# Adaptive Backstepping Nonsingular Terminal Sliding-Mode Attitude Control of Flexible Airships with Actuator Faults

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A backstepping nonsingular terminal sliding-mode control scheme is proposed for attitude tracking control of the flexible airship with actuator faults, actuator saturation, and uncertainties of stiffness.
- A wind observer with an adaptive disturbance observer is designed to reject variable external bounded disturbances and cope with model parameter uncertainties.
- An anti-windup compensator based on proportion to saturation errors is used to compensate actuator saturation.
- An adaptive fault estimator is incorporated into the BNTSM control to implement fault estimation and fault tolerant control.

## 2. Airship Modeling and Problem Formulation

#### 2.1. Airship Kinematics Model

**F**fixed on the undeformed airship with its origin, O

_{b}{O_{b}x_{b}y_{b}z_{b}}_{b}, at the center of volume (CV), see Figure 1. The motion of the airship is described as the translation and rotation of this body frame with respect to the inertial frame

**F**, plus the deformation of the material points on the body relative to the body frame [21]. A centerline frame

_{I}{O_{I}x_{I}y_{I}z_{I}}**F**is introduced to calculate the aerodynamics forces of the flexible airship. The elastic displacement

_{p}{O_{p}x_{p}y_{p}z_{p}}**u**of a material point on the airship can be written by a summation of the shape functions, as

**r**is the position vector of the material point from the origin, O

_{b}. The associated velocity distribution over the elastic body is:

^{T}is the translational velocity vector in

**F**

_{b}, $\mathsf{\Omega}={\left[\begin{array}{ccc}p& q& r\end{array}\right]}^{T}$ is the angular rate vector, and the × denotes cross-product operation. The first two terms of Equation (2) are related to the rigid body translation and rotation, while the last two terms reflect the influence of the elasticity on the local velocity.

**ξ**= [x, y, z]

^{T}is the position vector, $\eta ={\left[\begin{array}{ccc}\varphi & \theta & \psi \end{array}\right]}^{T}$ is the attitude vector, ${\upsilon}_{a}$ = [u, v, w]

^{T}is the translational velocity vector in

**F**

_{b}, and ${\mathbf{\upsilon}}_{w}$ is the wind velocity in fixed frame. From Equation (4), it can be obtained that the local velocity is:

**R**(η) denotes the direction cosine matrix [3], and the transformation matrix

**J**is:

**F**denotes the inertia force vector,

_{I}**F**

_{G}denotes the gravity vector,

**F**denotes the aerostatic force vector,

_{AS}**F**denotes the aerodynamics force vector, and

_{AD}**F**denotes the control force vector; ${M}_{i}$ and ${Q}_{i}$ denote the associated moments and general elastic forces of the flexible airship,

_{C}**i**=

**I**,

**G**,

**AS**,

**AD**and

**C**. The local velocity is as Equation (5); $\mathit{q}={[{q}_{1},{q}_{2},\cdots ,{q}_{2N}]}^{T}$ denotes the generalized elastic coordinate vector. ${S}_{E}$ denotes the internal elastic force vector,

**K**meets

^{3}ET

_{0}, E is the elastic modulus of the hull envelope, T

_{0}is its thickness, and R is the hull radius. Equation (7) can be rewritten as

_{T}, y

_{T}, z

_{T}) and (x

_{r}, y

_{r}, z

_{r}) denote positions of the thrust and control surface in the body frame,

**F**, respectively; x

_{b}_{m}is the position of the aerostatic center along the x-axis, k

_{T}is control gain for thruster. ${C}_{z{\delta}_{e}}$ and ${C}_{y{\delta}_{r}}$ denote the aerodynamic derivative coefficients of the elevator and rudder, respectively, $\overline{q}=\frac{1}{2}\rho {V}^{2}$ denotes the dynamic pressure, V is the airspeed, and ρ is the air density.

