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Fault-tolerant control of hypersonic morphing vehicle based on the predefined-time disturbance observer
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Yiheng LI1, Wenjie ZHANG2, *, Mingkai WANG3, Qunli XIA1, Yangxin LIU1
Journal of Systems Engineering and Electronics | 2026, 37(3) : 1019 - 1029
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Journal of Systems Engineering and Electronics | 2026, 37(3): 1019-1029
• CONTROL THEORY AND APPLICATION •
Fault-tolerant control of hypersonic morphing vehicle based on the predefined-time disturbance observer
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Yiheng LI1, Wenjie ZHANG2, *, Mingkai WANG3, Qunli XIA1, Yangxin LIU1
Affiliations
  • 1School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
  • 2Beijing Institute of Electronic System Engineering, Beijing 100854, China
  • 3Hangzhou Innovation Institute, Beihang University, Hangzhou 311115, China
Published: 2026-06-18 doi: 10.23919/JSEE.2026.000125
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To address the attitude control problem under the uncertainty, external disturbance, and actuator failure, a predefined-time fault-tolerant control method based on a predefined time disturbance observer is proposed. First, the dynamics model of hypersonic morphing vehicle (HMV) is established, and the control system is designed as an outer-loop attitude angle control loop and an inner-loop angular rate control loop considering the actuator failure problem. Secondly, a predefined-time disturbance observer is designed to estimate the comprehensive disturbances, and compensate in the control law. By integrating back-stepping control with predefined-time theory, a predefined-time attitude tracking control method is proposed, enabling the convergence time of the attitude tracking error to be designed through a simple parameter. Rigorous Lyapunov function analysis has demonstrated that the attitude tracking error can converge to an arbitrarily small neighborhood around the origin within a predefined time, and all signals in the closed-loop system are bounded. Finally, comparative simulations validate the effectiveness of the proposed method.

hypersonic morphing vehicle (HMV)  /  actuator failure  /  predefined-time disturbance observer  /  predefined-time fault-tolerant control
Yiheng LI, Wenjie ZHANG, Mingkai WANG, Qunli XIA, Yangxin LIU. Fault-tolerant control of hypersonic morphing vehicle based on the predefined-time disturbance observer[J]. Journal of Systems Engineering and Electronics, 2026 , 37 (3) : 1019 -1029 . DOI: 10.23919/JSEE.2026.000125
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1.　Introduction
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Hypersonic morphing vehicles are capable of altering their aerodynamic configurations according to mission requirements to improve the performance [1]. However, with the change of the aerodynamic configurations of hypersonic morphing vehicle (HMV), its moment of inertia, mass distribution, aerodynamic force, and aerodynamic moment will also change significantly, which imposes strong nonlinearities, disturbances, and coupling on HMV [2,3]. Meanwhile, due to the complex structure and harsh flight conditions, HMV are more susceptible to actuator failures leading to performance degradation or even loss of control. Therefore, designing a high-reliability attitude controller with high tracking precision and fast convergence for HMV is of great significance.
The existing attitude control methods of HMV are roughly categorized into linear and nonlinear control methods [4]. Linear control methods encompass proportional-integral-differential (PID) [5], linear parameter varying (LPV) [6–8], gain scheduled control [9], etc. These methods are widely employed in early research due to their advantages of simple structure and ease in parameter tuning. Nevertheless, most linear control methods need to linearize the dynamic model around the equilibrium point and omit the nonlinear characteristics, which leads to the limitations of such methods in solving the attitude control problem of HMV with strong nonlinearity. The back-stepping control method is a commonly-used nonlinear control method for solving the attitude control problem of morphing aircraft [10–13]. To deal with the HMV attitude control problem, back-stepping control methods are usually combined with disturbance observers [14–17]. The disturbance observer can accurately estimate the disturbance and make compensations in the control law by reconstructing the external uncertain information, which is one of the most effective methods to suppress the uncertain disturbance and improve the robustness of the control system.
To ensure the rapid convergence of tracking errors for HMV during high-dynamic flight and morphing processes, the finite-time control theory [18–20] and fixed-time control [21–25] are proposed based on the asymptotically stable method, which can enhance the transient performance of the control system. However, the convergence time of the finite-time control depends on the initial states of the system, and the convergence time of fixed-time control methods is complicated to design, making it impossible to predefine through a simple parameter. Therefore, in recent years, predefined-time control theory has received widespread attention. In [26], Lyapunov-like conditions were studied to ensure a class of dynamical systems exhibit predefined-time stability, and effective proofs of the predefined-time stability and its equivalent conditions were provided. In [27], a novel predefined-time sliding mode control method based on the predefined-time neural network disturbance observer was proposed for attitude control during the large-scale morphing flight of the morphing vehicle. The proposed control method can track the attitude to the desired values within the predefined time.
The harsh flight environment and complex structure of HMV can easily lead to actuator failures, thus reducing control performance and system stability. In order to solve this problem, a variety of studies on fault-tolerant control for aircraft have been proposed [28–33]. Liang et al. [28] designed a back-stepping fault-tolerant controller based on the fixed-time observer to deal with the problem of the attitude control of the morphing aircraft, which enables stable control of a closed-loop system in case of partial actuator failure. Li et al. [29] proposed a back-stepping fault-tolerant control method based on the fixed-time neural network observer to address the tracking control problem of morphing aircraft with model uncertainties, external disturbances, and actuator faults, and the designed controller was proved to be more adaptive and robust to various uncertainties and disturbances. However, regarding the control problem of morphing aircraft, the existing fault-tolerant control methods rarely manage to simultaneously address the rapid convergence of tracking errors and the robustness of control when facing complex disturbances from multiple sources, and they also falls short of accurately predefining the convergence time for the errors of controllers and observers.
Motivated by the preceding discussion, a fault-tolerant control method based on a predefined-time disturbance observer is proposed for the attitude control of HMV with uncertainty, external disturbances, and actuator failure. The main contributions of this paper are as follows.
(i) A predefined-time fault-tolerant control method is developed based on predefined-time disturbance observer. Compared with the control methods in [22] and [28], the tracking error of the proposed controller is guaranteed to converge to an arbitrarily small neighborhood of the origin within a predefined-time, and the convergence time is simple to design and is determined by a parameter.
