed controller is too conservative and energy-consuming.
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 of practical application 1 、x 2 And the desired output signal y d Establishing 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 z 1 ＝x 1 -y d ，z 2 ＝x 2 -α 1 And z 3 ＝x 3 -α 2 Wherein α is 1 And alpha 2 Representing a virtual control function to be designed, a state variable x 3 U, u is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two 1 ，z 2 And z 3 Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Step five, according to the first derivative of the Lyapunov function
Design of virtual control function alpha using backstepping method and instruction filter 1 And alpha 2 And 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 y d The 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:
wherein x is 1 ，x 2 Representing the state variables of a non-linear second-order system,
denotes x 2 B is a constant, f (x) 1 ,x 2 ) For a practically known non-linear function representing the non-linearity of the system, d (t) represents the disturbance term of the non-linear second-order system, u represents the control input signal of the non-linear second-order system, y represents 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 y d 。
Preferably, in the present invention, the constant b is not 0.
Preferably, in the present invention, the nonlinear function f (x) 1 ,x 2 ) 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:
wherein the content of the first and second substances,
denotes x 3 The first derivative of (a) is,
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 three 1 ，z 2 And z 3 The designed Lyapunov function V is as follows:
further, in the fourth step, the first derivative of the lyapunov function V in the third step with respect to time is:
wherein the content of the first and second substances,
representing the desired output signal y d The first derivative of (a);
and
respectively representing virtual control functions alpha 1 And alpha 2 The first derivative of (a);
representing the first derivative of the control input u.
Further, in the present invention, in the fifth step, the obtained virtual control function α 1 And alpha 2 And the system control input u is:
wherein k is 1 ,k 2 ,k 3 Is a constant number of times, and is,
representing the desired output signal y d The first derivative of (a);
is alpha 1 The first derivative of (a) is,
and
the outputs of the instruction filters are:
wherein λ is 1 ，λ 2 Is a constant number of times, and is,
and
respectively, the state variables of the instruction filter.
Preferably, in the present invention, k is 1 ，k 2 And k 3 Are both greater than 0.
Preferably, in the present invention, λ 1 And λ 2 Are both 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 bound d The 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 unknown disturbance and a disturbance upper bound estimation using a conventional method;
FIG. 6 is a graph of a system control input signal when the method of the present invention is employed;
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 unknown disturbance and an upper-bound disturbance estimation curve of a linear motor system in 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 of the present invention 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 practice 1 、x 2 And the desired output signal y d Establishing 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 z 1 ＝x 1 -y d ，z 2 ＝x 2 -α 1 And z 3 ＝x 3 -α 2 Wherein α is 1 And alpha 2 Representing a virtual control function to be designed, a state variable x 3 U, u is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two 1 ，z 2 And z 3 Designing a Lyapunov function V;
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Step five, according to the first derivative of the Lyapunov function
Design of virtual control function alpha using backstepping method and instruction filter 1 And alpha 2 And a system control input u; obtaining a self-adaptive backstepping controller based on instruction filtering disturbance estimation to realize the desired output signal y d The 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 the systemSystem to expected output signal y d The 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:
wherein x is 1 ，x 2 Represents the state variables of a non-linear second-order system,
denotes x 2 B is a constant, f (x) 1 ,x 2 ) For a practically known non-linear function representing the non-linearity of the system, d (t) represents the disturbance term of the non-linear second-order system, u represents the control input signal of the non-linear second-order system, y represents 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 y d 。
Further, in the present embodiment, the constant b is not 0.
Further, in the present embodiment, the nonlinear function f (x) 1 ,x 2 ) 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:
wherein, the first and the second end of the pipe are connected with each other,
denotes x 3 The first derivative of (a) is,
the first derivative of u is indicated.
Further, the instant foodIn the embodiment, the error variable z set in the second step is used in the third step 1 ，z 2 And z 3 The designed Lyapunov function V is as follows:
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:
wherein the content of the first and second substances,
representing the desired output signal y d The first derivative of (a);
and
respectively representing virtual control functions alpha 1 And alpha 2 The first derivative of (a);
representing the first derivative of the control input u.
Further, in this embodiment, in step five, the obtained virtual control function α is obtained 1 And alpha 2 And the system control input u is:
wherein k is 1 ,k 2 ,k 3 Is a constant number of times, and is,
representing the desired output signal y d The first derivative of (a);
is alpha 1 The first derivative of (a) is,
and
the outputs of the instruction filter are:
wherein λ is 1 ，λ 2 Is a constant number of times, and is,
and
respectively, the state variables of the instruction filter.
Further, in the present invention, k is 1 ，k 2 And k 3 Are both greater than 0.
