# Anti-Unwinding Attitude Control with Fixed-Time Convergence for a Flexible Spacecraft

Authors: Chutiphon Pukdeboon; Anuchit Jitpattanakul
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

This paper investigates the fixed-time attitude tracking control problem for flexible spacecraft with unknown bounded disturbances. First, with the knowledge of norm upper bounds of external disturbances and the coupling effect of flexible modes, a novel robust fixed-time controller is designed to deal with this problem. Second, the controller is further enhanced by an adaptive law to avoid the knowledge of norm upper bounds of external disturbances and coupling effect of flexible modes. This control law guarantees the convergence of attitude tracking errors in fixed time where the settling time is bounded by a constant independent of initial conditions. Moreover, the proposed controllers can prevent the unwinding phenomenon. Simulation results are presented to demonstrate the performance of the proposed control scheme.


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## Body

## 1. Introduction

With the development of modern satellite technology, space missions are expected to ensure rapid, high-precision and global response such as navigation, communication, astronomy, and earth observations. Modern spacecraft often requires the structure of a rigid hub with flexible appendages. Owing to strong coupling between hub and flexible appendages attitude maneuver is affected by vibration of flexible appendages. This may degrade pointing accuracy and dynamic performance of the attitude control system. Furthermore, in practical control systems, there inevitably exist model uncertainty and nonlinearity, inertia uncertainty, and various external disturbance torques, such as aerodynamic torque, radiation torque, and gravity gradient torque. These factors make the design of an attitude control law with rapid and precise maneuver performance for a flexible spacecraft very difficult and even pose a great challenge to space missions. In recent years, a variety of control methods have been proposed to solve the attitude control problem, like proportional derivative control [1], adaptive control [2], H∞ control [3], passivity-based control [4, 5], sliding mode control [6–8], active disturbance rejection control [9], disturbance observer based control [10], and so forth [11–13]. Although these nonlinear control laws have offered sufficient and reliable effectiveness and robustness in spacecraft attitude control systems, they require infinite time to accomplish an attitude maneuver. However, the ability of rapid maneuver is highly required in many real-time space missions. In other words, infinite-time attitude control is inadequate in some space missions.In fact, finite-time stabilization [14–17] can offer faster convergence to the origin and better disturbance rejection than asymptotic stabilization. Thus, finite-time control approaches are desirable to be considered in spacecraft attitude controller designs. Terminal sliding mode control (TSMC) [18, 19] method has provided a practical way to design a finite-time controller. For TSMC, nonlinear sliding surface was proposed and the finite-time convergence was analyzed based on the concept of terminal attractor. TSMC has been applied to spacecraft attitude control problems [20–22]. Later, a nonsingular terminal sliding mode based robust finite-time control law was proposed in [23, 24] to deal with the singularity problem, which exists in traditional TSMC. More recently, nonsingular fast terminal sliding mode control laws for a rigid spacecraft were designed in [25, 26]. In [27, 28] nonsingular fast terminal sliding mode control laws have been developed for attitude motions of flexible spacecraft.Apart from the above attention to achieve finite-time convergence, actuator fault is another key issue in spacecraft attitude control which may cause the performance degradation or even result in control system instability. To handle this issue, fault-tolerant control (FTC) is a widely used scheme to enhance the capability of maintaining control stability and high performance despite casual actuator failures. Recently, several FTC schemes have been developed for spacecraft attitude control [29–32]. Xiao et al. [33] proposed a fault-tolerant controller for a flexible spacecraft in the presence of possible additive fault and partial loss of actuator effectiveness fault. Zhang et al. [34] presented a robust fault-tolerant control scheme to achieve attitude control of flexible spacecraft with disturbances and actuator failures. These control algorithms can tolerate partial loss of actuator effectiveness. In [35] a fault-tolerant sliding mode attitude controller for flexible spacecraft with inertia uncertainty, external disturbance, actuator misalignment, and input saturation has been developed. Recently, FTC design for spacecraft attitude control with consideration of different kinds of actuator faults and failures is still an open problem.Although control methods mentioned above can provide good results of attitude tracking control of spacecraft, the initial system state must be known to estimate the settling time. It would be very useful if the settling time can be determined without knowledge of initial conditions. In [36, 37] Polyakov and his colleagues have proposed the concept of the fixed-finite-time stability. For the fixed-finite-time stability concept, the upper bound of the settling time can be estimated and it is independent of initial conditions. For the spacecraft attitude control system it is desirable to estimate the settling time independent of initial conditions. Moreover, these control methods may cause the unwinding phenomenon encountered in unit quaternion based attitude systems since they consider only one of two equilibrium points of unit quaternion [38].In this paper, the fixed-time control problem associated with attitude tracking of flexible spacecraft in the presence of bounded external disturbances and coupling effect of flexible modes is investigated. The aim is to design a robust fixed-time attitude controller so that rapid attitude maneuver can be achieved with high pointing accuracy. By extending the concepts of fixed-finite stability by Polyakov and his colleagues [36, 37], a new result of fixed-finite stability is introduced. This result is also used to develop our proposed controllers. So far, to the best our knowledge, there were no results of fixed-time sliding control methods for attitude maneuver control of a flexible spacecraft in presence of bounded disturbances with an unknown boundary. Inspired by the above discussion, two fixed-time controllers are presented in this paper and the main contributions lie in the following aspects:(1)
A new result of fixed-time stability is proposed by extending concepts of fixed-finite stability by Polyakov and his colleagues [36, 37]. For the case that the upper bounds of disturbances and the coupling effect of flexible modes are available in advance, a novel fixed-time controller is proposed for flexible spacecraft attitude maneuver. This controller ensures the convergence of attitude tracking errors in fixed time where the settling time is bounded by a constant independent of initial conditions. Moreover, this controller can eliminate the unwinding phenomenon.(2)
A new adaptive law is designed to estimate the upper bounds of the disturbances and the coupling effect in fixed time. Then, a new adaptive fixed-time controller without requiring prior knowledge of their boundaries is designed. By Lyapunov stability theory, rigorous fixed-time stability is analyzed and the expression of an accurate upper bound of convergent regions is provided.The remaining part of this paper is organized as follows. Section2 presents the description of a flexible spacecraft model and some key lemmas. Then, a robust fixed-time controller is designed in Section 3. The fixed-time stability proof is also given. In Section 4, a novel robust adaptive fixed-time attitude controller for flexible spacecraft is designed. The developed attitude controller makes the system states converge into a small neighborhood of the designed sliding mode in fixed time. To verify the effectiveness of the proposed control methods, simulation results are given in Section 5. In Section 6, we present conclusions.