(ii) The predefined-time control theory is introduced to design the disturbance observers for estimating the comprehensive disturbances of the HMV under model uncertainties, external disturbances, and actuator failures. Compared to the method in [15], this approach significantly enhances the accuracy of disturbance estimation and control robustness, while also ensuring that the observer error converges within the predefined-time.
The rest of this paper is organized as follows. Section 2 introduces the attitude dynamics model of HMV with actuator failure. Section 3 designs the predefined-time disturbance observer, and the predefined-time fault-tolerant controller, and proves the stability of the system. Section 4 demonstrates the superiority of the proposed control method by comparative simulation. Section 5 presents the conclusive remarks and discusses the future works.
2.　Modeling and preliminaries
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In this section, firstly, the attitude dynamics model of HMV is established and converted into control-oriented state space equations. Secondly, a fault-tolerant control model of HMV is established by considering the problem of actuator failure during the flight. Finally, the relevant lemmas and theorems used in the subsequent controller design and proof are given.
2.1　Attitude dynamics model of HMV
The configuration schematic of the HMV with a variable sweep angle investigated in this paper is shown in Fig. 1, where $ \zeta $ represents the sweep angle.
Considering the attitude control problem during the re-entry phase, the dynamics model of the HMV can be expressed as follows:
$ \left\{ \begin{gathered} {{{\boldsymbol{\dot x}}}_1} = {{\boldsymbol{f}}_1} + {{\boldsymbol{g}}_1}{{\boldsymbol{x}}_2} + {{\boldsymbol{d}}_1} \\ {{{\boldsymbol{\dot x}}}_2} = {{\boldsymbol{f}}_2} + {{\boldsymbol{g}}_2}{\boldsymbol{u}} + {{\boldsymbol{d}}_2} \\ \end{gathered} \right. $
where $ {{\boldsymbol{x}}_1} = {\left[ {\alpha ,\beta ,\gamma } \right]^\text{T}} $ denotes the attitude angle vector consisting of angle of attack, sideslip angle and bank angle; $ {{\boldsymbol{x}}_2} = {\left[ {{\omega _{x1}},{\omega _{y1}},{\omega _{z1}}} \right]^\text{T}} $ denotes angular rate vector consisting of the angular rates of the roll, pitch, and yaw channels. $ {\boldsymbol{u}} = {\left[ {{\delta _x},{\delta _y},{\delta _z}} \right]^\text{T}} $ denotes the system control input vector consisting of the roll elevator deflection, yaw elevator deflection, and pitch elevator deflection. d1, d2 denote comprehensive disturbances caused by factors such as model uncertainties and external disturbances. The rest of the components f1, g1, f2, g2 in (1) can be expressed as $ {{\boldsymbol{f}}_1} = {\left[ {\begin{array}{*{20}{c}} {{f_{11}}},&{{f_{12}}},&{{f_{13}}} \end{array}} \right]^\text{T}} $, $ {f_{11}} $, $ {f_{12}} $, $ {f_{13}} $ can be expressed as
$ \left\{\begin{aligned}& {f_{11}} = \frac{{ - Y + mg\cos\;\theta \cos\;\gamma }}{{mV\cos\; \beta }} \\& {f_{12}} = \frac{{N + mg\cos\;\theta \sin\; \gamma }}{{mV}} \\& {f_{13}} = \frac{{Y(\tan\; \theta \sin\;\gamma + \tan\;\beta ) + N\tan\;\theta \cos\;\gamma }}{{mV}} - \\&\qquad\frac{{g\cos\;\theta \cos\; \gamma \tan\; \beta }}{V}\end{aligned} \right. ,$
$ \begin{split}& {{\boldsymbol{f}}_2} = \left[ \begin{gathered} \frac{{(J_y^2 + J_{xy}^2 - {J_y}{J_z})}}{{{J_x}{J_y} - J_{xy}^2}}{\omega _y}{\omega _z} + \frac{{({J_z} - {J_x} - {J_y}){J_{xy}}}}{{{J_x}{J_y} - J_{xy}^2}}{\omega _x}{\omega _z} \\ \frac{{({J_x}{J_z} - J_x^2 - J_{xy}^2)}}{{{J_x}{J_y} - J_{xy}^2}}{\omega _x}{\omega _z} + \frac{{({J_x} + {J_y} - {J_z}){J_{xy}}}}{{{J_x}{J_y} - J_{xy}^2}}{\omega _y}{\omega _z} \\ \qquad\quad \frac{{({J_x} - {J_y}){\omega _x}{\omega _y} + {J_{xy}}(\omega _x^2 - \omega _y^2)}}{{{J_z}}} \\ \end{gathered} \right] + \\&\qquad\quad \left[ \begin{gathered} \frac{{{J_y}{C_{mx0}}\bar q{S _\text{ref}}{L_\text{ref}} + {J_{xy}}{C_{my0}}\bar q{S _\text{ref}}{L_\text{ref}}}}{{{J_x}{J_y} - J_{xy}^2}} \\ \frac{{{J_{xy}}{C_{mx0}}\bar q{S _\text{ref}}{L_\text{ref}} + {J_x}{C_{my0}}\bar q{S _\text{ref}}{L_\text{ref}}}}{{{J_x}{J_y} - J_{xy}^2}} \\ \qquad\qquad\frac{{{C_{mz0}}\bar q{S _\text{ref}}{L_\text{ref}}}}{{{J_z}}} \\ \end{gathered} \right],\end{split} $
$ {{\boldsymbol{g}}_1} = \left[ {\begin{array}{*{20}{c}} { - \cos\; \alpha \tan\; \beta }&{\sin\; \alpha \tan\; \beta }&1 \\ {\sin\; \alpha }&{\cos\; \alpha }&0 \\ {\cos\; \alpha \sec\; \beta }&{ - \sin\; \alpha \sec\; \beta }&0 \end{array}} \right], $
$ {{\boldsymbol{g}}_2} = \bar q{S _\text{ref}}{L_\text{ref}}{\boldsymbol{JC}} $
where $ {\boldsymbol{J}} $, $ {\boldsymbol{C}} $ can be expressed as
$ {\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} {\dfrac{1}{{{J_x}{J_y} - J_{xy}^2}}}&0&0 \\ 0&{\dfrac{1}{{{J_x}{J_y} - J_{xy}^2}}}&0 \\ 0&0&{\dfrac{1}{{{J_z}}}} \end{array}} \right], $
