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A self-tuning robust observer for marine surface vessels

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Abstract

The current work aims at developing a nonlinear, self-tuning, robust observer to estimate state variables pertinent to the control of under-actuated marine surface vessels. The state estimator combines the advantages of the variable structure systems theory with those of the self-tuning fuzzy logic algorithm. It does not require an exact knowledge of the system dynamics or the construction of a rule-based expert fuzzy inference system. However, the upper bounds of both modeling imprecision and external disturbances must be known. The convergence of the estimation process is guaranteed by forcing the tuning parameters to satisfy inequalities stemming from the sliding conditions. The observer has been implemented herein to provide accurate estimate of the state variables that are needed by an integrated guidance and control system for the autonomous operation of an under-actuated marine surface vessel. The observer along with the integrated guidance and control system have been tested on a six degree-of-freedom ship model that considers numerous uncertainties associated with wave excitation, nonlinear restoring forces, retardation forces, sea-current and wind resistive loads. The theoretical results prove that the proposed self-tuning observer can produce accurate estimates of the state variables in the presence of significant modeling imprecision and environmental disturbances. In addition, they illustrate the use of estimated state variables in the guidance and control system with minimal impact on the close-loop performance of the marine vessel.

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Abbreviations

\(F_{\mathrm{rud}_{y}}\) :

Combined \(y\)-components of the lift and drag forces on the rudder

\(F_{\mathrm{rud}_{y}}\) :

Combined \(y\)-components of the lift and drag forces on the rudder

\(F_\mathrm{th\_cont}\) :

Propeller control thrust

\(H_{1/3}\) :

Average height of the highest one-third peaks of the wave

\(\mathop {i}\limits _{\widetilde{}}, \mathop {j}\limits _{\widetilde{}}, \mathop {k}\limits _{\widetilde{}}\) :

Unit vectors along the \(x_o, y_o\) and \(z_o\) axes, respectively

\(\mathop {I}\limits _{\widetilde{}}, \mathop {J}\limits _{\widetilde{}}, \mathop {K}\limits _{\widetilde{}}\) :

Unit vectors along the \(X, Y\) and \(Z\) axes, respectively

\(m_\mathrm{ship}\) :

Mass of the ship

\(p,q,r\) :

Angular velocity of the ship around the \(\mathop {i}\limits _{\widetilde{}}, \mathop {j}\limits _{\widetilde{}}\) and \(\mathop {k}\limits _{\widetilde{}}\) directions, respectively

\(r_i\) :

Singleton fuzzy set of the \(i\)th rule for the control action

\(\hbox {sgn}\) :

Sign function

\(T_\mathrm{rud\_cont}\) :

Rudder control torque

\(u,\;v,w\) :

Translational velocity components of the ship along the \(\mathop {i}\limits _{\widetilde{}}, \mathop {j}\limits _{\widetilde{}}\) and \(\mathop {k}\limits _{\widetilde{}}\) directions, respectively

\(w_i\) :

Firing weight of the \(i\)th rule of the fuzzy inference system

\(\left\{ {x,y,z} \right\} \) :

Body-fixed coordinate system

\(\left\{ {X,Y,Z} \right\} \) :

Inertial reference system

\(\Delta _1\) :

Distance from the origin of the body-fixed coordinate system to the pivot of the rudder

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

Roll, pitch, and yaw angular displacements of the ship with respect to the inertial frame, respectively

\(( \, )_e\) :

A subscript “e” denotes an estimated value of the variable

\(( \, )_d \) :

A subscript “d” denotes a desired value of the controlled variable

\(( \, )_{\sup } \) :

A subscript “sup” denotes a supremum value of the variable

\(\mathop {( \, )}\limits ^\wedge \) :

A “\(^{\wedge }\)” symbol indicates a nominal value of the variable

\(\mathop {( \, )}\limits _\sim \) :

A “\(\sim \)” symbol indicates a vector quantity

\(\mathop {( \, )}\limits ^\sim \) :

A “\(\sim \)” symbol indicates the estimation error

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Acknowledgments

This research is supported by a grant from the Office of Naval Research (ONR) under Award No: N00014-11-1-0803. Ms. Kelly B. Cooper is the Program Director.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Wayne State University, 5050 Anthony Wayne Dr., Detroit, MI, 48202, USA

    Nassim Khaled & Nabil G. Chalhoub

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  1. Nassim Khaled
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  2. Nabil G. Chalhoub
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Correspondence to Nassim Khaled.

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Khaled, N., Chalhoub, N.G. A self-tuning robust observer for marine surface vessels. Nonlinear Dyn 79, 937–951 (2015). https://doi.org/10.1007/s11071-014-1713-6

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