CN109557933A - A kind of rigid aircraft state constraint control method based on imperial Burger observer - Google Patents
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Abstract
A kind of rigid aircraft state constraint control method based on imperial Burger observer proposes that imperial Burger observer estimates unknown state amount, there is no need to know the angular speed of aircraft for there are external disturbance and without the rigid aircraft of angular velocity measurement.Having used realizes state constraint with the modified obstacle liapunov function of unconstrained situation suitable for constraining, and devises rigid aircraft state constraint control method in conjunction with Reverse Step Control.The present invention external interference and and in the case where without angular velocity measurement, ensure that attitude of flight vehicle observation error and tracking error can reach uniform ultimate bounded, and state variable suffers restraints.
Description
Technical field
The present invention relates to a kind of rigid aircraft state constraint control methods based on imperial Burger observer, this is to be directed to deposit
In the total state constraint output feedback attitude tracking and controlling method of external disturbance and the rigid aircraft design without angular velocity measurement.
Background technique
Rigid aircraft one kind is non-linear, close coupling, multiple-input and multiple-output complication system, awing have many outsides
The disturbance torque moment affects aircraft, such as radiation torque, gravity gradient torque and geomagnetic torque.And in many situations
Under, the angular velocity signal of aircraft may can all lead to the letter of its angular speed containing very big noise or even sensor degradation
It number can not be accurately obtained.Therefore, a kind of attitude control method not depending on angular velocity information has very strong realistic meaning.
And with the raising of the task of execution fining degree, it is inadequate for being solely focused on the stable state accuracy of aircraft.For
The mapping and stability of guarantee system, it will usually which system mode and the amplitude of output are constrained.And it is run in system
In the process, if violating constraint condition, it may result in system performance decline or even safety problem occur.Obstacle Liapunov
Functional based method is a kind of about beam control method, the basic principle is that when variable approaches zone boundary, liapunov function
Value tends to be infinitely great, to guarantee the constraint of variable.Traditional logarithm obstacle liapunov function is not particularly suited for unconstrained
The case where, however modified obstacle liapunov function can but be suitable for constraint and unconstrained situation simultaneously.Use improvement
Variable not only can be constrained in type obstacle liapunov function, can also be effectively improved the transient state and steady-state performance of system.
Backstepping control method is a kind of iterative design control method based on Lyapunov theorem, Feedback Control Laws and Lee
Ya Punuofu function can design together in the process gradually recursive.Backstepping can higher order controller design when by by
Step recurrence reduces controller and sets characteristic meter difficulty.One major advantage of Reverse Step Control is that it can be some useful to avoid eliminating
It is non-linear and realize high-precision control performance.Imperial Burger observer is one proposed by imperial Burger and Kalman and cloth west et al.
Kind state observation, the vehicle rate information that can not be obtained can be estimated using observer, be not necessarily to angle so as to realize
The Design of Feedback Controller of velocity information.
Summary of the invention
In order to overcome the rigid aircraft posture restraint control problem of no angular velocity information, the present invention provides a kind of based on dragon
The rigid aircraft state constraint control method of Burger observer, the case where system is there are external disturbance and without angular velocity information
Under, realize that the posture observation error of rigid aircraft system and tracking error can reach uniform ultimate bounded.
