1 Introduction

Reusable launch vehicles (RLVs) have attracted tremendous attention by researchers due to their excellent traits, such as low flight cost and reusability. They are expected to play an important role in military and civil applications in the future [1]. However, great challenge are encountered in the reentry phase, in which the vehicles suffer significant uncertainties, severe external disturbances and poor aerodynamic maneuverability [2]. Thus, rendering the control strategy more robust to the uncertainties, disturbances and actuator faults and causing it to stabilize more rapidly and accurately becomes an essential research topic.

Fault tolerant control (FTC) strategies are designed to maintain safe operating limits and to mitigate the effects of system/component malfunctions [3]. FTC for the reentry vehicle has been extensively studied in recent years, and many types of robust control approaches have been used as the basis for developing FTC [4]. The approaches include \(H_{\infty }\) control [5, 6], back-stepping control [7, 8], sliding mode control (SMC) [9, 10] and others. Among these approaches, the SMC is an attractive choice because of its useful properties, including superior robustness to disturbances, finite-time convergence and higher control precision, etc. The rotational equation for RLVs in the reentry phase is of second order. For stabilizing the second-order system, the terminal sliding manifold (TSM) is a suitable choice, since it has advantageous properties including fast and finite-time convergence [11, 12]. However, there exists a singularity problem for the early TSM methods. To overcome this problem, many researches on the non-singular TSM (NTSMC) have been carried out [13,14,15,16].In addition, the traditional TSM does not demonstrate uniform convergence speed when the system states are at different ranges from the equilibrium. Hence, the so-called fast TSM (FTSM) strategies were proposed to compensate for this shortcoming [17, 18]. In the FTSM, the convergence is rapid even when the states are far from the equilibrium. An advanced version of the FTSM is the fixed-time TSM, which drives the system to the origin in fixed time. In recent years, extensive research on the nonsingular fast/fixed-time TSM control (NFTSMC) has been conducted [19,20,21]. In particular, a singularity-free fixed time TSM structure that possesses a faster convergence speed and retains a higher control precision than its predecessors was proposed in [19].

On the other hand, in spite of the advances in the NFTSMC, the control laws proposed are always discontinuous. This can give rise to large undesirable chattering as was shown in the simulation tests in reference [19]. To solve this problem, saturation functions and boundary-layer functions were employed [22]. While the chattering is reduced by that, the system’s trajectories are not constrained exactly on the sliding surface but rather to its vicinity. Consequently, the robustness of the controller is weakened. The designers still have to make the trade-off between the control accuracy and the chattering reduction. The approach to solve this dilemma is the higher order SMC strategy. However, the difficulty of designing a higher order NFTSMC lies in its special structure. Specifically, the derivative of the sliding manifold is an implicit function for the control law \(u\) [as is shown in Eq. (27)]. Moreover, the upper bounds of the disturbances and uncertainties are always assumed to be known a priori. However, they are not easily obtained in real applications. In addition, any overestimation of the disturbances leads to additional unwanted chattering. To resolve these two problems, two major types of methods have been studied recently. One resorts to a sliding manifold on the basis of the properties of the homogeneous system [23], which we will refer to as the homogeneity-based TSM (HBTSM) here. The derivative of such a terminal sliding manifold is an explicit function for the control \(u\). Based on the HBTSM, an adaptive continuous nonsingular control law can be developed [24,25,26,27]. In [24], the equivalent control based adaptive super-twisting algorithm was combined with the HBTSM, which results in a continuous controller and mitigates the chattering. In [25, 26, 28], the multivariable version of the HBTSM was established and used to develop a continuous fault tolerant control algorithm. However, compared with the NFTSMC, the HBTSM has several disadvantages. For instance, for the HBTSM, it is hard to estimate the upper bound for the convergence time and the property of fast/fixed-time convergence vanishes. Another type of method is to use a disturbance observer, so that the effect of disturbances can be offset by their estimation [27, 29,30,31,32,33]. Without the influence of the disturbances, the discontinuous part (sign function) in the control law can be removed. However, the introduction of an observer increases the redundancy and complexity of the controller. Considering the shortcomings of the above methods, some researchers have made attempts to develop a continuous NFTSMC without the usage of an observer. In [34], a higher order version of the NFTSMC was proposed by combining a super-twisting SMC. However, the stability of the system has not been proved in this design and the stability of the system may not be guaranteed under all conditions [35].

Moreover, due to the unique structure of the traditional discontinuous NFTSMC, the magnitude of its output value during the initial stage of the control process is sensitive to the change of the disturbance value [19]. As a consequence, the initial control signal magnitude increases dramatically when the disturbance becomes slightly larger. This effect tends to give rise to large oscillations, which can lead to instability when there exists control saturation.

Motivated by the aforementioned problems, a novel adaptive second-order nonsingular fast terminal sliding mode controller (A2-NFTSMC) is proposed and applied to fault tolerant control of the RLV attitude in this paper. The main contributions are summarized as follows:

  1. a)

    The proposed controller is developed on the basis of the well-researched NFTSMC, instead of the HBTSM. Thus, the properties of fast/fixed-time convergence and easy estimation of convergence time are inherited from the NFTSMC.

  2. b)

    The proposed controller represents a second-order sliding mode algorithm. As a result, the chattering is greatly attenuated compared with the existing discontinuous NFTSMC without sacrificing the algorithm’s accuracy and robustness to disturbances and faults. In addition, the proposed controller does not rely on observers to estimate the external disturbances. Therefore, the structure of the controller is simpler and more concise.

  3. c)

    Unlike most of the existing control laws based on the NFTSMC, the proposed controller does not require any prior knowledge of the external disturbances, modeling uncertainties or actuator faults. In addition, the shortcoming of the traditional NFTSMC that its output magnitude is sensitive to the change of the disturbance value is overcome in the proposed control strategy. Consequently, the hazard of introducing large oscillations during the initial stage of the control process is eliminated.

This paper is organized as follows. Section 1 gives an introduction of this research. Section 2 presents the modeling of the RLV dynamics during the reentry phase and the statement of the control problem. Section 3 is dedicated to the proposed adaptive second-order nonsingular fast terminal sliding mode controller design. Section 4 provides simulation tests to demonstrate the efficacy of the proposed method. The conclusions are given in Sect. 5.

Notation 1: In this paper, the following notations are used. \(\left\| {\, \cdot \,} \right\|\) denotes the Euclidean norm of vectors or the induced norm of matrices.

