Attitude output feedback control for rigid spacecraft with finite-time convergence

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Research article

Attitude output feedback control for rigid spacecraft with finite-time convergence Qinglei Hu n, Guanglin Niu School of Automation Science and Electrical Engineering, Beihang University, XueYuan Road No.37, HaiDian District, Beijing 100191, China

art ic l e i nf o

a b s t r a c t

Article history: Received 31 May 2016 Received in revised form 12 July 2017 Accepted 21 July 2017

The main problem addressed is the quaternion-based attitude stabilization control of rigid spacecraft without angular velocity measurements in the presence of external disturbances and reaction wheel friction as well. As a stepping stone, an angular velocity observer is proposed for the attitude control of a rigid body in the absence of angular velocity measurements. The observer design ensures finite-time convergence of angular velocity state estimation errors irrespective of the control torque or the initial attitude state of the spacecraft. Then, a novel finite-time control law is employed as the controller in which the estimate of the angular velocity is used directly. It is then shown that the observer and the controlled system form a cascaded structure, which allows the application of the finite-time stability theory of cascaded systems to prove the finite-time stability of the closed-loop system. A rigorous analysis of the proposed formulation is provided and numerical simulation studies are presented to help illustrate the effectiveness of the angular-velocity observer for rigid spacecraft attitude control. & 2017 Published by Elsevier Ltd. on behalf of ISA.

Keywords: Attitude control Rigid spacecraft Output feedback Finite-time stability Reaction wheel friction

1. Introduction Accurate and reliable attitude control is one of the most important problems and widely studied application in current spacecraft attitude control system design, meanwhile, there are many valuable researches in this area. Many studies related to attitude control scheme have been published based on several inspiring approaches, such as proportional-derivative (PD) control algorithm which has been widely applied in practical projects [1], optimal control [2], adaptive control [3], but there are few works that have explicitly dealt with the running characteristics of reaction wheel. In practical spacecraft attitude control implementation, reaction wheel is often used as actuator to provide effective and continuous control torque. However, reaction wheel always suffers the friction torque inevitably, which will influence the control performance to some extent [4]. One method to deal with the influence caused by reaction wheel friction torque is adding a prior friction torque versus speed of reaction wheel to control torque command in a feed-forward way. But it is difficult to measure friction torque accurately due to measurement error of wheel speed and friction uncertainties. Furthermore, the friction torque increases due to aging of reaction wheel. Thus, this method could hardly achieve accurate attitude control in some sense. However, n

Corresponding author. E-mail address: [email protected] (Q. Hu).

except for literature [5], there are few research works found for this related area, which was acknowledged by the author. Note that most of the existing attitude control laws are asymptotically stable, which means state errors converge to the equilibriums as time goes to infinity [6]. Obviously, such stability performance requiring infinite settling time is not an optimal option during critical mission phases of some high-value real-time missions. Finite-time control theory provides fast convergence rate and high-precision performance. Thus, the research on finite-time stabilization is an interesting and challenging problem. Existing finite time control methods can be broadly classified into two categories: the Lyapunov-based approach [7–9] and the homogeneous domination approach [10,11]. Some significant recent researches have been done using finite time control-based strategies to guarantee finite time stability for spacecraft attitude. In Ref. [12], a robust sliding mode controller was developed to guarantee the spacecraft attitude system states can be forced to be attained in the small set of a sliding surface in finite time. The terminal sliding mode controller was developed in [13] and consisted of the estimation of inertia uncertainty and disturbance by adaptive method. Lu and Xia [14] proposed a finite-time nonsingular terminal sliding mode control law associated with adaptation which provides finite-time convergence and higher precision. By employing terminal sliding mode and some properties of dual quaternion in [15], the practical finite-time stability of the closed-loop system for spacecraft formation flying is guaranteed through the designed control law. In Ref. [16], a sliding mode based finite-time control scheme is presented to address the

http://dx.doi.org/10.1016/j.isatra.2017.07.023 0019-0578/& 2017 Published by Elsevier Ltd. on behalf of ISA.

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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problem of attitude stabilization for rigid spacecraft in the presence of actuator faults and external disturbances. A typical feature in all the above-mentioned attitude control schemes is in dependence on angular velocity measurements. Unfortunately, this requirement is not always satisfied in reality since the cost limitations and the implementation constraints. Thus, a common practice is to approximate the angular velocity signal through an ad hoc numerical differentiation of the attitude angles. Aiming at the condition that completely independent of angular velocity measurement, a nonlinear output feedback attitude control is designed by involving a linear passivity filter, which is derived without explicit differentiation of attitude to synthesize angular velocity-like signals [17]. Furthermore, an angular velocity observer was developed in Ref. [18] to achieve global convergence of angular velocity estimation error. In [19–21], the quaternionbased output feedback control scheme was designed for velocityfree attitude control of spacecraft. However, the convergence rate of angular velocity estimation error couldn’t be guaranteed by above techniques. With this in mind, Sun [22] established a switched saturated feedback controller developed based on a state observer, which could ensure the closed-loop system is semiglobal finite-time stable in the presence of constraints on control input magnitude, but the external disturbances was not considered. Zou [23] proposed the finite-time observer, and designed an attitude stabilizing control law to guarantee that the attitude states of the spacecraft converge to the equilibriums in finite time. While the above two corresponding finite-time controllers are designed via non-smooth controller construct. Hu [24] et al. investigated the finite-time relative position coordinated tracking problem by output feedback for spacecraft formation flying in the absence of velocity measurement. Ref. [25] introduced the power integrator method and homogeneous system to demonstrate the finite time stabilization of velocity-free attitude control system. To obtain a more realistic control performance, the finite-time observer combined with output feedback controller is studied in the presence of saturation and external disturbances [26]. The main contribution of this paper is designing a class of finite-time control algorithm respect to the rigid spacecraft attitude stabilization that explicitly takes account of velocity-free and external disturbances as well as reaction wheel friction, and assures a fast response. More specifically, by exploiting structural properties of the spacecraft model, a nonlinear finite time observer is proposed to estimate the angular velocity in finite time, in which both the external disturbances and friction torque of reaction wheel are explicitly accounted for. Then, based on the results obtained from the finite time observer, a finite time attitude-control law from the use of a velocity estimator is developed. The proposed control strategy is analytically verified and also validated via a simulation study. The rest of this paper is organized as follows: Section 2 provides the rigid spacecraft kinematics and dynamics models in attitude control system along with some lemmas which could be applied to this research. In Section 3, a finite time observer design scheme is proposed and we provide the associated stability analysis for the estimation error dynamics. In Section 4, based on the finite-time observer, an output feedback control law is given by introducing a power integrator technique, meanwhile, Lyapunov theorem provides a rigorous proof of finite-time stabilization on closed-loop system. Section 5 illustrates the performance of the presented control scheme which is analyzed combined with the results of digital simulation.

