# Finite-Time Attitude Tracking Control for Hypersonic Flight Vehicles with Actuator Saturation

Authors: Teng Cao; Huajun Gong; Huiyunuo Xiao; Yixuan Xue
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This paper investigates finite-time attitude tracking control strategies for hypersonic flight vehicles (HFVs) with parameter uncertainties, external disturbances, and actuator saturations by applying sliding model control, adaptive mechanism, and nonlinear disturbance observer techniques. A nonlinear dynamic model of HFV attitude system in reentry flight phase is established. Then, a basic attitude control method of the HFV system is designed based on a terminal sliding mode control (TSMC) scheme to accommodate the system-lumped disturbance torques and guarantee the finite-time stability. To relax the prior knowledge of bounded lumped disturbance of the TSMC-based HFV attitude system, an adaptive TSMC (ATSMC) scheme is proposed. In order to relax the limit of compound uncertainties and attenuate chattering phenomenon of the TSMC-based HFV attitude system, a nonlinear disturbance observer-based TSMC (DO-TSMC) scheme is presented, which enhances the disturbance attenuation and robust tracking performance. Finally, simulation results of a generic X-33 nonlinear model exhibit the effectiveness of the proposed TSMC, ATSMC, and DO-TSMC schemes.


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## Body

## 1. Introduction

Research on hypersonic flight vehicles (HFVs) is widely concerned in recent years due to its high-speed transportation, ability to accomplish modern space missions, and broad application in military and civil fields [1–4]. However, there exists many challenges to design control law for the HFV attitude system containing complex coupling terms, unknown disturbances, and uncertainty of external environment [5, 6]. Under the influence of inaccurate modeling and external disturbances, the control system may not be able to complete the scheduled task, or even out of control. Therefore, it is essential for us to propose efficient approaches to design the attitude system.In recent years, there have been a lot of research results on hypersonic vehicle controller design with external disturbances and uncertainties. Sliding mode control (SMC) [7, 8], backstepping control [9], twisting control [10], adaptive control [11], coupling effect-triggered control (CETC) [12], and compound control methods of the ways mentioned above have been largely used to enhance the attitude control performance. In [13], a tracking control scheme with quantization mechanism, which uses an interval type-2 fuzzy neural network (IT2FNN) is proposed for HFVs. A coordinate-free, finite-time, attitude control is employed for thrust-vectoring spacecraft to guarantee the required thrust vector exactly at the predefined time [14]. In [15], under the condition of the inaccuracy of measured flight path angle, external disturbances, and actuator saturation, a backstepping-based control is developed for the tracking control of HFVs. In [16], the predefined-time adaptive fuzzy tracking control problem of HFVs is solved by a novel fuzzy adaptive control strategy based on fuzzy approximation and backstepping control techniques. To cope with tracking performance with uncertainties of HFVs, the article [17] explores a new adaptive fuzzy nonsmooth backstepping output-feedback control scheme.On account of its fast global convergence, simple algorithm, high robustness against external perturbation, and system uncertainties, SMC has been extensively applied to compensate for the lumped disturbances. In order to improve the robust performance and obtain finite-time convergence, the terminal sliding mode (TSM) controller based on the backstepping frame is designed [18]. However, by reason of the difficulty to obtain the upper bound of uncertainty or disturbance in practice, compound control methods may lead to large chattering phenomenon and energy loss. In [19], a method combined with TSMC and adaptive techniques is raised to overcome actuator faults, which ensures the system stable in the fixed time even under the condition of actuator faults and model uncertainties. An adaptation strategy is employed to estimate unknown information. In [20], the synthetic neural learning combined with the nonsingular second-order terminal sliding mode is raised to deal with model uncertainties for hypersonic reentry vehicle (HRV). In this paper, the influence value of system fault is obtained by RBF neural network mechanism. However, the disturbance in the system is not estimated. The disadvantage of this controller design method is that it needs to select a larger gain to ensure the disturbance and fault estimation error in the system, which will cause energy waste. In [21], a singular free fast terminal sliding mode controller is proposed for velocity subsystem of HFVs with multisource uncertainty and actuator fault. In order to attenuate chattering and compensate for the disturbances, the article [22] presents a novel fixed-time sliding mode disturbance observer (SMDO). In [23], a novel fixed-time convergent nonsmooth backstepping control scheme for air-breathing hypersonic vehicles through augmented sliding mode observers (ASMOs) is raised to solve uncertainty and measurement noise. It is worth noting that although references [21–23] use the scheme of disturbance observer to compensate the lumped disturbance value, the design of disturbance observer has certain limitations. For example, the disturbance should be differentiable and some observers require the disturbance to be constant or slowly varying. In reality, the disturbance or fault may not satisfy these assumptions. In [24], a novel learning observer-based control strategy is proposed for a faulty rigid spacecraft attitude system. The observer can estimate actuator fault that are constant, periodic, or aperiodically time-varying. In addition, because the actuator is constrained by its own physical conditions, we need to consider the case of actuator input saturation in the controller design process. Input saturation will seriously affect the performance of the control system, leading to system instability [25, 26]. In [27], the difference between the required control input and the actual control input is expressed as a known continuous function multiplied by an unknown constant vector. The adaptive technique is used to estimate the unknown vector, and then, a backstepping controller is designed to compensate the error value of control saturation. In order to further solve the problem of input saturation, reference [28] proposed a fast adaptive terminal sliding mode controller with antisaturation by introducing hyperbolic tangent function and auxiliary system, which can not only meet the physical constraints of the actuator but also ensure that the sliding mode manifold is finite-time stable. In [29], an adaptive fixed-time antisaturation control (AFAC) algorithm is proposed for FHV with actuator constraints. The adaptive law in controller is updated according to the deviation value of the control signal to improve the ability of the controller to suppress actuator saturation. However, the introduction of adaptive update increases the complexity of the controller. To the best of our knowledge, the results about the integrated attitude tracking control design for HFV system with time-varying disturbance and actuator saturation are limited, which remain challenging and motivate us to do this study.In this paper, we focus on the attitude control problem of HFVs in reentry phase, where the model uncertainty, external disturbance, and actuator saturation are taken into account. In order to achieve the finite-time stability, the terminal sliding mode manifold is introduced. Assuming that there exists a prior knowledge of the bounded external disturbances and inertia uncertainties, a basic TSMC-based attitude control scheme is proposed. Further, an adaptive mechanism and a nonlinear disturbance observer are, respectively, designed for the TSMC-based attitude control system, which ensure the finite-time convergence in both reaching phase and sliding phase. Comparing with the results in the literature, the main contributions of this paper are as follows:(1)
A basic TSMC scheme is proposed for the HFV attitude system in the presence of model uncertainties, external disturbance, and actuator saturation. Compared with the conventional control methods, such as backstepping control [30] and integrated SMC [31, 32], the basic TSMC method can ensure the closed-loop attitude system to be asymptotically stable and track the reference signals in a finite time. Inspired by reference [32, 33], the TSMC control strategies with the low-pass filter are proposed to attenuate the chattering phenomenon caused by switching function. The discontinuous signal can be smoothed by the low-pass filter so the system can be stable with no chattering in the finite time. The stability of closed-loop attitude control system is analyzed using the Lyapunov method, and moreover, the system can be guaranteed to be finite-time stable(2)
To overcome the requirement of a prior knowledge of bounded lumped disturbance of the basic TSMC-based HFV attitude system, an adaptive law is introduced to adapt the switching gains, yielding an adaptive TSMC (ATSMC) scheme which is proposed. Compared with the conventional methods [32, 33], a suitable switching gain is selected according to the adaptive mechanism, which can reduce the large chattering phenomenon caused by the large gain(3)
To solve the limits of lumped disturbance and attenuate chattering phenomenon of the basic TSMC-based HFV attitude system, a nonlinear disturbance observer-based TSMC (DO-TSMC) is presented, which enhances the disturbance attenuation and robust performance. Compared with the existing adaptive SMC [31] and observer-based methods [34], the proposed DO-TSMC scheme relaxes the limits of uncertainties and disturbance and achieves higher precision, less chattering, faster finite-time convergence, and no need for prior knowledge of uncertainties. The class of disturbance considered in this paper can be much larger than the existing disturbance observer method [34]This paper is organized as follows. In Section2, the attitude control problem of the HFVs is formulated. In Section 3, the TSMC, ATSMC, and DO-TSMC-based attitude control systems are designed, respectively. Simulation studies are given in Section 4 to demonstrate the effectiveness of the proposed schemes, followed by the conclusion of this study in Section 5.


## 2. Attitude Model and Control System Framework of HFVs

In this section, a dynamic model of HFVs with parameter uncertainty, external disturbance, and actuator saturation is given, and the attitude control system framework is designed.


### 2.1. HFV Attitude Dynamic Model

#### 2.1.1. Generic Attitude Dynamic Model

The attitude dynamic equations of HFVs are given by(1)p=IzzMxIxxIzz−Ixz2+IxzMzIxxIzz−Ixz2+Ixx−Iyy+IzzIxzIxxIzz−Ixz2pq+Iyy−IzzIzz−Ixz2IxxIzz−Ixz2qr,̇q=MyIyy+IxzIyyr2−p2+Izz−IxxIyypr,ṙ=IxzMxIxxIzz−Ixz2+IxxMzIxxIzz−Ixz2+Ixx−IyyIxx+Ixz2IxxIzz−Ixz2pq+Iyy−Ixx−IzzIxzIxxIzz−Ixz2qr,and the kinematic equations of HFVs are given by
(2)α̇=−pcosαtanβ+q−rsinαtanβ−cosϕcosβγ̇−ξ̇cosχ−τ̇+ΩEcosξcosχ+sinϕcosβχ̇cosγ−ξ̇sinχsinγ+τ̇+ΩEcosξcosχsinγ−sinξcosγ,β̇=psinα−rcosα+sinϕγ̇−ξ̇cosχ+τ̇+ΩEcosξsinχ+cosϕχ̇cosγ−ξ̇sinχsinγ−τ̇+ΩEcosξcosχsinγ−sinξcosγ,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ+α̇sinβ−χ̇sinγ−ξ̇sinχcosγ+τ̇+ΩEcosξcosχcosγ+sinξsinγ,where ϕ, α, and β are bank, attack, and sideslip angles; p, q, and r are roll, pitch, and yaw angular rates; χ, ξ, and τ are heading, latitude, and longitude angles; ΩE is the Earth’s angular rate; and the definitions of other variables are given in [31].


#### 2.1.2. Attitude Dynamic Model in Reentry Phase

In the reentry flight phase, because the rotational motion of the HFVs in reentry phase is much faster than the rotational motion of the Earth, the angular velocity of the EarthΩE can be neglected, and the translational motion can hardly affect the rotational motion. The derivatives of both position and direction of velocity and the Earth’s angular velocity are negligible with respect to the rotational motion. Therefore, Equation (2) can be simplified as
(3)α=−pcosαtanβ+q−rsinαtanβ,β=psinα−rcosα̇,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ.


