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Research article Special Issues

Transformation of PET raw data into images for event classification using convolutional neural networks

  • Received: 09 March 2023 Revised: 09 June 2023 Accepted: 25 June 2023 Published: 12 July 2023
  • In positron emission tomography (PET) studies, convolutional neural networks (CNNs) may be applied directly to the reconstructed distribution of radioactive tracers injected into the patient's body, as a pattern recognition tool. Nonetheless, unprocessed PET coincidence data exist in tabular format. This paper develops the transformation of tabular data into n-dimensional matrices, as a preparation stage for classification based on CNNs. This method explicitly introduces a nonlinear transformation at the feature engineering stage and then uses principal component analysis to create the images. We apply the proposed methodology to the classification of simulated PET coincidence events originating from NEMA IEC and anthropomorphic XCAT phantom. Comparative studies of neural network architectures, including multilayer perceptron and convolutional networks, were conducted. The developed method increased the initial number of features from 6 to 209 and gave the best precision results (79.8) for all tested neural network architectures; it also showed the smallest decrease when changing the test data to another phantom.

    Citation: Paweł Konieczka, Lech Raczyński, Wojciech Wiślicki, Oleksandr Fedoruk, Konrad Klimaszewski, Przemysław Kopka, Wojciech Krzemień, Roman Y. Shopa, Jakub Baran, Aurélien Coussat, Neha Chug, Catalina Curceanu, Eryk Czerwiński, Meysam Dadgar, Kamil Dulski, Aleksander Gajos, Beatrix C. Hiesmayr, Krzysztof Kacprzak, Łukasz Kapłon, Grzegorz Korcyl, Tomasz Kozik, Deepak Kumar, Szymon Niedźwiecki, Szymon Parzych, Elena Pérez del Río, Sushil Sharma, Shivani Shivani, Magdalena Skurzok, Ewa Łucja Stępień, Faranak Tayefi, Paweł Moskal. Transformation of PET raw data into images for event classification using convolutional neural networks[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14938-14958. doi: 10.3934/mbe.2023669

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  • In positron emission tomography (PET) studies, convolutional neural networks (CNNs) may be applied directly to the reconstructed distribution of radioactive tracers injected into the patient's body, as a pattern recognition tool. Nonetheless, unprocessed PET coincidence data exist in tabular format. This paper develops the transformation of tabular data into n-dimensional matrices, as a preparation stage for classification based on CNNs. This method explicitly introduces a nonlinear transformation at the feature engineering stage and then uses principal component analysis to create the images. We apply the proposed methodology to the classification of simulated PET coincidence events originating from NEMA IEC and anthropomorphic XCAT phantom. Comparative studies of neural network architectures, including multilayer perceptron and convolutional networks, were conducted. The developed method increased the initial number of features from 6 to 209 and gave the best precision results (79.8) for all tested neural network architectures; it also showed the smallest decrease when changing the test data to another phantom.



    Table 1.  Nomenclature.
    b earth's magnetic induction field b0 earth's magnetic induction field in coordinate system
    er unit vector of orbital coordinate system Ji main central inertia moments, i=1,2,3
    Mc magnetic control moment m intrinsic magnetic moment
    r orbital radius ω absolute angular velocity
    ω0 orbital angular velocity θi attitude Euler angles,i=1,2,3
    μE earth's magnetic field constant β inclination angle of orbital plane to the equatorial plane
    Θ direction cosine matrix

     | Show Table
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    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]

    Jdωdt+ω×Jω=3ω20(er×Jer)+Mc, (1.1)

    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

    Mc=m(t)×b(t),

    where m(t)=[m1,m2,m3]T and b(t)=Θb0(t), b0(t) can be approximated by the direct magnetic dipole as [11],

    b0(t)=μEr3[cosω0tsinβcosβ2sinω0tsinβ].

    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.

    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.