Notation 2: For any vector \(x = [x_{1} ,x_{2} , \cdots ,x_{n} ]^{T}\) and positive constant \(r\), denote

\( sign^{r} (x) = \Big[| {x_{1}} |^{r} sign(x_{1}),| {x_{2} } |^{r} sign(x_{2}), \cdots ,| {x_{n}} |^{r} sign(x_{n})\Big]\).

2 Dynamics Modeling and Problem Statement

During the reentry phase, the motion of RLVs can be decomposed into rotational motion and translational motion. The equations of rotational motion are described as [4, 36]

$$ \begin{aligned} \dot{p} &= \frac{{J_{zz} M_{x} }}{{J_{xx} J_{zz} - J_{xz}^{2} }} + \frac{{J_{xz} M_{z} }}{{J_{xx} J_{zz} - J_{xz}^{2} }} + \frac{{(J_{xx} - J_{yy} + J_{zz} )J_{xz} }}{{J_{xx} J_{zz} - J_{xz}^{2} }}pq \hfill \\ &\quad + \frac{{(J_{yy} - J_{zz} )J_{zz} - J_{xz}^{2} }}{{J_{xx} J_{zz} - J_{xz}^{2} }}qr, \hfill \\ \dot{q} &= \frac{{M_{x} }}{{J_{yy} }} + \frac{{J_{xz} }}{{J_{yy} }}(r^{2} - p^{2} ) + \frac{{J_{zz} - J_{xx} }}{{J_{yy} }}pr, \hfill \\ \dot{r} &= \frac{{J_{xz} M_{x} }}{{J_{xx} J_{zz} - J_{xz}^{2} }} + \frac{{J_{xx} M_{z} }}{{J_{xx} J_{zz} - J_{xz}^{2} }} + \frac{{(J_{xx} - J_{yy} )J_{xx} + J_{xz}^{2} }}{{J_{xx} J_{zz} - J_{xz}^{2} }}pq\hfill \\ &\quad + \frac{{(J_{yy} - J_{xx} - J_{zz})J_{xz} }}{{J_{xx} J_{zz} - J_{xz}^{2} }}qr, \end{aligned} $$
(1)
$$ \begin{aligned} \dot{\alpha} &= - p\cos \alpha \tan \beta + q - r\sin \alpha \tan \beta\\ &\quad + \frac{\sin \sigma }{{\cos \beta }}\left[ {\dot{\psi }\cos \gamma - \dot{\phi }\sin \psi \sin \gamma } \right. \\ &\quad \left. { + (\dot{\theta } + \omega_{e} )(\cos \phi \cos \psi \sin \gamma - \sin \phi \cos \gamma )} \right] \\ &\quad - \frac{\cos \sigma }{{\cos \beta }}\left[ {\dot{\gamma } - \dot{\phi }\cos \psi - (\dot{\theta } + \omega_{e} )\cos \phi \sin \psi } \right], \\ \dot{\beta } &= p\sin \alpha - r\cos \alpha + \sin \sigma \left[ {\dot{\gamma } - \dot{\phi }\cos \psi - (\dot{\theta } + \omega_{e} )\cos \phi \sin \psi } \right] \\ &\quad \cos \sigma \left[\dot{\psi }\cos \gamma - \dot{\phi }\sin \psi \sin \gamma \right.\\ & \quad + \left. {(\dot{\theta } + \omega_{e} )(\cos \phi \cos \psi \sin \gamma - \sin \phi \cos \gamma )} \right], \\ \dot{\sigma } &= - p\cos \alpha \cos \beta - q\sin \beta - r\sin \alpha \cos \beta + \dot{\alpha }\sin \beta - \dot{\psi }\sin \gamma \\ &\quad - \dot{\phi }\sin \psi \cos \gamma + (\dot{\theta } + \omega_{e} )(\cos \phi \cos \psi \cos \gamma - \sin \phi \sin \gamma ), \end{aligned} $$
(2)

where \(p\), \(q\), \(r\) are the roll, pitch and yaw angular rates; \(\alpha\), \(\beta\), \(\sigma\) represent the angle of attack, the sideslip angle and bank angle; \(\gamma\) is the flight path angle; \(\psi\) is the heading angle; \(\omega_{e}\) is the Earth’s angular speed; \(\theta\) and \(\phi\) are the longitude and latitude; \(J_{ij} \;(i = x,y,z;\;j = x,y,z)\) is the moment of inertia; \(M_{x}\), \(M_{y}\), \(M_{z}\) are the roll, pitch yaw moments, respectively.

Since the rotational motions of RLVs are much faster than their translational motions, the time derivatives of both the position and the velocity are considered to be negligible with respect to the rotational motion. Equations (1) and (2) can be simplified as [37]

$$ \dot{\Omega } = R\omega + \Delta f, $$
(3)
$$ \dot{\omega } = - J^{ - 1} \omega^{ \times } J\omega + J^{ - 1} M + \Delta d, $$
(4)

where \( M = {\text{ }}[M_{x} \;M_{y} \;M_{z} ]^{T} \), \(\Omega = [\alpha \;\beta \;\sigma ]^{T}\), and \(\omega = [p\;q\;r]^{T}\); \(\Delta f = [\Delta f_{x} \;\Delta f_{y} \;\Delta f_{z} ]^{T}\) is the bounded differentiable unmatched disturbance vector caused by simplified modeling; \(\Delta d \in \Re^{3}\) is the bounded unknown disturbance vector caused by model uncertainty and external disturbances. \(J\), \(R\), \(\omega^{ \times } \in \Re^{3 \times 3}\) are the nominal inertia matrix, the coordinate transformation matrix and the skew–symmetric matrix operator on vector \(\omega\), respectively. They are defined as follows:

$$ \begin{gathered} J = \left[ {\begin{array}{*{20}c} {J_{xx} } & 0 & { - J_{xz} } \\ 0 & {J_{yy} } & 0 \\ { - J_{xz} } & 0 & {J_{zz} } \\ \end{array} } \right],\;\omega^{ \times } = \left[ {\begin{array}{*{20}c} 0 & { - r} & q \\ r & 0 & { - p} \\ { - q} & p & 0 \\ \end{array} } \right] \hfill \\ R = \left[ {\begin{array}{*{20}c} { - \cos \alpha \tan \beta } & 1 & { - \sin \alpha \tan \beta } \\ {\sin \alpha } & 0 & { - \cos \alpha } \\ { - \cos \alpha \cos \beta } & { - \sin \alpha } & { - \sin \alpha \cos \beta } \\ \end{array} } \right], \hfill \\ \end{gathered} $$
(5)

where \(\Delta d\) can be expressed as \(\Delta d = J^{ - 1} [ - \Delta J\dot{\omega } - \omega^{ \times } \Delta J\omega + \Delta M]\), where \(\Delta J \in \Re^{3 \times 3}\) is the uncertain part of the inertia matrix and \(\Delta M \in \Re^{3}\) is a vector of unknown external disturbances.