## 2. Nonlinear Model of Spacecraft and Problem Formulation

### 2.1. Mathematical Model of a Flexible Spacecraft

The flexible spacecraft is characterized by a central rigid body with attached appendages. The mathematical model of the flexible spacecraft can be described in the body-fixed frame as [39](1)Jω˙+ω×Jω+δTη˙+δTη¨=ut+τdt,(2)η¨+Cη˙+Kη=-δω˙,(3)Q˙=12-qTq0I3+q×ω,where J∈R3×3 is the symmetric inertia matrix of the whole structure, ω∈R3 is the angular velocity of the spacecraft in the body frame, τd∈R3 denotes external disturbances, u∈R3 is the control torque acting on the hub, η∈Rn is the modal coordinate vector of the flexible appendages, n is the number of flexible modes considered, and δ∈R3×n is the coupling matrix between the flexible appendage and the hub. In (2), the damping matrix C and stiffness matrix K are given by (4)C=2ζ1Λ1⋯0⋮⋱⋮0⋯2ζnΛn,K=Λ12⋯0⋮⋱⋮0⋯Λn2,where ζi and Λi denote the damping ration and natural frequency of the ith order mode. The operator ·× denotes a 3×3 symmetric matrix such as(5)ω×=0-ω3ω2ω30-ω1-ω2ω10,for ω=ω1ω2ω3T. The vector Q=q0qTT represents the attitude quaternion of the spacecraft subject to the unity length constraint, that is, Q=1, where q0 and q=q1q2q3 denote the scalar and the vector components of the unit quaternion, respectively. I3 is the 3×3 identity matrix. In (2), we define an auxiliary variable ϑ=δω+η˙ and one has(6)ϑ˙=δω˙+η¨=-Cϑ+Cδω-Kη.Substituting (6) into (1) yields(7)J-δTδω˙=-ω×Jω+ut+ξt+τdt,where the total coupling effect term ξ(t) denotes(8)ξt=δTKCηϑ-δTCδω-ω×δTϑ-δω.LetQd=qd0qdTT with qd=qd1qd2qd3T be the unit quaternion representing the desired attitude and satisfying Qd=1. Let ωr∈R3 be the desired angular velocity. The quaternion error Qe=e0eTT with e=e1e2e3T and the angular velocity error ωe are defined as follows:(9)e=qd0q-qd×q-q0qd,e0=qTqd+q0qd0,ωe=ω-ωr.The unit quaternion Qe satisfies Qe=1.In fact, due to onboard payload motion, rotation of solar arrays, fuel consumption, and out-gassing during operation, the inertial matrixJ of spacecraft may be time-varying. Here, we assume that it consists of two parts; that is, J=Jn+ΔJ, where Jn and ΔJ represent the nominal value component and the parameter perturbation component of the inertial matrix J, respectively. Both of the nominal value component of Jn and the perturbation matrix ΔJ are symmetric since J is always a symmetric matrix.If the termsΔJω˙-ω×ΔJω are considered as the disturbance, then (1) becomes(10)Jn-δTδω˙e=-Jnω˙r-ω×Jnω+ut+ξt+dt,where d(t)=-ΔJω˙-ω×ΔJω+τd(t).Under the coordinate given in (9), (1) and (3) can be written as(11)J0ω˙e=-J0ω˙r-ω×Jnω+ut+ξt+dt,(12)Q˙e=12-eTe0I3+e×ωe,where J0=Jn-δTδ.Throughout the paper, it is assumed that the influence caused by the external disturbances and the flexible modes is bounded in the following sense.Assumption 1.
The total external disturbanced(t) in (11) is bounded and satisfies d≤θ0, where θ0>0 is an unknown constant.Assumption 2.
The coupling effect termξ(t) in (11) satisfies ξ(t)≤b1+b2ω2 with b1>0 and b2>0. In other words, ξ(t)≤θ1Φ1 for Φ1=1+ω2 and θ1=max⁡(b1,b2).


