Open access peer-reviewed chapter - ONLINE FIRST

Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft

By Yuichi Ikeda

Submitted: March 16th 2019Reviewed: June 2nd 2019Published: July 30th 2019

DOI: 10.5772/intechopen.87191

Downloaded: 52

Abstract

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system. Considerable research has been done about the nonlinear attitude tracking control of spacecraft, and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework. This chapter considers discrete-time nonlinear attitude tracking control problem of spacecraft. To this end, a Euler approximation system with respect to tracking error is first derived. Then, we design a discrete-time nonlinear attitude tracking controller so that the closed-loop system consisting of the Euler approximation system becomes input-to-state stable (ISS). Furthermore, the exact discrete-time system with a derived controller is indicated semiglobal practical asymptotic (SPA) stable. Finally, the effectiveness of proposed control method is verified by numerical simulations.

Keywords

  • spacecraft
  • attitude tracking control
  • discrete-time nonlinear control

1. Introduction

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy [1, 2, 3]. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system.

Figure 1.

Switching maneuver.

Figure 2.

Time histories of MRPs σ t and σ e t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, σ d t ).

Figure 3.

Time histories of attitude angles θ t and θ e t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, θ d t ).

Figure 4.

Time histories of angular velocities ω t and ω e t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ω d t ).

Figure 5.

Time histories of control input u t (solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).

Considerable research has been done about the nonlinear attitude tracking control of spacecraft [4, 5, 6, 7, 8, 9, 10, 11, 12], and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework.

Although a sampled-data control method for nonlinear system did not advance because it is difficult to discretize a nonlinear system, a control method based on the Euler approximate model has been proposed in recent years [13, 14] and is applied to ship control [15]. Although our research group has proposed a sampled-data control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period Tas the control law is of the form u=ax+bx/T.

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period T. The effectiveness of proposed control method is verified by numerical simulations.

The following notations are used throughout the chapter. Let Rand Ndenote the real and the integer numbers. Rn and Rn×mare the sets of real vectors and matrices. For real vector aRn, aTis the vector transpose, adenotes the Euclidean norm, and a×R3×3is the skew symmetric matrix

a×=0a3a2a30a1a2a10

derived from vector aR3. For real symmetric matrix A, A>0means the positive definite matrix. The identity matrix of size 3×3is denoted by I3. λAmaxRand λAminRare the maximal and the minimal eigenvalues of a matrix A, respectively.

2. Relative equation of motion and discrete-time model for spacecraft

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame i, the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft?s body-fixed frame bare given by the following equations:

σ̇t=Gσtωt,E1
Gσt=121σt22I3+σtσtT+σt×,
ω̇t=J1ωt×t+ut+wt,E2

where Eq. (1) is the kinematics that represents the attitude of bwith respect to the i, Eq. (2) is the rotation dynamics, σtR3[−] is the MRPs, ωtR3[rad/s] is the angular velocity, utR3[Nm] is the control torque (input), wtR3[Nm] is the disturbance input, and JR3×3[kg m2] is the moment of inertia.

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) σdtR3and angular velocity ωdtR3in fixed frame d. The MRPs of the relative attitude σetR3and the relative angular velocity ωetR3in the frame bare given by

σet=Net1+σt2σdt2+2σdtTσt,E3
Net=1σdt2σt1σt2σdt+2σt×σdt,
ωet=ωtCtωdt,E4

where CtR3×3is the direction cosine matrix from bto dthat expresses the following Eq. [7]:

Ct=I3+8σet×241σet2σet×1+σet22.E5

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity Ċt=ωet×Ctyields the following relative motion equations:

σ̇et=Gσetωet,E6
ω̇et= J1[ωet+Ctωdt×Jωet+CtωdtJCtω̇dtωet×Ctωdt+ut+wtE7

Hereafter, we assume that the variables of spacecraft σtand ωtare directly measurable and Jis known. In addition, regarding the desired states σdt, ωdt, ω̇dt, and the disturbance wt, the following assumption is made.

Assumption 1: the desired states σdt, ωdt, and ω̇dtare uniformly continuous and bounded t0. The disturbance wtis uniformly bounded t0.

