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Abstract

In this article, an adaptive fault tolerant control strategy is proposed to solve the trajectory tracking problem of a generic hypersonic vehicle subjected to actuator fault, external disturbance, and input saturation. The longitudinal model of generic hypersonic vehicle is divided into velocity subsystem and altitude subsystem, in which dynamic inversion and backstepping are applied, respectively, to track the desired trajectories. For the unknown maximum disturbance upper bound, actuator fault, and input saturation constraint, adaptive laws are proposed to estimate these information online. Finally, numeric simulation is conducted in the cruise phase for generic hypersonic vehicle. Simulation results show that the controllers designed in this article can make generic hypersonic vehicle track the desired trajectories in the presence of actuator fault, external disturbance, and input saturation.

Keywords

Hypersonic vehicle actuator fault dynamic inversion backstepping

Introduction

Hypersonic flight vehicles (HFVs), which possess good advantage in military and civil application owing to its high flight speed and cost-effective capability in space access, have become a research hot spot. However, designing controllers for HFV poses numerous challenges mainly caused by its highly coupled and nonlinear nature as well as flight conditions. The control system of HFVs is sensitive to the changes of flight conditions and its physical or aerodynamic parameters since they are always flying in near space areas at a high Mach numbers. Although there are great amounts of research on HFVs,^1
–3 it is still enormously worth to study control strategy for HFVs.

Scholars have made numerous studies on the longitude dynamics of HFVs based on nonlinear and adaptive control in the last decade. Input output linearization technology is firstly applied to simplify the complex coupled longitude model as a two-input and two-output system, and a sliding mode surface with respect to tracking error is then proposed with vehicle mass uncertainty, aerodynamic parameters uncertainty, wing area uncertainty, and so on considered.⁴ Controllers based on sliding mode technology can efficiently reject system disturbance; however, chattering is an inevitable phenomenon causing great damage to actuators. For chattering reduction purpose, a high-order sliding mode control strategy is proposed by Wang et al.⁵ To deal with the external disturbances for HFVs, sliding mode controller combined with an observer is implemented by Chen et al.⁶ Recently, increasing attention is paid on finite time controllers^7,8 for their advantage of higher control accuracy and enhanced convergence speed over other control methods. In the study of Sun et al.,⁹ a novel finite time controller with disturbance observer is designed to cope with tracking problem while there are unknown disturbance and uncertainty in generic hypersonic vehicle (GHV) system. Besides the issues mentioned above, dynamic surface control is widely applied for HFVs.^10
–12 Trajectory tracking problem is analyzed by using backstepping, in which intelligent algorithm with much less computation burden is applied by Xu et al.^13,14 Neural network-based dynamic surface control strategy is applied in the HFV system.^15,16 Compared with the traditional dynamic surface control, neural network can estimate the complex control gain and system uncertain online, which need less information about the longitude dynamics. Dead-zone input nonlinearity is such common trouble that the HFV system is always trapped in, which makes the key factors considered in controller designing procedures greatly different. In the study of Xu,¹⁷ a novel neural network method is used to compensate for dead-zone input nonlinearity. Moreover, the output magnitude of actuators is under some certain constraints because of the characteristics of the mechanical structure. In the study of Tian et al.,¹⁸ a robust adaptive controller is proposed with input constraints, external disturbance, and system uncertainty taken into consideration, in which the effect of input saturation is eliminated by an adaptive law. Inspired by the work of Chen et al.,¹⁹ different methods coping with the saturation problem for HFVs have been adopted.^20
–22

However, the aforementioned controllers are designed based on the assumption that HFVs are working without actuator faults. As the flight conditions always change rapidly, the dynamics of HFVs inevitably suffer from actuator faults. It is extremely urgent to design controllers considering actuator faults. An adaptive controller dealing with tracking problem for GHV encountered actuator efficiency loss is constructed, so that the system can complete flight mission perfectly.²³ However, only actuator effectiveness loss fault is considered.

In this article, adaptive dynamic inversion and backstepping method are exploited, respectively, to fulfill the need of tracking problem of GHV with actuator effectiveness loss fault, actuator drift fault, external disturbance, and input saturation constraint. This article can be divided as follows: The longitudinal model of GHV is established in the following section. In “Controller design” section, dynamic version controller and fault tolerant controller are developed for velocity subsystem as well as altitude subsystem, respectively. Furthermore, the feasibility of the controllers is proved as well. Numerical simulations are exhibited in “Simulation” section. Finally, this article ends up with some concluding remarks.

