# Finite-Time Reentry Attitude Control Using Time-Varying Sliding Mode and Disturbance Observer

Authors: Xuzhong Wu; Shengjing Tang; Jie Guo; Yao Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper presents the finite-time attitude control problem for reentry vehicle with redundant actuators in consideration of planet uncertainties and external disturbances. Firstly, feedback linearization technique is used to cancel the nonlinearities of equations of motion to construct a basic mode for attitude controller. Secondly, two kinds of time-varying sliding mode control methods with disturbance observer are integrated with the basic mode in order to enhance the control performance and system robustness. One method is designed based on boundary layer technique and the other is a novel second-order sliding model control method. The finite-time stability analyses of both resultant closed-loop systems are carried out. Furthermore, after attitude controller produces the torque commands, an optimization control allocation approach is introduced to allocate them into aerodynamic surface deflections and on-off reaction control system thrusts. Finally, the numerical simulation results demonstrate that both of the time-varying sliding mode control methods are robust to uncertainties and disturbances without chattering phenomenon. Moreover, the proposed second-order sliding mode control method possesses better control accuracy.


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## Body

## 1. Introduction

Covering from outer space into earth’s atmosphere, reentry flight is a critical phase of the operation for reentry vehicles (RVs) [1]. Since the flight conditions change rapidly in the reentry phase, reentry attitude control is always in face of wide range of planet uncertainties and external disturbances. On the other hand, aerodynamic surfaces come into the eyes of engineers firstly with their advantage of saving energy. However, the density of atmosphere can be so low in beginning of reentry flight that the desired control torque may be unachievable with the employment of aerodynamic surfaces alone because of poor aerodynamic maneuverability. As a result, RVs have to rely on reaction control system (RCS) jets in addition to aerodynamic surfaces. In this case, control allocation among redundant actuators becomes necessary, which further raises the difficulties in attitude control design. Meanwhile, a robust attitude control system for RVs with redundant actuators is desirable.The conventional attitude control method for RVs is gain scheduling (GS) [2, 3]. This method linearizes the system with a set of trimmed points, designs individual gains at each point, and then interpolates those gains online with respect to system parameters such as dynamic pressure or Mach number. Nevertheless, the conventional GS involves the lack of guaranteed global robustness and stability [4]. The reentry flight conditions change rapidly, which makes this method impractical [5]. Moreover, the point designs of gain scheduling are manpower intensive and highly time consuming [6]. To conclude, GS is weak at performing linearity.Compared to GS, feedback linearization (FBL) [7–10] can exactly cancel the model nonlinearities and replace undesirable dynamics with desirable dynamic using nonlinear coordinate transformation. However, FBL relies on the knowledge of the exact model dynamics, which severely influences FBL’s practicality because uncertainties and disturbances exist inevitably. To improve the flight control performance systematically on the basis of FBL, Rahideh et al. [7] incorporated neural network (NN) based compensation in the FBL design; Van Soest et al. [8] combined FBL with constrained linear model predictive control (MPC) method; Xu et al. [9] utilized the combination of FBL and adaptive sliding mode control (SMC) method.Among the various upgraded nonlinear control methods, SMC outstands with many advantages, such as simplicity of implementation, fast dynamic response, good transient behavior, exponential stability, insensitivity to parameter variations, and robustness to plant uncertainties and external disturbances [11–14]. Therefore, SMC has been successfully applied to a variety of complex engineering systems [15]. Barambones Caramazana et al. [14] develop a sliding mode position control incorporating a flux estimator for high-performance real-time applications of induction motors. Wu et al. [15] investigate the key problems of SMC of Markovian jump singular time-delay systems. Shtessel et al. [16–19] studied the application of SMC method to reusable launch vehicle (RLV) in launch and reentry mode, and a multiple-time-scale SMC strategy is proposed in [18] for RLV in ascent phase.Generally, SMC design consists of two steps [20, 21]: (1) select a sliding surface as a function of the system states so that the system trajectories along the surface meet the desired performance, such as stability and tracking capability; (2) design a suitable control law to drive the states onto the predefined sliding surface in finite time. When it comes to the design of conventional SMC, there are two major problems concerned. One is its unguaranteed global robustness and the other is chattering phenomenon. The conventional sliding surfaces [16–18] employ linear function of tracking errors, which results in the fact that the transient dynamics of SMC consists of reaching phase and sliding phase. However, the SMC method can only ensure the robustness against planet uncertainties and external disturbances in sliding phase. Therefore, the conventional sliding surfaces do not possess the property of global robustness. Several studies are dedicated to global robustness of SMC. Sun et al. [22] introduced an integral sliding mode control (ISMC) method to solve the longitudinal control problem of air-breathing hypersonic vehicle (AHV). Shtessel et al. [19] proposed a two-loop controller that utilized a time-varying sliding mode control (TVSMC) method to achieve fault tolerance for RLV attitude control. With the elimination of reaching phase, both ISMC and TVSMC can keep the system states on the sliding surface from the initial time, so that global robustness against planet uncertainties and external disturbances is guaranteed.As to the chattering phenomenon, it is assumed that the control can be switched from one structure to another infinitely fast in the design of SMC [23]. However, it is impossible to achieve high-speed switching control because of the inevitable switching delay computation and the limitation of the physical actuators. The existence of time delay introduces instability, oscillation, and poor performance [24]. High control gains of SMC lead to high frequency oscillations known as chattering phenomenon. This harmful phenomenon may erode the performance to gain robustness, decrease the control accuracy, and damage the actuators [25]. There are essentially two ways to alleviate the chattering phenomenon [23]: one way is boundary layer method [17–19, 26] and the other is higher order sliding mode control (HOSMC). The boundary layer method replaces the sign function (discontinuous control) with smooth approximations, such as high-gain saturation function or sigmoid function. Nevertheless, this method no longer drives the system state to the origin and cannot guarantee the robustness and accuracy within the boundary layer [23]. HOSMC was proposed by Levant [27]. Instead of influencing the first order time derivative, the discontinuous control acts on sliding variable’s higher order derivative. As a special case of HOSMC, second-order sliding mode control (SOSMC) is the most popular approach in engineering. There are many kinds of SOSMC, such as twisting algorithm [28], super-twisting algorithm [29], suboptimal algorithm [30, 31], and prescribed convergence law algorithm [28].Disturbance observer (DO) is an effective way to enhance system robustness. The disturbance estimation is used for compensation. DO was first proposed by Ohishi et al. [32]. Hall and Shtessel [33] combined SMC and sliding mode disturbance observer (SMDO), which estimates the bounded uncertainties and disturbances effectively to improve RLV attitude control. Shtessel et al. [34] proposed a homogeneous DO based on the standard robust exact differentiator to solve the missile guidance problem.Inspired by previous work, this paper proposes two TVSMC methods to solve the finite-time attitude control problem by incorporating the disturbance observer. One is BTVSMC/DO which is the abbreviation for boundary layer method based time-varying sliding mode controller with disturbance observer, and the other is SOTVSMC/DO which means the second-order time-varying sliding mode controller with disturbance observer. With the same dedication to systematically enhance robustness and suppress control chattering, the two methods adopt different ways to alleviate chattering. The former is designed based on boundary layer technique, and the latter utilizes a novel SOSMC. The main contributions of this paper are summarized as follows.(1)
This paper incorporates a novel reaching law based on SOSMC with the time-varying sliding function. In order to enhance the robustness of the method, a DO based on the standard robust exact differentiator is employed to estimate the system’s uncertainties and disturbances in finite time. In addition, the finite-time convergence of time-varying sliding function for resulted method is proved via Lyapunov theory, and consequently the asymptotical stability of the closed-loop nonlinear system is proved according to the definition of the time-varying sliding function.(2)
Since RVs deploy both aerodynamic surfaces and RCS jets, this paper introduces a control allocation approach to assign control responsibility amongst redundant actuators. The nonlinear programming problem is established and solved by optimization method, and the pulse-width-pulse-frequency (PWPF) is employed to modulate the on-off thrusters.(3)
The proposed control methods are applied to finite-time attitude control problem for RVs. Numerical simulation results confirm the validity and superior performance of the proposed control methods by comparing them with other conventional control methods. The comparison between boundary layer method and SOSMC is also presented.The major contents of the following part in this paper are as follows. Section2 describes the rotational equations of motion and formulates problems of attitude controller and control allocation. In Section 3, feedback linearization technique is employed to the equations of motion. Section 4 presents two TVSMC methods as well as the corresponding stability analysis. A control allocation method is introduced in Section 5. In Section 6, the performances of proposed control methods are assessed by numerical tests. Finally, Section 7 summarizes and lists the conclusions.


