An adaptive attitude control method for flexible aircraft based on internal model

Abstract

The present invention discloses a method for adaptive attitude control of a flexible aircraft based on an internal model, and relates to the field of flexible aircraft system control; the method comprises: step 1, establishing a kinematic and dynamic model of a flexible aircraft system; step 2, designing an internal model; step 3, designing an adaptive state feedback controller based on the internal model. The present invention designs an adaptive state feedback controller based on an internal model for a flexible aircraft system that is subjected to external interference and has an unknown interference frequency, which can not only realize interference suppression and vibration suppression of the flexible aircraft, but also ensure that the estimated frequency converges to its true value.

Description

Technical Field

[0001] The invention relates to the field of flexible aircraft system control, and in particular to an internal model-based flexible aircraft adaptive attitude control method. Background Art

[0002] In order to reduce the launch cost and extend the life of spacecraft in orbit, modern spacecraft generally carry flexible accessories, such as large solar panels and large flexible antennas, which make the spacecraft have the structural characteristics of large structure, low stiffness and weak damping. There are many factors in the space environment that excite the vibration of flexible accessories, such as the deployment of flexible accessories, the adjustment of the spacecraft orbit, etc. These vibrations are not only difficult to measure, but also reduce the stability of the attitude. In addition, flexible aircraft are also affected by various space interference torques, such as gravity gradient torque, aerodynamic torque, etc. These interferences may cause the control accuracy of the control system to decrease or even become unstable. Based on the above background, how to design a controller to achieve high-precision attitude control of flexible aircraft is of great research significance.

[0003] On the one hand, the theory of nonlinear output regulation has made great progress in the past few decades, and its internal model control method can easily solve the system's trajectory tracking and interference suppression problems, and has good robustness. On the other hand, since adaptive control technology can estimate the uncertain parameters of the controlled object online, it is widely used in the field of nonlinear system control. When the flexible aircraft system is subject to external interference and the interference frequency is unknown, combining internal model control with adaptive control technology can not only achieve interference suppression and vibration suppression of the flexible aircraft, but also ensure that the estimated frequency converges to its true value. Summary of the invention

[0004] Based on the technical problems existing in the background technology, the present invention proposes an adaptive attitude control method for a flexible aircraft based on an internal model. When the flexible aircraft system is subjected to external interference and the interference frequency is unknown, an adaptive state feedback controller based on the internal model is designed, which can not only realize the interference suppression and vibration suppression of the flexible aircraft system, but also ensure that the estimated frequency converges to its true value.

[0005] The technical solution adopted by the present invention is:

[0006] A method for adaptive attitude control of a flexible aircraft based on an internal model, characterized in that it comprises the following steps:

[0007] Step 1: Establish the kinematic and dynamic models of the flexible aircraft system;

[0008] Step 2: Design the inner mold;

[0009] Step 3: Design an adaptive state feedback controller based on the internal model.

[0010] Furthermore, the method for adaptive attitude control of a flexible aircraft based on an internal model is characterized in that, in step 1, a kinematic and dynamic model of the flexible aircraft system is established, and the process is as follows:

[0011] 1.1, the kinematic model of the flexible aircraft based on quaternions is established as follows:

[0012]

[0013] in The attitude quaternion of the aircraft satisfies q T q=1, is the angular velocity of the flexible aircraft in the body coordinate system, for q v The skew-symmetric matrix is

[0014] 1.2, the dynamic model of the flexible aircraft is established as follows:

[0015]

[0016] in is the inertia matrix of the aircraft, is the coupling matrix between the rigid body and the flexible attachment, is the modal coordinate vector of the flexible attachment, n is the number of modal coordinate vectors considered, is the control torque acting on the rigid body, is external interference, C=diag{2ξ i Λ i ,i=1,…n}and is a symmetric positive definite matrix, where ξ i , Λ i are the damping ratio and natural frequency respectively.