2. Background and preliminaries 2.1. Definitions and lemmas Consider a general system as followings:

ẋ = f (x, t ), f (0, t ) = 0, x ∈ U ⊂ Rn

(1)

where f is continuous function on an open neighborhood U of the origin. Lemma 1 ([27]): Suppose a Lyaponov function V ( x, t ) is defined as:

V̇ ( x, t ) ≤ − lV α( x, t ), ∀ x ∈ U1\{ 0}

(2)

where U1 is a neighborhood of the origin, and l > 0, 0 < α < 1. Then, the origin of system is locally finite-time stable. The settling time satisfies

V1 − a x( t0, t0)

(

T≤

) , x( t ) ∈ U 0

l( 1 − a )

1

(3)

Then, suppose there exists a Lyapunov function V ( x, t ) on U1, and

V̇ ( x, t ) ≤ − lV α( x, t ) + kV ( x, t ), ∀ x ∈ U1\{ 0}

(4)

where l, k > 0,and 0 < α < 1. For an initial condition x( t0), the origin of system is locally finite-time stable if x( t0) ∈ { U1 ∩ U2}, where U2 =

{ x|V

1−α

( x, t ) < kl } is a neighborhood of the origin and

satisfies that U1 ⊆ U2 or U2 ⊆ U1. The settling time satisfies

V1 − α x( t0), t0

(

T≤

)

( l − kV ( x( t ), t ))( 1 − α) 0

(5)

0

Lemma 2 ([28]): If 0 < κ = κ1/κ2 ≤ 1, where κ1 and κ2 are positive odd integers, then x κ − y κ ≤ 21 − κ x − y κ , ∀ x, y ∈ R . Lemma 3 ([29]): For any x ∈ R, y ∈ R, c > 0, d > 0 and γ > 0, there is an inequality which satisfies x c y d ≤ cγ x c + d /( c + d) + d y c + d / γ c / d( c + d ) .

(

)

Lemma 4 ([30]): For any x i ∈ R, i = 1, 2, ⋅⋅⋅, n, and a real number κ ∈ ( 0, 1⎤⎦,

(∑

n

i=1

xi

κ

)

≤

n

∑i = 1

κ

xi ≤ n1 − κ

(∑

n

i=1

xi

κ

).

Given a vector x ∈ Rn and a scalar α ≥ 0, define sig α ( x )= α ⎡⎣ sig α ( x1) , sig α ( x2) , …, sig α ( x n)⎤⎦T , where sig α ( x i ) = sgn( x i ) x i

( i = 1, 2, …, n) , and sgn( ⋅) denotes the signum function. 2.2. Rigid spacecraft model The rigid spacecraft attitude control system with disturbances and reaction wheel friction can be described as following kinematics and dynamics models [4]:

(

)

Jω̇ = − S( ω) Jω + Jrw Ω + u + d + df

q̇ =

(6)

1 E( q)ω 2

(7) T

where ω = [ ω1, ω2, ω3] denotes the angular velocity vector of T spacecraft body-fixed reference frame, and q = ⎡⎣ q , q ⎤⎦ = 0

v

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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⎡⎣ q , q , q , q ⎤⎦T 0 1 2 3

From Eqs. (9), (11) and (12), it can be derived that

denotes the unit quaternion in spacecraft body 3×3

is the posiframe respect to an inertial reference frame. J ∈ R tive-definite inertia matrix of spacecraft, where Jrw = diag Jrw,1 , Jrw,2 , Jrw,3 , and Jrw, i , i = 1, 2, 3 denotes moment of inertia of the ith reaction wheel mounted in the spacecraft, Ω is the rotation speed vector of reaction wheels. The actual control torque τ provided by the reaction wheel will be used in simulation later and is given as

{

}

−Jrw Ω̇ = τ

Jrw Ω̇ = Km

⎡ 0 −ω ω ⎤ 3 2 ⎥ ⎢ S ( ω) = ⎢ ω3 0 −ω1⎥ ⎢ −ω ω1 0 ⎥⎦ ⎣ 2 ⎡ ⎤ − qv T ⎥. Additionally, Further, E ( q) is defined as E ( q) = ⎢ ⎢q I + S q ⎥ 3 ⎣ 0 v ⎦ the friction torque of reaction wheel df is modeled as [5]

( )

⎡ 2 ⎤ df = βd Ω + ⎢⎣ βk + βs / 1 + ( Ω/Ωs ) ⎦⎥sgn( Ω)

(

)

(8)

where, βd , βk , and βs respectively denotes unknown viscous friction coefficient, unknown Coulomb friction torque and unknown Stribeck friction torque, Ωs denotes characteristic speed of Stribeck friction. The friction torque df is upper bounded since the rotation speed vector of reaction wheels Ω is bounded. In practical spacecraft attitude control system, the dynamic property of actuator is one of the most significant factor influencing the overall system. On account of the brushless direct current motor which is employed for driving reaction wheel, the winding current can be given as

Ud − 2NpKeΩ 2( R d + R e )

(9)

Pem = 2NpKeΩid

(10)

The torque Tem satisfies

Pem = Kmid Ω

(11)

where Km = 2NpKe denotes the coefficient representing the motor torque. Thus, if the brushless direct current motor speeds up from the static, and note that reaction wheel always works in a high speed, it is rational to ignore Stribeck effect as well as Coulomb friction torque, and regard the viscous friction as the most significant part in the whole friction torque, thus, the friction torque could be simplified as df = βdΩ . Furthermore, reaction wheel dynamics can be described as

Jrw Ω̇ = τ − βd Ω

(12)

where Jrw denotes the whole moment of inertia of the reaction wheel rotor, τ is the driving torque generated by motor, and df is the friction torque of the reaction wheel.

− βd Ω

(13)

Ω( s ) Kw = Ud( s ) Tws + 1

Tw =

(14) Km

2NpKmKe + 2βd( Rd + Re)

Jw ( Rd + Re) NpKmKe + βd( Rd + Re)

is the gain constant of actuator, and

denotes the time constant of actuator.

Furtherly, suppose that βd → 0, and from Eqs. (12) and (14), it can be retrieved that

J K ws Tem( s ) = w Ud( s ) Tws + 1

(15)

The model of motor used here presents the relation between the armature current and other elements directly, so that we need to design a current feedback loop to obtain accurate control torques, and it can be expressed as followings

∫0

t

⎛ KUc ⎞ ⎜ − Kid⎟dt = Ud ⎝ K ⎠

(16)

where K denotes the forward magnification of feedback loop. From Eqs. (11), (15) and (16), the model of actual control torque with control commanding signal is obtained that

Jw KK wKm Tem( s ) = 2 2 Uc ( s ) KmTws + Km + Jw KK wKm

(17)

Select the gain K which satisfies K ≫

Km , KwJw

Eq. (17) will be

approximate to a first-order inertia link as follows

Tem( s ) 1 = Uc ( s ) Tns + 1 where Tn =

where Ud denotes total voltage of the two phase winding, Np is the number of pole-pairs of motor, Ke is the coefficient of counter electromotive force, Ω is the rotation speed vector of reaction wheels, Rd and Re is winding resistance and the inverter resistance, respectively. Then, the electromagnetic power of brushless direct current motor is as followings:

Tem =

2( R d + R e )

where K w =

T ques provided by controller, d = ⎡⎣ d1, d2, d3⎤⎦ represents the external disturbance from the space environment. Here, the notation S ( ω) is used to denote the skew-symmetric matrix:

iḋ =

Ud − 2NpKeΩ

which indicates the reaction wheel dynamics. By means of the Laplace transform, it yields that

T

In addition, u = ⎡⎣ u1, u2 , u3⎤⎦ denotes the vector of control tor-

3

TwKm Jw KKw

(18) represents a time constant which is a significant

performance index of reaction wheel dynamics. Thus, the reaction wheel driving model can be equivalent to a first-order inertia link. Remark 1: Reaction wheel friction torque may occur in practical spacecraft-regulation maneuvers. Accordingly, the dynamic model of the reaction wheel considering friction torque is added in simulation studies for verifying the robustness and effectiveness of proposed algorithm. Assumption 1: The external disturbance d is unknown but bounded with ‖d‖ ≤ dmax , where the upper bound dmax on the magnitude of the disturbance is known. Similarly, the friction torque df is bounded with ‖df ‖ ≤ Df , where the upper bound of the reaction wheel friction torque is known as Df .

