1.
Introduction
Satellite attitude control system, which is an important guarantee system for satellite normal operation, affects the working performance and on-orbit lifetimes directly. Because of low cost and high reliability, magnetic control technology has been widely applied in the early stages of satellite development [1,2,3]. White, Shigemoyo and Bourquin firstly provided the concept of satellite attitude control, making it possible to achieve orientation of satellites with magnetictorquers. Soon, the first satellite magnetic attitude control method was proposed by Ergin and Wheeler [4], and they gave some advantages of magnetic control. Later, Renard [5] presented method based on averaged models to study the issue of attitude control with magnetic moments. Recently, wide application of periodic control system brings the research and development of magnetic control theory to a new height [6,7,8]. In order to predict the trajectory of a satellite with magnetic moments, numerous efforts have been made to investigate the nonlinear system's stability and controllability, which is based on the following Euler dynamic equation [9,10]
where J=diag(J1,J2,J3), ω=[ω1,ω2,ω3]T, ω0=[ω01,ω02,ω03]T, the unit vector er=[0,0,−1], and × is the familiar operation of cross product. The control moment Mc is
where m(t)=[m1,m2,m3]T and b(t)=Θb0(t), b0(t) can be approximated by the direct magnetic dipole as [11],
Many scholars focused on controllability of satellite magnetic control system based on control theories for nonlinear and linear time-varying continuous systems [12,13,14,15]. However, exact solutions and fundamental solution matrix of these time-varying continuous systems are hard to obtain, although controllability conditions of systems are satisfied. Therefore, the trajectory of satellites can not be predicted correctly. Therefore, numerous studies have been done to deal with the nonlinear second-order time-varying system (1.1) by linearization method [13,15]. With the development of computer technology, quite a number of research works regarding discrete-time systems have been reported in the literature [16,17,18] and the controllability and observability of discrete linear systems have attracted a lot of interest [19,20,21,22,23]. Witczak etc. provided a necessary and sufficient condition for the observability of first-order discrete time-varying linear systems. Mahmudov proposed the controllability and observability conditions of second-order discrete linear time-varying systems in a matrix form. The controllability and observability conditions for the problem of discrete satellite magnetic attitude control have not been presented to the best of our knowledge. Usually, the difference methods based on Taylor series are widely used for approximation discretization of the continuous-time systems [23]. A brief review of the possible approach to discretize linear and nonlinear time-varying systems so far has been presented in [24]. Here, we investigate controllability and observability property of the linearized form of equation (1.1) by transforming it into a discrete time-varying system with second derivation by the forward and backward Euler method. Then, the controllability and observability conditions of a new discrete second-order linear time-varying system are proposed, which are applied to investigate the controllability and observability of the discretized satellite magnetic control system. Different periods τ are chosen to investigate the effect on controllability and observability of the resulting discrete system.
The rest of this paper is structured as follows. In Section 2, two different discrete schemes of the second-order linear time-varying system represented by the linearized satellite magnetic attitude control motion equation are obtained by Euler method. The linearized satellite magnetic attitude control system is changed into a discrete second-order time-varying system. In Section 3, the controllability conditions of a new discrete second-order linear time-varying system are proposed, which are applied to investigate the controllability of the discretized satellite magnetic control system. Section 4 investigates the observability of the discrete second-order satellite magnetic control system based on corresponding observability conditions. We give concluding remarks in the final section.
2.
Transformation of satellite attitude control system
In this section, the nonlinear second-order time-varying system represented by satellite attitude magnetic control motion equation (1.1) is linearized. We obtain two different discrete schemes of the second-order linear time-varying system by Euler method and transform the linearized satellite magnetic attitude control motion equation into a discrete second-order time-varying linear system.
2.1. Linearized equation of satellite magnetic control
We assume the mass center of the satellite moving in earth's gravitational field and in a circular orbit. To satellite magnetic attitude control motion equation, the coordinate systems are described as follows:
(1) orbital system (X,Y,Z). The origin is satellite mass center. The Z-axis points in the direction of the radius vector; the Y-axis is normal to the satellite orbit plane; and the X-axis forms the right-hand trial.
(2) satellite body frame (x,y,z). The axes are assumed to coincide with the body's principle inertia axes and their origin are still at the center of satellite mass.