## 2. Preliminary

### 2.1. The Rotational Equations of Motion

Reentry guidance is concerned with steering the vehicle from entry interface (EI) to the designated target point in prescribed condition while satisfying necessary path constraints such as heating rate constraint, aerodynamic load constraint, and dynamic pressure constraint [35, 36]. The steering commands are defined in terms of angle of attack (AOA) αc, sideslip angle βc, and bank angle σc. Furthermore, to prevent excessive heat buildup, βc is kept around zero under the application of back-to-turn (BTT) control policy [37]. The subsequent reentry control system tracks these three attitude commands. And the objective of the reentry control system is to determine the actuator command vector δ so that the reentry vehicle can follow the attitude commands that are specified by guidance system.The motion of reentry vehicle can be divided into translational motion and rotational motion. Since the focus of this paper is about control system, the translational equations of motion utilized in guidance system are not presented. The reentry dynamics are governed by a group of nonlinear differential equations [38]. The kinematic equations of reentry vehicle are defined as [37](1)dαdt=-pcos⁡αtanβ+q-rsinαtanβ+sinσcos⁡βθ˙ψ˙cos⁡γ-ϕ˙sinψsinγ+θ˙+ωecos⁡ϕcos⁡ψsinγ-sinϕcos⁡γψ-cos⁡σcos⁡βγ˙-ϕ˙cos⁡ψ-θ˙+ωecos⁡ϕsinψ,dβdt=psinα-rcos⁡α+sinσγ˙-ϕ˙cos⁡ψ-θ˙+ωecos⁡ϕsinψ+cos⁡σθ˙ψ˙cos⁡γ-ϕ˙sinψsinγ+θ˙+ωecos⁡ϕcos⁡ψsinγ-sinϕcos⁡γψ,dσdt=-pcos⁡αcos⁡β-qsinβ-rsinαcos⁡β+α˙sinβ-ψ˙sinγ-ϕ˙sinψcos⁡γ+θ˙+ωecos⁡ϕcos⁡ψcos⁡γ-sinϕsinγ,where α, β, and σ are AOA, sideslip angle, and bank angle, respectively. p, q, and r are the rates of roll, pitch, and yaw, respectively. γ denotes flight path angle and ψ denotes heading angle. θ and ϕ are longitude and latitude of reentry vehicle. ωe is the angular rate of Earth rotation.In order to simplify the online calculation, this paper obtains the kinetics of reentry vehicle under the following assumption.Assumption 1.
(
1
) The reentry vehicle is a rigid body; the terms impacted by elastic effects are not considered. (2) The reentry vehicle has a longitudinal symmetry plane, which means the products of inertia Ixy=Iyz=0. (3) The vehicle is unpowered during reentry.Hence, the kinetics of reentry vehicle can be expressed as [37](2)dpdt=IzMl+IzxMn+IzxIz+Ix-Iypq+IyIz-Iz2-Izx2qrIxIz-Izx2,dqdt=Mm+Iz-Ixpr+Izxr2-p2Iy,drdt=IzxMl+IxMn+-IxIy+Ix2+Izx2pq+Izx-Iz-Ix+IyqrIxIz-Izx2,where Ml, Mm, and Mn are three control torques defined in the body frame roll pitch and yaw, respectively. Ix, Iy, and Iz denote the moments of inertia, and Izx denotes the product of inertia.The control-oriented model can be developed for control design based on (1) and (2). Since the rotational motions are much faster than translational motions and the motion of Earth, the translational terms and angular velocity of earth can be neglected; that is, γ˙=ψ˙=θ˙=ϕ˙=0, ωe=0. Therefore, the rotational equations of motions (1) and (2) can be further simplified as follows:(3)Ω˙=Rω+Δfω˙=-I-1ω×Iω+I-1Mc+Δd,where Ω=α,β,σT is the attitude angle vector, ω=p,q,rT is the attitude angular rate vector, and Mc=Ml,Mm,MnT is the command control torque vector. R∈R3×3 is the coordinate-transformation matrix, Δf=Δf1,Δf2,Δf3,T denotes the unknown bounded uncertainties caused by the model reduction, ω×∈R3×3 stands for the skew-symmetric matrix operator on vector ω,I∈R3×3 denotes the symmetric positive definite inertia matrix of reentry vehicle, and Δd∈R3 denotes the bounded uncertain term. R, ω×, I, and Δd are given by(4)R=-cos⁡αtanβ1-sinαtanβsinα0-cos⁡α-cos⁡αcos⁡β-sinβ-sinαcos⁡β,ω×=0-rqr0-pqp0,I=Ix0-Izx0Iy0-Izx0Iz,(5)Δd=I-1-ΔIω˙-ω×ΔIω+ΔM,where ΔI∈R3×3 denotes unknown bounded inertia variations and ΔM∈R3 stands for the bounded external disturbance moment.


### 2.2. Problem Formulation

As shown in Figure1, the control problem for reentry vehicle with redundant actuators can be solved in two steps. They areFigure 1
Control architecture for reentry vehicle.(
1
) specifying the control torque vector Mc=Ml,Mm,MnT in equation set (3), which leads the output vector Ω to track the attitude command Ωc=αc,βc,σcT in a finite time:(6)limt>tFe=limt>tFΩ-Ωc=0,where e=Ω-Ωc is the tracking error;(
2
) designing a control allocation method fCA· that maps the command control torque vector Mc to actuator deflection commands δc [39]:(7)δc=fCAMc.The actual torque produced by control allocation may not exactly equal the torque command. Assume thatΔMca is the bounded disturbance caused by the process of control allocation, and the torque vector produced by actuators can be expressed as Mc+ΔMca. Hence, the bounded uncertain term of (5) can be rewritten as (8)Δd=I-1-ΔIω˙-ω×ΔIω+ΔM+ΔMca.