### 2.2. Definition and Lemma

Consider the following autonomous system:(13)x˙t=fxt,x0=0,f0=0,x∈Rn,where f:U0→Rn is continuous on an open neighborhood U0 of the origin x=0.Definition 3 (see [16]).
The equilibriumx=0 of system (13) is finite-time convergent if there exist an open neighborhood U⊂U0 of the origin and a function and the function T:U∖{0}→(0,∞) such that the solution of the system (13) defined as x(t,x0) with x0∈U∖{0}⊂Rn satisfies x(t,x0)∈U∖{0}⊂Rn for t∈[0,T(x0)] and limt→T(x0)x(t,x0)=0 with x(t,x0)=0 for t>T(x0). The zero solution of the system (13) is finite-time stable if it is Lyapunov stable and finite-time convergent in a U⊂U0 of the origin. If U0=U=Rn, then the zero solution is globally finite-time stable.Lemma 4 (see [16]).
For system (13), suppose that there exists a continuous positive definite function V(x):U→R such that (14)V˙x+βVxp≤0,x∈U0∖0,where p∈(0,1),β∈R+, and U0⊂U is an open neighborhood of the origin. Then, the origin is a finite-time stable equilibrium of system (13). Furthermore, T(x0) is the settling time that is defined as the time needed to reach V(x)≡0 and satisfies (15)Tx0≤1β1-pVx01-p,where V(x0) is the initial value of V(x).Definition 5 (see [36]).
The equilibriumx=0 of system (13) is said to be fixed-finite-time stable if it is globally finite-time stable and the settling-time function T(x0) is bounded, that is, a positive constant Tmax can be found in such a way that T(x0)≤Tmax, ∀x0∈Rn.Lemma 6 (see [22]).
For system (13), suppose there exists a positive definite and radially unbounded function V(x):Rn→R such that V(x)=0⇒x=0 and(16)V˙x≤-αVpx-βVxq,where α>0, β>0, p∈(0,1), and q>1. Then, the origin of system (13) is fixed-time stable and (17)Tx0≤1α1-p+1βq-1,∀x0∈Rn.Lemma 7 (see [40]).
Letxi∈R,i=1,2,…,n be real numbers and p∈(0,1); then the following inequality holds: (18)x1+⋯+xnp≤x1p+⋯+xnp.Lemma 8 (see [41]).
For any numbersxi≥0,i=1,2,…,n, and p>1, then (19)∑i=1nxip≥n1-p∑i=1nxip.In the following proposition, we give a new result of fixed-finite stability. This result is an extended version of fixed-finite stability presented in Lemma6 and will be used in controller designs in later section.Proposition 9.
Consider system (13). If there is a positive definite and radially unbounded function V(x):Rn→R such that V(x)=0⇒x=0 and(20)V˙x≤-γVx-αVpx-βVxq,where α>0, β>0, p∈(0,1), and q>1, then, the origin of system (13) is fixed-time stable (21)Tx0≤1βq-1+1β1-pln⁡γ+αα,∀x0∈Rn.Proof.
Owing to (20), we can obtain(22)V˙xt≤-γVx-βVpx,Vxt≤1,-βVqx,Vxt>1.Hence, for any x(t) such that V(x(0))>1, the last inequality ensures V(x(t0))≤1. This means V(x(0)) is reduced such that V(x(t))=1 at the time t0. Integrating both sides of the last inequality yields(23)V1-qt≤-1-qβt+V1-qx0,and consequently we can obtain t0 as(24)t0≤1-V1-qx0βq-1.As is known that V(x(0))>1, one has(25)t0≤1βq-1.Next, the first inequality ensures that V(x(t0))=1 will approach V(x(t1))=0. Now, t1 can be calculated from the first inequality. The first inequality can be rearranged as(26)V-pxtV˙xt≤-γV1-pxt-α,and consequently(27)dt≤-V-pxtγV1-pxt+αdVxt.Now, integrating both sides of the above inequality from t0 to t1 yields(28)t1-t0≤-∫Vxt00V-pxtγV1-pxt+αdVxt=-1γ1-pln⁡α-ln⁡γV1-pxt0+α=1γ1-pln⁡γV1-pxt0+αα.Using V(x(t0))=1, one obtains(29)t1≤t0+1γ1-pln⁡γ+ααort1≤1βq-1+1γ1-pln⁡γ+αα.This completes the proposition.


### 2.3. Problem Statement

In this paper,qd, ωd, and ω˙d are assumed to be bounded, and the quaternion and angular velocity measurements are always available. The main objective is to design a control law which forces the attitude and angular velocity errors to a small region around the origin in finite time in the sense of a fixed-time concept. This can be expressed as(30)limt→T⁡et,ωet=0,where T is the convergence time that can be estimated even if information of initial values of system states is unavailable.


