CN109557933B - Rigid aircraft state constraint control method based on Longberger observer - Google Patents
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Abstract
A rigid aircraft state constraint control method based on a Roberter observer is provided, aiming at a rigid aircraft with external interference and no angular velocity measurement, the Roberter observer estimates unknown state quantity, and therefore the angular velocity of the aircraft does not need to be known. An improved barrier Lyapunov function suitable for constrained and unconstrained conditions is used for realizing state constraint, and a rigid aircraft state constraint control method is designed by combining with backstepping control. The method ensures that the attitude observation error and the tracking error of the aircraft can be consistent and finally bounded under the conditions of external interference and no angular velocity measurement, and the state variable is constrained.
Description
Technical Field
The invention relates to a rigid aircraft state constraint control method based on a Luenberger observer, which is an all-state constraint output feedback attitude tracking control method designed for a rigid aircraft with external interference and no angular velocity measurement.
Background
A rigid aircraft is a nonlinear, strong-coupling, multi-input and multi-output complex system, and a plurality of external disturbance moments affect the aircraft at any moment in flight, such as radiation moment, gravity gradient moment, geomagnetic moment and the like. And in many cases, the angular velocity signal of the aircraft may contain a lot of noise, and even sensor damage may result in the angular velocity signal not being accurately obtained. Therefore, the attitude control method independent of the angular velocity information has strong practical significance.
As the level of task refinement performed increases, it is not sufficient to focus solely on the steady-state accuracy of the aircraft. To ensure transient performance and stability of the system, the system state and the amplitude of the output are usually constrained. During the operation of the system, if the constraint condition is violated, the performance of the system may be reduced and even a safety problem may occur. The barrier lyapunov function method is a constraint control method, and the basic principle is that when a variable approaches a boundary of a region, the value of the lyapunov function tends to be infinite, so that the constraint of the variable is ensured. The conventional logarithmic barrier lyapunov function is not suitable for the unconstrained case, whereas the modified barrier lyapunov function may be suitable for both the constrained and unconstrained cases. The improved barrier Lyapunov function is used for not only restraining variables, but also effectively improving transient and steady-state performance of the system.
The backstepping control method is a recursion design control method based on the Lyapunov theorem, and a feedback control law and a Lyapunov function can be designed together in the process of gradual recursion. The backstepping method can reduce the difficulty of designing the controller by gradually recursion when designing the high-order controller. One of the main advantages of the backstepping control is that it avoids eliminating some of the useful non-linearities and achieves high accuracy control performance. The lunberg observer is a state observation proposed by lunberg, kalman and buchsi, etc., and can estimate the angular velocity information of the aircraft that cannot be obtained by using the observer, thereby realizing the design of a feedback controller without angular velocity information.
Disclosure of Invention
In order to solve the problem of attitude constraint control of a rigid aircraft without angular velocity information, the invention provides a rigid aircraft state constraint control method based on a Robert observer, which can realize that attitude observation errors and tracking errors of a rigid aircraft system can be consistent and finally bounded under the condition that the system has external interference and no angular velocity information.