## 2.1. The Rotational Equations of Motion

Reentry guidance is concerned with steering the vehicle from entry interface (EI) to the designated target point in prescribed condition while satisfying necessary path constraints such as heating rate constraint, aerodynamic load constraint, and dynamic pressure constraint [35, 36]. The steering commands are defined in terms of angle of attack (AOA) αc, sideslip angle βc, and bank angle σc. Furthermore, to prevent excessive heat buildup, βc is kept around zero under the application of back-to-turn (BTT) control policy [37]. The subsequent reentry control system tracks these three attitude commands. And the objective of the reentry control system is to determine the actuator command vector δ so that the reentry vehicle can follow the attitude commands that are specified by guidance system.The motion of reentry vehicle can be divided into translational motion and rotational motion. Since the focus of this paper is about control system, the translational equations of motion utilized in guidance system are not presented. The reentry dynamics are governed by a group of nonlinear differential equations [38]. The kinematic equations of reentry vehicle are defined as [37](1)dαdt=-pcos⁡αtanβ+q-rsinαtanβ+sinσcos⁡βθ˙ψ˙cos⁡γ-ϕ˙sinψsinγ+θ˙+ωecos⁡ϕcos⁡ψsinγ-sinϕcos⁡γψ-cos⁡σcos⁡βγ˙-ϕ˙cos⁡ψ-θ˙+ωecos⁡ϕsinψ,dβdt=psinα-rcos⁡α+sinσγ˙-ϕ˙cos⁡ψ-θ˙+ωecos⁡ϕsinψ+cos⁡σθ˙ψ˙cos⁡γ-ϕ˙sinψsinγ+θ˙+ωecos⁡ϕcos⁡ψsinγ-sinϕcos⁡γψ,dσdt=-pcos⁡αcos⁡β-qsinβ-rsinαcos⁡β+α˙sinβ-ψ˙sinγ-ϕ˙sinψcos⁡γ+θ˙+ωecos⁡ϕcos⁡ψcos⁡γ-sinϕsinγ,where α, β, and σ are AOA, sideslip angle, and bank angle, respectively. p, q, and r are the rates of roll, pitch, and yaw, respectively. γ denotes flight path angle and ψ denotes heading angle. θ and ϕ are longitude and latitude of reentry vehicle. ωe is the angular rate of Earth rotation.In order to simplify the online calculation, this paper obtains the kinetics of reentry vehicle under the following assumption.Assumption 1.
(
1
) The reentry vehicle is a rigid body; the terms impacted by elastic effects are not considered. (2) The reentry vehicle has a longitudinal symmetry plane, which means the products of inertia Ixy=Iyz=0. (3) The vehicle is unpowered during reentry.Hence, the kinetics of reentry vehicle can be expressed as [37](2)dpdt=IzMl+IzxMn+IzxIz+Ix-Iypq+IyIz-Iz2-Izx2qrIxIz-Izx2,dqdt=Mm+Iz-Ixpr+Izxr2-p2Iy,drdt=IzxMl+IxMn+-IxIy+Ix2+Izx2pq+Izx-Iz-Ix+IyqrIxIz-Izx2,where Ml, Mm, and Mn are three control torques defined in the body frame roll pitch and yaw, respectively. Ix, Iy, and Iz denote the moments of inertia, and Izx denotes the product of inertia.The control-oriented model can be developed for control design based on (1) and (2). Since the rotational motions are much faster than translational motions and the motion of Earth, the translational terms and angular velocity of earth can be neglected; that is, γ˙=ψ˙=θ˙=ϕ˙=0, ωe=0. Therefore, the rotational equations of motions (1) and (2) can be further simplified as follows:(3)Ω˙=Rω+Δfω˙=-I-1ω×Iω+I-1Mc+Δd,where Ω=α,β,σT is the attitude angle vector, ω=p,q,rT is the attitude angular rate vector, and Mc=Ml,Mm,MnT is the command control torque vector. R∈R3×3 is the coordinate-transformation matrix, Δf=Δf1,Δf2,Δf3,T denotes the unknown bounded uncertainties caused by the model reduction, ω×∈R3×3 stands for the skew-symmetric matrix operator on vector ω,I∈R3×3 denotes the symmetric positive definite inertia matrix of reentry vehicle, and Δd∈R3 denotes the bounded uncertain term. R, ω×, I, and Δd are given by(4)R=-cos⁡αtanβ1-sinαtanβsinα0-cos⁡α-cos⁡αcos⁡β-sinβ-sinαcos⁡β,ω×=0-rqr0-pqp0,I=Ix0-Izx0Iy0-Izx0Iz,(5)Δd=I-1-ΔIω˙-ω×ΔIω+ΔM,where ΔI∈R3×3 denotes unknown bounded inertia variations and ΔM∈R3 stands for the bounded external disturbance moment.


## 2.2. Problem Formulation

As shown in Figure1, the control problem for reentry vehicle with redundant actuators can be solved in two steps. They areFigure 1
Control architecture for reentry vehicle.(
1
) specifying the control torque vector Mc=Ml,Mm,MnT in equation set (3), which leads the output vector Ω to track the attitude command Ωc=αc,βc,σcT in a finite time:(6)limt>tFe=limt>tFΩ-Ωc=0,where e=Ω-Ωc is the tracking error;(
2
) designing a control allocation method fCA· that maps the command control torque vector Mc to actuator deflection commands δc [39]:(7)δc=fCAMc.The actual torque produced by control allocation may not exactly equal the torque command. Assume thatΔMca is the bounded disturbance caused by the process of control allocation, and the torque vector produced by actuators can be expressed as Mc+ΔMca. Hence, the bounded uncertain term of (5) can be rewritten as (8)Δd=I-1-ΔIω˙-ω×ΔIω+ΔM+ΔMca.


## 3. Feedback Linearization

By the selection of control input as control torque vectorM and the output as attitude angle vector Ω, the nonlinear attitude equations (3) can be expressed as [40](9)x˙=fx+gxu+d,y=hx,where x=α,β,σ,p,q,rT is the state vector, y=α,β,σT is the output vector, u=Ml,Mm,MnT is the control vector, and d=ΔfT,ΔdTT stands for the system uncertain term. fx and gx can be obtained by (10) and (11), respectively:(10)fx=f1xf2xf3xf4xf5xf6x=-pcos⁡αtanβ+q-rsinαtanβpsinα-rcos⁡α-pcos⁡αcos⁡β-qsinβ-rsinαcos⁡βIzxIz+Ix-Iypq+IyIz-Iz2-Izx2qrIxIz-Izx2Iz-Ixpr+Izxr2-p2Iy-IxIy+Ix2+Izx2pq+Izx-Iz-Ix+IyqrIxIz-Izx2,(11)gx=g1xg2xg3x=000000000IzIxIz-Izx20IzxIxIz-Izx201Iy0IzxIxIz-Izx20IxIxIz-Izx2.The vector relative degree of system (9) is 2,2,2. After differentiating output vector y twice, the control input vector u appears:(12)y¨=Ku+B+RI-1Δd+FΔf,where K, B, and F are given by(13)K=Lg1Lfh1xLg2Lfh1xLg3Lfh1xLg1Lfh2xLg2Lfh2xLg3Lfh2xLg1Lfh3xLg2Lfh3xLg3Lfh3x,B=Lf2h1x,Lf2h2x,Lf2h3xT,F=psinαtanβ-rcos⁡αtanβ+1-pcos⁡αsec2β-rsinαsec2β0pcos⁡α+rcos⁡α10psinαcos⁡β-rcos⁡αcos⁡βpcos⁡αsinβ-qcos⁡β+rsinαsinβ1.According to (12), the total relative degree of system equals the order of the system. Furthermore, since the sideslip angle β≈0 during reentry,(14)det⁡K=cos⁡β-sinβtanβIxIz-Izx2≈1IxIz-Izx2≠0.Thus, the system (9) can be linearized completely without zero dynamics by using the following feedback control law:(15)u=K-1-B+v,where v=v1,v2,v3T is selected as the new control input in this paper.Define the bounded uncertainty termsΔv=RI-1Δd+FΔf as the lumped uncertainty, and substitute (15) into (12), and the basic model for attitude controller design can be obtained by(16)y¨=v+Δv.


