# Adaptive-Gain Second-Order Sliding Mode Control of Attitude Tracking of Flexible Spacecraft

Authors: Chutiphon Pukdeboon
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper investigates the robust finite-time control problem for flexible spacecraft attitude tracking maneuver in the presence of model uncertainties and external disturbances. Two robust attitude tracking controllers based on
finite-time second-order sliding mode control algorithms are presented to solve this problem. For the first controller, a novel second-order sliding mode control scheme is developed to achieve high-precision tracking performance. For the second control law, an adaptive-gain second-order sliding mode control algorithm combing an adaptive law with second-order sliding mode control strategy is designed to relax the requirement of prior knowledge of the bound of the system uncertainties. The rigorous proofs show that the proposed controllers provide finite-time convergence of the attitude and angular velocity tracking errors. Numerical simulations on attitude tracking control are presented to demonstrate the performance of the developed controllers.


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## Body

## 1. Introduction

In recent years, considerable attention has been focused on spacecraft attitude control problems. Attitude control systems are required to offer the present generation of spacecraft with attitude maneuver, tracking, and pointing capabilities. Since the attitude dynamics of spacecraft is coupled and highly nonlinear, the attitude controller designs are usually difficult. Besides, the effect of motion of the elastic appendages makes the control problem more complicated. In the absence of external disturbances, the modal-independent proportional-derivative (PD) controller proposed in [1] can achieve asymptotical stability of both the attitude and angular velocity. In practical situations, the model parameters of the spacecraft may not be exactly known and the spacecraft is always subject to external disturbances. Thus, the attitude control problem with uncertainties and external disturbance has also attracted a great deal of attention. Various nonlinear robust control approaches [2–4] have been proposed for solving the attitude tracking control problem of flexible spacecraft. These control schemes include adaptive control [5, 6], sliding mode control [7, 8], output feedback control [9, 10], optimal control [11, 12], and intelligent control [13].Among these methods, sliding mode control (SMC) has been shown to be a potential approach when applied to a system with disturbances which satisfy the matched uncertainty condition [14]. Robust attitude controllers of flexible spacecraft based on the SMC scheme have been proposed in [15, 16]. These control laws can achieve global asymptotic stability and provide good tracking results. However, these controllers were designed based on an asymptotic stability analysis which implies that the system trajectories converge to the equilibrium with infinite settling time. It is well known that finite-time stabilization of dynamical systems may provide a faster disturbance attenuation besides giving faster convergence to the required orientation. A recently developed technique for finite-time stabilization is the terminal sliding mode (TSM) method [17, 18] which can be used to design a controller that will guarantee a finite-time convergence to the origin. In [19, 20], the attitude motion of flexible spacecraft has been studied and the TSM method was used to design finite- time controllers. However, the TSM method usually provides lower tracking precision when compared with second-order sliding mode control (SOSMC) schemes [21, 22]. This technique preserves the robustness ability of SMC and also yields improved accuracy and performance. Various real-life applications have been controlled in a practical implementation of SOSMC schemes (e.g., see [23–25]). The SOSMC strategies have been successfully applied to attitude tracking controller designs for a rigid spacecraft in [26, 27], but these schemes have been rarely used for attitude tracking control of flexible spacecraft.In this paper, by virtue of SOSMC designs, the proposed attitude tracking control laws can guarantee the convergence of attitude tracking errors in finite-time. First, a novel finite-time SOSMC scheme is designed to achieve fast and accurate tracking responses. Next, to obtain the second control law, we combine the adaptive law with the SOSMC scheme proposed in [27]. This controller relaxes the requirement of prior knowledge on the bound of uncertainties. With the time scaling approach [28], the control parameters can be tuned by using only one variable.The main contributions of this paper are as follows.(I)
Robust finite-time control algorithms based on SOSMC schemes have been rarely studied for attitude tracking maneuver of a flexible spacecraft. The finite-time stability of the proposed control laws is analyzed using Lyapunov stability concepts.(II)
A novel adaptive law for the gains of the SOSMC algorithm has been rarely developed by using the time scaling approach. The presented control method does not needa priori knowledge of the uncertainty and disturbance bounds.This paper is organized as follows. In Section2, the dynamic and kinematic equations governing the attitude model [29, 30] are described and the control design problem is formulated. Section 3 presents a novel SOSMC control algorithm for a flexible spacecraft. The sliding manifold is chosen and the sliding control law is studied and a proof of finite-time convergence of this controller is given. Section 4 proposes an adaptive-gain SOSMC law. The stability of the closed-loop system is analyzed. A numerical example of spacecraft tracking maneuvers is presented in Section 5 to verify the usefulness of the proposed controllers. In Section 6, we present conclusions.