## 2.1. Mathematical Model of a Flexible Spacecraft

The flexible spacecraft is characterized by a central rigid body with attached appendages. The mathematical model of the flexible spacecraft can be described in the body-fixed frame as [39](1)Jω˙+ω×Jω+δTη˙+δTη¨=ut+τdt,(2)η¨+Cη˙+Kη=-δω˙,(3)Q˙=12-qTq0I3+q×ω,where J∈R3×3 is the symmetric inertia matrix of the whole structure, ω∈R3 is the angular velocity of the spacecraft in the body frame, τd∈R3 denotes external disturbances, u∈R3 is the control torque acting on the hub, η∈Rn is the modal coordinate vector of the flexible appendages, n is the number of flexible modes considered, and δ∈R3×n is the coupling matrix between the flexible appendage and the hub. In (2), the damping matrix C and stiffness matrix K are given by (4)C=2ζ1Λ1⋯0⋮⋱⋮0⋯2ζnΛn,K=Λ12⋯0⋮⋱⋮0⋯Λn2,where ζi and Λi denote the damping ration and natural frequency of the ith order mode. The operator ·× denotes a 3×3 symmetric matrix such as(5)ω×=0-ω3ω2ω30-ω1-ω2ω10,for ω=ω1ω2ω3T. The vector Q=q0qTT represents the attitude quaternion of the spacecraft subject to the unity length constraint, that is, Q=1, where q0 and q=q1q2q3 denote the scalar and the vector components of the unit quaternion, respectively. I3 is the 3×3 identity matrix. In (2), we define an auxiliary variable ϑ=δω+η˙ and one has(6)ϑ˙=δω˙+η¨=-Cϑ+Cδω-Kη.Substituting (6) into (1) yields(7)J-δTδω˙=-ω×Jω+ut+ξt+τdt,where the total coupling effect term ξ(t) denotes(8)ξt=δTKCηϑ-δTCδω-ω×δTϑ-δω.LetQd=qd0qdTT with qd=qd1qd2qd3T be the unit quaternion representing the desired attitude and satisfying Qd=1. Let ωr∈R3 be the desired angular velocity. The quaternion error Qe=e0eTT with e=e1e2e3T and the angular velocity error ωe are defined as follows:(9)e=qd0q-qd×q-q0qd,e0=qTqd+q0qd0,ωe=ω-ωr.The unit quaternion Qe satisfies Qe=1.In fact, due to onboard payload motion, rotation of solar arrays, fuel consumption, and out-gassing during operation, the inertial matrixJ of spacecraft may be time-varying. Here, we assume that it consists of two parts; that is, J=Jn+ΔJ, where Jn and ΔJ represent the nominal value component and the parameter perturbation component of the inertial matrix J, respectively. Both of the nominal value component of Jn and the perturbation matrix ΔJ are symmetric since J is always a symmetric matrix.If the termsΔJω˙-ω×ΔJω are considered as the disturbance, then (1) becomes(10)Jn-δTδω˙e=-Jnω˙r-ω×Jnω+ut+ξt+dt,where d(t)=-ΔJω˙-ω×ΔJω+τd(t).Under the coordinate given in (9), (1) and (3) can be written as(11)J0ω˙e=-J0ω˙r-ω×Jnω+ut+ξt+dt,(12)Q˙e=12-eTe0I3+e×ωe,where J0=Jn-δTδ.Throughout the paper, it is assumed that the influence caused by the external disturbances and the flexible modes is bounded in the following sense.Assumption 1.
The total external disturbanced(t) in (11) is bounded and satisfies d≤θ0, where θ0>0 is an unknown constant.Assumption 2.
The coupling effect termξ(t) in (11) satisfies ξ(t)≤b1+b2ω2 with b1>0 and b2>0. In other words, ξ(t)≤θ1Φ1 for Φ1=1+ω2 and θ1=max⁡(b1,b2).