_{B}denotes the volume of the airship hull and S is the hull cross-sectional area:

**d**, and the actuator saturation are introduced, thus the system (10) can be modified, as

**d**denotes disturbances except for winds and model uncertainties, such as aerodynamic coefficients and structural stiffness; $\mathrm{sat}(\cdot )$ denotes saturation function. As the airship has a large inertia and its motion is slow and sedate, d can be assumed to be a slow-varying disturbance, which can be estimated on-line by using an adaptive law. Let $\widehat{\mathit{d}}$ be the estimated value of the uncertain parameter

**d**; $\tilde{\mathit{d}}=\widehat{\mathit{d}}-\mathit{d}$ is the associated estimated error. Finally,

**Г**denotes the disturbance coefficient matrix. The faults can be modeled as abrupt changes of the nominal control action from ${\mathit{u}}_{a}$ [26,27],

**I**is the m×m identity matrix; the fault vector ${\mathit{f}}_{a}=\left[{f}_{a,1},{f}_{a,2},\cdots ,{f}_{a,m}\right]\in {R}^{m}$ denotes the control action from the failed or un-manipulated actuators; ${u}_{h}={\overline{\rho}}_{a}{\mathit{u}}_{a}(t)$ denotes the remaining control of the health actuators; ${\mathit{u}}_{f}=(\mathit{I}-{\overline{\rho}}_{a}(t)){\mathit{f}}_{a}(t)$ denotes the fault of the inputs.

**Assumption**

**1.**

**Remark**

**1.**

_{sy}= 0); thus, the sway velocity, v, cannot be directly controlled. If the wind is in the presence in this case, then the airship can align against the wind through the yaw motion, reducing the lateral forces requirement to a low and acceptable value; thus, the lateral force input can vanish in stationary conditions. Case 2. The airship works in Case 1 without ailerons or differential actuators (i.e., δ

_{eL}= δ

_{eR}, δr

_{U}= δ

_{rB}, and the roll control moment, M

_{Tx}≈ 0). Sway velocity, v, and bank angle, φ, cannot be directly controlled. In this case, the disturbance of the roll moment resulting from the wind can be attenuated by the airship roll damp; thus, the roll moment input can vanish in stationary conditions.

#### 2.2. Fault Tolerant Trajectory Tracking Problem

_{r}, with $\underset{t\to \infty}{\mathrm{lim}}{\Vert e(t)\Vert}_{2}<{\epsilon}_{0}$, even in a specified model parameter uncertainty, unknown wind disturbances, and control surface faults; where the tracking error is $\mathit{e}(t)=\eta (t)-{\eta}_{r}(t)$, ε

_{0}is a prescribed constant, and 2-norm is ${\Vert \mathit{e}(t)\Vert}_{2}=\sqrt{{\mathit{e}}^{T}\mathit{e}}$.

## 3. BNTSM-Based Trajectory Tracking Design

_{n}denotes the undamped nature frequency of the filter. To reduce the filter error $e={x}_{r}^{0}-{x}_{r}$, the filter frequency, ω

_{n}, is the bandwidth of ${X}_{r}^{0}(s)$, which is generally less than that of G

_{r}(s), and then ω

_{n}can be selected.

**e**

_{4}can be obtained as

_{3}is the unit matrix with a dimension of 3 × 3. Thus, there exists a positive definite symmetrical matrix, ${\mathit{P}}_{e}$, such that

_{1}= s(φ), s

_{2}= s(θ), s

_{3}= s(ψ) and 1 < p/q < 2, β > 0; sgn(·) is the sign function. In addition, the derivative of the sliding-mode surface, s, is

**η**, will converge to the desired value of

**η**

_{r}. The attitude controller is derived in two steps, and the detailed design is presented in Appendix A.

**u**as the difference between the desired control input

**u**and the actuator output, i.e., ∆

**u**=

**u**

_{a}− sat(

**u**

_{a}), an anti-windup compensator is designed to compensate for the actuator saturation, as follows:

**Remark**

**2.**

## 4. Simulation and Analysis

**F**

_{g}of the airship is ${\xi}_{0}$ = [0, 0, 0 m]

^{T}, the initial velocity in

**F**

_{b}is υ

_{0}= [8.42 m/s, 0, 0]

^{T}, the initial attitude is η

_{0}= [0, 0, 0]

^{T}, and the initial angular velocity is ω

_{0}= [0, 0, 0]

^{T}. The position constraints for the elevator and rudder are [−30°, 30°]. Three scenarios of bounded wind disturbances, control surface faults, and the variable stiffness of the flexible airship envelope are simulated to illustrate the BNTSM controller performances. The software of MATLAB and Simulink are employed to solve the problems.