## 2. Nonlinear Mode and Problem Formulation

### 2.1. Spacecraft Attitude Dynamics and Kinematics

The unit quaternion is adopted to describe the attitude of the spacecraft for global representation without singularities [29]. The unit quaternion Q is defined by
(1)Q=[nsin⁡(ϕ2)cos⁡⁡(ϕ2)]=[qq4],
where n∈R3 is a unit vector called the Euler axis, ϕ∈R denotes the magnitude of Euler axis rotation and q∈R3 and q4∈R are the vector components and the scalar of the unit quaternion, respectively. They are subject to the constraint qTq+q42=1. Consider the first time derivative of Q. The kinematic equations are given by [29, 30]
(2)q˙=12(q4I3+q×)ω,q˙4=-12qTω,
where I3 is a 3×3 identity matrix, and q× is a skew-symmetric matrix:
(3)q×=[0-q3q2q30-q1-q2q10].


### 2.2. Relative Attitude Error Kinematics

We explain briefly the attitude error using quaternions. We define here the desired quaternionQr=[qrTq4r]T∈R3×R with  qr=[q1rq2rq3r]T. Also the attitude error Qe=[qeTq4e]T∈R3×R with qe=[q1eq2eq3e]T. Using the quaternion multiplication law, we obtain
(4)Qe=[q4rq-q4qr-qr×qq4q4r+qTqr]
subject to the constraint
(5)QeTQe=(qTq+q42)(qrTqr+q4r2)=1.The kinematic equation for the attitude error is expressed as [29, 30]
(6)Q˙e=12[qe×+q4eI3×3-qeT]ωe.


### 2.3. Flexible Spacecraft Dynamics

The equation governing a flexible spacecraft is expressed as [3]
(7)Jω˙+δTη¨=-ω×(Jω+δTη˙)+u+d,η¨+C-η˙+K-η=-δω˙,
where J=JT∈R3×3 is the total inertia matrix of the spacecraft, η∈R4 is the modal displacement, and δ∈R4×3 is the coupling matrix between the central rigid body and the flexible attachments. ω=[ω1ω2ω3]T∈R3 represents the angular velocity vector and ω× is a skew-symmetric matrix with a formula similar to q×. u=[u1u2u3]T∈R3 is the control input, d=[d1d2d3]T∈R3 represents the external disturbance torque, and K- and C- denote the stiffness and damping matrices, respectively, which are defined as
(8)K-=diag⁡(ωni2,i=1,2,…,N),C-=diag⁡(2ζiωni,i=1,2,…,N)
with damping ζi and natural frequency ωni.If termsδTη¨ and ω×δTη˙ are considered as lumped perturbations to rigid body dynamics, (7) can be obtained as
(9)Jω˙=-ω×Jω+u+d+Δf(η˙,η¨,ω),
where Δf(η˙,η¨,ω)=-δTη¨-ω×δTη˙ may be considered as the lumped perturbation. Let the angular velocity error be defined as ωe=ω-ωd; then the error dynamics are given by
(10)Jω˙e=-(ωe+ωd)×J(ωe+ωd)-Jω˙d+u+Td,
where Td=d+Δf(η˙,η¨,ω).Using (6), one has
(11)q˙e=Ξ(qe,q4e)ωe
with Ξ(qe,q4e)=(1/2)(qe×+q4eI3).DefineP≜Ξ-1 and  J*≜PTJP and consider the time derivative of (11). One can obtain [3]
(12)J*q¨e+C*q˙e+N*=τu+τd,
where
(13)C*=-J*P˙-1P-2PT(JPq˙e)×P,τu=12PTu,τd=12PTTd,N*=PT[(Pq˙e)×Jωd]+PT[(ωd)×JPq˙e]+12PT[(ωd)×Jωd]+12PTJω˙d.Throughout the remainder of this paper, the following are assumed.Assumption 1.
The elastic oscillation and its rate are supposed to be bounded; that is to say,∥η(t)∥ and ∥η˙(t)∥ are bounded during the whole attitude tracking process.Letx1=qe and x2=q˙e; the spacecraft model (12) can be written as
(14)x˙1=x2,x˙2=F+B0u+d~,
where F=-J*(C*x2+N*), B0=(1/2)(PTJP)-1PT, and d~=(PTJP)-1τd.Assumption 2.
The first time derivative of the disturbance vectord~ to the spacecraft system in (12) is assumed to be bounded and it satisfies the following condition:
(15)|d~˙i(t)|≤ϱ,
where ϱ is a positive constant.


### 2.4. Problem Statement

In this work, we consider tracking maneuvers. The control objective is to realize desired rotations of flexible spacecraft in the presence of external disturbances. In other words, we shall find a controlleru subject to (14) such that, for all initial conditions, the desired rotations are achieved as follows:
(16)lim⁡t→Tqe=0,lim⁡t→Tq4e=1,lim⁡t→Tωe=0,
where T is a positive constant. Note that, when qe→0, we have q4e→1, due to the constraint relation (5).