## 2.2. Definition and Lemma

Consider the following autonomous system:(13)x˙t=fxt,x0=0,f0=0,x∈Rn,where f:U0→Rn is continuous on an open neighborhood U0 of the origin x=0.Definition 3 (see [16]).
The equilibriumx=0 of system (13) is finite-time convergent if there exist an open neighborhood U⊂U0 of the origin and a function and the function T:U∖{0}→(0,∞) such that the solution of the system (13) defined as x(t,x0) with x0∈U∖{0}⊂Rn satisfies x(t,x0)∈U∖{0}⊂Rn for t∈[0,T(x0)] and limt→T(x0)x(t,x0)=0 with x(t,x0)=0 for t>T(x0). The zero solution of the system (13) is finite-time stable if it is Lyapunov stable and finite-time convergent in a U⊂U0 of the origin. If U0=U=Rn, then the zero solution is globally finite-time stable.Lemma 4 (see [16]).
For system (13), suppose that there exists a continuous positive definite function V(x):U→R such that (14)V˙x+βVxp≤0,x∈U0∖0,where p∈(0,1),β∈R+, and U0⊂U is an open neighborhood of the origin. Then, the origin is a finite-time stable equilibrium of system (13). Furthermore, T(x0) is the settling time that is defined as the time needed to reach V(x)≡0 and satisfies (15)Tx0≤1β1-pVx01-p,where V(x0) is the initial value of V(x).Definition 5 (see [36]).
The equilibriumx=0 of system (13) is said to be fixed-finite-time stable if it is globally finite-time stable and the settling-time function T(x0) is bounded, that is, a positive constant Tmax can be found in such a way that T(x0)≤Tmax, ∀x0∈Rn.Lemma 6 (see [22]).
For system (13), suppose there exists a positive definite and radially unbounded function V(x):Rn→R such that V(x)=0⇒x=0 and(16)V˙x≤-αVpx-βVxq,where α>0, β>0, p∈(0,1), and q>1. Then, the origin of system (13) is fixed-time stable and (17)Tx0≤1α1-p+1βq-1,∀x0∈Rn.Lemma 7 (see [40]).
Letxi∈R,i=1,2,…,n be real numbers and p∈(0,1); then the following inequality holds: (18)x1+⋯+xnp≤x1p+⋯+xnp.Lemma 8 (see [41]).
For any numbersxi≥0,i=1,2,…,n, and p>1, then (19)∑i=1nxip≥n1-p∑i=1nxip.In the following proposition, we give a new result of fixed-finite stability. This result is an extended version of fixed-finite stability presented in Lemma6 and will be used in controller designs in later section.Proposition 9.
Consider system (13). If there is a positive definite and radially unbounded function V(x):Rn→R such that V(x)=0⇒x=0 and(20)V˙x≤-γVx-αVpx-βVxq,where α>0, β>0, p∈(0,1), and q>1, then, the origin of system (13) is fixed-time stable (21)Tx0≤1βq-1+1β1-pln⁡γ+αα,∀x0∈Rn.Proof.
Owing to (20), we can obtain(22)V˙xt≤-γVx-βVpx,Vxt≤1,-βVqx,Vxt>1.Hence, for any x(t) such that V(x(0))>1, the last inequality ensures V(x(t0))≤1. This means V(x(0)) is reduced such that V(x(t))=1 at the time t0. Integrating both sides of the last inequality yields(23)V1-qt≤-1-qβt+V1-qx0,and consequently we can obtain t0 as(24)t0≤1-V1-qx0βq-1.As is known that V(x(0))>1, one has(25)t0≤1βq-1.Next, the first inequality ensures that V(x(t0))=1 will approach V(x(t1))=0. Now, t1 can be calculated from the first inequality. The first inequality can be rearranged as(26)V-pxtV˙xt≤-γV1-pxt-α,and consequently(27)dt≤-V-pxtγV1-pxt+αdVxt.Now, integrating both sides of the above inequality from t0 to t1 yields(28)t1-t0≤-∫Vxt00V-pxtγV1-pxt+αdVxt=-1γ1-pln⁡α-ln⁡γV1-pxt0+α=1γ1-pln⁡γV1-pxt0+αα.Using V(x(t0))=1, one obtains(29)t1≤t0+1γ1-pln⁡γ+ααort1≤1βq-1+1γ1-pln⁡γ+αα.This completes the proposition.


## 2.3. Problem Statement

In this paper,qd, ωd, and ω˙d are assumed to be bounded, and the quaternion and angular velocity measurements are always available. The main objective is to design a control law which forces the attitude and angular velocity errors to a small region around the origin in finite time in the sense of a fixed-time concept. This can be expressed as(30)limt→T⁡et,ωet=0,where T is the convergence time that can be estimated even if information of initial values of system states is unavailable.