3. Finite time angular velocity observer design In this section, the design of an angular velocity observer in the presence of external disturbance as well as friction torque of reaction wheel simultaneously is presented. Besides, the parameter selection is discussed and the observer performance is also investigated. The finite time observer is proposed as followings

⎡ 2γ30qq˜v −1 ̇ 1 q^ = E q^ C ( q˜ ) ⎢ ω^ + θγ1P −1sig α q˜v + ⎢ 2 q˜0 1 − q˜v T q˜v ⎣

()

( )

(

)

⎤ ⎥ ⎥ ⎦

(19)

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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4

^̇ = − S ω ^ Jω ^ + J Ω + u + θ 2γ Jsig α1 q˜ Jω rw 2 v

( )(

)

( )

(20) −1

The quaternion estimation error is defined as q˜ = q^ ⊗ q , ⎡ ⎤ ^ Tq q^0q0 − q −1 ⎢ ⎥ v v ^ where q ⊗ q = ⎢ ⎥. The observer gain θ , γ1, ^ ^ ^ ⎢⎣ q0qv + q0qv + S qv qv ⎥⎦ γ2, γ3 in observer dynamics (19) and (20) are some positive constants. Some other parameters are defined with the satisfaction of

( )

α∈

( , 1), α = 2α − 1, 1 2

1

(

( ))

P = q˜0I3 + S q˜v /2. In addition, C ( q˜ ) de-

notes the direction cosine matrix corresponding to the quaternion q˜ which can be defined by

(

)

T

T

( )

C ( q˜ ) = q˜0 − q˜v q˜v I3 + 2q˜vq˜v − 2q˜0S q˜v

(21)

^ is the angular velocity estimation error. Further, where ω˜ = ω − ω taking the time derivative of C ( q˜ ) yields

(

))

(

Ċ ( q˜ ) = − S ω˜ − θγ1P −1sig α q˜v − 2γ3q˜v /q˜0 1 − q˜v T q˜v C ( q˜ )

( )

(22)

From the observer in Eqs. (19) and (20), the corresponding estimation error dynamics could be derived as

⎡ 2γ3q˜v 1 q˜ ̇ = E( q˜ )⎢ ω˜ − θγ1P −1sig α q˜v − ⎢ 2 q˜0 1 − q˜v T q˜v ⎣

( )

(

⎤ ⎥ ⎥ ⎦

)

(23)

ω˜ ̇ = − J −1 ⎡⎣ S( ω) Jω + Jrw Ω − S ω^ Jω^ + Jrw Ω ⎤⎦ − θ 2γ2sig α1 q˜v

(

(

)

+ J − 1 d + df

( )(

)

( )

)

(24)

Based on Eq. (23), a Lyapunov candidate function is defined as

V = q˜v T q˜v /2

(25)

T

V̇ = q˜v Pω˜ − θγ1q˜v sig ( q˜ ) −

≤

θγ1q˜v T q˜v 2

+

α

Proof: The dynamic equations relate to q˜v and ω˜ can be rewritten as follows:

q˜v̇ =

1 ω˜ − θγ1sig α ( q˜ ) + f1( q˜ , ω˜ ) 2 = − θ 2γ2sig α1 q˜v + f2 ω, ω^ + J −1 d + df

( )

ω˜ ̇

(

(

)

(

)

(30)

)

^ in Eq. (30) are both ˜ ω˜ ) and f2 ω, ω where the two terms f1( q, inhomogeneous and can be defined as

⎧ I3 ⎡ ⎤ T ⎪ f1( q˜ , ω˜ ) = P − 2 ω˜ − 2γ3Pq˜v /⎣ q˜0 1 − q˜v q˜v ⎦ ⎨ ⎪ f ω, ω^ = − J −1 ⎡ S( ω) Jω + J Ω − S ω^ Jω^ + J Ω ⎤ ⎣ ⎦ rw rw ⎩ 2

(

(

)

(

)

(

)

)

( )(

)

In order to analyze the homogenous characteristic of system (30), consider the following coordinate transformation: ζ1 = q˜v /θ θ1,

ζ2 = ω˜ /θ1 + θ1, where θ1 is a positive scalar and satisfies 0 < θ1 < 1. Further, Eq. (30) could be rewritten

⎧ f ( q˜ , ω˜ ) ⎪ ζ1̇ = θ ζ2 − θ1 + ( α − 1)θ1γ sig α ( ζ1) + 1 1 ⎪ 2 θ θ1 ⎨ f ( ω, ω˜ ) ⎪ 1 + ( α1− 1)θ1 γ2sig α1( ζ1) + 2 1 + θ + d¯ ⎪ ζ2̇ = − θ ⎩ θ 1

(31)

where d¯ = J −1 d + df /θ1 + θ1

(

)

T T T⎤ ⎡ Define ζ˜ = ⎢ sig1/ α2( ζ1) , sig1/ αα2( ζ2) ⎥ with the following ⎣ ⎦ dilation:

The differentiation of V in (25) respect to time yields T

Theorem 1: Consider the observer designed in (19) and (20), there ^ ‖ ≤ δ , and the parameter θ exists a positive constant δ , if ‖ω‖ ≤ δ , ‖ω in observer is large enough, the estimation error system can be proven to be semi-globally finite-time stable.

(

γ3q˜v T q˜v 1 − q˜v T q˜v

) (

)

δλr ( ζ ) = ( λζ11, λζ12, λζ13, λ αζ21, λ αζ22, λ αζ23)

γ q˜ T q˜ ‖Pω˜‖2 − θγ1q˜v T sig α ( q˜ ) − 3 v Tv 2θγ1 1 − q˜v q˜v

(26)

(32)

where α2 = αα1. Then select a Lyapunov candidate function:

Using the following inequalities

≤

2 1 2 q˜0 2

(

T Vα( ζ ) = ζ˜ Nζ˜

1 q˜0 + ‖q˜v‖ 2 2 + ‖q˜v‖2 = 2

−θγ1q˜v T sig α ( q˜ ) ≤ − θγ1q˜v T q˜v‖P‖ <

(

)

)

(27)

Eq. (27) can be rewritten as

V̇ ≤ −

θγ1q˜v T q˜v 2

γ3q˜v T q˜v

δ2 − + ≤ T 2 θγ1 1 − q˜v q˜v

⎛ −⎜ γ3 + ⎝

δ2 ⎞ T ⎟q ˜ q˜ θγ1 ⎠ v v

1 − q˜v T q˜v

+

δ2 θγ1

(28)

q˜v will reduce when V̇ < 0, hence it can be obtained from inequality (28) that q˜v could be restrained in a sphere

‖q˜v‖ ≤

Ωq˜v

δ

2

δ 2 + θγ1γ3

= δ1. Define a compact set Ωq˜ :