### 2.5. Finite-Time Stability

We now restate the concepts related to finite-time stability [31, 32].Definition 3 (see [31]).
Consider a time invariant system in the form of(17)x˙=f(x),f(0)=0,x∈Rn,
where f:U^0→Rn is continuous on an open neighborhood U^0 of the origin. The equilibrium x=0 of the system (17) is (locally) finite-time stable if (i) it is Lyapunov stable, in U^, an open neighborhood of the origin, with U^⊂U^0, (ii) it is finite-time convergent in U^, that is, for any condition x0∈U^∖{0}, there is a settling time T>0, such that every solution x(t,x0) of the system (17) is defined with x(t,x0)∈U^{0} for t∈[0,T] and it satisfies
(18)lim⁡t→T(x0)x(t,x0)=0
and x(t,x0)=0 if t≥T. Moreover, if U^=Rn, the origin is globally finite-time stable.Definition 4.
Consider a controlled system(19)x˙=f(x)+g(x)u,x∈Rn,u∈Rm
with g(x)≠0. It is finite-time stabilizable if there is a feedback law u(x) such that x=0 is a (locally) finite-time stable equilibrium of closed-loop system.Lemma 5 (see [32]).
Suppose that the origin is a finite-time stable equilibrium of (17) and the settling time function Tf is continuous at zero, where f(·) is continuous. Let U^ be defined as in Definition 3 and let ι∈(0,1). There exists a continuous scalar function V such that (i)  V is positive definite and (ii)  V˙ is real valued and continuous on U^ and there exists c∈R+ such that
(20)V˙+cVι≤0,foranyι∈(0,1).Lemma 6 (Feng et al. [18]).
For any numbersλ1>0, λ2>0, and 0<ϖ<1, an extended Lyapunov contion of finite-time stability can be given in the form of fast terminal sliding mode as
(21)V˙(x)+λ1V(x)+λ2V2ϖ(x)≤0,
where the settling time can be estimated by
(22)Tr≤1λ1(1-ϖ)ln⁡λ1V1-ϖ(x0)+λ2λ2.


## 2.1. Spacecraft Attitude Dynamics and Kinematics

The unit quaternion is adopted to describe the attitude of the spacecraft for global representation without singularities [29]. The unit quaternion Q is defined by
(1)Q=[nsin⁡(ϕ2)cos⁡⁡(ϕ2)]=[qq4],
where n∈R3 is a unit vector called the Euler axis, ϕ∈R denotes the magnitude of Euler axis rotation and q∈R3 and q4∈R are the vector components and the scalar of the unit quaternion, respectively. They are subject to the constraint qTq+q42=1. Consider the first time derivative of Q. The kinematic equations are given by [29, 30]
(2)q˙=12(q4I3+q×)ω,q˙4=-12qTω,
where I3 is a 3×3 identity matrix, and q× is a skew-symmetric matrix:
(3)q×=[0-q3q2q30-q1-q2q10].


## 2.2. Relative Attitude Error Kinematics

We explain briefly the attitude error using quaternions. We define here the desired quaternionQr=[qrTq4r]T∈R3×R with  qr=[q1rq2rq3r]T. Also the attitude error Qe=[qeTq4e]T∈R3×R with qe=[q1eq2eq3e]T. Using the quaternion multiplication law, we obtain
(4)Qe=[q4rq-q4qr-qr×qq4q4r+qTqr]
subject to the constraint
(5)QeTQe=(qTq+q42)(qrTqr+q4r2)=1.The kinematic equation for the attitude error is expressed as [29, 30]
(6)Q˙e=12[qe×+q4eI3×3-qeT]ωe.


## 2.3. Flexible Spacecraft Dynamics

The equation governing a flexible spacecraft is expressed as [3]
(7)Jω˙+δTη¨=-ω×(Jω+δTη˙)+u+d,η¨+C-η˙+K-η=-δω˙,
where J=JT∈R3×3 is the total inertia matrix of the spacecraft, η∈R4 is the modal displacement, and δ∈R4×3 is the coupling matrix between the central rigid body and the flexible attachments. ω=[ω1ω2ω3]T∈R3 represents the angular velocity vector and ω× is a skew-symmetric matrix with a formula similar to q×. u=[u1u2u3]T∈R3 is the control input, d=[d1d2d3]T∈R3 represents the external disturbance torque, and K- and C- denote the stiffness and damping matrices, respectively, which are defined as
(8)K-=diag⁡(ωni2,i=1,2,…,N),C-=diag⁡(2ζiωni,i=1,2,…,N)
with damping ζi and natural frequency ωni.If termsδTη¨ and ω×δTη˙ are considered as lumped perturbations to rigid body dynamics, (7) can be obtained as
(9)Jω˙=-ω×Jω+u+d+Δf(η˙,η¨,ω),
where Δf(η˙,η¨,ω)=-δTη¨-ω×δTη˙ may be considered as the lumped perturbation. Let the angular velocity error be defined as ωe=ω-ωd; then the error dynamics are given by
(10)Jω˙e=-(ωe+ωd)×J(ωe+ωd)-Jω˙d+u+Td,
where Td=d+Δf(η˙,η¨,ω).Using (6), one has
(11)q˙e=Ξ(qe,q4e)ωe
with Ξ(qe,q4e)=(1/2)(qe×+q4eI3).DefineP≜Ξ-1 and  J*≜PTJP and consider the time derivative of (11). One can obtain [3]
(12)J*q¨e+C*q˙e+N*=τu+τd,
where
(13)C*=-J*P˙-1P-2PT(JPq˙e)×P,τu=12PTu,τd=12PTTd,N*=PT[(Pq˙e)×Jωd]+PT[(ωd)×JPq˙e]+12PT[(ωd)×Jωd]+12PTJω˙d.Throughout the remainder of this paper, the following are assumed.Assumption 1.
The elastic oscillation and its rate are supposed to be bounded; that is to say,∥η(t)∥ and ∥η˙(t)∥ are bounded during the whole attitude tracking process.Letx1=qe and x2=q˙e; the spacecraft model (12) can be written as
(14)x˙1=x2,x˙2=F+B0u+d~,
where F=-J*(C*x2+N*), B0=(1/2)(PTJP)-1PT, and d~=(PTJP)-1τd.Assumption 2.
The first time derivative of the disturbance vectord~ to the spacecraft system in (12) is assumed to be bounded and it satisfies the following condition:
(15)|d~˙i(t)|≤ϱ,
where ϱ is a positive constant.