The attitude of system (x,y,z) relative to the orbital system (X,Y,Z) is given by Euler angles θ1,θ2,θ3. Then the components of ω have form [13]
where
and the direction cosine matrix Θ=[Θij] is represented using a 1-2-3 Euler angle rotation sequence as follows
In [15], Morozov and Kalenova gave the following linearized special second-order time-varying system of equation (1.1)
where
and
The complete linearized derivation of equation (1.1) can be found in [13].
With certain assumptions, the equation of the measurement yielded by the magnetometer is represented as [25]
where
and αj(j=1,2,3) are constant quantities determining the position of the orbit in space, x(t)∈R3,y(t)∈R2 are state vector and output vector respectively.
Then we have the following linear satellite magnetic attitude control system with measurement, corresponding coefficient matrices are in accordance with matrices in equation (2.4) and (2.5)
2.2. Discretization of the linearized satellite control system
Firstly, letting t=kτ(k=0,1,2...), we realize an approximate form of ˙x(t) and ¨x(t) using Taylor forward expansion as
and
Substituting equation (2.7) and (2.8) into system (2.6), we have
Then we can get
where I3 denotes the identity matrix of 3 dimension and
Noting that
and
which means the matrix I3+τD is invertible. Therefore, we can rewrite the system (2.10) as the following discrete system
where
Next, by analogy, the system (2.6) can also be discretized with backward Euler method. We can obtain an approximate form of ˙x(t) and ¨x(t) using Taylor backward expansion as
and
Substituting equation (2.15) and (2.16) into system (2.6), letting t=kτ(k=0,1,2...),
Based on equation (2.11) and (2.17) we get
Therefore, we can rewrite the system (2.17) as the following discrete system with backward Euler method,
where
In general, when the sampling period τ is about one tenth of the minimum time constant of the system, the approximation is satisfactory enough. In fact, the error between the exact solution and the numerical solution will be lager if the period is too big, which makes the mathematical precision lower. On the contrary, if the period is too small, then step size increases and the computation is huge. Then, based on the similar form of discrete system (2.14) and (2.19), we can rewrite them as the following general discrete system
where xk∈Rn,yk∈Rr,uk∈Rm(m≤n) are the state vector, the output vector and the control vector respectively. ˆA0,ˆA1∈Rn×n, ˆBk∈Rn×m,Ck∈Rr×n are coefficient matrices.
3.
Controllability of the discrete system
3.1. Controllability analysis
According to the definition of controllability of discrete linear time-varying systems in [26], we present the definition of controllability and uncontrollability of the second-order discrete time-varying linear system (2.20).
Definition 1. The second-order discrete time-varying linear system (2.20) is said to be controllable to final state xn=xf in finite n steps if it exists an input sequence U={u0,u1,...,un−1} which brings the initial state (x−1,x0) to a final state xn=xf in the finite discrete time interval [0,n]. Otherwise the system (2.20) is uncontrollable.
Theorem 1. The linear discrete-time varying system (2.20) is controllable if and only if rankC=n, and the controllability matrix C is defined by
where
Proof Based on the form of the state vector in the system (2.20), we derive the following set of equations
Thus, by iteration, it is finally not difficult to find that in equations (3.3) the k−th instant state vector xk starting from initial time k=0 and k=1 is
where Q(k),P(k),M(k)i are polynomial functions consisted of matrices ˆA0,ˆA1, and they satisfy the following important iteration equations
In a general case, the state vector at final time n can be written as
Therefore
If we denote
as the controllability matrix in the sequel, then the equation (3.7) determines the input sequence which transfers the initial state (x−1,x0) to the desired state xf=xn in n steps. Thus, these equations will have a solution for any given vector xf if and only if the matrix has full rank, i.e. rankC=n, the discrete system (2.20) is controllable.
In order to further verify the correctness of the theoretical results, some numerical examples are given.
Example.1 Let us investigate the controllability of the following system [20]:
Firstly, it is noticed that there is no ˙x(t) in system (3.8), which means the discrete forms with forward and backward method of system (3.8) are equivalent, that is
where
Letting τ=0.1s, according to Theorem 1, we have
Then
here,
and
Therefore, discrete system (3.9) is controllable.