In order to solve the above-mentioned technical problem the technical solution proposed is as follows:
A kind of rigid aircraft state constraint control method based on imperial Burger observer, comprising the following steps:
Step 1, the kinematics and dynamics modeling of the rigid aircraft based on modified discrete chirp-Fourier transform is established, process is such as
Under:
The kinematical equation of 1.1 rigid aircraft systems are as follows:
Wherein σ=[σ1,σ2,σ3]TFor modified discrete chirp-Fourier transform, which depict the posture features of aircraft；It is leading for σ
Number, σTIt is the transposition of σ；ω∈R3It is the angular speed of rigid aircraft；I3It is R3×3Unit matrix；σ×Form are as follows:
G form isIt has property| | G | | it is two models of G
Number；
The kinetics equation of 1.2 rigid aircraft systems are as follows:
Wherein J ∈ R3×3It is the moment of inertia matrix of rigid aircraft；It is the derivative of ω, indicates rigid aircraft
Angular acceleration；u∈R3With d ∈ R3Respectively control moment and external disturbance；ω×Form are as follows:
1.3 rightDerivation simultaneously substitutes into formula (3), obtains
Wherein L=G-1,It is the derivative of L；J-1It is the inverse matrix of J；A×Form are as follows:
D '=GJ-1D and satisfaction | | d ' | |≤dm, wherein dmIt is a normal number；
Step 2, for the rigid aircraft system with external disturbance and without angular velocity measurement, controller, process are designed
It is as follows:
2.1 design imperial Burger state observer, enable x1=[x11,x12,x13]T=σ, Flight
The output of device is y=σ, and formula (1) and (3) are rewritten are as follows:
It enablesThen formula (9) is changed to state space form
Wherein
k1,k2It is two normal numbers；According to Lyapunov theorem, as long as matrix A is Hull dimension thatch matrix, for appointing
Anticipate symmetrical matrix Q, certainly exists a positive definite matrix P and following formula is set up:
ATP+PA=-2Q (12)
The imperial Burger observer form of design is as follows:
Wherein Respectively x1And x2Estimated value,It isDerivative；Being will be in E (x)
Variable replacement x isWhen value；H is the gain matrix of observer, form are as follows:
Wherein h1,h2, δ is normal number；
It enablesDefinitionFor observer observation error, formula is subtracted with formula (10)
(12) it obtains
WhereinIt is xeDerivative,Meet It isTwo norms, M=
[m1,m2,m3]T, mi, i=1,2,3 be normal number constant, | | M | | it is two norms of M, | | xe| | it is xeTwo norms；
2.2 designing controller, dummy variable is defined first:
Wherein σdIt is expectation posture；α is virtual controlling rule, and form is
Whereinkb1It is normal number, meets kb1≥||z1(0)||2, and | | z1(0) | | it is z1Initially
Two norms of value,It is z1Transposition；c1It is normal number；It is σdDerivative；
Controller design are as follows:
Wherein c2It is normal number；kb2It is normal number, meets kb2≥||z2(0)||2, and | | z2
(0) | | it is z2Two norms of initial value,It is z2Transposition；It is the derivative of α；
Step 3, rigid aircraft attitude system stability proves that process is as follows:
3.1 prove the posture observation error and tracking error uniform ultimate bounded of rigid aircraft system, design modified
Obstacle liapunov function is following form:
Wherein ln is natural logrithm；E natural constant；
It is substituted into formula (218) derivation and by formula (12), (14), (17) and (18):
Wherein η is normal number；||H2| | it is H2Two norms；| | P | | it is two norms of P；
Formula (20) abbreviation is obtained:
Whereinλmax(P) be matrix P maximum feature
Value；
Therefore, according to Lyapunov theorem, rigid aircraft posture observation error and tracking error be can be realized
Uniform ultimate bounded；
3.2 proof rigid aircraft quantity of states suffer restraints:
It enablesSolution formula (28) obtains such as lower inequality:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
Wherein V (0) is the output valve of V；
Convolution (19) and (22), obtain
By solving inequality (23), z is obtained1Finally converge to following neighborhood:
By identical derivation, z is obtained2Finally converge to following neighborhood:
Find out from formula (24) and (25), z1And z2Respectively by kb1And kb2Constraint obtain rigidity and fly in conjunction with System describe
All quantity of states of row device all suffer restraints.
The present invention is observed in the case where rigid aircraft is there are external disturbance and without angular velocity measurement in conjunction with imperial Burger
Device, Reverse Step Control method and modified obstacle liapunov function design a kind of rigid aircraft based on imperial Burger observer
State constraint control method realizes high-precision control and the state constraint of system.