From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative motion equations is obtained as

σe,k+1=σe,k+kTk+1TGσesωesds,E8
ωe,k+1=ωe,k+kTk+1Tωes+Csωds×Jωes+CsωdsJCsω̇dsωes×Csωds+uk+wkdsE9

and the Euler approximate model of relative motion equations are obtained as

σe,k+1=σe,k+TGσe,kωe,k,E10
ωe,k+1=ωe,kTJ1ωe,k+Ckωd,k×Jωe,k+Ckωd,kJCkω̇d,kωe,k×Ckωd,k+uk+wk.E11

3. Discrete-time nonlinear attitude tracking control

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system xk=σe,kTωe,kTTsatisfies the following equation:

xk+1ρx0k+γwk,xkR3,wkR3,

where ρ·is the class KL function and γ·is the class K function. To this end, assume that ωe,kis the virtual input to subsystem (10), and derive the stabilizing function αkthat σe,kis asymptotic convergence to zero. Then, derive the control input ukthat closed-loop system becomes ISS. Here, regarding the variable σe,k, the following assumption is made.

Assumption 2:σe,klies in the region that satisfies the following equation:

0σe,k1,k.

Remark 1: from the relational expression

σe,k=εe,k1+ηe,k,

where εe,kR3and ηe,kRare the quaternion (εe,kTηe,kT=1,εe,k1,ηe,k1,k). Assumption 2 is equivalent to ηe,k01.

In addition, Lemmas when using the derivation of the control law are shown below.

Lemma 1: for all σR3, the following equations hold [5]:

σTGσ=bσT,GσTGσ=b2I3,b=1+σ24>0.

Lemma 2: when the quadratic equation

ax2+bx+c=0abcR

has two distinct real roots x=α,βα<β, if a>0, then the solution of the quadratic inequality

ax2+bx+c<0

is α<x<β.

3.1. Derivation of virtual input αk

Assume that ωe,kis the virtual input to subsystem (10), and define the stabilizing function such that

ωe,k=αk=f1σe,k,E12

where f1Ris the feedback gain. The candidate Lyapunov function for (10) is defined as

V1k=σe,k2.E13

From Lemma 1, the difference of Eq. (13) along the trajectories of the closed-loop system is given by

ΔV1k=V1k+1V1k=Tf1bk22Tf1bkσe,k2.E14

From Lemma 2, ΔV1kbecomes negative, i.e., the range of f1that holds the following equation

Tf1bk22Tf1bk<0E15

is obtained as

0<f1<2Tbk.E16

In addition, since 21/bk4under Assumption 2, the range of f1that holds Eq. (15) is obtained as

0<f1<4T.E17

Therefore, if f1satisfies Eq. (17) and ωe,kαkk, then σe,k0.

3.2. Derivation of control input uk

The error variable between the state ωe,kand αkis defined as

zkωe,kαk.E18

The control input ukthat makes the closed-loop system becomes ISS is derived.

From Eq. (18), subsystem (10) becomes

σe,k+1=σe,k+TGσe,kzk+αk.E19

From Eqs. (18) and (19) and the following equation

αkαk+1=Tf1Gσe,kzkf1bkσe,k,

the discrete-time equation with respect to zkis

zk+1=zk+Tf1Gσe,kzkf1bkσe,k+ TJ1zk+α,k+Ckωd,k×Jzk+α,k+Ckωd,k JCkω̇d,kzk+α,k×Ckωd,k+uk+wk.E20

Now, by setting ukto

uk= zk+α,k+Ckωd,k×Jzk+α,k+Ckωd,k+ JCkω̇d,kzk+α,k×Ckωd,k f1JGσe,kzkf1bkσe,kf2Jzk,

Eq. (20) becomes

zk+1=1Tf2zk+TJ1wk,E21

where f2Ris the feedback gain. The candidate Lyapunov function for Eqs. (19) and (21) is defined as

V2k=V1k+zk2=Xk2,Xk=σe,kTzkTT.E22

As Eq. (14) is given by

ΔV1k=Tbk2zk2+Tf1bk22Tf1bkσe,k2+2Tbk1Tf1bkzkTσe,kT

from Eq. (18), by using completing square, the difference of Eq. (22) along the trajectories of the closed-loop system is given by

