CN112130584A - Finite time self-adaptive control method of four-rotor aircraft based on command filtering - Google Patents
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Abstract
The invention relates to a finite time self-adaptive control method of a four-rotor aircraft based on command filtering, aiming at the four-rotor aircraft with unknown nonlinear dynamics and external disturbance, a position and attitude trajectory tracking controller is designed by using a finite time command filtering backstepping method, so that the four-rotor aircraft can be quickly and accurately controlled; a finite time command filter is introduced to realize the rapid approximation of the derivative of the virtual control signal, thereby effectively avoiding the problem of dimension explosion; a new fractional order error compensation mechanism is designed to quickly remove the influence of filtering errors, so that the control performance of the four-rotor aircraft is further improved; strictly proving that all signals in a closed-loop system are bounded in limited time by using a finite time stability theory, and the position and attitude tracking errors are converged to a neighborhood near an origin in the limited time; the effectiveness of the control scheme was verified by a simulation comparative example.
Description
Technical Field
The invention relates to a finite time self-adaptive control method of a four-rotor aircraft based on command filtering.
Background
The four-rotor aircraft has the characteristics of simple structure, high deployment efficiency, flexible control and the like, is widely concerned by researchers, and is widely applied to the fields of aerial photography, intelligent transportation, urban fire protection, cargo transportation and the like. However, the problems of parameter uncertainty, underactuation, strong coupling characteristics and the like exist in a four-rotor aircraft system, and how to design and realize high-quality flight is a challenging problem in the control field.
In order to improve the control performance of a four-rotor aircraft, a great deal of research has been carried out by some scholars and various effective nonlinear control algorithms have been proposed. When the aircraft is influenced by parameter uncertainty and air resistance, part of scholars provide a control algorithm of the four-rotor aircraft by using a sliding mode control technology; but the control algorithm has the phenomenon that a discontinuous switch control item is easy to generate buffeting.
Fortunately, the backstepping design method has significant advantages in the controller design of the structural uncertainty system, and in recent years, some researchers have succeeded in applying the backstepping method to the four-rotor flight controller design and have achieved results. However, when a control algorithm of the four-rotor aircraft is designed by adopting a backstepping recursion method, the virtual control signal needs to be repeatedly derived, and the problem of dimension explosion is easily caused. To solve this problem, a dynamic plane control technique and a command filter back-stepping method are successively proposed. Some scholars give their dynamic surface flight control algorithms for aircraft with lumped unknown non-linearities. Based on the dynamic surface control technology, the four-rotor aircraft cooperative fault-tolerant control is also researched. In essence, both the dynamic surface control technology and the command filtering step-back method utilize a filter to obtain the derivative of a virtual control signal, so as to reduce the computational complexity, but compared with the former method, only first-order filtering is considered, and an error compensation mechanism is also introduced into the command filtering step-back method to remove the influence of a filtering error on the control performance, so as to obtain better control performance.
It is worth noting that the above control scheme only guarantees progressive convergence and does not allow limited time tracking control of a quad-rotor aircraft. Considering that the finite time control has the advantages of high convergence rate, high tracking precision, strong robustness and the like, the research on the finite time control of the four-rotor aircraft has important practical significance.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a finite time self-adaptive control method of a four-rotor aircraft based on command filtering.
