Station-keeping Control of an Underactuated Stratospheric Airship

  • Weixiang ZhouEmail author
  • Pingfang Zhou
  • Yueying Wang
  • Ning Wang
  • Dengping Duan


This paper studies the station-keeping control of underactuated stratospheric airships in the presence of model uncertainties and wind field, and a T–S fuzzy model-based adaptive backstepping SMC (sliding mode controller) is proposed. Firstly, a fuzzy dynamics model is constructed to represent the local dynamic behaviors of the given 6-DOF nonlinear dynamics of an airship. And different from the traditional algorithm, the station-keeping control is resorted to path-following control. Then, the guidance-based path-following principle is adopted to obtain the guidance law, and the backstepping technique is used to obtain the desired velocities. In order to solve the problem of model uncertainties between T–S fuzzy model and nominal model, the sliding mode control approach is adopted. Adaptive terms are designed to estimate the upper bound of the uncertainties. Besides, a wind field observer is designed to estimate the speed and direction of the wind. The stabilization of the system is discussed using Lyapunov stability theory. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed method.


Stratospheric airship Station-keeping control T–S fuzzy model Backstepping control Adaptive sliding mode control 


\(\varvec{\chi }\)

Error variable vector

\(\varvec{\Delta A}\)

Model uncertainties between nominal model and T–S fuzzy model

\(\varvec{\Delta },\varvec{C\Delta D}\)

Parameter variations

\(\varvec{\eta }\)

Position and attitude of the airship

\(\hat{\varvec{\kappa }}_{\varvec{i}}\)

Estimated upper bound of uncertainties

\(\varvec{\kappa }_{\varvec{i}}\)

Unknown upper bound of uncertainties

\(\varvec{\varOmega }_{\varvec{c}}\)

Desired attitude vector

\(\varvec{\omega }_{\varvec{c}}\)

Desired angular velocities

\(\varvec{\varOmega }_\mathbf{{e}}\)

Tracking error vector of attitude

\(\varvec{\omega }_\mathbf{{e}}\)

Error vector of angular velocities

\(\varvec{\varOmega }\)

Attitude of the airship

\(\varvec{\omega }=[p;q;r{]}\)

Angular velocity of the airship

\(\varvec{\tau =[\tau _v;\tau _\omega ]}\)

Control signals

\(\varvec{\upsilon }_\mathbf{{a}}\)

Airspeed velocity of the airship


Centrifugal and Coriolis matrix


Damping and aerodynamic matrix


Vector of disturbance force and torque


Position error vector

\(\varvec{h(\eta )}\)

Restoring forces and moments


Rotation matrix


Transformation matrix from ERF to PPF


Mass matrix


Position of the airship in horizontal plane in wind field

\({\varvec{P}}_\mathbf{{h}}^{\varvec{c}}=[{x}_{{c}}({\mu });{y}_{{c}}({\mu }){]}\)

The desired path


The hovering point


Position of the airship


Control forces and control moments vector




Velocity vector of the airship


Wind field

\(\hat{\psi }_\mathrm{w}\)

Estimated wind direction

\(\hat{\psi }_\mathrm{w}\)

Wind direction


Estimated wind speed


Wind speed

\(\mu \)

Path parameter

\(\phi ,\theta ,\psi \)

Attitude angles of the airship

\(\psi _c\)

Desired yaw angle


Desired forward speed


Position of the airship



The authors thank the editor and anonymous reviewers for their valuable comments and suggestions that enabled us to clarify the analysis and improve the readability of the paper. This work was supported by the National Natural Science Foundation of China [Grant Nos. 61773258 and 61703275].


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Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Weixiang Zhou
    • 1
    Email author
  • Pingfang Zhou
    • 1
  • Yueying Wang
    • 1
  • Ning Wang
    • 2
  • Dengping Duan
    • 1
  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.School of Marine Electrical EngineeringDalian Maritime UniversityDalianChina

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