The technical scheme proposed for solving the technical problems is as follows:
a rigid aircraft state constraint control method based on a Longberger observer comprises the following steps:
1.1 the kinematic equation for a rigid aircraft system is:
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;is the derivative of σ, σTIs the transpose of σ; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
1.2 the kinetic equation for a rigid aircraft system is:
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;is the derivative of ω, representing the angular acceleration of the rigid vehicle; u is an element of R3And d ∈ R3Respectively control moment and external disturbance; omega×The form is as follows:
Wherein L ═ G-1,Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;A×in the form of:
2.1 design the Lorberg State observer, let x1=[x11,x12,x13]T=σ, The output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Wherein
k1,k2Are two normal numbers; according to the lyapunov theorem, as long as the matrix a is a helvets matrix, for any symmetric matrix Q there must be a positive definite matrix P such that the following holds:
ATP+PA=-2Q (12)
the form of the designed lunberger observer is as follows:
wherein Are respectively x1And x2Is determined by the estimated value of (c),is thatA derivative of (a);replacing x with the variable in E (x)The value of time; h is the gain matrix of the observer, of the form:
order toDefinition ofSubtracting the formula (12) from the formula (10) to obtain the observation error of the observer
WhereinIs xeThe derivative of (a) of (b),satisfy the requirement of Is thatTwo norm of (M ═ M)1,m2,m3]T,miI ═ 1,2,3 is the constant of normal numbers, | | | M | | | is the two-norm of M, | | | x |eIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
wherein sigmadIs the desired pose; α is a virtual control law of the form
Whereinkb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,is z1Transposing; c. C1Is a normal number;is σdA derivative of (a);
the controller is designed as follows:
wherein c is2Is a normal number;kb2is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,is z2Transposing;is the derivative of α;
3.1 proving that the attitude observation error and the tracking error of the rigid aircraft system are consistent and finally bounded, and designing an improved obstacle Lyapunov function into the following form:
wherein ln is a natural logarithm; e, natural constant;
the derivation of equation (218) and the substitution of equations (12), (14), (17) and (18) yields:
wherein η is a normal number; i H2Is H2A second norm of (d); p is the two-norm of P;
equation (20) is simplified to:
therefore, according to the Lyapunov theorem, the attitude observation error and the tracking error of the rigid aircraft system can be consistent and finally bounded;
3.2 proving that the rigid aircraft state quantities are constrained:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
wherein V (0) is the output value of V;
combining formulae (19) and (22) to obtain
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
by the same derivation, z is obtained2Eventually converging to the following neighborhood:
as seen from formulae (24) and (25), z1And z2Are respectively subjected to kb1And kb2And (4) combining the system description to make all state quantities of the rigid aircraft be restrained.
Under the conditions that external interference exists in the rigid aircraft and no angular velocity measurement exists, the state constraint control method of the rigid aircraft based on the Longberg observer is designed by combining the Longberg observer, the backstepping control method and the improved barrier Lyapunov function, and high-precision control and state constraint of the system are achieved.
The technical conception of the invention is as follows: aiming at a rigid aircraft with external interference and no angular velocity measurement, a Longberg observer is provided for estimating unknown state quantity, and a state constraint control method is designed by combining backstepping control and an improved barrier Lyapunov function, so that the attitude observation error and the tracking error of the rigid aircraft can be consistent and bounded finally.
The invention has the advantages that: under the conditions that external interference exists in the system and no angular velocity measurement exists, the observation error and the tracking error of the system can be consistent and finally bounded, and the state quantity of the aircraft can be restrained.
Drawings
FIG. 1 is a diagram of the effect of attitude tracking of a rigid aircraft of the present invention;
FIG. 2 is a schematic representation of the rigid aircraft attitude tracking error of the present invention;
FIG. 3 is a schematic illustration of the rigid aircraft control input torque of the present invention;