The purpose of the invention is realized by the following technical scheme:
a four-rotor aircraft finite time self-adaptive control method based on command filtering is characterized in that: the method comprises the following steps:
establishing a four-rotor aircraft dynamic model:
wherein x, y and z are the positions of the four-rotor aircraft in an inertial coordinate system; phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle; m is the mass of the organism; g is the acceleration of gravity; l is the distance from the center of mass of the machine body to the rotating shaft of the motor; j. the design is a squarex,Jy,JzThe rotational inertia of the four-rotor aircraft about three axes x, y and z is respectively; g(·)Is the air resistance coefficient of the system; d(·)Is the external disturbance to the system;
the following state variables are defined The simplified model of quadrotor aircraft dynamics is then as follows:
(II) designing a controller:
first, a tracking error variable is defined:
χi+1=Ξi+1-Λi+1,c (4)
wherein i is 1, 3, 5, 7, 9, 11,is xiiCorresponding reference signal, [ y ]1,y2,y3,y4,y5,y6]=[φd,θd,ψd,zd,xd,yd];Λi+1,cIs a virtual control signal ΛiAs a filtered output signal at the filtering input, wherein the finite time command filtering is in the form:
wherein phiiAnd phii+1Is a state variable; a isi,1,ai,2And eiFor design parameters;Λi+1,c=φi,For the finite time command filtering shown in equation (5), there is a constantAnd ρ > 0 such that the following equation holds
defining a tracking error compensation variable:
κi=χi-ηi (7)
κi+1=χi+1-ηi+1 (8)
wherein etai,ηi+1To compensate the signal;
2.1) design attitude controller
The attitude subsystem is divided into a roll angle subsystem, a pitch angle subsystem and a yaw angle subsystem, and a controller is designed for each subsystem to realize attitude tracking control of the four-rotor aircraft;
for the attitude sub-system (i ═ 1, 3, 5)
Firstly, the following virtual control signals and controllers are designed:
wherein c isi,ci+1,si,si+1,liIs a normal number; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;
is a vector of basis functions of the fuzzy logic system;is an unknown constantWherein, inIs a weight vector for the fuzzy logic system,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
For eliminating filtering error lambdai+1,c-ΛiThe following fractional order error compensation signal is introduced:
wherein constant hi,hi+1Greater than 0, ηi(0)=ηi+1(0)=0;
2.2) design of the position controller
The position subsystem is divided into a z-height subsystem, an x-position subsystem and a y-position subsystem, and a controller is designed for each subsystem, so that the position trajectory tracking control of the four-rotor aircraft is realized;
for the location subsystem (i ═ 7, 9, 11)
The virtual control signals and the controller are designed as follows:
wherein constant ci,ci+1,si,si+1,liGreater than 0; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;is a vector of basis functions of the fuzzy logic system;is an unknown constantWherein, inIs a weight vector for the fuzzy logic system,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
the following fractional order error compensation signal is constructed:
wherein h isi,hi+1Is a normal number, ηi(0)=ηi+1(0)=0;
2.3) inverse solution of the desired Signal
Control input using position subsystemInverse solution to obtain the information [ phi ] needed by the attitude subsystemd,θd]Thereby realizing the four-rotor aircraft tracking reference signal [ x ]d,yd,zd,ψd]Meanwhile, the roll angle and the pitch angle are automatically stabilized;
according to the simplified model formula (2) of the dynamics of the four-rotor aircraft, it can be known that:
further obtain
Further, the finite time adaptive control method of the four-rotor aircraft based on command filtering further comprises:
(III) stability analysis:
according to the designed virtual control signal, the controller, the parameter update rate and the error compensation signal, the finite time stability theory analysis is utilized to prove that all signals of the closed-loop system are bounded within finite time;
step1, for i 1, 3, 5, 7, 9, 11, according to formula (3), formula (4) and formula (7), for κiDerived by derivation
Step2 based on equations (2), (4) andequation (8), for κi+1Derived to obtain
Model uncertainty termUsing fuzzy logic systems for unknown non-linear functionsTo pairIt approaches, for any givenExist of
Scaled according to an inequality
Step3, selecting a Lyapunov function asAccording to the formula (30) andthe derivative of V may be arranged as
By scaling according to the inequalities and combining equation (6)
Equation (31) can be converted to equation (32) to equation (34)
Further obtain the
or
According to equation (37), ifThenThus, kappa can be obtainedi,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
According to the formula (38), whenThenLet us know thati,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
As can be seen from the equations (39) and (41), when i is 1, 3, 5, 7, 9, 11, κiAnd ηiWill eventually converge to the following set
Convergence time of
When T is more than or equal to T, the compound can be obtained
Therefore, χiConverge to a neighborhood near the origin in a finite time, and all signals in a closed loop system are bounded in finite time.