    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]

    ω=[ω1ω2ω3]=Cθ3θ2[˙θ1˙θ2˙θ3]+Θ[ω01ω02ω03], (2.1)

    where

    Cθ3θ2=[cosθ2cosθ3sinθ30cosθ2sinθ3cosθ30sinθ201], (2.2)

    and the direction cosine matrix Θ=[Θij] is represented using a 1-2-3 Euler angle rotation sequence as follows

    {Θ11=cosθ2cosθ3,Θ12=cosθ1sinθ3+sinθ1sinθ2cosθ3,Θ13=sinθ1sinθ3cosθ1sinθ2cosθ3,Θ21=cosθ2sinθ3,Θ22=cosθ1cosθ3sinθ1sinθ2sinθ3,Θ23=sinθ1cosθ3+cosθ1sinθ2sinθ3,Θ31=sinθ2,Θ32=sinθ1cosθ2,Θ33=cosθ1cosθ2. (2.3)

    In [15], Morozov and Kalenova gave the following linearized special second-order time-varying system of equation (1.1)

    ¨x(t)+D˙x(t)+Kx(t)=B(t)u(t), (2.4)

    where

    D=[00J1+J32J1ω0000J21J3J3ω000]=[00d1ω0000d3ω000],
    K=[4ω20J32J10003ω20J13J2000ω20J21J3]=[k1ω2000k2ω20000k3ω20],
    B(t)=ω20ϵ[02b1sinω0tb42b2sinω0t0b2cosω0tb5b3cosω0t0],

    and

    x(t)=[θ1,θ2,θ3]Tisthestatevector,ϵ=μEω20r3,d=J2J1+J3,di=dJi(i=1,3),Jij=JiJj(i,j=1,2,3),k1=4J32J1,k2=3J31J2,k3=J12J3,bj=sinβJj(j=1,2,3),b4=cosβJ1,b5=cosβJ3u(t)=m(t)=[m1,m2,m3]Tiscontrolvector,whichallowsthesatelliteattitudepositiontobestabilized.

    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]

    y(t)=C(t)x(t) (2.5)

    where

    C(t)=[α20α1sinω0tα3cosω0t0α2α1cosω0tα3sinω0t]

    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)

    {¨x(t)+D˙x(t)+Kx(t)=B(t)u(t),y(t)=C(t)x(t). (2.6)

    Firstly, letting t=kτ(k=0,1,2...), we realize an approximate form of ˙x(t) and ¨x(t) using Taylor forward expansion as

    ˙x(t)x((k+1)τ)x(kτ)τ (2.7)

    and

    ¨x(t)x((k+1)τ)2x(kτ)+x((k1)τ)τ2. (2.8)

    Substituting equation (2.7) and (2.8) into system (2.6), we have

    {x((k+1)τ)2x(kτ)+x((k1)τ)τ2+Dx((k+1)τ)x(kτ)τ+Kx(kτ)=B(kτ)u(kτ),y(kτ)=C(kτ)x(kτ). (2.9)

    Then we can get

    {(I3+τD)xk+1+(τ2KτD2I3)xk+xk1=τ2Bkuk,yk=Ckxk, (2.10)

    where I3 denotes the identity matrix of 3 dimension and

    xk=x(kτ),uk=u(kτ),yk=y(kτ),Ck=C(kτ),Bk=B(kτ). (2.11)

    Noting that

    I3+τD=[10d1ω0τ010d3ω0τ01] (2.12)

    and

    det(I3+τD)=1+d1d3ω20τ2>0, (2.13)

    which means the matrix I3+τD is invertible. Therefore, we can rewrite the system (2.10) as the following discrete system