To further include possible actuator faults, Eq. (4) can be expressed as

$$ \begin{gathered} \dot{\omega } = - J^{ - 1} \omega^{ \times } J\omega + J^{ - 1} [\Lambda M + F_{\delta } ] + \Delta d \\ = - J^{ - 1} \omega^{ \times } J\omega + J^{ - 1} M + J^{ - 1} [\left( {\Lambda - I} \right)M + F_{\delta } ] + \Delta d, \\ \end{gathered} $$
(6)

where vector \(\Lambda = diag[\Lambda_{x} \;\Lambda_{y} \;\Lambda_{z} ]\) is the actuator effectiveness with \(0 \le \Lambda_{i} \; \le 1,\,\;i = x,y,z\) and \( F_{\delta } \in [F_{\delta _{x} }\;F_{\delta _{y}} \;F_{\delta _{z}} ] \) is the actuator bias fault.

The aim of this paper is to design a continuous sliding mode controller such that the angle vector \(\Omega\) tracks the guidance command \(\Omega_{ref} = [\alpha_{ref} \;\beta_{ref} \;\sigma_{ref} ]^{T}\) accurately in finite time in the presence of uncertainties \(\Delta f\) and disturbances \(\Delta d\).

For the sake of the controller design, feedback linearization of the dynamic system is carried out first. Assume that \(X_{1} = \Omega - \Omega_{ref}\) and \(X_{2} = \dot{\Omega } - \dot{\Omega }_{ref}\), which denote the tracking errors of the attitude angles and the angular rates, respectively. By differentiating \(X_{1}\) and \(X_{2}\), we obtain

$$ \dot{X}_{1} = X_{2} , $$
(7)
$$ \dot{X}_{2} = F + GM + \Delta D, $$
(8)

where \(X_{1} = [x_{1x} \;x_{1y} \;x_{1z} ]\), \(X_{2} = [x_{2x} \;x_{2y} \;x_{2z} ]\); \(G = RJ^{ - 1}\); \(F = [F_{x} \;F_{y} \;F_{z} ]\) is the known normal part of the system; the disturbance \(\Delta D = [\Delta D_{x} \;\Delta D_{y} \;\Delta D_{z} ]\) is the unknown part which is induced by uncertainties, disturbances and actuator faults. Here, \(F\) and \(\Delta D\) can be expressed as

$$ F = \left( {\dot{R} - RJ^{ - 1} \omega^{ \times } J} \right)\omega - \ddot{\Omega }_{ref} , $$
(9)
$$ \Delta D = RJ^{ - 1} [\left( {\Lambda - I} \right)M + F_{\delta } ] + R\Delta d + \Delta \dot{f}. $$
(10)

Since, during the reentry phase, angle \(\beta\) is kept near 0, we have

$$ \det (G) = \frac{\cos \beta - \sin \beta \tan \beta }{{J_{xx} J_{zz} - J_{xz}^{2} }} \approx \frac{1}{{J_{xx} J_{zz} - J_{xz}^{2} }} \ne 0. $$
(11)

Therefore, \(G^{ - 1}\) exists. Introducing a control vector \(U = [u_{x} ,u_{y} ,u_{z} ]\) and defining it as \(U = GM\), Eq. (8) can be expressed as

$$ \dot{X}_{2} = F + U + \Delta D. $$
(12)

Assumption 1

[26]: It is assumed that uncertainty \(\Delta D\) in (10) is differentiable and bounded with \(\left\| {\Delta D} \right\| \le L_{1}\); its derivative is also bounded with \(\left\| {\Delta \dot{D}} \right\| \le L_{2}\), where \(L_{1}\) and \(L_{2}\) are unknown positive constants.

Assumption 2

[26]: Assume that the condition \(\left\| {J^{ - 1} (\Lambda - I)J} \right\| < 1\) holds.

Remark 1 Eq. (6) can be rewritten as

$$ \dot{\omega } = - J^{ - 1} \omega^{ \times } J\omega + \left( {I + J^{ - 1} \left( {\Lambda - I} \right)J} \right)J^{ - 1} M + J^{ - 1} F_{\delta } + \Delta d. $$
(13)

In the second term in (13), i.e., the control term, \(J^{ - 1} \left( {\Lambda - I} \right)J\) denotes the uncertain portion of the matrix and the identity matrix \(I\) represents the nominal portion. According to [38], for the nominal part of the control signal to dominate the uncertain part, condition \(\left\| {\left( {J^{ - 1} \left( {\Lambda - I} \right)J} \right) \cdot I^{ - 1} } \right\| < 1\) must be satisfied.

3 Controller Design

3.1 Preliminaries

Lemma 1

[19] Consider the following scalar system:

$$ \dot{y} = - l_{1} sign^{{m_{1} }} y - l_{2} sign^{{m_{2} }} y,\;y(0) = y_{0},$$
(14)

where \(l_{1} > 0\), \(l_{2} > 0\), \(m_{1} \ge 1\) and \(0 < m_{2} < 1\). The equilibrium point of the above system is finite time stable, and the convergence time \(T\) is bounded by

$$ T \le T_{\max } = \frac{1}{{l_{1} (m_{1} - 1)}} + \frac{1}{{l_{2} (1 - m_{2} )}}. $$
(15)

Remark 2

Note that the upper bound of convergence time does not depend on the initial value \(y_{0}\). Then system (14) can be seen as being fixed-time stable [39].

Lemma 2

[40] (Jensen’s inequality) For a vector \(v = [v_{1} \;v_{2} \cdots \,v_{n} ]\) with \(v_{i} \ge 0\), there holds inequality.

$$ \left( {\sum\limits_{i = 0}^{n} {v_{i}^{q} } } \right)^{1/q} \le \left( {\sum\limits_{i = 0}^{n} {v_{i}^{p} } } \right)^{1/p} , $$
(16)

if \(0 < p < q.\)

Lemma 3

[40] For a continuous convex function \(f\), if numbers \(v_{1} ,v_{2} \cdots \,v_{n}\) are in its domain, then there holds inequality:

$$ f\left( {\sum\limits_{i = 0}^{n} {v_{i} } /n} \right) \le \left( {\sum\limits_{i = 0}^{n} {f(v_{i} )} } \right)/n. $$
(17)

