CN105468007B - A kind of flexible satellite track linearisation attitude control method based on interference observer - Google Patents
A kind of flexible satellite track linearisation attitude control method based on interference observer Download PDFInfo
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Abstract
A kind of flexible satellite track linearisation attitude control method based on interference observer, the present invention relates to the flexible satellite tracks based on interference observer to linearize attitude control method.The present invention is not influenced to solve the problem of that single track linearization method of controlling is not strong to the rejection ability of interference, robustness is poor in view of external disturbance and flexible appendage.The present invention describes spacecraft attitude with Eulerian angles, using the thought of equivalent interference, establishes flexible spacecraft dynamics and kinematical equation；The pseudoinverse of controlled device is sought in the case of ignoring equivalent interference, designs the quasi- differentiator of particular form, obtains the name control of desired trajectory；With ratio-integration control design linear time-varying adjuster.Consider the influence of equivalent interference, design interference observer, ensure the tracking error asymptotic convergence of Spacecraft.The present invention improves the antijamming capability of system, enhances the robustness of system.The present invention is applied to the gesture stability field of flexible satellite.
Description
Technical Field
The invention relates to a flexible satellite trajectory linearization attitude control method based on a disturbance observer.
Background
With the development of the times and the progress of the society, the exploration of outer space by human beings has risen to a new height. China has crossed the lines of the spacious kingdom. The aerospace technology plays a vital role in national economy, national defense construction, cultural education and scientific research, and is the centralized embodiment of national comprehensive strength.
The aerospace technology is a comprehensive engineering technology formed by applying the theory of aerospace to the engineering and aerospace engineering practices of research, design, manufacture, test, launching, flight, return, control, management and the like of spacecrafts and carriers. The satellite system comprises seven parts: position and attitude control systems, antenna systems, transponder systems, telemetry command systems, power systems, temperature control systems, and tracking and propulsion systems. The attitude control system determines the tracking performance of the satellite, and is an important guarantee for the satellite to smoothly complete a space task.
As the field of space exploration is widened continuously, the difficulty of an exploration task is increased continuously, the structure of a spacecraft also shows a complicated trend, and the spacecraft is inevitably influenced by various interference moments and parameter uncertainty. The external and self interference greatly influences the working performance of the spacecraft, and the difficulty of attitude control is increased. And mathematical models of the interference are not easily clearly described. Therefore, the problem of interference suppression of the flexible spacecraft is a research hotspot in the aerospace field, and the control accuracy of the satellite is directly determined.
Aiming at the problem of interference suppression of a spacecraft, domestic and foreign scholars carry out deep research and provide a plurality of control algorithms, and part of the control algorithms are introduced as follows:
hua Liu et al [1] (Liu H, Guo L, Zhang Y. Ananti-disturbance PD control and stabilization of flexible space [ J ]. Nonlinerar Dynamics,2012,67(3):2081-2088) designs an interference observer and a PD controller aiming at the interference problem of a flexible spacecraft, inhibits two different interferences and improves the control precision and attitude stability of the spacecraft. Although the control algorithm based on the PID is good for the control performance of the linear system, the control algorithm has great limitation for the complex nonlinear system and the complex signal tracking, is not strong in robustness in the aspect of interference suppression, and must be combined with other algorithms to meet the control requirement.
Qian Yong et al [2] (Qian Yong, full order and strong, analysis [ J ] analysis of influence of scanning mirror motion on satellite attitude based on variable structure control [ 2013,29(6):7-10) utilizes a decoupling variable structure controller to control a stationary orbit satellite, considers the coupling between the scanning mirror and the satellite body of the satellite load, respectively designs sliding modes for four components of a quaternion, and avoids the influence of a singular problem state. According to simulation results, the tracking accuracy and stability of the attitude angle can be improved by adopting variable structure control, and the interference of the motion of the scanning mirror on the satellite is reduced to a great extent. However, since the variable structure control is equivalent to a switching function, the control is discontinuous, buffeting is easily caused, and the buffeting easily excites unmodeled characteristics of the system, so that the control performance of the system is influenced.
Zhuliang et al [3] (Shao X, Wang H.A Novel Method of route linearization Control Based on Disturbance emission [ J ]. physical Problemin Engineering,2014,2014) designed a direct adaptive TLC Control scheme using a Track Linearization Control (TLC) Method and a neural network technique, and it can be seen through simulation that the performance of the system is improved by the action of the neural network, making up for the deficiencies of the previous TLC. However, many control algorithms based on trajectory linearization do not consider external interference and self-parameter uncertainty influence, and thus the robustness of the system is poor.
Zheng Zhu et al [4] (army. spacecraft control principle. northwest university of industry press, 2001) designed an extended state observer to achieve the observation of rigid body spacecraft interference. Due to the fact that fuel consumption exists in the star body, the mass is time-varying, the rotational inertia of the star body cannot be determined, and meanwhile external interference cannot be ignored. The sliding mode controller designed by Zheng Zhu et al ensures convergence of the reference attitude state. As can be seen from the simulation result, the control method has good control performance and higher attitude tracking precision. However, this document is designed for a rigid body satellite, and does not consider the influence of a flexible member.