## 2.4. Problem Statement

In this work, we consider tracking maneuvers. The control objective is to realize desired rotations of flexible spacecraft in the presence of external disturbances. In other words, we shall find a controlleru subject to (14) such that, for all initial conditions, the desired rotations are achieved as follows:
(16)lim⁡t→Tqe=0,lim⁡t→Tq4e=1,lim⁡t→Tωe=0,
where T is a positive constant. Note that, when qe→0, we have q4e→1, due to the constraint relation (5).


## 2.5. Finite-Time Stability

We now restate the concepts related to finite-time stability [31, 32].Definition 3 (see [31]).
Consider a time invariant system in the form of(17)x˙=f(x),f(0)=0,x∈Rn,
where f:U^0→Rn is continuous on an open neighborhood U^0 of the origin. The equilibrium x=0 of the system (17) is (locally) finite-time stable if (i) it is Lyapunov stable, in U^, an open neighborhood of the origin, with U^⊂U^0, (ii) it is finite-time convergent in U^, that is, for any condition x0∈U^∖{0}, there is a settling time T>0, such that every solution x(t,x0) of the system (17) is defined with x(t,x0)∈U^{0} for t∈[0,T] and it satisfies
(18)lim⁡t→T(x0)x(t,x0)=0
and x(t,x0)=0 if t≥T. Moreover, if U^=Rn, the origin is globally finite-time stable.Definition 4.
Consider a controlled system(19)x˙=f(x)+g(x)u,x∈Rn,u∈Rm
with g(x)≠0. It is finite-time stabilizable if there is a feedback law u(x) such that x=0 is a (locally) finite-time stable equilibrium of closed-loop system.Lemma 5 (see [32]).
Suppose that the origin is a finite-time stable equilibrium of (17) and the settling time function Tf is continuous at zero, where f(·) is continuous. Let U^ be defined as in Definition 3 and let ι∈(0,1). There exists a continuous scalar function V such that (i)  V is positive definite and (ii)  V˙ is real valued and continuous on U^ and there exists c∈R+ such that
(20)V˙+cVι≤0,foranyι∈(0,1).Lemma 6 (Feng et al. [18]).
For any numbersλ1>0, λ2>0, and 0<ϖ<1, an extended Lyapunov contion of finite-time stability can be given in the form of fast terminal sliding mode as
(21)V˙(x)+λ1V(x)+λ2V2ϖ(x)≤0,
where the settling time can be estimated by
(22)Tr≤1λ1(1-ϖ)ln⁡λ1V1-ϖ(x0)+λ2λ2.