Lemma 4

For a vector \(v = [v_{1} \;v_{2} \cdots \,v_{n} ]\) with \(v_{i} \ge 0\) , it can be proven from Lemma 2 and Lemma 3 that there hold inequalities:

$$ \sum\limits_{i = 0}^{n} {v_{i} } \le \left( {\sum\limits_{i = 0}^{n} {v_{i}^{p} } } \right)^{1/p} ,{\text{ if}}\;p < 1, $$
(18)
$$ \sum\limits_{i = 0}^{n} {v_{i} } \le n^{p - 1} \left( {\sum\limits_{i = 0}^{n} {v_{i}^{p} } } \right)^{1/p} ,\;{\text{if}}\;p > 1. $$
(19)

3.2 Design of Adaptive Second-Order Nonsingular Fast Terminal Sliding Mode Controller

In order for the angle vector \(\Omega\) to track the guidance command \(\Omega_{ref}\), the controller to be designed should cause state vector \(X_{1}\) in (7) to converge to the origin in finite time. Thus, in this section, a novel A2-NFTSMC that achieves this goal will be proposed. First, assume that state vector \(X_{2}\) is obtainable. The terminal sliding manifold is chosen as [19]

$$ S_{i} = sign^{{a_{1i} }} x_{1i} + \frac{{k_{2i} a_{2i} }}{{2a_{2i} - 1}}sign^{{2 - 1/a_{2i} }} \left( {x_{2i} + k_{1i} sign^{{a_{1i} }} x_{1i} } \right),\;i = x,y,z, $$
(20)

where control parameters \(k_{1i} > 0\), \(k_{2i} > 0\), \(a_{2i} > 1\) and \(1 < a_{1i} < 2 - 1/a_{2i}\) are preselected. Then, define \(S = [S_{x} ,S_{y} ,S_{z} ]\).

Theorem 1

When the sliding manifold satisfies \(S = 0\), state vectors \(X_{1}\) and \(X_{2}\) converge to the origin in fixed time, and the upper bound for the convergence time for each pair of their elements is

$$ T_{si} = \frac{1}{{\overline{k}_{1i} (\overline{a}_{1i} - 1)}} + \frac{1}{{\overline{k}_{2i} (1 - \overline{a}_{2i} )}},\;\quad i = x,y,z, $$

where \(k_{1i} = \overline{k}_{1i}\), \(\overline{k}_{2i} = \left( {(2a_{2i} - 1)/k_{2i} a_{2i} } \right)^{{a_{2i} /\left( {2a_{2i} - 1} \right)}}\), \(\overline{a}_{1i} = a_{1i}\) and \(\overline{a}_{2i} = a_{1i} a_{2i} /(2a_{2i} - 1).\)

Proof

As \(\dot{X}_{1} = X_{2}\), when \(S = 0\), Eq. (20) can be transformed into

$$ \dot{x}_{1i} = - \overline{k}_{1i} sign^{{\overline{a}_{1i} }} x_{1i} - \overline{k}_{2i} sign^{{\overline{a}_{2i} }} x_{1i} ,\quad i = x,y,z, $$
(21)

By Lemma 1, we can obtain that the upper bound of the convergence time for each \(x_{1i}\) equals \(T_{si}\). Then, as \(x_{1i}\) is stable and constant, the state \(x_{2i}\) also converges to the origin correspondingly.

To drive the sliding manifold \(S\) to the origin in finite time while resisting disturbance \(\Delta D\) in the meantime, an adaptive continuous control law is presented as

$$ u_{i} = - F_{i} - k_{1i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} \left( {\frac{{\phi_{i} }}{{k_{1i} }} + x_{2i} } \right) - k_{3i} sign^{{c_{1} }} (S_{i} ) - k_{4i} sign^{{c_{2} }} (S_{i} ) + v_{i} , $$
(22)
$$ \dot{v}_{i} = - r_{i} v_{i} - k_{di} sign(\chi_{i} ),\;i = x,y,z, $$
(23)

with

$$ \phi_{i} = \frac{1}{{k_{2i} }}sign^{{1/a_{2i} }} (x_{2i} + k_{1i} sign^{{a_{1i} }} x_{1i} ) + \frac{{k_{1i} a_{2i} }}{{2a_{2i} - 1}}(x_{2i} + k_{1i} sign^{{a_{1i} }} x_{1i} ),$$
(24)

and adaptive gain \(k_{di}\) chosen as

$$ \dot{k}_{di} = \left\{ \begin{gathered} l_{0i} \left( {\left| {S_{i} } \right|^{{m_{1i} }} + \left| {S_{i} } \right|^{{m_{2i} }} } \right),\quad \left| {S_{i} } \right| > \varepsilon_{0} \hfill \\ 0,\quad \left| {S_{i} } \right| \le \varepsilon_{0} \hfill \\ \end{gathered} \right. $$
(25)

where \(\chi_{i} = v_{i} + \Delta D_{i}\); \(k_{3i}\), \(k_{4i}\), \(r_{i}\), \(l_{0i}\), \(\gamma_{0}\) and \(\varepsilon_{0}\) are preselected positive parameters; parameters \(c_{1}\), \(c_{2}\), \(m_{1i}\), \(m_{2i}\) are chosen as \(0 < c_{1} < 1\), \(c_{2} > 1\), \(0 < m_{1i} < 1\), \(m_{2i} > 1\). We can note that, according to Eq. (23), \(sign(\chi_{i} )\) is required to implement the control law. How to obtain \(sign(\chi_{i} )\) is explained Remark 3.

Theorem 2

Consider the dynamic system of Eqs. (7) and (8), where the disturbance vector \(\Delta D\) satisfies Assumption 1. If the controller is designed as in (22) to (25), then there exists a finite time in which the state vectors \(X_{1}\) and \(X_{2}\) converge to the origin.

Proof

In the following proof, subscript \(i\) is used to represent the three different channels \(x,y\) and \(z\). Define a Lyapunov function

$$ V_{0} = \frac{1}{2}S^{T} S. $$
(26)