ΔV2k= T2f222Tf2+T2bk2zk2+Tf1bk22Tf1bkσe,k2+2Tbk1Tf1bkzkTσe,kT+T2wkTJ2wk+2T1Tf2wkTJ1zk2T2f224Tf2+T2bk2+1zk2+Tf1bk22Tf1bkσe,k2+2Tbk1Tf1bkzkTσe,kT+2TλJ2wk2= XkTQkXk+2TλJ2wk2,E23

where

λJ=J,Qk=Q11,kQ12,kQ12,kTQ22,k,Q11,k=Tf1bk22Tf1bkI3,
Q12,k=Tbk1Tf1bkI3,Q22,k=2T2f224Tf2+T2bk2+1I3.

In Eq. (23), if Qk<0, then

ΔV2kλQkminXk2+2TλJ2wk2,

where λQkmin<0Ris the minimum eigenvalue of Qkand the condition of ISS holds [18]. Hereafter, conditions of f1and f2which the matrix Qkholds Qk<0are derived under Assumption 2.

From Schur complement, condition Qk<0is equivalent to the following equations:

Tf1bk22Tf1bk<0,E24
2T2f224Tf2+ck<0ck=Tbkf122f1TbkTbkf122f1.E25

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From Lemma 2, the range of f2that holds for Eq. (25) is obtained as

222ck2T<f2<2+22ck2T,E26

and the following Eq.

2ck>0Tbkf122f1+TbkTbkf122f1>0E27

must hold true in order to obtain a real number. As the denominator of Eq. (27) is the same as Eq. (24), the following equation must hold

Tbkf122f1+Tbk<0E28

in order to hold Eq. (27). From Lemma 2, the range of f1that holds for Eq. (28) is obtained as

11Tbk2Tbk<f1<1+1Tbk2Tbk,E29

and the following Eq.

1Tbk2>00<T<1bkE30

must hold in order to have the real number. As 21/bk4under Assumption 2, Tmust satisfy the condition

0<T<2.E31

In addition, since

maxbk11Tbk2Tbk=24T2T,
minbk1+1Tbk2Tbk=2+4T2T

under Assumption 2, the condition (29) is given by

24T2T<f1<2+4T2T0<T<2.E32

Therefore, if f1satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold. Furthermore, since

maxbk222ck2T=1TTf124f1+T2T2f1Tf14,
minbk2+22ck2T=1T+Tf124f1+T2T2f1Tf14,

under Assumption 2, the condition (26) is given by.

1TTf124f1+T2T2f1Tf14<f2<1T+Tf124f1+T2T2f1Tf140<T<2.E33

Therefore, if f1and f2satisfy Eqs. (32) and (33) under Assumption 2, then Qk<0.

Summarizing the above, the following theorem can be obtained.

Theorem 1: if sampling period Tand feedback gains f1and f2satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with the following control law

uk= zk+α,k+Ckωd,k×Jzk+α,k+Ckωd,k+JCkω̇d,kzk+α,k×Ckωd,kf1JGσe,kzkf1bkσe,kf2Jzk= ωk×Jωk+JCkω̇d,kzk+α,k×Ckωd,kf1f2Jσe,kJf1Gσe,k+f2I3ωe,kE34

becomes ISS.

Then, we show that the pair ukV2kis semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period Tand feedback gains f1and f2satisfy Eqs. (31), (32), and (33) under Assumption 2. By using the following coordinate transformation

Xk=10f11σe,kωe,k=ZX¯k,

Lyapunov function V2kand its difference ΔV2kcan be rewritten as

V2k=X¯kTZTZX¯k=X¯kTRX¯k,
ΔV2k=X¯kTZTQkZX¯k+2TλJ2wk2=X¯kTQ¯kX¯k+2TλJ2wk2.