$ {\boldsymbol{C}} = \left[ {\begin{array}{*{20}{c}} {{J_y}}&{{J_{xy}}}&0 \\ {{J_{xy}}}&{{J_x}}&0 \\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {C_{mx}^{{\delta _x}}}&{C_{mx}^{{\delta _y}}}&{C_{mx}^{{\delta _z}}} \\ {C_{my}^{{\delta _x}}}&{C_{my}^{{\delta _y}}}&{C_{my}^{{\delta _z}}} \\ {C_{mz}^{{\delta _x}}}&{C_{mz}^{{\delta _y}}}&{C_{mz}^{{\delta _z}}} \end{array}} \right], $
where $ m $ denotes the mass of HMV, $ V $ denotes the flight velocity, $ \theta $ denotes the flight path angle, $ g $ denotes the gravitational acceleration. $ Y = L + {F_{s{y_v}}} $, where $ L = \bar q{S _\text{ref}}{C_L} $ denotes the lift and $ {F_{s{y_v}}} $ denotes the component of the deformation additional force in the y-axis of the velocity coordinate system. $ N = Z + {F_{s{z_v}}} $, where $ Z = \bar q{S _\text{ref}}{C_Z} $ denotes the lateral force and $ {F_{s{z_v}}} $ denotes the component of the deformation inertia force in the z-axis of the velocity coordinate system. $ \bar q = 0.5\rho {V^2} $ denotes the dynamic pressure, $ \rho $ denotes the atmosphere density. $ {S _\text{ref}} $ denotes the reference area. $ {J_x} $, $ {J_y} $, $ {J_z} $ denote the roll, yaw and pitch moments of inertia. $ {J_{xy}} $ denotes the inertial product of HMV. $ {M_x} $, $ {M_y} $, $ {M_z} $ denote roll, yaw, and pitch aerodynamic moments respectively, which can be expressed as $ {M_x} = {C_{mx}}\bar q{S _\text{ref}}{L_\text{ref}} $, $ {M_y} = {C_{my}}\bar q{S _\text{ref}}{L_\text{ref}} $, $ {M_z} = {C_{mz}}\bar q{S _\text{ref}}{L_\text{ref}} $, where $ {L_\text{ref}} $ denotes the reference length. $ {{\boldsymbol{M}}_s} = {[\begin{array}{*{20}{c}} {{M_{sx}}},&{{M_{sy}}},&{{M_{sz}}} \end{array}]^\text{T}} $ denotes the component of the inertia moment due to deformation in each axis of the elastic body coordinate system. The inertia forces and moments generated during deformation can be expressed as
$ \left\{\begin{aligned}&{{\boldsymbol{F}}}_{s}=-\sum _{i=1}^{2}{m}_{i}\frac{{{\mathrm{d}}}^{2}{{\boldsymbol{s}}}_{i}}{{\mathrm{d}}{t}^{2}}\\&{{\boldsymbol{M}}}_{s}=-\sum _{i=1}^{2}\left({m}_{i}{{\boldsymbol{s}}}_{i}\times \frac{{\mathrm{d}}{\boldsymbol{V}}}{{\mathrm{d}}t}+{m}_{i}{{\boldsymbol{s}}}_{i}\times \frac{{{\mathrm{d}}}^{2}{{\boldsymbol{s}}}_{i}}{{\mathrm{d}}{t}^{2}}\right)+\sum _{i=1}^{2}{{\boldsymbol{s}}}_{i}\times {m}_{i}{\boldsymbol{g}}\end{aligned}\right. $
where si denotes the position vector from the aircraft’s center of mass to the wing’s center of mass. $ {m_i}(i = 1,2) $ denotes the mass of the wings on both sides of the HMV. The velocity vector V will be decomposed in the body coordinate system. The change in the sweep angle of HMV not only causes the inertial forces and moments but also leads to the change in aerodynamic forces and moments due to the transformation of the aerodynamic configuration.
The aircraft model and aerodynamic parameters in this paper utilize the data from [34]. The coefficients of aerodynamic force and aerodynamic moment can be expressed as
$ \begin{split}& {C_i} = {C_{i0}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ) + C_i^{{\delta _x}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ){\delta _x} + \\&\quad C_i^{{\delta _y}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ){\delta _y} + C_i^{{\delta _z}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ){\delta _z} \end{split} $
where i=L, Z, mx, my, mz. The aerodynamic coefficients and aerodynamic moment coefficients are fitted to formulas related to seven variables including Mach $ {\mathrm{Ma}} $, angle of attack $ \alpha $, sideslip angle $ \beta $, roll elevator deflection $ {\delta _x} $, yaw elevator deflection $ {\delta _y} $, pitch elevator deflection $ {\delta _z} $ and sweep angle $ \zeta $. For example, $ i = mz $ for the pitch moment coefficient, $ {C_{mz0}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ) $ denotes the pitch moment coefficient at $ {\delta _x} = {\delta _y} = {\delta _z} = 0 $, $ C_{mz}^{{\delta _x}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ) $ denotes the partial derivative of the pitch moment coefficient with respect to $ {\delta _x} $, $ C_{mz}^{{\delta _y}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ) $ denotes the partial derivative of the pitch moment coefficient with respect to $ {\delta _y} $, $ C_{mz}^{{\delta _z}}({\mathrm{Ma}},\alpha ,\beta ,\zeta ) $ denotes the partial derivative of the pitch moment coefficient with respect to $ {\delta _z} $.
The singularities of the matrices g1 and g2 in (1) are analyzed below. Taking the determinant of matrix g1, it can be obtained that when $ \beta \ne 90^\circ $, g1 is a non-singular matrix. In matrix $ {{\boldsymbol{g}}_2} $, the absolute values of the elements $ C_{mx}^{{\delta _x}} $, $ C_{my}^{{\delta _y}} $, and $ C_{mz}^{{\delta _z}} $ along the main diagonal are much larger than the absolute values of the other elements, hence matrix g2 can be approximated as a diagonal matrix. For HMV under normal flight conditions, g2 is a non-singular matrix.