## 4. Sliding Mode Attitude Controller Design

This section develops two TVSMC attitude controllers to solve the finite-time control problem by incorporating disturbance observer. The first controller is BTVSMC/DO and the second controller is SOTVSMC/DO. This part elaborates on the design of sliding surface and reaching law of the controllers. Moreover, the design of disturbance observer is presented, too.


### 4.1. Time-Varying Sliding Surface Design

The time-varying sliding surface is selected as [41](17)s=e˙+ce+ae-λt,where s=s1,s2,s3T, the tracking error vector e=y-yc=e1,e2,e3T,c=diagc1,c2,c3 is the sliding function gain matrix, and the element ci>0,i=1,2,3,λ∈R+,a=[a1,a2,a3]T is the coefficient vector to guarantee the existence of sliding mode from the beginning of motion. Hence, a is defined as (18)a=-e˙0-ce0.Lemma 2.
If the sliding modest=03×1 is satisfied, the system (9) is globally asymptotically stable.Proof.
According to (17), st=03×1 can be rewritten in scalar form:(19)e˙i+ciei+aie-λt=0,i=1,2,3.
Ifci≠λ, the differential equations can be solved as (20)eit=ciei0ci-λe-λt-λei0ci-λe-cit≤ciei0ci-λ-λei0ci-λe-ςt,where ς=min⁡ci,λ, and thus the system (9) is globally exponentially stable.
Ifci=λ, the differential equation can be solved as (21)eit=e-citcit+1ei0.Because limt→∞eit→0, the system (9) is globally asymptotically stable.
In conclusion, the asymptotic stability of the system (9) is guaranteed when sliding mode st=03×1 is satisfied. This completes the proof.Remark 3.
To simplify the selection procedure, the four parametersc1, c2, c3, and λ in (17) are set equal so that each of them is able to determine the sliding surface. As the parameters become larger, the rate of tracking error is faster and the control input is required to be larger. However, control input in real situation could not always be bigger as a faster convergence rate requires. As a result, a trade-off between control input and convergence rate is necessary, which can be achieved by trial-and-error method.


### 4.2. Disturbance Observer Design

The first order derivative of the sliding surfaces is (22)s˙=e¨+ce˙-λae-λt=-y¨c+y¨+ce˙-λae-λt=-b+v+Δv,where b=y¨c-ce˙+λae-λt.Hence, the control vectorv can be expressed as(23)v=b-Δv+s˙.The sliding variable dynamics (22) is sensitive to the unknown bounded term Δv. However, the detailed information of Δv in (23) is unavailable. To estimate the lumped uncertainty, the robust differentiator technique [34] is employed.Assumption 4.
s
1
,
s
2
,
s
3 are measured by Lebesgue-measurable noise bounded ε1>0, ε2>0, ε3>0, respectively. Furthermore, v,b are assumed to be bounded and Lebesgue measurable respectively, and the lumped uncertainty Δv is 2 times differentiable and bounded.
Considerz0=z01,z02,z03T, z1=z11,z12,z13T, and z2=z21,z22,z23T as the estimated values of state variables, and the observer can be expressed as [34](24)z˙0=v-b+ν0ν0=-λ0L1/3z01-s12/3000z02-s22/3000z03-s32/3×sgnz0-s+z1,z˙1=ν1,ν1=-λ1L1/2z11-ν011/2000z12-ν021/2000z13-ν031/2×sgnz1-ν0+z2,z˙2=-λ2Lsgnz2-ν1,where ν0=ν01,ν02,ν03T, ν0=ν01,ν02,ν03T. λ0, λ1, λ2 and L=diag(l1,l2,l3) are the parameters to be selected.Lemma 5 (see [34]).
Suppose Assumption4 is satisfied. DO (24) is finite-time stable. The following inequalities can be established in finite time:(25)z0i-si≤μ0iεi,z1i-Δvi≤μ1iεi2/3,z2i-Δv˙i≤μ2iεi1/3,hhhhhhhi=1,2,3,where μ0i,μ1i,μ2i,i=1,2,3, are positive constants.Remark 6.
The proof of Lemma5 is similar to the studies of Shtessel et al. [34] and is not presented in this paper. The parameters λ0,λ1,λ2 can be chosen recursively, and the simulation-checked set 8, 5, 3 is suitable for the observer design [34, 42].
In absence of measurement noise, the exact equalities can be established in a finite time:(26)z0=s,z1=Δv,z2=Δv˙.
After DO is constructed, the control vectorv can be modified as(27)v=b-z1+s˙.