By differentiating function \(V_{0}\) and taking into account the control \(U\) in (22), we obtain

$$ \begin{aligned} \dot{V}_{0} &= \sum\limits_{i = x,y,z} {S_{i} \left( {a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} x_{2i} + \psi_{i} (\dot{x}_{2i} + a_{1i} k_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} x_{2i} )} \right)} \\ &= \sum\limits_{i = x,y,z} S_{i} \big(a_{1i} | {x_{1i} } |^{{a_{1i} - 1}} x_{2i}\\ &\quad + \psi_{i} (F_{i} + u_{i} + \Delta D_{i} + a_{1i} k_{1i} | {x_{1i} } |^{{a_{1i} - 1}} x_{2i})\big) \\ &= \sum\limits_{i = x,y,z} S_{i} \big(a_{1i} | {x_{1i} } |^{{a_{1i} - 1}} x_{2i} + \psi_{i} (- a_{1i} | {x_{1i}}|^{{a_{1i} - 1}} \phi_{i}\\ &\quad - k_{3i} sign^{{c_{1} }} (S_{i} ) - k_{4i} sign^{{c_{2}}} (S_{i} ) + v_{i} + \Delta D_{i})\big), \\ \end{aligned} $$
(27)

where \(\psi_{i} = k_{2i} \left| {x_{2i} + k_{1i} sign^{{a_{1i} }} x_{1i} } \right|^{{1 - 1/a_{2i} }}\). Since, with regard to Eq. (24), there exists relation

$$ \psi_{i} \phi_{i} = x_{2i} + k_{1i} S_{i} , $$
(28)

we further obtain

$$ \begin{aligned} \dot{V}_{0} &= \sum\limits_{i = x,y,z} S_{i} \left( - k_{1i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} S_{i} + \psi_{i} ( - k_{3i} sign^{{c_{1i} }} (S_{i} )\right.\\ &\quad\left.- k_{4i} sign^{{c_{2i} }} (S_{i} ) + v_{i} + \Delta D_{i} )\right) \\ &\le \sum\limits_{i = x,y,z} \left(- k_{1i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} S_{i}^{2} + \psi_{i} ( - k_{3i} \left| {S_{i} } \right|^{{c_{1i} }}\right.\\ &\quad\left. - k_{4i} \left| {S_{i} } \right|^{{c_{2i} }} ) + S_{i} \psi_{i} \chi_{i}\right). \\ \end{aligned} $$
(29)

We can notice that, as \(\psi_{i} \ge 0\), the terms in (29) are all negative except for \(S_{i} \psi_{i} \chi_{i}\). Now, prove that there exists a value of \(k_{di}\) for which \(\chi_{i}\) converges to the origin in finite time. In view of Eq. (23), the derivative of \(\chi_{i}\) is

$$ \frac{{{\text{d}}\chi_{i} }}{{{\text{d}}t}} = - k_{di} sign(\chi_{i} ) - r_{i} v_{i} + \Delta \dot{D}_{i} . $$
(30)

Consider a Lyapunov function candidate \(V_{{\chi_{i} }} = {{\chi_{i}^{2} } \mathord{\left/ {\vphantom {{\chi_{i}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}\). Combining with (30), we obtain the derivative of \(V_{{\chi_{i} }}\):

$$ \begin{gathered} \dot{V}_{{\chi_{i} }} = - k_{di} \left| {\chi_{i} } \right| - r_{i} v_{i} \chi_{i} + \Delta \dot{D}_{i} \chi_{i} \\ \le - k_{di} \left| {\chi_{i} } \right| - r_{i} \chi_{i}^{2} + \left| {r_{i} \Delta D_{i} + \Delta \dot{D}_{i} } \right|\left| {\chi_{i} } \right| \\ \le - k_{di} \left| {\chi_{i} } \right| - r_{i} \chi_{i}^{2} + \left( {r_{i} L_{1} + L_{2} } \right)\left| {\chi_{i} } \right|. \\ \end{gathered} $$
(31)

If \(k_{di}\) satisfies the condition

$$ k_{di} > r_{i} L_{1} + L_{2} + \varepsilon_{i} , $$
(32)

where \(\varepsilon_{i}\) is a predefined positive constant, Eq. (31) can be rewritten as

$$ \dot{V}_{{\chi_{i} }} \le - \varepsilon_{i} \sqrt {V_{{\chi_{i} }} } - r_{i} V_{{\chi_{i} }}, $$
(33)

Thus, according to Lemma 1, \(V_{{\chi_{i} }}\) converges to the origin in finite time, which further establishes the finite-time convergence of \(\chi_{i}\).

In addition, since \(a_{2i} > 1\), \(\psi_{i}\) is nonsingular. If both \(X_{1}\) and \(X_{2}\) are bounded, then \(S_{i}\) and \(\psi_{i}\) are also bounded. Therefore, the convergence of \(S_{i} \psi_{i} \chi_{i}\) to 0 in finite time can be concluded. Then, since \((c_{1} + 1)/2 < 1\) and \((c_{2} + 1)/2 > 1\), in view of Lemma 4, we obtain

$$ \begin{aligned} \dot{V}_{0} &\le \sum\limits_{i = x,y,z} {\left( { - k_{2i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} S_{i}^{2} + \psi_{i} ( - k_{3i} \left| {S_{i} } \right|^{{c_{1} + 1}} - k_{4i} \left| {S_{i} } \right|^{{c_{2} + 1}} )} \right)} \\ &\le \sum\limits_{i = x,y,z} {\psi_{i} \left( { - k_{3i} \left( {\left| {S_{i} } \right|^{2} } \right)^{{(c_{1} + 1)/2}} - k_{4i} \left( {\left| {S_{i} } \right|^{2} } \right)^{{(c_{2} + 1)/2}} } \right)} \\ &\le - 2^{{(c_{1} + 1)/2}} \mathop {\min }\limits_{i = x,y,z} (\psi_{i} k_{3i} )V_{0}^{{(c_{1} + 1)/2}} - 3^{{(1 - c_{2}^{2} )/4}} {\kern 1pt} 2^{{(c_{2} + 1)/2}}\\ &\quad \mathop {\min }\limits_{i = x,y,z} (\psi_{i} k_{4i} )V_{0}^{{(c_{2} + 1)/2}}. \end{aligned} $$
(34)

For the convenience of discussing the state of \(\psi_{i}\), define two groups of areas \(\Gamma_{1i}\) and \(\Gamma_{2i}\)

$$ \begin{gathered} \Gamma_{1i} = \left\{ {\psi_{i} \ge 1} \right\},\;i = x,y,z, \hfill \\ \Gamma_{2i} = \left\{ {\psi_{i} < 1} \right\},\;i = x,y,z, \hfill \\ \end{gathered} $$
(35)

and \(\kappa_{i} = x_{2i} + k_{1i} sign^{{a_{1i} }} x_{1i}\).

Case 1: When the states are all in the first group of areas \(\Gamma_{1i}\)\(\;(i = x,y,z)\), it can be shown that

$$ \dot{V}_{0} \le - \underbrace {{2^{{(c_{1} + 1)/2}} \mathop {\min }\limits_{i = x,y,z} (k_{3i} )}}_{{K_{a} }}V_{0}^{{(c_{1} + 1)/2}} - \underbrace {{3^{{(1 - c_{2}^{2} )/4}} {\kern 1pt} 2^{{(c_{2} + 1)/2}} \mathop {\min }\limits_{i = x,y,z} (k_{4i} )}}_{{K_{b} }}V_{0}^{{(c_{2} + 1)/2}} . $$
(36)

As \(K_{a} > 0\) and \(K_{b} > 0\), according to Lemma 1 and Theorem 1, states \(X_{1}\) and \(X_{2}\) can be driven to the origin in finite time.