## 3. SOSMC Algorithm

In this section, a novel SOSMC algorithm is proposed to accomplish attitude tracking control of a flexible spacecraft. Under Assumptions1 and 2, this control scheme is designed such that the attitude tracking error qe and the angular velocity error ωe approach zero in finite time although uncertainties and external disturbances are taken into account. This new SOSMC law is achieved by a Lyapunov function approach, so the finite-time stability of the closed-loop system of the spacecraft model (14) is ensured. The goal is to enforce the sliding mode on the manifold
(23)S=x2+κx1,
where S=[S1S2S3]T∈R3 and κ is a positive constant. From (23), the first time derivative of S is obtained as
(24)S˙=x˙2+κx˙1,=F+B0u+d~+κx2.
Then, it is followed by choosing a SOSMC law to force the system trajectory onto the sliding surface. Let s1=S and s2=S˙; the proposed control law is defined as
(25)u=(B0)-1(-F-κx2-μ1sign⁡α(s1)-μ2s1∫0t‍hhhhhhh-∫0t‍(μ3sign⁡α(s2(τ))+μ4s2(τ))dτ),
where 0<α<1. The gain matrices μ1=diag⁡(μ11,μ12,μ13), μ2=diag⁡(μ21,μ22,μ23), μ3=diag⁡(μ31,μ32,μ33), and μ4=diag⁡(μ41,μ42,μ43), with μ1i, μ2i, μ3i and μ4i(i=1,2,3), are positive gains. The function sign⁡γ(y) is defined as
(26)sign⁡γ(y)=[|y1|γsign⁡(y1)|y2|γsign⁡(y2)⋯|ym|γsign⁡(ym)]T
with 0<γ<1 and y∈Rm.Substituting (25) into (24), the closed-loop dynamics is obtained as
(27)s1˙=μ1sign⁡α(s1)-μ2s1-∫0t‍(μ3sign⁡α(s2(τ))+μ4s2(τ))dτ+d~.
Let d~˙i(t)=δi(t),  z1i=s1i and z2i=-∫0t‍(μ3isign⁡α(s2i(τ))-μ4is2i(τ))dτ+d~i. The system (27) can be written in scalar form (i=1,2,3) as
(28)z˙1i=-μ1i|z1i|αsign⁡(z1i)+z2i-μ2iz1i,z˙2i=-μ3i|z2i|αsign⁡(z2i)-z2i-μ4iz1i+δi.
Next, the proof of finite-time convergence to the origin is given.Theorem 7.
Under Assumptions1 and 2 and the action of the control law (25), the state trajectories z1i and z2i converge in finite time to the region
(29)∥ε∥≤Δ=min⁡⁡{Δ1,Δ2},Δ1=2ϱϵ2,Δ2=2(2ϱϵ1)1/α,
where ε=[z1iz2i]T and Δ1, Δ2>0 are the results from the chosen gains.Proof.
We choose the Lyapunov functionV1=(1/2)ε2 for analyzing the closed-loop system dynamics of (28). The time derivative of V1 is given by
(30)V˙1=z1i(-μ1i|z1i|αsign⁡(z1i)+z2i-μ2iz1i)+z2i(-μ3i|z2i|αsign⁡(z2i)-z1i-μ4iz2i+δi)=-μ1i|z1i|α+1-μ2iz1i2-μ3i|z2i|α+1-μ4iz2i2+δiz2i,
which can be further written as
(31)V˙1≤-μ1m|z1i|α+1-μ2mz1i2-μ3m|z2i|α+1-μ4mz2i2+ϱ|z2i|,
where μ1m=min⁡(μ1i), μ2m=min⁡(μ2i), μ3m=min⁡(μ3i), and μ4m=min⁡(μ4i), i=1,2,3.
Lettingμ-1=min⁡(μ1m,μ3m) and μ-2=min⁡(μ2m,μ4m), one obtains
(32)V˙1≤-μ-1(|z1i|α+1+|z2i|α+1)-μ-2(z1i2+z2i2)+ϱ∥ε∥V˙1≤-μ-12(α+1)/2V1(α+1)/2-2μ-2V1+2ϱV1.
Letting ϵ1=μ-12(α+1)/2 and ϵ2=2μ-2, we can change (32) into the following forms:
(33)V˙1≤-(ϵ2-2ϱV1)V1-ϵ1V1(α+1)/2,(34)V˙1≤-ϵ2V1-(ϵ1-2ϱV1α/2)V1(α+1)/2.
From (33), if ϵ2-2ϱ/V1>0 then the finite-time stability is still ensured, and hence, by Lemma 5, the state trajectories z1i and z2i of the system (28) converge to the region
(35)∥ε∥≤2ϱϵ2
in finite time. From (33), if ϵ1-2ϱ/V1α/2>0, then the finite-time stability is still ensured, and hence, by Lemma 5, the state trajectories z1i and z2i of the system (28) converge to the region
(36)∥ε∥≤2(2ϱϵ1)1/α
in finite time. Therefore, the state trajectories z1i and z2i of the system (28) converge in finite-time to the region
(37)∥ε∥≤Δ=min⁡{Δ1,Δ2},
where Δ1=2ϱ/ϵ2 and Δ2=2(2ϱ/ϵ1)1/α.Remark 8.
The accuracy of tracking errorsz1i and z2i is determined by the control parameters μ1i, μ2i, μ3i, and μ4i, i=1,2,3. To obtain the desired accuracy, it is necessary to reduce the size of the region (37) by increasing the values of the control parameters μ1i, μ2i, μ3i, and μ4i. However, using large values of these parameters leads to high magnitudes of control torques.


