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Station-keeping Control of an Underactuated Stratospheric Airship

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Abstract

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.

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Abbreviations

\(\varvec{\chi }\) :

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

\(\varvec{C(V)}\) :

Centrifugal and Coriolis matrix

\({\varvec{D}}_\mathbf{{diss}}\) :

Damping and aerodynamic matrix

\(\varvec{d}\) :

Vector of disturbance force and torque

\({\varvec{e}}\) :

Position error vector

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

Restoring forces and moments

\(\varvec{J}=\varvec{diag}({\varvec{J}}_{\mathbf{1}},{\varvec{J}}_{\mathbf{2}})\) :

Rotation matrix

\({\varvec{J}}_{\mathbf{3}}\) :

Transformation matrix from ERF to PPF

\({\varvec{M}}\) :

Mass matrix

\({\varvec{P}}_\mathbf{{h}}^\mathbf{{a}}\) :

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

\({\varvec{P}}_\mathbf{{h}}^{\varvec{d}}\) :

The hovering point

\({\varvec{P}}\) :

Position of the airship

\(\varvec{u}\) :

Control forces and control moments vector

\(\varvec{v}_\mathbf{{a}}=[u_a;v_a;w_a{]}\) :

Airspeed

\({\varvec{V}}\) :

Velocity vector of the airship

\(\varvec{W}\) :

Wind field

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

Estimated wind direction

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

Wind direction

\(\hat{V}_\mathrm{w}\) :

Estimated wind speed

\(\hat{V}_\mathrm{w}\) :

Wind speed

\(\mu \) :

Path parameter

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

Attitude angles of the airship

\(\psi _c\) :

Desired yaw angle

\(u_c\) :

Desired forward speed

x, y, z :

Position of the airship

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Acknowledgements

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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Authors and Affiliations

  1. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Weixiang Zhou, Pingfang Zhou, Yueying Wang & Dengping Duan

  2. School of Marine Electrical Engineering, Dalian Maritime University, Dalian, 116026, China

    Ning Wang

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Correspondence to Weixiang Zhou.

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Zhou, W., Zhou, P., Wang, Y. et al. Station-keeping Control of an Underactuated Stratospheric Airship. Int. J. Fuzzy Syst. 21, 715–732 (2019). https://doi.org/10.1007/s40815-018-0566-4

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