#### 2.1.3. Uncertain Model with Disturbance

The reentry attitude dynamics of HFVs with parameter uncertainty and external disturbance can be described as(4)I+ΔIω̇=−ω×I+ΔIω+M+d,whereω=p,q,rT is the angular rate vector, I∈R3×3 is the inertia matrix, and ΔI∈R3×3 is an uncertain part of the inertia matrix, which is caused by the fuel consumption and variations of particular payloads. d=d1,d2,d3T is the external disturbance vector. M=Mx,My,MzT is the control torque vector, which is calculated by
(5)M=Dtu,where Dt∈R3×m, m is the number of the control surfaces, and u=u1,u2,⋯,umT is the vector of aerodynamic surface deflections. Then, the attitude dynamic model can be rewritten as
(6)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Du+f1,where Ω=ϕ,α,βT is attitude angle vector, f1=I−1d−ΔIω̇−ω×ΔIω, and the matrices R, I, and ω× are as
(7)R=−cosαcosβ−sinβ−sinαcosβ−cosαtanβ1−sinαtanβsinα0−cosα,I=Ixx−Ixy−Ixz−IxyIyy−Iyz−Ixz−IyzIzz,ω×=0−rqr0−p−qp0.


#### 2.1.4. Uncertain Model with Disturbance and Actuator Saturation

Since there are input constraints on the control surfaces of HFVs, thus the attitude dynamic model (6) is rewritten as
(8)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Dsatu+f1,where satui, i=1,2,⋯,m is defined as
(9)satui=u¯i,ui>u¯i,ui,u¯i≤ui≤u¯i,u¯i,ui<u¯i.Further, define an auxiliary variableδ=δ1,δ2,⋯,δmT as
(10)δi=u¯i−ui,ui>u¯i,0,u¯i≤ui≤u¯i,u¯i−ui,ui<u¯i,then (9) and (10) yield to
(11)satu=δ+u.Substituting (11) into (8), the HFV attitude dynamic model with uncertainties, disturbances, and actuator input saturations can be expressed by
(12)Ω̇=Rω,ω=̇−I−1ω×Iω+I−1Du+f,where f=I−1d−ΔIω̇−ω×ΔIω+I−1Dδ and it can be considered as a lumped disturbance for HFV attitude dynamics.


### 2.2. Attitude Control Problem and Control System Framework

#### 2.2.1. Control Objective

The control objective is to solve the attitude control problem of HFV system (12) with the parameter uncertainties, external disturbances, and control input constraints. An advanced control method is necessary to make the attitude system output Ω track a reference input Ωr in the finite time.


#### 2.2.2. Control Method Motivation

As a robust nonlinear control method, the SMC theory is applied in this control problem. Therefore, a basic TSMC scheme is designed, an ATSMC scheme is designed in which an adaptive law is used to estimate the upper bounds of the lumped disturbance, and finally, a DO-TSMC is designed in which a nonlinear disturbance observer is used to online estimate the value of lumped disturbances.


#### 2.2.3. Control System Framework

A block diagram of the HFV attitude control system is designed as shown in Figure1. It consists of two parts; the inner loop is the adaptive mechanism and disturbance estimation observer, while the outer loop is the basic TSMC-based attitude controller.Figure 1
Block diagram of the HFVs attitude control system.


## 2.1. HFV Attitude Dynamic Model

### 2.1.1. Generic Attitude Dynamic Model

The attitude dynamic equations of HFVs are given by(1)p=IzzMxIxxIzz−Ixz2+IxzMzIxxIzz−Ixz2+Ixx−Iyy+IzzIxzIxxIzz−Ixz2pq+Iyy−IzzIzz−Ixz2IxxIzz−Ixz2qr,̇q=MyIyy+IxzIyyr2−p2+Izz−IxxIyypr,ṙ=IxzMxIxxIzz−Ixz2+IxxMzIxxIzz−Ixz2+Ixx−IyyIxx+Ixz2IxxIzz−Ixz2pq+Iyy−Ixx−IzzIxzIxxIzz−Ixz2qr,and the kinematic equations of HFVs are given by
(2)α̇=−pcosαtanβ+q−rsinαtanβ−cosϕcosβγ̇−ξ̇cosχ−τ̇+ΩEcosξcosχ+sinϕcosβχ̇cosγ−ξ̇sinχsinγ+τ̇+ΩEcosξcosχsinγ−sinξcosγ,β̇=psinα−rcosα+sinϕγ̇−ξ̇cosχ+τ̇+ΩEcosξsinχ+cosϕχ̇cosγ−ξ̇sinχsinγ−τ̇+ΩEcosξcosχsinγ−sinξcosγ,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ+α̇sinβ−χ̇sinγ−ξ̇sinχcosγ+τ̇+ΩEcosξcosχcosγ+sinξsinγ,where ϕ, α, and β are bank, attack, and sideslip angles; p, q, and r are roll, pitch, and yaw angular rates; χ, ξ, and τ are heading, latitude, and longitude angles; ΩE is the Earth’s angular rate; and the definitions of other variables are given in [31].


### 2.1.2. Attitude Dynamic Model in Reentry Phase

In the reentry flight phase, because the rotational motion of the HFVs in reentry phase is much faster than the rotational motion of the Earth, the angular velocity of the EarthΩE can be neglected, and the translational motion can hardly affect the rotational motion. The derivatives of both position and direction of velocity and the Earth’s angular velocity are negligible with respect to the rotational motion. Therefore, Equation (2) can be simplified as
(3)α=−pcosαtanβ+q−rsinαtanβ,β=psinα−rcosα̇,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ.


### 2.1.3. Uncertain Model with Disturbance

The reentry attitude dynamics of HFVs with parameter uncertainty and external disturbance can be described as(4)I+ΔIω̇=−ω×I+ΔIω+M+d,whereω=p,q,rT is the angular rate vector, I∈R3×3 is the inertia matrix, and ΔI∈R3×3 is an uncertain part of the inertia matrix, which is caused by the fuel consumption and variations of particular payloads. d=d1,d2,d3T is the external disturbance vector. M=Mx,My,MzT is the control torque vector, which is calculated by
(5)M=Dtu,where Dt∈R3×m, m is the number of the control surfaces, and u=u1,u2,⋯,umT is the vector of aerodynamic surface deflections. Then, the attitude dynamic model can be rewritten as
(6)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Du+f1,where Ω=ϕ,α,βT is attitude angle vector, f1=I−1d−ΔIω̇−ω×ΔIω, and the matrices R, I, and ω× are as
(7)R=−cosαcosβ−sinβ−sinαcosβ−cosαtanβ1−sinαtanβsinα0−cosα,I=Ixx−Ixy−Ixz−IxyIyy−Iyz−Ixz−IyzIzz,ω×=0−rqr0−p−qp0.


### 2.1.4. Uncertain Model with Disturbance and Actuator Saturation

Since there are input constraints on the control surfaces of HFVs, thus the attitude dynamic model (6) is rewritten as
(8)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Dsatu+f1,where satui, i=1,2,⋯,m is defined as
(9)satui=u¯i,ui>u¯i,ui,u¯i≤ui≤u¯i,u¯i,ui<u¯i.Further, define an auxiliary variableδ=δ1,δ2,⋯,δmT as
(10)δi=u¯i−ui,ui>u¯i,0,u¯i≤ui≤u¯i,u¯i−ui,ui<u¯i,then (9) and (10) yield to
(11)satu=δ+u.Substituting (11) into (8), the HFV attitude dynamic model with uncertainties, disturbances, and actuator input saturations can be expressed by
(12)Ω̇=Rω,ω=̇−I−1ω×Iω+I−1Du+f,where f=I−1d−ΔIω̇−ω×ΔIω+I−1Dδ and it can be considered as a lumped disturbance for HFV attitude dynamics.


## 2.1.1. Generic Attitude Dynamic Model

The attitude dynamic equations of HFVs are given by(1)p=IzzMxIxxIzz−Ixz2+IxzMzIxxIzz−Ixz2+Ixx−Iyy+IzzIxzIxxIzz−Ixz2pq+Iyy−IzzIzz−Ixz2IxxIzz−Ixz2qr,̇q=MyIyy+IxzIyyr2−p2+Izz−IxxIyypr,ṙ=IxzMxIxxIzz−Ixz2+IxxMzIxxIzz−Ixz2+Ixx−IyyIxx+Ixz2IxxIzz−Ixz2pq+Iyy−Ixx−IzzIxzIxxIzz−Ixz2qr,and the kinematic equations of HFVs are given by
(2)α̇=−pcosαtanβ+q−rsinαtanβ−cosϕcosβγ̇−ξ̇cosχ−τ̇+ΩEcosξcosχ+sinϕcosβχ̇cosγ−ξ̇sinχsinγ+τ̇+ΩEcosξcosχsinγ−sinξcosγ,β̇=psinα−rcosα+sinϕγ̇−ξ̇cosχ+τ̇+ΩEcosξsinχ+cosϕχ̇cosγ−ξ̇sinχsinγ−τ̇+ΩEcosξcosχsinγ−sinξcosγ,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ+α̇sinβ−χ̇sinγ−ξ̇sinχcosγ+τ̇+ΩEcosξcosχcosγ+sinξsinγ,where ϕ, α, and β are bank, attack, and sideslip angles; p, q, and r are roll, pitch, and yaw angular rates; χ, ξ, and τ are heading, latitude, and longitude angles; ΩE is the Earth’s angular rate; and the definitions of other variables are given in [31].


## 2.1.2. Attitude Dynamic Model in Reentry Phase

In the reentry flight phase, because the rotational motion of the HFVs in reentry phase is much faster than the rotational motion of the Earth, the angular velocity of the EarthΩE can be neglected, and the translational motion can hardly affect the rotational motion. The derivatives of both position and direction of velocity and the Earth’s angular velocity are negligible with respect to the rotational motion. Therefore, Equation (2) can be simplified as
(3)α=−pcosαtanβ+q−rsinαtanβ,β=psinα−rcosα̇,ϕ̇=−pcosαcosβ−qsinβ−rsinαcosβ.


## 2.1.3. Uncertain Model with Disturbance

The reentry attitude dynamics of HFVs with parameter uncertainty and external disturbance can be described as(4)I+ΔIω̇=−ω×I+ΔIω+M+d,whereω=p,q,rT is the angular rate vector, I∈R3×3 is the inertia matrix, and ΔI∈R3×3 is an uncertain part of the inertia matrix, which is caused by the fuel consumption and variations of particular payloads. d=d1,d2,d3T is the external disturbance vector. M=Mx,My,MzT is the control torque vector, which is calculated by
(5)M=Dtu,where Dt∈R3×m, m is the number of the control surfaces, and u=u1,u2,⋯,umT is the vector of aerodynamic surface deflections. Then, the attitude dynamic model can be rewritten as
(6)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Du+f1,where Ω=ϕ,α,βT is attitude angle vector, f1=I−1d−ΔIω̇−ω×ΔIω, and the matrices R, I, and ω× are as
(7)R=−cosαcosβ−sinβ−sinαcosβ−cosαtanβ1−sinαtanβsinα0−cosα,I=Ixx−Ixy−Ixz−IxyIyy−Iyz−Ixz−IyzIzz,ω×=0−rqr0−p−qp0.