⎧ ⎪ q˜ 0 ≤ ‖q˜v‖ ≤ =⎨ ⎪ v ⎩

where N is any positive definite symmetric matrix. Furthermore, ^ = 0 and ˜ ω˜ ) = 0, f2 ω, ω specific to the particular case that f1( q, ¯ d = 0 as well, then Eq. (31) can be rewritten as

(

⎧ θ 1+ α − 1 θ α ⎪ ζ1̇ = ζ2 − θ ( ) 1γ1sig ( ζ1) 2 ⎨ ⎪ ̇ 1 + ( α1− 1)θ1 γ2sig α1( ζ1) ⎩ ζ2 = − θ

)

(34)

Let fα represents the vector field of Eq. (34) with homogeneous of degree α − 1, and LfαVα( ζ ) denotes the Lie bracket of the vector field q˜ and ω˜ [11]. Thus, Vα( ζ ) and LfαVα( ζ ) are homogeneous of

v

⎫ ⎪ δ2 ⎬ 1 = δ < 1 ⎪ δ 2 + θγ1γ3 ⎭

(33)

(29)

which ensures that ‖q˜v‖ will decrease as long as q˜v is beyond the compact set Ωq˜ . What’s more, since the initial value of ‖q˜v‖ can be v selected as zero, for 0 ≤ t ≤ ∞, ‖q˜v‖ is always driven in δ1. Then the following statement can be concluded as:

degree 2/α2 and 2/α2 + α − 1 respectively. By Lemma 4.2 in [11], the inequality can be derived as following:

−c1( α, θ )Vα β( ζ ) ≤ L fαVα( ζ ) ≤ − c2( α, θ )Vα β( ζ ) where c1(α, θ ) = − min{z: Vα(z ) = 1}LfαVα(z ), c2( α, θ ) = −

(35) max{ z: Vα( z ) = 1}LfαVα( z ),

and β = α2( 2/α2 + α − 1)/2.

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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Referring to [16] and [17], limc2( α , θ ) ≥ θ /λ max( N ) can be proved. α→ 1

Taking the time derivative of Vα and substituting Eq. (35) obtains ⎡ 1 − 1⎞ ⎛ 1 ⎢ diag ⎜ ζ1 α 2 ⎟ f1( q˜ , ω˜ ) θ1 ⎝ ⎠ α θ T ⎢ 2 Vα̇ ( ζ ) ≤ − c2( α, θ )Vα β( ζ ) + 2ζ˜ N ⎢ 1 ⎛ 1 − 1⎞ ⎢ ^ diag ⎜ ζ2 αα 2 ⎟ f2 ω, ω ⎢⎣ αα θ 1 + θ1 ⎝ ⎠ 2

(

)

5

{

}

where ρ = max ρ1/α2, ρ2 /( αα2) and c3 = 6 6 ρλ max( N )/λ max( N ). The term respect to disturbance in (36) can be simplified:

⎤ ⎡0 ⎥ ⎢ T ¯ 1 ˜ ˜ ⎛ 2ζ Nd ≤ 2‖ζ ‖λ max( N )⎢ 1 − 1⎞ diag ⎜ ζ2 αα2 ⎟d¯ ⎥ ⎢⎣ αα2 ⎠ ⎥⎦ ⎝

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

T

+ 2ζ˜ Nd¯

≤

(36)

Select the proper parameter θ which satisfies that θ > δ 2, then

≤

(37)

(

αα2 + 1 2

‖q˜v( t )‖ ≤ δ1 ≤ δ2 = 1 − δ22 , t ∈ ( 0, ∞). There are some inequalities can be used:

q˜ − 1 + ‖q˜v‖ ⎛ 1 ⎞ ⎜ P − I3⎟ω˜ ≤ ‖ω‖ 0 ≤ 2δ‖q˜v‖ ⎝ 2 ⎠ 2

1 ⎡ ⎤ αα2 − 1 2‖ζ˜‖λ max( N ) dmax + Df ⎢ 3 ⎥ ζ ⎢ ∑ 2, i ⎥ αα2θ1 + θ1λ min( J ) ⎢⎣ i = 1 ⎥⎦

2⋅3

)

(

αα2θ

where c4 =

2⋅3

)

λ max( N ) dmax + Df ‖ζ˜‖2 − αa2 1 + θ1

λ min( J )

αα2+ 1 2 λmax N

( )( dmax + Df )

αα2θ1 + θ1λmin( J )

≤ c4Vα1 −

αα2 2

( ζ)

(44)

.

Substituting (43) and (44) to (36) derives

−

2γ3Pq˜v

(

T

q˜0 1 − q˜v q˜v

)

≤

2 γ3 3/2 δ22

(1 − )

Vα̇ ( ζ ) ≤ − c2Vα β( ζ ) + c3Vα( ζ ) + c4Vα1 −

‖q˜v‖ (38)

( )

^ Jω ^ + J Ω ‖ ≤ 4δλmax J −1 λmax ( J )‖ω˜ ‖ + R‖ω˜ ‖ ‖S ( ω) Jω + Jrw Ω − S ω (39) rw

(

( )(

)

)

where R is the upper bound of reaction wheel angular momentum Jrw Ω . Using above three inequalities, the second term of (36) could be rewritten as

⎤ ⎡ 1 1 diag ζ α2 − 1 f1 q˜v , ω˜ ⎥ ⎢ θ1 α θ T ⎢ 2 ⎥ 2ζ˜ N ⎢ ⎥ 1 −1 1 ^ αα diag ζ 2 f2 ω, ω ⎥ ⎢ 1 + θ1 αα θ ⎦ ⎣ 2

(

)

(

(

)

(45)

Due to 0 < 1 − < 1, and according to the value of Vα( ζ ), there are two cases will be discussed. Case 1: Vα( ζ ) > 1 In this case, Eq. (45) can be retrieved as follows:

Vα̇ ( ζ ) ≤ − c2Vα β( ζ ) + ( c3 + c4 )Vα( ζ )

(46)

By using Lemma 1, it is concluded that Vα( ζ ) will convergence Case 2: Vα( ζ ) ≤ 1

)

In this case, if the parameter θ is large enough so that c2 > c3, then (45) becomes

⎡ρ ⎛ 3 ⎞ 1 − 1⎞⎛ 3 ≤ 2λ max( N )‖ζ˜‖⎢ 1 ⎜⎜ ∑ ζ1, i α2 ⎟⎟⎜⎜ ∑ ζ1, i ⎟⎟ ⎢⎣ α2 ⎝ i = 1 ⎠⎝ i = 1 ⎠ +

( ζ)

αα2 2

to the region Vα( ζ ) ≤ 1 in finite time.

