Nomenclature

$\bar c$

mean aerodynamic chord

${c_i}$

controller gain parameter, $i = 1,2$

${C_D}$

drag coefficient

${C_L}$

lift coefficient

${C_M}$

pitching moment coefficient

${C_T}$

thrust coefficient

$d$

external disturbance

$D$

drag force

FDD

fault detection and diagnosis

FPA

flight-path angle

FTC

fault-tolerant control

$h$

flight altitude

${h_d}$

desired flight altitude

${h_{df}}$

filtered desired flight altitude

HFV

hypersonic flight vehicle

${I_{yy}}$

moment of inertia about pitch axis

${k_1}$

controller design parameter

${k_2}$

controller design parameter

${k_h}$

flight-path angle command proportional coefficient

${k_i}$

flight-path angle command integral coefficient

$L$

lift force

$m$

mass of aircraft

${M_{yy}}$

pitch moment

PID

proportional integral derivative

$q$

pitch rate

$\bar q$

dynamic pressure

$r$

the radial distance from earth’s centre

$S$

reference area

$T$

thrust force

$u$

system control input

$v$

desired control input

$V$

mass-centre velocity

$x_i^F$

actual sensor measured values of system state

${z_i}$

state error, $i = 1,2,3$

$\alpha $

angle-of-attack

${\alpha _1}$

designed virtual control variable

${\alpha _2}$

designed virtual control variable

$\gamma $

flight-path angle

${\gamma _d}$

desired flight-path angle

${\gamma _i}$

adaptive law parameter, $i = 1,2$

${\delta _e}$

elevator deflection

$\varepsilon $

robust term parameter

$\theta $

pitch angle

$\mu $

the gravitational constant

$\rho $

air density

${\rho _i}$

effectiveness loss fault coefficient, $i = 0,1,2,3$

$\sigma $

control gain direction

$\varphi $

bias fault coefficient

2.0 Problem statement

2.1 HFV basic model

In this paper, the HFV basic model under study is developed from the NASA Langley Research Centre [Reference Marrison and Stengel52]. We assume that HFV is a rigid body and ignore its elastic modes. The key angles of simplified model are shown in Fig. 1.

Figure 1. Simplified HFV model.

Where $O{x_g}{y_g}$ and $O{x_b}{y_b}$ denote the global inertial coordinate system and body axis coordinate system, respectively. $V$ represents the mass-centre velocity of aircraft. $\gamma ,\alpha ,\theta $ are flight-path angle (FPA), angle-of-attack and pitch angle. $q,{\delta _e}$ denote the pitch rate and the elevator deflection of HFV. Combined with the flight altitude $h$ of HFV, the longitudinal dynamic model can be described by the set of differential equations for $V,h,\gamma ,\alpha ,q$ as follows

(1) \begin{align} \dot V = \frac{{T\cos \alpha - D}}{m} - \frac{{\mu \sin \gamma }}{{{r^2}}}\end{align}

(2) \begin{align}\dot h = V\sin \gamma \end{align}

(3) \begin{align}\dot \gamma = \frac{{L + T\sin \alpha }}{{mV}} - \frac{{\left( {\mu - {V^2}r} \right)\cos \gamma }}{{V{r^2}}}\end{align}

(4) \begin{align}\dot \alpha = q - \dot \gamma \end{align}

(5) \begin{align}\dot q = \frac{{{M_{yy}}}}{{{I_{yy}}}}\end{align}

where $m$ , $\mu $ , $r$ , ${I_{yy}}$ are the mass of HFV, the gravitational constant, the radial distance from Earth’s centre, and the moment of inertia about pitch axis, respectively. The lift $L$ , drag $D$ , thrust $T$ and pitch moment ${M_{yy}}$ can be formulated as follows

(6) \begin{align}L = \bar qS{C_L}\end{align}

(7) \begin{align}D = \bar qS{C_D}\end{align}

(8) \begin{align}T = \bar qS{C_T}\end{align}

(9) \begin{align}{M_{yy}} = \bar qS\bar c\left( {{C_M}\left( \alpha \right) + {C_M}\left( q \right) + {C_M}\left( {{\delta _e}} \right)} \right)\end{align}

where $\bar q = \rho {V^2}/2$ denotes the dynamic pressure of HFV. $\rho $ and $S$ represent the air density and the reference area, respectively. Applying the polynomial fitting of the aero coefficients near the equilibrium point, the aerodynamic forces and moments coefficients can be described as follows [Reference Shaughnessy, Pinckney, McMinn, Cruz and Kelley53]

(10) \begin{align}{C_L} = C_L^\alpha \alpha = 0.6203\alpha \end{align}

(11) \begin{align}{C_D} = 0.645{\alpha ^2} + 0.0043378\alpha + 0.003772\end{align}

(12) \begin{align}{C_T} = \left\{ \begin{array}{l}0.02576\beta \qquad\qquad\qquad {\rm{ if\ }}\beta \lt 1\\0.0224 + 0.00336\beta \qquad {\rm{ if\ }}\beta \gt 1{\rm{ }}\end{array} \right.\end{align}

(13) \begin{align}{C_M}(\alpha ) = - 0.035{\alpha ^2} + 0.036617\alpha + 5.3261 \times {10^{ - 6}}\end{align}

(14) \begin{align}{C_M}(q) = (\bar c/2V)q\left( { - 6.796{\alpha ^2} + 0.3015\alpha - 0.2289} \right)\end{align}

(15) \begin{align}{C_M}\left( {{\delta _e}} \right) = {c_e}\left( {{\delta _e} - \alpha } \right)\end{align}

where $\bar c$ denotes the mean aerodynamic chord. More parameter values and specific coefficient explanations about this model can be found in Ref. [Reference Xu, Mirmirani and Ioannou54].