## 4. Adaptive-Gain SOSMC Algorithm

Next, the proposed adaptive-gain SOSMC scheme is designed by modifying the smooth SOSMC law presented in [27]. In stead of using the constant controller gains as presented in [27], the adaptive gains are considered in this paper. A novel adaptive law to tune these gains is presented. The main advantages of the proposed method are that only one parameter has to be tuned and this method relaxes the requirement of prior knowledge of the bound of system uncertainties.The proposed control law is given by(38)u=B0-1(∫0t‍-F-κx2-λ(t)sign⁡(p-1)/p(S)-ϵ(t)S-α(t)∫0t‍sign⁡(p-2)/p(S(τ))dτ-K(t)∫0t‍S(τ)dτ),
where p>2 and the adaptive gains λ(t), α(t), ϵ(t), and K(t) are the design parameters defined as
(39)λ(t)=λ0diag⁡(L(p-1)/p(t),L(p-1)/p(t),L(p-1)/p(t)),α(t)=α0diag⁡(L(t),L(t),L(t)),ϵ(t)=ϵ0diag⁡(L(t),L(t),L(t)),K(t)=K0diag⁡(L2(t),L2(t),L2(t))
with λ0, α0, ϵ0, K0 being positive constants. The adaptive law of the time-varying function L(t) is given by
(40)L˙(t)={l,for|Si|≠0,0,for|Si|=0,
where l and the initial value L(0) are positive constants.Substituting (38) into (14), we obtain
(41)S˙=-λ(t)sign⁡(p-1)/p(S)-ϵ(t)S-α(t)×∫0t‍sign⁡(p-2)/pS(τ)dτ-K(t)∫0t‍S(τ)dτ+d~.
Next, the same procedure as presented in Section 3 is recalled. We can have the scalar form of (41) as
(42)z˙1i=-λ(t)|z1i|(p-1)/psign⁡(z1i)-ϵ(t)z1i+z2i,z˙2i=-α(t)|z1i|(p-2)/psign⁡(z1i)-K(t)z1i+δi,1111111111111111111111111111111i=1,2,3.Theorem 9.
Suppose thatλ0, α0, ϵ0, K0>0 are selected such that
(43)α0K0>[α0pp-1+(2p-1p)2λ02]ϵ02,
and Assumptions 1 and 2 hold. Then, all trajectories of the system (42) converge in finite time to the region
(44)Ω={∥ξ∥≤(γ3L(t)γ1)(p-1)/(p-2)},
where ξ=[L(p-1)/p|z1i|(p-1)/psign⁡(z1i)Lz1iz2i]T and the positive scalars γ1 and γ3 will be defined later.Proof.
Let the Lyapunov function be chosen as(45)V2(z)=α0pp-1L2(p-1)/p|z1i|2(p-1)/p+K0L2z1i2+12z2i2+12(λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)+ϵ0Lz1i-z2i)2.
The selected Lyapunov function can be expressed as
(46)V2(z)=(α0pp-1+12λ02)L2(p-1)/p|z1i|2(p-1)/p+z2i2+(K0+12ϵ02)L2z1i2+λ0ϵ0|z1i|L(p-1)/p|z1i|(p-1)/p-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)z2i-ϵ0Lz1iz2i,
which can be written as
(47)V2(z)=ξTΠ1ξ,
where
(48)Π1=12[2α0pp-1+λ02λ0ϵ0-λ0λ0ϵ0(2K0+ϵ02)-ϵ0-λ0-ϵ02].
It satisfies
(49)σmin⁡(Π1)∥ξ∥2≤V2≤σmax⁡(Π1)∥ξ∥2,
where ∥ξ∥2=L2(p-1)/p|z1i|2(p-1)/p+L2z1i2+z2i2. The time derivative of V2 is obtained as
(50)V˙2≤12(4α0+λ02p-1p)L(2p-2)/p|z1i|(p-2)/psign⁡(z1i)z˙1i+2z2iz˙2i+(2K0+ϵ02)L2z1iz˙1i+λ0ϵ0(2p-1p)L(p-1)/p|z1i|(p-1)/psign⁡(z1i)z˙1i-λ0L(p-1)/p|z1i|-1/pz1iz˙2i-λ0L(p-1)/p(p-1p)|z1i|-1/pz2iz˙1i-ϵ0Lz1iz˙2i-ϵ0Lz2iz˙1i+12(4α0+λ02p-1p)|z1i|2(p-1)/pL(p-2)/pL˙+(2K0+ϵ02)LL˙z1i2+λ0ϵ0((2p-1)p)L(p-1)/p|z1i|(p-1)/pL˙+λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)z2iL-1/pL˙-ϵ0z1iz2iL˙.
Substituting z˙1i and z˙2i into (50), we obtain