The first scheme is as follows:
document [6] (Zhu Z, Xia Y, Fu M, et al. Attitution tracking of a vertical spaced based on extended state observer [ C ]. Systems and Control in Aeronoutics and satellites (ISSCAA),20103rd International Symposium on. IEEE,2010:621-626) proposes a satellite attitude Control method based on an extended state observer. Firstly, a satellite kinetic equation is established by using quaternions, and a sliding mode control algorithm is designed based on the model. A state observer is designed according to the uncertainty of the rotational inertia and the external interference, and the estimated value of the interference is brought into a previously designed control law, so that the tracking control of the satellite is realized. The specific contents of the scheme are as follows:
(1) satellite dynamics model:
and a satellite dynamic model is represented by a quaternion, so that the influence of singular points is reduced. And introducing an error quaternion, and converting the control target into the error quaternion which converges to 0in a limited time. Definition of
x＝ω+Kev(71)
Wherein evRepresenting the error quaternion vector portion. By the equation (71), the control target is changed within a limited timeThe quantity x converges to 0. Considering uncertainty of the satellite moment of inertia, the moment of inertia J is expressed as J ═ J0+. DELTA.J, wherein J0the method adopts the idea of equivalent error, combines the external error and the uncertain part of the moment of inertia, and uses the methodAnd (4) showing. The satellite dynamics model can be simplified to
(2) Design sliding mode control law
Selecting a sliding mode surface S ═ C2x, wherein S ═ S1,S2,S3]T∈R3,C2∈R3×3. Selection of approximation law
The sliding mode arrival condition is ensured. Due to the fact that
Determination of control law
(3) Design observer
Due to equivalent interferenceIs unknown and therefore needs to be estimated by an observer. Introducing a new variable x2Representative systemTotal interference of systemEquation (37) can be written as follows:
wherein g (t) represents total interferenceThe differential of (a) is still unknown. The second order extended state observer is designed as follows:
in the formula E1Representing the estimated error of the observer, Z1And Z2is the output vector of the observer, β01and beta02Is the gain of the observer, and the function fal (-) is defined as follows
Wherein
In the formula 0<α1<1,δ>0。
The output Z of the state observer can be ensured by selecting proper parameters1Equal to state x, output Z2Is equal to
Therefore, the control law (75) is further improved
uESO(t)＝(C2B0)-1(-τS-σsgn(S)-C2F-C2Z2) (80)
The system has stronger anti-interference capability and robustness under the action of the control law, and can well realize attitude tracking.
The disadvantages of the scheme are described as follows:
according to a dynamic model of the system, the system does not consider the influence of the flexible vibration of the satellite sailboard, has no modal equation and carries out control law design by taking the satellite as a rigid body. But the actual satellite flexures have a large impact on the system, undermining the dynamic performance of the system, and even causing system instability.
Scheme II:
document [7] (Zhi W, band-hua l. compound control system Design based on feedback mapping techniques and neural network tracking mode for flexible satellite [ C ]. Computer Design and Applications (ICCDA),2010international reference on. ieee,2010,2: V2-418-V2-422) aims at a spacecraft Design variable structure control law with flexible parts, and effectively suppresses external interference by using three-level sliding mode control, so that a spacecraft tracking error is 0. Meanwhile, the uncertainty factor is estimated by utilizing the neural network, so that the jitter generated by discontinuous control is well weakened. The specific contents of the scheme are as follows:
(1) satellite kinematics and dynamics modeling
The method comprises the steps of considering flexible accessories such as a satellite antenna and a solar panel, adopting a triple-orthogonal flywheel as an execution element, respectively modeling dynamics of the whole satellite, the antenna and the solar panel, and considering vibration modes of the solar panel and the antenna. For the convenience of controller design, the vibration of the flexible attachment is considered as an external disturbance acting on the satellite, simplifying the satellite dynamics model.
(2) Sliding mode control law design
Based on a backstepping control technology, three layers of sliding mode control are adopted to reversely deduce a control input variable.
Firstly, designing a first layer of sliding mode surface, wherein the target is to quickly and accurately track a target quaternion and define angular velocity virtual controlAnd given its concrete form, construct the V function of the first layer; then designing a second layer of sliding mould surface, and aiming at quickly and accurately trackingDefining moment virtual controlSelecting proper exponential approximation law and deducingConstructing a V function of the second layer; and finally, designing a third layer of sliding mode surface, and obtaining higher control precision by using a dynamic model of a servo system. The trace instruction isThe control input being flywheel angular velocity omegacSelecting proper exponential approximation law to determine omegacThe specific expression of (1).
Derived by analysis, the control input omegacThe system tracking error can be guaranteed to be 0.
(3) Perfection of control law
In order to prevent the differential explosion phenomenon, low-pass filters are adopted to respectively carry out the pairAndand (6) filtering. At the same time, in order to suppress the influence of parameter uncertaintyAnd (5) estimating uncertain parameters by adopting an RBF neural network when the system vibrates.
According to the simulation result, the system can realize the tracking of the attitude in a short time, and the tracking precision is still high under the conditions of external interference and parameter disturbance. However, the controller of the system is too complex, and the realization of the control algorithm can only be completed on a computer with enough high calculation speed, so that the control algorithm is difficult to apply in engineering. Besides, the uncertainty of the rotational inertia of the spacecraft is not considered in the scheme, and the influence of the uncertainty on the control precision is ignored.
Disclosure of Invention
The invention provides a flexible satellite trajectory linearization attitude control method based on a disturbance observer, which aims to solve the problems that in the research of the disturbance suppression problem of the flexible spacecraft at present, a single trajectory linearization control method has weak disturbance suppression capability and poor robustness, and does not consider the influence of external disturbance and flexible accessories.