2.2 Control-oriented model and faults analysis

Assumption 2.1 The trust term $T\sin \alpha $ in Equation (3) is generally neglected because it is much smaller than $L$ .

During straight and level flight, the nonlinear dynamics equations (1)–(5) can be linearised at the trim point. Based on Assumption 2.1, the HFV system can be divided into a velocity subsystem and an altitude subsystem. However, the velocity subsystem is a first-order system that can be controlled by a simple dynamic inversion approach, which is not the focus of control algorithms research in this paper.

Assumption 2.2 The reference altitude $h$ and its high derivates are known and bounded.

Assumption 2.3 Since $\gamma $ is quite small during the real hypersonic flight, we take $\sin \gamma \approx \gamma $ , $\cos \gamma \approx 1$ for simplification in the modelling.

For the altitude subsystem (2)–(5), it is noted that the mathematical relationship between altitude $h$ and FPA $\gamma $ is analytical and unambiguous without any uncertainty. Therefore, the altitude control command can be converted into the FPA control command, which simplifies the control-oriented modelling and controller design. According to Assumption 2.3, we define the reference altitude command ${h_d}\left( t \right)$ , error $\tilde h\left( t \right) = h\left( t \right) - {h_d}\left( t \right)$ . And the reference FPA command can be given as Ref. [Reference Xu, Yang and Pan55]

(16) \begin{align}{\gamma _d}\left( t \right) = \frac{{ - {k_h}\tilde h\left( t \right) - {k_i}\int {\tilde h\left( t \right) + {{\dot h}_d}\left( t \right)} }}{V}\end{align}

where ${k_h}$ and ${k_i}$ are the positive design constants. Thus, there is $h\left( t \right) \to {h_d}\left( t \right)$ when $\gamma \left( t \right) \to {\gamma _d}\left( t \right)$ .

According to the angular interrelations among various coordinate systems, there are $\theta = \alpha + \gamma $ that can be observed in Fig. 1. Consequently, based on the subsystem (3)–(5), we choose FPA $\gamma $ , pitch angle $\theta $ , and pitch rate $q$ as the system state variables to establish the control-oriented model, which are expressed as ${x_1} = \gamma $ , ${x_2} = \theta $ , ${x_3} = q$ . Considering the external disturbance during the flight, the state space model can be transformed into the following from

(17) \begin{align}{\dot x_1} = {f_1}({x_1}) + {g_1}{x_2}\end{align}

(18) \begin{align}{\dot x_2} = {f_2} + {g_2}{x_3}\end{align}

(19) \begin{align}{\dot x_3} = {f_3}({x_1},{x_2},{x_3}) + {g_3}\left( {u(t) + d(t)} \right)\end{align}

where ${f_1}({x_1}) = {\varsigma _1}{x_1} + {\varsigma _2}\cos \left( {{x_1}} \right)$ , ${\varsigma _1} = - C_L^\alpha \bar qS/mV \lt 0$ , ${\varsigma _2} = - \left( {\mu - {V^2}r} \right)/V{r^2} \lt 0$ , ${g_1} = C_L^\alpha \bar qS/mV$ , ${f_2} = 0$ , ${g_2} = 1$ , ${f_3}({x_1},{x_2},{x_3}) = \bar qS\bar c\left( {{C_M}\left( \alpha \right) + {C_M}\left( q \right) - {c_e}\alpha } \right)/{I_{yy}}$ , ${g_3} = \bar qS\bar c{c_e}/{I_{yy}}$ . The system control input $u\left( t \right) = {\delta _e}\left( t \right)$ . $d\left( t \right)$ denotes the continuous and bounded disturbance.

In this paper, to more comprehensively consider the possible actuator fault in the real unknown flight environment, the following fault model is assumed

(20) \begin{align}u\left( t \right) = \sigma {\rho _0}\left( t \right)v\left( t \right) + \varphi \left( t \right)\end{align}

where $v\left( t \right)$ is the control input to be designed. $\sigma $ is a nonzero constant that denotes the unknown control gain direction. ${\rho _0}\left( t \right) \in \mathbb{R}$ and $\varphi \left( t \right) \in \mathbb{R}$ are the unknown effectiveness loss fault and the bias fault degree of actuator. When $\sigma = 1$ , ${\rho _0}\left( t \right) = 1$ and $\varphi \left( t \right) = 0$ , there is no fault in actuator.

In terms of the sensor faults of HFV, effectiveness loss fault is assumed for every system state variable measurement as

(21) \begin{align}x_i^F = {\rho _i}{x_i},\quad{\rm{ }}i = 1,2,3\end{align}

where $x_i^F$ denotes the actual measured value of the ${i_{th}}$ sensor. ${\rho _i}$ is the unknown effectiveness loss degree for the ${i_{th}}$ state variable. When ${\rho _i} = 1$ , no fault occurs in the ${i_{th}}$ sensor.

Assumption 2.4 For the faults in actuator and sensors, we assume that the unknown fault degrees ${\rho _0}(t)$ and ${\rho _i}$ are bounded and there exist positive constants ${\rho _{00}}$ and ${\rho _{i0}}$ satisfying $0 \lt {\rho _{00}} \le {\rho _0}(t) \le 1$ and $0 \lt {\rho _{i0}} \le {\rho _i} \le 1$ , respectively. And $\varphi (t)$ is also bounded.

Assumption 2.5 It is assumed that there exists a positive constant $D$ satisfying $\left| {{g_3}{\rho _3}\left( {\varphi \left( t \right) + d\left( t \right)} \right)} \right| \le D$ with bias fault degree $\varphi \left( t \right)$ and varying disturbance $d\left( t \right)$ are bounded.

Remark 2.1 During hypersonic flight, the most common component faults in HFV system can be divided into four categories: effectiveness loss faults, bias faults, saturation faults and drift faults. It can be noted that the actuator fault type (Equation (20)) considered in this paper is a combination of varying effectiveness loss fault and varying bias fault, and the sensor fault type (Equation (21)) is effectiveness loss fault. These types of faults frequently occur in the actual complex HFV flight environment and are the research emphasis of this study.

