A method for designing a state observer for a space tether system

Abstract

The application discloses a state observer design method of a space tether system, a rigid body tether with a tensioning effect is used as the tether; firstly, a dynamics equation of the whole tether satellite system is established by using a Lagrange energy method; then, a criterion for the observability of the system is derived by using a linearization method when the velocity term state of the system is unobservable; and a nonlinear system state observer satisfying a linear observation convergence condition is further designed under the observability condition; the obtained observer can guarantee local asymptotic stability around a linearization point. The application can guarantee that the observation error converges in a dynamically changing neighborhood of an equilibrium point, and the application can be designed separately from a controller to realize the stability of the system.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of satellites, and particularly relates to a state observer design method for a space tether system. BACKGROUND

[0002] The space tether can be used in space capture, formation configuration and space "elevator" and other space special tasks which have been tested or imagined. The tension provided by the tether can offset the external force or Coriolis force received by the satellite, which helps to save the energy consumption generated by the active variable configuration and on-orbit operation. The position and attitude of the space body are easy to obtain through sensor measurement, but the rate of change of the state (usually the variable of system analysis) is relatively difficult to accurately measure. Therefore, designing the velocity item state observer of the space tether has great significance for the dynamics analysis and control of the tether. SUMMARY

[0003] In order to overcome the deficiencies of the prior art, the application provides a state observer design method for a space tether system. The tether adopts a rigid body tether with a tension effect. First, the Lagrange energy method is used to establish the dynamic equation of the entire tether satellite system. Then, when the velocity item state of the system is unmeasurable, the linearization method is used to derive the observability criterion of the system. Under the observability condition, a nonlinear system state observer is further designed to meet the linear observation convergence condition. The obtained observer can ensure local asymptotic stability around the linearization point. The application can ensure that the observation error converges in a dynamically changing neighborhood of the equilibrium point, and the controller can be designed separately to realize the stability of the system.

[0004] The technical solution adopted by the application to solve the technical problems comprises the following steps:

[0005] Step 1: definition of coordinate system;

[0006] Define E-XYZ as the earth-centered inertial coordinate system, the Z axis is perpendicular to the equatorial plane, the X axis and the Y axis are located in the equatorial plane, and the X axis points to the direction of the vernal equinox, and the Y axis direction is determined by the right-hand rule;

[0007] Define o-xyz as the system mass center orbit coordinate system, wherein the x axis points from the earth center to the system mass center, the z axis is perpendicular to the orbit plane, and the y axis is determined by the right-hand rule;

[0008] Define o-xyz as the system mass center coordinate system, which is obtained by rotating the orbit system by the y axis by an angle of β clockwise, wherein β is the out-of-plane angle of the system motion;

[0009] Step 2: According to the kinetic characteristics of the tether: rigid tether with pull-up effect, the whole system is approximately a rigid body, and the energy of the system includes the kinetic energy and potential energy of the end-point agent and the elastic potential energy of the tether; the tether system is unfolded by using the Lagrange equation as shown below:

[0010]

[0011] Where L is the Lagrange variable, X is the variable, and Q is the non-conservative generalized force received by the system;

[0012] Thus, the dynamics equation of the tether system is obtained, which is a second-order nonlinear system, expressed as:

[0013]

[0014] Where x represents the state vector of the system, M, N, and P are matrices of corresponding dimensions, and M is non-singular, Q x represents the generalized force corresponding to the state variable;

[0015] Step 3: State space transformation of the dynamics equation of the tether system, and order reduction of the system, so that the system is transformed into an affine nonlinear system;

[0016] Step 4: The derivatives of position and angle in the state space dynamics equation of the tether are linear functions of their rates of change, and the derivatives of their rates of change are composed of nonlinear functions of the whole system variables and linear functions of the input, and their specific expressions are as follows:

[0017]

[0018] Where x and y represent position and angle, a and b are constants, u is the control variable, and f(x, y) represents the nonlinear term in the system dynamics equation;

[0019] Step 5: The output vector of the tether system is c i = [1 0], where i corresponds to the i-th subsystem composed of the i-th variable and its rate of change, i.e., the output represents the length or angle of the tether, and the length rate and angle rate are unobservable; the nonlinear state observer designed by output feedback is:

[0020]

[0021] Where the nonlinear term is:

[0022]

[0023] is the nonlinear function term in the i-th subsystem, and C = [c1 … c m ] TLet x be the state vector of the system, and let y be the output vector matrix, respectively, represents the estimated state of the i-th observer subsystem, H is the observer coefficient matrix, and B represents the input state matrix;

[0024] The observation error vector approximately satisfies:

[0025]

[0026] wherein, is the Jacobian matrix of the nonlinear term F, and

[0027]

[0028] The linearization error of the nonlinear term F is a structural disturbance;

[0029] Step 6: Since the Jacobian matrix of the nonlinear term F i is:

[0030]

[0031] Therefore,

[0032]

[0033] For the linearized system, (c i J i ) is a pair of observable pairs; according to the observability criterion of the linear system, there is an observation matrix H such that J-HC is a stable matrix, so that a linear observer applicable in a neighborhood near the linearization point is obtained.