[0017] 1.3, assuming that the external disturbance d to the system is a combination of a finite number of constant signals and sinusoidal signals, that is, it has the following form:

[0018]

[0019] Among them C i0 ,C ij ,γ ij ,i=1,2,3,j=1,2,…,n i are all unknown constants, C ij ,γ ij are the amplitude and phase of the sinusoidal signal, b ij is the known or unknown frequency of the sinusoidal signal.

[0020] 1.4, consider the static-to-static attitude maneuver problem of a flexible aircraft, that is, the desired angular velocity ω d =0, then the angular velocity error can be defined as ω e =ω. Consider the following standard coordinates and input transformations:

[0021]

[0022] in The desired attitude quaternion satisfies ||q d ||=1, is the quaternion error, k1 is a positive number.

[0023] 1.5, through coordinate and input transformation, the attitude error system of the flexible aircraft can be written as follows:

[0024]

[0025]

[0026]

[0027] 1.6, therefore, the control objective of this paper is to design a state feedback controller for the flexible aircraft so that all states of the closed-loop system are bounded and satisfy

[0028]

[0029] Furthermore, the method for adaptive attitude control of a flexible aircraft based on an internal model is characterized in that, in step 2, the internal model is designed, and the process is as follows:

[0030] 2.1, for D i By differentiating (t), we can obtain:

[0031]

[0032] in is a real number containing the unknown quantity σ, r i is a normal number, i=1,2,3.

[0033] 2.2, let Then D i (t) can be generated by the following linear system:

[0034]

[0035] in

[0036] 2.3, select a pair of controllable matrices (M i ,Ni ),in is the Hurwitz matrix, Then for any σ, there exists a non-singular matrix T i σ The following Sylvester equation is satisfied:

[0037]

[0038] 2.4, define the following notation:

[0039]

[0040] According to the formula, we can get:

[0041]

[0042] 2.5, design the inner mold as follows:

[0043]

[0044] in Combining the system and we get the following augmented system:

[0045]

[0046] 2.6, perform the following coordinate and input transformations:

[0047]

[0048] Where T ο is the matrix T σ The nominal value of , the system is converted into the following form:

[0049]

[0050] in is the uncertainty of the system.

[0051] Furthermore, the method for adaptive attitude control of a flexible aircraft based on an internal model is characterized in that, in step 3, an adaptive state feedback controller based on an internal model is designed, and the process is as follows:

[0052] 3.1, define x2 = η, the system can be written in the following state space equation form:

[0053]

[0054] in Since both K and C are symmetric positive definite matrices, G is a Hurwitz matrix.

[0055] 3.2, substituting formula and into the system, we can get

[0056]

[0057]

[0058] 3.3, let h = [z, x] T , we can get the following system about h:

[0059]

[0060] in Since both M and G are Hurwitz matrices, it is obvious that A is also a Hurwitz matrix. Therefore, for any p>0, there exists a symmetric positive definite matrix Meet PA+A T P=-pI (r+2n) .

[0061] 3.4, the system can be written as follows:

[0062]

[0063] in

[0064] ζ1=[Ψ(T σ ) -1 -(Ψ(T σ ) -1 NδK+δCK)-(Ψ(T σ ) -1 NδC+δCC-δK)]

[0065] 3.5, for the system, perform the following dynamic coordinate transformation:

[0066]

[0067] get

[0068]

[0069]

[0070] in

[0071] ζ2=[Ψ(T o ) -1 -(Ψ(T o ) -1 NδK+δCK)-(Ψ(T o ) -1 NδC+δCC-δK)]

[0072] ζ3=-(Ψ(To ) -1 Nδδ T +δCδ T )

[0073]

[0074] 3.6, let Γ = J - δδ T , since Γ is a non-singular matrix, the system can be written as follows:

[0075]

[0076] where ζ4 = Γ -1 ζ1,ζ5=Γ -1 Ψ(T o ) -1 NJ.

[0077] 3.7, Matrix E σ The uncertainty in can be linearly parameterized as follows:

[0078]

[0079] where l≥1, is a constant matrix, is a smooth function, is the Kronecker product.

