Nonlinear Observer Design for Dynamic Positioning

Jeroen Snijders, Jacob van der WoudeDelft University of Technology

Jaap-Harm WesthuisGusto BV

Control Systems I

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Contents

• Introduction• Mathematical model • Observer design• Simulations• Results• Conclusions• Further research

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Introduction

• Total vessel motion superposition of low frequency (LF) vessel motions and wave frequency (WF) motions

• For station keeping only LF motions should be used to prevent unnecessary actuator use

• Separation of LF and WF motions: Wave filtering

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Introduction

• First use of Kalman filtering in 1970’s– Drawbacks: Necessity of linearization yielding lack of

guaranteed global stability and time consuming tuning• Examination of the possibilities of nonlinear observer

design from 1990’s– Globally stable, optimality not yet considered

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Introduction

• Objectives of this project– Providing a theoretical basis for nonlinear optimal observer

design– Presentation of new nonlinear suboptimal observer– Comparison of the new observer with an existing observer

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Mathematical model

• LF displacement due to both actuator forces and slowly-varying wind, wave and current forces

• WF motions approximated by a second-order linear model

• The total vessel displacement is considered to be a superposition of LF and WF induced displacement

• Only position and heading measurements are available

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Mathematical model

• These assumptions lead to the following mathematical representation of the system

vCxyEwBuxxAx

+=++= )(&

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Observer design

• General nonlinear system

• with performance indicator

Objective: Find an estimate minimizing J and obeying the system dynamics and measurements

),(),,(

vxgytwxfx

==&

∫+=t

t

dwxLttxtwxJ0

),,()),((),,( 00 ττφ

x̂

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Observer design

• Equivalent derivation as for the Hamilton-Jacobi-Bellman equation in optimal control leads to the equation for the nonlinear optimal observer

where• Drawback: Exact solution to the HJB equation not known

for most problems• Here suboptimal solution using a state-dependent Riccati

equation (SDRE) approach

),,,(min **

tJwxHt

Jxw

=∂∂

−

fLtwxH Tλλ +−=),,,(

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Observer design

• Why an SDRE approach?– Ease of implementation and applicability for online

use– Form of the considered problem– Promising results in other areas of research

• Drawback: Global stability can not be guaranteed

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Observer design

• Basic principles SDRE• Rewrite

asvxgyEwxfx

+=+=

)()(&

vxxCyEwxxAx

+=+=

)()(&

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Observer design

• Solve the state-dependent Riccati equation

with Q and R weighting matrices• Compute the gain matrix as for the Kalman filter

• The observer now has the form

0)()()()()()()()( 1 =+−+ − TTT EQExPxCRxCxPxPxAxAxP

)ˆ)(ˆ(ˆ)ˆ(ˆ xCyxKxxAx −+=&

1)()()( −= RxCxPxK T

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Observer design

• Comparison with existing observer• The observer is proven to be globally stable• The chosen configuration is as in [1]

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Simulations

• Considered ship is supply vessel, length 80 m• JONSWAP spectrum with Hs = 5m, Tp = 12s and

= 3.3• WF motions and LF forces are generated separately• LF motions due to LF forces and control input• Measurements superposition of LF and WF motions• Estimate of LF position used for controller feedback to PID

controller

γ

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SimulationsDesired trajectory for surge and sway

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SimulationsDesired heading

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ResultsLF estimation and measured position in surge

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ResultsLF SDRE estimation and measured position in surge

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ResultsLF estimation and measured position in sway

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ResultsLF SDRE estimation and measured position in sway

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ResultsLF estimation and measured heading

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ResultsLF SDRE estimation and measured heading

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ResultsZoom in on LF estimation and measured surge position

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ResultsZoom in on LF SDRE estimation and measured surge position

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Conclusions

• New nonlinear observer design– Promising simulation results– Flexibility in modeling– Drawback: loss of global stability guarantee

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Further research

• Extended stability analysis • Algorithms for adjusting the weighting matrices • Usage of different models and methods

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Reference

[1] G. Torsetnes. Nonlinear Control and Observer Design forDynamic Positioning using Contraction Theory. Master’sthesis, Norwegian University of Science and Technology,2004