Since R>0and Q¯k<0, V2kand ΔV2ksatisfy following equations:

λRminX¯k2V2kλRmaxX¯k2,E35
ΔV2kλQ¯kminX¯k2+2TλJ2wk2.E36

In addition, X¯kis bounded, and V2kis radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assumption 1:

ukM,E37

where Mis a positive constant. Furthermore, V2kalso satisfies the following equation for all x,zR6with maxxzΔ:

V2xV2z=xTRxzTRz=x+zTRxz=λRmaxx+zxz2ΔλRmaxxz,E38

where Δis a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function V2kand control input uksatisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair ukV2kbecomes SPA stabilizing pair for the Euler approximate systems (10) and (11). Then, the following theorem can be obtained by Theorem A.1 in Appendix.

Theorem 2: control input (34) is SPA stabilizing for exact discrete-time systems (8) and (9).

4. Numerical simulation

The properties of the proposed method are discussed in the numerical study. For this purpose, parameter setting of simulation is as follows:

J=7050.00.53643.90.53623901640.043.91640.06130.0kgm2,σ0=000,ω0=000rad/s
T=1.0:Case10.5:Case20.1:Case3,f1=0.6,f2=0.8.

The moment of inertia Jis from [1]. The initial values σ0correspond to Euler angles of 1–2-3 system of θ0=θ10θ20θ30T=000Tdeg. The feedback gains f1and f2satisfy Eqs. (25) and (28) for all cases of T. The desired states σdt, ωdt, and ω̇dtin this simulation are the switching maneuver as shown in Figure 1.

The results of the numerical simulation are shown in Figures 25. The relative attitude σetand relative angular velocity ωetconverge to the neighborhood of σetωet=00, and the control input amplitude utdoes not depend on the sampling period Talthough there is a slight difference in the maximal value of ut.

5. Conclusion

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period T. The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as sampled-data control system will be subject to future work.

This section shows preliminary results for nonlinear sampled-data control [13, 14, 19].

Let us consider the following nonlinear system:

ẋt=fxtut,x0=x0,f00=0,EA1

where xtRnis the state variable and xtRmis the control input. The function fxtutin Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some intervals of x0τ.

The nonlinear system (A1) is assumed to be between a sampler (A/D converter) and zero-order hold (D/A converter), and the control signal is assumed to be piecewise constant, that is,

ut=ukTuk,tkTk+1T,k0N,EA2

where T>0is a sampling period. In addition, assume that the state variable

xkxkTEA3

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are expressed as follows, respectively:

xk+1=xk+kTk+1TfxsukdsFTexkuk,EA4
xk+1=xk+TfxkukFTEulerxkuk,EA5

where we abbreviate xkand ukto xkand uk. For the stability of the exact discrete-time model (A4) (FTe) and Euler approximate model (A5) (FTEuler), the following definitions are used [13, 14, 19].

Definition 1: consider the following discrete-time nonlinear system:

xk+1=FTxkuTxk,EA6

where xkRnis the state variable and uTxkRmis a control input. The family of controllers uTxkSPA stabilizes the system (A6) if there exists a class KL function β·such that for any strictly positive real numbers Dν, there exists T>0, and such that for all T0Tand all initial state x0with x0D, the solution of the system satisfies

xkβx0kT+ν,k0N.EA7

Definition 2: let T̂>0be given, and for each T0T̂, let functions VT:RnRand uT:RnRmbe defined. The pair of families uTVTis a SPA stabilizing pair for the system (A7) if there exist a class Kfunctions α1, α2, and α3such that for any pair of strictly positive real numbers Δδ, there exists a triple of strictly positive real numbers TLMTT̂such that for all x,zRnwithmaxxzΔ, and T0T:

α1xVTxα2x,EA8
VTFTxuTxVTxα3x+,EA9
VTxVTzLxz,EA10
uTxM.EA11

In addition, if there exists T>0such that Eqs. (A8)–(A11) with δ=0hold for all xRnand T0T, then the pair uTVTis globally asymptotic (GA) stabilizing pair for the system (A6).

Using the above definitions, the following theorem is obtained by literatures [13, 14, 19].

Theorem A.1: if the pair uTVTis SPA stabilizing for FTEuler, then uTis SPA stabilizing forFTe.

Hence, if we can find a family of pairs of uTVTthat is a GA or SPA stabilizing pair for FTEuler, then the controller uTwill stabilize the exact model FTe.

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Yuichi Ikeda (July 30th 2019). Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft [Online First], IntechOpen, DOI: 10.5772/intechopen.87191. Available from:

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