2.2　Actuator failure modelling
In this paper, the actuator model of HMV with the gain and bias faults can be represented as follows:
$ {{\boldsymbol{u}}_{\mathrm{F}}}(t) = {\boldsymbol{\eta }}(t){\boldsymbol{u}}(t) + {\boldsymbol{\varLambda }}(t),\;\;{t \geqslant {t_{\mathrm{f}}}} $
where $ {{\boldsymbol{u}}_{\mathrm{F}}}(t) $ is the control input under actuator failure, $ {\boldsymbol{u}}(t) $ is the control inputs to be designed, $ {t_{\mathrm{f}}} $ is the occurrence time of the actuator faults, $ {\boldsymbol{\eta }}(t) = {\text{diag}}\left( {\eta _1}(t), {\eta _2}(t),{\eta _3}(t) \right) $ is the contribution efficiency of the actuator satisfying $ 0 \leqslant {\eta _i}(t) \leqslant 1 $ $ \left( {i = 1,2,3} \right) $. $ {\boldsymbol{\varLambda }}(t) = \left[ {\varLambda _1}(t),{\varLambda _2}(t), {\varLambda _3}(t) \right]^\text{T} $ is the actuator fault biases satisfying $ \left|{\varLambda }_{i}(t)\right|\le \bar{\varLambda }(t) \left(i=1,2,3\right) $, $ \bar \varLambda (t) $ is a positive constant. Therefore, the fault-tolerant model of HMV considering actuator failures is represented as follows:
$ \left\{ \begin{gathered} {{{\boldsymbol{\dot x}}}_1} = {{\boldsymbol{f}}_1} + {{\boldsymbol{g}}_1}{{\boldsymbol{x}}_2} + {{\boldsymbol{d}}_1} \\ {{{\boldsymbol{\dot x}}}_2} = {{\boldsymbol{f}}_2} + {{\boldsymbol{g}}_2}{\boldsymbol{u}} + {{\boldsymbol{D}}_2} \\ \end{gathered} \right. $
where $ {{\boldsymbol{D}}_2} = {\boldsymbol{g}}_2({\boldsymbol{\eta}}-{\boldsymbol{I}}){\boldsymbol{u}}+{{\boldsymbol{g}}_2}{\boldsymbol{\varLambda }} + {{\boldsymbol{d}}_2} $.
Remark 1　The actuator fault models employed for the exemplary HMV are also commonly seen in, for instance, [28–32]. This paper treats the effects of actuator faults on aircraft, model uncertainty, and disturbances generated by morphing process as comprehensive disturbances, which is observed based on predefined-time disturbance observer and compensated in control law.
Remark 2　Predefined-time, in essence, is a pre-designed convergence time that utilizes the predefined-time stability theory to achieve the pre-design of the convergence time of tracking error. In this paper, this theory is applied to the HMV control system, enabling the tracking error and the disturbance estimation error to converge to an arbitrarily small neighborhood within the predefined-time. As an important extension of the finite-time stability theory and the fixed-time stability theory, the predefined-time stability theory not only effectively overcomes the problem of initial dependence in traditional finite-time control, but also breaks through the limitation of the implicit correlation between the convergence time and parameters in fixed-time control.
3.　Controller design
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In this section, a predefined-time fault-tolerant controller based on predefined-time disturbance observer is designed for HMV to achieve accurate tracking of attitude angle commands. The controller scheme is illustrated in Fig. 2, where the control system is decomposed into an outer-loop attitude angle tracking control system and an inner-loop attitude angular rate tracking control system. It can be seen from Fig 2 that the output of the outer-loop attitude angle tracking control system is processed through a first-order command filter, which subsequently serves as the input for the inner-loop angular velocity tracking control system. The inner-loop system generates the final control law, establishing a closed-loop control system through the HMV dynamic system.
The design approach of the controller is as follows. Firstly, a predefined-time disturbance observer is designed based on the predefined-time convergence control theory, and the convergence proof of the observer error is given. Then, a predefined-time fault-tolerant controller is designed based on the predefined-time theory, and a predefined-time disturbance observer is used to observe the comprehensive disturbances generated by the model uncertainty, the external disturbances and actuator failures, and to compensate them in the control law. Finally, the stability of the control system is proved based on Lyapunov stability theory and predefined-time control theory.
3.1　Design of predefined-time disturbance observer
According to (11), the general form of the dynamical model of HMV can be expressed as $ {\boldsymbol{\dot x}} = {\boldsymbol{f}} + {\boldsymbol{gu}} + {\boldsymbol{d}} $, where u is the control input, f and g are the known terms in the dynamics model, and d is the disturbance to be observed. The predefined-time disturbance observer is designed as
$ \left\{\begin{aligned}& {{\dot {\hat {\boldsymbol{x}}}}} = {\boldsymbol{f}} + {\boldsymbol{gu}} + {\boldsymbol{\hat d}} + \frac{{({2^{ - p}}{{\boldsymbol{e}}^3} + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}^{2p - 1}})}}{{(1 - p){T_c}\lambda }} \\& {{\dot {\hat {\boldsymbol{d}}}}} = \frac{a}{{(1 - p){T_c}\lambda }}({2^{ - p}}{{\boldsymbol{e}}^3} + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}^{2p - 1}})\end{aligned} \right. $
where $ {\boldsymbol{\hat x}} $ is an estimate of the system state $ {\boldsymbol{x}} $, $ {\boldsymbol{\hat d}} $ is an estimate of the disturbance d. $ a \gt 0 $ is the gain of the predefined-time disturbance observer. $ {\boldsymbol{e}} = {\boldsymbol{x}} - {\boldsymbol{\hat x}} $ is the estimation error of the system state, and the time derivative of e is
$ {\boldsymbol{\dot e}} = {\boldsymbol{\dot x}} - {{\dot {\hat {\boldsymbol{x}}}}} = {\boldsymbol{d}} - {\boldsymbol{\hat d}} - \frac{{({2^{ - p}}{{\boldsymbol{e}}^3} + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}^{2p - 1}})}}{{(1 - p){T_c}\lambda }}. $
The stability of the predefined-time disturbance observer and the convergence of the disturbance observer errors will be proved subsequently.