Remark 2.2 In order to express the formulas and paper more concisely, the long variables like ${f_1}({x_1})$ , ${f_3}({x_1},{x_2},{x_3})$ , ${\rho _0}(t)$ are replaced with ${f_1}$ , ${f_3}$ , ${\rho _0}$ .

Remark 2.3 [Reference Nussbaum47] If the function ${\rm N}\left( \chi \right)$ satisfies the following two-sided properties, it is called the Nussbaum function.

(22) \begin{align}\mathop {\lim }\limits_{k \to \pm \infty } {\rm{sup}}\frac{1}{k}\int\limits_0^k {N\left( \chi \right)} {\rm{ d}}\chi = \infty \end{align}

(23) \begin{align}\mathop {\lim }\limits_{k \to \pm \infty } {\rm{inf}}\frac{1}{k}\int\limits_0^k {N\left( \chi \right)} {\rm{ d}}\chi = - \infty \end{align}

Through this note, there are different expressions for ${\rm N}\left( \chi \right)$ , such as ${e^{{\chi ^2}}}\cos \left( {\pi \chi /2} \right)$ , ${\chi ^2}\cos \left( \chi \right)$ , ${\chi ^2}\sin \left( \chi \right)$ .

Lemma 2.1 [Reference Wen, Zhou, Liu and Su56] $V\left( t \right)$ and $k\left( t \right)$ are the smooth functions defined on $[0,{t_f})$ with $V\left( t \right) \gt 0,{\rm{ }}\forall t \in [0,{t_f})$ , and $N(k)$ is a smooth Nussbaum type function. If the following inequality holds,

(24) \begin{align}V\left( t \right) \le {c_0} + \frac{{{e^{ - {c_1}t}}}}{b}\int_0^t {\left( {\xi N\left( k \right) - 1} \right)\dot k{e^{{c_1}\tau }}d\tau } ,{\rm{ }}\forall t \in [0,{t_f})\end{align}

where ${c_1},b$ are positive constants and ${c_0},\xi $ are nonzero constants, then $V\left( t \right)$ , $k\left( t \right)$ and $\int_0^t {\left( {\xi N\left( k \right) - 1} \right)\dot k{e^{{c_1}\tau }}d\tau } $ must be bounded on $[0,{t_f})$ .

3.0 Controller design and analysis

In this section, the control objective is to design the appropriate FTC law and adaptive law to achieve altitude tracking and ensure system stability for HFV in the presence of simultaneous actuator and sensor faults, unknown control direction and varying external disturbance. To provide a clear overview of the entire design scheme, the functional decomposition diagram is shown in Fig. 2.

Figure 2. Structure diagram of proposed FTC design scheme.

In the adaptive backstepping control strategy, the entire recursive design procedure can be divided into four steps. The first two steps involve designing the virtual control variables ${\alpha _1}$ and ${\alpha _2}$ , while Step 3 and Step 4 provide the final control law $v$ and associated adaptive law based on the Nussbaum function. The detailed design steps are presented below.

Step 1 In order for forcing $x_1^F$ to track the FPA command ${y_d}$ . Define the FPA tracking error ${z_1}$ as

(25) \begin{align}{z_1} = x_1^F - {y_d} = {\rho _1}{x_1} - {y_d}\end{align}

where ${y_d} = {\gamma _d}$ , $x_1^F$ denotes the measured value of ${x_1}$ . The dynamics of ${z_1}$ is written as

(26) \begin{align}{\dot z_1} = \dot x_1^F - {\dot y_d} = {\rho _1}({f_1} + {g_1}{x_2}) - {\dot y_d}\end{align}

where ${\dot y_d} = {\dot \gamma _d}$ . The designed virtual control variable ${\alpha _1}$ is introduced as

(27) \begin{align}{\alpha _1} = \frac{{ - {c_1}{z_1} - f_1^F + {{\dot y}_d}}}{{{g_1}}}\end{align}

where ${c_1} \gt 0$ is the gain parameter. According to the Assumption 2.3, ${f_1}$ can be simplified to ${f_1} = {\varsigma _1}{x_1} + {\varsigma _2}$ . Thus, $f_1^F = {\varsigma _1}x_1^F + {\varsigma _2}$ . Similarly, aiming at forcing ${\alpha _1}$ to track $x_2^F$ , we define the error ${z_2}$ between $x_2^F$ and ${\alpha _1}$ as

(28) \begin{align}{z_2} = x_2^F - {\alpha _1} = {\rho _2}{x_2} - {\alpha _1}\end{align}

And ${x_2}$ is obtained as

(29) \begin{align}{x_2} = \frac{1}{{{\rho _2}}}\left( {{z_2} + {\alpha _1}} \right)\end{align}

Applying Equations (27)–(29) into Equation (26), ${\dot z_1}$ can be rewritten as

(30) \begin{align}{{\dot z}_1} &= {\rho _1}\left( {{f_1} + \frac{{{g_1}}}{{{\rho _2}}}\left( {{z_2} + {\alpha _1}} \right)} \right) - {{\dot y}_d} = {\rho _1}\left( {{f_1} + \frac{{{g_1}}}{{{\rho _2}}}\left( {{z_2} + \frac{{ - {c_1}{z_1} - f_1^F + {{\dot y}_d}}}{{{g_1}}}} \right)} \right) - {{\dot y}_d}\nonumber \\ & = \left({\rho _1}{f_1} - \frac{{{\rho _1}}}{{{\rho _2}}}f_1^F\right) + \frac{{{\rho _1}}}{{{\rho _2}}}({g_1}{z_2} - {c_1}{z_1} + {{\dot y}_d}) - {{\dot y}_d} \end{align}