[0080] 3.8, according to the formula, we can get:

[0081]

[0082] where ξ(x e ,υ,s,ω e )=NJx e -υ+[I r ,-NδK,-NδC]s-Nδδ T ω e , operator is the Tracy-Singh product, that is make The system can be further written as the following linear parameterized form:

[0083]

[0084] 3.9, Next, by combining the system and we can get the following flexible aircraft attitude error system:

[0085]

[0086] 3.10, design the following stabilization controller for the system:

[0087]

[0088] Where k and k2 are both positive numbers, is the estimated value of the uncertain parameter μ(σ).

[0089] 3.11, let Then there is

[0090]

[0091] in σ5 is the maximum eigenvalue of ζ5, and τ1 is a positive number. By choosing appropriate p, τ1 and k2, we can ensure By Barbalat's lemma, we can get Then there is According to the dynamic coordinate transformation, we can get Right now Therefore, through the above analysis, the control target has been achieved. In addition, if θ i (t) satisfies the continuous excitation condition, that is, there exists ε,t0,T0>0, such that For any unit vector If true, the estimated frequency can converge to its true value.

[0092] 3.12, the adaptive state feedback controller is obtained as follows:

[0093]

[0094]

[0095]

[0096] The advantages of the present invention are:

[0097] The present invention proposes an adaptive attitude control method for a flexible aircraft based on an internal model. When the flexible aircraft system is subjected to external interference and the interference frequency is unknown, an adaptive state feedback controller based on the internal model is designed. This method can not only realize interference suppression and vibration suppression of the flexible aircraft, but also ensure that the estimated frequency converges to its true value. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 This is the overall system control block diagram;

[0099] Figure 2 is the attitude Euler angle error curve;

[0100] Figure 3 is the attitude angular velocity error curve;

[0101] Figure 4 is the modal coordinate vector curve;

[0102] Figure 5 is the control input curve;

[0103] Figure 6 To estimate the frequency curve; DETAILED DESCRIPTION

[0104] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments.

[0105] Example 1.

[0106] A method for adaptive attitude control of a flexible aircraft based on an internal model, the specific steps are as follows:

[0107] Step 1: Establish the kinematic and dynamic models of the flexible aircraft system. The process is as follows:

[0108] 1.1, the kinematic model of the flexible aircraft based on quaternions is established as follows:

[0109]

[0110] in The attitude quaternion of the aircraft satisfies q T q=1, is the angular velocity of the flexible aircraft in the body coordinate system, for q v The skew-symmetric matrix is

[0111] 1.2, the dynamic model of the flexible aircraft is established as follows:

[0112]

[0113] in is the inertia matrix of the aircraft, is the coupling matrix between the rigid body and the flexible attachment, is the modal coordinate vector of the flexible attachment, n is the number of modal coordinate vectors considered, is the control torque acting on the rigid body, is external interference, C=diag{2ξ i Λ i ,i=1,…n}and is a symmetric positive definite matrix, where ξ i , Λ i are the damping ratio and natural frequency respectively.

[0114] 1.3, assuming that the external disturbance d to the system is a combination of a finite number of constant signals and sinusoidal signals, that is, it has the following form:

[0115]

[0116] Among them C i0 ,C ij ,γ ij ,i=1,2,3,j=1,2,…,n i are all unknown constants, C ij ,γ ij are the amplitude and phase of the sinusoidal signal, b ij is the known or unknown frequency of the sinusoidal signal.

[0117] 1.4, consider the static-to-static attitude maneuver problem of a flexible aircraft, that is, the desired angular velocity ω d =0, then the angular velocity error can be defined as ω e =ω. Consider the following standard coordinates and input transformations:

[0118]

[0119] in The desired attitude quaternion satisfies ||q d ||=1, is the quaternion error, k1 is a positive number.

[0120] 1.5, through coordinate and input transformation, the attitude error system of the flexible aircraft can be written as follows:

[0121]

[0122]

[0123]

[0124] 1.6, therefore, the control objective of this paper is to design a state feedback controller for the flexible aircraft so that all states of the closed-loop system are bounded and satisfy

[0125]

[0126] Step 2, design the inner mold, the process is as follows:

[0127] 2.1, for D i By differentiating (t), we can obtain:

[0128]

[0129] in is a real number containing the unknown quantity σ, r i is a normal number, i=1,2,3.