3.2　Design of predefined-time fault-tolerant controller
Based on the back-stepping control method, the control system is divided into the outer-loop attitude angle tracking control system and the inner-loop angular rate tracking control system. $ {{\boldsymbol{x}}_{1d}} = {[ {\begin{array}{*{20}{c}} {{\alpha _d}},&{{\beta _d}},&{{\gamma _d}} \end{array}} ]^\text{T}} $ is the attitude angle command from the guidance system. The tracking error of the outer-loop attitude angle is $ {{\boldsymbol{e}}_1} = {{\boldsymbol{x}}_1} - {{\boldsymbol{x}}_{1d}} $, the tracking error of the inner-loop angular rate is $ {{\boldsymbol{e}}_2} = {{\boldsymbol{x}}_2} - {{\boldsymbol{x}}_{2d}} $, where $ {{\boldsymbol{x}}_{2d}} $ is the output of the first-order filter associated with the outer-loop virtual control law. Design the first order filter of the virtual control law as
$ {\tau _2}{{\boldsymbol{\dot x}}_{2d}} + {{\boldsymbol{x}}_{2d}} = {{\boldsymbol{x}}_{2c}} $
where $ {\tau _2} $ is the filter parameter. $ {{\boldsymbol{x}}_{2c}} $ is the virtual control law, which will be designed subsequently. In this paper, we define $ {{\boldsymbol{x}}_{2d}}{\text{(0)}} = {{\boldsymbol{x}}_{2c}}{\text{(0)}} $ and ignore the filter error. The design of the predefined-time fault-tolerant controller is described below.
Step 1　For the outer-loop attitude angle control system, the time derivative of e1 is
$ {{\boldsymbol{\dot e}}_1} = {{\boldsymbol{f}}_1} + {{\boldsymbol{g}}_1}{{\boldsymbol{x}}_2} + {{\boldsymbol{d}}_1} - {{\boldsymbol{\dot x}}_{1d}}. $
Design the virtual control law $ {{\boldsymbol{x}}_{2c}} $ of the outer-loop attitude angle control system as
$ \begin{split}&\qquad {{\boldsymbol{x}}_{2c}} = {\boldsymbol{g}}_1^{ - 1}( - {{\boldsymbol{f}}_1} - {{{\boldsymbol{\hat d}}}_1} + {{{\boldsymbol{\dot x}}}_{1d}} - \\& \frac{1}{{(1 - p){T_c}\lambda }}({2^{ - p}}{{\boldsymbol{e}}_1}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_1}^{2p - 1})) \end{split} $
where $ \lambda \gt 0 $, $ 0 \lt p \lt 1 $. $ {{\boldsymbol{\hat d}}_1} $ is an estimate of the comprehensive disturbance of the outer loop. According to (12), the predefined-time disturbance observer for the outer-loop attitude angle tracking control is designed as
$ \left\{\begin{aligned}& {{{{\dot {\hat {\boldsymbol{x}}}}}}_1} = {{\boldsymbol{f}}_1} + {{\boldsymbol{g}}_1}{{\boldsymbol{x}}_2} + {{{\boldsymbol{\hat d}}}_{{1}}} + \frac{{({2^{ - p}}{{\boldsymbol{e}}_{10}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{10}}^{2p - 1})}}{{(1 - p){T_c}\lambda }} \\& {{{{\dot {\hat {\boldsymbol{d}}}}}}_1} = \frac{{{a_1}}}{{(1 - p){T_c}\lambda }}({2^{ - p}}{{\boldsymbol{e}}_{10}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{10}}^{2p - 1}) \end{aligned} \right. $
where $ {a_1} $ is the gain of the predefined-time disturbance observer, $ {{\boldsymbol{e}}_{10}} = {{\boldsymbol{x}}_1} - {{\boldsymbol{\hat x}}_1} $ is the state observer error. Substituting (16) into (15) yields
$ {{{\boldsymbol{\dot e}}}_1} = - \frac{{({2^{ - p}}{{\boldsymbol{e}}_1}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_1}^{2p - 1})}}{{(1 - p){T_c}\lambda }} + {{{\tilde {\boldsymbol{d}}}}_1} + {{\boldsymbol{g}}_1}{{\boldsymbol{e}}_2} $
where $ {{\tilde {\boldsymbol{d}}}_1} = {{\boldsymbol{d}}_1} - {{\boldsymbol{\hat d}}_1} $ is the disturbance estimation error.
Step 2　For the inner-loop angular rate control system, the time derivative of $ {{\boldsymbol{e}}_2} $ is
$ {{\boldsymbol{\dot e}}_2} = {{\boldsymbol{f}}_2} + {{\boldsymbol{g}}_2}{\boldsymbol{u}} + {{\boldsymbol{D}}_2} - {{\boldsymbol{\dot x}}_{2d}}. $
The control law u can be constructed as follows:
$ {\boldsymbol{u}} = {\boldsymbol{g}}_2^{ - 1}\left(- \frac{{({2^{ - p}}{{\boldsymbol{e}}_2}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_2}^{2p - 1})}}{{(1 - p){T_c}\lambda }} - {{\boldsymbol{f}}_2} - {{{{\hat {\boldsymbol{D}}}}}_2} + {{{{\dot {\boldsymbol{x}}}}}_{2d}} - {\boldsymbol{g}}_1^\text{T}{{\boldsymbol{e}}_1} \right) $
where $ \lambda \gt 0 $, $ 0 \lt p \lt 1 $. $ {{{\hat {\boldsymbol{D}}}}_2} $ is an estimate of the comprehensive disturbance of the inner-loop angular rate control system. According to (12), the predefined-time disturbance observer for the inner-loop attitude angle tracking control is designed as
$ \left\{ \begin{aligned}& {{{{\dot {\hat {\boldsymbol{x}}}}}}_2} = {{\boldsymbol{f}}_2} + {{\boldsymbol{g}}_2}{\boldsymbol{u}} + {{{{\hat {\boldsymbol{D}}}}}_2} + \frac{{({2^{ - p}}{{\boldsymbol{e}}_{20}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{20}}^{2p - 1})}}{{(1 - p){T_c}\lambda }} \\& {{{{\dot {\hat {\boldsymbol{D}}}}}}_2} = \frac{{{a_2}}}{{(1 - p){T_c}\lambda }}({2^{ - p}}{{\boldsymbol{e}}_{20}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{20}}^{2p - 1}) \end{aligned}\right. $
where $ {a_2} $ is the gain of the predefined-time disturbance observer, $ {{\boldsymbol{e}}_{20}} = {{\boldsymbol{x}}_2} - {{\boldsymbol{\hat x}}_2} $ is the state observer error. Substituting (20) into (19) yields
$ {{\boldsymbol{\dot e}}_2} = - \frac{{({2^{ - p}}{{\boldsymbol{e}}_2}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_2}^{2p - 1})}}{{(1 - p){T_c}\lambda }} + {{{\tilde {\boldsymbol{D}}}}_2} - {\boldsymbol{g}}_1^\text{T}{{\boldsymbol{e}}_1} $
where $ {{{\tilde {\boldsymbol{D}}}}_2} = {{\boldsymbol{D}}_2} - {{{\hat {\boldsymbol{D}}}}_2} $ is the disturbance estimation error.