    {xk+1=A0xk1+A1xk+˜Bkuk,yk=Ckxk, (2.14)

    where

    A0=(I3+τD)1=[1d1d3ω20τ2+10d1ω0τd1d3ω20τ2+1010d3ω0τd1d3ω20τ2+101d1d3ω20τ2+1],
    A1=(I3+τD)1(τD+2I3τ2K)=[1+k1ω20τ2+1d1d3ω20τ2+10d1τω0(k3ω20τ2+1)d1d3ω20τ2+10k2ω20τ2+20d3ω0τ(k1ω20τ2+1)d1d3ω20τ2+101+k3ω20τ2+1d1d3ω20τ2+1],
    ˜Bk=(I3+τD)1τ2Bk=τ2ω20ϵ[b5d1ω0τd1d3ω20τ2+12b1sinω0kτb3d1ω0τcosω0kτd1d3ω20τ2+1b4d1d3ω20τ2+12b2sinω0kτ0b2cosω0kτb5d1d3ω20τ2+1(b3cosω0kτ+2b1d3ω0τsinω0kτ)d1d3ω20τ2+1b4d3ω0τd1d3ω20τ2+1].

    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

    ˙x(t)x(kτ)x((k1)τ)τ (2.15)

    and

    ¨x(t)x((k+1)τ)2x(kτ)+x((k1)τ)τ2. (2.16)

    Substituting equation (2.15) and (2.16) into system (2.6), letting t=kτ(k=0,1,2...),

    {x((k+1)τ)2x(kτ)+x((k1)τ)τ2+Dx((kτ)x((k1)τ)τ+Kx(kτ)=B(kτ)u(kτ),y(kτ)=C(kτ)x(kτ), (2.17)

    Based on equation (2.11) and (2.17) we get

    {xk+1+(Kτ2+Dτ2I3)xk+(I3Dτ)xk1=τ2Bkuk,yk=Ckxk. (2.18)

    Therefore, we can rewrite the system (2.17) as the following discrete system with backward Euler method,

    {xk+1=˜A0xk1+˜A1xk+˜˜Bkuk,yk=Ckxk, (2.19)

    where

    ˜A0=τDI3=[10d1ω0τ010d3ω0τ01],
    ˜A1=2I3τDτ2K=[k1ω20τ2+20d1ω0τ0k2ω20τ2+20d3ω0τ0k3ω20τ2+2],
    ˜˜Bk=τ2ω20ϵ[02b1sinω0kτb42b2sinω0kτ0b2cosω0kτb5b3cosω0kτ0],k=0,1,2....

    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

    {xk+1=ˆA0xk1+ˆA1xk+ˆBkuk,yk=Ckxk, (2.20)

    where xkRn,ykRr,ukRm(mn) are the state vector, the output vector and the control vector respectively. ˆA0,ˆA1Rn×n, ˆBkRn×m,CkRr×n are coefficient matrices.

    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,...,un1} which brings the initial state (x1,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

    C=[M(n)n1ˆBn1,M(n)n2ˆBn2,...,M(n)1ˆB1,M(n)0ˆB0], (3.1)

    where

    M(n)i2=M(n)i1ˆA1+M(n)iˆA0,i=2,...,n1,M(n)n2=ˆA1,M(n)n1=In. (3.2)

    Proof Based on the form of the state vector in the system (2.20), we derive the following set of equations

    x1=ˆA0x1+ˆA1x0+ˆB0u0,x2=ˆA0x0+ˆA1x1+ˆB1u1=ˆA0x1+ˆA1(ˆA0x1+ˆA1x0+ˆB0u0)+ˆB1u1=ˆA1ˆA0x1+(ˆA0+ˆA21)x0+ˆA1ˆB0u0+ˆB1u1. (3.3)

    Thus, by iteration, it is finally not difficult to find that in equations (3.3) the kth instant state vector xk starting from initial time k=0 and k=1 is

    xk=Q(k)x1+P(k)x0+k1i=0M(k)iˆBiui, (3.4)

    where Q(k),P(k),M(k)i are polynomial functions consisted of matrices ˆA0,ˆA1, and they satisfy the following important iteration equations