[0130] 2.2, let Then D i (t) can be generated by the following linear system:

[0131]

[0132] in

[0133] 2.3, select a pair of controllable matrices (M i ,N i ),in is the Hurwitz matrix, Then for any σ, there exists a non-singular matrix T i σ The following Sylvester equation is satisfied:

[0134]

[0135] 2.4, define the following notation:

[0136]

[0137] According to the formula, we can get:

[0138]

[0139] 2.5, design the inner mold as follows:

[0140]

[0141] in Combining the system and we get the following augmented system:

[0142]

[0143] 2.6, perform the following coordinate and input transformations:

[0144]

[0145] Where T ο is the matrix T σ The nominal value of , the system is converted into the following form:

[0146]

[0147] in is the uncertainty of the system.

[0148] Step 3: Design an adaptive state feedback controller based on the internal model. The process is as follows:

[0149] 3.1, define x2 = η, the system can be written in the following state space equation form:

[0150]

[0151] in Since both K and C are symmetric positive definite matrices, G is a Hurwitz matrix.

[0152] 3.2, substituting formula and into the system, we can get

[0153]

[0154]

[0155] 3.3, let h = [z, x] T , we can get the following system about h:

[0156]

[0157] in Since both M and G are Hurwitz matrices, it is obvious that A is also a Hurwitz matrix. Therefore, for any p>0, there exists a symmetric positive definite matrix Meet PA+A T P=-pI (r+2n) .

[0158] 3.4, the system can be written as follows:

[0159]

[0160] in

[0161] ζ1=[Ψ(T σ ) -1 -(Ψ(T σ ) -1 NδK+δCK)-(Ψ(T σ ) -1 NδC+δCC-δK)]

[0162] 3.5, for the system, perform the following dynamic coordinate transformation:

[0163]

[0164] get

[0165]

[0166]

[0167] in

[0168] ζ2=[Ψ(T o ) -1 -(Ψ(T o ) -1 NδK+δCK)-(Ψ(T o ) -1 NδC+δCC-δK)]

[0169] ζ3=-(Ψ(T o ) -1 Nδδ T +δCδ T )

[0170]

[0171] 3.6, let Γ = J - δδ T , since Γ is a non-singular matrix, the system can be written as follows:

[0172]

[0173] where ζ4 = Γ -1 ζ1,ζ5=Γ -1 Ψ(T o ) -1 NJ.

[0174] 3.7, Matrix E σ The uncertainty in can be linearly parameterized as follows:

[0175]

[0176] where l≥1, is a constant matrix, is a smooth function, is the Kronecker product.

[0177] 3.8, according to the formula, we can get:

[0178]

[0179] where ξ(x e ,υ,s,ω e )=NJx e -υ+[I r ,-NδK,-NδC]s-Nδδ T ω e , operator is the Tracy-Singh product, that is make The system can be further written as the following linear parameterized form:

[0180]

[0181] 3.9, Next, by combining the system and we can get the following flexible aircraft attitude error system:

[0182]

[0183] 3.10, design the following stabilization controller for the system:

[0184]

[0185] Where k and k2 are both positive numbers, is the estimated value of the uncertain parameter μ(σ).

[0186] 3.11, let Then there is

[0187]

[0188] in σ5 is the maximum eigenvalue of ζ5, and τ1 is a positive number. By choosing appropriate p, τ1 and k2, we can ensure By Barbalat's lemma, we can get Then there is According to the dynamic coordinate transformation, we can get Right now Therefore, through the above analysis, the control target has been achieved. In addition, if θ i (t) satisfies the continuous excitation condition, that is, there exists ε,t0,T0>0, such that For any unit vector If true, the estimated frequency can converge to its true value.

[0189] 3.12, the adaptive state feedback controller is obtained as follows:

[0190]

[0191] The model parameters of the flexible aircraft in this embodiment are as follows:

[0192]

[0193]

[0194] ξ1=0.005607, ξ2=0.008620, ξ3=0.01283, ξ4=0.02516,

[0195] Λ1=0.7681rad / s, Λ2=1.1038rad / s,

[0196] Λ3=1.8733rad / s, Λ4=2.5496rad / s.