FIG. 4 is a schematic view of the rigid aerial vehicle observation error of the present invention;
FIG. 5 is a control flow diagram of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 5, a rigid aircraft state constraint control method based on a lunberger observer includes the following steps:
1.1 the kinematic equation for a rigid aircraft system is:
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;is the derivative of σ, σTIs the transpose of σ; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
1.2 the kinetic equation for a rigid aircraft system is:
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;is the derivative of ω, representing the angular acceleration of the rigid vehicle; u is an element of R3And d ∈ R3Respectively control moment and external disturbance; omega×The form is as follows:
Wherein L ═ G-1,Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;A×in the form of:
2.1 design the Lorberg State observer, let x1=[x11,x12,x13]T=σ, The output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Wherein
k1,k2Are two normal numbers; according to the lyapunov theorem, as long as the matrix a is a helvets matrix, for any symmetric matrix Q there must be a positive definite matrix P such that the following holds:
ATP+PA=-2Q (12)
the form of the designed lunberger observer is as follows:
wherein Are respectively x1And x2Is determined by the estimated value of (c),is thatA derivative of (a);replacing x with the variable in E (x)The value of time; h is the gain matrix of the observer, of the form:
order toDefinition ofSubtracting the formula (12) from the formula (10) to obtain the observation error of the observerTo
WhereinIs xeThe derivative of (a) of (b),satisfy the requirement of Is thatTwo norm of (M ═ M)1,m2,m3]T,miI ═ 1,2,3 is the constant of normal numbers, | | | M | | | is the two-norm of M, | | | x |eIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
wherein sigmadIs the desired pose; α is a virtual control law of the form
Whereinkb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,is z1Transposing; c. C1Is a normal number;is σdA derivative of (a);
the controller is designed as follows:
wherein c is2Is a normal number;kb2is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,is z2Transposing;is the derivative of α;
3.1 proving that the attitude observation error and the tracking error of the rigid aircraft system are consistent and finally bounded, and designing an improved obstacle Lyapunov function into the following form:
wherein ln is a natural logarithm; e, natural constant;
the derivation of equation (218) and the substitution of equations (12), (14), (17) and (18) yields:
wherein η is a normal number; i H2Is H2A second norm of (d); p is the two-norm of P;
equation (20) is simplified to:
therefore, according to the Lyapunov theorem, the attitude observation error and the tracking error of the rigid aircraft system can be consistent and finally bounded;
3.2 proving that the rigid aircraft state quantities are constrained:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
wherein V (0) is the output value of V;
combining formulae (19) and (22) to obtain
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
by the same derivation, z is obtained2Finally converge toThe following neighborhoods:
as seen from formulae (24) and (25), z1And z2Are respectively subjected to kb1And kb2And (4) combining the system description to make all state quantities of the rigid aircraft be restrained.
To illustrate the effectiveness of the proposed method, the invention provides a numerical simulation experiment of a rigid aircraft system, with a rotational inertia matrix of
External interference is
d=1.5×10-3[3cos(0.8t)+1,1.5sin(0.8t)+3cos(0.8t),3sin(0.8t)+1]TCattle-rice (27)
The initial value of the attitude is
The desired attitude trajectory is σd=[sin(2t),sin(2t+π),cos(2t)]T. The control parameters are selected as follows: k is a radical of1=20,k2=4,h1=1,h2=4.5,δ=20,c1=2,c2=3,kb1=0.4,kb2=1.5。
Fig. 1 shows the attitude tracking effect of the rigid aircraft without angular velocity measurement, and fig. 2 shows the attitude tracking error of the rigid aircraft. As can be seen from fig. 1 and 2, the controller of the present invention can realize highly accurate attitude tracking control and does not require angular velocity information. Fig. 3 shows the control input torque u of the method according to the invention. Fig. 4 shows the observation error of the lunberger observer, and it can be seen from the figure that the observer of the method can realize accurate estimation of the state quantity.
While the foregoing has described a preferred embodiment of the invention, it will be appreciated that the invention is not limited to the embodiment described, but is capable of numerous modifications without departing from the basic spirit and scope of the invention as set out in the appended claims.