A flexible satellite trajectory linearization attitude control method based on a disturbance observer is realized by the following steps:
the method comprises the following steps: the flexible spacecraft dynamics modeling obtains a model of
Step two: performing nominal control design on the spacecraft by using the model obtained in the step one;
when the inverse of the controlled object can not be solved, obtaining the nominal control of the tracking system by solving the pseudo-inverse of the state variable;
to general will, etcEffect interferenceNeglecting, finding the kinematic and kinetic model of the system with respect to the nominal state variablesAndby inversion of
Wherein saidIs a nominal value of the angular velocity of the rotation of the spacecraft body coordinate system relative to the orbit coordinate system,for nominal control of moments in roll, pitch and yaw axes, Jxx、Jyy、JzzThe moment of inertia of the spacecraft along the rolling axis, the pitching axis and the yawing axis;
equations (15) and (16) can accurately represent the inverse of the control variable;
step three: the kinematic and dynamic model of the system relative to the nominal state variable is obtained by using the step twoAndinverse of (1), designing linear time-varying of tracking errorRegulator, obtaining state feedback control law u1And u2；
Step four: according to the state feedback control law u obtained in the step three2Designing a non-linear disturbance observer, and outputting the observerAs part of a system control law; obtaining the control law of the whole system
The invention has the following effects:
the invention designs a control algorithm for interference suppression and attitude tracking of a flexible spacecraft. Describing the attitude of the spacecraft by using an Euler angle, and establishing a flexible spacecraft dynamics and kinematics equation by adopting an equivalent interference idea; solving the pseudo-inverse of the controlled object under the condition of neglecting equivalent interference, and designing a quasi-differentiator in a specific form to obtain nominal control of the expected track; a linear time-varying regulator is designed by proportional-integral control. And considering the influence of equivalent interference, designing an interference observer and ensuring the asymptotic convergence of the tracking error of the flexible spacecraft.
Compared with the prior art, the invention has the following advantages:
1. the control algorithm is easy to realize and has stronger engineering practicability;
2. the uncertainty of the rotational inertia and the influence brought by a flexible accessory are considered, and the system interference is restrained by using the nonlinear interference observer;
3. by utilizing a track linearization method, the special pseudo-inverse does not influence the stability of a closed-loop system, and the method is suitable for a non-minimum phase system;
4. the linear time-varying regulator designed by utilizing the PD spectrum characteristic principle improves the anti-interference capability of the system and enhances the robustness of the system;
5. by adopting a method of combining the nonlinear disturbance observer and the trajectory linearization, the time of the system reaching the expected attitude is shortened, and the tracking precision is improved;
6. the steady state amplitude of the control moment and the modal vibration amplitude of the flexible windsurfing board are relatively small.
Drawings
FIG. 1 is a schematic diagram of a trajectory-based control systemand y represents the desired value and the actual value of the system output, respectively, η represents the linear time-varying feedback control law,and u represents the system nominal control and the overall control law.
FIG. 2 is a block diagram of TLC control based on a non-linear disturbance observer, where r represents the desired output of the system and y represents the actual output of the system; e represents the error between the expected output and the actual output of the system; u represents a linear time-varying feedback control law;represents a nominal control of the system;representing an estimate of the observer for the disturbance; d represents the total interference of the system.
FIG. 3 is a graph of expected and actual Euler angles of a spacecraft in 0-400 s; the upper graph, the middle graph and the lower graph respectively represent expected value and actual value change curve graphs of the rolling angle, the pitch angle and the yaw angle of the spacecraft within 0-400 s; in the drawings Representing the Euler angle of the spacecraft body coordinate system relative to the orbit coordinate system;representing the expected values of the roll, pitch and yaw of the spacecraft.
FIG. 4 is a graph of expected and actual Euler angles of a spacecraft in 0-10 s; the upper graph, the middle graph and the lower graph respectively represent expected value and actual value change curves of a rolling angle, a pitching angle and a yaw angle of the spacecraft within 0-10 s; in the drawingsRepresenting the Euler angle of the spacecraft body coordinate system relative to the orbit coordinate system;representing the expected values of the roll, pitch and yaw of the spacecraft.
FIG. 5 is a graph of expected and actual angular velocities of the spacecraft within 0-400 s; the upper graph, the middle graph and the lower graph respectively represent expected value and actual value change curves of the rolling angular velocity, the pitch angular velocity and the yaw angular velocity of the spacecraft within 0-400 s; in the figure ω1,ω2,ω3Representing the actual values of the rolling angular velocity, the pitch angular velocity and the yaw angular velocity of the spacecraft; omegad1,ωd2,ωd3Representing the expected values of the roll, pitch and yaw rates of the spacecraft.
FIG. 6 is a graph of expected and actual angular velocities of the spacecraft within 0-10 s; the upper graph, the middle graph and the lower graph respectively represent expected value and actual value change curves of the rolling angular velocity, the pitch angular velocity and the yaw angular velocity of the spacecraft within 0-10 s; in the figure ω1,ω2,ω3Representing the actual values of the rolling angular velocity, the pitch angular velocity and the yaw angular velocity of the spacecraft; omegad1,ωd2,ωd3Representing the expected values of the roll, pitch and yaw rates of the spacecraft.
FIG. 7 is a graph of Euler angle error of a spacecraft; in the drawingsRespectively representing the errors of the rolling angle, the pitch angle and the yaw angle of the spacecraft.
FIG. 8 is a graph of spacecraft angular velocity error; in the figure ωe1,ωe2,ωe3Respectively representing the errors of the rolling angular velocity, the pitch angular velocity and the yaw angular velocity of the spacecraft.
FIG. 9 is a control torque graph; in the figure Tc1,Tc2,Tc3Representing the control moments of the roll, pitch and yaw axes of the spacecraft, respectively.
FIG. 10 is a graph of modal vibration of a spacecraft sailboard, wherein eta is1,η2,η3,η4Representing four modal coordinates.
FIG. 11 is a graph of system actual and estimated interference; wherein the upper, middle and lower graphs represent actual and estimated disturbances on the roll, pitch and yaw axes, respectively; in the figure d1,d2,d3Respectively representing the actual interference on a rolling axis, a pitching axis and a yawing axis of the spacecraft; d1',d2',d3' represents the estimated disturbances on the spacecraft roll, pitch and yaw axes, respectively.