[0197] The external interference is in the form of D i (t) = C i sin(b i t+γ i ), i = 1, 2, 3, where C1 = 1, C2 = 2, C3 = 5, b1 = 0.1, b2 = 0.2, b3 = σ, γ1 = 0.1, γ2 = 0.2, γ3 = 0.3. Initial value υ(0)=[0 0 0 0 00]. The matrix used to construct the internal model is as follows:

[0198]

[0199] Available Assuming the nominal value of σ is 0, then

[0200] E σ =Ψ(T σ ) -1 -Ψ(T o ) -1 =Lμ(σ)

[0201]

[0202] Since the attitude variables expressed by Euler angles are easier to understand than quaternions, Euler angles are used in simulation. represents the attitude trajectory. The attitude Euler angle error can be defined as χ e = x d -χ, where χ d Indicates the desired posture Euler angle. In particular, the conversion relationship between Euler angle and quaternion is as follows:

[0203]

[0204]

[0205]

[0206]

[0207] In addition, in order to further optimize transient performance, sinusoidal angular acceleration path planning is also introduced in the simulation. Taking a single axis as an example, the angular acceleration path curve It can be expressed as follows:

[0208]

[0209]

[0210] in Δφ=φ d -φ0,φ d and φ0 are the desired and initial angles respectively, is the acceleration and deceleration time, is the uniform speed period, V max and a max are the maximum angular velocity and angular acceleration respectively. Finally, by performing a second integration on the angular acceleration path, a smooth and continuous attitude maneuvering path can be obtained. The path planning for the other two axes is similar.

[0211] The initial and expected values ​​of the attitude Euler angles are [0°, 30°, 0°] T and [40°,0°,30°] T , the initial value and expected value of the attitude angular velocity are [0° / s, 0° / s, 0° / s] T In addition, the maximum angular velocity and angular acceleration are selected as V max =3.5° / s and a max =0.2° / s 2 , the unknown frequency is selected as σ = 0.7, and the controller parameter is selected as k = 3 × 10 4 , k1=1.5 and k2=30.

[0212] Figure 1 is the overall system block diagram. Based on the above controller parameters, the simulation results are as follows Figure 2-6 shown. Figure 2 is the attitude Euler angle error curve, Figure 3 is the attitude angular velocity error curve, Figure 4 is the modal coordinate vector curve, Figure 5 To control the input curve, Figure 6 is the estimated frequency curve. According to the simulation results, when subjected to external interference and the interference frequency is unknown, the designed controller can achieve interference suppression and vibration suppression of the flexible aircraft, and the estimated frequency can converge to its true value.