### 4.3. Reaching Law Design

Before giving the reaching law design, three lemmas to be used are presented.Lemma 7 (see [43]).
Consider the system of differential equations:(28)x˙t=fxt,where x∈Rn, f:D0→Rn is continuous on an open neighborhood D0 containing the origin, f0=0.
Suppose there exists a continuous positive definite functionV:D0→R. In addition, there exist real numbers k>0, α∈0,1, and an open neighborhood of the origin D1⊆D0 satisfies(29)V˙+kVα≤0,x∈D1∖0.Then the origin is a finite-time stable equilibrium of system (28). The settling time t is depended on the initial value x0: (30)tx0≤Vx01-αk1-α.Furthermore, if D=D0=Rn, the origin is a globally finite-time stable equilibrium of system (28).Lemma 8 (see [44]).
Suppose there exists a continuous positive definite functionV:D0→R. In addition, there exist real numbers k,l>0, α∈0,1, and an open neighborhood of the origin D1⊆D0 satisfies (31)V˙+kVα+lV≤0,x∈D1∖0.Then the origin is a finite-time stable equilibrium of system (28). The settling time t is depended on the initial value x0: (32)tx0≤ln⁡1+l/kVx01-αk1-α.Furthermore, if D=D0=Rn, the origin is a globally finite-time stable equilibrium of system (28).Lemma 9 (see [45]).
Forxi∈R, i=1,…,n, 0<p≤1 is a real number, and the inequality holds:(33)x1+⋯+xnp≤x1p+⋯+xnp.
Consider the reaching law with saturation function:(34)s˙=-ηsat⁡s,where η=diagη1,η2,η3, ηi∈R+,i=1,2,3, sat⁡s=sat⁡s1,sat⁡s2,sat⁡s3T stands for the saturation function that is used to attenuate the chattering problem, and sat⁡si,i=1,2,3, is defined as(35)sat⁡si=hi-1sisi≤hisgn(si)si>hii=1,2,3,where h=diag(h1,h2,h3) and hi is the boundary layer thickness.
Substitute (34) into (27); the control algorithm of BTVSMC/DO can be expressed as(36)v=b-z1-ηsat⁡s.Theorem 10.
Based on Assumption4, the attitude control problem described in (9) can be solved by BTVSMC/DO (36). Furthermore, the attitude tracking error e is asymptotically stable if the exact estimate of Δv is available through the DO.Proof.
Consider the Lyapunov function candidate:(37)V1=12sTs.
According to (37) and (22), the time derivative of V1 is (38)V˙1=sTs˙=sT-b+v+Δv.
Substituting (36) into (38) gives(39)V˙1=sT-ηsat⁡s-z1+Δv.
According to Lemma5, DO (24) is finite-time stable; hence, we suppose there exists a moment t=tob, which satisfies z1=Δv, t≥tob.
Whent≥tob, (40)V˙1=sT-ηsat⁡(s)=∑i=13ηisisat⁡si.
In view of (35), consider the following two cases.(1)
Ifsi>hi,i=1,2,3, we can get ηisisat⁡(si)=ηisisgn(si)=ηisi.(2)
Ifsi≤hi,i=1,2,3, we can get ηisisat⁡(si)=ηisihi-1si=ηisi2hi-1≤ηisi.
Hence, it is obvious that(41)V˙1≤∑i=13ηisi≤-χV1,where χ=min⁡η1,η2,η3; according to Lemma 7, the trajectory of system will be driven into the related sliding surface s=03×1 in a finite time tr:(42)tr≤2V1tobχ+tob,where V1tob is the value of V1 at t=tob.
According to Lemma2, once the slide mode s=03×1 is established, the system (9) is globally asymptotically stable. This completes the proof.Generally, a thicker boundary layer (larger values ofhi,i=1,2,3) contributes to smaller chattering; however, the static error inside the boundary layer may be large. Since the boundary layer method may result in the erosion of robustness and precision, a novel second-order SMC is proposed in this paper.Consider the reaching law:(43)s˙=-k1sigm-1/mssgns-k2s+τ,τ˙=k3sigm-2/mssgns-k4s,where m>2, k1=diagk11,k12,k13, k2=diagk21,k22,k23, k3=diagk31,k32,k33, and k4=diagk41,k42,k43 with k1i,k2i,k3i,k4i∈R+,i=1,2,3. And sigms is defined as(44)sigms=sgns1s1m,sgns2s2m,sgns3s3mT.Substitute (43) into (27); the related control algorithm of SOTVSMC/DO is given by(45)v=b-z1-k1sigm-1/mssgns-k2s+∫k3sigm-2/mssgns-k4s.Remark 11.
In view of (43), when the system’s initial state is close to equilibrium point (si=0,i=1,2,3), the convergence speed mainly depends on the nonlinear terms. Otherwise, when the system’s initial state is far from equilibrium point, the convergence speed mainly depends on the linear terms. Hence, the system can hold a fast convergence speed whether the initial state is close to equilibrium point or not.Theorem 12.
Suppose that the following inequality holds:(46)m2k3ik4i-m3m-1k3i+4m2-4m+1k1i2k2i2>0,i=1,2,3.Based on Assumption 4, the attitude control problem described in (9) can be solved by the SOTVSMC/DO (45). Furthermore, the attitude tracking error e is asymptotically stable if the exact estimate of Δv is available through the DO.Proof.
The Lyapunov function can be expressed as(47)V2=∑i=13V2i,i=1,2,3.
According to [46], V2i is defined as(48)V2i=12-k1isim-1/msgnsi-k2isi+τi2+k3imm-1si2m-1/m+k4isi2+12τi2.
Construct the vectorξi=sim-1/msgnsi,si,τiT; then V2i and the derivative V˙2i can be expressed as(49)V2i=ξiTΛiξi,V˙2i=-si-1/mξiTΓ1iξi+ξiTΓ2iξi,where (50)Λi=122m/m-1k3i+k1i2k1ik2i-k1ik1ik2i2k4i+k2i2-k2i-k1i-k2i2,Γ1i=k1im×mk3i+m-1k1i20-m-1k1i0mk4i+3m-1k2i2-2m-1k2i-m-1k1i-2m-1k2im-1,Γ2i=k2ik3i+3m-2/mk1i2000k4i+k2i2-k2i0-k2i1.
V
2
i is continuous positive function:(51)λmin⁡Λiξi2≤V2i≤λmax⁡Λiξi2,where · denotes the Euclidean norm. λmin⁡· and λmax⁡· denote the minimum and maximum eigenvalues of the related matrix, respectively:(52)sim-1/m≤si2m-1/m+si2+τi2=ξi≤V2iλmin⁡Λisi1/m≤V2iλmin⁡Λi1/2m-1.
According to (46), Γ1i and Γ2i are positive definite matrixes; hence, V˙2i is negative:(53)V˙2i≤-si-1/mλmin⁡Γ1iξi2-λmin⁡Γ2iξi2≤-λmin⁡ΛiV2i1/2(m-1)×λmin⁡Γ1iV2iλmax⁡Λi-λmin⁡Γ2iV2iλmax⁡Λi=-χ1iV2i(2m-3)/(2m-2)-χ2iV2i,where χ1i=λmin⁡Λi1/2(m-1)λmin⁡Γ1i/λmax⁡Λi, χ2i=λmin⁡Γ2i/λmax⁡Λi,(54)V2=∑i=13-χ1iV2i(2m-3)/(2m-2)-χ2iV2i≤-χ1∑i=13V2i(2m-3)/(2m-2)-χ2V2,where χ1=max⁡χ11,χ12,χ13, χ2=max⁡χ21,χ22,χ23.
Sincem>2, 0<(2m-3)/(2m-2)<1. According to Lemma 9(55)V2+χ1V2(2m-3)/(2m-2)+χ2V2≤0.
Similar to the proof of Theorem10, according to Lemma 8, the trajectory of system will be driven into the related sliding surface s=03×1 in a finite time tr:(56)tr≤2m-1ln⁡1+χ2/χ1V2tob1/2m-1χ1+tob,where tob is the moment which satisfies z1=Δv, t≥tob. V2tob is the value of V2 at t=tob.
According to Lemma2, once the slide mode s=03×1 is established, the system (9) is globally asymptotically stable. This completes the proof.


## 4.1. Time-Varying Sliding Surface Design

The time-varying sliding surface is selected as [41](17)s=e˙+ce+ae-λt,where s=s1,s2,s3T, the tracking error vector e=y-yc=e1,e2,e3T,c=diagc1,c2,c3 is the sliding function gain matrix, and the element ci>0,i=1,2,3,λ∈R+,a=[a1,a2,a3]T is the coefficient vector to guarantee the existence of sliding mode from the beginning of motion. Hence, a is defined as (18)a=-e˙0-ce0.Lemma 2.
If the sliding modest=03×1 is satisfied, the system (9) is globally asymptotically stable.Proof.
According to (17), st=03×1 can be rewritten in scalar form:(19)e˙i+ciei+aie-λt=0,i=1,2,3.
Ifci≠λ, the differential equations can be solved as (20)eit=ciei0ci-λe-λt-λei0ci-λe-cit≤ciei0ci-λ-λei0ci-λe-ςt,where ς=min⁡ci,λ, and thus the system (9) is globally exponentially stable.
Ifci=λ, the differential equation can be solved as (21)eit=e-citcit+1ei0.Because limt→∞eit→0, the system (9) is globally asymptotically stable.
In conclusion, the asymptotic stability of the system (9) is guaranteed when sliding mode st=03×1 is satisfied. This completes the proof.Remark 3.
To simplify the selection procedure, the four parametersc1, c2, c3, and λ in (17) are set equal so that each of them is able to determine the sliding surface. As the parameters become larger, the rate of tracking error is faster and the control input is required to be larger. However, control input in real situation could not always be bigger as a faster convergence rate requires. As a result, a trade-off between control input and convergence rate is necessary, which can be achieved by trial-and-error method.