3.3　Stability analysis
In this subsection, we first prove the convergence of the state observation errors and disturbance estimation errors of the predefined-time disturbance observer in Theorem 1. Subsequently, in Theorem 2, we establish the stability and convergence of the proposed control method, which relies on the results of Theorem 1.
Theorem 1　If the control system of the HMV employs predefined-time disturbance observers, the predefined-time disturbance observer can guarantee the closed loop system bounded, then the state observer errors $ {{\boldsymbol{e}}_{10}} $, $ {{\boldsymbol{e}}_{20}} $ and the disturbance estimation error $ {{\tilde {\boldsymbol{d}}}_1} $, $ {{\tilde {\boldsymbol{D}}}_2} $ can converge to the vicinity of zero within the predefined time.
Proof　Consider the following Lyapunov function candidate:
$ V = \frac{1}{2}{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}} + \frac{1}{2}{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}. $
Take the first-order derivative of $ V $.
$ \dot V = {\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{\dot e}}_{10}} + {\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{\dot e}}_{20}} $
By combining (15) and (17), $ {{\boldsymbol{\dot e}}_{10}} $ can be expressed as
$ {{{\boldsymbol{\dot e}}}_{10}} = {{{\boldsymbol{\dot x}}}_1} - {{{{\dot {\hat {\boldsymbol{x}}}}}}_1} = {{{\tilde {\boldsymbol{d}}}}_1} - \frac{{({2^{ - p}}{{\boldsymbol{e}}_{10}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{10}}^{2p - 1})}}{{(1 - p){T_c}\lambda }}. $
By combining (19) and (21), $ {{\boldsymbol{\dot e}}_{20}} $ can be expressed as
$ {{{\boldsymbol{\dot e}}}_{20}} = {{{\boldsymbol{\dot x}}}_2} - {{{{\dot {\hat {\boldsymbol{x}}}}}}_2} = {{{{\tilde {\boldsymbol{D}}}}}_2} - \frac{{({2^{ - p}}{{\boldsymbol{e}}_{20}}^3 + {2^{1 - 2p}}{\lambda ^{2 - p}}{{\boldsymbol{e}}_{20}}^{2p - 1})}}{{(1 - p){T_c}\lambda }}. $
Substituting (25) and (26) into (24) yields
$ \begin{split}& \dot V = {\boldsymbol{e}}_{10}^\text{T}{{{\tilde {\boldsymbol{d}}}}_1} - \dfrac{{{2^{2 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}}}}{2}} \right)}^2} + {2^{1 - p}}{\lambda ^{2 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}}}}{2}} \right)}^p}}}{{(1 - p){T_c}\lambda }} + \\&\quad {\boldsymbol{e}}_{20}^\text{T}{{{\tilde {\boldsymbol{D}}}}_2} - \dfrac{{{2^{2 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}}}{2}} \right)}^2} + {2^{1 - p}}{\lambda ^{2 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}}}{2}} \right)}^p}}}{{(1 - p){T_c}\lambda }}. \end{split}$
From Lemma 3 in [27], it can be derived that
$ \begin{split}& \dot V \leqslant - \dfrac{{{2^{1 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}}}}{2} + \dfrac{{{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}}}{2}} \right)}^2}}}{{(1 - p){T_c}\lambda }} + {\boldsymbol{e}}_{10}^\text{T}{{{\tilde {\boldsymbol{d}}}}_1} - \\&\quad \dfrac{{{2^{1 - p}}{\lambda ^{2 - p}}{{\left( {\dfrac{{{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}}}}{2} + \dfrac{{{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}}}{2}} \right)}^p}}}{{(1 - p){T_c}\lambda }} + {\boldsymbol{e}}_{20}^\text{T}{{{\tilde {\boldsymbol{D}}}}_2}. \end{split} $
According to Lemma 4 in [24] (Young’s inequality), the following inequalities hold true:
$\left\{ \begin{split}& {\boldsymbol{e}}_{10}^\text{T}{{{\tilde {\boldsymbol{d}}}}_1} \leqslant \frac{{{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}}}}{2} + \frac{{{\tilde {\boldsymbol{d}}}_1^\text{T}{{{\tilde {\boldsymbol{d}}}}_1}}}{2} \\& {\boldsymbol{e}}_{20}^\text{T}{{{\tilde {\boldsymbol{D}}}}_2} \leqslant \frac{{{\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}}}}{2} + \frac{{{\tilde {\boldsymbol{D}}}_2^\text{T}{{{\tilde {\boldsymbol{D}}}}_2}}}{2}\end{split} \right., $
$ \vartheta = \frac{{\left( {{\boldsymbol{e}}_{10}^\text{T}{{\boldsymbol{e}}_{10}} + {\tilde {\boldsymbol{d}}}_1^\text{T}{{{\tilde {\boldsymbol{d}}}}_1} + {\boldsymbol{e}}_{20}^\text{T}{{\boldsymbol{e}}_{20}} + {\tilde {\boldsymbol{D}}}_2^\text{T}{{{\tilde {\boldsymbol{D}}}}_2}} \right)}}{2}. $
According to Lemma 3 in [27] it follows that
$ \dot V \leqslant - \frac{{({V^p}{{\left( {V + \lambda } \right)}^{^{2 - p}}})}}{{(1 - p){T_c}\lambda }} + \vartheta. $
According to the Proposition 1 in [27], it can be proved that the state observer errors e10, e20 can converge to the vicinity of zero within the predefined time. Therefore e10, e20 have upper bounds. Then the following formula can be satisfied: $ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{\boldsymbol{\dot e}}}_{10}}} \right\| = 0 $, $ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{\boldsymbol{\dot e}}}_{20}}} \right\| = 0 $. According to the Proposition 1 in [27] and (13), when $ t \geqslant {T_c} $, the disturbance estimation error $ {{\boldsymbol{\dot e}}_{10}} = {{\tilde {\boldsymbol{d}}}_1} $, $ {{\boldsymbol{\dot e}}_{20}} = {{\tilde {\boldsymbol{D}}}_2} $ stabilizes to zero asymptotically. □
Theorem 2　Considering the dynamics model of HMV in (11), with the predefined-time fault-tolerant controllers in (16) and (20), and the predefined-time disturbance observers in (17) and (21), all signals of the closed-loop control system can be stabilized and bounded, and the attitude tracking error e1 and e2 will also finally converge into the small regions within the predefined time.