## 3. Fixed-Time Attitude Controller

In this section, a new fixed-time sliding surface is introduced. The fixed-time convergence of this sliding surface is analyzed. Then, a new fixed-time based sliding mode attitude controller is developed to achieve rapid maneuver and high-precise attitude control performance for a flexible spacecraft in presence of external disturbances.Now, we define a new variablez(ωe,e0,e) as(31)zωe,qe0,qe=ωe+sign⁡e0Ke,where K=diag(k1,k2,k3) is a positive diagonal matrix.Inspired by [22], we developed a nonsingular terminal sliding surface for the spacecraft attitude system described by (11) and (12) as follows:(32)s=z+C1∫0tsign⁡zβdτ+C2∫0tsign⁡zγdτ,where C1=diag(c11,c12,c13) and C2=diag(c21,c22,c23) are constant matrices with c1i,c2i>0(i=1,2,3), 0<β<1, and γ>1. For any vector x=x1x2x3T and constant p>0, the function of sign⁡(x)p is defined as (33)sign⁡xp=x1psign⁡x1x2psign⁡x2x3psign⁡x3.For the sake simplicity, the sliding surface can be rewritten in the scalar form (i=1,2,3) as(34)si=zi+c1i∫0tziβsign⁡zidτ+c2i∫0tziγsign⁡zidτ.When the sliding mode is established under a suitably designed controller, one can obtain(35)si=zi+c1i∫0tziβsign⁡zidτ+c2i∫0tziγsign⁡zidτ=0,s˙i=z˙i+c1iziβsign⁡zi+c2iziγsign⁡zi=0.Therefore, the dynamics of the associate sliding mode can be obtained as(36)z˙i=-c1iziβsign⁡zi-c2iziγsign⁡zi,i=1,2,3.Theorem 10.
The zero solutionzi=0(i=1,2,3), of the sliding mode dynamics (36), is globally fixed-time stable and the settling time is given by(37)T≤1c1i1-β+1c2iγ-1.Proof.
The Lyapunov function candidate is chosen as(38)V1=zi.Its first time derivative is(39)V˙1=sign⁡ziz˙i=sign⁡zi-c1iziβsign⁡zi-c2iziγsignzi=-c1iziβ-c2iziγ=-c1iV1β-c2iV1γ.By Lemma 6, it can be concluded that zi=0(i=1,2,3) are fixed-time stabilized. This completes the proof.Once the fixed-time-based sliding surface is designed, it is followed by designing a control lawu∈R3 as follows:(40)u=ω×Jnω+J0ω˙r+J0Ksign⁡e012e0I3+e×ωe-J0C1sign⁡zβ-J0C2sign⁡zγ-μ1s-μ2sign⁡s-μ3sign⁡sρ,where ρ>1, μ1=diag(μ11,μ12,μ13), μ2=diag(μ21,μ22,μ23), and μ3=diag(μ31,μ32,μ33) are diagonal matrices with μ1i,μ2i,μ3i>0(i=1,2,3). For any vector x=x1x2x3T, the function of sign(x) is defined as (41)sign⁡x=sign⁡x1sign⁡x2sign⁡x3,where γ1∈(0,1) and k1 is a positive number; the larger the value of k1, the faster the reaching speed. The term μ2sign(s) is employed to cope with external disturbances and coupling effect.The fixed-time stability of the closed-loop system under the action of the controller (40) is analyzed in the following theorem.Theorem 11.
Consider the system described by (11) and (12) and let Assumptions 1 and 2 hold. If the control law is designed as (40), then the sliding surface s=0 is achieved in a fixed time.Proof.
According to (32), we can obtain the time derivative of J0s along the trajectory of the system consisting of (11) and (12):(42)J0s˙=J0ω˙e+J0Ksign⁡e0+J0C1sign⁡zβ+J0C2sign⁡zγ.
Then, substituting (11) into (42) yields(43)J0s˙=J0z˙+J0C1sign⁡zβ+J0C2sign⁡zγ=-ω×Jnω-J0ω˙r+u+d+ξ+J0Ksign⁡e012e0I3+e×ωe+J0C1sign⁡zβ+J0C2sign⁡zγ.Substituting (40) into (43), we obtain(44)J0s˙=-μ1s-μ2sign⁡s-μ3sign⁡sρ+d+ξ.
Consider the candidate Lyapunov function:(45)V2=12sTJ0s,which satisfies(46)λminJ0s2≤2V2≤λmaxJ0s2,where λmin(J0) and λmax(J0) denote the minimum and maximum singular values of the matrix J0.
According to (44), the derivative of V2 along the trajectory of the system described by (11) and (12) is(47)V˙2=sTJ0s˙=sT-μ1s-μ2sign⁡s-μ3sign⁡sρ+d=-λminμ1∑i=13si-λminμ2∑i=13si2-λminμ3∑i=13si1+ρ+sd≤-λminμ1-θ0s-λminμ2s2-31-ρ/2λminμ3s1+ρ.Letting ε=λmin(μ1)-θ0>0, one obtains(48)V˙2≤-2ελmaxJV21/2-2λminμ2λmaxJV2-31-ρ/2λminμ32λmaxJ1+ρ/2V21+ρ/2=-κ1V21/2-κ2V2-κ3V21+ρ/2,where κ1=2ε/λmax(J0), κ2=2λmin(μ2)/λmax(J0), and κ3=λmin(μ3)(31-ρ/2)2/λmax(J0)1+ρ/2. Hence, the sliding variable s(t) reaches zero in a fixed time that can be calculated by using Proposition 9.Remark 12.
To avoid chattering in practical applications, owing to the imperfect implementation of the sign function, a saturation function is a simple choice to replace the sign function in the control law (40). Thus, in (40), for s=s1s2s3T, sign⁡(s) is replaced by sats=sat(s1)sat(s2)sat(s3)T. The function sat(·) is a saturation function (49)satx=1,x>ϕsx,x<ϕs-1,x<-ϕs,where ϕs is a small positive constant.