Longitudinal model of hypersonic vehicle and problem formulation

Vehicle model with actuator fault

In this article, the longitudinal model of a GHV can be established as follows¹

\dot{V} = \frac{T \cos α - D}{m} - \frac{μ sin γ}{r^{2}}

\dot{γ} = \frac{L + T sin α}{m V} - \frac{(μ - V^{2} r) \cos γ}{V r^{2}}

\dot{h} = V sin γ

\dot{α} = q - \dot{γ}

\dot{q} = \frac{M_{y y}}{I_{y y}}

where

L = \bar{q} S C_{L}

D = \bar{q} S C_{D}

T = \bar{q} S C_{T}

M_{y y} = \bar{q} S \bar{c} [C_{M} (α) + C_{M} (δ) + C_{M} (q)]

r = h + r_{e}

C_{L} = 0.6203 α

C_{D} = 0.6450 α^{2} + 0.0043378 α + 0.003772

C_{T} = {\begin{cases} 0.022576 β β < 1 \\ 0.0224 + 0.00336 β β > 1 \end{cases}

C_{M} (α) = - 0.035 α^{2} + 0.036117 α + 5.3261 \times 10^{- 6}

C_{M} (q) = \frac{\bar{c} q (- 0.6796 α^{2} + 0.3015 α - 0.2289)}{2 V}

C_{M} (δ_{e}) = c_{e} (δ_{e} - α)

c_{e} = 0.0292

with $\bar{q}$ denoting the dynamic pressure which satisfies $\bar{q} = (1 / 2) ρ V^{2}$ ; V, γ, h, α, and q denoting velocity, flight path angle, altitude, angle of attack, and pitch rate of GHV, respectively²⁴; T, D, L, and Myy denoting the thrust, drag, lift, and pitching moment, respectively²⁵; m, S, ρ, I_yy, μ, and r_e denoting the mass, reference area, density of air, moment of inertia, the gravitational constant, and radius of the Earth, respectively²⁶; and $\bar{c}$ and c_e denoting constants.

Most of the existing works on GHV are laying focus on nonlinear output tracking control with the assumption that the actuators work under nominal conditions, which ignores the actuator faults posing huge possibility to cause some catastrophic failures. Therefore, increasing emphasis is put on accurate and reliable control for GHV in the presence of actuator faults. In this article, actuator effectiveness loss fault, actuator drift failure, external disturbance, and control input saturation constraint are taken into consideration, which can be expressed as

δ_{e} = ρ_{1} u_{c} + d_{1} (t) + Δ_{1}

where

u_{c} = sat (u_{1}) = χ_{1} (u_{1}) u_{1}

χ_{1} (u_{1}) = {\begin{cases} \frac{u_{1 max}}{u_{1}} sign (u_{1}) | u_{1} | > u_{1 max} \\ 1 | u_{1} | \leq u_{1 max} \end{cases}

u₁ is the control input, u_{1 max} is the maximum control input satisfying u_{1 max} > 0, ρ₁ represents the unknown effectiveness factor of actuator satisfying $0 < ι_{1} \leq ρ_{1} \leq 1$ , $ι_{1} \leq 1$ denotes the minimum value of ρ₁, d₁(t) represents the unknown external disturbance, and Δ₁ is additive fault.

Problem formulation

The control objective of this article can be described as designing control laws for GHV such that the desired velocity and altitude trajectories could be tracked in the presence of actuator fault, external disturbance, and input saturation. The following assumptions are applied for the convenience of controllers design.

Assumption 1

The external disturbances and additive actuator fault are supposed to satisfy $| d_{1} (t) + Δ_{1} | \leq λ_{1}$ , where λ₁ is an unknown positive constant.

Controller design

Based on functional decomposition, the longitudinal model can be divided into velocity subsystem and altitude subsystem. For velocity subsystem, dynamic inversion is applied to design controller as described by Xu.¹⁷ However, adaptive fault tolerant controller is developed for the altitude subsystems. By transforming a given altitude command h_d into the desired flight path angle, it can be obtained

γ_{d} = \arcsin (\frac{- k_{h} \tilde{h} + {\dot{h}}_{d}}{V})

where k_h is a positive scalar. Note that the flight path angle $γ \in (- 90 °, 90 °)$ in practice.

To simplify the design of fault tolerant control law, one can transform the altitude subsystems (2) to (5) into a strict-feedback form by defining

x = {[x_{1}, x_{2}, x_{3}]}^{T} = {[γ, ϑ_{p}, q]}^{T}

where ϑ_p = α + γ. Therefore, the altitude subsystem can be formulated as follows²³

\begin{array}{l} {\dot{x}}_{1} = f_{1} (x_{1}, V) + g_{1} (V) x_{2} \\ {\dot{x}}_{2} = f_{2} + g_{2} x_{3} \\ {\dot{x}}_{3} = f_{3} (x_{1}, x_{2}, x_{3}, V) + g_{3} (V) δ_{e} \\ = f_{3} (x_{1}, x_{2}, x_{3}, V) + g_{3} (V) (ρ_{1} χ_{1} (u_{1}) u_{1} + d_{1} (t) + Δ_{1}) \end{array}

where

\begin{array}{l} g_{1} (V) = \frac{0.31015 ρ V S}{m} \\ f_{1} (x_{1}, V) = \frac{T sin α}{m V} - \frac{(μ - V^{2} r) \cos γ}{V r^{2}} \\ f_{2} = 0, g_{2} = 1 \end{array}

\begin{array}{l} f_{3} (x_{1}, x_{2}, x_{3}, V) = \frac{0.5 ρ V^{2} S (C_{M} (α) + C_{M} (q) - c_{e} α)}{I_{y y}} \\ g_{3} (V) = \frac{\bar{q} S \bar{c} c_{e}}{I_{y y}} \end{array}

Assumption 2

There is an upper bound for g1(V), that is, ${\bar{g}}_{1} > | g_{_{1}} (V) | > 0$ .