[0034] The beneficial effects of the present application are as follows:

[0035] The nonlinear state observer of the present application is aimed at the case that the velocity term state of the second-order nonlinear system of the space tether is unobservable, adopts a linearization method to prove the convergence condition and realizability of the observer error, and uses the stability condition of the observation error equation to push out the design scheme of the observer coefficient matrix, which, when applied to an actual system, can guarantee that the observation error converges in a dynamically changing neighborhood of the equilibrium point, and can be designed separately with the controller to realize the stability of the system. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 is a schematic diagram of the coordinate system of the space tether satellite system of the present application.

[0037] Figure 2 is a structural diagram of the state observer of the space tether designed by the present application.

[0038] Figure 3The response schematic diagram of the in-plane angular velocity under the action of the observer of the application.

[0039] Figure 4 The response schematic diagram of the in-plane angular velocity under the action of the observer of the application. DETAILED DESCRIPTION

[0040] The application is further illustrated below with reference to the accompanying drawings and examples.

[0041] The application aims to provide a state estimation method of a space tether system, and a design principle of a tether state observer satisfying a condition is obtained through a convergence condition of a state observer.

[0042] 1. Definition of coordinate system

[0043] The coordinate schematic diagram of the system under study is shown in Figure 1 , wherein E-XYZ is an earth-centered inertial coordinate system, the Z axis is perpendicular to the equatorial plane, the X axis and the Y axis are located in the equatorial plane, and the X axis points to the direction of the vernal equinox, and the direction of the Y axis is determined by the right-hand rule. o-xyz is a system mass center orbit coordinate system, wherein the x axis points from the earth center to the system mass center, the z axis is perpendicular to the orbit plane, and the y axis is determined by the right-hand rule. In addition, o-xyz is a system mass center body system, which is obtained by rotating the orbit system by the y axis by an angle of β clockwise, and β is the out-of-plane angle of the system motion. The out-of-plane rotation motion of the system can be tracked in real time by using the body system, and the three-dimensional motion characteristics of the system are given by combining the in-plane motion law of the system in the orbit system.

[0044] The three-dimensional coordinates of the end point satellite are obtained in the body system, and the obtained three-satellite coordinates are

[0045]

[0046] The conversion matrix between the body system and the orbit system is utilized

[0047]

[0048] The coordinates in the body system above can be converted to the orbit system, and the obtained coordinates reflect the relationship between a group of system variables l i , θ i , β and the coordinates. The kinetic energy and the potential energy of the system are conveniently constructed by using the coordinates, and then the dynamic equation about the system variables l i , θ i , β can be obtained by the Lagrange method.

[0049] Specific method:

[0050] A) According to the dynamic characteristics of the tether—rigid tether with the effect of pull-up, the whole system is approximately a rigid body, and the energy of the system includes the kinetic energy and potential energy of the end-point agent, and the elastic potential energy of the tether. The Lagrange equation is expanded as follows:

[0051]

[0052] where L is the Lagrange variable, X is the variable, Q is the non-conservative generalized force suffered by the system, and the dynamics equation of the tether system is obtained, which is a second-order nonlinear system, expressed as:

[0053]

[0054] where x represents the state vector of the system, M, N, P are matrices of corresponding dimensions, and M is non-singular, Q x represents the generalized force corresponding to the state variable.

[0055] B) The obtained dynamics equation contains a set of basic variables, including the length of the tether, the in-plane angle, the out-of-plane angle, and the rate of change of the length and angle. In order to facilitate system analysis, the equation containing the above variables is transformed into state space, and the purpose of the transformation is to reduce the order of the system, so that the whole system behaves as an affine nonlinear system.

[0056] C) In the state space dynamics equation of the tether, the derivative of the position and angle is a linear function of the rate of change, and the derivative of the rate of change is composed of a nonlinear function of the whole system variables and a linear function of the input, and its specific expression is as follows:

[0057]

[0058] where x, y represent the position / angle and the corresponding rate of change, respectively, and a, b are constants.

[0059] D) The output vector is c i = [1 0], where i corresponds to the i-th variable and its rate of change to form a subsystem, i.e., the output represents the length or angle of the tether, and the length rate and angle rate are not observable. The nonlinear state observer designed by output feedback is

[0060]

[0061] Nonlinear term:

[0062]

[0063] where is the nonlinear function of the i-th subsystem, and C = [c1 … c m ] TThis is the output vector matrix. The observation error vector obtained from the original system and the observer system approximately satisfies the following in a small neighborhood of the equilibrium point:

[0064]

[0065] in, Let F be the Jacobian matrix of the nonlinear term F, and:

[0066]

[0067] The linearization error of the nonlinear term is a structural perturbation.

[0068] E) The condition for convergence of observation error is that J-HC is the Hurwitz matrix. By designing an observer coefficient matrix that meets the requirements based on this condition, the designed nonlinear observer can guarantee that the observation error is asymptotically stable in the neighborhood of the equilibrium point.