Based on the FPA tracking error ${z_1}$ (Equation (25)), the first Lyapunov function is chosen as

(31) \begin{align}{V_1} = \frac{1}{2}z_1^2\end{align}

The derivative of ${V_1}$ can be given as

(32) \begin{align}{{\dot V}_1} &= {z_1}{{\dot z}_1} = {z_1}\left({\rho _1}{f_1} - \frac{{{\rho _1}}}{{{\rho _2}}}f_1^F\right) + \frac{{{\rho _1}}}{{{\rho _2}}}{z_1}({g_1}{z_2} - {c_1}{z_1} + {{\dot y}_d}) - {z_1}{{\dot y}_d}\nonumber \\ &= \left( {1 - \frac{{{\rho _1}}}{{{\rho _2}}}} \right){\varsigma _1}{z_1}\left( {{z_1} + {y_d}} \right) + \left( {{\rho _1} - \frac{{{\rho _1}}}{{{\rho _2}}}} \right){\varsigma _2}{z_1} + \frac{{{g_1}{\rho _1}}}{{{\rho _2}}}{z_1}{z_2} - {c_1}\frac{{{\rho _1}}}{{{\rho _2}}}z_1^2 + \left( {\frac{{{\rho _1}}}{{{\rho _2}}} - 1} \right){z_1}{{\dot y}_d}\end{align}

Step 2 The derivative of ${z_2}$ is

(33) \begin{align}{\dot z_2} = \dot x_2^F - {\dot \alpha _1} = {\rho _2}{x_3} - {\dot \alpha _1}\end{align}

Based on the Assumption 2.2 and Assumption 2.3, ${\dot \alpha _1}$ is bounded as

(34) \begin{align}{\dot \alpha _1} = \frac{{ - {c_1}{{\dot z}_1} - \dot f_1^F + {{\ddot y}_d}}}{{{g_1}}}\end{align}

where $\dot f_1^F = {\varsigma _1}\dot x_1^F$ . The virtual control ${\alpha _2}$ is designed as

(35) \begin{align}{\alpha _2} = - {c_2}{z_2} + {\dot \alpha _1}\end{align}

where ${c_2} \gt 0$ is the gain parameter. To get ${\alpha _2}$ to track $x_3^F$ , the error ${z_3}$ is defined as

(36) \begin{align}{z_3} = x_3^F - {\alpha _2} = {\rho _3}{x_3} - {\alpha _2}\end{align}

Then ${x_3}$ can be rewritten as

(37) \begin{align}{x_3} = \frac{1}{{{\rho _3}}}\left( {{z_3} + {\alpha _2}} \right)\end{align}

Based on the error ${z_2}$ (Equation (28)) the second Lyapunov function can be chosen as

(38) \begin{align}{V_2} = \frac{1}{2}z_2^2\end{align}

Combining Equations (33)–(37) into Equation (38), the derivative of ${V_2}$ is given as

(39) \begin{align}{{\dot V}_2}& = {z_2}{{\dot z}_2} = {z_2}\left( {\frac{{{\rho _2}}}{{{\rho _3}}}\left( {{z_3} - {c_2}{z_2} + {{\dot \alpha }_1}} \right) - {{\dot \alpha }_1}} \right) \nonumber \\ &= \frac{{{\rho _2}}}{{{\rho _3}}}{z_2}{z_3} - \frac{{{\rho _2}}}{{{\rho _3}}}{c_2}z_2^2 + {z_2}\left( {\frac{{{\rho _2}}}{{{\rho _3}}} - 1} \right)\left( {\frac{{ - {c_1}{{\dot z}_1} - \dot f_1^F + {{\ddot y}_d}}}{{{g_1}}}} \right)\end{align}

Step 3 The derivative of ${z_3}$ is

(40) \begin{align}{\dot z_3} = \dot x_3^F - {\dot \alpha _2} = {\rho _3}\left( {{f_3} + {g_3}\left( {u + d} \right)} \right) - {\dot \alpha _2}\end{align}

where ${\dot \alpha _2}$ is bounded. Combining Equations (20), (34) and (35) into Equation (40), we can get the derivative of ${z_3}$ considering the actuator fault as

(41) \begin{align}{{\dot z}_3} &= {\rho _3}{f_3} + \sigma {g_3}{\rho _3}{\rho _0}v + {g_3}{\rho _3}\left( {\varphi + d} \right) + {c_2}{{\dot z}_2} - {{\ddot \alpha }_1} \nonumber\\ &= {\rho _3}{f_3} + \mu v + {g_3}{\rho _3}\left( {\varphi + d} \right) + \phi x_3^F - {c_2}{{\dot \alpha }_1} - {{\ddot \alpha }_1}\end{align}

where $\mu = \sigma {g_3}{\rho _3}{\rho _0}$ , $\phi = {\rm{ }}{c_2}{\rho _2}/{\rho _3}$ are the unknown parameters for the simplification of subsequent controller design and system stability analysis.