FIG. 12 is a spacecraft interference estimation error graph; in the figure de1,de2,de3Representing the interference estimation errors on the roll, pitch and yaw axes of the spacecraft, respectively.
Detailed Description
The first embodiment is as follows: a flexible satellite trajectory linearization attitude control method based on a disturbance observer comprises the following steps:
the key steps of the invention are as follows:
1. satellite attitude control
Attitude control of the satellite includes attitude determination, attitude stabilization control, and attitude maneuver control. The attitude determination is to research the azimuth or direction of the spacecraft relative to a reference datum so as to obtain attitude angle parameters, and the accuracy of the attitude angle parameters depends on the accuracy of an attitude sensor and an attitude algorithm. Attitude stabilization control is to maintain the attitude of the aircraft at a desired set direction and at a specified value. Attitude maneuver control is a reorientation process that transitions an aircraft from one attitude to another. The invention mainly carries out deep research on attitude tracking control under the condition that the flexible satellite is interfered.
The attitude control system ensures that the satellite is operating on the orbit with a predetermined attitude accuracy. The posture stabilization mode is divided into two basic types: passive stabilization systems and active stabilization systems. The combination of these two systems in turn leads to three types of semi-passive, semi-active and hybrid stabilization systems. From the control concept, the passive stabilization system belongs to an open-loop control system, and the active stabilization system belongs to a closed-loop negative feedback control system. The passive stabilization system controls the attitude of the satellite by using natural environment torque or physical torque sources, such as spin, gravity gradient, geomagnetic field, solar radiation pressure torque and aerodynamic torque. The active control is a three-degree-of-freedom attitude closed-loop control system in terms of control principle, does not depend on the intervention of a ground command center, and completely realizes the attitude control process by equipment carried by an aircraft. The stabilization method is mainly divided into spin stabilization and triaxial stabilization. The invention mainly researches the condition of triaxial stability.
2. Trajectory linearization
The track linearization control is suitable for solving the nonlinear tracking problem, the essence of the track linearization control is decoupling control, and the open-loop nonlinear dynamic inverse and linear time-varying feedback linearization control are combined, so that the exponential stability of the system output on the nominal track is ensured. The design idea of trajectory linearization is: firstly, a track tracking problem is converted into a tracking error regulation problem by using a nonlinear dynamic inverse method, and then a state feedback control law is designed by using a PD spectrum theory of a linear time-varying system, so that the tracking error of the system is converged to zero. The controller structure is shown in fig. 1.
Consider a multiple-input multiple-output nonlinear system of the form:
x∈Rn，u∈Rm，y∈Rmthe state, input and output of the system, respectively; f (x), g (x) and h (x) are smooth bounded functions of the appropriate number of bits. Order toRespectively representing the nominal state, output and control input of the system, there are:
the following state tracking error is defined:
and construct a control law of
wherein eta is the linear time-varying feedback control rate required to be designed, and is given later, the corresponding nonlinear tracking error dynamic system is
At this time, the original nonlinear system tracking problem is converted into a nonlinear regulation problem, and the controller comprises two parts:
a pseudo dynamic inverse controller for an open-loop controlled object based on a desired system output valueGenerating a nominal control input
a closed loop linear time varying feedback regulator η (e) is used to stabilize the system and to provide the system with certain response characteristics.
Taking into accountCan be regarded as a time-varying parameter of the system, therefore the formula (4) can be abbreviated as
Consider a linear time-varying system as follows:
consider a linear time-varying system as follows:
assume that 1: e-0 is an isolated balance point of formula (5), andcontinuous differentiable, Jacobian matrixWith respect to t being consistently bounded, the Lipschitz condition is satisfied on D.
Assume 2: a (t) and B (t) in the formula (6) are completely controllable.
A linear time-varying feedback controller can be designed from hypothesis 2:
η＝K(t)e (7)
the equilibrium point e of the linear time-varying system (6) is exponentially stabilized and recorded as 0
AC＝A(t)+B(t)K(t) (8)
From document [3] (Shao X, Wang H.A Novel Method of road trajectory Based on Disturbance Rejection [ J ]. physical Problemin Engineering,2014,2014), the linear time-varying feedback Control law (7) ensures that the system (4) is exponentially stable at the equilibrium point e of 0.
3. Non-linear interference observer
The nonlinear disturbance observer is used for observing external disturbance in a nonlinear system model and suppressing the influence of the disturbance through feedforward.
A typical nonlinear system is described as follows:
wherein x ∈ RnU ∈ R, d ∈ R respectively represent the state vector, the system input and the external interference.
In order to estimate and suppress the external disturbance, a disturbance observer needs to be designed. Assuming that the system interference originates from a linear exogenous system
To estimate the unknown disturbance d, an observer is designed as follows:
according to the document [8] (Chen W H. Disturbance based control for nonlinear systems [ J ]. Mechantronics, IEEE/ASME Transactions on,2004,9(4):706-710.), it is known that the estimation error of the observer converges asymptotically to 0.
4. Flexible spacecraft attitude model
The sailboard is used as a main flexible part of the spacecraft, and the movement of the sailboard is mutually coupled with the attitude movement of the spacecraft body. The motion of the flexible attachment is described by a distribution parameter of infinite degrees of freedom.