## 2.1.4. Uncertain Model with Disturbance and Actuator Saturation

Since there are input constraints on the control surfaces of HFVs, thus the attitude dynamic model (6) is rewritten as
(8)Ω̇=Rω,ω̇=−I−1ω×Iω+I−1Dsatu+f1,where satui, i=1,2,⋯,m is defined as
(9)satui=u¯i,ui>u¯i,ui,u¯i≤ui≤u¯i,u¯i,ui<u¯i.Further, define an auxiliary variableδ=δ1,δ2,⋯,δmT as
(10)δi=u¯i−ui,ui>u¯i,0,u¯i≤ui≤u¯i,u¯i−ui,ui<u¯i,then (9) and (10) yield to
(11)satu=δ+u.Substituting (11) into (8), the HFV attitude dynamic model with uncertainties, disturbances, and actuator input saturations can be expressed by
(12)Ω̇=Rω,ω=̇−I−1ω×Iω+I−1Du+f,where f=I−1d−ΔIω̇−ω×ΔIω+I−1Dδ and it can be considered as a lumped disturbance for HFV attitude dynamics.


## 2.2. Attitude Control Problem and Control System Framework

### 2.2.1. Control Objective

The control objective is to solve the attitude control problem of HFV system (12) with the parameter uncertainties, external disturbances, and control input constraints. An advanced control method is necessary to make the attitude system output Ω track a reference input Ωr in the finite time.


### 2.2.2. Control Method Motivation

As a robust nonlinear control method, the SMC theory is applied in this control problem. Therefore, a basic TSMC scheme is designed, an ATSMC scheme is designed in which an adaptive law is used to estimate the upper bounds of the lumped disturbance, and finally, a DO-TSMC is designed in which a nonlinear disturbance observer is used to online estimate the value of lumped disturbances.


### 2.2.3. Control System Framework

A block diagram of the HFV attitude control system is designed as shown in Figure1. It consists of two parts; the inner loop is the adaptive mechanism and disturbance estimation observer, while the outer loop is the basic TSMC-based attitude controller.Figure 1
Block diagram of the HFVs attitude control system.


## 2.2.1. Control Objective

The control objective is to solve the attitude control problem of HFV system (12) with the parameter uncertainties, external disturbances, and control input constraints. An advanced control method is necessary to make the attitude system output Ω track a reference input Ωr in the finite time.


## 2.2.2. Control Method Motivation

As a robust nonlinear control method, the SMC theory is applied in this control problem. Therefore, a basic TSMC scheme is designed, an ATSMC scheme is designed in which an adaptive law is used to estimate the upper bounds of the lumped disturbance, and finally, a DO-TSMC is designed in which a nonlinear disturbance observer is used to online estimate the value of lumped disturbances.


## 2.2.3. Control System Framework

A block diagram of the HFV attitude control system is designed as shown in Figure1. It consists of two parts; the inner loop is the adaptive mechanism and disturbance estimation observer, while the outer loop is the basic TSMC-based attitude controller.Figure 1
Block diagram of the HFVs attitude control system.


## 3. Finite-Time Attitude Controller Design

In this section, three finite-time attitude controllers based on the TSMC, ATSMC, and DO-TSMC schemes are designed for the HFV attitude system in the presence of model uncertainties, external disturbances, and actuator saturations.


### 3.1. Preliminaries

Assumption 1.
The reference signalΩr=ϕr,αr,βrT is bounded and continuously differentiable, and its derivative Ω̇r is bounded.Assumption 2.
The system-lumped disturbance torquef is bounded, and it satisfies f≤ld, where ld>0 is a constant.Assumption 3.
The derivative of the system-lumped disturbance torque is bounded, and it satisfiesḟ≤kd, where kd>0 is a constant.Lemma 4 [35].
Consider the nonlinear system described asẋ=fx,u,f0=0,x∈Rn. Suppose Vx is a continuous positive definite function (defined on D∈R) and V̇x+λVρx is negative semidefinite on D for 0<ρ<1 and λ>0, then there exists an area D0∈R such that any Vx which starts from D0∈R can reach Vx≡0 in the finite time. Moreover, if Treach is the time to reach Vx≡0, then
(13)Treach≤V1−ρxoλ1−ρ,where Vx0 is the initial value of Vx.Lemma 5 [36].
Consider the nonlinear system described asẋ=fx,u,f0=0,x∈Rn and there exists a continuous function Vx, scalars ε>0, 0<κ<1, and 0<η<∞ such that
(14)V̇x≤−εVκx+η.
Then, the trajectory of the nonlinear system is finite-time stable. Therefore, the trajectory of the closed-loop system is bounded in finite time aslimθ⟶θ0x∈Vκx≤η/1−θε, here 0<θ0<1. And the time to reach such a neighborhood is bounded as
(15)T≤V1−κx0εθ01−κ,where Vx0 is the initial value of Vx.Lemma 6 [36].
For any real numberli∈R,i=1,⋯,n, there exists 0<p≤1 such that
(16)l1+l2+⋯+lnp≤l1p+l2p+⋯+lnp.


### 3.2. TSMC-Based Attitude Control System Design

A finite-time attitude control system is designed based on the basic TSMC scheme.


#### 3.2.1. Outer-Loop Controller

A tracking error of outer loop of attitude system is defined as(17)e1=Ω−Ωr,where e1∈R3 and Ωr∈R3 is reference input which can be constant or time-varying. The first fast terminal sliding mode surface is defined as
(18)s1=ė1+c1e1+c2e1q1/p1,where s1∈R3, c1>1, c2 is a positive scalar, and q1 and p1 are odd integers satisfying p1>q1>0. Then, a virtual control law ωc can be designed as
(19)ωc=R−1ωeq+ωn,ωeq=Ω̇r−c1e1−c2e1q1/p1,ω̇n=v1,v1=−k1s1−η1sgns1,where ωn0=0; k1 and η1 are two positive constants.


#### 3.2.2. Inner-Loop Controller

Similarly, a tracking error of system inner loop is defined as(20)e2=ω−ωc,where e2∈R3. The second fast terminal sliding mode surface is defined as
(21)s2=ė2+c3e2+c4e2q2/p2,where s2∈R3, c3>1, c4 is a positive scalar, and p2 and q2 are odd integers satisfying p2>q2>0. Then, the TSMC-based attitude control law is designed as
(22)u=I−1D†ueq+unueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=v2,v2=−k2s2−kd+kT+η2sgns2,where I−1D†=I−1DTI−1DI−1DT−1; un0=0; kT, k2, and η2 are positive constants; kd is a constant defined in Assumption 3; and T≥0 and kT are selected to satisfy kT≥Tld.Theorem 7.
Consider the HFV attitude system (12) with uncertainties, disturbances, and actuator saturations, suppose that three assumptions are satisfied, the TSMC-based attitude controller (22) guarantees that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time. Furthermore, the convergent time is calculated by
(23)Treach≤V1/20η2.Proof.
Choose a Lyapunov function candidate for the closed-loop attitude control system as(24)V=12s1Ts1+12s2Ts2.
Substituting the virtual control law (19) into the sliding manifold (18) and differentiating s1, we have
(25)ṡ1=Rė2+Ṙe2−k1s1−η1sgns1.
Considering the system model (12), the sliding mode manifold (21) can be rewritten as
(26)s2=−I−1ω×Iω+I−1Du+f−ω̇c+c3e2+c4e2q2/p2.
Substituting the control law (22) into (26) gives
(27)s2=−I−1ω×Iω+ueq+un+f−ω̇c+c3e2+c4e2q2/p2=f+un.
The solution of (22) is given by
(28)unt=unt0+1Tkd+kT+η2sgns+k2s2et−t0−1Tkd+kT+η2sgns2+k2s2.
Thus, the following relationship under the conditionun0=0 can be obtained.
(29)kT≥Tld≥Tuntmax≥Tunt.
For the sliding manifold (21), its derivative with respect to t along system (12) can be obtained as
(30)ṡ2=ddtf+u̇n+Tun−Tun=ddtf+v2−Tun.
Substituting (22) into (30) gives
(31)ṡ2=ddtf−kd+kT+η2sgns2−k2s2−Tun.
Considering (25) and (31), the time derivative of V is given by
(32)V̇=s1TRė2+Ṙe2−η1sgns1−k1s1+s2Tddtf−kd+kT+η2sgns2−k2s2−Tun≤−η1s1−k1s1Ts1+s2Tddtf−kds2−Tuns2+kTs2−η2s2−k2s2Ts2+s1TṘe2+Rė2≤−η1s1−η2s2−k1s1Ts1−k2s2Ts2+s1TṘe2+Rė2.
According to the Young inequality [37], the following inequality can be obtained:
(33)s1TṘe2≤12s1TṘṘTs1+12e2Te2≤12s1TṘṘTs1+12s2Ts2.
Substituting (33) into (32) and assuming the term Rė2 is bounded satisfying Rė2≤ρ, where ρ is a positive constant, then
(34)V̇≤−η1−ρs1−η2s2−s1Tk1I−12ṘṘTs1−k2−12s2Ts2.
Letη=minη1−ρ,η2. By selecting the appropriate k1, k2, η1, and η2, such that k2>1/2, η1−ρ>0, and k1I−1/2ṘṘT>0. Considering Lemma 6, we have
(35)V̇≤−η1−ρs1−η2s2≤−η2V1/2<0.
According to Lemma4, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zeros in the finite time. Meanwhile, the convergent time is calculated by
(36)Treach≤2V1/20η2.
This completes the proof of Theorem7.


### 3.3. ATSMC-Based Attitude Control System Design

The TSMC-based attitude controller (22) is designed based on the assumption that the bounds of the uncertainties and their derivatives are known in advance. However, it is difficult to obtain the bound values in advance in some situations, and the switching gain needs to be chosen as a large value to compensate the impact of uncertainties. Unfortunately, the large switching gain may cause large chattering on the sliding surface. To enhance the performance of the HFV attitude control system, an adaptive strategy is employed into the TSMC scheme.


#### 3.3.1. ATSMC-Based Attitude Controller

The ATSMC-based attitude controller is designed as(37)u=I−1D†ueq+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=va,va=−k2s2−k^+η2sgns2,where the gain k^ is the predicted value of the gain kd+kT.