Assumption 3

χ1(u1) always satisfies $0 < κ < χ_{1} (u_{1}) \leq 1$ , where κ is a positive constant with η₂ = 1/κ.

For the strict-feedback system (21), the procedures of the controller design can be expressed as follows:
Step 1. For x₁, ${\tilde{x}}_{1} = x_{1} - x_{1 d}$ is introduced with x_1d = γ_d; from equation (21), one can obtain
${\dot{\tilde{x}}}_{1} = {\dot{x}}_{1} - {\dot{x}}_{1 d} = f_{1} + g_{1} x_{2} - {\dot{x}}_{1 d}$
23

Introduce a virtual control x_2c, which is defined as

$x_{2 c} = \frac{1}{g_{1}} (- f_{1} - k_{1} {\tilde{x}}_{1} - k_{2} sign ({\tilde{x}}_{1}) {| {\tilde{x}}_{1} |}^{r} + {\dot{x}}_{1 d})$
24

where k₁ and k₂ are positive constants; r = r₁/r₂, and r₁ and r₂ are positive odd integers with r₁<r₂. A new state variable x_2d is developed to obtain the differentiation of x_2c by introducing a first-order filter

$ε_{2} {\dot{x}}_{2 d} + x_{2 d} = x_{2 c}, x_{2 d} (0) = x_{2 c} (0)$
25

where ε₂ > 0 is a constant.
Step 2. For x₂, by introducing ${\tilde{x}}_{2} = x_{2} - x_{2 d}$ and y₂ = x_2d − x_2c, one has
${\dot{\tilde{x}}}_{2} = {\dot{x}}_{2} - {\dot{x}}_{2 d} = x_{3} - {\dot{x}}_{2 d}$
26

To stabilize equation (26), the virtual control x_3c can be chosen as

$x_{3 c} = {\dot{x}}_{2 d} - k_{3} {\tilde{x}}_{2} - k_{4} sign ({\tilde{x}}_{2}) {\tilde{x}}_{2}^{r}$
27

where k₃ and k₄ are positive constants. A first-order filter is defined as

$ε_{3} {\dot{x}}_{3 d} + x_{3 d} = x_{3 c}, x_{3 d} (0) = x_{3 c} (0)$
28

where ε₃ > 0 is a positive constant.
Step 3. Without considering control input saturation constraint, by defining ${\tilde{x}}_{3} = x_{3} - x_{3 d}$ and y₃ = x₃d − x_3c, the control input u_c can be obtained as
$u_{c} = - g_{3}^{- 1} (u_{nom 1} + u_{n 1})$
29

$u_{nom 1} = (f_{3} + k_{5} {\tilde{x}}_{3} + k_{6} sign ({\tilde{x}}_{3}) {\tilde{x}}_{3}^{r} + λ_{1} | g_{3} | sign ({\tilde{x}}_{3}) - {\dot{x}}_{3 d})$
30

$u_{n 1} = \frac{1 - ρ_{1}}{ρ_{1}} | u_{nom 1} | sign ({\tilde{x}}_{3})$
31

where k₅ and k₆ are positive constants.

With the basic fault tolerant control laws (29) to (31), adaptive law is constructed to estimate the unknown parameters. Adaptive fault tolerant controller is presented as

$u_{1} = - g_{3}^{- 1} (u_{nom 1} + u_{n 1})$
32

$u_{nom 1} = (f_{3} + k_{5} {\tilde{x}}_{3} + k_{6} sign ({\tilde{x}}_{3}) {\tilde{x}}_{3}^{r} + {\hat{λ}}_{1} | g_{3} | sign ({\tilde{x}}_{3}) - {\dot{x}}_{3 d})$
33

$u_{n 1} = {\hat{η}}_{2} {\hat{η}}_{1} | u_{nom 1} | sign ({\tilde{x}}_{3})$
34

${\dot{\hat{λ}}}_{1} = k_{7} (| {\tilde{x}}_{3} | | g_{3} | - k_{8} {\hat{λ}}_{1})$
35

${\dot{\hat{η}}}_{1} = k_{9} (| {\tilde{x}}_{3} | | u_{nom} | - k_{10} {\hat{η}}_{1})$
36

${\dot{\hat{η}}}_{2} = k_{11} ({\hat{η}}_{_{2}}^{3} {\hat{η}}_{1} | u_{nom 1} | | {\tilde{x}}_{3} | + k_{12} {\hat{η}}_{2})$
37

where ${\hat{λ}}_{1}$ , ${\hat{η}}_{1}$ , and ${\hat{η}}_{2}$ are the estimate values of λ₁, η₁, and η₂, respectively; ${\tilde{λ}}_{1} = λ_{1} - {\hat{λ}}_{1}$ and ${\tilde{η}}_{1} = η_{1} - {\hat{η}}_{1}$ , with $η_{1} = [1 - (ρ_{1} κ)] / ρ_{1},$ $k_{7}, k_{8}, k_{9}, k_{10}, k_{11}, a n d k_{12}$ being positive scalars.