Technical concept of the invention are as follows: for there are external interference and without the rigid aircraft of angular velocity measurement, propose dragon
Burger observer estimates unknown state amount, constrains in conjunction with Reverse Step Control and modified obstacle liapunov function design point
Control method, it is final to realize that rigid aircraft posture observation error and tracking error reach uniform ultimate bounded.
Advantages of the present invention are as follows: in the case where system is there are external interference and without angular velocity measurement, realize the sight of system
Uniform ultimate bounded can be reached by surveying error and tracking error, and can guarantee that flight state amount suffers restraints.
Detailed description of the invention
Fig. 1 is rigid aircraft Attitude Tracking effect picture of the invention；
Fig. 2 is rigid aircraft Attitude Tracking error schematic diagram of the invention；
Fig. 3 is that rigid aircraft of the invention controls input torque schematic diagram；
Fig. 4 is rigid aircraft observation error schematic diagram of the invention；
Fig. 5 is control flow schematic diagram of the invention.
Specific embodiment
The present invention will be further described with reference to the accompanying drawing.
Referring to figs. 1 to Fig. 5, a kind of rigid aircraft state constraint control method based on imperial Burger observer, the control
Method processed the following steps are included:
Step 1, the kinematics and dynamics modeling of the rigid aircraft based on modified discrete chirp-Fourier transform is established, process is such as
Under:
The kinematical equation of 1.1 rigid aircraft systems are as follows:
Wherein σ=[σ1,σ2,σ3]TFor modified discrete chirp-Fourier transform, which depict the posture features of aircraft；It is leading for σ
Number, σTIt is the transposition of σ；ω∈R3It is the angular speed of rigid aircraft；I3It is R3×3Unit matrix；σ×Form are as follows:
G form isIt has property| | G | | it is two models of G
Number；
The kinetics equation of 1.2 rigid aircraft systems are as follows:
Wherein J ∈ R3×3It is the moment of inertia matrix of rigid aircraft；It is the derivative of ω, indicates rigid aircraft
Angular acceleration；u∈R3With d ∈ R3Respectively control moment and external disturbance；ω×Form are as follows:
1.3 rightDerivation simultaneously substitutes into formula (3), obtains
Wherein L=G-1,It is the derivative of L；J-1It is the inverse matrix of J；A×Form are as follows:
D '=GJ-1D and satisfaction | | d ' | |≤dm, wherein dmIt is a normal number；
Step 2, for the rigid aircraft system with external disturbance and without angular velocity measurement, controller, process are designed
It is as follows:
2.1 design imperial Burger state observer, enable x1=[x11,x12,x13]T=σ, Flight
The output of device is y=σ, and formula (1) and (3) are rewritten are as follows:
It enablesThen formula (9) is changed to state space form
Wherein
k1,k2It is two normal numbers；According to Lyapunov theorem, as long as matrix A is Hull dimension thatch matrix, for appointing
Anticipate symmetrical matrix Q, certainly exists a positive definite matrix P and following formula is set up:
ATP+PA=-2Q (12)
The imperial Burger observer form of design is as follows:
Wherein Respectively x1And x2Estimated value,It isDerivative；Being will be in E (x)
Variable replacement x isWhen value；H is the gain matrix of observer, form are as follows:
Wherein h1,h2, δ is normal number；
It enablesDefinitionFor observer observation error, formula is subtracted with formula (10)
(12) it obtains
WhereinIt is xeDerivative,Meet It isTwo norms, M=
[m1,m2,m3]T, mi, i=1,2,3 be normal number constant, | | M | | it is two norms of M, | | xe| | it is xeTwo norms；
2.2 design controllers, define dummy variable first:
Wherein σdIt is expectation posture；α is virtual controlling rule, and form is
Whereinkb1It is normal number, meets kb1≥||z1(0)||2, and | | z1(0) | | it is z1Initially
Two norms of value,It is z1Transposition；c1It is normal number；It is σdDerivative；
Controller design are as follows:
Wherein c2It is normal number；kb2It is normal number, meets kb2≥||z2(0)||2, and | | z2
(0) | | it is z2Two norms of initial value,It is z2Transposition；It is the derivative of α；
Step 3, rigid aircraft attitude system stability proves that process is as follows:
3.1 prove the posture observation error and tracking error uniform ultimate bounded of rigid aircraft system, design modified
Obstacle liapunov function is following form:
Wherein ln is natural logrithm；E natural constant；
It is substituted into formula (218) derivation and by formula (12), (14), (17) and (18):
Wherein η is normal number；||H2| | it is H2Two norms；| | P | | it is two norms of P；
Formula (20) abbreviation is obtained:
Whereinλmax(P) be matrix P maximum feature
Value；
Therefore, according to Lyapunov theorem, rigid aircraft posture observation error and tracking error be can be realized
Uniform ultimate bounded；
3.2 proof rigid aircraft quantity of states suffer restraints:
It enablesSolution formula (28) obtains such as lower inequality:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
Wherein V (0) is the output valve of V；
Convolution (19) and (22), obtain
By solving inequality (23), z is obtained1Finally converge to following neighborhood:
By identical derivation, z is obtained2Finally converge to following neighborhood:
Find out from formula (24) and (25), z1And z2Respectively by kb1And kb2Constraint obtain rigidity and fly in conjunction with System describe
All quantity of states of row device all suffer restraints.