[0213] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A method for adaptive attitude control of a flexible aircraft based on an internal model, characterized in that it comprises the following steps: Step 1: Establish the kinematic and dynamic models of the flexible aircraft system; Step 2: Design the inner mold; Step 3: Design an adaptive state feedback controller based on the internal model; In step 1, the kinematic and dynamic models of the flexible aircraft system are established, and the process is as follows: 1.1, the kinematic model of the flexible aircraft based on quaternions is established as follows: in The attitude quaternion of the aircraft satisfies q T q=1, is the angular velocity of the flexible aircraft in the body coordinate system, Q v The skew-symmetric matrix is 1.2, the dynamic model of the flexible aircraft is established as follows: in is the inertia matrix of the aircraft, is the coupling matrix between the rigid body and the flexible attachment, is the modal coordinate vector of the flexible attachment, n is the number of modal coordinate vectors considered, is the control torque acting on the rigid body, is external interference, C=diag{2ξ i Λ i ,i=1,…n}and is a symmetric positive definite matrix, where ξ i , Λ i are the damping ratio and natural frequency respectively; 1.3, assuming that the external disturbance d to the system is a combination of a finite number of constant signals and sinusoidal signals, that is, it has the following form: Among them C i0 ,C ij ,Υ ij ,i=1,2,3,j=1,2,…,n i are all unknown constants, C ij ,γ ij are the amplitude and phase of the sinusoidal signal, b ij is the known or unknown frequency of the sinusoidal signal; 1.4, consider the static-to-static attitude maneuver problem of a flexible aircraft, that is, the desired angular velocity ωd =0, then the angular velocity error can be defined as ω e =ω; Consider the following standard coordinates and input transformations: in The desired attitude quaternion satisfies ||q d ||=1, is the quaternion error, k1 is a positive number; 1.5, through the coordinate and input transformation (4), the attitude error system of the flexible aircraft can be written as follows: 1.6, therefore, the control objective of this paper is to design a state feedback controller for the flexible aircraft so that all states of the closed-loop system are bounded and satisfy In step 2, the inner mold is designed, and the process is as follows: 2.1, for D i By differentiating (t), we can obtain: in is a real number containing the unknown quantity σ, r i is a positive constant, i = 1, 2, 3; 2.2, let Then D i (t) can be generated by the following linear system: in 2.3, select a pair of controllable matrices (M i ,N i ),in is the Hurwitz matrix, Then for any σ, there exists a non-singular matrix T i σ The following Sylvester equation is satisfied: 2.4, define the following notation: According to formula (10), we can get: 2.5, design the inner mold as follows: in Combining systems (5) and (14) we obtain the following augmented system: 2.6, perform the following coordinate and input transformations: Where T ο is the matrix T σ The nominal value of , then system (15) is converted into the following form: in is the uncertainty of the system; In step 3, an adaptive state feedback controller based on the internal model is designed. The process is as follows: 3.1, define x2 = η, the system (6) can be written in the form of the following state space equation: in Since K and C are both symmetric positive definite matrices, G is a Hurwitz matrix; 3.2, substituting formulas (6) and (18) into system (17), we can obtain 3.3, let h = [z, x] T , we can get the following system about h: in Since both M and G are Hurwitz matrices, it is clear that A is also a Hurwitz matrix; therefore, for any p>0, there exists a symmetric positive definite matrix Meet PA+A T P=-pI (r+2n) ; 3.4, then the system (20) can be written as follows: in ζ1=[Ψ(T σ ) -1 -(Ψ(T σ ) -1 NδK+δCK)-(Ψ(T σ ) -1 NδC+δCC-δK)] 3.5, for system (21), perform the following dynamic coordinate transformation: get in ζ2=[Ψ(T o ) -1 -(Ψ(T o ) -1 NδK+δCK) -(Ψ(T o ) -1 NδC+δCC-δK)] ζ3=-(Ψ(T o ) -1 Nδδ T +δCδ T ) 3.6, let Γ = J - δδ T , since Γ is a non-singular matrix, the system (25) can be written as follows: where ζ4 = Γ -1 ζ1,ζ5=Γ -1 Ψ(T o ) -1 NJ; 3.7, Matrix E σ The uncertainty in can be linearly parameterized as follows: where l≥1, is a constant matrix, is a smooth function, is the Kronecker product; 3.8, according to formula (27), we can get: where ξ(x e ,υ,s,ω e )=NJx e -υ+[I r ,-NδK,-NδC]s-Nδδ T ω e , operator is the Tracy-Singh product, that is make Then the system (26) can be further written in the following linear parameterized form: 3.9, Next, by combining systems (24) and (29), we can obtain the following flexible aircraft attitude error system: 3.10, design the following stabilization controller for system (30): Where k and k2 are both positive numbers, is the estimated value of the uncertain parameter μ(σ); 3.11, let Then there is in σ5 is the maximum eigenvalue of ζ5, and τ1 is a positive number; by choosing appropriate p, τ1 and k2, we can ensure By Barbalat's lemma, we can get Then there is According to the dynamic coordinate transformation (23), we can get Right now Therefore, through the above analysis, the control objective (8) has been achieved; in addition, if θ i (t) satisfies the continuous excitation condition, that is, there exists ε,t0,T0>0, such that For any unit vector If true, the estimated frequency can converge to its true value; 3.12, the adaptive state feedback controller is obtained as follows:

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