## 4.2. Disturbance Observer Design

The first order derivative of the sliding surfaces is (22)s˙=e¨+ce˙-λae-λt=-y¨c+y¨+ce˙-λae-λt=-b+v+Δv,where b=y¨c-ce˙+λae-λt.Hence, the control vectorv can be expressed as(23)v=b-Δv+s˙.The sliding variable dynamics (22) is sensitive to the unknown bounded term Δv. However, the detailed information of Δv in (23) is unavailable. To estimate the lumped uncertainty, the robust differentiator technique [34] is employed.Assumption 4.
s
1
,
s
2
,
s
3 are measured by Lebesgue-measurable noise bounded ε1>0, ε2>0, ε3>0, respectively. Furthermore, v,b are assumed to be bounded and Lebesgue measurable respectively, and the lumped uncertainty Δv is 2 times differentiable and bounded.
Considerz0=z01,z02,z03T, z1=z11,z12,z13T, and z2=z21,z22,z23T as the estimated values of state variables, and the observer can be expressed as [34](24)z˙0=v-b+ν0ν0=-λ0L1/3z01-s12/3000z02-s22/3000z03-s32/3×sgnz0-s+z1,z˙1=ν1,ν1=-λ1L1/2z11-ν011/2000z12-ν021/2000z13-ν031/2×sgnz1-ν0+z2,z˙2=-λ2Lsgnz2-ν1,where ν0=ν01,ν02,ν03T, ν0=ν01,ν02,ν03T. λ0, λ1, λ2 and L=diag(l1,l2,l3) are the parameters to be selected.Lemma 5 (see [34]).
Suppose Assumption4 is satisfied. DO (24) is finite-time stable. The following inequalities can be established in finite time:(25)z0i-si≤μ0iεi,z1i-Δvi≤μ1iεi2/3,z2i-Δv˙i≤μ2iεi1/3,hhhhhhhi=1,2,3,where μ0i,μ1i,μ2i,i=1,2,3, are positive constants.Remark 6.
The proof of Lemma5 is similar to the studies of Shtessel et al. [34] and is not presented in this paper. The parameters λ0,λ1,λ2 can be chosen recursively, and the simulation-checked set 8, 5, 3 is suitable for the observer design [34, 42].
In absence of measurement noise, the exact equalities can be established in a finite time:(26)z0=s,z1=Δv,z2=Δv˙.
After DO is constructed, the control vectorv can be modified as(27)v=b-z1+s˙.


## 4.3. Reaching Law Design

Before giving the reaching law design, three lemmas to be used are presented.Lemma 7 (see [43]).
Consider the system of differential equations:(28)x˙t=fxt,where x∈Rn, f:D0→Rn is continuous on an open neighborhood D0 containing the origin, f0=0.
Suppose there exists a continuous positive definite functionV:D0→R. In addition, there exist real numbers k>0, α∈0,1, and an open neighborhood of the origin D1⊆D0 satisfies(29)V˙+kVα≤0,x∈D1∖0.Then the origin is a finite-time stable equilibrium of system (28). The settling time t is depended on the initial value x0: (30)tx0≤Vx01-αk1-α.Furthermore, if D=D0=Rn, the origin is a globally finite-time stable equilibrium of system (28).Lemma 8 (see [44]).
Suppose there exists a continuous positive definite functionV:D0→R. In addition, there exist real numbers k,l>0, α∈0,1, and an open neighborhood of the origin D1⊆D0 satisfies (31)V˙+kVα+lV≤0,x∈D1∖0.Then the origin is a finite-time stable equilibrium of system (28). The settling time t is depended on the initial value x0: (32)tx0≤ln⁡1+l/kVx01-αk1-α.Furthermore, if D=D0=Rn, the origin is a globally finite-time stable equilibrium of system (28).Lemma 9 (see [45]).
Forxi∈R, i=1,…,n, 0<p≤1 is a real number, and the inequality holds:(33)x1+⋯+xnp≤x1p+⋯+xnp.
Consider the reaching law with saturation function:(34)s˙=-ηsat⁡s,where η=diagη1,η2,η3, ηi∈R+,i=1,2,3, sat⁡s=sat⁡s1,sat⁡s2,sat⁡s3T stands for the saturation function that is used to attenuate the chattering problem, and sat⁡si,i=1,2,3, is defined as(35)sat⁡si=hi-1sisi≤hisgn(si)si>hii=1,2,3,where h=diag(h1,h2,h3) and hi is the boundary layer thickness.
Substitute (34) into (27); the control algorithm of BTVSMC/DO can be expressed as(36)v=b-z1-ηsat⁡s.Theorem 10.
Based on Assumption4, the attitude control problem described in (9) can be solved by BTVSMC/DO (36). Furthermore, the attitude tracking error e is asymptotically stable if the exact estimate of Δv is available through the DO.Proof.
Consider the Lyapunov function candidate:(37)V1=12sTs.
According to (37) and (22), the time derivative of V1 is (38)V˙1=sTs˙=sT-b+v+Δv.
Substituting (36) into (38) gives(39)V˙1=sT-ηsat⁡s-z1+Δv.
According to Lemma5, DO (24) is finite-time stable; hence, we suppose there exists a moment t=tob, which satisfies z1=Δv, t≥tob.
Whent≥tob, (40)V˙1=sT-ηsat⁡(s)=∑i=13ηisisat⁡si.
In view of (35), consider the following two cases.(1)
Ifsi>hi,i=1,2,3, we can get ηisisat⁡(si)=ηisisgn(si)=ηisi.(2)
Ifsi≤hi,i=1,2,3, we can get ηisisat⁡(si)=ηisihi-1si=ηisi2hi-1≤ηisi.
Hence, it is obvious that(41)V˙1≤∑i=13ηisi≤-χV1,where χ=min⁡η1,η2,η3; according to Lemma 7, the trajectory of system will be driven into the related sliding surface s=03×1 in a finite time tr:(42)tr≤2V1tobχ+tob,where V1tob is the value of V1 at t=tob.
According to Lemma2, once the slide mode s=03×1 is established, the system (9) is globally asymptotically stable. This completes the proof.Generally, a thicker boundary layer (larger values ofhi,i=1,2,3) contributes to smaller chattering; however, the static error inside the boundary layer may be large. Since the boundary layer method may result in the erosion of robustness and precision, a novel second-order SMC is proposed in this paper.Consider the reaching law:(43)s˙=-k1sigm-1/mssgns-k2s+τ,τ˙=k3sigm-2/mssgns-k4s,where m>2, k1=diagk11,k12,k13, k2=diagk21,k22,k23, k3=diagk31,k32,k33, and k4=diagk41,k42,k43 with k1i,k2i,k3i,k4i∈R+,i=1,2,3. And sigms is defined as(44)sigms=sgns1s1m,sgns2s2m,sgns3s3mT.Substitute (43) into (27); the related control algorithm of SOTVSMC/DO is given by(45)v=b-z1-k1sigm-1/mssgns-k2s+∫k3sigm-2/mssgns-k4s.Remark 11.
In view of (43), when the system’s initial state is close to equilibrium point (si=0,i=1,2,3), the convergence speed mainly depends on the nonlinear terms. Otherwise, when the system’s initial state is far from equilibrium point, the convergence speed mainly depends on the linear terms. Hence, the system can hold a fast convergence speed whether the initial state is close to equilibrium point or not.Theorem 12.
Suppose that the following inequality holds:(46)m2k3ik4i-m3m-1k3i+4m2-4m+1k1i2k2i2>0,i=1,2,3.Based on Assumption 4, the attitude control problem described in (9) can be solved by the SOTVSMC/DO (45). Furthermore, the attitude tracking error e is asymptotically stable if the exact estimate of Δv is available through the DO.Proof.
The Lyapunov function can be expressed as(47)V2=∑i=13V2i,i=1,2,3.
According to [46], V2i is defined as(48)V2i=12-k1isim-1/msgnsi-k2isi+τi2+k3imm-1si2m-1/m+k4isi2+12τi2.
Construct the vectorξi=sim-1/msgnsi,si,τiT; then V2i and the derivative V˙2i can be expressed as(49)V2i=ξiTΛiξi,V˙2i=-si-1/mξiTΓ1iξi+ξiTΓ2iξi,where (50)Λi=122m/m-1k3i+k1i2k1ik2i-k1ik1ik2i2k4i+k2i2-k2i-k1i-k2i2,Γ1i=k1im×mk3i+m-1k1i20-m-1k1i0mk4i+3m-1k2i2-2m-1k2i-m-1k1i-2m-1k2im-1,Γ2i=k2ik3i+3m-2/mk1i2000k4i+k2i2-k2i0-k2i1.
V
2
i is continuous positive function:(51)λmin⁡Λiξi2≤V2i≤λmax⁡Λiξi2,where · denotes the Euclidean norm. λmin⁡· and λmax⁡· denote the minimum and maximum eigenvalues of the related matrix, respectively:(52)sim-1/m≤si2m-1/m+si2+τi2=ξi≤V2iλmin⁡Λisi1/m≤V2iλmin⁡Λi1/2m-1.
According to (46), Γ1i and Γ2i are positive definite matrixes; hence, V˙2i is negative:(53)V˙2i≤-si-1/mλmin⁡Γ1iξi2-λmin⁡Γ2iξi2≤-λmin⁡ΛiV2i1/2(m-1)×λmin⁡Γ1iV2iλmax⁡Λi-λmin⁡Γ2iV2iλmax⁡Λi=-χ1iV2i(2m-3)/(2m-2)-χ2iV2i,where χ1i=λmin⁡Λi1/2(m-1)λmin⁡Γ1i/λmax⁡Λi, χ2i=λmin⁡Γ2i/λmax⁡Λi,(54)V2=∑i=13-χ1iV2i(2m-3)/(2m-2)-χ2iV2i≤-χ1∑i=13V2i(2m-3)/(2m-2)-χ2V2,where χ1=max⁡χ11,χ12,χ13, χ2=max⁡χ21,χ22,χ23.
Sincem>2, 0<(2m-3)/(2m-2)<1. According to Lemma 9(55)V2+χ1V2(2m-3)/(2m-2)+χ2V2≤0.
Similar to the proof of Theorem10, according to Lemma 8, the trajectory of system will be driven into the related sliding surface s=03×1 in a finite time tr:(56)tr≤2m-1ln⁡1+χ2/χ1V2tob1/2m-1χ1+tob,where tob is the moment which satisfies z1=Δv, t≥tob. V2tob is the value of V2 at t=tob.
According to Lemma2, once the slide mode s=03×1 is established, the system (9) is globally asymptotically stable. This completes the proof.