The motion of the spacecraft with the flexible attachment is described by adopting a hybrid coordinate method, namely, a central rigid body is described by using coordinates (such as Euler angles) which are usually used for describing the posture of the rigid body, and the flexible attachment is described by using discrete modal coordinates. Therefore, a dynamic model which can describe the motion of the spacecraft accurately enough and is convenient for the analysis and design of a spacecraft control system is established. Irrespective of the rotation of the windsurfing board relative to the spacecraft body, the kinematic equation for a flexible spacecraft is as follows:
wherein ω is [ ω ═ ω [ [ ω ]xωyωz]T∈R3Is the angular velocity of the spacecraft body, T ═ TxTyTz]T∈R3Control moment of rolling axis, pitching axis and yawing axis of spacecraft, d ═ dxdydz]T∈R3For disturbance moments acting on the aircraft, J ∈ R3×3Is the moment of inertia of the spacecraft, in the form,
η∈Rnrepresenting the flexible modal coordinate, n representing the modal order, δ ∈ R3×na matrix of coupling coefficients representing the flexible attachment and the spacecraft body, C ═ diag {2 ξiΩi,i＝1,2,…n},representing the vibration damping coefficient and the frequency coefficient matrix, xi, of the flexible attachment respectivelyi,ΩiRespectively representing the ith order modal damping ratio and modal frequency of the flexible attachment.
For analysis, the idea of equivalent interference is adopted, and the rigid-flexible coupling terms are usedAnd interference d are combined intoThe system model is simplified by considering the total interference of the system, and the flexible spacecraft dynamic model is
Neglecting the moment of inertia matrix J ∈ R3×3The coupling term in (1) does not need to consider a vibration equation of the sailboard, and a system kinematic model and a dynamic model are as follows:
5. design of tracking error linear time-varying regulator
Regulator design principles
The tracking error linear time-varying regulator adopts a state feedback PI control method, and the controller gain design method comprises the following steps:
(1) and controlling the outer ring Euler angle, and defining new state variables and input variables as follows:
wherein,θint＝∫θdt,ψintobtaining a new nonlinear state space model
Wherein
Obtaining a linearized tracking system
Wherein
(2) Using the same approach for inner loop angular velocity control, new state variables and input variables are defined as follows:
obtaining a linearized tracking system
Wherein
Wherein
(3) Designing a state feedback control law to enable the tracking error of the Euler angle to be asymptotically stable, and controlling the input u1＝-K1γaug，
determining desired damping and desired bandwidth ξ for a closed-loop system as a function of design requirements1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed loop system is
λ2+α1j2λ+α1j1＝0,j＝1,2,3 (57)
Closed loop system matrix
Euler angle tracking control system gain matrix K1Satisfy the requirement of
Acl1＝A1-B1K1(59)
K1The values can be determined from the equations (48), (49), (50), (58) and (59).
(4) Similarly, in order to realize the control of the attitude angular velocity of the spacecraft, a state feedback control law u is designed2＝-K2ωaugdetermining the desired damping and the desired bandwidth ξ for a closed-loop system according to design requirements1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed loop system is
λ2+α2j2λ+α2j1＝0,j＝1,2,3 (60)
Closed loop system matrix
Attitude angular velocity control gain matrix K2Satisfy the requirement of
Acl2＝A2-B2K2(62)
From the equations (54), (55), (56), (61) and (62), a control gain matrix K is determined2。
Proof of error convergence
Assume that 1: e is 0F: [0, ∞) x De→RnContinuously can be micro, De＝{e∈Rn|||e||<ReThe Jacobian matrixIs bounded and is at DeThe Lipschitz condition is satisfied.
Assume 2: system matrix pair (A)i(t),Bi(t)) are consistent and fully controllable.
From the above assumed conditions, when the system is equivalent to the errorTime-varying, linear time-varying state feedback control law ui＝Ki(t)ei,(ei＝γaug,ωaug) According to the document [3]It can be seen that the solution of the system is exponentially stable at the origin.
To simplify the design, let Acli＝Ai+BiKi，AcliIs the Hurwitz momentThe parameters of the array can be selected by PD characteristic spectrum principle.
6. Nonlinear disturbance observer design
Observer design principle
Considering equivalent interferenceThe influence on the system, the dynamic equation of the flexible spacecraft is in the form:
wherein f is2(ω)＝J-1·(-ω×Jω),g2＝J-1. The following non-linear disturbance observer is designed for unknown disturbances:
wherein,representing an estimation of an unknown equivalent disturbance, z is an internal state variable of a non-linear observer, p (ω) is a non-linear vector function to be designed, and the gain of the observer is defined as
Proof of convergence of estimation error
For simplicity, it is assumed that the equivalent interference is slowly time-varying, i.e.
Let the estimation error
Then
Substitution of formula (64) intoIn (1) obtaining
Due to the fact that
By substituting the formulae (63) and (69) into the formula (68) to obtain
P (ω) is designed such that equation (70) is globally stable. Then the error is estimatedApproaching 0.
In order to inhibit the interference problem in the spacecraft tracking process, the method of combining the nonlinear interference observer and the trajectory linearization control is adopted, the advantage that the trajectory linearization is suitable for solving the nonlinear tracking problem is utilized, the nonlinear interference observer is combined for interference inhibition, and the robustness of the system is improved. The method mainly includes the steps that on the premise that the rotary inertia of the flexible spacecraft, a flexible sailboard modal damping matrix, a flexible sailboard coupling coefficient, an initial attitude and a target attitude are given, a coupling item in the rotary inertia is ignored, an equivalent interference thought is adopted, equivalent errors are firstly ignored, nominal control of the spacecraft is obtained, then parameters of a linear time-varying regulator are determined according to a PD characteristic spectrum principle, a nonlinear system is decoupled, and tracking errors of the system are enabled to converge gradually. And finally, considering equivalent interference, and designing a nonlinear interference observer to compensate the interference. The structure diagram of the system control is shown in fig. 2, and the control law consists of three parts:
whereinNominal control, time-varying feedback control and disturbance compensation control are indicated, respectively.