**Scenario**

**1.**

**ζ**

_{n}= 0.9, ω

_{n}= 20 rad/s. The mode number, N, selects N = 2,

**P**

_{e}= 0.1, $\mathit{\Gamma}=1$. The gain of the actuator saturation compensator is K

_{s}= 1, and control gain for thruster k

_{T}= 90,000. A doublet command is predefined as the desired attitude.

**Scenario**

**2.**

**Scenario**

**3.**

_{i}will become a smaller and smaller amplitude of oscillation with the increasing stiffness. The general coordinate, q

_{1}, for the first bending mode, is increased in amplitude by 30% for the −50% stiffness variant, and the general coordinate, q

_{1}, reduced in amplitude by 8.3% for the +30% stiffness case. The same case happens for the generalized coordinate velocity responses. This demonstrates that stiffness can suppress structural mode oscillation. Figure 16 shows that the control inputs change little when the stiffness of the flexible airship is variable, where * denotes multiplication sign.

## 5. Conclusions

## 6. Future Recommendation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BISMC | Backstepping integral sliding mode control |

BNTSM | Backstepping nonsingular terminal sliding mode |

CLFs | Control Lyapunov functions |

CV | Center of volume |

EI | Bending stiffness, E means the elastic modulus, I means the area moment of inertia |

NTSM | Nonsingular terminal sliding mode |

SMC | Sliding-mode control |

RBFNN | Radial basis function neural network |

## Appendix A

**z**

_{1}.

_{1}, as

**J**is in Equation (4). Then,

_{1}, a new signal, ${\mathit{x}}_{2r}^{0}$, is defined as

_{2}will be designed in Step 2. The command signal, ${\mathit{x}}_{2r}^{0}$, is filtered to produce the reference signal, ${\mathit{x}}_{2r}$, and its derivative, ${\dot{\mathit{x}}}_{2r}$. It can be implemented to enforce magnitude and rate limits through the command filter (37). By the design of the command filter (37), the signal $\left({\mathit{x}}_{2,r}-{\mathit{x}}_{2,r}^{0}\right)$ is bounded and small.

**z**

_{1}is estimated by the following stable linear filter

_{1}> 0. To remove the effect of filtering the stabilizing functions from the tracking error, the compensated tracking error is defined as

_{1}as a quadratic function of the compensated tracking error is

**z**

_{2}, is estimated by the following stable linear filter:

**P**

_{e}and

**P**

_{f}are positive definite weight matrices. Considering Equation (A9) and substituting Equations (35), (45), (A4), (A6) and (A10) into the derivative of V

_{2}, yields

**d**is an unknown constant or a slow-varying disturbance, then $\dot{\tilde{\mathit{d}}}=\dot{\widehat{\mathit{d}}}-\dot{\mathit{d}}\approx \dot{\widehat{\mathit{d}}}$. If

**d**is a fast-varying process, a RBF-neural network-based backstepping control can be used [33]. The stabilizing function is defined as

_{2}> 0, h, ς are sliding-mode surface parameters with h > 0, ς > 0. The control input of

**u**

_{h}is generated by using the above Equation (A13), and then

**Q**

_{ξ}and

**Q**${\upsilon}_{w}$, respectively. Using (A16), it is obtained that

_{k}= min{2Q, 2ς, –γ

_{f}+1, $2{q}_{\xi i},2{q}_{wi}\}>0$ (i = 1, 2, 3), then

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Parameter | Value | Parameter | Value (kg) |
---|---|---|---|