For the validity for illustrating proposition method, The present invention gives the numerical simulation of rigid aircraft system experiment, rotations
Inertia matrix is
External disturbance is
D=1.5 × 10-3[3cos(0.8t)+1,1.5sin(0.8t)+3cos(0.8t),3sin(0.8t)+1]TOx rice
(27)
Posture initial value is
Desired posture track is σd=[sin (2t), sin (2t+ π), cos (2t)]T.Control parameter is chosen are as follows: k1=
20,k2=4, h1=1, h2=4.5, δ=20, c1=2, c2=3, kb1=0.4, kb2=1.5.
Fig. 1 shows Attitude Tracking effect of the rigid aircraft in no angular velocity measurement, and Fig. 2 shows that rigidity flies
The Attitude Tracking error of row device.Can be seen that the controller that the present invention is mentioned from Fig. 1 and Fig. 2 may be implemented high-precision posture
Tracing control and angular velocity information is not needed.Fig. 3 shows the control input torque u of the mentioned method of the present invention.Fig. 4 is shown
The observation error of imperial Burger observer, as can be seen from the figure accurately estimating to quantity of state may be implemented in the observer of this method
Meter.
Described above is the excellent effect of optimization that one embodiment that the present invention provides is shown, it is clear that the present invention is not only
It is limited to above-described embodiment, without departing from essence spirit of the present invention and without departing from the premise of range involved by substantive content of the present invention
Under it can be made it is various deformation be implemented.
Claims (1)
1. a kind of rigid aircraft state constraint control method based on imperial Burger observer, which is characterized in that the controlling party
Method the following steps are included:
Step 1, the kinematics and dynamics modeling of the rigid aircraft based on modified discrete chirp-Fourier transform is established, process is as follows:
The kinematical equation of 1.1 rigid aircraft systems are as follows:
Wherein σ=[σ1,σ2,σ3]TFor modified discrete chirp-Fourier transform, which depict the posture features of aircraft；It is the derivative of σ,
σTIt is the transposition of σ；ω∈R3It is the angular speed of rigid aircraft；I3It is R3×3Unit matrix；σ×Form are as follows:
G form isIt has property| | G | | it is two norms of G；
The kinetics equation of 1.2 rigid aircraft systems are as follows:
Wherein J ∈ R3×3It is the moment of inertia matrix of rigid aircraft；It is the derivative of ω, indicates the angle of rigid aircraft
Acceleration；u∈R3With d ∈ R3Respectively control moment and external disturbance；ω × form are as follows:
1.3 rightDerivation simultaneously substitutes into formula (3), obtains
Wherein L=G-1,It is the derivative of L；J-1It is the inverse matrix of J；A×Form are as follows:
D '=GJ-1D and satisfaction | | d ' | |≤dm, wherein dmIt is a normal number；
Step 2, for the rigid aircraft system with external disturbance and without angular velocity measurement, controller is designed, process is as follows:
2.1 design imperial Burger state observer, enable x1=[x11,x12,x13]T=σ, Aircraft
Output is y=σ, and formula (1) and (3) are rewritten are as follows:
It enablesThen formula (9) is changed to state space form
Wherein
k1,k2It is two normal numbers；According to Lyapunov theorem, as long as matrix A is Hull dimension thatch matrix, for any right
Claim matrix Q, certainly exist a positive definite matrix P and following formula is set up:
ATP+PA=-2Q (12)
The imperial Burger observer form of design is as follows:
Wherein Respectively x1And x2Estimated value,It isDerivative；It is by the variable in E (x)
Replacing x isWhen value；H is the gain matrix of observer, form are as follows:
Wherein h1,h2, δ is normal number；
It enablesDefinitionFor observer observation error, formula (12) are subtracted with formula (10)
It obtains
WhereinIt is xeDerivative,Meet It isTwo norms, M=[m1,
m2,m3]T, mi, i=1,2,3 be normal number constant, | | M | | it is two norms of M, | | xe| | it is xeTwo norms；
2.2 design controllers, define dummy variable first:
Wherein σdIt is expectation posture；α is virtual controlling rule, and form is
Whereinkb1It is normal number, meets kb1≥||z1(0)||2, and | | z1(0) | | it is z1Initial value
Two norms,It is z1Transposition；c1It is normal number；It is σdDerivative；
Controller design are as follows:
Wherein c2It is normal number；kb2It is normal number, meets kb2≥||z2(0)||2, and | | z2(0) | | be
z2Two norms of initial value,It is z2Transposition；It is the derivative of α；
Step 3, rigid aircraft attitude system stability proves that process is as follows:
3.1 prove the posture observation error and tracking error uniform ultimate bounded of rigid aircraft system, design modified obstacle
Liapunov function is following form:
Wherein ln is natural logrithm；E natural constant；
It is substituted into formula (218) derivation and by formula (12), (14), (17) and (18):
Wherein η is normal number；||H2| | it is H2Two norms；| | P | | it is two norms of P；
Formula (20) abbreviation is obtained:
Whereinλmax(P) be matrix P maximum eigenvalue；
Therefore, according to Lyapunov theorem, rigid aircraft posture observation error can be realized consistent with tracking error
Ultimate boundness；
3.2 proof rigid aircraft quantity of states suffer restraints:
It enablesSolution formula (28) obtains such as lower inequality:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
Wherein V (0) is the output valve of V；
Convolution (19) and (22), obtain
By solving inequality (23), z is obtained1Finally converge to following neighborhood:
By identical derivation, z is obtained2Finally converge to following neighborhood:
Find out from formula (24) and (25), z1And z2Respectively by kb1And kb2Constraint obtain rigid aircraft in conjunction with System describe
All quantity of states all suffer restraints.
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Cited By (3)
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| CN111442786A (en) * | 2020-05-21 | 2020-07-24 | 合肥工业大学 | Zero drift deviation and attitude estimation method of aircraft gyroscope |
| CN113050417A (en) * | 2021-02-26 | 2021-06-29 | 中国地质大学（武汉） | Design method of rapid finite time controller of all-state constraint mechanical arm |
| CN114329943A (en) * | 2021-12-23 | 2022-04-12 | 哈尔滨工业大学（深圳） | Control performance boundary design method, device and medium based on attitude rotation matrix |
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| CN114329943B (en) * | 2021-12-23 | 2023-01-24 | 哈尔滨工业大学（深圳） | Control performance boundary design method, device and medium based on attitude rotation matrix |
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Application publication date: 20190402 Assignee: Tongxiang Tujian Intelligent Technology Co.,Ltd. Assignor: JIANG University OF TECHNOLOGY Contract record no.: X2023980037543 Denomination of invention: A State Constraint Control Method for Rigid Aircraft Based on Longberg Observer Granted publication date: 20210803 License type: Common License Record date: 20230705 |