Based on the error ${z_3}$ (Equation (36)), we choose the third Lyapunov function as

(42) \begin{align}{V_3} = \frac{1}{2}z_3^2\end{align}

Applying Equation (41) into Equation (42), the derivative of ${V_3}$ is given as

(43) \begin{align}{\dot V_3} = {z_3}\left( {{\rho _3}{f_3} + \mu v + {g_3}{\rho _3}\left( {\varphi + d} \right) + \phi x_3^F} \right) + {\dot V^{\prime}_3}\end{align}

(44) \begin{align}{\dot V^{\prime}_3} = {z_3}\left( { - {c_2}{{\dot \alpha }_1} - {{\ddot \alpha }_1}} \right)\end{align}

Step 4 It can be noticed that the control gain $\mu $ is a variable including unknown parameters, so the Nussbaum function $N\left( k \right)$ is adopted, which is chosen as $N\left( k \right) = {k^2}\cos \left( k \right)$ . The final control input is designed as follows

(45) \begin{align}v = N\left( k \right)\bar v\end{align}

where

(46) \begin{align}\bar v = - {k_2}{z_3} - \beta \end{align}

(47) \begin{align}\beta = {k_1}{z_3} + \hat \phi x_3^F + \eta \tanh \left( {{z_3}} \right) + {\rho _3}f_3^F\end{align}

(48) \begin{align}\dot k = {\gamma _1}{z_3}\bar v\end{align}

where $\beta $ is the designed auxiliary variable. $\hat \phi $ denotes the estimated value of $\phi $ . $f_3^F$ is a function about the measured system state variables, which can be expressed as $f_3^F = \bar qS\bar c\big( {{C_m}\left( {x_2^F - x_1^F} \right) + {C_m}\left( {x_3^F} \right) - {c_e}\left( {x_2^F - x_1^F} \right)} \big)/{I_{yy}}$ . ${k_1}$ , ${k_2}$ and ${\gamma _1}$ are the positive constants. $\eta $ is designed as $\eta \geqslant \varepsilon + D$ where $\varepsilon $ is a suitable positive constant. $\tanh ( \bullet )$ is a smooth hyperbolic tangent function that can be obtained from the hyperbolic sine function and the hyperbolic cosine function as $\tanh (x) = \sinh (x)/\cosh (x) = \left( {{e^x} - {e^{ - x}}} \right)/\left( {{e^x} + {e^{ - x}}} \right)$ , which can be employed in the control law to avoid the chattering problem in terms of eliminating varying external disturbance.

The adaptive law of the estimated parameter $\hat \phi $ is designed as

(49) \begin{align}\dot {\hat \phi} = {\gamma _2}{z_3}x_3^F\end{align}

where ${\gamma _2}$ is the positive constant. In order for approximating the estimated value $\hat \phi $ to the true value $\phi $ , we define the error $\tilde \phi = \hat \phi - \phi $ , and the fourth Lyapunov function is defined as

(50) \begin{align}{V_4} = \frac{1}{{2{\gamma _2}}}{\tilde \phi ^2}\end{align}

And we can get $\dot {\tilde \phi} = \dot {\hat \phi} $ . The derivate of ${V_4}$ is given as

(51) \begin{align}{\dot V_4} = \frac{1}{{{\gamma _2}}}{\tilde \phi} \dot {\hat \phi} = {z_3}\tilde \phi x_3^F \end{align}

Proof On the basis of Equations (31), (38), (42) and (50), four Lyapunov functions are combined to obtain a new Lyapunov function as shown below, which is related to the errors ${z_1},{z_2},{z_3}\ {\rm{ and\ }}\tilde \phi $

(52) \begin{align}V = \sum\limits_{i = 1}^4 {{V_i}} = \frac{1}{2}z_1^2 + \frac{1}{2}z_2^2 + \frac{1}{2}z_3^2 + \frac{1}{{2{\gamma _2}}}{\tilde \phi ^2}\end{align}

Based on Assumption 2.4, and applying the Young’s inequality, Equations (32), (39) and (44) can be rewritten as follows

(53) \begin{align}{{\dot V}_1} &= \left( {1 - \frac{{{\rho _1}}}{{{\rho _2}}}} \right){\varsigma _1}{z_1}\left( {{z_1} + {y_d}} \right) + \left( {{\rho _1} - \frac{{{\rho _1}}}{{{\rho _2}}}} \right){\varsigma _2}{z_1} + \frac{{{g_1}{\rho _1}}}{{{\rho _2}}}{z_1}{z_2} - {c_1}\frac{{{\rho _1}}}{{{\rho _2}}}z_1^2 + \left( {\frac{{{\rho _1}}}{{{\rho _2}}} - 1} \right){z_1}{{\dot y}_d} \nonumber\\ &\le \left( {\varsigma _1^2 + \frac{{2 + {g_1}}}{{2{\rho _{20}}}} - {c_1}{\rho _{10}} + \frac{1}{2}} \right)z_1^2 + \left( {\frac{{{g_1}}}{{2{\rho _{20}}}}} \right)z_2^2 + \frac{1}{2}y_d^2 + \frac{1}{{2{\rho _{20}}}}\dot y_d^2 + \frac{1}{{2{\rho _{20}}}}\varsigma _2^2\end{align}

(54) \begin{align}{{\dot V}_2} &\le \frac{1}{{{\rho _{30}}}}\left( {\frac{{z_2^2}}{2} + \frac{{z_3^2}}{2}} \right) - {c_2}{\rho _{20}}z_2^2 + {z_2}\left( {\frac{{{\rho _2}}}{{{\rho _3}}} - 1} \right)\left( {\frac{1}{{{g_1}}}\left( { - {c_1}{{\dot z}_1} - {\varsigma _1}\left( {{{\dot z}_1} + {{\dot y}_d}} \right) + {{\ddot y}_d}} \right)} \right) \nonumber\\ &\le \frac{{1 + {c_1}}}{{2{g_1}}}\left( {1 - {c_1}{\rho _{10}}} \right)z_1^2 + \left( {\frac{1}{{2{\rho _{30}}}} + \frac{{1 + \varsigma _1^2}}{{2{g_1}}} + \frac{{1 + {c_1}}}{{2{\rho _{20}}}} + \frac{{\varsigma _1^2 + {c_1}}}{{{g_1}}}\left( {\varsigma _1^2 + \frac{{2 + {g_1}}}{{2{\rho _{20}}}} - \frac{{{c_1}{\rho _{10}}}}{2}} \right) - {c_2}{\rho _{20}}} \right)z_2^2 \nonumber\\ &+ \frac{1}{{2{\rho _{30}}}}z_3^2 + \frac{{1 + {c_1}}}{{2{g_1}}}y_d^2 + \left( {\frac{{1 + {c_1}}}{{2{g_1}{\rho _{20}}}} + \frac{1}{{2{g_1}}}} \right)\dot y_d^2 + \frac{1}{{2{g_1}}}\ddot y_d^2 + \frac{{1 + {c_1}}}{{2{g_1}{\rho _{20}}}}\varsigma _2^2 \end{align}