Theorem 1

Consider the system described in equations (2) to (4). By applying the proposed control schemes (32) to (37), the state error $\tilde{x} = {[{\tilde{x}}_{1}, {\tilde{x}}_{2}, {\tilde{x}}_{3}]}^{T}$ is uniformly ultimately bounded.

Proof

Lyapunov function candidate is selected as

$S_{0} = \sum_{i = 1}^{3} S_{i}$
38

with S_i satisfying

$\begin{array}{l} S_{1} = \frac{1}{2} {\tilde{x}}_{1}^{2} + \frac{1}{2} y_{2}^{2} \\ S_{2} = \frac{1}{2} {\tilde{x}}_{2}^{2} + \frac{1}{2} y_{3}^{2} \\ S_{3} = \frac{1}{2} {\tilde{x}}_{3}^{2} + \frac{1}{2 k_{7}} {\tilde{λ}}_{1}^{2} + \frac{ρ_{1}}{2 k_{9}} {\tilde{η}}_{_{1}}^{2} + \frac{ρ_{1}}{2 k_{11}} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2} \end{array}$

The differential of y_i, i = 2,3 can be written as

$\begin{array}{l} {\dot{y}}_{i} = {\dot{x}}_{i d} - {\dot{x}}_{i c} = - \frac{y_{i}}{ε_{i}} + B_{i} (\cdot) \\ B_{i} (\cdot) = - {\dot{x}}_{i c} \end{array}$
39

It can be obtained from the study of Xu et al.²⁷ that there exist constants o_i > 0, i = 2, 3 with

$| B_{i} (\cdot) | \leq o_{i}$
40

where o_i are unknown positive constants. The time derivative of S_i results in

$\begin{array}{l} {\dot{S}}_{1} = {\tilde{x}}_{1} {\dot{\tilde{x}}}_{1} + y_{2} {\dot{y}}_{2} \\ \leq {\tilde{x}}_{1} (f_{1} + g_{1} x_{2} - {\dot{x}}_{1 d}) - \frac{y_{2}^{2}}{ε_{2}} + | y_{2} | | o_{2} | \\ = g_{1} {\tilde{x}}_{1} {\tilde{x}}_{2} + g_{1} {\tilde{x}}_{1} y_{2} - k_{1} {\tilde{x}}_{1}^{2} - k_{2} {| {\tilde{x}}_{1} |}^{r + 1} - \frac{y_{2}^{2}}{ε_{2}} + | y_{2} | | o_{2} | \\ \leq - (k_{1} - {\bar{g}}_{1}) {\tilde{x}}_{1}^{2} - k_{2} {| {\tilde{x}}_{1} |}^{r + 1} - (\frac{1}{ε_{2}} - \frac{{\bar{g}}_{1}}{2} - \frac{1}{2}) y_{2}^{2} \\ + \frac{{\bar{g}}_{1}}{2} {\tilde{x}}_{2}^{2} + \frac{o_{2}^{2}}{2} \end{array}$
41

$\begin{array}{l} {\dot{S}}_{2} = {\tilde{x}}_{2} ({\dot{x}}_{2} - {\dot{x}}_{2 d}) + y_{3} {\dot{y}}_{3} \\ \leq {\tilde{x}}_{2} {\tilde{x}}_{3} + {\tilde{x}}_{2} y_{3} - k_{3} {\tilde{x}}_{2}^{2} - k_{4} {| {\tilde{x}}_{2} |}^{r + 1} - \frac{y_{3}^{2}}{ε_{3}} + | y_{3} | | o_{3} | \\ \leq - (k_{3} - 1) {\tilde{x}}_{2}^{2} - k_{4} {| {\tilde{x}}_{2} |}^{r + 1} - (\frac{1}{ε_{3}} - 1) y_{3}^{2} + \frac{1}{2} {\tilde{x}}_{3}^{2} + \frac{1}{2} o_{3}^{2} \end{array}$
42

With the application of equations (32) to (37), the time derivative of S₃ results in