    Q(k)=ˆA0Q(k2)+ˆA1Q(k1),Q(1)=On×n,Q(0)=In,P(k)=ˆA0P(k2)+ˆA1P(k1),P(0)=On×n,P(1)=In,k=2,3,...,M(k)i2=M(k)i1ˆA1+M(k)iˆA0,i=2,...,k1,M(k)k2=ˆA1,M(k)k1=In. (3.5)

    In a general case, the state vector at final time n can be written as

    xn=Q(n)x1+P(n)x0+n1i=0M(n)iˆBiui. (3.6)

    Therefore

    xnQ(n)x1P(n)x0=n1i=0M(n)iˆBiui=M(n)0ˆB0u0+M(n)1ˆB1u1+...+M(n)n1ˆBn1un1=[M(n)n1ˆBn1,M(n)n2ˆBn2,...,M(n)1ˆB1,M(n)0ˆB0][un1un2...u1u0]. (3.7)

    If we denote

    C=[M(n)n1ˆBn1,M(n)n2ˆBn2,...,M(n)1ˆB1,M(n)0ˆB0]

    as the controllability matrix in the sequel, then the equation (3.7) determines the input sequence which transfers the initial state (x1,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]:

    ¨x(t)=[101231452]x(t)+[103]u(t). (3.8)

    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

    xk+1=ˆA0xk1+ˆA1xk+ˆBuk (3.9)

    where

    ˆA1=2I3+τ2A,ˆA0=I3,ˆB=τ2B,A=[101231452],B=[103].

    Letting τ=0.1s, according to Theorem 1, we have

    n=3,M(3)2=I3,M(3)1=ˆA1=[2.0100.010.022.030.010.040.052.02],
    M(3)0=M(3)1ˆA1+M(3)2ˆA0=[3.03970.00050.04030.08123.12040.04030.16020.20253.0795].

    Then

    C=[M(3)2ˆB,M(3)1ˆB,M(3)0ˆB]=τ2[M(3)2B,M(3)1B,M(3)0B]=τ2[B,ˆA1B,(ˆA21+ˆA0)B], (3.10)

    here,

    ˆA1B=[1.980.056.1],(ˆA21+ˆA0)B=[2.91880.20219.3987],

    and

    detC=0.0002210,rankC=3=n. (3.11)

    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]

    ¨x(t)+2D˙x(t)+Kx(t)=B(t)u(t) (3.12)

    Here,

    D=[0110],K=[α100α2],B(t)=[costsint],α1,α2=const.

    For forward Euler method, the discrete form of system is as follows:

    (I+2τD)xk+1+(τ2K2I2τD)xk+xk1=τ2Bkuk. (3.13)

    Then, choosing τ=0.1s, we have

    I+2τD=[12τ2τ1]=[10.20.21],det(I+2τD)=1+4τ2=1.040,

    which means the inverse matrix of I+2τD exits and

    (I+2τD)1=[11+4τ22τ1+4τ22τ1+4τ211+4τ2][0.96150.19230.19230.9615].

    The discrete system can be represented as

    xk+1=ˆA0xk1+ˆA1xk+ˆBkuk (3.14)

    where

    ˆA0=(I+2τD)1,ˆA1=(I+2τD)1(2I+2τDτ2K),ˆBk=τ2(I+2τD)1B(kτ).

    According to Theorem 1, we have

    n=2,M(2)1=I2,M(2)0=ˆA1.

    Then

    C=[M(2)1ˆB1,M(2)0ˆB0]=[ˆB1,ˆA1ˆB0], (3.15)

    here

    ˆB1[0.009370.00287],ˆB0[0.009620.00192],ˆA1=[2.04+0.01×α11.040.2+0.002×α21.040.20.002×α11.042.04+0.01×α21.04].

    Because of α1=1+2b,α2=1b,0<b<<1, then

    detC0.0001120.0000021α20,rankC=2=n. (3.16)

    Therefore, discrete system (3.13) is controllable. Similarly, we can prove the controllability of system (3.12) with forward Euler discretized form.