## 4. Adaptive Fixed-Time Attitude Control

In the previous section, the proposed fixed-time attitude controller (40) can force the states of the system described by (11) and (12) to the origin. However, it is required to know the upper bounds of external disturbances and coupling effect. In practice, it is difficult to know these upper bounds, which implies that the values of θ0 and θ1 may not be available. Without prior knowledge of the upper bounds of external disturbances and coupling effect, the gain of sign function needs to be chosen sufficiently large to satisfy the existence condition of sliding mode, which may reduce the steady-state performance of the attitude control system due to undesirable chattering in control torque and require large control input magnitudes.In order to achieve more efficiently performance, an adaptive scheme is constructed to estimate the upper bound information. Then, an adaptive law is designed to guarantee fixed-time reachability of given desired attitude motion. Here, we design an adaptive fixed-time-based control law as(50)u=ω×Jnω+J0ω˙r+J0Ksign⁡e012e0I3+e×ωe-J0C1sign⁡zβ-J0C2sign⁡zγ-μ1s-μ2sign⁡s-μ2sign⁡sρ+ua,where ua is a continuous adaptive control term designed as follows:(51)ua=-θ^02ε02+θ^1Φ2ε12s,where(52)θ^˙0=β0s22ε02-k0θ^0,θ^˙1=β1Φs22ε12-k1θ^1.Theorem 13.
Consider the motion equations (11) and (12) that satisfy Assumptions 1 and 2, but with d¯ and θ1 unknown. If the control law is designed as (50), then the attitude error e and the angular velocity error ωe can be stabilized in fixed time into a neighborhood of the sliding mode s(t)=0.Proof.
Consider the following candidate Lyapunov function:(53)V3=12sTJ0s+1β0θ~02+1β1θ~12,where θ~0 and θ~1 are the parameter estimation errors defined as θ~0≔θ0-θ^0 and θ~1≔θ1-θ^1, respectively. The derivative of V3 along the trajectory of system (11) and (12) is(54)V˙3=sTJ0s˙+1β0θ~0θ^˙0+1β1θ~1θ^˙1=sT-μ1s-μ2sign⁡s-μ3sin⁡zρ-θ^02ε02s-θ^1Φ2ε12s+dt+ξt-s22ε02-k0θ^0θ0-θ^0-Φs22ε12-k1θ^1θ1-θ^1≤-sTμ1s-sTμ2sign⁡s-sTμ3sign⁡sρ+sd+sξt-θ02ε02s2-θ12ε12s2Φ+k0θ^0θ~0+k1θ^1θ~1≤-λminμ1s2+sd+θ1sΦ-θ02ε02s2-θ12ε12s2Φ+k0θ^0θ~0+k1θ^1θ~1.Note that(55)k0θ^0θ~0=k0θ0θ~0-k0θ~02≤k0-θ~02+12θ~02+12θ02=-k02θ~02+k02θ02,and similarly(56)k1θ^1θ~1≤-k12θ~12+k12θ12.Moreover, letting θ0=(d¯)2 and applying Young’s inequality, one obtains(57)sd≤θ02ε02s2+ε022,θ1sΦ≤θ12ε12s2Φ+ε122.Now V˙3 becomes(58)V˙3≤-λminμ1s2+ε022+ε122-k02θ~02-k12θ~12+ε02θ02+ε12θ12≤-2λminμ1λmaxJ012sTJ0s-k0β012β0θ~02-k1β112β1θ~12+Υ,where Υ=ε02/2+ε12/2+k0/2θ02+k1/2θ12. As a result, one has(59)V˙3≤-αV3+Υ,where α=min(2λminμ1/λmaxJ0,k0β0,k1β1).
Using the boundedness theorem,s, θ~0, θ~1 are uniformly ultimate bounded (UUB). Therefore, one can assume that θ~0≤θ¯0 and θ~1≤θ¯1. To prove the fixed-time stability, we define a new Lyapunov function(60)V4=12sTJ0s.Differentiating V4 with respect to time, one obtains(61)V˙4≤-sTμ1s-sTμ2sign⁡s-sTμ3sign⁡sρ-θ^02ε02s2-θ^1Φ2ε12s2+sd+sξ≤-λminμ1s2-λminμ2s-λminμ3∑i=13siρ-θ^02ε02s2-θ^12ε12Φs2+θ02ε02s2+θ12ε12Φs2+ε022+ε122≤-λminμ1s2-λminμ2s-λminμ3∑i=13siρ-θ~02ε02s2-θ~12ε12Φs2+ε022+ε122≤-λminμ1-θ¯02ε02-θ¯1Φ2ε12s2-λminμ2s-31-ρ/2λminμ3sρ+1+ϖ,where ϖ=ε02/2+ε12/2.
Ifλmin(μ1) is chosen such that λmin(μ1)=θ¯0/2ε02+θ¯1Φ/2ε12+ϵ where ϵ is a positive constant, then V˙4 can be written in the following two forms:(62)V˙4≤-ϵs2-λminμ2-ϖss-31-ρ/2λminμ3sρ+1,orV˙4≤-ϵs2-λminμ2s-31-ρ/2λminμ3-ϖsρ+1sρ+1.With the process similar to Theorem 11, the trajectory of the system is fixed-time stable. When λmin(μ2)>ϖ/s or 31-ρ/2λmin(μ3)>ϖ/sρ+1, the trajectories of the solution of the system converge to the residual set given by(63)D=limt→Tf⁡st∣s≤min⁡ϖλminμ2,ϖ31-ρ/2λminμ31/1+ρ.Hence, the proof of Theorem 13 is completed.Remark 14.
In particular,θ^0 and θ^1 are the estimated values of θ0 and θ1, in Assumptions 1 and 2, respectively. In [13], the upper bound of d(t) was estimated by traditional adaptive update law. The sign function was combined with the adaptive update law and it can only ensure uniformly ultimate boundedness of θ^0 and θ^1. In this paper, the design parameters k0 and k1 are used to determine the convergence region, and these parameters can be selected to be sufficiently small. However, these parameters also determine the convergence speed of the estimated bounds θ^0 and θ^1. If values k0 and k1 are chosen too small, then a slow convergence rate is obtained. Therefore, a compromise is made between the convergence region and the convergence speed. Moreover, the values of k0 and k1 cannot be selected too small [42]. Based on the above adaptive law, the trajectories of θ^0 and θ^1 are reduced significantly and then converge to a value near zero.