${\dot{S}}_{3} = {\tilde{x}}_{3} (f_{3} + g_{3} (ρ_{1} χ_{1} u_{1} + d_{1} (t) + Δ_{1}) - {\dot{x}}_{3 d}) - \frac{1}{k_{7}} {\tilde{λ}}_{1} {\dot{\hat{λ}}}_{1} - \frac{ρ_{1}}{k_{9}} {\tilde{η}}_{1} {\dot{\hat{η}}}_{1} - \frac{ρ_{1}}{k_{11} {\hat{η}}_{2}^{2}} ({\hat{η}}_{2}^{- 1} - η_{2}^{- 1}) {\dot{\hat{η}}}_{2} \leq {\tilde{x}}_{3} (f_{3} + (1 - ρ_{1} χ_{1}) u_{nom 1} - u_{nom 1} - ρ_{1} χ_{1} u_{n 1} - {\dot{x}}_{3 d}) + | g_{3} | | {\tilde{x}}_{3} | λ_{1} - \frac{1}{k_{7}} {\tilde{λ}}_{1} {\dot{\hat{λ}}}_{1} - \frac{ρ_{1}}{k_{9}} {\tilde{η}}_{1} {\dot{\hat{η}}}_{1} - \frac{ρ_{1}}{k_{11} {\hat{η}}_{2}^{2}} ({\hat{η}}_{2}^{- 1} - η_{2}^{- 1}) {\dot{\hat{η}}}_{2} \leq - k_{5} {\tilde{x}}_{3}^{2} - k_{6} {| {\tilde{x}}_{3} |}^{r + 1} + (1 - ρ_{1} κ) | {\tilde{x}}_{3} | | u_{nom 1} | - ρ_{1} κ {\hat{η}}_{2} {\hat{η}}_{1} | u_{nom 1} | | {\tilde{x}}_{3} | + {\tilde{λ}}_{1} | g_{3} | | {\tilde{x}}_{3} | - \frac{ρ_{1}}{k_{9}} (\frac{1 - ρ_{1} κ}{ρ_{1}} - {\hat{η}}_{1}) {\dot{\hat{η}}}_{1} - \frac{1}{k_{7}} {\tilde{λ}}_{1} {\dot{\hat{λ}}}_{1} - ρ_{1} (1 - κ {\hat{η}}_{2}) ({\hat{η}}_{1} | u_{nom 1} | | {\tilde{x}}_{3} |) - k_{12} ρ_{1} {\hat{η}}_{2}^{- 1} ({\hat{η}}_{2}^{- 1} - η_{2}^{- 1}) = - k_{5} {\tilde{x}}_{3}^{2} - k_{6} {| {\tilde{x}}_{3} |}^{r + 1} + k_{8} {\tilde{λ}}_{1} {\hat{λ}}_{1} + ρ_{1} k_{10} {\tilde{η}}_{1} {\hat{η}}_{1} - k_{12} ρ_{1} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2} + k_{12} ρ_{1} η_{2}^{- 1} (η_{2}^{- 1} - {\hat{η}}_{2}^{- 1})$
43

Noting that

$\begin{array}{l} k_{8} {\tilde{λ}}_{1} {\hat{λ}}_{1} = k_{8} (- {\tilde{λ}}_{1}^{2} + \tilde{λ} 1 λ_{1}) \\ \leq k_{8} (- {\tilde{λ}}_{1}^{2} + \frac{1}{2 δ_{2}} {\tilde{λ}}_{1}^{2} + \frac{δ_{2}}{2} λ_{1}^{2}) \\ \leq - \frac{k_{8} (2 δ_{2} - 1)}{2 δ_{2}} {\tilde{λ}}_{1}^{2} + \frac{k_{8} δ_{2}}{2} λ_{1}^{2} \\ ρ_{1} k_{10} {\tilde{η}}_{1} {\hat{η}}_{1} \leq \frac{- ρ_{1} k_{10} (2 δ_{3} - 1)}{2 δ_{3}} {\tilde{η}}_{1}^{2} + \frac{ρ_{1} k_{10} δ_{3}}{2} η_{1}^{2} \\ k_{12} ρ_{1} η_{2}^{- 1} (η_{2}^{- 1} - {\hat{η}}_{2}^{- 1}) \leq \frac{k_{12} ρ_{1} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2}}{2} + \frac{k_{12} ρ_{1}}{2 η_{2}^{2}} \end{array}$

where δ₂, δ₃₁ > 0.5. Therefore, equation (43) can be expressed as

$\begin{array}{l} {\dot{S}}_{3} \leq - χ_{2} (\frac{{\tilde{x}}_{3}^{2}}{2} + \frac{1}{2 k_{7}} {\tilde{λ}}_{1}^{2} + \frac{ρ 1}{2 k_{9}} {\tilde{η}}_{1}^{2} + \frac{ρ_{1}}{2 k_{11}} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2}) \\ + \frac{k_{8} δ_{2}}{2} λ_{1}^{2} + \frac{ρ_{1} k_{10} δ_{3}}{2} η_{1}^{2} + \frac{k_{12} ρ_{1}}{2 η_{2}^{2}} - k_{6} {| {\tilde{x}}_{3} |}^{r + 1} \\ \leq - χ_{2} S_{3} + o_{4} \end{array}$

where

$χ_{2} = min (2 k_{5}, \frac{k_{8} k_{7} (2 δ_{2} - 1)}{δ_{2}}, \frac{k_{9} k_{10} (2 δ_{3} - 1)}{δ_{3}}, k_{11} k_{12})$