    To the specific discrete system (2.19) with backward difference, we have

    M(3)2=I3,
    M(3)1=ˆA1=˜A1=[k1ω20τ2+20d1τω00k2ω20τ2+20d3τω00k3ω20τ2+2],
    M(3)0=M(3)1ˆA1+M(3)2ˆA0=˜A21+˜A0=[(k1ω20τ2+2)2d1d3τ2ω2010d1τω0(k1ω20τ2+k3ω20τ2+3)0(k2ω20τ2+2)210d3τω0(k1ω20τ2+k3ω20τ2+3)0d1d3τ2ω20+(k3ω20τ2+2)21],
    ˆB2=˜˜B2=τ2ω20ϵ[02b1sin2ω0τb42b2sin2ω0τ0b2cos2ω0τb5b3cos2ω0τ0],
    ˆB1=˜˜B1=τ2ω20ϵ[02b1sinω0τb42b2sinω0τ0b2cosω0τb5b3cosω0τ0].
    ˆB0=˜˜B0=τ2ω20ϵ[00b400b2b5b30].
    M(3)1ˆB1=˜A1˜˜B1=τ2ω20ϵ[b5d1ω0τl12b4(k1ω20τ2+2)2b2sinω0τ(k2ω20τ2+2)0b2cosω0τ(k2ω20τ2+2)b5(k3ω20τ2+2)l32b4d3ω0τ]

    where

    {l12=2b1sinω0τ(k1ω20τ2+2)b3d1ω0τcosω0τ,l32=2b1d3ω0τsinω0τb3cosω0τ(k3ω20τ2+2).

    It can be shown

    det(M(3)2ˆB2)=det(ˆB2)=b2(b3b4b1b5)sin4ω0τ=0 (3.17)

    where

    b3b4b1b5=sinβJ3cosβJ1sinβJ1cosβJ3=0. (3.18)

    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

    rankC=rank[M(3)2ˆB2,M(3)1ˆB1,M(3)0ˆB0]=3. (3.19)

    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.

    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 kβN(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

    S=[C0OOC1C2ˆA0C2ˆA1C3ˆA1ˆA0C3(ˆA0+ˆA21)......C2n1Q(2n1)C2n1P(2n1)]

    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

    y0=C0x0,y1=C1x1,y2=C2x2=C2(ˆA0x0+ˆA1x1+ˆB1u1)=C2ˆA0x0+C2ˆA1x1+C2ˆB1u1,y3=C3x3=C3ˆA1ˆA0x0+C3(ˆA0+ˆA21)x1+C3ˆA1ˆB1u1+C3ˆB2u2.

    Then the measurement yk according to the equation (3.4) of state vector xk is

    yk=CkQ(k)x0+CkP(k)x1+k1i=1CkM(k)iˆBiui, (4.1)

    In general, we have

    y2n1=C2n1Q(2n1)x0+C2n1P(2n1)x1+2n2i=1C2n1M(2n1)iˆBiui, (4.2)

    As consequence, equation (4.2) can be rewritten in the following relevant matrix form

    [y0y1y2C2ˆB1u1y3C3ˆA1ˆB1u1C3ˆB2u2......y2n12n2i=1C2n1M(2n1)iˆBiui]=[C0OOC1C2ˆA0C2ˆA1C3ˆA1ˆA0C3(ˆA0+ˆA21)......C2n1Q(2n1)C2n1P(2n1)][x0x1].