Compared with the prior art, the invention has obvious advantages and beneficial effects, and is embodied in the following aspects:
aiming at a position subsystem and an attitude subsystem of a four-rotor aircraft, a position and attitude trajectory tracking controller is designed by respectively using a finite time command filtering backstepping method, so that the four-rotor aircraft is quickly and accurately controlled; the finite time command filtering can not only quickly approximate the derivative of the virtual control signal, but also effectively avoid buffeting caused by a sign function and further weaken the limit condition of the virtual control signal;
introducing finite time command filtering, designing a fractional order error compensation mechanism based on non-smooth signals and combining a reverse step design method, and providing a four-rotor aircraft finite time control technical scheme, wherein a designed controller ensures that all signals in a closed loop system are bounded within finite time, and position and attitude tracking errors are converged into the vicinity of an origin within the finite time;
thirdly, the invention is based on the fractional order error compensation mechanism of the non-smooth signal, thereby ensuring that the influence of the filtering error is quickly compensated and having more effectiveness in practical application;
and fourthly, the obvious superiority of the finite time control scheme provided by the invention can be verified through a simulation comparison experiment.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by the practice of the invention. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are required to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and for those skilled in the art, other related drawings can be obtained according to the drawings without inventive efforts.
FIG. 1: the flow chart of the control method of the invention is shown schematically;
FIG. 2: a two-dimensional tracking curve graph of an actual gesture track and an expected signal;
FIG. 3: two-dimensional tracking curve graphs of the actual track and the expected track of the position;
FIG. 4: an attitude trajectory tracking error curve graph;
FIG. 5: position trajectory tracking error plot.
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. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the invention without making creative efforts, shall fall within the protection scope of the invention.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, it need not be further defined and explained in subsequent figures. Meanwhile, in the description of the present invention, the directional terms and the sequence terms, etc. are used only for distinguishing the description, and are not to be construed as indicating or implying relative importance.
Aiming at the problem of trajectory tracking control of a four-rotor aircraft with parameter uncertainty and external interference, the invention provides a finite time self-adaptive control method based on command filtering, and by introducing a finite time command filter, the rapid approximation of a virtual control signal derivative is realized, so that the problem of dimension explosion existing in the traditional backstepping design method is effectively avoided; a new fractional order error compensation mechanism is designed to remove the influence of filtering errors, so that the control performance is further improved; strictly proving that all signals in a closed-loop system are bounded in limited time by using a finite time stability theory, and the position and attitude tracking errors are converged to a neighborhood near an origin in the limited time; the effectiveness of the control scheme was verified by a simulation comparative example.
As shown in fig. 1, the finite time adaptive control method of the quad-rotor aircraft based on command filtering specifically includes the following steps:
establishing a four-rotor aircraft dynamic model:
wherein x, y and z are the positions of the four-rotor aircraft in an inertial coordinate system; phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle; m is the mass of the organism; g is the acceleration of gravity; l is the distance from the center of mass of the machine body to the rotating shaft of the motor; j. the design is a squarex,Jy,JzThe rotational inertia of the four-rotor aircraft about three axes x, y and z is respectively; g(·)Is the air resistance coefficient of the system; d(·)Is the external disturbance to the system;
the following state variables are defined The simplified model of quadrotor aircraft dynamics is then as follows:
(II) designing a controller:
first, a tracking error variable is defined:
χi+1=Ξi+1-Λi+1,c (4)
wherein i is 1, 3, 5, 7, 9, 11,is xiiCorresponding reference signal, [ y ]1,y2,y3,y4,y5,y6]=[φd,θd,ψd,zd,xd,yd];Λi+1,cIs a virtual control signal ΛiAs a filtered output signal at the filtering input, wherein the finite time command filtering is in the form:
wherein phiiAnd phii+1Is a state variable; a isi,1,ai,2And eiIs a design parameter; lambdai+1,c=φi,For the finite time command filtering shown in equation (5), there is a constantAnd ρ > 0 such that the following equation holds
defining a tracking error compensation variable:
κi=χi-ηi (7)
κi+1=χi+1-ηi+1 (8)
wherein etai,ηi+1To compensate the signal;
2.1) design attitude controller
The attitude subsystem is divided into a roll angle subsystem, a pitch angle subsystem and a yaw angle subsystem, and a controller is designed for each subsystem to realize attitude tracking control of the four-rotor aircraft;
for the attitude sub-system (i ═ 1, 3, 5)
Firstly, the following virtual control signals and controllers are designed:
wherein c isi,ci+1,si,si+1,liIs a normal number; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;is a vector of basis functions of the fuzzy logic system;is an unknown constantWherein, inIs a weight vector for the fuzzy logic system,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
For eliminating filtering error lambdai+1,c-ΛiThe following fractional order error compensation signal is introduced:
wherein constant hi,hi+1Greater than 0, ηi(0)=ηi+1(0)=0;
2.2) design of the position controller
The position subsystem is divided into a z-height subsystem, an x-position subsystem and a y-position subsystem, and a controller is designed for each subsystem, so that the position trajectory tracking control of the four-rotor aircraft is realized;
for the location subsystem (i ═ 7, 9, 11)
The virtual control signals and the controller are designed as follows:
wherein constant ci,ci+1,si,si+1,liGreater than 0; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;is a vector of basis functions of the fuzzy logic system;is an unknown constantWherein, inIs a weight vector for the fuzzy logic system,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
the following fractional order error compensation signal is constructed:
wherein h isi,hi+1Is a normal number, ηi(0)=ηi+1(0)=0;
2.3) inverse solution of the desired Signal
Control input using position subsystemInverse solution to obtain the information [ phi ] needed by the attitude subsystemd,θd]Thereby realizing the four-rotor aircraft tracking reference signal [ x ]d,yd,zd,ψd]Meanwhile, the roll angle and the pitch angle are automatically stabilized;
according to the simplified model formula (2) of the dynamics of the four-rotor aircraft, it can be known that:
further obtain
(III) stability analysis:
according to the designed virtual control signal, the controller, the parameter update rate and the error compensation signal, the finite time stability theory analysis is utilized to prove that all signals of the closed-loop system are bounded within finite time;
step1, for i 1, 3, 5, 7, 9, 11, according to formula (3), formula (4) and formula (7), for κiDerived by derivation
Step2 based on equation (2), equation (4) and equation (8), for kappai+1Derived to obtain
Model uncertainty termUsing fuzzy logic systems for unknown non-linear functionsTo pairIt approaches, for any givenExist of
Scaled according to an inequality
Selecting Lyapunov function ofTo Vi+1The derivation is obtained by combining equations (25) to (28):
Step3, selecting a Lyapunov function asAccording to the formula (30) andthe derivative of V may be arranged as
By scaling according to the inequalities and combining equation (6)
Equation (31) can be converted to equation (32) to equation (34)
Further obtain the
or
According to equation (37), ifThenThus, kappa can be obtainedi,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
According to the formula (38), whenThenLet us know thati,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
As can be seen from the equations (39) and (41), when i is 1, 3, 5, 7, 9, 11, κiAnd ηiWill eventually converge to the following set
Convergence time of
When T is more than or equal to T, the compound can be obtained
Therefore, χiConverge to a neighborhood near the origin in a finite time, and all signals in a closed loop system are bounded in finite time.
(IV) simulation result and analysis:
the Matlab/Simulink software is used for carrying out simulation verification on the proposed finite time control scheme, and the model parameters of the four-rotor aircraft are selected as follows:
m=2kg,g=9.8m/s2,l=0.325m,
Jx=0.082kg·m2,Jy=0.082kg·m2,
Jz=0.149kg·m2.