#### 3.3.2. Adaptive Law of the Gain

The gaink^ is online regulated by an adaptive law:
(38)k^̇=1εs2,whereε>0 is the adaptation coefficient. The smaller ε will provide a faster convergence but may generate a bigger value than the desired one.Theorem 8.
Considering the HFV attitude control system (12) with uncertainties, external disturbances, and input saturations, suppose the assumptions are satisfied, the ATSMC-based attitude controller (37) with adaptive law (38) guarantees the system trajectory to reach the sliding surface and remain on it in the finite time, which means that the control system is finite-time stable.Proof.
Define a Lyapunov function candidate as(39)V=12s1Ts1+12s2Ts2+12εk~Tk~,where the adaptation error k~=k^−kd+kT. Differentiating (39), we have
(40)V̇=s1Tṡ1+s2Tṡ2+εk~Tk~̇≤−η1−ρs1−k1s1Ts1+s1TṘe2−k^+η2s2+s2Tddtf−s2TTun−k2s2Ts2+εk^−kd+kTk.^̇
Substituting the adaptive law (38) into (40) yields
(41)V̇=−η1−ρs1+s2Tddtf−kd+kT+η2s2−s2TTun−k1s1Ts1+s1TṘe2−k2s2Ts2≤−η1−ρs1+s2Tddtf−kds2+−s2TTun−kTs2−η2s2≤−η1−ρs1−η2s2.
By selecting the appropriate positive constantsη1 and η2, it is found that V̇<0. Therefore, the attitude conntrol system is asymptotically stable. This implies that the trajectory reaches the sliding surface and remains on it in the finite time. This completes the proof.


### 3.4. DO-TSMC-Based Attitude Control System Design

A nonlinear disturbance observer is designed to enhance the disturbance attenuation ability and robustness performance against the uncertainties of the inertia parameters and disturbances. It estimates and compensates for the uncertainties through feedforward, which is no need for prior information of uncertainties.A new stateωat∈R3 is preliminarily introduced with its dynamic satisfying
(42)ω̇a=−I−1ω×Iω+I−1Du+Fωe,where ωe=ω−ωa and F∈R3×3 is a constant matrix determined by the design.


#### 3.4.1. Nonlinear Disturbance Observer

A nonlinear disturbance observer is given by(43)x^̇=−Lx^+LFωe−Pωe,f^=x^+Pωe,where f^ is the estimation of the compound disturbance, x^ is the internal state of the nonlinear observer, Pωe is a vector-valued function designed as Pωe=Lωe, and L is the positive observer gain matrix defined by L=LT∈R3×3.Theorem 9.
Considering the HFV attitude dynamics (12) and the nonlinear disturbance observer (43), by selecting a sufficiently large matrix L, the following results can be achieved for all f~0.
Result 1: the disturbance estimation errorf~=f−f^ is globally exponentially stable if ḟ=0
Result 2: ifḟ≠0 and the rate of change of f is bounded, i.e., there exists a positive scalar kd∈R such that ḟt≤kd for all t≥0, then the disturbance estimation error f~ converges with an exponential rate, equal to 1−ςlmin−1/4, where 0<ς<1Proof.
From (12) to (42), it can be obtained that the dynamics of ωe is such that
(44)ω̇e=ω̇−ω̇a=f−Fωe.
In accordance, it follows from (44) that the estimation error f~ is such that
(45)f~̇=ḟ+Lx^−LFωe+LPωe−Lf−Fωe=ḟ−Lf~.
Considering the following candidate Lyapunov function asV1=1/2f~Tf~, one has
(46)V̇1=f~Tf~̇=−f~TLf~+f~Tḟ.
Then, the following two cases are discussed to analyze the stability off~.
Case 1: ifḟ=0, then (46) can be further simplified as
(47)V̇1=−f~TLf~≤−2lminV1,where lmin>0 is the minimum eigenvalue of L
Solving (47) yields V1t≤V10e−2lmint or
(48)f~t≤2V10e−lmint,which implies that the estimation error f~t will be globally exponential stabilized for any initial observer state, i.e., limt⟶∞f~t=0.
Case 2: ifḟt≠0 and ḟt≤kd, one can get from (46) that
(49)V̇1≤−lminf~t2+f~tkd
At this stage, by applying the following well-known Young inequality2aTb≤aTa+bTb,∀a,b∈Rn, it can be formulated that
(50)V̇1≤−lmin−14f~t2+kd2=−1−ςlmin−14f~t2−ςlmin−14f~t2+kd2,where 0<ς<1 is a positive constant. Therefore,
(51)V̇1≤−1−ςlmin−14f~t2,∀ft≥2kdς4lmin−1.
To this end, it can be concluded from (51) and the uniform ultimate boundedness theorem that the estimation error f~t is globally, uniformly, and ultimately bounded.
Moreover, solving the inequality (51), one has
(52)V1t≤V10e−21−ςlmin−1/4t,∀ft≥2kdς4lmin−1,(53)f~t≤2V10e−1−ςlmin−1/4t,∀ft≥2kdς4lmin−1.
Hence,(54)f~t≤2V10e−1−ςlmin−1/4t+2kdς4lmin−1,∀t≥0.
Then, one can conclude that the estimation errorf~t converges with an exponential rate to the ball with radius 2kd/ς4lmin−1 for all f~0.


#### 3.4.2. Composite Control Law

The DO-TSMC-based attitude controller is designed as(55)u=I−1D†ueq−f^+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n=vb,vb=−k3s2−η3sgns2,where f^ is the estimate of f which can be obtained by the observer (43), un0=0, and η3 and k3 are positive constants.Theorem 10.
Considering the HFV attitude system (12), the DO-TSMC-based attitude controller (55) with the nonlinear disturbance observer (43) and the virtual controller (19) can guarantee that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time.Proof.
Choose the Lyapunov function candidate for the closed-loop attitude control systems as(56)V=12s1Ts1+12s2Ts2.
Differentiating (56), substituting (25), (31), and (55), it leads to
(57)V̇=s1T−η1sgns1−k1s1+Rė2+Ṙe2+s2Tf~̇−η3sgns2−k3s2.
According to the Young inequality, the following inequalities can be obtained:(58)s2Tf~̇≤12s2Ts2+12f~̇Tf~̇=12s22+f~̇2.
Substituting (58) and (33) into (57) and selecting the appropriate k1, k3, η1, and η2, such that k3>1, η1−ρ>0, and k1I−1/2ṘṘT>0, it further has
(59)V̇≤−η1−ρs1−η3s2+ϖ,where ϖ=1/2f~̇2>0. Let χ=minη1−ρ,η3. 6, we have
(60)V̇≤−χ2V1/2+ϖ.
According to Lemma5, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zero in the finite time. This completes the proof of Theorem 10.Remark 11.
In terms of singular perturbation theory, the inner-loop sliding mode dynamics in (21) must be much faster than the outer-loop sliding mode dynamics in (18) to preserve sufficient time-scale separation between two loops.Remark 12.
In the HFV attitude control system, since the control system keeps the sideslip angleβ=0∘ during the reentry phase, it is assumed that the singular situation β=±90∘ will not occur. Moreover, only v2, va, and vb contain switch terms, while the actual control u does not contain these terms. In the designed controller (22), u̇n+Tun=v2 is equal to a low-pass filter with the bandwidth of T, where v2 is input signal and un is output signal. The Laplace transformation of it is given by uns/v2s=1/s+T. Although v2 is chattering because of the switch function, un can be smoothed due to the low-pass filter, which can eliminate the impact of chattering on the system. In particular case, T=0, such as the designed controller (55), u̇n=vb is same as a pure integrator, which can also soften the sgns2 signal. In addition, the introduction of adaptive mechanism and disturbance observer can make the sign function gain selection very small. Therefore, the proposed control methods are chattering-free.Remark 13.
Most disturbance observers are designed on the assumption that the disturbance value is a constant or a slowly varying [34], i.e., ḟ=0, ḟ≈0, or limt⟶∞ḟ=0. The designed nonlinear disturbance observer can release the restrictions on the change speed of the lumped disturbance and prove that the estimation error of the lumped disturbance is exponentially convergent.


## 3.1. Preliminaries

Assumption 1.
The reference signalΩr=ϕr,αr,βrT is bounded and continuously differentiable, and its derivative Ω̇r is bounded.Assumption 2.
The system-lumped disturbance torquef is bounded, and it satisfies f≤ld, where ld>0 is a constant.Assumption 3.
The derivative of the system-lumped disturbance torque is bounded, and it satisfiesḟ≤kd, where kd>0 is a constant.Lemma 4 [35].
Consider the nonlinear system described asẋ=fx,u,f0=0,x∈Rn. Suppose Vx is a continuous positive definite function (defined on D∈R) and V̇x+λVρx is negative semidefinite on D for 0<ρ<1 and λ>0, then there exists an area D0∈R such that any Vx which starts from D0∈R can reach Vx≡0 in the finite time. Moreover, if Treach is the time to reach Vx≡0, then
(13)Treach≤V1−ρxoλ1−ρ,where Vx0 is the initial value of Vx.Lemma 5 [36].
Consider the nonlinear system described asẋ=fx,u,f0=0,x∈Rn and there exists a continuous function Vx, scalars ε>0, 0<κ<1, and 0<η<∞ such that
(14)V̇x≤−εVκx+η.
Then, the trajectory of the nonlinear system is finite-time stable. Therefore, the trajectory of the closed-loop system is bounded in finite time aslimθ⟶θ0x∈Vκx≤η/1−θε, here 0<θ0<1. And the time to reach such a neighborhood is bounded as
(15)T≤V1−κx0εθ01−κ,where Vx0 is the initial value of Vx.Lemma 6 [36].
For any real numberli∈R,i=1,⋯,n, there exists 0<p≤1 such that
(16)l1+l2+⋯+lnp≤l1p+l2p+⋯+lnp.


## 3.2. TSMC-Based Attitude Control System Design

A finite-time attitude control system is designed based on the basic TSMC scheme.


### 3.2.1. Outer-Loop Controller

A tracking error of outer loop of attitude system is defined as(17)e1=Ω−Ωr,where e1∈R3 and Ωr∈R3 is reference input which can be constant or time-varying. The first fast terminal sliding mode surface is defined as
(18)s1=ė1+c1e1+c2e1q1/p1,where s1∈R3, c1>1, c2 is a positive scalar, and q1 and p1 are odd integers satisfying p1>q1>0. Then, a virtual control law ωc can be designed as
(19)ωc=R−1ωeq+ωn,ωeq=Ω̇r−c1e1−c2e1q1/p1,ω̇n=v1,v1=−k1s1−η1sgns1,where ωn0=0; k1 and η1 are two positive constants.