## 5. Control Allocation

With low atmosphere density, reentry vehicle suffers poor aerodynamic maneuverability at high altitude. In such case, the combination of RCS jets and aerodynamic surfaces is considered to meet the control performances. To ensure that the command control torqueMc can be produced jointly by the actuators input δc, a control allocation approach is designed in this paper.The core of the control allocation problem is to solve the nondeterministic system equations with typical constraints. Suppose that the number of aerodynamic surface isn1, and the number of RCS jets is n2: (57)Mc=D·δc,where δc=δA;δRCS, δA=δA1,δA2,…,δAn1T denotes the vector of aerodynamic surface deflection, and δRCS=δRCS1,δRCS2,…,δRCSn2 stands for the vector of RCS thruster states. The matrix D· can be expressed as(58)D·=DA·,DRCS·,where DA· and DRCS· stand for aerodynamic torque matrix and RCS torque matrix, respectively.The typical constraints for the control allocation problem are commonly defined as(59)δAimin⁡≤δAi≤δAimax⁡,δd-Aimin⁡≤δ˙Ai≤δd-Aimax⁡,i=1,…,n1,0≤δRCSj≤1,j=1,…,n2,where δAimin⁡ and δAimax⁡ are the lower boundary and upper boundary of aerodynamic surface δAi, respectively. δd-Aimin⁡ and δd-Aimax⁡ are the lower boundary and upper boundary of deflection rate δ˙Ai, respectively.The optimization method can be used to solve the nondeterministic system equations. The primary object of the control allocation is to minimize the difference between command control torqueMc and the torque produced by actuators [39]. Moreover, another objective is to minimize the use of RCS jets. Hence, the cost function can be expressed as(60)min⁡J1=W1Mc-D·δc+W2δRCS,where W1∈R3, W2∈Rn2 are the weights to be designed.Therefore, the control allocation problem is transformed into optimization problem to minimize the cost function (60) subject to (59).In practical, on-off RCS jets can only provide the maximum torque or zero torque. Thus,δRCS should be defined as binary variables. This paper employs the PWPF modulator to convert the continuous signal into on-off RCS commands. As shown in Figure 2, PWPF modulator consists of a low pass filer and a Schmitt trigger inside a feedback loop. Km and Tm are the low pass filer gain. uon⁡ and uoff⁡ are the on-value and off-value of Schmitt trigger.Figure 2
PWPF modulator.