Gx=Gy=Gz=0.6kg/s,
Gφ=Gθ=Gψ=0.6kg/rad.
in the simulation, the desired reference signal for the quad-rotor aircraft was set toThe external interference is selected asThe initial conditions of the system are [ phi (0), theta (0), psi (0), x (0), y (0), z (0)]=[0,0,0,1,0,0](ii) a The control design parameters are selected as follows:
∈1=∈3=∈5=5×10-4,
∈7=∈9=∈11=2.5×10-3.
c2i-1=0.6,c2i=0.8,li=mi=2,
s2i-1=h2i-1=0.8,s2i=h2i=1.2,
ri=0.8,ai,1=8,ai,2=5,i=1,...,6.
and further, a finite time self-adaptive tracking control simulation result of the four-rotor aircraft can be obtained.
Simulation results of the four-rotor aircraft of the finite time command filtering backstepping method and the traditional command filtering backstepping method are shown in fig. 2-5, fig. 2 illustrates a two-dimensional tracking curve of an actual attitude track and an expected signal, fig. 3 illustrates a two-dimensional tracking curve of an actual position track and an expected position track, fig. 4 illustrates an attitude track tracking error curve, and fig. 5 illustrates a position track tracking error curve. It can be seen that the tracking error of the limited time tracking control of the present invention is not only smaller than that of the asymptotic tracking control, but also has a faster convergence rate, and the tracking error is well maintained at a smaller degree.
In conclusion, the position and attitude trajectory tracking controller is designed by using a finite time command filtering backstepping method aiming at the position subsystem and the attitude subsystem of the four-rotor aircraft, so that the four-rotor aircraft can be quickly and accurately controlled; the finite time command filtering can not only quickly approximate the derivative of the virtual control signal, but also effectively avoid buffeting caused by a sign function and further weaken the limit condition of the virtual control signal; the method is based on a fractional order error compensation mechanism of the non-smooth signal, ensures that the influence of the filtering error is quickly compensated, and has higher effectiveness in practical application.
Finite time command filtering is introduced, a fractional order error compensation mechanism based on non-smooth signals is designed, a backstepping design method is combined, and a four-rotor aircraft finite time control technical scheme is provided; the significant superiority of the limited time control scheme provided by the invention can be verified through simulation comparison experiments.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention. It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, it need not be further defined and explained in subsequent figures.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present invention, and shall be covered by the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
It is noted that, herein, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in a process, method, article, or apparatus that comprises the element.
Claims (2)
1. Four-rotor aircraft finite time self-adaptive control method based on command filtering is characterized in that: the method comprises the following steps:
establishing a four-rotor aircraft dynamic model:
wherein x, y and z are the positions of the four-rotor aircraft in an inertial coordinate system; phi, theta and psi are respectively a roll angle, a pitch angle and a yaw angle; m is the mass of the organism; g is the acceleration of gravity; l is the distance from the center of mass of the machine body to the rotating shaft of the motor; j. the design is a squarex,Jy,JzThe rotational inertia of the four-rotor aircraft about three axes x, y and z is respectively; g(·)Is the air resistance coefficient of the system; d(·)Is the external disturbance to the system; tau isF,τφ,τθ,τψIs a control input;
the following state variables are defined The simplified model of quadrotor aircraft dynamics is then as follows:
wherein Indicating external disturbance to the four-rotor aircraft, [ tau ]1,τ2,τ3,τ4]=[τφ,τθ,τψ,τF]And τ5=cosφsinθcosψ+sinφsinψ,τ6=cosφsinθsinψ-sinφcosψ;
(II) designing a controller:
first, a tracking error variable is defined:
χi+1=Ξi+1-Λi+1,c (4)
wherein i is 1, 3, 5, 7, 9, 11,is xiiCorresponding reference signal, [ y ]1,y2,y3,y4,y5,y6]=[φd,θd,ψd,zd,xd,yd];Λi+1,cIs a virtual control signal ΛiAs a filtered output signal at the filtering input, wherein the finite time command filtering is in the form:
wherein phiiAnd phii+1Is a state variable; a isi,1,ai,2And eiIs a design parameter; lambdai+1,c=φi,For the finite time command filtering shown in equation (5), the constants τ > 0 and ρ > 0 exist such that the following holds
Wherein O isi(∈ρτ) Is represented byiPhi and phii+1The degree of approximation therebetween;
defining a tracking error compensation variable:
κi=χi-ηi (7)
κi+1=χi+1-ηi+1 (8)
wherein etai,ηi+1To compensate the signal;
2.1) design attitude controller
The attitude subsystem is divided into a roll angle subsystem, a pitch angle subsystem and a yaw angle subsystem, and a controller is designed for each subsystem to realize attitude tracking control of the four-rotor aircraft;
for the attitude sub-system (i ═ 1, 3, 5)
Firstly, the following virtual control signals and controllers are designed:
wherein c isi,ci+1,si,si+1,liIs a normal number; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;being basis functions of fuzzy logic systemsVector quantity;is an unknown constantWherein, inIs a weight vector for the fuzzy logic system,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
For eliminating filtering error lambdai+1,c-ΛiThe following fractional order error compensation signal is introduced:
wherein constant hi,hi+1Greater than 0, ηi(0)=ηi+1(0)=0;
2.2) design of the position controller
The position subsystem is divided into a z-height subsystem, an x-position subsystem and a y-position subsystem, and a controller is designed for each subsystem, so that the position trajectory tracking control of the four-rotor aircraft is realized;
for the location subsystem (i ═ 7, 9, 11)
The virtual control signals and the controller are designed as follows:
wherein constant ci,ci+1,si,si+1,liGreater than 0; 1/2 < gamma ═ gamma1/γ2<1,γ1,γ2Is positive odd;is a vector of basis functions of the fuzzy logic system;is an unknown constantWherein, inAs a fuzzy logic systemThe weight vector of (2) is calculated,estimating an error for the parameter; rate of parameter updateThe structure is as follows:
the following fractional order error compensation signal is constructed:
wherein h isi,hi+1Is a normal number, ηi(0)=ηi+1(0)=0;
2.3) inverse solution of the desired Signal
Using control input [ tau ] of the position subsystem5,τ6]Inverse solution to obtain the information [ phi ] needed by the attitude subsystemd,θd]Thereby realizing the four-rotor aircraft tracking reference signal [ x ]d,yd,zd,ψd]Meanwhile, the roll angle and the pitch angle are automatically stabilized;
according to the simplified model formula (2) of the dynamics of the four-rotor aircraft, it can be known that:
further obtain
2. The command-filtering-based finite-time adaptive control method for a quad-rotor aircraft according to claim 1, wherein: further comprising:
(III) stability analysis:
according to the designed virtual control signal, the controller, the parameter update rate and the error compensation signal, the finite time stability theory analysis is utilized to prove that all signals of the closed-loop system are bounded within finite time;
step1: for i ═ 1, 3, 5, 7, 9, 11, according to formula (3), formula (4) and formula (7), for κiDerived by derivation
Step2: based on formula (2), formula (4) and formula (8), for ki+1Derived to obtain
Model uncertainty termUsing fuzzy logic systems for unknown non-linear functionsTo pairIt approaches, for any givenExist of
Scaled according to an inequality
Step3: choosing Lyapunov function asAccording to the formula (30) andthe derivative of V may be arranged as
By scaling according to the inequalities and combining equation (6)
Equation (31) can be converted to equation (32) to equation (34)
Further obtain the
or
According to equation (37), ifThenThereby obtaining ki,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
According to the formula (38), whenThenLet us know thati,ηiAndat a finite time TrInner convergence to the following set
Convergence time TrIs composed of
As can be seen from the equations (39) and (41), when i is 1, 3, 5, 7, 9, 11, κiAnd ηiWill eventually converge to the following set
Convergence time of
When T is more than or equal to T, the compound can be obtained
Therefore, χiConverge to a neighborhood near the origin in a finite time, and all signals in a closed loop system are bounded in finite time.
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