(51)V˙2≤12(4α0+λ02p-1p)L(2p-2)/p|z1i|(p-2)/psign⁡(z1i)×(z2i-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)-ϵ0Lz1i)+2z2i(-α0L|z1i|(p-2)/psign⁡(z1i)1111111-K0L2z1i+δi(|z1i|(p-2)/p))+(2K0+ϵ02)L2z1i×(-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)-ϵ0Lz1i+z2iL(p-1^)/p)+(2p-1p)λ0ϵ0L(2p-1)/p|z1i|(p-1)/psign⁡(z1i)×(-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)-ϵ0Lz1i+z2i|z1i|(p-1)/p)-λ0L(p-1)/p|z1i|-1/pz1i×(-α0L|z1i|(p-2)/psign⁡(z1i)-K0L2z1i+δi)-λ0L(p-1)/p(p-1p)|z1i|-1/pz2i×(-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)-ϵ0Lz1i+z2i|z1i|(p-2)/p)-ϵ0Lz1i(-α0L|z1i|(p-2)/psign⁡(z1i)-K0L2z1i+δi|z1i|(p-2)/p)-ϵ0Lz2i(-λ0L(p-1)/p|z1i|(p-1)/psign⁡(z1i)-ϵ0Lz1i+z2iL(p-1)/p|z1i|(p-1)/p)+12(4α0+λ02p-1p)|z1i|2(p-1)/pL(p-2)/pL˙+(2K0+ϵ02)LL˙z1i2+λ0ϵ0((2p-1)p)L-1/p|z1i|(p-1)/pL˙+λ0p-1p|z1i|(p-1)/psign⁡(z1i)z2iL-1/pL˙-ϵ0z1iz2iL˙.
After lengthy algebraic manipulation, one obtains(52)V˙2≤|z1i|-1/pL-1/p×([α0λ0+λ03(p-1)p]|z1i|2(p-1)/pL2(p-1)/p11111-2λ02(p-1)pL(p-1)/p|z1i|(p-1)/psign⁡(z1i)z2i111111+[λ0K0+12λ0ϵ02(6p-2)p]z1i2L2111111-λ0ϵ0(4p-2)pLz1iz2i+(p-1)pλ0z2i2[α0λ0+λ03(p-1)p])-([ϵ0α0+ϵ03(3p-2)p]|z1i|2(p-1)/pL2(p-1)/p+(ϵ0K0+ϵ03)z1i2L2+2ϵ02Lz1iz2i-ϵ0z2i2[ϵ0α0+ϵ03(3p-2)p])-δi|z1i|(p-1)/pL(p-1)/psign⁡(z1i)-δiϵ0Lz1i+2δiz2i+L˙(t)L[(2α0+λ02p-1p)L(2p-2)/p|z1i|(2p-2)/p]+(2K0+ϵ02)L2L˙z1i2+λ0ϵ02p-1p|z1i|(2p-1)/pL(p-1)/pL˙-λ0p-1pL(p-1)/p|z1i|(p-1)/psign⁡(z1i)z2i-ϵ0Lz1iz2i,
which can be written as
(53)V˙2=-|z1i|-1/pL-1/pL(t)ξTΩ1ξ-L(t)ξTΩ2ξ+δiΓ1ξ+L˙(t)LΓ2Π2ξ,
where Γ1=[-λ0-ϵ02]T, Γ2=[L(p-1)/p|z1i|(p-1)/psign⁡(z1i)Lz1i0]T,
(54)Ω1=λ02×[2α0+λ02(2p-2p)0-λ0(2p-2p)02K0+ϵ02(6p-2p)-ϵ0(4p-2p)-λ0(2p-2p)-ϵ0(4p-2p)2(p-1)p],Ω2=ϵ0[α0+λ02(3p-2p)000K0+ϵ0-ϵ00-ϵ01],Π2=12[4α0+2λ02(p-1p)λ0ϵ0(2p-1p)-λ0λ0ϵ0(2p-1p)(2K0+ϵ02)-ϵ0-λ0-ϵ02].
By Assumption 2 and using ∥ξ∥≥∥Γ2∥, one obtains
(55)V˙2=-|z1i|-1/pL-1/pL(t)σmin⁡(Ω1)∥ξ∥2-L(t)σmin⁡(Ω2)∥ξ∥2+|δi|∥Γ1∥∥ξ∥+L˙(t)Lσmin⁡(Π2)∥ξ∥2.
Using|z1i|-1/pL-1/p≥∥ξ∥-1/(p-1), we have
(56)V˙2=-L(t)σmin⁡(Ω1)∥ξ∥(2p-3)/(p-1)-L(t)σmin⁡(Ω2)∥ξ∥2+ϱ∥Γ1∥∥ξ∥+L˙(t)Lσmax⁡(Π2)∥ξ∥2.
Using (49), the expression (56) becomes
(57)V˙2=-L(t)σmin⁡(Ω1)σmax⁡(Π1)∥ξ∥(p-2)/(p-1)V21/2-L(t)σmin⁡(Ω2)σmax⁡(Π1)V2+ϱ∥Γ1∥σmin⁡(Π1)V21/2+L˙(t)2L(t)σmax⁡(Π2)σmin⁡(Π1)V2.
From simplicity, we define(58)γ1=σmin⁡(Ω1)σmax⁡(Π1),γ2=∥σmin⁡(Ω2)∥σmax⁡(Π1),γ3=ϱ∥Γ1∥σmin⁡(Π1),γ4=σmax⁡(Π2)2σmin⁡(Π1),
where γ1, γ2, γ3, and γ4 are positive constants. Thus, (57) can be written as
(59)V˙2≤-(L(t)γ1∥ξ∥(p-2)/(p-1)-γ3)V21/2-(L(t)γ2-L˙(t)Lγ4)V2.
BecauseL˙(t)≥0 such that the term L(t)γ2-(L˙(t)/L)γ4 is positive in finite time, it follows from (59) that, if L(t)γ1∥ξ∥(p-2)/(p-1)>γ3, the error system (42) will converge in finite time to the region
(60)Ω={∥ξ∥≤(γ3L(t)γ1)(p-1)/(p-2)}.
In fact, we can choose λ1i, λ2i, λ3i, and λ4i such that γ3/L(t)γ1<1. With p>2, (p-1)/(p-2) being sufficiently large, so ξ is sufficiently small in finite time.Remark 10.
The control law (38) includes an adaptive law which is used to compensate for the derivative of total disturbances d~. Normally, the bound of d~˙ is much smaller than the bound of d~, so this supports the high accuracy of our proposed method.