The method comprises the following steps: the flexible spacecraft dynamics modeling obtains a model of
Step two: performing nominal control design on the spacecraft by using the model obtained in the step one;
when the inverse of the controlled object can not be solved, obtaining the nominal control of the tracking system by solving the pseudo-inverse of the state variable;
to the interference equivalentNeglecting, finding the kinematic and kinetic model of the system with respect to the nominal state variablesAndby inversion of
Wherein saidIs a nominal value of the angular velocity of the rotation of the spacecraft body coordinate system relative to the orbit coordinate system,for nominal control of moments in roll, pitch and yaw axes, Jxx、Jyy、JzzThe moment of inertia of the spacecraft along the rolling axis, the pitching axis and the yawing axis;
equations (15) and (16) can accurately represent the inverse of the control variable;
step three: the kinematic and dynamic model of the system relative to the nominal state variable is obtained by using the step twoAnddesigning a linear time-varying regulator for tracking error to obtain a state feedback control law u1And u2；
Step four: according to the state feedback control law u obtained in the step three2Designing a non-linear disturbance observer, and outputting the observerAs part of a system control law; obtaining the control law of the whole system
ωx，ωy，ωzThe system of the spacecraft projects in the directions of the rolling axis, the pitching axis and the yawing axis of the system of the spacecraft body relative to the angular velocity of the inertial system;
the spacecraft system projects in the directions of a rolling axis, a pitching axis and a yawing axis of the spacecraft system relative to the angular acceleration of the inertial system;
the system of the spacecraft is projected in the directions of the roll axis, the pitch axis and the yaw axis of the system of the spacecraft in relation to the angular velocity of the inertial system;
the spacecraft system projects nominal state variables in the directions of the rolling axis, the pitching axis and the yawing axis of the system coordinate system relative to the angular acceleration of the inertial system;
-euler angles of the spacecraft body coordinate system with respect to the orbit coordinate system;
-nominal values of angular velocities of rotation of the spacecraft body coordinate system with respect to the orbital coordinate system;
Tx，Ty，Tz-roll axis direction, pitch axis direction and yaw axis direction control moments;
-nominal control moments in roll axis direction, pitch axis direction and yaw axis direction;
dx，dy，dzroll axis direction, pitch axis direction and yaw axis direction disturbing moments;
η∈Rn-flexible modal coordinates, n being the modal order;
Jx(Jxx)，Jy(Jyy)，Jz(Jzz) The moment of inertia of the spacecraft along the roll, pitch and yaw axes;
δ∈R3×n-a matrix of coupling coefficients of the flexible appendage and the spacecraft body;
ξi-the ith order modal damping ratio of the flexible attachment;
Ωi-a modal frequency;
ωn,diff-the bandwidth of a low-pass filter attenuating the high-frequency gain;
-an euler angle integral variable;
ξ1j-desired damping of the outer loop closed loop system;
ωn1jbandwidth factor of the outer loop closed loop system；
ei-a tracking error;
-estimation of unknown equivalent interference;
z-the internal state variable of the non-linear observer;
p (ω) -the nonlinear vector function to be designed;
l (ω) -the gain of the observer.
d＝[dxdydz]T∈R3For disturbance moments acting on the aircraft;
-nominal control input, feedback control input and observer disturbance compensation.
The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is: the specific process of flexible spacecraft dynamics modeling in the first step is as follows:
the motion of the spacecraft with the flexible attachment is described by adopting a hybrid coordinate method, namely, a central rigid body is described by using coordinates (such as Euler angles) which are usually used for describing the posture of the rigid body, and the flexible attachment is described by using discrete modal coordinates. Therefore, a dynamic model which can describe the motion of the spacecraft accurately enough and is convenient for the analysis and design of a spacecraft control system is established.
Irrespective of the rotation of the windsurfing board relative to the spacecraft body, the kinematic equation for a flexible spacecraft is as follows:
for analysis, the idea of equivalent interference is adopted, and the rigid-flexible coupling terms are usedAnd interference d are combined intoThe system model is simplified by considering the total interference of the system, and the flexible spacecraft dynamic model is
The third concrete implementation mode: the present embodiment differs from the first or second embodiment in that: in the second step, the derivative form of the control variable appears in the nominal control, and a quasi-differentiator is needed to obtain the derivative of the nominal state, and the invention adopts a first-order quasi-differentiator with the form as follows:
where s is the Laplace operator, ωn,diffThe bandwidth of the low pass filter determines the ability of the differentiator to reject high frequency components.