(55) \begin{align} {{\dot V^{\prime}}_3} &= {{{c_2}} \over {{g_1}}}\left( {{c_1} + {\varsigma _1}} \right){z_3}{{\dot z}_1} + {1 \over {{g_1}}}\left( {{c_1} + {\varsigma _1}} \right){z_3}{{\ddot z}_1} + {{{\varsigma _1}{c_2}} \over {{g_1}}}\left( {{z_3}{{\dot y}_d}} \right) - \left( {{c_2} + {{{\varsigma _1}} \over {{g_1}}}} \right){z_3}{{\ddot y}_d} - {1 \over {{g_1}}}{z_3}{{\dddot y}_d} \nonumber \\ &\le {{{c_1}{c_2}} \over {2{g_1}}}\left( {1 - {c_1}{\rho _{10}}} \right)z_1^2 + {{{c_1}{c_2}} \over {2{\rho _{20}}}}z_2^2 + \left( {{{{c_1}{c_2}} \over {{g_1}}}\left( {\varsigma _1^2 + {{2 + {g_1}} \over {2{\rho _{20}}}} - {{{c_1}{\rho _{10}}} \over 2}} \right) + {{{c_1} + 2{g_1}{c_1}} \over {2{g_1}{\rho _{30}}}} + {{{c_2}\varsigma _1^2 - {\varsigma _1} + 1} \over {2{g_1}}}} \right)z_3^2 \nonumber \\ &+ {{{c_1}{c_2}} \over {2{g_1}}}y_d^2 + \left( {{{{c_1}{c_2}} \over {2{g_1}{\rho _{20}}}} + {{{c_2}} \over {2{g_1}}}} \right)\dot y_d^2 + \left( {{{{c_1}} \over {2{g_1}{\rho _{30}}}} - {{{\varsigma _1}} \over {2{g_1}}}} \right)\ddot y_d^2 + {1 \over {2{g_1}}}\dddot y_d^2 + {{{c_1}{c_2}} \over {2{g_1}{\rho _{20}}}}\varsigma _2^2 \end{align}

The unknown effectiveness loss fault degrees of sensors have the properties as ${\rho _1}/{\rho _2} \geqslant {\rho _{10}}$ , ${\rho _1}/{\rho _3} \geqslant {\rho _{10}}$ , and ${\rho _2}/{\rho _1} \geqslant {\rho _{20}}$ , ${\rho _3}/{\rho _1} \geqslant {\rho _{30}}$ . Besides, there are ${\rho _1}/{\rho _2} \le 1/{\rho _{20}}$ , ${\rho _1}/{\rho _3} \le 1/{\rho _{30}}$ .

Combining with Equations (43), (51)–(55), the derivate of $V$ can be rewritten as

(56) \begin{align}\dot V &= {{\dot V}_1} + {{\dot V}_2} + {{\dot V^{\prime}}_3} + {z_3}\left( {{\rho _3}{f_3} + \mu v + {g_3}{\rho _3}\left( {\varphi + d} \right) + \phi x_3^F} \right) + {{\dot V}_4} \nonumber\\& \le - {\Gamma _1}z_1^2 - {\Gamma _2}z_2^2 - {\Gamma _3}z_3^2 + {C_1} + {z_3}\left( {{\rho _3}{f_3} + \mu v + {g_3}{\rho _3}\left( {\varphi + d} \right) + \phi x_3^F} \right) + {z_3}\tilde \phi x_3^F\end{align}

where

(57) \begin{align}{\Gamma _1} = {c_1}{\rho _{10}} - \varsigma _1^2 - \frac{{2 + {g_1}}}{{2{\rho _{20}}}} - \frac{{{c_1}{c_2} + {c_1} + 1}}{{2{g_1}}}\left( {1 - {c_1}{\rho _{10}}} \right) - \frac{1}{2}\end{align}

(58) \begin{align}{\Gamma _2} = {c_2}{\rho _{20}} - \frac{1}{{2{\rho _{30}}}} - \frac{{1 + \varsigma _1^2}}{{2{g_1}}} - \frac{{\varsigma _1^2 + {c_1}}}{{{g_1}}}\left( {\varsigma _1^2 + \frac{{2 + {g_1}}}{{2{\rho _{20}}}} - \frac{{{c_1}{\rho _{10}}}}{2}} \right) - \frac{{{c_1}{c_2} + {c_1} + {g_1} + 1}}{{2{\rho _{20}}}}\end{align}

(59) \begin{align}{\Gamma _3} = - \frac{1}{{2{\rho _{30}}}} - \frac{{{c_1}{c_2}}}{{{g_1}}}\left( {\varsigma _1^2 + \frac{{2 + {g_1}}}{{2{\rho _{20}}}} - \frac{{{c_1}{\rho _{10}}}}{2}} \right) - \frac{{{c_1} + 2{g_1}{c_1}}}{{2{g_1}{\rho _{30}}}} - \frac{{{c_2}\varsigma _1^2 - {\varsigma _1} + 1}}{{2{g_1}}}\end{align}