)

(

αα2 2

Vα̇ ( ζ ) ≤ − ( c2 − c3)Vα( ζ ) + c4Vα1 −

⎞⎤ 1 − 1⎞⎛ 3 ρ2 ⎛ 3 ⎜ ∑ ζ2, i αα2 ⎟⎜ ∑ ζ2, i ⎟⎥ ⎜ ⎟ ⎜ ⎟⎥ αα2 ⎝ i = 1 ⎠⎝ i = 1 ⎠⎦

αα2 2

( ζ)

⎡ α2( 2α − 1) ⎤ αα2 c ⎥Vα1 − 2 ( ζ ) = − ( c2 − c3)⎢ Vα 2 ( ζ ) − 1 + θ 4 1 ⎢⎣ θ ( c2 − c3) ⎥⎦

(40)

(47)

2

where ρ1 and ρ2 can be defined: ρ1 = 2δ +

(

3/2

)

2 γ3/ 1 − δ22

and

( )

ρ2 = 4δλ max J −1 λ max( J ) + R‖ω˜ ‖ In view of Lemma 3, it follows that

⎞ α2( 2α − 1) ⎛ c ⎟ . Thus, Vα( ζ ) Vα( ζ ) reduces as long as Vα( ζ ) > ⎜ 1 + θ 4 ⎝ θ 1( c2 − c3) ⎠ will converge to a region as following in finite time. 2

⎛ ⎞ α2( 2α − 1) c ⎟⎟ Vα( ζ ) ≤ ⎜⎜ 1 + θ 4 ⎝ θ 1( c2 − c3) ⎠

1 1 ⎞ ⎛ 3 ⎛ 3 ⎞ α2 − 1 ⎛ 3 α2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ∑ ζ1, i ⎟⎜⎜ ∑ ζ1, i ⎟⎟ ≤ 3⎜ ∑ ζ1, i ⎟ ≤ 3 3 ‖ζ˜2‖ ⎜ i=1 ⎟ ⎜ i=1 ⎟⎝ i = 1 ⎠ ⎝ ⎠ ⎝ ⎠

(41)

(48)

Furthermore, ‖ζ˜‖ will be constrained in the region: 1

Similarly, it yields that

‖ζ˜‖ ≤

1 1 ⎞ ⎛ 3 ⎛ 3 ⎞ αα2 − 1 ⎛ 3 αα2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎟ ≤ 3 3 ‖ζ˜2‖ ⎜ ∑ ζ2, i ⎟⎜⎜ ∑ ζ2, i ⎟⎟ ≤ 3⎜ ∑ ζ2, i ⎜ i=1 ⎟ ⎜ i=1 ⎟⎝ i = 1 ⎠ ⎝ ⎠ ⎝ ⎠

Substituting (41), (42) into ‖ζ˜1‖ + ‖ζ˜2‖ ≤ 2 ‖ζ˜‖, it obtains that ⎡ ⎛ 1 ⎞ 1 diag ⎜ ζ α 2 − 1⎟f1 q˜v , ω˜ ⎢ θ1 ⎝ ⎠ α θ T ⎢ 2 2ζ˜ N ⎢ ⎛ αα1 − 1⎞ 1 ^ ⎢ diag ⎜ ζ 2 ⎟f2 ω , ω ⎝ ⎠ ⎢⎣ αα 2θ1 + θ1

(

)

(

)

(40),

and

using

(42) inequality

⎤ ⎥ ⎥ ˜ 2 ⎥ ≤ 6 6 ρλmax( N )‖ζ ‖ ≤ c3Vα( ζ ) ⎥ ⎥⎦

(43)

⎛ ⎞ α2( 2α − 1) c4 ⎜⎜ ⎟⎟ = δ3 1 + θ1 ( c2 − c3) ⎠ λ max( N ) ⎝ θ 1

(49)

Select the parameter θ sufficiently large so that δ3 can be ensured smaller than 1 or even a tiny range. Thus, estimation error could be limited in the region ‖ζ˜‖ ≤ 1. Remark 2: The nonlinear finite time observer is proposed to estimate the angular velocity in finite time by exploiting structural properties of the spacecraft model, in which both the external disturbances and friction torque of reaction wheel are explicitly accounted for. In practice, the friction torque is very small, and hence it is reasonable to compensate for the friction effect by using

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6

the similar processing method as the external disturbances. On account of the homogeneous method, preceding part of the text discusses two cases whether the current value of observer Lyapunov candidate is in the unit sphere, then demonstrates the angular velocity estimation error could be convergent to a region in finite time in both cases.

( ) = − k 1/ κq̇

∂ ω^d1

1

∂t

=−

1

k11/ κ q0ω1 − q3ω2 + q2ω3 2

(

)

3

k11/ κ k 1/ κ ω1 + ω2 + ω3 ≤ 1 2 2

(

≤

∑(

)

ω˜ j + ω^j

j=1

)

(59)

(

)

^1/ κ di /∂t can be With the same handling used in Eq. (59), ∂ ω retrieved as

4. Finite-time stabilizing controller design In order to design the finite-time controller, firstly, define the following Lyapunov function:

(

)

V0 = 2 1 − q0

(50)

(

)

(51)

^ is In view of back-stepping method, the virtual controller for ω designed as

ω^d = − k1qv κ

^ = q T ω˜ + q T ω^ − ω^d − q T ω^ V0̇ = qv T ω = qv T ω˜ + ω v v v

(

)

(

)

3

) ∑

∑ 21− κ qi η κ

^i − ω^di ≤ qi ω

i=1 3

≤21 − κ ηi comes

Vi̇ ≤

(

κ

qi η + κ ηi

3

V0̇ ≤ − k1qv T qv κ +

1+ κ

)

21 − κ +κ 1 i=1

∑

(

qi η κ + κ ηi

1+ κ

)

+ qv T ω˜

∑ Vi

(55)

(56)

i=1

1 1− κ

( 2 − κ )2

3

∑(

)η

ω˜ j + ω^j

1−κ

)

1

+

i

( 2 − κ )21− κ k11+ 1/ κ

j=1

(k

1

≤

1+ κ

)

+ 21 − κ ηi

k1κ qi

+

1+κ

^i − ω ^di η 1 − κ ds ≤ ω i

ηi2 − κ u¯ i

(61)

1+ κ

k11 + 1/ κ

^di1/ κ −ω

∫ω^

ω^ i

(s

1/ κ − ω^di

1/ κ

di

)

1+κ

21 − κ κ 1 + κ η 1+κ i

(62)

1 2k1

ω˜ j ηi +

)

1+ κ

3

∑

ηi

1+ κ

+

j=1

3

+

2k1( 1 + κ )

j=1

21 − κ κ 2k1( 1 + κ )

+

(

3 k1 + 21 − κ ηi

3

∑

κ ∑ j = 1 qi

1+ κ

2( 1 + κ )

1

( 2 − κ )21− κ k11+ 1/ κ

ηi2 − κ u¯ i

(63)

^i , q˜ u¯ i = J −1 ui + h ω i

(

)

(64)

^ i , q˜ = J −1 ⎡⎣ −S ω ^ i Jω ^ i + θ 2γ Jsig α1 q˜ ⎤⎦. where h ω 2 i i The finite-time controller with angular velocity estimation is given as

(

)

( )

( )

)

^1/ κ di ∂ ω

21 − κ k11 + 1/ κ

∂t

1

( 2 − κ )21− κ k11+ 1/ κ

ω^ i

ω^di

ηi

2− κ

where u¯ i is given by u¯ i = obtain

u¯ i

^i ∂ω . ∂t

1/ κ

(65)

(57)

ds is a differentiable as well as positive part

) ∫ (s

)

Substituting Eq. (64) into (63) yields

ds

2−κ

(

+

κ

Substituting the inequality (62) to (61), it follows that

(

1

+

^1/ κ di −ω

1/ κ

2− κ

of Vi . Differentiating the above Lyapunov candidate Vi with respect to time yields