    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:

    rank[C0OOC1C2ˆA0C2ˆA1C3ˆA1ˆA0C3(ˆA0+ˆA21)......C2n1Q(2n1)C2n1P(2n1)]=2n. (4.3)

    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]

    ¨x(t)=[2134]x(t)+[12]u(t),y(t)=[13]x(t). (4.4)

    The discrete system has the form

    xk+1=ˆA0xk1+ˆA1xk+ˆBuk,yk=Cxk (4.5)

    where

    ˆA0=I2,ˆA1=2I2+τ2A,ˆB=τ2B,A=[2134],B=[12],C=[13]. (4.6)

    Letting τ=0.1s, according to Theorem 2, we have

    n=2,ˆA1=[2.020.010.032.04],ˆA21=[4.08070.04060.12184.1619], (4.7)
    S=[COOCCˆA0CˆA1CˆA1ˆA0C(ˆA0+ˆA21)]=[13000013132.116.132.116.134.446112.5263], (4.8)
    detS=0.040,rankS=4=2n. (4.9)

    Therefore, the discrete system (4.5) is observable.

    To system (2.19), according to equations (3.5), we also have

    Q(2)=ˆA0Q(0)+ˆA1Q(1)=ˆA0,Q(3)=ˆA0Q(1)+ˆA1Q(2)=ˆA1ˆA0,Q(4)=ˆA0Q(2)+ˆA1Q(3)=ˆA20+ˆA21ˆA0,Q(5)=ˆA0Q(3)+ˆA1Q(4)=ˆA0ˆA1ˆA0+ˆA1(ˆA20+ˆA21ˆA0),P(2)=ˆA0P(0)+ˆA1P(1)=ˆA1,P(3)=ˆA0P(1)+ˆA1P(2)=ˆA0+ˆA21,P(4)=ˆA0P(2)+ˆA1P(3)=ˆA0ˆA1+ˆA1ˆA0+ˆA31,P(5)=ˆA0P(3)+ˆA1P(4)=ˆA0(ˆA0+ˆA1ˆA0)+ˆA1(ˆA0ˆA1+ˆA1ˆA0+ˆA31), (4.10)

    and

    ˆA0=˜A0=[10d1ω0τ010d3ω0τ01],
    ˆA1=˜A1=[k1ω20τ2+20d1ω0τ0k2ω20τ2+20d3ω0τ0k3ω20τ2+2],

    and we can rewrite the measurement matrix Ck as the following form

    Ck=[α20α1sinω0kτα3cosω0kτ0α2α1cosω0kτα3sinω0kτ],k=0,1,...,5

    letting

    α1sinω0kτα3cosω0kτ=α21+α23sin(ω0kτ+ϕ)ηk(ϕ),α1cosω0kτα3sinω0kτ=α21+α23sin(ω0kτ+φ)ηk(φ),

    where

    ϕ=arctanα3α1,φ=arctan(α1α3),

    we get

    C0=[α20α30α2α1],C1=[α20η1(ϕ)0α2η1(φ)],C2=[α20η2(ϕ)0α2η2(φ)],

    and

    C2Q(2)=[u510u53u61α2u63],C2P(2)=[u540u56u64u65u66],

    where

    u51=α2+d3ω0τη2(ϕ),u53=α2d1ω0τη2(ϕ),u61=d3ω0τη2(φ),u63=η2(φ),u54=α2(k1ω20τ2+2)d3ω0τη2(ϕ),u56=α2d1ω0τ+(k3ω20τ2+2)η2(ϕ),u64=d3ω0τη2(φ),u65=α2(k2ω20τ2+2),u66=(k3ω20τ2+2)η2(φ).

    The following matrix U is chosen as the first six columns of the observability matrix S

    U=[α20α30000α2α1000000α20η1(ϕ)0000α2η1(φ)u510u53u540u56u61α2u63u64u65u66]. (4.11)

    The determination of matrix U is

    (α22)(α2(u53(u66+(k2ω20τ2+2)η1(φ))u56(α1+u63))+η1(ϕ)(u53u64u54(α1+u63))+α3((α2)(u51(u66+(k2ω20τ2)η1(φ)u56u61)+η1(ϕ)(u51u64u61u54)))).

    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

    rankU=rankS=6. (4.12)

    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.

    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.

    This work was supported in part by the National Natural Science Foundation of China (12171196).

    All authors declare no conflicts of interest in this paper.



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