(60) \begin{align} {C_1} = \left( { - {1 \over 2} - {{{c_1}{c_2} + {c_1} + 1} \over {2{g_1}}}} \right)y_d^2 + \left( { - {1 \over {2{\rho _{20}}}} - {{{c_1}{c_2} + {c_1} + 1} \over {2{g_1}{\rho _{20}}}} - {{{c_2} + 1} \over {2{g_1}}}} \right)\dot y_d^2 \nonumber \\ + \left( {{{{\varsigma _1} - 1} \over {2{g_1}}} - {{{c_1}} \over {2{g_1}{\rho _{30}}}}} \right)\ddot y_d^2 - {1 \over {2{g_1}}}\dddot y_d^2 + \left( { - {1 \over {2{\rho _{20}}}} - {{{c_1}{c_2} + {c_1} + 1} \over {2{g_1}{\rho _{20}}}}} \right)\varsigma _2^2 \end{align}

Based on Equations (13), (14) and (19), applying designed control law (Equations (45)–(48)) into Equation (56), $\dot V$ can be given as

(61) \begin{align}{\dot V} &\le - {\Gamma _1}z_1^2 - {\Gamma _2}z_2^2 - {\Gamma _3}z_3^2 + {z_3}\left( {\mu v + {g_3}{\rho _3}\left( {\varphi + d} \right) + {\rho _3}{f_3} + \phi x_3^F} \right) + \frac{1}{{{\gamma _2}}}{\tilde \phi} \dot {\hat \phi} + {C_1} \nonumber \\ &= - \sum\limits_{i = 1}^3 {{\Gamma _i}z_i^2} + {z_3}\left( {\beta - \left( {{k_1}{z_3} + \hat \phi x_3^F + \eta {\mathop{\rm sgn}} \left( {{z_3}} \right) + {\rho _3}f_3^F} \right) + \mu N\left( k \right)\bar v + {g_3}{\rho _3}\left( {\varphi + d} \right) + {\rho _3}{f_3} + \phi x_3^F} \right)\nonumber \\ &+ {z_3}\tilde \phi x_3^F - \frac{{\dot k}}{{{\gamma _1}}} + \frac{{\dot k}}{{{\gamma _1}}} + {C_1}\nonumber \\ &\le - \sum\limits_{i = 1}^3 {{\Gamma _i}z_i^2} + {z_3}\left( {\beta - {k_1}{z_3} + \tilde \phi x_3^F - \hat \phi x_3^F + \mu N\left( k \right)\bar v + \phi x_3^F} \right) + \frac{1}{2}z_3^2 - \frac{{\dot k}}{{{\gamma _1}}} + {z_3}\left( { - {k_2}{z_3} - \beta } \right) + {C_2}\nonumber \\ & \le - \sum\limits_{i = 1}^3 {{\Gamma _i}z_i^2} - \left( {{k_1} + {k_2} - \frac{1}{2}} \right)z_3^2 + \left( {\mu N\left( k \right) - 1} \right)\frac{{\dot k}}{{{\gamma _1}}} + {C_2}\nonumber \\ &\le - \lambda V + \left( {\mu N\left( k \right) - 1} \right)\frac{{\dot k}}{{{\gamma _1}}} + {C_2}\end{align}

where $\left| {{\rho _3}\left( {{f_3} - f_3^F} \right)} \right| \le \bar \omega $ , ${C_2} = {C_1} + {\bar \omega ^2}/2$ , $\lambda = \min \left\{ {{\Gamma _1},{\Gamma _2},{\Gamma _3} + {k_1} + {k_2} - 1/2} \right\} \gt 0$ . Multiplying by ${e^{\lambda t}}$ and integrating it, we have

(62) \begin{align}V\left( t \right) &\le \left( {V\left( 0 \right) - \frac{{{C_2}}}{\lambda }} \right){e^{ - \lambda t}} + \frac{{{C_2}}}{\lambda } + \frac{{{e^{ - \lambda t}}}}{{{\gamma _1}}}\int_0^t {\left( {\mu N\left( k \right) - 1} \right)\dot k{e^{\lambda \tau }}d\tau } \nonumber \\ &\le \left| {V\left( 0 \right) - \frac{{{C_2}}}{\lambda }} \right| + \left| {\frac{{{C_2}}}{\lambda }} \right| + \frac{{{e^{ - \lambda t}}}}{{{\gamma _1}}}\int_0^t {\left( {\mu N\left( k \right) - 1} \right)\dot k{e^{\lambda \tau }}d\tau } \nonumber \\ &= \xi + \frac{{{e^{ - \lambda t}}}}{{{\gamma _1}}}\int_0^t {\left( {\mu N\left( k \right) - 1} \right)\dot k{e^{\lambda \tau }}d\tau } \end{align}

where $\xi = \left| {V\left( 0 \right) - \frac{{{C_2}}}{\lambda }} \right| + \left| {\frac{{{C_2}}}{\lambda }} \right|$ . According to Lemma 2.1, we can obtain that $V\left( t \right)$ is bounded on $[0,{t_f})$ , which means the tracking errors ${z_1},{z_2},{z_3}$ and estimated error $\tilde \phi $ have upper bounds. Therefore, with the appropriate parameters designed, the proposed controller can guarantee that all state variables of the HFV altitude subsystem are ultimately bounded stable considering the unknown actuator and sensor faults, unknown control direction and varying external disturbance.

4.0 Simulation

In this section, the effectiveness of the proposed controller is verified through numerical simulations. The flight environment and initial state variables are set as $h = 110000\,{\rm{ ft}}$ , $V = 15060{\rm{ ft/s}}$ , ${\gamma _0} = 0{\rm{ rad}}$ , ${\theta _0} = 0.0315{\rm{ rad}}$ , ${q_0} = 0{\rm{ rad/s}}$ . The time-varying external disturbance is chosen as $d = 0.02 + 0.01\sin \left( t \right)$ . In the whole process of simulation, the reference altitude command is $112000\,{\rm{ ft}}$ in first $50{\rm{ s}}$ and will be switched back to initial altitude $110000\,{\rm{ ft}}$ as follows:

(63) \begin{align}{h_d} = \left\{ {\begin{array}{*{20}{l}}{112000\,{\rm{ ft, \quad if\ 0\ s}} \le t{\rm{ \lt 50\ s}}}\\{110000\,{\rm{ ft,\quad if\ 50\ s}} \le t \le {\rm{100\ s }}}\end{array}} \right.\end{align}

To avoid a differential explosion in the step response, the desired altitude command ${h_{df}}$ is generated from the output of the following two-order filter.