The fourth concrete implementation mode: the difference between this embodiment mode and one of the first to third embodiment modes is: designing a linear time-varying regulator for tracking error in the third step to obtain a state feedback control law u1And u2The specific process comprises the following steps:
the method comprises the following steps: novel nonlinear state space modelAnd a linearized tracking systemObtaining;
and controlling the outer ring Euler angle, and defining new state variables and input variables as follows:
whereinθint＝∫θdt,ψintObtaining a new nonlinear state space model
Wherein
The linear tracking system is obtained as follows:
wherein
Step two: novel nonlinear state space modelAnd a linearized tracking systemObtaining:
using the same approach for inner loop angular velocity control, new state variables and input variables are defined as follows:
the linear tracking system is obtained as follows:
wherein
Wherein
Step three: law of state feedback control u1Obtaining;
design of the State feedback control law, control input u1＝-K1γaug，
determining desired damping and desired bandwidth ξ for a closed-loop system as a function of design requirements1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed-loop system is:
λ2+α1j2λ+α1j1＝0,j＝1,2,3 (30)
closed loop system matrix
Euler angle tracking control system gain matrix K1Satisfy the requirement of
Acl1＝A1-B1K1(32)
K1Can be determined from the formulae (21), (22), (23), (31), (32);
due to u1＝-K1γaugFrom K by1And gammaaugCan find u1；
Step four: law of state feedback control u2Obtaining;
design of the State feedback control law, control input u2＝-K2ωaugdetermining the desired damping and the desired bandwidth ξ for a closed-loop system according to design requirements1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed-loop system is:
λ2+α2j2λ+α2j1＝0,j＝1,2,3 (33)
the closed-loop system matrix is:
attitude angular velocity control gain matrix K2Satisfy the requirement of
Acl2＝A2-B2K2(35)
From the equations (26), (27), (28), (34) and (35), the control gain matrix K can be determined2；
Due to u2＝-K2ωaugFrom K by2And ωaugCan find u2。
The fifth concrete implementation mode: the difference between this embodiment and one of the first to fourth embodiments is: the specific process of designing the nonlinear disturbance observer in the fourth step is as follows:
considering equivalent interferenceThe influence on the system, the dynamic equation of the flexible spacecraft is in the form:
wherein f is2(ω)＝J-1·(-ω×Jω),g2＝J-1；
Designing a nonlinear disturbance observer aiming at unknown disturbance as follows:
wherein saidRepresenting the estimation of the unknown equivalent disturbance, z is the internal state variable of a non-linear observer, p (ω) is the non-linear vector function to be designed, the gain of the observer is defined as:
assuming that the equivalent interference is slowly time-varying, i.e.
Defining estimation errorWhen it is assumed thatThen e if p (ω) is chosen to satisfy equation (39) as globally stabled(t) approaches 0; the nonlinear vector function p (omega) to be designed meets the condition;
output of observerAs part of the system control law, to compensate for external disturbances. From document [9](Zhuliang. based on a non-linear disturbance observer)Aerospace vehicle trajectory linearization control [ J]The university of aerospace journal of Nanjing 2007,39(4):492-494) knows that compensating external interference with the output of the observer can avoid the influence of the interference on the system.
To sum up, the control law of the whole systemAccording to the document [9 ]]Therefore, the error dynamic characteristic local index of the composite closed loop system is stable.
The first embodiment is as follows:
1. simulation parameters:
the invention adopts the following simulation parameters;
flexible solar panel modal frequency matrix: omega ═ diag (0.7681; 1.1038; 1.8733; 2.5490). times.2 pi (rad/s)
Flexible solar panel modal damping matrix xi biag (0.056; 0.0086; 0.013; 0.025)
The initial attitude of the spacecraft is as follows:θ＝0.1°,ψ＝0.1°，
2. the controller parameters are as follows:
when nominal control is designed, a quasi-differentiator is used for estimating the derivative of a nominal state, and the parameters of the first-order quasi-differential regulator of an inner loop and an outer loop, namely the bandwidth of a low-pass filter are respectively as follows: omegan,diff_i＝15(rad/s),ωn,diff_oAt 5(rad/s), the high frequency gain is effectively attenuated.
the damping frequency of the system determines the regulation time and overshoot of the system, and further determines the stability of the system1j＝0.707,ξ2j＝0.707,j＝1,2,3
The bandwidth of the system also affects the dynamic performance of the closed-loop system, and the expected bandwidth of the inner loop and the outer loop in the patent is designed as follows:
ωn11＝0.01(rad/s),ωn12＝0.01(rad/s),ωn13＝0.01(rad/s)
ωn21＝1(rad/s),ωn22＝1(rad/s),ωn23＝0.5(rad/s)
the gain matrix of the observer can configure a closed-loop pole of a system and adjust the steady-state performance, and the gain of the nonlinear disturbance observer in the patent is as follows: l ═ 101010]T。
3. Simulation analysis
The control target of the invention is to enable the flexible spacecraft to track the expected attitude without error. In consideration of the practical situation, the execution capacity of the actuator is limited, so that the control torque is limited in amplitude in the simulation so as not to exceed 2 (N.m). The simulation results are shown in FIGS. 3-10:
as can be seen from the simulation results, the time for the spacecraft to reach the expected attitude is 4s, and the Euler angle steady-state error amplitude is 5×10-5rad, steady state error of angular velocity of 2X 10-4rad/s, the steady state amplitude of the control moment is 0.1Nm, and the maximum amplitude of the flexible mode vibration is 0.018.
In conclusion, the spacecraft attitude tracking can be realized by reasonably selecting the parameters of the error regulator and the gain of the observer based on the trajectory linear control of the nonlinear disturbance observer, the precision is high, and the flexible sailboard modal vibration amplitude is small. In addition, the non-linear disturbance observer is used for estimating and suppressing the uncertain disturbance, and a relatively ideal effect is achieved.