Proof　Consider the following Lyapunov function candidate:
$ {V_1} = \frac{1}{2}{\boldsymbol{e}}_1^\text{T}{{\boldsymbol{e}}_1} + \frac{1}{2}{\boldsymbol{e}}_2^\text{T}{{\boldsymbol{e}}_2}. $
Take the first-order derivative of $ {V_1} $.
$ {\dot V_1} = {\boldsymbol{e}}_1^\text{T}{{\boldsymbol{\dot e}}_1} + {\boldsymbol{e}}_2^\text{T}{{\boldsymbol{\dot e}}_2} $
Substituting (24) and (28) into (39) yields
$ \begin{split}& {{\dot V}_1} = - \dfrac{{{2^{2 - p}}\left( {{{\left( {\dfrac{{{\boldsymbol{e}}_1^\text{T}{{\boldsymbol{e}}_1}}}{2}} \right)}^2} + {{\left( {\dfrac{{{\boldsymbol{e}}_2^\text{T}{{\boldsymbol{e}}_2}}}{2}} \right)}^2}} \right)}}{{(1 - p){T_c}\lambda }} + {\boldsymbol{e}}_1^\text{T}{{{\tilde {\boldsymbol{d}}}}_1} - \\&\quad \dfrac{{{2^{1 - p}}{\lambda ^{2 - p}}\left( {{{\left( {\dfrac{{{\boldsymbol{e}}_1^\text{T}{{\boldsymbol{e}}_1}}}{2}} \right)}^p} + {{\left( {\dfrac{{{\boldsymbol{e}}_2^\text{T}{{\boldsymbol{e}}_2}}}{2}} \right)}^p}} \right)}}{{(1 - p){T_c}\lambda }} + {\boldsymbol{e}}_2^\text{T}{{{\tilde {\boldsymbol{D}}}}_2}.\end{split} $
According to Lemma 3 in [27], the following inequalities hold true:
$ {\dot V_1} \leqslant - \frac{{\left( {{V_1}^p{{({V_1} + \lambda )}^{^{2 - p}}}} \right)}}{{(1 - p){T_c}\lambda }} + {\vartheta _1} $
where $ {\vartheta _1} = {\boldsymbol{e}}_1^\text{T}{{\tilde {\boldsymbol{d}}}_1} + {\boldsymbol{e}}_2^\text{T}{{\tilde {\boldsymbol{D}}}_2} $. According to the Proposition 1 in [27] and Theorem 1, the attitude tracking error e1 and e2 will also finally converge into the small regions within the predefined time. Thereby the proof of Theorem 2 is completed. □
4.　Simulation
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In this section, the effectiveness of the predefined-time fault-tolerant control method for HMV based on the predefined-time disturbance observer is verified by comparative simulation. The simulation comprised two scenarios. Scenario 1 is the simulation of the attitude tracking control when the actuator is working normally and without external disturbances. Scenario 2 is the comparison simulation of attitude tracking control considering actuator faults and external disturbances. Scenario 2 is developed on the basis of Scenario 1, where the predefined-time $ {T_c} $ for the HMV is initially established. In both simulation scenarios, amplitude limits are imposed on the rudder deflection angle and its rate to ensure practical engineering applicability. Specifically, the rudder deflection angle is limited to $ \left| {{\delta _z}} \right| \leqslant 30^\circ $, $ \left| {{\delta _y}} \right| \leqslant 20^\circ $, $ \left| {{\delta _x}} \right| \leqslant 20^\circ $.
4.1　Scenario 1: nominal cases
In order to verify the effectiveness of the predefined-time fault-tolerant controller, several predefined-times are set in the simulation to analyze the effect of the predefined-times on the control effect. The predefined-times is set as $ {T_c} = 2\;{\mathrm{s}} $, $ {T_c} = 3\;{\mathrm{s}} $, $ {T_c} = 5\;{\mathrm{s}} $, $ {T_c} = 10\;{\mathrm{s}} $, $ {T_c} = 15\;{\mathrm{s}} $, $ {T_c} = 20\;{\mathrm{s}} $. Other parameters are assigned as $ p = 0.9 $, $ \lambda = 0.5 $ throughout simulations.
The initial values for the simulation of the attitude angle of the HMV are set to $ {\left[ {{\alpha _0},{\beta _0},{\gamma _0}} \right]^\text{T}} = {[15^\circ ,2^\circ ,0^\circ ]^\text{T}} $, and the initial values of the angular rate of the three channels are set to $ 0^\circ /{\mathrm{s}} $. The altitude and velocity of the HMV are set to $ {h_0} = 30\;{\mathrm{km}} $, $ {V_0} = 2\;500{\text{ m/s}} $. The attitude angle command is set to $ {\left[ {{\alpha _d},{\beta _d},{\gamma _d}} \right]^\text{T}} = {[10^\circ ,0^\circ ,10^\circ ]^\text{T}} $. Simulation results of 0−20 s are shown in Fig. 3 and Fig. 4.
The above simulation results indicate that the predefined-time fault-tolerant control methods can guarantee the attitude angle effectively tracks the commands under the different predefined-times, while the convergence time of the tracking error changes with the predefined-times.