(64) \begin{align}\frac{{{h_{df}}}}{{{h_d}}} = \frac{1}{{0.5{s^2} + 2s + 1}}\end{align}

The FPA command parameters are set as ${k_h} = 0.5$ , ${k_i} = 0.1$ . The controller gain parameters and adaptive law parameters are selected as ${c_1} = 6$ , ${c_2} = 1$ , ${k_1} = 8$ , ${k_2} = 9$ , ${\gamma _1} = 20$ , ${\gamma _2} = 1$ . And the robust term parameter is chosen as $\varepsilon = 0.1$ . To demonstrate the effectiveness of the proposed controller, we take PID controller with their gain parameters set to $[300,40,500]$ as the comparison.

For more comprehensive fault types analysis of actuator and sensors, we simulate the following cases:

1. (1) In Case 1, we assume that the fault types of actuator and sensors both are fixed, that is, the values of fault degrees are constants. For sensors, the effectiveness loss fault degrees are considered as ${\rho _1} = 0.8$ , ${\rho _2} = 0.9$ , ${\rho _3} = 0.7$ . For actuator, the control direction is set as a positive constant $\sigma = 0.65$ , the effectiveness loss fault degree is set as ${\rho _0} = 0.5$ , and the bias fault degree is chosen as $\varphi = 0.3$ .

2. (2) In Case 2, to better simulate the change of actuator fatigue degree during a real hypersonic flight, the time-varying effectiveness loss fault type of actuator is introduced. Keeping all conditions unchanged, the different actuator effectiveness loss fault degrees are selected in different time periods as follows

(65) \begin{equation}{\rho _0} = \left\{ \begin{array}{l}0.5 + 0.3\sin \left( t \right),\quad{\rm{ if\ 30\ s }} \le {\rm{ }}t \lt 50\ {\rm{ s}}\\0.5 - 0.2\sin \left( t \right),\quad{\rm{ if\ }}t \geqslant 80\ {\rm{ s}}\\0.5,\qquad\qquad\qquad\,\,{\rm{ if\ others}}\end{array} \right.\end{equation}

1. (3) In Case 3, we assume that the degree of actuator bias fault is changeable, which considers the potential impacts of the harsh and volatile external environment on actuator stuck. The varying bias fault is more likely to occur in the hypersonic flight phase. Therefore, with the remaining conditions in Case 2 unchanged, the bias fault of actuator is given as

   (66) \begin{equation}\varphi = \left\{ \begin{array}{l}0.3 - 0.4\cos \left( t \right),\quad{\rm{ if\ 30\ s }} \le {\rm{ }}t \lt 50\ {\rm{ s}}\\0.3 + 0.6\cos \left( t \right),\quad{\rm{ if\ }}t \geqslant 80\ {\rm{ s}}\\0.3,\qquad\qquad\qquad\,\,{\rm{ if\ others}}\end{array} \right.\end{equation}

2. (4) In Case 4, we simulate the opposite signs of the unknown control direction, which is set to $\sigma = - 5$ while keeping all other conditions unchanged from Case 3.

The simulation results of the above cases are shown in Figs. 3–15.

As shown in Fig. 3, which presents simulations of the fixed fault type of actuator and sensors in Case 1. The proposed controller has the basic fault-tolerant characteristic to track the reference altitude command including both climbing and descending stages within $14{\rm{s}}$ . Figs 4–7 show the altitude tracking comparison between the proposed FTC approach and the PID controller under Cases 2, 3 and 4. It can be clearly seen that the fluctuation of the proposed controller is significantly smaller than that of the PID controller, particularly when the actuator’s gain fault or bias fault degree varies with the changing flight environment. Furthermore, when control direction changes to a negative value in Case 4, the tracking control is still effective and stable under the proposed controller, while it fails under the PID controller. In Figs. 8–12, we can see that the state variables of altitude subsystem are uniformly bounded and stable with the proposed controller, but violent oscillation with PID control. The control input of the system in the presence of unknown faults of actuator and sensors, unknown control direction and external disturbance are provided in Figs 13–15. With the proposed controller, the varying faults and tracking commands lead to minor fluctuations of input at $30{\rm{ s}}$ , $50{\rm{ s}}$ and $80{\rm{ s}}$ . Moreover, the control input is smaller and more practical compared to the PID controller in the actual hypersonic flight.

Figure 3. Altitude tracking response in Case 1.

Figure 4. Altitude tracking response with the PID controller in Case 2.

Figure 5. Altitude tracking response with the PID controller in Case 3.

Figure 6. Altitude tracking response in Case 4.

Figure 7. Altitude tracking response of the PID controller in Case 4.

Figure 8. System state variables in Case 1.

Figure 9. System state variables in Case 2.

Figure 10. System state variables in Case 3.

Figure 11. System state variables of the PID controller in Case 3.

Figure 12. System state variables in Case 4.

Figure 13. Elevator deflection in Case 3.

Figure 14. Elevator deflection of PID controller in Case 3.

Figure 15. Elevator deflection in Case 4.

Under the various conditions of component faults and varying disturbance, which specifically consider the common fault types of the actuator and sensors during the hypersonic flight, these simulation results demonstrate that the proposed controller has the better control performance, including reliable robustness and stronger stability in comparison with the PID control method. Notably, the proposed control strategy effectively suppresses the fluctuations caused by unknown faults of actuator and sensors under varying external disturbance. Furthermore, it successfully addresses the issue of unknown control direction of the actuator.