$o_{4} = \frac{k_{8} δ_{2}}{2} λ_{1}^{2} + \frac{ρ_{1} k_{10} δ_{3}}{2} η_{1}^{2} + \frac{k_{12} ρ_{1}}{2 η_{2}^{2}}$

For the differential of S₀, one obtains

$\begin{array}{l} {\dot{S}}_{0} \leq - (k_{1} - {\bar{g}}_{1}) {\tilde{x}}_{1}^{2} - (\frac{1}{ε_{2}} - \frac{{\bar{g}}_{1}}{2} - \frac{1}{2}) y_{2}^{2} + \frac{{\bar{g}}_{1}}{2} {\tilde{x}}_{2}^{2} + \frac{o_{2}^{2}}{2} \\ - (k_{3} - 1) {\tilde{x}}_{2}^{2} - (\frac{1}{ε_{3}} - 1) y_{3}^{2} + \frac{{\tilde{x}}_{3}^{2}}{2} + \frac{o_{3}^{2}}{2} - k_{5} {\tilde{x}}_{3}^{2} \\ + k_{8} {\tilde{λ}}_{1} {\hat{λ}}_{1} + ρ_{1} k_{10} {\tilde{η}}_{1} {\hat{η}}_{1} - k_{12} ρ_{1} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2} \\ + k_{12} ρ_{1} η_{2}^{- 1} (η_{2}^{- 1} - {\hat{η}}_{2}^{- 1}) - k_{2} {| {\tilde{x}}_{1} |}^{r + 1} - k_{4} {| {\tilde{x}}_{2} |}^{r + 1} \\ - k_{6} {| {\tilde{x}}_{3} |}^{r + 1} \\ \leq - (k_{1} - {\bar{g}}_{1}) {\tilde{x}}_{1}^{2} - (k_{3} - 1 - \frac{{\bar{g}}_{1}}{2}) {\tilde{x}}_{2}^{2} - (k_{5} - \frac{1}{2}) {\tilde{x}}_{3}^{2} \\ - (\frac{1}{ε_{2}} - \frac{{\bar{g}}_{1}}{2} - \frac{1}{2}) y_{2}^{2} - (\frac{1}{ε_{3}} - 1) y_{3}^{2} + \frac{o_{2}^{2}}{2} + \frac{o_{3}^{2}}{2} \\ + k_{8} {\tilde{λ}}_{1} {\hat{λ}}_{1} + ρ_{1} k_{10} {\tilde{η}}_{1} {\hat{η}}_{1} - k_{12} ρ_{1} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2} \\ + k_{12} ρ_{1} η_{2}^{- 1} (η_{2}^{- 1} - {\hat{η}}_{2}^{- 1}) \\ \leq - ϑ_{1} \frac{{\tilde{x}}_{1}^{2}}{2} - ϑ_{2} \frac{{\tilde{x}}_{2}^{2}}{2} - ϑ_{3} \frac{{\tilde{x}}_{3}^{2}}{2} - ϑ_{4} \frac{y_{2}^{2}}{2} - ϑ_{5} \frac{y_{3}^{2}}{2} - ϑ_{6} \frac{{\tilde{λ}}_{1}^{2}}{2 k_{7}} \\ - ϑ_{7} \frac{ρ_{1} {\tilde{η}}_{1}^{2}}{2 k_{9}} - ϑ_{8} \frac{ρ_{1}}{2 k_{11}} {({\hat{η}}_{2}^{- 1} - η_{2}^{- 1})}^{2} + o_{5} \\ \leq - ϑ_{9} S_{0} + o_{5} \end{array}$
44

where

$ϑ_{1} = 2 (k_{1} - {\bar{g}}_{1}), ϑ_{2} = 2 (k_{3} - 1 - \frac{{\bar{g}}_{1}}{2}), ϑ_{3} = 2 (k_{5} - \frac{1}{2})$

$ϑ_{4} = 2 (\frac{1}{ε_{2}} - \frac{{\bar{g}}_{1}}{2} - \frac{1}{2}), ϑ_{5} = 2 (\frac{1}{ε_{3}} - 1), ϑ_{6} = \frac{k_{8} k_{7} (2 δ_{2} - 1)}{δ_{2}}$

$ϑ_{7} = \frac{k_{9} k_{10} (2 δ_{3} - 1)}{δ_{3}}, ϑ_{8} = k_{11} k_{12}, ϑ_{9} = min (ϑ_{j}), j = 1, 2, \dots, 8$