Claims (1)
1. A rigid aircraft state constraint control method based on a Longberger observer is characterized by comprising the following steps:
step 1, establishing a kinematics and dynamics model of the rigid aircraft based on the modified rodgers parameter, wherein the process is as follows:
1.1 the kinematic equation for a rigid aircraft system is:
wherein σ ═ σ1,σ2,σ3]TTo correct the rodriger parameter, it describes the attitude characteristics of the aircraft;is the derivative of σ, σTIs the transpose of σ; g represents a directional cosine matrix; omega epsilon to R3Is the angular velocity of the rigid aircraft; i is3Is R3×3An identity matrix; sigma×In the form of:
1.2 the kinetic equation for a rigid aircraft system is:
wherein J ∈ R3×3Is a rotational inertia matrix of the rigid aircraft;is the derivative of ω, representing the angular acceleration of the rigid vehicle; u is an element of R3And d ∈ R3Respectively control moment and external disturbance; omega×The form is as follows:
ω1,ω2,ω3three normal numbers are indicated;
Wherein E represents a new state quantity, L ═ G-1,Is the derivative of L; j. the design is a square-1Is the inverse matrix of J;Λ×in the form of:
step 2, aiming at a rigid aircraft system with external interference and no angular velocity measurement, designing a controller, and carrying out the following process:
2.1 design the Lorberg State observer, let x1=[x11,x12,x13]T=σ,x1Representing an observer first variable; x is the number of2Representing an observer second variable; the output of the aircraft is y ═ σ, and equations (1) and (3) are rewritten as:
Wherein
k1,k2Are two normal numbers; according to the lyapunov theorem, as long as the matrix a is a helvets matrix, for any symmetric matrix Q there must be a positive definite matrix P such that the following holds:
ATP+PA=-2Q (12)
the form of the designed lunberger observer is as follows:
wherein Are respectively x1And x2Is determined by the estimated value of (c),is thatA derivative of (a);replacing x with the variable in E (x)The value of time; h is the gain matrix of the observer, of the form:
wherein h is1,h2δ is a normal number;H1representation is a gain sub-matrix 1, H2Denoted gain sub-matrix 2;
order toDefinition ofFor observer error, the first equation of equation (10) is used to subtract equation (13) to obtain
WhereinIn order to observe the error by the observer,is xeThe derivative of (a) of (b), represents an estimation error, satisfies Is thatM is a two norm ofm1,m2,m3]TM denotes a positive vector, MiIs a normal number, i ═ 1,2,3, | | | M | | | is the two-norm of M, | | | xeIs xeA second norm of (d);
2.2 design controller, first define the virtual variables:
wherein z is1Representing virtual variables 1, z2Representing virtual variables 2, σdIs the desired pose; α is a virtual control law of the form
WhereinΦ1Representing controller parameters 1, kb1Is a normal number, satisfies kb1≥||z1(0)||2And | z |1(0) Is z1The two-norm of the initial value is,is z1Transposing; c. C1Is a normal number;is σdA derivative of (a);
the controller is designed as follows:
wherein c is2Is a normal number;Φ2representing controller parameters 2, kb2Is a normal number, satisfies kb2≥||z2(0)||2And | z |2(0) Is z2The two-norm of the initial value is,is z2Transposing;is the derivative of α;
step 3, proving the stability of the attitude system of the rigid aircraft, wherein the process is as follows:
3.1 proving that the attitude observation error and the tracking error of the rigid aircraft system are consistent and finally bounded, and designing an improved obstacle Lyapunov function into the following form:
wherein ln is a natural logarithm; e, natural constant; p represents a positive definite matrix;
derivation of equation (19) and substitution of equations (12), (14), (17), and (18) yields:
wherein η is a normal number; i H2Is H2A second norm of (d); p is the two-norm of P;
equation (20) is simplified to:
whereinC represents a normal quantity, λmax(P) is the maximum eigenvalue of matrix P;μ represents a normal amount;
therefore, according to the Lyapunov theorem, the attitude observation error and the tracking error of the rigid aircraft system can be consistent and finally bounded;
3.2 proving that the rigid aircraft state quantities are constrained:
0≤V≤μ0+(V(0)-μ0)e-Ct (22)
wherein V (0) is the output value of V;
combining formulae (19) and (22) to obtain
By solving the inequality (23), z is obtained1Eventually converging to the following neighborhood:
by the same derivation, z is obtained2Eventually converging to the following neighborhood:
as seen from formulae (24) and (25), z1And z2Are respectively subjected to kb1And kb2And (4) combining the system description to make all state quantities of the rigid aircraft be restrained.
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