Example.2 The equations of controlling the motion of a spacecraft between the earth and the moon have the form [25]
Here,
For forward Euler method, the discrete form of system is as follows:
Then, choosing τ=0.1s, we have
which means the inverse matrix of I+2τD exits and
The discrete system can be represented as
where
According to Theorem 1, we have
Then
here
Because of α1=1+2b,α2=1−b,0<b<<1, then
Therefore, discrete system (3.13) is controllable. Similarly, we can prove the controllability of system (3.12) with forward Euler discretized form.
3.2. Numerical calculations of controllability of discrete magnetic attitude control system
To the specific discrete system (2.19) with backward difference, we have
where
It can be shown
where
Based on the results in equation (3.17), the condition of full rank of C is not satisfied when the determinant of matrix C composed of the first three columns. However, if the other three independent columns (such as, 1, 2, 5) are selected, for convenience, we choose τ=0.1 as the period of the discrete system without loss of generality, it is easy to see that det(C)≠0 on the basic of satellite and its orbit parameters in [27]. Then
According to Theorem 1, the system (2.19) by backward Euler method is controllable.
Moreover, we make τ=0.2,0.05 respectivly to compute the rank of C and the results show different τ have no effect on the controllability of system (2.19). In the same way, we can also calculate the rank of C for the system (2.14) with Matlab and then prove the discretized system (2.14) by forward Euler method is controllable.
4.
Observability of the discrete system
4.1. Observability analysis
Based on the definition of observability of discrete linear time-varying systems in [28], we can also present the following definition of observability of system (2.20).
Definition 2. The second order discrete linear time-varying system (2.20) is observable if for any unknown initial state (x0,x1), there exists a finite kβ∈N(kβ>0) such that (x0,x1) can be determined uniquely from the knowledge of output yk and input uk,k∈[0,kβ]. Otherwise the system is said to be unobservable.
Theorem 2. The second-order linear discrete-time system (2.20) is observable if and only if the observability matrix S has rank equal to 2n and
where the definitions of M(k),P(k) are same as above, and O denotes the relative dimensions of zero matrix.
Proof Taking k=0,1,... in system (2.20) and equations (3.3), we generate the following sequence
Then the measurement yk according to the equation (3.4) of state vector xk is
In general, we have
As consequence, equation (4.2) can be rewritten in the following relevant matrix form
We know from linear algebra that the system of linear algebra equations with 2n unknowns, equation (4.1) has a unique solution (x0,x1) if and only if the system matrix has rank 2n:
The matrix in (4.3) we denote by S. Then the initial values x0,x1 are determined uniquely, if and only if rankS=2n.
Analogously, a numerical example is given as follows to verify the validity of observability analysis.
Example.3 Considering the following system with discrete method [20]
The discrete system has the form
where
Letting τ=0.1s, according to Theorem 2, we have
Therefore, the discrete system (4.5) is observable.
4.2. Numerical calculations of observability of discrete magnetic attitude control system
To system (2.19), according to equations (3.5), we also have
and
and we can rewrite the measurement matrix Ck as the following form
letting
where
we get
and
where
The following matrix U is chosen as the first six columns of the observability matrix S
The determination of matrix U is
Generally speaking, the determination of matrix U is not equal to 0 if and only if αj(j=1,2,3)≠0, which is valid according to the definition of αj. In addition, using Matlab, choosing τ=0.1, we also have
Based on Theorem 2, the system (2.19) with backward Euler method is observable.
Moreover, we make τ=0.2,0.05 respectivly to compute the rank of S and the results show different τ have no effect on the observability of system (2.19). The observability of system (2.14) with forward Euler method can be proven similarly.
5.
Conclusions
In this paper, two different discrete schemes of the second-order linear time-varying system represented by the linearized satellite magnetic attitude control motion equation are obtained by Euler method. Subsequently, the controllability and observability conditions of a new discrete second-order linear time-varying system are proposed, which are applied to investigate the controllability and observability of the discretized satellite magnetic control system. Some numerical examples are given to further verify the correctness of theoretical results. Research results show that, generally speaking, different periods τ and parameters in coefficient matrices have no effect on the controllability and observability of the resulting discrete system.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (12171196).
Conflict of interest
All authors declare no conflicts of interest in this paper.