## 5. Simulations

Numerical simulations on the flexible spacecraft have been conducted to verify the performance of the proposed adaptive fixed-time control law (50) and PD-based sliding mode control (PD-SMC) method in [5]. For this, the same model parameters used in [39] are applied in this study. The inertia matrix and the flexible coupling matrix are as follows: (64)J=35034327010410190kg·m2,and coupling matrices (65)δ=6.456371.278142.15629-1.256190.91756-1.672641.116782.48901-0.836741.23637-2.6581-1.12503kg1/2·m/s2,respectively. The natural frequencies and damping are provided by(66)Λ1=0.7681,Λ2=1.1038,Λ3=1.8733,Λ4=2.5496rad/secζ1=0.0056,ζ2=0.0086,ζ3=0.013,ζ1=0.025.In this numerical simulation, we assume that the desired angular velocity is given by(67)ωrt=0.05sinπt100sin2πt100sin3πt100rad/s.For the initial conditions of the unit quaternion and the target unit quaternion, we set Q0=-0.17365-0.26320.7896-0.5264T and Qd0=0001T, respectively. The initial value of the angular velocity is supposed to be ω(0)=[000]T rad/s. We bounded the magnitude of the control torque as |ui|≤3.0N-m,i=1,2,3. The external disturbance torque that includes constant disturbance and periodic disturbance of two different frequencies is described as follows: (68)τdt=3cos⁡t-10+4sin⁡0.3t3cos⁡0.5t+15-1.5sin⁡0.2t3sin⁡t+10+8sin⁡0.4t×10-2Nm.The parameters for the controller (50) are chosen as K=0.2I3,  C1=I3,  C1=0.6I3,  β=7/9,  γ=1.1,  μ1=5I3,  μ2=3I3,  μ3=1.2I3,  ρ=5/3,  η0=10,  η1=10,  ε0=0.01, and  ε1=0.01. For the PD-SMC method in [5], simulations are performed with the parameters given as kP=3, kd=8, ke=3.5, and Γ=20.The performance of the PD-SMC method in [5] is shown in Figures 1–7. From Figure 1 one can see that the time responses of quaternion errors converge to zero after 60 seconds. Figure 2 depicts the unwinding phenomenon encountered in unit quaternion based attitude systems. Since e0(0)=-0.17365, the scalar quaternion error should finally converge to the equilibrium point e0(t)=-1 instead of the equilibrium point e0(t)=1. As shown in Figure 3, the time responses of angular velocity tracking errors are stabilized to zero but they are not smooth due to coupling effect of flexible modes. Also, Figure 4 shows that the time responses of sliding variables si  (i=1,2,3) converge to zero with low accuracy. As shown in Figure 5, the time responses of control torques have much variation during the first 60 seconds. Figures 6 and 7 illustrate the time responses of modal displacements which are ultimately bounded by small values near zero. For the tracking accuracy, the Euclidean norm of steady errors of e(t), ωe(t), and s(t) can be listed as qe≤5.7×10-3 and ωe≤7.7×10-4 and s≤1.77×10-4 with sampling time h=0.005.Figure 1
Quaternion errors under PD-SMC.Figure 2
Scalar component of quaternion errors under PD-SMC.Figure 3
Angular velocity errors under PD-SMC.Figure 4
Sliding surface under PD-SMC.Figure 5
First and second modes under PD-SMC.Figure 6
First and second modes under PD-SMC.Figure 7
Third and fourth modes under PD-SMC.Figures8–14 show the performance of the proposed adaptive fixed-time control law (50). As can be seen from Figure 8, time responses of quaternion errors can be stabilized to zero. Figure 9 shows that the controller (50) can prevent the unwinding phenomenon. When e0(0)=-0.17365, the scalar quaternion errors can properly converge to the equilibrium point e0(t)=-1. As shown in Figure 10, the time responses of angular velocity errors are smoother when compared with those obtained by the PD-SMC method in [5]. From Figure 11 one can see that the sliding surface s=0 is achieved with higher precision than those of the PD-SMC method. Figure 12 shows the time responses of control torques which have small variation. As shown in Figures 13 and 14 the flexible mode vibration is reduced greatly at the beginning and then decreased to a small value. In fact these modal coordinates converge to a small amplitude of the flexible mode vibration (less than 0.05) after 50 seconds. Practically, it is sufficiently good for some spacecraft missions. However, sufficiently large control torques can be applied to keep the flexible mode vibration at an acceptable amplitude of the flexible mode vibration in a small amount of time. Regarding the tracking accuracy, the Euclidean norm of steady errors of e(t), ωe(t), and s(t) can be listed as qe≤4.6×10-5 and ωe≤2.06×10-4 and s≤1.87×10-4 with sampling time h=0.005.Figure 8
Quaternion errors under controller (50).Figure 9
Scalar component of quaternion errors under controller (50).Figure 10
Angular velocity errors under controller (50).Figure 11
Sliding surface under controller (50).Figure 12
Control inputs under controller (50).Figure 13
First and second modes under controller (50).Figure 14
Third and fourth modes under controller (50).From these simulation results, the proposed adaptive fixed-time control law (50) provides offers smoother time responses of attitude and angular velocity errors and higher accuracy of attitude and angular velocity errors than the PD-SMC method in [5]. Furthermore, the controller (50) effectively prevents the unwinding phenomenon. In view of the simulation, the proposed control law seems to be a more effective control approach for general case of attitude tracking maneuvers of a flexible spacecraft.


## 6. Conclusion

In this paper, an adaptive fixed-time control law has been developed for the attitude tracking control problem of a flexible spacecraft. The first objective of this research has been aimed at providing a new result in fixed-time stability. Then, a fixed-time attitude tracking controller is proposed in the presence of external disturbances and coupling effect of flexible modes. Later, the controller is further enhanced by an adaptive law to relax the knowledge of norm upper bounds of external disturbances and coupling effect of flexible modes. The proposed adaptive fixed-time controller offers a fast transient process and high attitude tracking accuracy. This controller can avoid the unwinding phenomenon. Using Lyapunov stability theory, we have proved that the error dynamics converge to a desired region containing the origin in finite time in the sense of fixed-time convergence. Numerical simulations on attitude control of a flexible spacecraft model are also presented to demonstrate the performance of the proposed control methods.



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*Source: 5018323-2017-08-07.xml*