### 3.2.2. Inner-Loop Controller

Similarly, a tracking error of system inner loop is defined as(20)e2=ω−ωc,where e2∈R3. The second fast terminal sliding mode surface is defined as
(21)s2=ė2+c3e2+c4e2q2/p2,where s2∈R3, c3>1, c4 is a positive scalar, and p2 and q2 are odd integers satisfying p2>q2>0. Then, the TSMC-based attitude control law is designed as
(22)u=I−1D†ueq+unueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=v2,v2=−k2s2−kd+kT+η2sgns2,where I−1D†=I−1DTI−1DI−1DT−1; un0=0; kT, k2, and η2 are positive constants; kd is a constant defined in Assumption 3; and T≥0 and kT are selected to satisfy kT≥Tld.Theorem 7.
Consider the HFV attitude system (12) with uncertainties, disturbances, and actuator saturations, suppose that three assumptions are satisfied, the TSMC-based attitude controller (22) guarantees that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time. Furthermore, the convergent time is calculated by
(23)Treach≤V1/20η2.Proof.
Choose a Lyapunov function candidate for the closed-loop attitude control system as(24)V=12s1Ts1+12s2Ts2.
Substituting the virtual control law (19) into the sliding manifold (18) and differentiating s1, we have
(25)ṡ1=Rė2+Ṙe2−k1s1−η1sgns1.
Considering the system model (12), the sliding mode manifold (21) can be rewritten as
(26)s2=−I−1ω×Iω+I−1Du+f−ω̇c+c3e2+c4e2q2/p2.
Substituting the control law (22) into (26) gives
(27)s2=−I−1ω×Iω+ueq+un+f−ω̇c+c3e2+c4e2q2/p2=f+un.
The solution of (22) is given by
(28)unt=unt0+1Tkd+kT+η2sgns+k2s2et−t0−1Tkd+kT+η2sgns2+k2s2.
Thus, the following relationship under the conditionun0=0 can be obtained.
(29)kT≥Tld≥Tuntmax≥Tunt.
For the sliding manifold (21), its derivative with respect to t along system (12) can be obtained as
(30)ṡ2=ddtf+u̇n+Tun−Tun=ddtf+v2−Tun.
Substituting (22) into (30) gives
(31)ṡ2=ddtf−kd+kT+η2sgns2−k2s2−Tun.
Considering (25) and (31), the time derivative of V is given by
(32)V̇=s1TRė2+Ṙe2−η1sgns1−k1s1+s2Tddtf−kd+kT+η2sgns2−k2s2−Tun≤−η1s1−k1s1Ts1+s2Tddtf−kds2−Tuns2+kTs2−η2s2−k2s2Ts2+s1TṘe2+Rė2≤−η1s1−η2s2−k1s1Ts1−k2s2Ts2+s1TṘe2+Rė2.
According to the Young inequality [37], the following inequality can be obtained:
(33)s1TṘe2≤12s1TṘṘTs1+12e2Te2≤12s1TṘṘTs1+12s2Ts2.
Substituting (33) into (32) and assuming the term Rė2 is bounded satisfying Rė2≤ρ, where ρ is a positive constant, then
(34)V̇≤−η1−ρs1−η2s2−s1Tk1I−12ṘṘTs1−k2−12s2Ts2.
Letη=minη1−ρ,η2. By selecting the appropriate k1, k2, η1, and η2, such that k2>1/2, η1−ρ>0, and k1I−1/2ṘṘT>0. Considering Lemma 6, we have
(35)V̇≤−η1−ρs1−η2s2≤−η2V1/2<0.
According to Lemma4, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zeros in the finite time. Meanwhile, the convergent time is calculated by
(36)Treach≤2V1/20η2.
This completes the proof of Theorem7.


## 3.2.1. Outer-Loop Controller

A tracking error of outer loop of attitude system is defined as(17)e1=Ω−Ωr,where e1∈R3 and Ωr∈R3 is reference input which can be constant or time-varying. The first fast terminal sliding mode surface is defined as
(18)s1=ė1+c1e1+c2e1q1/p1,where s1∈R3, c1>1, c2 is a positive scalar, and q1 and p1 are odd integers satisfying p1>q1>0. Then, a virtual control law ωc can be designed as
(19)ωc=R−1ωeq+ωn,ωeq=Ω̇r−c1e1−c2e1q1/p1,ω̇n=v1,v1=−k1s1−η1sgns1,where ωn0=0; k1 and η1 are two positive constants.


## 3.2.2. Inner-Loop Controller

Similarly, a tracking error of system inner loop is defined as(20)e2=ω−ωc,where e2∈R3. The second fast terminal sliding mode surface is defined as
(21)s2=ė2+c3e2+c4e2q2/p2,where s2∈R3, c3>1, c4 is a positive scalar, and p2 and q2 are odd integers satisfying p2>q2>0. Then, the TSMC-based attitude control law is designed as
(22)u=I−1D†ueq+unueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=v2,v2=−k2s2−kd+kT+η2sgns2,where I−1D†=I−1DTI−1DI−1DT−1; un0=0; kT, k2, and η2 are positive constants; kd is a constant defined in Assumption 3; and T≥0 and kT are selected to satisfy kT≥Tld.Theorem 7.
Consider the HFV attitude system (12) with uncertainties, disturbances, and actuator saturations, suppose that three assumptions are satisfied, the TSMC-based attitude controller (22) guarantees that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time. Furthermore, the convergent time is calculated by
(23)Treach≤V1/20η2.Proof.
Choose a Lyapunov function candidate for the closed-loop attitude control system as(24)V=12s1Ts1+12s2Ts2.
Substituting the virtual control law (19) into the sliding manifold (18) and differentiating s1, we have
(25)ṡ1=Rė2+Ṙe2−k1s1−η1sgns1.
Considering the system model (12), the sliding mode manifold (21) can be rewritten as
(26)s2=−I−1ω×Iω+I−1Du+f−ω̇c+c3e2+c4e2q2/p2.
Substituting the control law (22) into (26) gives
(27)s2=−I−1ω×Iω+ueq+un+f−ω̇c+c3e2+c4e2q2/p2=f+un.
The solution of (22) is given by
(28)unt=unt0+1Tkd+kT+η2sgns+k2s2et−t0−1Tkd+kT+η2sgns2+k2s2.
Thus, the following relationship under the conditionun0=0 can be obtained.
(29)kT≥Tld≥Tuntmax≥Tunt.
For the sliding manifold (21), its derivative with respect to t along system (12) can be obtained as
(30)ṡ2=ddtf+u̇n+Tun−Tun=ddtf+v2−Tun.
Substituting (22) into (30) gives
(31)ṡ2=ddtf−kd+kT+η2sgns2−k2s2−Tun.
Considering (25) and (31), the time derivative of V is given by
(32)V̇=s1TRė2+Ṙe2−η1sgns1−k1s1+s2Tddtf−kd+kT+η2sgns2−k2s2−Tun≤−η1s1−k1s1Ts1+s2Tddtf−kds2−Tuns2+kTs2−η2s2−k2s2Ts2+s1TṘe2+Rė2≤−η1s1−η2s2−k1s1Ts1−k2s2Ts2+s1TṘe2+Rė2.
According to the Young inequality [37], the following inequality can be obtained:
(33)s1TṘe2≤12s1TṘṘTs1+12e2Te2≤12s1TṘṘTs1+12s2Ts2.
Substituting (33) into (32) and assuming the term Rė2 is bounded satisfying Rė2≤ρ, where ρ is a positive constant, then
(34)V̇≤−η1−ρs1−η2s2−s1Tk1I−12ṘṘTs1−k2−12s2Ts2.
Letη=minη1−ρ,η2. By selecting the appropriate k1, k2, η1, and η2, such that k2>1/2, η1−ρ>0, and k1I−1/2ṘṘT>0. Considering Lemma 6, we have
(35)V̇≤−η1−ρs1−η2s2≤−η2V1/2<0.
According to Lemma4, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zeros in the finite time. Meanwhile, the convergent time is calculated by
(36)Treach≤2V1/20η2.
This completes the proof of Theorem7.


## 3.3. ATSMC-Based Attitude Control System Design

The TSMC-based attitude controller (22) is designed based on the assumption that the bounds of the uncertainties and their derivatives are known in advance. However, it is difficult to obtain the bound values in advance in some situations, and the switching gain needs to be chosen as a large value to compensate the impact of uncertainties. Unfortunately, the large switching gain may cause large chattering on the sliding surface. To enhance the performance of the HFV attitude control system, an adaptive strategy is employed into the TSMC scheme.


### 3.3.1. ATSMC-Based Attitude Controller

The ATSMC-based attitude controller is designed as(37)u=I−1D†ueq+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=va,va=−k2s2−k^+η2sgns2,where the gain k^ is the predicted value of the gain kd+kT.


### 3.3.2. Adaptive Law of the Gain

The gaink^ is online regulated by an adaptive law:
(38)k^̇=1εs2,whereε>0 is the adaptation coefficient. The smaller ε will provide a faster convergence but may generate a bigger value than the desired one.Theorem 8.
Considering the HFV attitude control system (12) with uncertainties, external disturbances, and input saturations, suppose the assumptions are satisfied, the ATSMC-based attitude controller (37) with adaptive law (38) guarantees the system trajectory to reach the sliding surface and remain on it in the finite time, which means that the control system is finite-time stable.Proof.
Define a Lyapunov function candidate as(39)V=12s1Ts1+12s2Ts2+12εk~Tk~,where the adaptation error k~=k^−kd+kT. Differentiating (39), we have
(40)V̇=s1Tṡ1+s2Tṡ2+εk~Tk~̇≤−η1−ρs1−k1s1Ts1+s1TṘe2−k^+η2s2+s2Tddtf−s2TTun−k2s2Ts2+εk^−kd+kTk.^̇
Substituting the adaptive law (38) into (40) yields
(41)V̇=−η1−ρs1+s2Tddtf−kd+kT+η2s2−s2TTun−k1s1Ts1+s1TṘe2−k2s2Ts2≤−η1−ρs1+s2Tddtf−kds2+−s2TTun−kTs2−η2s2≤−η1−ρs1−η2s2.
By selecting the appropriate positive constantsη1 and η2, it is found that V̇<0. Therefore, the attitude conntrol system is asymptotically stable. This implies that the trajectory reaches the sliding surface and remains on it in the finite time. This completes the proof.


## 3.3.1. ATSMC-Based Attitude Controller

The ATSMC-based attitude controller is designed as(37)u=I−1D†ueq+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n+Tun=va,va=−k2s2−k^+η2sgns2,where the gain k^ is the predicted value of the gain kd+kT.


## 3.3.2. Adaptive Law of the Gain

The gaink^ is online regulated by an adaptive law:
(38)k^̇=1εs2,whereε>0 is the adaptation coefficient. The smaller ε will provide a faster convergence but may generate a bigger value than the desired one.Theorem 8.
Considering the HFV attitude control system (12) with uncertainties, external disturbances, and input saturations, suppose the assumptions are satisfied, the ATSMC-based attitude controller (37) with adaptive law (38) guarantees the system trajectory to reach the sliding surface and remain on it in the finite time, which means that the control system is finite-time stable.Proof.
Define a Lyapunov function candidate as(39)V=12s1Ts1+12s2Ts2+12εk~Tk~,where the adaptation error k~=k^−kd+kT. Differentiating (39), we have
(40)V̇=s1Tṡ1+s2Tṡ2+εk~Tk~̇≤−η1−ρs1−k1s1Ts1+s1TṘe2−k^+η2s2+s2Tddtf−s2TTun−k2s2Ts2+εk^−kd+kTk.^̇
Substituting the adaptive law (38) into (40) yields
(41)V̇=−η1−ρs1+s2Tddtf−kd+kT+η2s2−s2TTun−k1s1Ts1+s1TṘe2−k2s2Ts2≤−η1−ρs1+s2Tddtf−kds2+−s2TTun−kTs2−η2s2≤−η1−ρs1−η2s2.
By selecting the appropriate positive constantsη1 and η2, it is found that V̇<0. Therefore, the attitude conntrol system is asymptotically stable. This implies that the trajectory reaches the sliding surface and remains on it in the finite time. This completes the proof.