Case 2: If the states are not all in the group of areas \(\Gamma_{1i}\)\(\;(i = x,y,z)\), i.e., at least some states are in the second group of areas \(\Gamma_{2i}\), there is a possibility that \(\kappa_{i} = 0\). Here, it will be proved that the curve \(\kappa_{i} = 0\) is not attractive unless \(S_{i} = 0\). Supposing that \(\kappa_{i} = 0\), then the derivative of \(\kappa_{i}\) is

$$ \dot{\kappa }_{i} = - k_{3i} sign^{{c_{1} }} (S_{i} ) - k_{4i} sign^{{c_{2} }} (S_{i} ) + \chi_{i} . $$
(37)

As it has been shown that \(\chi_{i}\) converges to 0 in finite time, \(\dot{\kappa }_{i} > 0\) when \(S_{i} < 0\) and \(\dot{\kappa }_{i} < 0\) when \(S_{i} > 0\). Hence, the states pass through \(\Gamma_{2i}\) and reach \(\Gamma_{1i}\) in finite time. As a result, the states converge to the origin in finite time no matter which areas the states are in.

The above proof was achieved under the premises that inequality (32) is satisfied and states \(X_{1}\) and \(X_{2}\) are bounded. Note that \(k_{di}\) is supposed to increase in accordance with Eq. (25) if (32) is not satisfied and, consequently, the sliding manifold does not converge to the origin. Therefore, condition (32) must be met in finite time. Thus, the divergence time for the states is also finite and the states \(X_{1}\) and \(X_{2}\) are bounded. In addition, since the sliding mode converges to the origin in finite time, adaptive gain \(k_{di}\) is also bounded.

The proof of theorem 2 further leads to the conclusion that with the controller designed as in (22)–(25), the angle vector \(\Omega\) tracks the guidance command \(\Omega_{ref}\) in finite time in the presence of disturbances, uncertainties and actuator faults as described in (3) and (6).

Remark 3

In view of the control law in (23), \(\chi_{i}\) is not obtainable due to the unknown disturbance \(\Delta D_{i}\). Taking into account the control \(u_{i}\) in (22), \(\dot{S}_{i}\) is obtained as

$$ \dot{S}_{i} = - k_{2i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} S_{i} + \psi_{i} ( - k_{3i} sign^{{c_{1i} }} (S_{i} ) - k_{4i} sign^{{c_{2i} }} (S_{i} ) + \chi_{i} ). $$
(38)

Since the curve \(\kappa_{i} = 0\) is not an attractor except when \(S_{i} = 0\), there is \(\psi_{i} > 0\) almost all the time before the sliding manifold converges. Hence, resorting to the time delay method, \(sign(\chi_{i} )\) can be achieved as follows [27]. Integrate both sides of (38) and define

$$ \begin{aligned} g(t) &= \int_{0}^{t} {(\psi_{i} \chi_{i} ){\text{d}}\tau } \\ &= S_{i} + \int_{0}^{t} \left(k_{2i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} S_{i}+ \psi_{i}\right.\\ &\quad \left.\times \left({k_{3i} sign^{{c_{1i} }} (S_{i} ) + k_{4i} sign^{{c_{2i} }} (S_{i} )} \right) \right){\text{d}}\tau . \end{aligned} $$
(39)

There is \(\psi_{i} \chi_{i} = \mathop {\lim }\limits_{\tau \to 0} \left( {g(t) - g(t - \tau )} \right)/\tau\), which further leads to

$$ sign(\chi_{i} ) = sign(g(t) - g(t - \tau )), $$
(40)

where \(\tau\) is a short time delay.

Remark 4

According to the definition of \(k_{di}\) in (25), when \(\left| {S_{i} } \right| > \varepsilon_{0}\), \(k_{di}\) increases until condition (32) is satisfied before the system states start to converge to the origin. With Eqs. (25) and (32), it can be deduced that the time interval of this process is finite and bounded by \({{\left( {r_{i} L_{1} + L_{2} + \varepsilon_{i} } \right)} \mathord{\left/ {\vphantom {{\left( {r_{i} L_{1} + L_{2} + \varepsilon_{i} } \right)} {\left( {l_{0i} \left| {\varepsilon_{0} } \right|^{{m_{1i} }} + l_{0i} \left| {\varepsilon_{0} } \right|^{{m_{2i} }} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {l_{0i} \left| {\varepsilon_{0} } \right|^{{m_{1i} }} + l_{0i} \left| {\varepsilon_{0} } \right|^{{m_{2i} }} } \right)}}\).

Remark 5

Note that in Eq. (27), \(\dot{V}_{0}\) is an implicit function for control law \(u_{i}\). Therefore, many higher order sliding mode control strategies, such as the super-twisting algorithm, cannot be directly applied here. Although an attempt was made to combine a super-twisting control law with the NFTSMC in reference [34], there was no strict stability proof given and the system may not be stable under certain conditions [35]. The adaptive A2-NFTSMC strategy proposed in this paper represents a second-order sliding mode control law. It is an augmentation of the discontinuous version of the NFTSMC in [19]. Compared with the discontinuous one, the chattering is reduced, while the controller precision and robustness is guaranteed at the same time. As the gain \(k_{di}\) is adaptive, there is no need to know the upper bound of the disturbance. In addition, an accelerating term \(- r_{i} v_{i}\) is injected in (23) and the convergence speed is increased.

Remark 6

Another benefit of the proposed adaptive second-order structure of \(u_{i}\) in (22) is its insensitivity to the change of the magnitude of the disturbance as long as the initial values of \(v_{i}\) are set to 0. The reason for this is that \(v_{i}\) tracks \(- \Delta D_{i}\) rather fast without causing a large overshoot. However, for the traditional NFTSMC, a slight increase of the magnitude of the discontinuous term is amplified by the unique structure of NFTSMC, which further gives rise to a large overshoot (to be shown in the simulations below).

Remark 7

The design of this control strategy is partly motivated by reference [27]. Compared with the sliding manifold applied in [27], the sliding manifold selected in this paper (NFTSM) has advantageous features. The design of the control law in [27] is based on a sliding manifold with homogeneous property [23]. Although it was proven to be finite time convergent, the convergence time is hard to calculate. Meanwhile, compared with the NFTSM, not only has it faster converging speed than the other one and its convergence time is fixed instead of being only finite, but also its convergence time is easy to obtain according to Lemma 1.