$o_{5} = \frac{o_{2}^{2}}{2} + \frac{o_{3}^{2}}{2} + \frac{k_{8} δ_{2}}{2} λ_{1}^{2} + \frac{ρ_{1} k_{10} δ_{3}}{2} η_{1}^{2} + \frac{k_{12} ρ_{1}}{2 η_{2}^{2}}$

Also, k₁, k₂, k₃, k₄, k₅, k₆, k₇, k₈, k₉, k₁₀, k₁₁, and k₁₂ are chosen, so that ϑ_j satisfies ϑ_j > 0. It can be concluded that ${\tilde{x}}_{1}, {\tilde{x}}_{2}, a n d {\tilde{x}}_{3}$ are uniformly ultimately bounded. This completes the proof.

Remark 1

The proposed control laws can cause chattering problems as sign function is used. Control chattering can be dealt with by defining a thin boundary layer of width φ2 = 0.2, and boundary layer function is introduced to replace sign function $sign (\tilde{x} 3)$ in control laws (33) and (34), which is defined as

$sat ({\tilde{x}}_{3} / f_{2}) = {\begin{cases} {\tilde{x}}_{3} / φ_{2} | {\tilde{x}}_{3} / φ_{2} | \leq 1 \\ sign ({\tilde{x}}_{3} / f_{2}) | {\tilde{x}}_{3} / φ_{2} | > 1 \end{cases}$

Simulation

Effectiveness of the proposed controllers is verified by numerical simulations conducted on the longitudinal dynamics of the GHV in its cruising phase with the initial conditions: h = 110,000 ft, V = 15,060 ft/s, and $(γ, α, q) = (0.005 rad, 1.265 pi / 180 rad, 0 rad / s)$ . Control input constraint satisfies $| u_{1 max} | < 0.1 rad$ . The velocity and altitude commands are V_d = 15,160 ft/s and h_d = 110,100 ft respectively. An actuator fault occurs at 100s, which can be expressed as

$ρ_{1} = {\begin{cases} 1, t < 100 s \\ 0.6, t \geq 100 s \end{cases}$
45

$Δ_{1} = {\begin{cases} 0, t < 100 s \\ 0.01, t \geq 100 s \end{cases}$
46

The external disturbance can be presented as

$d_{1} = {\begin{cases} 0, t < 100 s \\ 0.01 \times sin (0.1 t), t \geq 100 s \end{cases}$
47

The initial value of states is chose as the same as cruising conditions while the initial adaption parameters and control gains are chosen as follows

${\hat{η}}_{1} (0) = 0.035, {\hat{η}}_{2} (0) = 0.1, {\hat{λ}}_{1} (0) = 0.1, k_{h} = 0.5$

$\begin{array}{l} k_{1} = 2.5, k_{2} = 0.01, k_{3} = 2, k_{4} = 0.01, k_{5} = 2, k_{6} = 0.01 \\ k_{7} = 0.1, k_{8} = 0.01, k_{9} = 0.1, k_{10} = 0.01, k_{11} = 0.1, k_{12} = 0.01 \end{array}$

$ε_{2} = ε_{3} = 0.1, r = 3 / 7$

Simulation results are elaborated in Figures 1 to 4. Figures 1 and 2 depict the tracking responses of velocity and altitude, from which it can be observed that altitude and velocity can track the step commands in spite of external disturbance, actuator fault, and input saturation. For control inputs throttle setting and elevator deflection in Figures 3 and 4, it is obviously observed that outputs of the controllers (32) to (37) are within the maximum allowable values of the actuator. Simulation results show that velocity and altitude can track the desired command trajectories with a good performance in spite of external disturbance, actuator fault, and input saturation.

Figure 1.
Response to a 15,160 ft/s step velocity command.

Figure 2.
Response to a 110,100 ft step altitude command.

Figure 3.
Throttle setting of engine β.

Figure 4.
Elevator deflection angle.

Conclusion

This article investigates adaptive control method for the cruising GHV to track desired trajectories when actuator fault occurs and external disturbance exists under input saturation. Based on functional decomposition, adaptive dynamic inversion controller is proposed for velocity subsystem while controller is designed based on backstepping for altitude subsystem. The adaption laws are utilized to estimate the values of actuator faults and external disturbances. Simulation results reveal that the designed controllers can finish tracking missions with good performance.

In the further research work, control scheme design for GHV will consider actuator faults with input saturation problem and multiple faults occur simultaneously.

Footnotes

Acknowledgement

The authors thank Yixin Cheng for his useful discussion on controller design and stability analysis.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (61622308), Aeronautical Science Foundation of China (20130753005, 2016ZC53019 and 2015ZA53003), Natural Science Basic Research Plan in Shaanxi Province (2016KJXX-86), Fundamental Research Funds of Shenzhen Science and Technology Project (JCYJ20160229172341417), and Xi’an Flight Automatic Control Research Institute.