## 5. Simulation Results

An example of attitude control of flexible spacecraft is presented with numerical simulations to demonstrate the performance of the developed controllers (25) and (38). The spacecraft is assumed to have the nominal inertia matrix [33]
(61)J=[35034327010410190]kg·m2
and coupling matrices
(62)δ=[6.456371.278142.15629-1.256190.91756-1.672641.116782.48901-0.836741.23637-2.6581-1.12503]kg1/2·m/s2,
respectively. The first four elastic modes that have been considered in the model used for simulating a spacecraft are ωn1=0.7681, ωn2=1.1038, ωn3=1.8733, and ωn4=2.5496 rad/sec with damping ξ1=0.0056, ξ2=0.0086, ξ3=0.013, and ξ1=0.025. The initial states of the rotation motion are given by
(63)Q(0)=[0.3-0.10.20.9274]T,ω(0)=03×1rad/sec,ϑ(0)=04×1.
For the controller (25), the chosen gains are given as μ1=diag⁡(15,15,15), μ2=diag⁡(10,10,10), μ3=diag⁡(5,5,5), and μ4=diag⁡(10,10,10). On the other hand, for controller (38), the control gains are selected to be λ0=15, α0=10, ϵ0=10, K0=15, and l=5×105. For both control algorithms, we use the same sliding manifold (23) with the constant κ=1.2.The attitude control problem is considered in the presence of external disturbanced(t). The external disturbances are described as
(64)d(t)=[0.3cos⁡(0.1t)+0.10.15sin(0.1t)+0.3cos⁡(0.1t)0.3sin(0.1t)+0.1]N-m.
And the desired angular velocity tracking is given by
(65)ωd(t)=[-0.04cos⁡(0.2t)-0.04sin(0.2t)0.05sin(0.2t)+cos⁡(0.2t)]rad/sec
together with
(66)qd(0)=[000]T.For simulation results with controllers (25) and (38), the attitude quaternion tracking errors are shown in Figures 1 and 7, and angular velocity tracking errors are illustrated in Figures 2 and 8. Controller (38) gives smoother attitude and angular velocity tracking outputs than controller (25). From Figures 3 and 9, it can be seen that the sliding vectors are on the sliding surface s=0 after 40 seconds. The control torques of the controllers (38) and (25) are shown in Figures 4 and 10, respectively. It can be seen that controller (38) gives smoother responses of control torques. With the larger magnitude, controller (38) makes the responses go to zero faster than controller (25), so it gives smoother responses during the first 40 seconds and this improves the transient performance. Thus, under the control law (38) the smoother velocity and attitude tracking error responses are obtained. The responses of modal displacements are presented in Figures 5, 6, 11, and, 12, in which a low vibration level is illustrated for controller (38) in comparison with controller (25). For controller (25), the boundary layer ∥s∥≤7.63×10-5 is achieved in finite time. Regarding the accuracy, the bounds on ∥qe∥ and ∥ωe∥ are ∥qe∥≤5.12×10-5 and ∥ωe∥≤4.28×10-5 with the sampling time h=0.005. Also, for controller (38), the boundary layer ∥s∥≤4.7×10-5 is reached in finite time. Regarding the accuracy, the bounds on ∥qe∥ and ∥ωe∥ are ∥qe∥≤9.35×10-6 and ∥ωe∥≤1.5×10-5 with the sampling time h=0.005.Figure 1
Quaternion tracking errors—controller (25).Figure 2
Angular velocity tracking errors—controller (25).Figure 3
Switching function—controller (25).Figure 4
Control torques—controller (25).Figure 5
Modal displacementsη1 and η2—controller (25).Figure 6
Modal displacementsη3 and η4—controller (25).Figure 7
Quaternion tracking errors—controller (38).Figure 8
Angular velocity tracking errors—controller (38).Figure 9
Switching function—controller (38).Figure 10
Control torques—controller (38).Figure 11
Modal displacementsη1 and η2—controller (38).Figure 12
Modal displacementsη3 and η4—controller (38).A comparison of the simulation results obtained by (25) and (38) shows the following. Both control laws give high accuracy of attitude tracking outputs. Controller (38) provides smoother responses of attitude and angular velocity tracking errors when compared with controller (25). Moreover, controller (38) offers a lower vibration level of modal displacements. In view of these simulation results, controller (38) seems to give better performance for practical attitude tracking control of a flexible spacecraft.


## 6. Conclusions

The proposed finite-time controllers have been successfully applied to attitude tracking maneuvers of a flexible spacecraft. The first controller based on the novel SOSMC algorithm is developed to deal with quaternion-based spacecraft-attitude-tracking maneuvers. For the second control law, an adaptive-gain second-order sliding mode control algorithm is designed to relax the requirement of prior knowledge on the bound of system uncertainties. Both control laws achieve high-precision tracking performance and strong robustness ability. The concepts of the Lyapunov stability are employed to ensure a finite-time property of the proposed controllers. Numerical simulations on attitude tracking control are provided to demonstrate the performance of the developed controllers.



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*Source: 312494-2014-05-21.xml*