Claims (1)
1. A flexible satellite trajectory linearization attitude control method based on a disturbance observer is characterized by comprising the following steps:
the method comprises the following steps: the flexible spacecraft dynamics modeling obtains a model of
Wherein ω is [ ω ]xωyωz]T∈R3Is the angular velocity, omega, of the spacecraft bodyx、ωy、ωzProjecting the angular speed of the spacecraft system relative to the inertial system in the directions of a rolling axis, a pitching axis and a yawing axis of the spacecraft system; omega×Is composed of For angular acceleration of spacecraft, T ═ TxTyTz]T∈R3Control moments for the rolling, pitching and yawing axes of a spacecraft, Tx、Ty、TzControlling the moment in the roll axis direction, pitch axis direction and yaw axis direction, J ∈ R3×3Is the moment of inertia of the spacecraft and is,for spacecraft equivalent interference, η ∈ RnRepresenting the flexible modal coordinate, n representing the modal order, δ ∈ R3 ×na matrix of coupling coefficients representing the flexible attachment and the spacecraft body, C ═ diag {2 ξiΩi,i＝1,2,…n},representing the vibration damping coefficient and the frequency coefficient matrix, xi, of the flexible attachment respectivelyi,ΩiRespectively representing the ith order modal damping ratio and modal frequency of the flexible accessory;
step two: performing nominal control design on the spacecraft by using the model obtained in the step one;
when the inverse of the controlled object can not be solved, obtaining the nominal control of the tracking system by solving the pseudo-inverse of the state variable;
to the interference equivalentNeglect, find the kinematic and dynamic model of the systemRelative to nominal state variableAndby inversion of
Wherein saidIs a nominal value of the angular velocity of the rotation of the spacecraft body coordinate system relative to the orbit coordinate system,for nominal control of moments in roll, pitch and yaw axes, Jxx、Jyy、JzzThe moment of inertia of the spacecraft along the rolling axis, the pitching axis and the yawing axis;
equations (15) and (16) can accurately represent the inverse of the control variable;
step three: the kinematic and dynamic model of the system relative to the nominal state variable is obtained by using the step twoAnddesigning a linear time-varying regulator for tracking error to obtain a state feedback control law u1And u2；
Step four: according to the state feedback control law u obtained in the step three2Designing a non-linear disturbance observerOutput of the observerAs part of a system control law; obtaining the control law of the whole system
The specific process of flexible spacecraft dynamics modeling in the first step is as follows:
irrespective of the rotation of the windsurfing board relative to the spacecraft body, the kinematic equation for a flexible spacecraft is as follows:
coupling rigid and flexibleAnd interference d are combined intod＝[dxdydz]T∈R3In order to act on the disturbance moment on the spacecraft, the dynamic model of the flexible spacecraft is
In the second step, a derivative form of the control variable appears in the nominal control, a quasi-differentiator is needed to obtain a derivative of the nominal state, and a first-order quasi-differentiator is adopted, wherein the form is as follows:
where s is the Laplace operator, ωn,diffThe bandwidth of the low-pass filter determines the capacity of the differentiator for suppressing high-frequency components;
the design in step threeLinear time-varying regulator for tracking error to obtain state feedback control law u1And u2The specific process comprises the following steps:
step three, firstly: novel nonlinear state space modelAnd a linearized tracking systemObtaining;
and controlling the outer ring Euler angle, and defining new state variables and input variables as follows:
whereinObtaining a new nonlinear state space model
The linear tracking system is obtained as follows:
wherein A is1System matrix, B1Is an input matrix;
wherein O is3×3Denotes a 3 th order 0 matrix, I3×3Representing a 3-order unit array;
step three: novel nonlinear state space modelAnd a linearized tracking systemObtaining:
for the control of the inner ring angular velocity, new state variables and input variables are defined as follows:
wherein, ω isxint＝∫ωxdt,ωyint＝∫ωydt,ωzint＝∫ωzdt, obtaining a new nonlinear state space model
The linear tracking system is obtained as follows:
wherein A is2System matrix, B2Is an input matrix;
wherein
Step three: law of state feedback control u1Obtaining;
design of the State feedback control law u1＝-K1γaug，
determining desired damping and desired bandwidth ξ for a closed-loop system as a function of design requirements1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed-loop system is:
λ2+α1j2λ+α1j1＝0,j＝1,2,3 (30)
where λ represents the characteristic root of the closed-loop system, α1j2＝2·ξ1j·ωn1j,
Closed loop system matrix
Euler angle tracking control system gain matrix K1Satisfy the requirement of
Acl1＝A1-B1K1(32)
K1Can be determined from the formulae (21), (22), (23), (31), (32);
due to u1＝-K1γaugFrom K by1And gammaaugCan find u1；
Step three and four: law of state feedback control u2Obtaining;
design of the State feedback control law u2＝-K2ωaugDetermining the desired damping and the desired damping of the closed loop system according to design requirementsbandwidth xi1j,ωn1jAnd j is 1,2,3, the characteristic equation of the closed-loop system is:
λ2+α2j2λ+α2j1＝0,j＝1,2,3 (33)
wherein alpha is2j2＝2·ξ2j·ωn2j,
The closed-loop system matrix is:
attitude angular velocity control gain matrix K2Satisfy the requirement of
Acl2＝A2-B2K2(35)
From the equations (26), (27), (28), (34) and (35), the attitude angular velocity control gain matrix K is obtained2；
Due to u2＝-K2ωaugFrom K by2And ωaugCan find u2；
The specific process of designing the nonlinear disturbance observer in the fourth step is as follows:
considering equivalent interferenceThe influence on the system, the dynamic equation of the flexible spacecraft is in the form:
wherein f is2(ω)＝J-1·(-ω×Jω),g2＝J-1；
Designing a nonlinear disturbance observer aiming at unknown disturbance as follows:
wherein saidRepresenting the estimation of the unknown equivalent disturbance, z is the internal state variable of a non-linear observer, p (ω) is the non-linear vector function to be designed, the gain of the observer is defined as:
assuming that the equivalent interference is slowly time-varying, i.e. Representing the rate of change of the equivalent interference;
defining estimation errorWhen in useThen, if the non-linear vector function p (ω) to be designed is chosen such that equation (39) is globally stable, ed(t) approaches 0;
output of observerAs part of the system control law, the output of the observer is used for compensating external interference so as to avoid the influence of the interference on the system;
law of control of the entire systemAnd the error dynamic characteristic of the composite closed loop system is stable in local index.
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