## 6. Numerical Simulation Results and Assessment

In order to verify the effectiveness of proposed control methods, the comparisons between proposed control methods and two conventional methods are presented. The two conventional methods are FBL and boundary layer method based time-varying sliding mode control (BTVSMC).The control algorithm of FBL can be expressed as [40](61)v=y¨c-kpe-kde˙,where the parameters kp and kd should be a positive value.As stated earlier, bound layer method is a conventional method to alleviate chattering phenomenon of SMC. Since DO is not employed in the control method, the control algorithm of BTVSMC is given by [17](62)v=b-ηsat⁡s.The numerical tests in this paper employ a reentry vehicle, whose moments of inertia areIx=588791 kg·m2, Iy=1303212 kg·m2, and Iz=1534164 kg·m2, and the products of inertia are Izx=Ixz=24242 kg·m2. In addition, the vehicle has a lifting-body configuration with 8 aerodynamic surfaces and 10 RCS jets. The aerodynamic surfaces include left outer elevon δelo, right outer elevon δero, left inner elevon δeli, right inner elevon δeri, left flap δfl, and right flap δfr, left rudder δrl, and right rudder δrr [47]. The constraints of the vector δA=δelo,δero,δeli,δeti,δfl,δfr,δrl,δrrT are given by(63)δAmax⁡=0,0,0,0,30,30,30,30T,δAmin⁡=-25,25,25,25,10,10,30,30T,δd-Amax⁡=-δd-Aimin⁡=10,10,10,10,10,10,10,10T,where δAmax⁡ and δAmin⁡ are measured in degree and δd-Amax⁡ and δd-Aimin⁡ are measured in degree per second.Each RCS jet of reentry vehicle can produce3559N of thrust. In addition, the RCS torque matrix DRCS· is defined as [48](64)DRCS·=0-204811625-691202054-11623691200-4980-946610944-4980-946511798-498-49819897-15723-9465-11798-1989715723946511798-1989719897.The initial conditions for reentry vehicle are taken as follows: the altitudeh=55.0 km, Mach number Ma=9.8, Ω0=32.0∘,2.0∘,58.0∘T, and w0=0.0∘/s,0.0∘/s,0.0∘/sT. And attitude angle commands are set to be Ωc=30.0∘,0.0∘,60.0∘T. The reentry vehicle suffers high structural stresses for high Mach number and bad aerodynamics for large AOA, and, hence, the rudders are not allowed to be used [10].Furthermore, additional constraints for control allocation problem should be satisfied:(65)δrl=δrr=0.The planet uncertainties are set in consideration of 5 percent bias conditions for moments of inertia and products of inertia, 10 percent bias conditions for aerodynamic coefficients, and 10 percent bias conditions for atmospheric density. In addition, the external disturbance torque vector takes the form of(66)ΔM=0.5+sin0.1t+sint0.5+sin0.1t+sint0.5+sin0.1t+sint×104N·m.To validate the robustness and the chattering reduction of the proposed methods, numerical simulations of FBL (61), BTVSMC (62), BTVSMC/DO (36), and SOTVSMC/DO (45) are presented. The overall attitude control system architecture for reentry is showed in Figure 1. And the integration step is specified as 0.01 seconds. In FBL (61), the control parameters are selected as kp=kd=3. In addition, the control parameters of the sliding mode controllers are specified in Table 1. After the command control torque vector, Mc, is produced by these controllers, the actuator command is obtained by control allocation algorithm as presented in Section 5. The parameters of control allocation problem are as shown in Table 2.Table 1
Sliding mode control parameters.
Parameter∖controller
BTVSMC
BTVSMC/DO
SOTVSMC/DO
Sliding surface parameters
c
=
diag
⁡
(
2,2
,
2
)  λ=2
c
=
diag
⁡
(
2,2
,
2
)  λ=2
c
=
diag
⁡
(
2,2
,
2
)  λ=2
Control parameters
η
=
diag
⁡
(
0.1,0.1,0.1
) h=diag⁡(0.015,0.015,0.015)
η
=
diag
⁡
(
0.1,0.1,0.1
) h=diag⁡(0.015,0.015,0.015)
k
1
=
diag
⁡
(
0.1,0.1,0.1
) k2=diag⁡(0.2,0.2,0.2) k3=diag⁡(0.1,0.1,0.1) k4=diag⁡(0.1,0.1,0.1) m=3
DO parameters
γ
1
=
8.2
,
γ
2
=
4.1
,
γ
3
=
2.0 L=diag⁡(0.005,0.005,0.01)
γ
1
=
8.2
,
γ
2
=
4.1
,
γ
3
=
2.0 L=diag⁡(0.005,0.005,0.01)Table 2
Control allocation parameters.
Parameter
Value
W
1
[
1,1
,
1
]
W
2
[
0.1
,
…
,
0.1
]
K
m
4.5
T
m
0.15
u
on
⁡
0.45
u
off
⁡
0.15The variations of the attitude angles including AOA, sideslip angle, and bank angle under FBL, BTVSMC, BTVSMC/DO, and SOTVSMC/DO are shown in Figure3. It is obvious that SMC has significant robustness performance in the presence of uncertainties and disturbances. Since FBL relies on the knowledge of the exact model dynamics, the tracking errors under FBL do not converge to zero. The second row of Figure 3 shows the attitude angle evolutions in steady-state region. Compared with FBL and BTVSMC, BTVSMC/DO and SOTVSMC/DO achieve the goals of tracking with higher accuracy.Figure 3
Comparison of attitude angle evolutions via FBL, BTVSMC, BTVSMC/DO, and SOTVSMC/DO.To further evaluate the tracking performance between BTVSMC/DO and SOTVSMC/DO, the local view of attitude angle evolutions via BTVSMC/DO and SOTVSMC/DO is shown in Figure4. The results suggest that all the attitude angles converge to the desired values within 4 seconds in the transient region for both methods. As shown in the local view of attitude angles in the steady-state region, the tracking errors under SOTVSMC/DO are much smaller than those under BTVSMC/DO.Figure 4
Local view of attitude angle evolutions via BTVSMC/DO and SOTVSMC/DO.The sliding surface responses via BTVSMC, BTVSMC/DO, and SOTVSMC/DO are shown in Figure5. It is obvious that the sliding surface under BTVSMC does not converge to zero because DO is not introduced. The system trajectories under BTVSMC/DO and SOTVSMC/DO move apart from zero after the start point, and move back to zero around 4 seconds later. The second row of Figure 5 shows the local view of steady-state region under BTVSMC/DO and SOTVSMC/DO. As boundary layer method is applied, the sliding surface stays inside of the boundary layer; however, the tracking accuracy within the boundary layer is not guaranteed. As a result, the sliding surface under SOTVSMC/DO converges to zero more precisely.Figure 5
Comparison of sliding surface responses via BTVSMC, BTVSMC/DO, and SOTVSMC/DO.The estimations of the sliding surface by BTVSMC/DO and SOTVSMC/DO are shown in Figures6(a) and 6(b) respectively. In addition, Figure 7 depicts the estimations of uncertainty terms. Clearly, all of sliding surfaces and lumped uncertainty can be estimated by DO effectively.Figure 6
The estimations of sliding surface by BTVSMC/DO and SOTVSMC/DO.
(a)
Comparison betweens and z0 under BTVSMC/DO
(b)
Comparison betweens and z0 under SOTVSMC/DOFigure 7
The estimations of the uncertainty terms by BTVSMC/DO and SOTVSMC/DO.
(a)
Comparison betweenΔv and z1 under BTVSMC/DO
(b)
Comparison betweenΔv and z1 under SOTVSMC/DOFigure8 illustrates the comparison of attitude angular rate evolutions via FBL, BTVSMC, BTVSMC/DO, and SOTVSMC/DO. And the produced command control torques are showed in Figure 9. Control chattering is undesirable in practice because it involves high control activity and may excite high frequency unmolded dynamics. As is shown in Figure 9, the problem of chattering phenomenon for SMC is eliminated in the cases of BTVSMC, BTVSMC/DO, and SOTVSMC/DO.Figure 8
Comparison of attitude angular rate evolutions via FBL, BTVSMC, BTVSMC/DO, and SOTVSMC/DO.Figure 9
Comparison of command control torque via FBL, BTVSMC, BTVSMC/DO, and SOTVSMC/DO.This paper formulates all the numerical simulations in consideration of the constraints for aerodynamic deflections and RCS thrusts, both of which are actuator command and can be obtained by control allocation algorithm presented in Section5. The aerodynamic deflections via BTVSMC/DO and SOTVSMC/DO are shown in Figures 10(a) and 11(a), respectively. The rudders are kept around zero. All the actuator limits in (59) and (65) are satisfied. The RCS thrust commands via BTVSMC/DO and SOTVSMC/DO are shown in Figures 10(b) and 11(b), respectively. The RCS thrusts compensate the torque errors caused by aerodynamic deflection saturation with high accuracy.Figure 10
Aerodynamic surface commands and RCS commands via BTVSMC/DO.
(a)
Elevon, flap, and rudder commands
(b)
RCS commandsFigure 11
Aerodynamic surface commands and RCS commands via SOTVSMC/DO.
(a)
Elevon, flap, and rudder commands
(b)
RCS commands


## 7. Conclusion

This paper investigates the finite-time control problem of reentry vehicle with aerodynamic surfaces and RCS jets and seeks for more reliable attitude controller design and the control allocation design. The numerical simulation results prove the tracking accuracy and robustness of the proposed attitude controller and control allocation method.For attitude controller design, two kinds of robust finite-time TVSMC controllers are proposed. Time-varying sliding surface is employed to eliminate the reaching phase of transit dynamics, and thus the global robustness is guaranteed. A DO is introduced to enhance the robustness against planet uncertainness and external disturbances. To alleviate the chattering, boundary layer method and second-order SMC method are employed, respectively. Both of the proposed methods can avoid chattering phenomenon effectively. Moreover, the tracking error under SOTVSMC/DO converges to zero more precisely.A control allocation approach is introduced to generate the actuator commands, including aerodynamic surface deflections and on-off RCS thrusts. The nondeterministic system equations are solved by optimization methods, and the proposed weighted cost function is capable of minimizing both the utilization of RCS jets and the differences between the desired torque and the torque produced by actuators.



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*Source: 262753-2015-05-20.xml*