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A Robust State Feedback Optimal Control Law with Backstepping Approach for Steering Control of an Autonomous Underwater Vehicle Using Semi-definite Programming

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Abstract

In this paper, the desired yaw orientation for an autonomous underwater vehicle is attained by acquiring a cascaded control structure based on a robust optimal control algorithm with a backstepping approach. A robust state feedback optimal control law is designed to control the yaw rate. Hence, the desired yaw rate is intended to be obtained by the backstepping controller by controlling the desired yaw orientation. The implementation of the proposed robust control algorithm is formulated by using semi-definite programming. A linear quadratic regulator in terms of linear matrix inequality is designed to address the control problem. The design of robust optimal control law in the steering plane is achieved by considering an uncertain polytopic AUV system. Realization of the proposed control algorithm is conducted in MATLAB/Simulink environment using the YALMIP tool. Robust behavior is ensured by the proposed control algorithm while tracking the desired yaw. The robustness analysis is extended by considering the various ranges of specific uncertain parameters to highlight the efficacies of the proposed control algorithm.

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Abbreviations

NED:

North, East and Down direction

\(\{Y\}\) :

Body fixed frame

\(\{N\}\) :

NED frame

\(\{F\}\) :

Serret–Frenet reference frame

m :

Mass of the AUV

\(u_\mathrm{s},v_\mathrm{s},r_\mathrm{s}\) :

Linear and angular velocities

\(x_\mathrm{s},y_\mathrm{s},\psi _\mathrm{s}\) :

Linear and angular positions

\(I_x,I_y,I_z\) :

Moments of inertia about x, y and z axes in body-fixed frame

(\(x_\mathrm{B}, y_\mathrm{B}, z_\mathrm{B}\)):

Center of buoyancy

(\(x_\mathrm{G}, y_\mathrm{G}, z_\mathrm{G}\)):

Center of gravity

\(T_\mathrm{d}\) :

Total thrust in vertical plane

\(\epsilon \) :

Lyapunov function

\(\delta _\mathrm{r}\) :

Rudder angle

\(d_{FN}\) :

Position of \(\{F\}\) frame relative to \(\{N\}\) frame

\(d_{YF}\) :

Position of \(\{Y\}\) frame relative to \(\{F\}\) frame

\(d_{YN}\) :

Position of \(\{Y\}\) frame relative to \(\{N\}\) frame

\(c_{\mathrm{f}}\) :

Curvilinear abscissa along the path

\(\psi _{\mathrm{f}}\) :

Yaw angle between \(\{N\}\) and \(\{F\}\) coordinate system

s:

Parameters of steering plane

D:

Desired values for path following

f:

Parameters of Serret–Frenet frame

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Author notes

  1. Siddhartha Vadapalli and Subhasish Mahapatra have contributed equally to this work.

Authors and Affiliations

  1. School of Electronics Engineering, VIT-AP University, Amaravati, Vijayawada, Andhra Pradesh, 522237, India

    Siddhartha Vadapalli & Subhasish Mahapatra

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  1. Siddhartha Vadapalli
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  2. Subhasish Mahapatra
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Correspondence to Subhasish Mahapatra.

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Vadapalli, S., Mahapatra, S. A Robust State Feedback Optimal Control Law with Backstepping Approach for Steering Control of an Autonomous Underwater Vehicle Using Semi-definite Programming. Arab J Sci Eng 48, 14449–14462 (2023). https://doi.org/10.1007/s13369-023-07689-w

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