## 3.4. DO-TSMC-Based Attitude Control System Design

A nonlinear disturbance observer is designed to enhance the disturbance attenuation ability and robustness performance against the uncertainties of the inertia parameters and disturbances. It estimates and compensates for the uncertainties through feedforward, which is no need for prior information of uncertainties.A new stateωat∈R3 is preliminarily introduced with its dynamic satisfying
(42)ω̇a=−I−1ω×Iω+I−1Du+Fωe,where ωe=ω−ωa and F∈R3×3 is a constant matrix determined by the design.


### 3.4.1. Nonlinear Disturbance Observer

A nonlinear disturbance observer is given by(43)x^̇=−Lx^+LFωe−Pωe,f^=x^+Pωe,where f^ is the estimation of the compound disturbance, x^ is the internal state of the nonlinear observer, Pωe is a vector-valued function designed as Pωe=Lωe, and L is the positive observer gain matrix defined by L=LT∈R3×3.Theorem 9.
Considering the HFV attitude dynamics (12) and the nonlinear disturbance observer (43), by selecting a sufficiently large matrix L, the following results can be achieved for all f~0.
Result 1: the disturbance estimation errorf~=f−f^ is globally exponentially stable if ḟ=0
Result 2: ifḟ≠0 and the rate of change of f is bounded, i.e., there exists a positive scalar kd∈R such that ḟt≤kd for all t≥0, then the disturbance estimation error f~ converges with an exponential rate, equal to 1−ςlmin−1/4, where 0<ς<1Proof.
From (12) to (42), it can be obtained that the dynamics of ωe is such that
(44)ω̇e=ω̇−ω̇a=f−Fωe.
In accordance, it follows from (44) that the estimation error f~ is such that
(45)f~̇=ḟ+Lx^−LFωe+LPωe−Lf−Fωe=ḟ−Lf~.
Considering the following candidate Lyapunov function asV1=1/2f~Tf~, one has
(46)V̇1=f~Tf~̇=−f~TLf~+f~Tḟ.
Then, the following two cases are discussed to analyze the stability off~.
Case 1: ifḟ=0, then (46) can be further simplified as
(47)V̇1=−f~TLf~≤−2lminV1,where lmin>0 is the minimum eigenvalue of L
Solving (47) yields V1t≤V10e−2lmint or
(48)f~t≤2V10e−lmint,which implies that the estimation error f~t will be globally exponential stabilized for any initial observer state, i.e., limt⟶∞f~t=0.
Case 2: ifḟt≠0 and ḟt≤kd, one can get from (46) that
(49)V̇1≤−lminf~t2+f~tkd
At this stage, by applying the following well-known Young inequality2aTb≤aTa+bTb,∀a,b∈Rn, it can be formulated that
(50)V̇1≤−lmin−14f~t2+kd2=−1−ςlmin−14f~t2−ςlmin−14f~t2+kd2,where 0<ς<1 is a positive constant. Therefore,
(51)V̇1≤−1−ςlmin−14f~t2,∀ft≥2kdς4lmin−1.
To this end, it can be concluded from (51) and the uniform ultimate boundedness theorem that the estimation error f~t is globally, uniformly, and ultimately bounded.
Moreover, solving the inequality (51), one has
(52)V1t≤V10e−21−ςlmin−1/4t,∀ft≥2kdς4lmin−1,(53)f~t≤2V10e−1−ςlmin−1/4t,∀ft≥2kdς4lmin−1.
Hence,(54)f~t≤2V10e−1−ςlmin−1/4t+2kdς4lmin−1,∀t≥0.
Then, one can conclude that the estimation errorf~t converges with an exponential rate to the ball with radius 2kd/ς4lmin−1 for all f~0.


### 3.4.2. Composite Control Law

The DO-TSMC-based attitude controller is designed as(55)u=I−1D†ueq−f^+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n=vb,vb=−k3s2−η3sgns2,where f^ is the estimate of f which can be obtained by the observer (43), un0=0, and η3 and k3 are positive constants.Theorem 10.
Considering the HFV attitude system (12), the DO-TSMC-based attitude controller (55) with the nonlinear disturbance observer (43) and the virtual controller (19) can guarantee that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time.Proof.
Choose the Lyapunov function candidate for the closed-loop attitude control systems as(56)V=12s1Ts1+12s2Ts2.
Differentiating (56), substituting (25), (31), and (55), it leads to
(57)V̇=s1T−η1sgns1−k1s1+Rė2+Ṙe2+s2Tf~̇−η3sgns2−k3s2.
According to the Young inequality, the following inequalities can be obtained:(58)s2Tf~̇≤12s2Ts2+12f~̇Tf~̇=12s22+f~̇2.
Substituting (58) and (33) into (57) and selecting the appropriate k1, k3, η1, and η2, such that k3>1, η1−ρ>0, and k1I−1/2ṘṘT>0, it further has
(59)V̇≤−η1−ρs1−η3s2+ϖ,where ϖ=1/2f~̇2>0. Let χ=minη1−ρ,η3. 6, we have
(60)V̇≤−χ2V1/2+ϖ.
According to Lemma5, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zero in the finite time. This completes the proof of Theorem 10.Remark 11.
In terms of singular perturbation theory, the inner-loop sliding mode dynamics in (21) must be much faster than the outer-loop sliding mode dynamics in (18) to preserve sufficient time-scale separation between two loops.Remark 12.
In the HFV attitude control system, since the control system keeps the sideslip angleβ=0∘ during the reentry phase, it is assumed that the singular situation β=±90∘ will not occur. Moreover, only v2, va, and vb contain switch terms, while the actual control u does not contain these terms. In the designed controller (22), u̇n+Tun=v2 is equal to a low-pass filter with the bandwidth of T, where v2 is input signal and un is output signal. The Laplace transformation of it is given by uns/v2s=1/s+T. Although v2 is chattering because of the switch function, un can be smoothed due to the low-pass filter, which can eliminate the impact of chattering on the system. In particular case, T=0, such as the designed controller (55), u̇n=vb is same as a pure integrator, which can also soften the sgns2 signal. In addition, the introduction of adaptive mechanism and disturbance observer can make the sign function gain selection very small. Therefore, the proposed control methods are chattering-free.Remark 13.
Most disturbance observers are designed on the assumption that the disturbance value is a constant or a slowly varying [34], i.e., ḟ=0, ḟ≈0, or limt⟶∞ḟ=0. The designed nonlinear disturbance observer can release the restrictions on the change speed of the lumped disturbance and prove that the estimation error of the lumped disturbance is exponentially convergent.


## 3.4.1. Nonlinear Disturbance Observer

A nonlinear disturbance observer is given by(43)x^̇=−Lx^+LFωe−Pωe,f^=x^+Pωe,where f^ is the estimation of the compound disturbance, x^ is the internal state of the nonlinear observer, Pωe is a vector-valued function designed as Pωe=Lωe, and L is the positive observer gain matrix defined by L=LT∈R3×3.Theorem 9.
Considering the HFV attitude dynamics (12) and the nonlinear disturbance observer (43), by selecting a sufficiently large matrix L, the following results can be achieved for all f~0.
Result 1: the disturbance estimation errorf~=f−f^ is globally exponentially stable if ḟ=0
Result 2: ifḟ≠0 and the rate of change of f is bounded, i.e., there exists a positive scalar kd∈R such that ḟt≤kd for all t≥0, then the disturbance estimation error f~ converges with an exponential rate, equal to 1−ςlmin−1/4, where 0<ς<1Proof.
From (12) to (42), it can be obtained that the dynamics of ωe is such that
(44)ω̇e=ω̇−ω̇a=f−Fωe.
In accordance, it follows from (44) that the estimation error f~ is such that
(45)f~̇=ḟ+Lx^−LFωe+LPωe−Lf−Fωe=ḟ−Lf~.
Considering the following candidate Lyapunov function asV1=1/2f~Tf~, one has
(46)V̇1=f~Tf~̇=−f~TLf~+f~Tḟ.
Then, the following two cases are discussed to analyze the stability off~.
Case 1: ifḟ=0, then (46) can be further simplified as
(47)V̇1=−f~TLf~≤−2lminV1,where lmin>0 is the minimum eigenvalue of L
Solving (47) yields V1t≤V10e−2lmint or
(48)f~t≤2V10e−lmint,which implies that the estimation error f~t will be globally exponential stabilized for any initial observer state, i.e., limt⟶∞f~t=0.
Case 2: ifḟt≠0 and ḟt≤kd, one can get from (46) that
(49)V̇1≤−lminf~t2+f~tkd
At this stage, by applying the following well-known Young inequality2aTb≤aTa+bTb,∀a,b∈Rn, it can be formulated that
(50)V̇1≤−lmin−14f~t2+kd2=−1−ςlmin−14f~t2−ςlmin−14f~t2+kd2,where 0<ς<1 is a positive constant. Therefore,
(51)V̇1≤−1−ςlmin−14f~t2,∀ft≥2kdς4lmin−1.
To this end, it can be concluded from (51) and the uniform ultimate boundedness theorem that the estimation error f~t is globally, uniformly, and ultimately bounded.
Moreover, solving the inequality (51), one has
(52)V1t≤V10e−21−ςlmin−1/4t,∀ft≥2kdς4lmin−1,(53)f~t≤2V10e−1−ςlmin−1/4t,∀ft≥2kdς4lmin−1.
Hence,(54)f~t≤2V10e−1−ςlmin−1/4t+2kdς4lmin−1,∀t≥0.
Then, one can conclude that the estimation errorf~t converges with an exponential rate to the ball with radius 2kd/ς4lmin−1 for all f~0.