[0069] Due to the nonlinear term F i The Jacobian matrix is:

[0070]

[0071] therefore,

[0072]

[0073] That is, for the linearized system, we have (c i J i ( ) are an observable pair. According to the observability criterion for linear systems, an observation matrix H can always be found such that J-HC is a stable matrix. In this case, for nonlinear systems, a linear observer applicable in a neighborhood of the linearization point is obtained. When the initialization error of the observer is within a certain range, the convergence of the observer can always overcome the error interference of the nonlinear term, allowing the observer to obtain an estimate of the unmeasurable term of the original system—the velocity term—for practical control and filtering applications. Furthermore, the linearization point can be adjusted in real time according to changes in the task scenario, and the small computational load of the observer design always meets the real-time requirements. Specific implementation examples:

[0075] like Figure 2 The diagram shown is a structural diagram of the designed space tethered state observer.

[0076] in, An estimate representing the length / angle variable. This represents the estimate of linear velocity / angular velocity. Unlike the state equation (1), the observer's equation includes feedback of the output error e. By adjusting the feedback coefficient h... iThe design of the observer guarantees the convergence of the observation error to zero. Moreover, F(X) contains the nonlinear terms of the system, and the use of F(X) in the observer design continues in the observer design, and the nonlinear uncertainty caused by it is mainly relied on the coefficients of the linear observer to make it converge, so it is locally stable.

[0077] As Figure 3 The response of the closed-loop system after the designed observer is adopted and the sliding mode feedback control is added to the system is shown. When the initial state of the observer is in the appropriate neighborhood of the system state, it can be seen that the true rope length variable and the estimated variable both asymptotically converge to the nominal value 1, and the convergence trajectories are basically consistent.

[0078] As Figure 4 The response of the in-plane angular velocity under the action of the observer is shown, and the two angular velocities and and their respective observations all converge to the nominal value 2 near the nominal value, which is specified by the sliding mode control law.

Claims

1. A method for designing a state observer for a space tethered system, characterized in that, Includes the following steps: Step 1: Define the coordinate system; Define E-XYZ as the Earth-centered inertial coordinate system, with the Z-axis perpendicular to the equatorial plane, the X-axis and Y-axis both located in the equatorial plane, the X-axis pointing towards the vernal equinox, and the Y-axis direction determined by the right-hand rule; Define o-xyz as the system's center-of-mass orbital coordinate system, where the x-axis points from the Earth's center to the system's center of mass, the z-axis is perpendicular to the orbital plane, and the y-axis is determined by the right-hand rule; Define o-xyz as the system's center of mass, obtained by rotating the orbital system clockwise around the y-axis by an angle β, where β is the out-of-plane angle of the system's motion; Step 2: Based on the dynamic characteristics of the tether: a rigid rope with a lifting effect, the entire system is approximated as a rigid body. The system's energy includes the kinetic and potential energy of the endpoint agents and the elastic potential energy of the tether; the tether system is expanded using the Lagrange equation shown below: Where L is a Lagrange variable, X is a variable, and Q is the non-conservative generalized force acting on the system; This yields the dynamic equation of the rope system, which is a second-order nonlinear system, expressed as: Where x represents the system's state vector, M, N, and P are matrices of corresponding dimensions, and M is non-singular, Q x This represents the generalized force corresponding to the state quantity; Step 3: Perform a state-space transformation on the dynamic equations of the rope system to reduce the order of the system and transform it into an affine nonlinear system; Step 4: In the tethered dynamic equations after state spatialization, the derivatives of position and angle are linear functions of their rates of change. The derivatives of their rates of change are composed of nonlinear functions of the entire system variables and linear functions of the inputs. Their specific expressions are as follows: Where x and y represent position and angle, respectively, a and b are constants, u is the control variable, and f(x,y) represents the nonlinear term in the system dynamics equation; Step 5: The output vector of the rope system is c i = [1 0], where i corresponds to the subsystem consisting of the i-th variable and its rate of change, i.e., the output is represented by the length or angle of the tether, while the rate of change of length and the rate of change of angle are unobservable; the nonlinear state observer designed using the output feedback method is: Among them, the nonlinear term: Let C be a nonlinear function term in the i-th subsystem, where C = [c1…c2] m ] T To output the vector matrix, Let H and B represent the estimated states of the i-th observer subsystem, respectively, where H is the observer coefficient matrix and B is the input state matrix. The observation error vector approximately satisfies the following in the neighborhood of the equilibrium point: in, Let F be the Jacobian matrix of the nonlinear term F, and The linearization error of the nonlinear term F is a structural perturbation; Step 6: Due to the nonlinear term F i The Jacobian matrix is: therefore, For the linearized system, (c i J i ) is an observable pair; according to the observability criterion of linear systems, there exists an observation matrix H such that J-HC is a stable matrix, thus obtaining a linear observer applicable in a neighborhood near the linearization point.

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