4 Simulation

In this section, the results of two simulation tests are presented to demonstrate the performance of the proposed controller. In the first test, the traditional NFTSMC method in [19] is compared with the proposed one. The simulation test is similar to the ones described in [4, 36].

The parameters of the RLV in the test are as follows. The moments of inertia are \(J_{xx} = 588791kg \cdot m^{2}\), \(J_{xz} = 24242kg \cdot m^{2}\), \(J_{yy} = 1303212kg \cdot m^{2}\) and \(J_{zz} = 1534164kg \cdot m^{2}\). The system is subjected to the initial conditions \(\alpha (0) = 32^{ \circ }\), \(\beta (0) = 2^{ \circ }\), \(\sigma (0) = 58^{ \circ }\), and \(p(0) = q(0) = r(0) = 0^{ \circ } /s\). The guidance commands are set as \(\alpha_{ref} = 30^{ \circ }\), \(\beta_{ref} = 0^{ \circ }\) and \(\sigma_{ref} = 60^{ \circ }\).

The external disturbance \(\Delta M\) is set as

$$ \Delta M = \left[ \begin{gathered} 1 + \sin (t) \hfill \\ 1 + \sin (t) \hfill \\ 1 + \sin (t) \hfill \\ \end{gathered} \right] \times 10^{5} N.m, $$
(41)

and the actuator faults are given as

$$ F_{\delta } = \frac{1}{2} \times \left[ \begin{gathered} 1 + \sin (0.4t) \hfill \\ 1 + \sin (0.4t) \hfill \\ 1 + \sin (0.4t) \hfill \\ \end{gathered} \right] \times 10^{5} N.m, $$
(42)
$$ \Lambda_{x} = \left\{ \begin{gathered} 1, \hfill \\ 0.75 + {0}{\text{.1sin(0}}{.5}t + \pi /3) \hfill \\ \end{gathered} \right., $$
(43)
$$ \Lambda_{y} = \left\{ \begin{gathered} 1, \hfill \\ 0.5 + {0}{\text{.1sin(0}}{.5}t + 2\pi /3) \hfill \\ \end{gathered} \right., $$
(44)
$$ \Lambda_{z} = \left\{ \begin{gathered} 1, \hfill \\ 0.6 + {0}{\text{.1sin(0}}{.5}t + \pi ) \hfill \\ \end{gathered} \right.. $$
(45)

In the simulation, fault \(\Lambda\) is exerted at 6 s and bias fault \(F_{\delta }\) is introduced at 9 s. The unmatched disturbance \(\Delta f\) in (3) is assumed to be 0 in the first test, so that no estimation of state \(X_{2}\) is needed. The parameters for the sliding manifold are set as \(k_{1} = [1.1,\;1.1,\;1.1]\), \(k_{2} = [4,\;4,\;4]\), \(a_{1} = [1.1,\;1.1,\;1.1]\), \(a_{2} = [3,\;3,\;3]\). For the A2-NFTSMC scheme in (22)–(25), the parameters are set as \(k_{3} = [1.2,\;1.2,\;1.2]\), \(k_{4} = [1.2,\;1.2,\;1.2]\), \(c_{1} = 0.5\), \(c_{2} = 1.2\), \(r = [1,\;1,\;1]\), \(l_{0} = [30,\;30,\;30]\), \(m_{1} = [0.6,\;0.6,\;0.6]\), \(m_{1} = [1.2,\;1.2,\;1.2]\), and \(\tau\) in (40) is chosen as the simulation sample step.

For the NFTSMC, the controller is expressed as

$$ u_{i} = - F_{i} - k_{1i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} \left( {\frac{{\phi_{i} }}{{k_{1i} }} + x_{2i} } \right) - k_{3i} sign^{{c_{1} }} (S_{i} ) - k_{4i} sign^{{c_{2} }} (S_{i} ) + k_{5i} sign(S_{i} ), $$
(46)

with \(k_{5} = [1,\;1,\;1]\) and the other parameters are the same as for the A2-NFTSMC. We notice that the NFTSMC is discontinuous and, to set the values of parameters \(k_{5}\), the upper bounds of \(\Delta D_{i}\) must be known in advance. Since using the sign function in (46) can give rise to unacceptably large chattering, the sign function \(sign(S_{i} )\) is often substituted by the function:

$$ \varpi_{i} = \frac{{e^{{\rho S_{i} }} - 1}}{{e^{{\rho S_{i} }} + 1}},\;\rho > 0,\;i = x,y,z. $$
(47)

In this test, \(\rho\) is chosen as 1000. The results of simulation test1 are demonstrated in Figs. 1, 2 and 3. As shown in Fig. 1, both controllers enforce the attitude angles to track the command in finite time. However, the tracking precision of the proposed controller is much superior to that of the NFTSMC, which indicates that the proposed controller exhibits better robustness. In addition, the proposed controller is much less influenced by the actuator faults introduced at 6 s and 9 s, respectively. This also can be observed in Fig. 2, where the controllers drive the state vector \(X_{2}\) to the origin. The reason for the difference between the performances of the proposed controller and the NFTSMC is that there has to be a tradeoff between the control accuracy and the chattering reduction for the parameter selection of the NFTSMC. In contrast, for the proposed controller, as it represents a higher order control strategy and it is inherently continuous, both chattering reduction and control precision can be guaranteed simultaneously. It can also be observed in Fig. 3 that the proposed control strategy does not produce a very large control torque magnitude at the initial phase as the NFTSMC does. The initial output value of the traditional NFTSMC is sensitive to changes of \(k_{5i}\), i.e., a small rise of the value of \(k_{5i}\) can lead to a large increase of the control torque value. The proposed controller eliminates this sensitivity. This feature is very beneficial in real applications when the magnitude of the controller output signal should not surpass the controller’s saturation value.