Vi̇ =

(s

u = J ⎡⎣ −k2η2κ− 1 − h ω^ , q˜ ⎤⎦

where Vi is defined:

(s

^i ω

∫ω^

With the observer dynamic (20), one can obtain

3

di

(60)

κ ω^j ηi ≤ ω^j − ω^dj ηi + ω^dj ηi ≤ 21 − κ ηi ηi + k1 ηi qj

(54)

To design the finite-time controller, Lyapunov candidate function is chosen as:

∫ω^

)

and substitute the inequality (60) into (58), then Vi̇ be-

1 2k1

Substitute Eq. (54) to Eq. (53), V0̇ will be rewritten as follows:

1/ κ

ω˜ j + ω^j

j=1

The following inequality is obtained by Lemma 2:

Vi̇ ≤

i=1

21 − κ ≤∑ +κ 1 i=1

^i ω

2

3

^ − ω^d ≤ qv T ω

Vi =

∑(

Use the inequality

(53)

^1/ κ − ω ^d1/ κ and by Lemma 2 and Lemma 3, the Define η = ω following inequality could be obtained

V = V0 +

3

1/ κ 1

(52)

where k1 is a positive parameter, and define 1/2 < κ = κ 1/κ2 < 1, κ1, κ2 should be positive odd integers. Substituting the virtual controller (52) to (51) yields:

(

∂t

)≤k

di

The differentiation of V0 in Eq. (50) is as follows:

V0̇ = qv T ω˜ + qv T ω^ − ω^d − qv T k1qv κ

(

1/ κ ∂ ω^ di

1/ κ − ω^ di

1 2k1

3

∑

ω˜ j ηi +

2 κ 2k1( 1 + κ )

)

3

∑

1 +κ

3

+

2k1( 1 + κ )

j =1 1 −κ

+

(

3 k1 + 21 − κ ηi

ηi

1 +κ

j =1

+

κ ∑ j = 1 qi

1 +κ

2( 1 + κ )

1

( 2 − κ )21 −κ k11 +1/ κ

(

^ i , q˜ ηi 2 − κ J −1 u i + h ω i

(

))

(66)

The last term of Eq. (66) can be further simplified as

1− κ

)

Vi̇ ≤

ds

1

( 2 − κ )21 − κ k11 + 1/ κ (58) ^di , one can With the definition of ω

(

^ i , q˜ ηi 2 − κ J −1ui + h ω i

(

)) = −

k2

( 2 − κ )21 − κ k11 + 1/ κ

ηi

1+κ

From the controller law given in (66), the virtual controller is defined as u¯ i = − k2ηi2κ− 1( i = 1, 2, 3), then Eq. (66) can be retrieved as followings

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Vi̇ ≤

1 2k1

+

3

∑

ω˜ j ηi +

j=1

Case 2: Vα( ζ ) ≤ 1, in this case, the time derivative of Eq. (69) is

1+ κ 3κ q 2( 1 + κ ) i

3k1 + 3( 1 + κ )21 − κ 2k1( 1 + κ )

3

∑

ηi

1+ κ

i=1

−

k2 1− κ

( 2 − κ )2

k11 + 1/ κ

ηi

3

1+ κ

Vṡ ≤ − k3 ∑ qi

+

3

k2 1− κ

( 2 − κ )2

∑

ηi

k11 + 1/ κ i = 1

2k1( 1 + κ )

3

ηi

1+ κ

(67)

i=1

The parameters k1and k2are selected by designer to satisfy ⎡ 3k + 3( 1 + κ )21 − κ + k1κ 22 − κ ⎤ 22 − κ + 3κ + k3⎥, k1 ≥ 2 1 + κ + k3, k2 ≥ ( 2 − κ )21 − κ k11 + 1/ κ ⎢ 1 2 k 1 κ + ( ) ( ) 1 ⎣ ⎦ where k3 is any positive constant. Then, substitute the above inequality of k1 and k2 into (67) yields 3

V̇ ≤ − k3 ∑ qi

1+ κ

3

− k3 ∑ ηi

i=1

1+ κ

3

+ qv T ω˜ +

i=1

3

1 ∑ ∑ ω˜ j ηi 2k1 i = 1 j = 1

1+ κ

− ( c2

i=1

3

3

⎡ 3 1+ κ ≤ − ε1⎢ ∑ qi + ⎢⎣ i = 1

∑

+ qv T ω˜ +

∑

3

− k3 ∑ ηi

⎤ ⎡ α2( 2α − 1) αα2 c ⎥Vα1 − 2 ( ζ ) − c3)⎢ Vα 2 ( ζ ) − 1 + θ 4 1 ⎢⎣ θ ( c2 − c3) ⎥⎦

1+ κ

3k1 + 3( 1 + κ )21 − κ + k1κ 22 − κ

1+ κ

i=1

3 3 3 ⎡ 1+ κ 22 − κ + 3κ ⎤ 1 ⎥ ∑ qi ≤ − ⎢ k1 − + qv T ω˜ + ∑ ∑ ω˜ j ηi ⎢⎣ 2k1 i = 1 j = 1 2( 1 + κ ) ⎥⎦ i = 1

−

7

(68)

Theorem 2: For the spacecraft attitude control system given by Eqs. (6) and (7), on the basis of the finite time controller as Eq. (65), and the initial state meets

1 ∑ ∑ ω˜ j ηi 2k1 i = 1 j = 1 3

ηi

1+ κ

+ Vα1 −

αα2 ⎤ 2 ⎥

i=1

⎥⎦

+ ε2

(71)

where parameters ε1 and ε2 can be defined as ε1 = min ⎛ ⎡ α2( 2α − 1) ⎤⎞ c4 ⎜ k3, ( c2 − c3)⎢ Vα 2 ( ζ ) − ⎥⎟⎟ , and ε2 = qv T ω˜ + 1 ∑i3= 1 1 + θ1 ⎜ 2k1 θ ( − ) c c ⎢ 2 3 ⎥ ⎣ ⎦⎠ ⎝ 3

∑ j = 1 ω˜ j ηi . Establish the following compact set:

ψ=

{ ( ζ , q, η)|V + Vα( ζ ) ≤ VM}

(72)

where VM is any positive constant, then ε2 is bounded with constraint of ε2 ≤ εM , εM here is a positive constant. Thus, the inequality (71) related to V̇ can be retrieved s

Vṡ ≤ − ε1V¯s + εM

(73) 1+κ

1+κ 3 3 where V¯s = ∑i = 1 qi + ∑i = 1 ηi + Vα β( ζ ). There must exist the lower bound of V¯s : V¯M = min V¯s when Vs = VM . Select the proper

{ }

V ( 0) + Vα( 0) ≤ VM where VM is any positive constant, then all signals in the closedloop system are uniformly ultimately bounded, and semi-global stabilization could be achieved as well as ω and qv converge to each small set which contains corresponding equilibrium point in finite time. Proof: Consider another Lyapunov function:

Vs = V + Vα( ζ )

(69)

Furthermore, take the time derivative of Vs along following two cases:

observer gain θ and the controller parameter k3 to ensure ε1 > εM /V¯M , further, the time derivative of Vs will achieve Vṡ < 0. Thus, ψ is verified to be an invariant set. If Vs( 0) satisfies Vs( 0) ≤ VM , then Vs ≤ VM for all the time, which indicates that ζ , η, q are ^ are also bounded. Define a posibounded. Furthermore, ω and ω ^ ≤ δ , meanwhile, by tive constant δ which satisfies ω ≤ δ and ω means of Theorem 1, it can be concluded that ω˜ converges to a set in finite time. Next, there exists a finite time t f 2 which ensures if t > t f 2, it is obtained that q0 ≥ 0 so that