## 3.4.2. Composite Control Law

The DO-TSMC-based attitude controller is designed as(55)u=I−1D†ueq−f^+un,ueq=I−1ω×Iω−c3e2−c4e2q2/p2+ω̇c,u̇n=vb,vb=−k3s2−η3sgns2,where f^ is the estimate of f which can be obtained by the observer (43), un0=0, and η3 and k3 are positive constants.Theorem 10.
Considering the HFV attitude system (12), the DO-TSMC-based attitude controller (55) with the nonlinear disturbance observer (43) and the virtual controller (19) can guarantee that the attitude angle error e1 and angular velocity error e2 converge to zeros in the finite time.Proof.
Choose the Lyapunov function candidate for the closed-loop attitude control systems as(56)V=12s1Ts1+12s2Ts2.
Differentiating (56), substituting (25), (31), and (55), it leads to
(57)V̇=s1T−η1sgns1−k1s1+Rė2+Ṙe2+s2Tf~̇−η3sgns2−k3s2.
According to the Young inequality, the following inequalities can be obtained:(58)s2Tf~̇≤12s2Ts2+12f~̇Tf~̇=12s22+f~̇2.
Substituting (58) and (33) into (57) and selecting the appropriate k1, k3, η1, and η2, such that k3>1, η1−ρ>0, and k1I−1/2ṘṘT>0, it further has
(59)V̇≤−η1−ρs1−η3s2+ϖ,where ϖ=1/2f~̇2>0. Let χ=minη1−ρ,η3. 6, we have
(60)V̇≤−χ2V1/2+ϖ.
According to Lemma5, it is not difficult to find that the sliding manifold s1 and sliding manifold s2 converge to zero in the finite time. This completes the proof of Theorem 10.Remark 11.
In terms of singular perturbation theory, the inner-loop sliding mode dynamics in (21) must be much faster than the outer-loop sliding mode dynamics in (18) to preserve sufficient time-scale separation between two loops.Remark 12.
In the HFV attitude control system, since the control system keeps the sideslip angleβ=0∘ during the reentry phase, it is assumed that the singular situation β=±90∘ will not occur. Moreover, only v2, va, and vb contain switch terms, while the actual control u does not contain these terms. In the designed controller (22), u̇n+Tun=v2 is equal to a low-pass filter with the bandwidth of T, where v2 is input signal and un is output signal. The Laplace transformation of it is given by uns/v2s=1/s+T. Although v2 is chattering because of the switch function, un can be smoothed due to the low-pass filter, which can eliminate the impact of chattering on the system. In particular case, T=0, such as the designed controller (55), u̇n=vb is same as a pure integrator, which can also soften the sgns2 signal. In addition, the introduction of adaptive mechanism and disturbance observer can make the sign function gain selection very small. Therefore, the proposed control methods are chattering-free.Remark 13.
Most disturbance observers are designed on the assumption that the disturbance value is a constant or a slowly varying [34], i.e., ḟ=0, ḟ≈0, or limt⟶∞ḟ=0. The designed nonlinear disturbance observer can release the restrictions on the change speed of the lumped disturbance and prove that the estimation error of the lumped disturbance is exponentially convergent.


## 4. Simulation Study

In this study, the proposed finite-time attitude tracking control algorithms are applied to a generic nonlinear model of X-33 HFV, and a MATLAB/Simulink-based thorough simulation study is conducted to show some insights in the nonlinear system.


### 4.1. Parameters of HFV Model and Attitude Control System

#### 4.1.1. HFV Model

In this section, we choose the X-33 HFV as the controlled plant and the parameters of this model refer to reference [34]. Figure 2 shows the configuration of the X-33 HFV [31, 38]. Its weight is 136078 kg, and it is equipped with four sets of control surfaces. Each control surface can be independently actuated with one actuator, i.e., rudders, body flaps, and inboard and outboard elevons, respectively, with left and right sides for each set. The selection of matrix D∈R3×8 refers to reference [34, 39]. Hence, the control input vector is u=u1,u2,u3,u4,u5,u6,u7,u8T, where u1 and u2 are the right and left inboard elevons, u3 and u4 are the right and left body flaps, u5 and u6 are the right and left rudders, and u7 and u8 are the right and left outboard elevons, respectively. The actuator position limits of the X-33 vehicle are listed in Table 1. The moment of inertia tensor and parameter uncertainties are given by
(61)I=5544860−23002011369490−2300201376852,ΔI=−500000−500000−500.Figure 2
The configuration of the X-33 HFV.Table 1
Position limits on the control surface of X-33.
ActuatorPosition limitMinimumMaximumUnitRight inboard elevons-25+25DegLeft inboard elevons-25+25DegRight body flaps-15+26DegLeft body flaps-15+26DegRight rudders-60+30DegLeft rudders-30+60DegRight outboard elevons-30+30DegLeft outboard elevons-30+30DegFurthermore, the external disturbances of the NSV are assumed as(62)dt=d1td2td3t=160000sin5t+0.2525000sin5t+0.2025000sin5t+0.10.


#### 4.1.2. Control Commands and Control Schemes

Assume that the HFV is flying with velocity of 2500 m/s and height of 40000 m. The initial attitude angles are thatϕ=1rad, α=0rad, β=0.2rad, and p=q=r=0rad/s. The HFV attitude tracking commands are given as ϕr=0.5rad, αr=0.3rad, and βr=0rad. The effectiveness of the TSMC scheme is verified by comparing with a backstepping control (BSC) method [30] and an integral sliding mode control (ISMC) method [32]. Furthermore, the effectiveness of the ATSMC and DO-TSMC is also verified. For the BSC, Λ1=diag25,25,25 and Λ2=diag25,25,25. For the ISMC, K1=diag0.4,0.4,0.4, K2=diag1.88,1.88,1.88, ρ=diag1,0.15,0.08, and ρ^=diag4.0×106,4.0×106,4.0×106. For the TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, kd+kT=15, and T=0.5. For the ATSMC, p1=3, q1=1, p2=4, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, T=0.5, and γ=0.4. For the DO-TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5η1=0.1, η3=0.2, and L=diag105,105,105.


### 4.2. Simulation Results and Discussion

The comparisons on the tracking performance of attitude angle and angular rate as well as actuator inputs using the BSC, ISMC, and TSMC schemes are shown in Figure3. The responses of the attitude angles and angular velocities by the ATMC method are shown in Figure 4. And Figure 5 shows the control surface deflections and adaptive value responses. Figure 6 shows the tracking responses of the attitude angles and angular rates under DO-TSMC scheme, and Figure 7 gives the control surface deflections and observer estimation error responses. Figure 8 shows the lumped disturbance estimation and estimation error responses under DO-TSMC.Figure 3
Attitude angle and angular rate responses under different methods.Figure 4
Attitude angle and angular rate responses under ATSMC.Figure 5
Control surface deflection and adaptive value responses under ATSMC.Figure 6
Attitude angle and its angular rate responses under DO-TSMC.Figure 7
Control surface deflection and observer error responses under DO-TSMC.Figure 8
Disturbance estimate and estimation error responses under DO-TSMC.


#### 4.2.1. Comparison between the TSMC, BSC, and ISMC Methods

The proposed TSMC method as a finite-time sliding mode control method is compared with the traditional BSC and ISMC methods for the NSV system. Viewing from Figures3 and 9, the attitude angle outputs cannot satisfactorily track the desired commands with large tracking errors and even cannot be stable by using the BSC method. Although the ISMC can make the system outputs track the desired commands, it needs a longer settling time than the TSMC method. As shown in Figure 9, affected by the disturbance, all control surfaces of the BSC system are in a large range of changes, which will cause a lot of energy loss. Due to the limited output of the actuator, it can be seen from the response of u3 that the right body flaps have been at the minimum deflection angle (−15deg) for about 0.2 seconds. After the adjustment of the TSMC controller, the output of the actuator has been in a reasonable small range. As shown in Figure 9, the left body flap u4 of BSC system is always at the minimum deflection angle (−15deg) at 0.25-0.4 second, resulting in actuator saturation. In all, the TSMC scheme obviously has the best performance than the BSC scheme and ISMC scheme, for the attitude control problem with model uncertainties, external disturbances, and actuator saturations.Figure 9
Control surface deflection responses.


#### 4.2.2. Comparison between the ATMC and TSMC Methods

The ATMC applies an adaptive mechanism to relax the bounded lumped disturbance information of the basic TSMC. Figure4 shows that the ATMC scheme can stabilize the attitude angles and angular rates in the finite time. For the TSMC, when the system lacks the upper bound information, a larger switching gain will be chosen that may lead to large chattering on the control surface. For the ATMC, the control chattering problem is effectively attenuated due to the adaptive mechanism. As shown in Figure 5, the predicted value of the gain kd+kT can converge to the constant 0.7 in a very short time. Therefore, the ATSMC scheme achieves a good performance even though the system uncertainty bounds are unknown in advance.


#### 4.2.3. Comparison between the DO-TMC and TSMC Methods

The DO-TMC applies a nonlinear disturbance observer to relax the restrictions of lumped disturbance and attenuate chattering phenomenon of the basic TSMC. For the DO-TMC, the HFV attitude system smoothly achieves stabilization with a settling time less than 1.5 s and a high accuracy, as shown in Figure6. By selecting the appropriate values of the disturbance observer parameters, it can be seen from Figure 7 that the observer error ωe can converge to a very small interval, i.e., ωei≤2×10−3. The estimated information of the observer is transmitted to the controller to adjust the output of the actuator in real time, so that each control surface can deflect in a small range. Figure 8 shows the estimated response curve and estimation error curve under the proposed disturbance observer. Figure 8 exhibits that the proposed observer has good performance on the estimated slow time varying, fast time varying, and periodic disturbance. Above all, the DO-TSMC scheme can significantly achieve a satisfied performance without any prior knowledge of the compound uncertainties of the HFV control system.


## 4.1. Parameters of HFV Model and Attitude Control System

### 4.1.1. HFV Model

In this section, we choose the X-33 HFV as the controlled plant and the parameters of this model refer to reference [34]. Figure 2 shows the configuration of the X-33 HFV [31, 38]. Its weight is 136078 kg, and it is equipped with four sets of control surfaces. Each control surface can be independently actuated with one actuator, i.e., rudders, body flaps, and inboard and outboard elevons, respectively, with left and right sides for each set. The selection of matrix D∈R3×8 refers to reference [34, 39]. Hence, the control input vector is u=u1,u2,u3,u4,u5,u6,u7,u8T, where u1 and u2 are the right and left inboard elevons, u3 and u4 are the right and left body flaps, u5 and u6 are the right and left rudders, and u7 and u8 are the right and left outboard elevons, respectively. The actuator position limits of the X-33 vehicle are listed in Table 1. The moment of inertia tensor and parameter uncertainties are given by
(61)I=5544860−23002011369490−2300201376852,ΔI=−500000−500000−500.Figure 2
The configuration of the X-33 HFV.Table 1
Position limits on the control surface of X-33.
ActuatorPosition limitMinimumMaximumUnitRight inboard elevons-25+25DegLeft inboard elevons-25+25DegRight body flaps-15+26DegLeft body flaps-15+26DegRight rudders-60+30DegLeft rudders-30+60DegRight outboard elevons-30+30DegLeft outboard elevons-30+30DegFurthermore, the external disturbances of the NSV are assumed as(62)dt=d1td2td3t=160000sin5t+0.2525000sin5t+0.2025000sin5t+0.10.


### 4.1.2. Control Commands and Control Schemes

Assume that the HFV is flying with velocity of 2500 m/s and height of 40000 m. The initial attitude angles are thatϕ=1rad, α=0rad, β=0.2rad, and p=q=r=0rad/s. The HFV attitude tracking commands are given as ϕr=0.5rad, αr=0.3rad, and βr=0rad. The effectiveness of the TSMC scheme is verified by comparing with a backstepping control (BSC) method [30] and an integral sliding mode control (ISMC) method [32]. Furthermore, the effectiveness of the ATSMC and DO-TSMC is also verified. For the BSC, Λ1=diag25,25,25 and Λ2=diag25,25,25. For the ISMC, K1=diag0.4,0.4,0.4, K2=diag1.88,1.88,1.88, ρ=diag1,0.15,0.08, and ρ^=diag4.0×106,4.0×106,4.0×106. For the TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, kd+kT=15, and T=0.5. For the ATSMC, p1=3, q1=1, p2=4, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, T=0.5, and γ=0.4. For the DO-TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5η1=0.1, η3=0.2, and L=diag105,105,105.