The rate of convergence of the tracking error decreases as the predefined-time increases. From the partial enlarged view in Fig. 4, it can be seen that the rudder deflection curves of Tc=2 s and Tc=3 s reach the limit value at the initial moment. However, considering the actual engineering situation of the actuator, the rudder deflection angular rate should be within a reasonable range, and in this paper, the maximum value of the absolute value of the rudder deflection angular rate is limited to 300°/s. In order to balance the rapid convergence speed of tracking error with the reasonableness of the rudder deflection angular rate, $ {T_c} = 5\;{\mathrm{s}} $ is chosen for subsequent simulation.
4.2　Scenario 2: simulation cases with actuator faults
In this scenario, actuator failures, external disturbances, and the change of sweep angles are considered simultaneously. Two comparative methods are set up to compare with the method proposed in this paper, in which comparison method 1 uses the predefined-time disturbance observer designed in [15], marked as ‘PTDO’ in the simulation results, comparison method 2 uses the fixed time disturbance observer-based back-stepping fault-tolerant control method designed in [28], marked as ‘FTDO’ in the simulation results. The sweep angle of the aircraft changes from 30° to 90° between 15 s and 17 s, the curve of sweep angle is shown in Fig. 5. The parameters for the proposed controller are set as $ p = 0.9 $, $ \lambda = 0.5 $, $ {T_c} = 5 $, $ {a_1} = 50 $, $ {a_2} = 50 $, $ {\tau _2} = 0.01. $ The uncertain external disturbances throughout the 50 s simulation are designed as follows:
$ \left\{\begin{aligned}&{{\boldsymbol{d}}}_{1}={\left[0.05\mathrm{sin}\frac{\text{π} t}{30},0.05\mathrm{sin}\frac{\text{π} t}{30},0.05\mathrm{sin}\frac{\text{π} t}{30}\right]}^\text{T}\\&{{\boldsymbol{d}}}_{2}={\left[2\mathrm{sin}\frac{\text{π} t}{5},12\mathrm{sin}\frac{\text{π} t}{5}\text{，}2\mathrm{sin}\frac{\text{π} t}{5}\right]}^\text{T}\end{aligned}\right. .$
Actuator failure at 25 s, the considered actuator failure are modeled as follows:
$ \begin{split}& {\boldsymbol{\eta }} = \left\{\begin{aligned}& {{{\left[ {1,1,1} \right]}^\text{T}}},\;\;{0 \leqslant {{t \lt }}25} \\ & {{{\left[ {0.8,0.8,0.7} \right]}^\text{T}}},\;\;{25 \leqslant {{t}} \leqslant 50} \end{aligned}\right., \\& {\boldsymbol{\varLambda }} = \left\{\begin{aligned}& {{\left[ {0,0,0} \right]}^\text{T}},\;\;{0 \leqslant {{t \lt }}25} \\ & {\Biggl[{38}\begin{aligned}&{0.02\sin \frac{{\text{π} t}}{2}} \\ & {0.03\sin \frac{{\text{π} t}}{2}} \\ & {0.02\sin \frac{{\text{π} t}}{2}} \end{aligned} \Biggl]{38}},\;\;{25 \leqslant {{t}} \leqslant 50} \end{aligned}\right.. \end{split} $
The simulation results of Scenario 2 are shown in Fig. 6–Fig. 11. All three control methods can track the command accurately at 0−25 s, as shown in Fig. 6. However, when the actuator failure occurrs at 25 s, the tracking error of the comparison methods increases significantly, and the tracking error of angle of attack fails to converge after the actuator failure, as shown in Fig. 7. The attitude angle tracking errors of the proposed method are significantly smaller than those of the comparison method when the actuator failure is considered, and the convergence time of the tracking error is significantly smaller than that of the comparison method. The attitude angular rate curve and control input for the three channels are shown in Fig. 8 and Fig. 9, during the morphing process at $ t = 15-17\;{\mathrm{s}} $, the pitch rudder deflection angle changes significantly. Outer-loop and inner-loop disturbance estimation values are shown in Fig. 10 and Fig. 11, it can be obtained that the inner loop disturbance observations are all markedly jittery when the actuator failure occurrs at 25 s.
Comparison of the simulation results between Scenario 1 and Scenario 2 shows that for HMV with uncertainties, external disturbances, and time-varying actuator faults, the predefined-time fault-tolerant control method based on the predefined-time disturbance observer designed in this paper has higher tracking accuracy and faster transient response. The proposed control method can regulate the tracking error convergence time with $ {T_c} $, which is superior regarding fast error convergence, robustness and stability.
5.　Conclusions
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This paper proposes a predefined-time fault-tolerant control method based on a predefined-time disturbance observer to address the attitude tracking control problem of HMV with model uncertainty, external disturbances and actuator failure. The advantages of the proposed method are demonstrated through numerical simulations. The novelties are as follows:
(i) A predefined-time fault-tolerant controller is proposed by integrating actual predefined-time convergence theory with backstepping control. This controller ensures that the attitude angle tracking error converges to an arbitrarily small neighborhood within a user-defined time, significantly improving the convergence speed of the control system. Moreover, the predefined-time parameter can be easily tuned through a single, straightforward parameter, greatly simplifying the control system design and reducing tuning complexity.
(ii) A predefined-time disturbance observer is developed based on predefined-time theory to accurately estimate composite disturbances arising from model uncertainties, external disturbances, and actuator faults. The observer guarantees that the estimation error converges within the predefined time, thereby enhancing the robustness and reliability of the control system in complex and uncertain environments.
In the future, research will be conducted on the jitter issues caused by the large gain of predefined-time disturbance observers, while also considering the errors of first-order filters in virtual control laws. In order to further improve the robustness of the control system under complex disturbances, adaptive neural networks or intelligent algorithms can also be introduced.
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Appendix
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Year 2026 volume 37 Issue 3
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doi: 10.23919/JSEE.2026.000125
  • Receive Date：2024-12-12
  • Online Date：2026-08-14
  • Published：2026-06-18
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  • Received：2024-12-12
Affiliations
    1School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
    2Beijing Institute of Electronic System Engineering, Beijing 100854, China
    3Hangzhou Innovation Institute, Beihang University, Hangzhou 311115, China

Corresponding:

ZHANG Wenjie
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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