1 − q0 ≤ 1 + q0 1 − q0 = qv T qv

(

)(

)

(74)

Hence, V0 is bounded by: Case 1: Vα( ζ ) > 1, in this case, the differentiation of Eq. (69) with respect to time yields

V0 ≤ 2qv T qv

(75)

Meanwhile, the upper bound of Vi on the basis of (57) is as follows: 3

Vṡ ≤ − k3 ∑ qi

1+ κ

3

− k3 ∑ ηi

i=1

Vi ≤

− c2Vα β( ζ ) + qv T ω˜

i=1 3

+

1+ κ

3

3

∑

ηi

1+ κ

i=1

where ε1 = min( k3, c2), ε2 =

3

⎤ + Vα β( ζ )⎥ + ε2 ⎥⎦

1 2k1

3 3 ∑i = 1 ∑ j = 1

ηi

2

(76)

Substitute (75) and (76), V is bounded by

1 ∑ ∑ ω˜ j ηi + ( c3 + c4)Vα( ζ ) 2k1 i = 1 j = 1

⎡ 3 1+ κ ≤ − ε1⎢ ∑ qi + ⎢⎣ i = 1

1

( 2 − κ )k11+ 1/ κ

ω˜ j ηi + ( c3 + c4 )Vα( ζ ).

V = V0 +

3

∑ Vi ≤ c4 ∑ ( qvi2 + ηi2) i=1

(70)

(77)

i=1

{

}

where c4 = max 2, 1/( 2 − κ )k11 + 1/ κ . Furthermore, substituting (77) to (68) yields that

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8

V̇ ≤ −

k3 c4 ( 1 + κ ) /2

V ( 1 + κ ) /2 + μ

(78)

T

where μ = qv ω˜ +

1 2k1

3 3 ∑i = 1 ∑ j = 1

ω˜ j ηi .

In terms of Theorem 1, ω˜ is bounded so that μ ≤ μM , μM is a positive constant which indicates the upper bound of μ , and (78) can be retrieved that

V̇ ≤ −

k3 V ( 1 + κ ) /2 + μM 1 + κ ) /2 ( c4

(79)

Table 1 Parameters of reaction wheel. Parameters

Value

Inertia Matrix

Jrw = [ 0.0063, 0, 0; 0, 0.0063, 0; 0, 0, 0.0063]kg⋅m2

Coulomb friction torque

βk = 0.004

Viscous friction coefficient Stribeck friction torque

βd = 0.0000318

Characteristic speed of Stribeck friction

βs = 0.005

Ωs = 2.5

V̇ is negative in the case of following condition:

V ( 1 + κ ) /2 ≤

c4 ( 1 + κ ) /2 + μM k3

(80)

which in turn concludes that semi-global stabilization is achieved as well as ω and qv converge to each small set which contains corresponding equilibrium point in finite time. Remark 3: In Theorem 1, suppose that there are no external disturbances, then ω˜ will converge to zero in finite time, furthermore, Eq. (78) can be rewritten as

V̇ ≤ −

k3 V ( 1 + κ ) /2 1 + κ ) /2 ( c4

(81)

Table 2 Parameters and initial conditions for simulation. Parameters

Value

Observer gains in proposed scheme Controller Parameters in proposed scheme The Initial Values of Attitude and Angular Velocity

γ1 = 1.5, γ2 = 1.2 , γ3 = 1.5, θ = 6 , α = 0.95 k1 = 10 , k2 = 15, κ = 13/15 T

ω( 0) = [ 0, 0, 0] , T

q( 0) = [ 0.663, − 0.6, 0.4, − 0.2]

The Initial Values of Atti^ ( 0) = [ 0.1, 0.1, 0.1]T , q^( 0) = q( 0) ω tude and Angular Velocity Estimation Parameters of scheme in θ = 10 , k1 = 0.2, k2 = 0.15, k3 = 0.1, k 4 = 0.1, Ref. [23] σ = 0.7, p1 = 0.6 , ρ1 = 0.1, ρ2 = 1

So ω˜ will converge to zero in finite time t f :

tf ≤

Vα1 − β( 0)

( c2 − c3Vα( 0))( 1 − β )

(82)

and the angular velocity ω will converge to zero in finite time t f 1:

tf1 ≤ tf +

V1 − ( 1 + κ ) /2 t f

( )

k3 c4( 1 + κ ) /2

( 1 − ( 1 + κ )/2)

(83)

Remark 4: Based on the results obtained from the finite time observer, a finite time attitude-control law from the use of a velocity estimator is developed. The finite time control law is established with the form of power integrator. Meanwhile, by respectively demonstrating the finite time stability in two cases about different values of controller Lyapunov candidate, the result of semi-global stability in finite time for closed-loop control system could be obtained.

5. Simulation and analysis To verify the effectiveness of proposed finite-time observer and controller, numerical simulations are provided in this section. The detailed response is simulated using the rigid spacecraft attitude dynamic model in Eq. (6) and Eq. (7) in conjunction with the proposed finite time observer in Eqs. (9) and (10) as well as the proposed finite time output feedback control law given by Eq. (55). The inertia matrix is obtained from Ref. [23], which is given as J = ⎡⎣ 20, 1.2, 0.9; 1.2, 17, 1.4; 0.9, 1.4, 15⎤⎦kg⋅m2. The friction torque of reaction wheel is studied in this work, and the dynamic model of reaction wheel should be introduced in simulation of Eq. (18). Referring to the modeling approach of reaction wheel friction provided by Ref. [5], Table 1 gives some parameters of reaction wheel as followings: For purpose of comparing the performance of the proposed scheme with other algorithm, actually there are another two control schemes involved in simulation: one is PD-like control,

the other is the finite time control scheme in Ref. [23]. With the controlling variable method, the simulation comparison should be divided into three cases on account of the proposed approach consisting of nonlinear observer and output feedback controller. Firstly, the proposed observer and another observer in literature [23] are combined with PD-like controller respectively. In this case, the estimation performance between the above two observers is obviously verified. Secondly, to illustrate the rapid maneuvering capability provided by proposed finite time controller, the PD-like controller is used for comparison, and the proposed observer is introduced simultaneously to both controllers. Thirdly, the scheme in our work is compared with approach in literature [23] to verify the overall performance. The detail analyses for those comparison results are carefully given below. Note that the attitude of spacecraft is expressed in Rodrigues parameters in Ref. [23], in order to facilitate the simulation contrast, Rodrigues parameters should be converted to unit quaternion. All computations and plots are performed by using the MATLAB/Simulink software package with the sampling time set to 0.1 s and the runtime set to 50 s. We can establish the attitude dynamic model of rigid spacecraft and take the actuator saturation as well as reaction wheel friction into account. The parameters of observer and controller in the following simulations were selected by trial and error until a good performance was obtained. This procedure resulted in the gains given in Table 2 for the preceding two control schemes, which are algorithm proposed in our work and approach in Ref. [23]. The main parameters used in simulation and some initial values of attitude as well as its estimation are chosen as followings: To demonstrate the control performance as well as conform to the actual environment, the external disturbance is given as T d = 0.01⎡⎣ sin( t ), cos( t ), sin( 2t )⎤⎦ . All of the numerical examples in this section use reaction wheels as actuators for the spacecraft attitude stabilization control system. In particular, note that the practical actuator saturation should be considered in this