## 4.1.1. HFV Model

In this section, we choose the X-33 HFV as the controlled plant and the parameters of this model refer to reference [34]. Figure 2 shows the configuration of the X-33 HFV [31, 38]. Its weight is 136078 kg, and it is equipped with four sets of control surfaces. Each control surface can be independently actuated with one actuator, i.e., rudders, body flaps, and inboard and outboard elevons, respectively, with left and right sides for each set. The selection of matrix D∈R3×8 refers to reference [34, 39]. Hence, the control input vector is u=u1,u2,u3,u4,u5,u6,u7,u8T, where u1 and u2 are the right and left inboard elevons, u3 and u4 are the right and left body flaps, u5 and u6 are the right and left rudders, and u7 and u8 are the right and left outboard elevons, respectively. The actuator position limits of the X-33 vehicle are listed in Table 1. The moment of inertia tensor and parameter uncertainties are given by
(61)I=5544860−23002011369490−2300201376852,ΔI=−500000−500000−500.Figure 2
The configuration of the X-33 HFV.Table 1
Position limits on the control surface of X-33.
ActuatorPosition limitMinimumMaximumUnitRight inboard elevons-25+25DegLeft inboard elevons-25+25DegRight body flaps-15+26DegLeft body flaps-15+26DegRight rudders-60+30DegLeft rudders-30+60DegRight outboard elevons-30+30DegLeft outboard elevons-30+30DegFurthermore, the external disturbances of the NSV are assumed as(62)dt=d1td2td3t=160000sin5t+0.2525000sin5t+0.2025000sin5t+0.10.


## 4.1.2. Control Commands and Control Schemes

Assume that the HFV is flying with velocity of 2500 m/s and height of 40000 m. The initial attitude angles are thatϕ=1rad, α=0rad, β=0.2rad, and p=q=r=0rad/s. The HFV attitude tracking commands are given as ϕr=0.5rad, αr=0.3rad, and βr=0rad. The effectiveness of the TSMC scheme is verified by comparing with a backstepping control (BSC) method [30] and an integral sliding mode control (ISMC) method [32]. Furthermore, the effectiveness of the ATSMC and DO-TSMC is also verified. For the BSC, Λ1=diag25,25,25 and Λ2=diag25,25,25. For the ISMC, K1=diag0.4,0.4,0.4, K2=diag1.88,1.88,1.88, ρ=diag1,0.15,0.08, and ρ^=diag4.0×106,4.0×106,4.0×106. For the TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, kd+kT=15, and T=0.5. For the ATSMC, p1=3, q1=1, p2=4, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5, η1=0.1, η2=0.4, T=0.5, and γ=0.4. For the DO-TSMC, p1=3, q1=1, p2=7, q2=5, c1=2, c2=3, c3=7, c4=10, k1=15, k2=10.5η1=0.1, η3=0.2, and L=diag105,105,105.


## 4.2. Simulation Results and Discussion

The comparisons on the tracking performance of attitude angle and angular rate as well as actuator inputs using the BSC, ISMC, and TSMC schemes are shown in Figure3. The responses of the attitude angles and angular velocities by the ATMC method are shown in Figure 4. And Figure 5 shows the control surface deflections and adaptive value responses. Figure 6 shows the tracking responses of the attitude angles and angular rates under DO-TSMC scheme, and Figure 7 gives the control surface deflections and observer estimation error responses. Figure 8 shows the lumped disturbance estimation and estimation error responses under DO-TSMC.Figure 3
Attitude angle and angular rate responses under different methods.Figure 4
Attitude angle and angular rate responses under ATSMC.Figure 5
Control surface deflection and adaptive value responses under ATSMC.Figure 6
Attitude angle and its angular rate responses under DO-TSMC.Figure 7
Control surface deflection and observer error responses under DO-TSMC.Figure 8
Disturbance estimate and estimation error responses under DO-TSMC.


### 4.2.1. Comparison between the TSMC, BSC, and ISMC Methods

The proposed TSMC method as a finite-time sliding mode control method is compared with the traditional BSC and ISMC methods for the NSV system. Viewing from Figures3 and 9, the attitude angle outputs cannot satisfactorily track the desired commands with large tracking errors and even cannot be stable by using the BSC method. Although the ISMC can make the system outputs track the desired commands, it needs a longer settling time than the TSMC method. As shown in Figure 9, affected by the disturbance, all control surfaces of the BSC system are in a large range of changes, which will cause a lot of energy loss. Due to the limited output of the actuator, it can be seen from the response of u3 that the right body flaps have been at the minimum deflection angle (−15deg) for about 0.2 seconds. After the adjustment of the TSMC controller, the output of the actuator has been in a reasonable small range. As shown in Figure 9, the left body flap u4 of BSC system is always at the minimum deflection angle (−15deg) at 0.25-0.4 second, resulting in actuator saturation. In all, the TSMC scheme obviously has the best performance than the BSC scheme and ISMC scheme, for the attitude control problem with model uncertainties, external disturbances, and actuator saturations.Figure 9
Control surface deflection responses.


### 4.2.2. Comparison between the ATMC and TSMC Methods

The ATMC applies an adaptive mechanism to relax the bounded lumped disturbance information of the basic TSMC. Figure4 shows that the ATMC scheme can stabilize the attitude angles and angular rates in the finite time. For the TSMC, when the system lacks the upper bound information, a larger switching gain will be chosen that may lead to large chattering on the control surface. For the ATMC, the control chattering problem is effectively attenuated due to the adaptive mechanism. As shown in Figure 5, the predicted value of the gain kd+kT can converge to the constant 0.7 in a very short time. Therefore, the ATSMC scheme achieves a good performance even though the system uncertainty bounds are unknown in advance.


### 4.2.3. Comparison between the DO-TMC and TSMC Methods

The DO-TMC applies a nonlinear disturbance observer to relax the restrictions of lumped disturbance and attenuate chattering phenomenon of the basic TSMC. For the DO-TMC, the HFV attitude system smoothly achieves stabilization with a settling time less than 1.5 s and a high accuracy, as shown in Figure6. By selecting the appropriate values of the disturbance observer parameters, it can be seen from Figure 7 that the observer error ωe can converge to a very small interval, i.e., ωei≤2×10−3. The estimated information of the observer is transmitted to the controller to adjust the output of the actuator in real time, so that each control surface can deflect in a small range. Figure 8 shows the estimated response curve and estimation error curve under the proposed disturbance observer. Figure 8 exhibits that the proposed observer has good performance on the estimated slow time varying, fast time varying, and periodic disturbance. Above all, the DO-TSMC scheme can significantly achieve a satisfied performance without any prior knowledge of the compound uncertainties of the HFV control system.


## 4.2.1. Comparison between the TSMC, BSC, and ISMC Methods

The proposed TSMC method as a finite-time sliding mode control method is compared with the traditional BSC and ISMC methods for the NSV system. Viewing from Figures3 and 9, the attitude angle outputs cannot satisfactorily track the desired commands with large tracking errors and even cannot be stable by using the BSC method. Although the ISMC can make the system outputs track the desired commands, it needs a longer settling time than the TSMC method. As shown in Figure 9, affected by the disturbance, all control surfaces of the BSC system are in a large range of changes, which will cause a lot of energy loss. Due to the limited output of the actuator, it can be seen from the response of u3 that the right body flaps have been at the minimum deflection angle (−15deg) for about 0.2 seconds. After the adjustment of the TSMC controller, the output of the actuator has been in a reasonable small range. As shown in Figure 9, the left body flap u4 of BSC system is always at the minimum deflection angle (−15deg) at 0.25-0.4 second, resulting in actuator saturation. In all, the TSMC scheme obviously has the best performance than the BSC scheme and ISMC scheme, for the attitude control problem with model uncertainties, external disturbances, and actuator saturations.Figure 9
Control surface deflection responses.


## 4.2.2. Comparison between the ATMC and TSMC Methods

The ATMC applies an adaptive mechanism to relax the bounded lumped disturbance information of the basic TSMC. Figure4 shows that the ATMC scheme can stabilize the attitude angles and angular rates in the finite time. For the TSMC, when the system lacks the upper bound information, a larger switching gain will be chosen that may lead to large chattering on the control surface. For the ATMC, the control chattering problem is effectively attenuated due to the adaptive mechanism. As shown in Figure 5, the predicted value of the gain kd+kT can converge to the constant 0.7 in a very short time. Therefore, the ATSMC scheme achieves a good performance even though the system uncertainty bounds are unknown in advance.


## 4.2.3. Comparison between the DO-TMC and TSMC Methods

The DO-TMC applies a nonlinear disturbance observer to relax the restrictions of lumped disturbance and attenuate chattering phenomenon of the basic TSMC. For the DO-TMC, the HFV attitude system smoothly achieves stabilization with a settling time less than 1.5 s and a high accuracy, as shown in Figure6. By selecting the appropriate values of the disturbance observer parameters, it can be seen from Figure 7 that the observer error ωe can converge to a very small interval, i.e., ωei≤2×10−3. The estimated information of the observer is transmitted to the controller to adjust the output of the actuator in real time, so that each control surface can deflect in a small range. Figure 8 shows the estimated response curve and estimation error curve under the proposed disturbance observer. Figure 8 exhibits that the proposed observer has good performance on the estimated slow time varying, fast time varying, and periodic disturbance. Above all, the DO-TSMC scheme can significantly achieve a satisfied performance without any prior knowledge of the compound uncertainties of the HFV control system.


## 5. Conclusion

In this study, three finite-time attitude tracking control schemes (TSMC, ATSMC, and DO-TSMC) have been designed for the attitude control problem of nonlinear HFV with model uncertainties, external disturbances, and actuator saturations. The ATSMC scheme is based on combination of the TSMC and an adaptation law, while the DO-TSMC scheme is based on the combination of the TSMC and a nonlinear disturbance observer. Meanwhile, the stability of the closed-loop attitude system is analyzed using the Lyapunov function theory. The simulation results of a nonlinear X-33 HFV model show that the TSMC scheme has a better performance than the traditional BSC and ISMC schemes, the ATSMC scheme achieves a satisfied performance in case of unknown bounds of the compound uncertainties, and the DO-TSMC scheme improves the robustness and disturbance rejection performance of the attitude control system.This work considers the uncertainties, disturbances, and actuator saturations of the HFV system, which will contribute to the practical applications and better control performance. However, the actuator faults such as stuck or loss of effectiveness are not considered, which will be one of the subjects for future research.



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*Source: 5419377-2022-11-23.xml*