I_{x} | 1.0834 × 10^{5} (kg·m^{2}) | m | 2.682 × 10^{3} |

I_{y} | 2.8819 × 10^{5} (kg·m^{2}) | m_{11} | 5.0575 × 10^{2} |

I_{z} | 2.1722 × 10^{5} (kg·m^{2}) | m_{22} | 5.0737 × 10^{3} |

I_{xz} | 0 | m_{33} | 5.0737 × 10^{3} |

L | 50 (m) | m_{44} | 6.8879 × 10^{4} |

(x_{G}, y_{G}, z_{G}) | (0, 0, 3.605) (m) | m_{55} | 5.8540 × 10^{5} |

V_{B} | 5.131 × 10^{3} (m^{3}) | m_{66} | 5.8540 × 10^{5} |

ET_{0} | 433,440 (N/m) |

Controller | Parameters | Value |
---|---|---|

PID | k_{p,z}_{j}, k_{i,z}_{j} (j = 1, 2) | 2, 0.04 |

BNTSM | c_{1}, c_{2} | diag(2, 2, 2), diag(0.2, 0.2, 0.2) |

h, ς, φ_{s} | 0.01 * diag(1,1,1), 0.01 * diag(1, 1.2, 1), 0.4 | |

p, q, β | 5, 3, 0.1 * diag(1, 1, 1) | |

α_{r}, γ_{d}, λ | 5, 1, 1. 8 * diag(1, 1, 1) | |

BISMC | c_{1}, c_{2} | 0.1 * diag(1, 1, 1), 0.01 * diag(1, 1, 1) |

h, ς | 0.1 * diag(1, 1, 1), diag(1, 1.2, 1) | |

λ_{1},λ_{2} | 0.5 * diag(1, 1, 1),0.05 * diag(1, 1, 1) | |

φ_{s}, k_{i,smc} | 0.4, 0.1 |

**Table 3.**Tracking errors and ET of the airship under the input of the controller without wind compensation.

Controller Error | E_{φ} (rad) | E_{θ} (rad) | E_{ψ} (rad) | ET (s) |
---|---|---|---|---|

PID | 0.0532 | 0.0843 | 0.0893 × 10^{−3} | 2153.6 |

BNTSM | 0.0525 | 0.0834 | 0.0887 × 10^{−3} | 2367.2 |

BISMC | 0.0528 | 0.0839 | 0.0890 × 10^{−3} | 2226.2 |

**Table 4.**Tracking errors and ET of the airship under the input of the controller with wind compensation.

Controller Error | E_{φ} (rad) | E_{θ} (rad) | E_{ψ} (rad) | ET (s) |
---|---|---|---|---|

PID | 0.0577 | 0.0786 | 0.0817 × 10^{−3} | 1993.2 |

BNTSM | 0.0574 | 0.0774 | 0.0805 × 10^{−3} | 2050.4 |

BISMC | 0.0575 | 0.0780 | 0.0811 × 10^{−3} | 2048.9 |

**Table 5.**Tracking errors and ET of the airship under the input of the controller without fault compensation.

Controller Error | E_{φ} (rad) | E_{θ} (rad) | E_{ψ} (rad) | ET (s) |
---|---|---|---|---|

PID | 0.0510 | 0.0810 | 0.0788 × 10^{−3} | 2171.4 |

BNTSM | 0.0511 | 0.0803 | 0.0775 × 10^{−3} | 2354.3 |

BISMC | 0.0510 | 0.0805 | 0.0780 × 10^{−3} | 2193.0 |

**Table 6.**Tracking errors and ET of the airship under the input of the controller with fault compensation.

Controller Error | E_{φ} (rad) | E_{θ} (rad) | E_{ψ} (rad) | ET (s) |
---|---|---|---|---|

PID | 0.0488 | 0.0804 | 0.0829 × 10^{−3} | 1821.0 |

BNTSM | 0.0490 | 0.0797 | 0.0817 × 10^{−3} | 2114.1 |

BISMC | 0.0489 | 0.0798 | 0.0819 × 10^{−3} | 2016.2 |

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**MDPI and ACS Style**

Liu, S.; Whidborne, J.F.; Song, S.; Lyu, W.
Adaptive Backstepping Nonsingular Terminal Sliding-Mode Attitude Control of Flexible Airships with Actuator Faults. *Aerospace* **2022**, *9*, 209.
https://doi.org/10.3390/aerospace9040209

**AMA Style**

Liu S, Whidborne JF, Song S, Lyu W.
Adaptive Backstepping Nonsingular Terminal Sliding-Mode Attitude Control of Flexible Airships with Actuator Faults. *Aerospace*. 2022; 9(4):209.
https://doi.org/10.3390/aerospace9040209

**Chicago/Turabian Style**

Liu, Shiqian, James F. Whidborne, Sipeng Song, and Weizhi Lyu.
2022. "Adaptive Backstepping Nonsingular Terminal Sliding-Mode Attitude Control of Flexible Airships with Actuator Faults" *Aerospace* 9, no. 4: 209.
https://doi.org/10.3390/aerospace9040209