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simulation, the maximum available torque is supposed as τi ≤ 1 N⋅m, where i = 1, 2, 3. Some simulation results are as follows. 5.1. Case A: proposed observer þ PD-like controller vs. finite-time observer in Ref. [23] þ PD-like controller In this case, the results of the simulation for the angular velocity estimation error, attitude, angular velocity, and control input are shown in Figs. 1–4. As shown in Fig. 1, it can be observed that the estimation errors of angular velocity for the observer in (19)

9

and (20) converge to the neighborhood of the equilibrium after roughly 0.2 s, whereas the observer in Ref. [23] takes around 0.5 s. This verifies the conclusions that the angular velocity can be precisely estimated by the proposed observer in finite time and also faster than the observer proposed in Ref. [23]. The angular velocity estimation and quaternion are also convergent to equilibrium with higher precision as well as faster transient response than the one in Ref. [23], this implies the performance of the proposed finite time observer is superior than that of the proposed observer in Ref. [23]. Meanwhile, the plots of control torques of different schemes are similar due to the same controller.

Fig. 1. The angular velocity estimation error.

Fig. 2. The angular velocity estimation.

Fig. 3. The response of attitude quaternion.

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Fig. 4. The response of actual control torque.

Fig. 5. The response of angular velocity estimation.

Fig. 6. The response of attitude quaternion.

5.2. Case B: proposed observer þ proposed controller vs. proposed observer þ PD-like controller In this case, the observer proposed in our work is employed to both strategies with the different controllers. Comparing with the angular velocity observation information of two control schemes shown in Figs. 5–8, it is found that the finite time controller guarantees that the angular velocity reaches stability in 17s, but the PD-like controller takes nearly 30s yet. As can be seen from Figs. 6 and 7, the finite time controller scheme provides faster

response as well as higher accuracy control performance even in the presence of external disturbances. In addition, to examine the effectiveness of the two aforementioned different controllers, we need define an energy index first, which is related to the amount of energy that actuators (reaction wheels) consume during the whole period. It can be defined as follows: E =

t

∫t ‖τ‖2 dt , this index 0

states how efficient the controller is during the overall simulation cycle. It can be apparently seen from Fig. 8 that the fuel consumed by the finite time control system is less than that of the PD control.

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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Fig. 7. The response of actual control torque.

Fig. 8. Energy consumption for the two control schemes.

5.3. Case C: proposed control scheme vs. control scheme in Ref. [23] Finally, two schemes are compared to examine the overall performance in this case. Fig. 9 shows that fast convergence within 0.2 s is achieved for estimation of the angular velocity using the proposed observer with high accuracy. Whereas the angular velocity estimation derived from the observer in Ref. [23] converges after 0.3 s but with severe oscillations. Figs. 10 and 11 show the

time responses for the two schemes, and it can be obviously descried that the proposed scheme has the faster transient response with higher accuracy. In Fig. 12, the energy consumption of proposed scheme is less than that of scheme in Ref. [23] during each period. And from Fig. 13, it is found that with proposed scheme, the Euler angle of the three-dimensional trajectory path is shorter, hence the proposed scheme achieves faster convergence rate and consume less energy. The sphere in Fig. 14 means that the angular velocity error will converge to a small neighborhood containing the equilibrium derived from Eq. (49). The red curve denotes observation error out of the neighborhood. On the contrary, the black curve represents when observation error arrives in the convergence set, it will be attracted and maintains in the sphere. Moreover, some facts are discovered by carrying out the simulation, the observer designed in Ref. [23] estimates the derivate of Rodrigues parameters, on account of the process of derivation, the sampling time should be less 0.01 s in simulation, otherwise, the angular velocity estimation will have a serious chattering, which reduces the control performance in some sense. However, in actual spacecraft attitude control implementation, it is quite difficult of setting the sampling time less than 0.01 s. Additionally, to examine the robustness of the proposed scheme subject to measurement noise, the measurement noise is added to q for both observer and control law feedback of all control schemes. Noisy measurements are generated by randomly perturbing the true unit-length eigenaxis, associated with q, which can be shown as [28]

Fig. 9. The angular velocity estimation error.

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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Fig. 10. The response of angular velocity estimation.

Fig. 11. The response of actual control torque.

Fig. 13. Attitude response of the spacecraft Euler angle in three-dimensional visualization. Fig. 12. Energy consumption for the two control schemes.

e=

1 sin

θ 2

qv ,

cos

θ = q0 2

where θ is the eigenangle, within a spherical cone of prescribed cone half-angle and uniform distribution centered around the true eigenaxis at each time t. The cone half-angle is specified as 0.05° for this simulation. As would be predicted, it is evident that the proposed scheme suffers in overall performance when measurement noise is present from Figs. 15 to 18. The magnitude of the attitude quaternion,

estimation of angular velocity and control torque are dictated by the magnitude of measurement noise, which are only minimized to a non-zero steady-state value. However, there is scarcely any influence of the estimation error of angular velocity caused by measurement noise. Thus, empirical evidence suggests that the proposed scheme is not adversely effected any worse in the presence of measurement noise than the situation that there is no noise. Summarizing all of the cases above, it is noted that the proposed scheme in our work can accomplish attitude stabilization mission successfully with high accuracy and rapid response speed in the presence of external disturbances, reaction wheel friction torque, and even actuator input constraints as well as measure noise in simulation.

Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i

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Fig. 17. The response of actual control torque.

Fig. 14. The convergence set of angular velocity estimation error.

Fig. 18. The response of angular velocity estimation.

Fig. 15. The response of attitude quaternion.

presence of external disturbances and reaction wheel friction simultaneously. Firstly, the reaction wheel dynamic model considering the friction torque is introduced. Afterwards, the controller was designed free of angular velocity measurement and a finite-time observer was employed to rapidly generate an angular velocity signal. Then, on the basis of angular velocity estimation, the semi-global finite time stability of closed-loop system was rigorously demonstrated and analyzed by using Lyapunov theorem and homogeneous method theoretically. The performance of the proposed finite-time scheme was verified through numerical simulation comparisons with PD-like controller and another finite time control strategy from the literature. Contrary to existing control schemes, it was shown that the proposed control scheme can provide a faster convergence rate, better disturbance rejection property, and higher accuracy control performance as well as less energy consumption, and even a good performance in the presence of measurement noise. In future work, the extension of the controller for fault tolerant control design with actuator faults and the measurement noise elimination will be investigated.

Fig. 16. The angular velocity estimation error.

Moreover, it is so convenient of selecting observer gains and controller parameters as to obtain the desired performance and strong robustness while satisfying the constraints on control magnitude.

6. Conclusion In this paper, the problem of attitude control with finite-time convergence was considered for the rigid spacecraft in the

Acknowledgments This work was supported partially by National Natural Science Foundation of China (Project No. 61522301, 61633003). The authors greatly appreciate the above financial support. The authors would also like to thank the associate editor and anonymous reviewers for their valuable comments and constructive suggestions that helped to improve the paper significantly.

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Please cite this article as: Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.023i