References

Parker

Serrani

Yurkovich

. Control-oriented modeling of an air-breathing hypersonic vehicle. J Guid Control Dyn 2007; 30(3): 856–869.

Bolender

Doman

. Nonlinear longitudinal dynamical model of an air-breathing hypersonic vehicle. J Spacecraft Rockets 2007; 44(2): 374–387.

Fiorentini

Serrani

. Adaptive restricted trajectory tracking for a non-minimum phase hypersonic vehicle model. Automatica 2012; 48(7): 1248–1261.

Mirmirani

Ioannou

. Adaptive sliding mode control design for a hypersonic flight vehicle. J Guid Control Dyn 2004; 27(5): 829–838.

Wang

Zong

. Continuous high order sliding mode controller design for a flexible air-breathing hypersonic vehicle. ISA Trans 2014; 53(3): 690–698.

Chen

Ren

. Anti-disturbance control of hypersonic flight vehicles with input saturation using disturbance observer. Sci China Inform Sci 2015; 58(7): 1–12.

Geng

Sheng

Liu

. Finite-time sliding mode attitude control for a reentry vehicle with blended aerodynamic surfaces and a reaction control system. Chin J Aeronaut 2014; 27(4): 964–976.

Wang

Zong

. Adaptive finite-time control for a flexible hypersonic vehicle with actuator fault. Math Probl Eng 2013; 2013

Sun

. Finite time integral sliding mode control of hypersonic vehicles. Nonlinear Dyn 2013; 73(1–2): 229–244.

10.

Butt

Yan

Kendrick

. Adaptive dynamic surface control of a hypersonic flight vehicle with improved tracking. Asian J Control 2013; 15(2): 594–605.

11.

Shi

. An overview on flight dynamics and control approaches for hypersonic vehicles. Sci China Inform Sci 2015; 58(7): 1–19.

12.

Lan

Wang

Liu

. Dynamic decoupling tracking control for the polytopic LPV model of hypersonic vehicle. Sci China Inform Sci 2015; 58(9): 1–14.

13.

Pan

Wang

. Discrete-time hypersonic flight control based on extreme learning machine. Neurocomputing 2014; 128: 232–241.

14.

Fan

Zhang

. Minimal-learning-parameter technique based adaptive neural control of hypersonic flight dynamics without back-stepping. Neurocomputing 2015; 164: 201–209.

15.

Wang

. Adaptive neural control of a hypersonic vehicle in discrete time. J Intell Robot Syst 2014; 73(1–4): 219–231.

16.

Zhang

Pan

. Neural network based dynamic surface control of hypersonic flight dynamics using small-gain theorem. Neurocomputing 2016; 173: 690–699.

17.

. Robust adaptive neural control of flexible hypersonic flight vehicle with dead-zone input nonlinearity. Nonlinear Dyn 2015; 80(3): 1509–1520.

18.

Tian

Fan

. Nonlinear robust adaptive deterministic control for flexible hypersonic vehicles in the presence of input constraint. Asian J Control 2015; 17(6): 2303–2316.

19.

Chen

Ren

. Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica 2011; 47(3): 452–465.

20.

Zong

Wang

Tian

. Robust adaptive dynamic surface control design for a flexible air-breathing hypersonic vehicle with input constraints and uncertainty. Nonlinear Dyn 2014; 78(1): 289–315.

21.

Wang

Gao

. Command filter based robust nonlinear control of hypersonic aircraft with magnitude constraints on states and actuators. J Intell Robot Syst 2014; 73(1–4): 233–247.

22.

Wang

Zhang

Jin

. Neural control of hypersonic flight dynamics with actuator fault and constraint. Sci China Inform Sci 2015; 58(7): 1–10.

23.

Huang

Jiang

. Adaptive backstepping control for a hypersonic vehicle with uncertain parameters and actuator faults. P I Mech Eng I-J Sys 2013; 227(1): 51–61.

24.

Sun

. Robust adaptive integral-sliding-mode fault-tolerant control for airbreathing hypersonic vehicles. P I Mech Eng I-J Sys 2012; 226(10): 1344–1355.

25.

Gao

Wang

. Adaptive neural control based on HGO for hypersonic flight vehicles. Sci China Inform Sci 2011; 54(3): 511–520.

26.

Chen

Wang

Tao

. Robust adaptive fault-tolerant control for hypersonic flight vehicles with multiple faults. J Aerospace Eng 2014; 28(4): 4014111.

27.

Huang

Wang

. Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation. Asian J Control 2014; 16(1): 162–174.

Adaptive fault tolerant control for hypersonic vehicle with external disturbance

Abstract

Keywords

Introduction

Longitudinal model of hypersonic vehicle and problem formulation

Vehicle model with actuator fault

Problem formulation

Assumption 1

Controller design

Assumption 2

Assumption 3

Theorem 1

Proof

Remark 1

Simulation

Conclusion

Footnotes

Acknowledgement

Declaration of conflicting interests

Funding

References