Fig. 1
figure 1

Comparison of angle of attack, sideslip angle and bank angle between NFTSMC and proposed controller

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Fig. 2
figure 2

Comparison of convergence of X2 between NFTSMC and proposed controller

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Fig. 3
figure 3

Comparison of control torque between NFTSMC and proposed controller

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Up to now, the discussion was based on the situation when the state vector \(X_{2}\) is obtainable. This is not practical in real applications as \(\dot{\Omega }\) contains an unknown part \(\Delta f\). In the second simulation test, \(\Delta f\) is taken into account:

$$ \Delta f = \left[ \begin{gathered} - \sin (t + 0.57) \hfill \\ - \sin (t + 0.57) \hfill \\ - \sin (t + 0.57) \hfill \\ \end{gathered} \right]N \cdot m. $$
(48)

The fixed-time extended state observer based non-singular fast terminal sliding mode controller (FESOB–NSFTSMC) in reference [31] is implemented for comparison purposes. The observer and controller of the FESOB–NSFTSMC can be expressed as

$$ \begin{aligned} \dot{Z}_{1} &= Z_{2} + e^{\eta + b} \upsilon {\kern 1pt} sign^{{\alpha_{1} }} (X_{1} - Z_{1} )\\ &\quad + e^{\eta + b} (1 - \upsilon {\kern 1pt} {\kern 1pt} )sign^{{\beta_{1} }} (X_{1} - Z_{1} ), \\ \dot{Z}_{2} &= Z_{3} + e^{2\eta + b} \upsilon {\kern 1pt} {\kern 1pt} sign^{{\alpha_{2} }} (X_{1} - Z_{1} )\\ &\quad + e^{2\eta + b} (1 - \upsilon {\kern 1pt} )sign^{{\beta_{2} }} (X_{1} - Z_{1} ) + U, \\ \dot{Z}_{3} &= e^{3\eta + b} \upsilon {\kern 1pt} {\kern 1pt} sign^{{\alpha_{3} }} (X_{1} - Z_{1} ) + e^{3\eta + b} (1 - \upsilon {\kern 1pt} {\kern 1pt} )\\ &\quad sign^{{\beta_{3} }} (X_{1} - Z_{1} ) + \Upsilon sign(X_{1} - Z_{1} ), \\ \end{aligned} $$
(49)
$$ \upsilon {\kern 1pt} = \left\{ \begin{gathered} 0,\quad {\text{if}}\,t < T_{u} \hfill \\ 1,\quad {\text{if}}\,t > T_{u} \hfill \\ \end{gathered} \right., $$
(50)

and

$$ u_{i} = - F_{i} - k_{1i} a_{1i} \left| {x_{1i} } \right|^{{a_{1i} - 1}} \left( {\frac{{\phi_{i} }}{{k_{1i} }} + z_{2i} } \right) - k_{3i} sign^{{c_{1} }} (S_{i} ) - k_{4i} sign^{{c_{2} }} (S_{i} ) - z_{3i} ,\;i = x,y,z, $$
(51)

where \(Z_{1} = [z_{1x} ,z_{1y} ,z_{1z} ]\), \(Z_{2} = [z_{2x} ,z_{2y} ,z_{2z} ]\), \(Z_{3} = [z_{3x} ,z_{3y} ,z_{3z} ]\), \(T_{u} = 0.1\), \(\alpha_{1} = 0.8\), \(\beta_{1} = 1.8\), \(\eta = 1.5\), \(b = 1.4\) and \(\Upsilon = 1\). \(\phi_{i}\) and \(S_{i}\) have the same form and parameters as in (20) and (24), only with the element \(x_{2i}\) substituted by \(z_{2i}\). It is worth noting that, although knowledge of the disturbance bounds is not required for the controller as stated in reference [31], the upper bounds must indeed be known in advance for the design of parameter \(\Upsilon\) in the observer. Therefore, the FESOB–NSFTSMC strategy still does not avoid the necessity to learn the disturbance bounds.

For the proposed controller, the parameters are selected as the same as in test 1. Instead of using state or disturbance observers, a time-delay estimation (TDE) scheme is applied to estimate \(X_{2}\). The benefit of the TDE is its simplicity and straight-forwardness in applications [41, 42]. Here, the TDE scheme is designed as follows.

Integrating both sides of equation \(\dot{\Omega } = R\omega + \Delta f\) yields

$$ h(t) = \int_{0}^{t} {\Delta f{\kern 1pt} {\text{d}}} \tau = \Omega - \Omega (0) - \int_{0}^{t} {R\omega {\kern 1pt} {\text{d}}} \tau . $$
(52)

The estimation of \(\Delta f\), \(\Delta \hat{f}\), then can be calculated as

$$ \Delta \hat{f} = \left( {h(t) - h(t - \tau_{0} )} \right)/\tau_{0} , $$
(53)

where \(\tau_{0}\) is a short time delay that is chosen as the simulation sample step. The estimation of \(X_{2}\) is further deduced as

$$ \hat{X}_{2} = R\omega - \dot{\Omega }_{ref} + \Delta \hat{f}. $$
(54)

The results of simulation test 2 can be found in Figs. 4, 5, 6, 7 and 8. From Fig. 4, we can see that both controllers have similar fault tolerances and the convergence times of the angles are the same. However, Fig. 5 (the zoomed version of Fig. 4) further illustrates that the proposed controller causes better tracking precision. From Figs. 6 and 7, it is obvious that the proposed controller does not cause large oscillations in the initial stage as the FESOB–NSFTSMC does. Furthermore, it does not cause as much chattering as the latter during the stable phase. The superiority of the proposed controller over the FESOB–NSFTSMC can also be observed from Fig. 8. The control torque of the FESOB–NSFTSMC suffers large oscillations during the initial stage compared with the proposed controller. Although the FESOB–NSFTSMC is also a continuous controller, there has to be an observer to estimate the disturbance \(\Delta D\). In contrast, the adaptive strategy of the proposed method enables the controller to dominate the disturbances without requiring additional control effort.

Fig. 4
figure 4

Comparison of angle of attack, sideslip angle and bank angle between FESOB–NSFTSMC and proposed controller

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Fig. 5
figure 5

Comparison of angle of attack, sideslip angle and bank angle between FESOB–NSFTSMC and proposed controller (zoomed)

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Fig. 6
figure 6

Comparison of convergence of X2 between FESOB–NSFTSMC and proposed controller

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Fig. 7
figure 7

Comparison of convergence of X2 between FESOB–NSFTSMC and proposed controller (zoomed)

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Fig. 8
figure 8

Comparison of control torque between FESOB–NSFTSMC and proposed controller

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5 Conclusions

In this paper, an A2-NFTSMC is proposed as the fault tolerant attitude control scheme for RLVs during their reentry phase. The proposed controller is established based on the NFTSMC which exhibits a fast/fixed-time convergence. By resorting to an adaptive second-order sliding mode control strategy, the traditional discontinuous term of the NFTSMC is avoided and the chattering is reduced, while the robustness of the controller is increased. The upper bounds of the disturbances are no longer needed to be known in advance. Moreover, the newly proposed NFTSMC output value is no longer sensitive to changes of the magnitude of the disturbance as the traditional NFTSMC. Simulation results demonstrate that, compared with the existing NFTSMC based controllers, the proposed one has better robustness and causes less chattering, while its control structure is kept simple and concise.