Further, in the present embodiment, λ 1 And λ 2 Are both 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 an error variable
Selecting the Lyapunov function as
To V 1 The first derivative can be found:
wherein delta 1 ，δ 2 Is a normal number of the blood vessel which is,
from formula (10):
wherein, constant c 1 And c 2 Satisfy V 1 (0)≥c 2 /c 1 。
From formula (11):
the following formulae (5) to (7) and formulae (12) to (13) may be substituted for formula (4):
xi therein 1 ，ξ 2 Is a normal number, and is,
from formula (14):
from formula (15):
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 x 1 (0)＝0.8，x 2 (0) = 0.3, constant b =2. To demonstrate the simulation, assume that the known system nonlinear function is
The system disturbance term d (t) =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 y d (t)＝1.2sin(0.5t)。
The parameters in the virtual control functions (5) and (6) and the control input (7) are taken to be k 1 ＝2，k 2 ＝2，k 3 =2; the parameter in the instruction filters (8) and (9) is lambda 1 ＝20，λ 2 =60; the initial value of the state variable of the instruction filters (8) and (9) is
For comparing results, the adaptive backstepping control method based on instruction filtering disturbance estimation (hereinafter referred to as the method of the invention) and the traditional adaptive control method based on disturbance upper bound estimation are adoptedThe comparison should be made by a reverse control method (hereinafter referred to as a conventional 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 y d The 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 the 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 inputs for a conventional method, shown in dotted lines; 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:
wherein x 1 (unit m) represents the motor coil position, x 2 The 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 x 1 (0)＝0.8m，x 2 (0) =0m/s, motor coil mass m =1.2kg,viscous friction coefficient σ =0.011. To demonstrate the simulation, assuming that the system disturbance term is d (t) =0.2sin0.25t, this 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 y d ＝0.6m。
The parameters in the virtual control functions (5) and (6) and the control input (7) are taken as b =0.83 1 ＝2，k 2 ＝2，k 3 =2; the parameter in the command filters (8) and (9) is lambda 1 ＝20，λ 2 =60; the initial value of the state variable of the instruction filters (8) and (9) is
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 method of the present invention and the conventional method, wherein the solid line represents the desired output signal y d The 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, shown in phantom, for a conventional method; FIG. 15 is a graph comparing the consumption curves of the system control input energy in the method of the present invention and the conventional method, wherein the dashed line represents the control input energy in the method of the present inventionThe dotted line controls the input energy consumption under conventional methods.
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 describe 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 (6)
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 of a nonlinear system applied in practice: variable motor coil position x of linear motor position control system 1 Motor coil speed x 2 And the desired output signal y d Establishing a nonlinear second-order system state space model containing a disturbance term;
the method comprises the following steps of establishing a state space model of a nonlinear second-order system containing a disturbance term:
wherein x is 1 ，x 2 Represents the state variables of a non-linear second-order system,
denotes x 2 B is a constant, f(x 1 ,x 2 ) For a practically known non-linear function representing the non-linearity of the system, d (t) represents the disturbance term of the non-linear second-order system, u represents the control input signal of the non-linear second-order system, y represents 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 y d ；
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 z 1 ＝x 1 -y d ，z 2 ＝x 2 -α 1 And z 3 ＝x 3 -α 2 Wherein α is 1 And alpha 2 Representing a virtual control function to be designed, a state variable x 3 = u, u is the system control input to be designed;
step three, utilizing the error variable z obtained in the step two 1 ，z 2 And z 3 Designing a Lyapunov function V;
wherein the designed Lyapunov function V is as follows:
step four, solving the first derivative of the Lyapunov function V in the step three to obtain the first derivative
Wherein, the first derivative of the Lyapunov function V in the third step with respect to time is:
wherein the content of the first and second substances,
representing the desired output signal y d The first derivative of (a);
and
respectively representing virtual control functions alpha 1 And alpha 2 The first derivative of (a);
represents the first derivative of the control input u;
step five, according to the first derivative of the Lyapunov function
Design of virtual control function alpha using backstepping method and instruction filter 1 And alpha 2 And a system control input u; obtaining a self-adaptive backstepping controller based on instruction filtering disturbance estimation to realize the desired output signal y d Tracking of (2);
wherein the obtained virtual control function alpha 1 And alpha 2 And the system control input u is:
wherein k is 1 ,k 2 ,k 3 Is a constant number of times, and is,
indicating a desired output signalNumber y d The first derivative of (a);
is alpha 1 The first derivative of (a) is,
and
the outputs of the instruction filter are:
wherein λ is 1 ，λ 2 Is a constant number of times, and is,
and
respectively, the state variables of the instruction filter.
2. The adaptive backstepping control method based on command filter disturbance estimation according to claim 1, wherein the constant b is not 0.
3. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 2, characterized in that the nonlinear function f (x) 1 ,x 2 ) Is a local Liphoz continuous function.
4. 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:
wherein the content of the first and second substances,
represents x 3 The first derivative of (a) is,
the first derivative of u is indicated.
5. The adaptive backstepping control method based on instruction filtering disturbance estimation according to claim 1, wherein k is 1 ，k 2 And k 3 Are all greater than 0.
6. The adaptive backstepping control method based on command filter disturbance estimation according to claim 1, wherein λ 1 And λ 2 Are all greater than 0.
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