Breu, Frequency Detuning of Parametric Roll

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1 Frequency Detuning of Parametric Roll Conference on CeSOS Highlights and AMOS Visions, May 29, 2013 Dominik Breu Department of Engineering Cybernetics, NTNU, Trondheim, Norway Centre for Ships and Ocean Structures, NTNU, Trondheim, Norway

2 3 Grand Voyager, 2005 Parametric roll resonance and what it might look like on the inside

3 4 APL China, 1998 Post-Panamax C11-class Route: Kaohsiung - Seattle Containers: 1/3 lost, 1/3 damaged Estimated damage: US $100 M France et al.: An Investigation on Head-Sea Parametric Rolling and Its Influence on Container Lashing Systems, Marine Technology,

4 5 Parametric Roll Resonance Resonance Phenomenon Differential equations with time-varying periodic coeff. No extern. excitation needed Small parametric excitation may lead to large response Parametric Roll Resonance Autoparametric resonance Container ships, cruise ships, and fishing vessels Nayfeh and Mook: Nonlinear Oscillations, Wiley, 2004 m φ + d φ + k (t)φ = τ k = k 0 + k t cos(ω e t) Parametric resonance when ω e 2 k 0 /m = ω φ Wave length Ship length Sufficient wave height Low roll damping

5 6 Effect of Waves Wave trough amidships Large water plane area Large τ τ Wave crest amidships Small water plane area Small τ, large h h Wave trough amidships Large water plane area Large τ τ Holden: Modeling and Control of Parametric Roll Resonance, PhD Thesis, NTNU, 2011

6 7 Avoid Parametric Roll Reducing the probability Change the ship design Additional damping to dissipate roll energy Modify the shape of the ship (hull form) Improved navigation (avoid conditions) Lessening the impact Opposing roll moment Fins Fluid tanks Moving mass Rudder deflection Violate the frequency cond.: ω e 2 k 0 /m = ω φ Frequency Detuning Control Change encounter frequency: ω e (u,ψ,ω 0,βw) n = ω 0 kucos(βw n ψ) Change heading and/or speed ω e : Encounter frequency u: Speed ψ: Heading angle ω 0 : Modal wave freq. k : Wave number β n w: Encounter angle

7 8 Modelling of Parametric Roll Model Verification 6-DOF Ship Model 1-DOF Roll Model 1-DOF Roll Model Control Model Ship panel model Pressure over instantaneous submerged hull First-order wave effects Consider heave, pitch, and roll motions Quasi-steady approach for heave and pitch Identification of the hydrostatic/hydrodynamical model coefficients Functional expressions of the coefficients

8 9 6-DOF Ship Model η = J (Θ) ν Mν + D (ν) ν + C (ν) ν + k (η, t) = τ c + τ e h i> Generalized position vector: η = pn>, Θ> R6 > Generalized velocity vector: ν = vb>, ω b> R6 Transformation matrix: J R6 6 Rigid-body inertia and added mass: M R6 6 Main characteristics Length 281 m Breadth 32.3 m Draft 11.8 m Displacement t Damping matrix: D R6 6 Coriolis/centripetal matrix: C R6 6 Control and environmental forces: τ c, τ e R6 Gravity and pressure forces: k (η, t) = kg (η) + kp (η, t) R6 Breu et al.: Ship Model for Parametric Roll Incorporating the Effects of Time-Varying Speed, Parametric Resonance in Dynamical Systems, Springer, 2012

9 10 Pressure Forces Assumptions Ship can be split into panels, each parametrized by (a,b) Known pressure field, unchanged by the passage of ship Integrate instantaneous pressure over each panel i of the instantaneously submerged hull S w,i to give force f p = f i = Ψ i (a,b)n(a,b) da db i S w,i Parameters a and b are functions of η and t Breu et al.: Ship Model for Parametric Roll Incorporating the Effects of Time-Varying Speed, Parametric Resonance in Dynamical Systems, Springer, 2012

10 11 1-DOF Roll Model Assumptions Waves regular and sinusoidal ζ = ζ 0 cos(ω 0 t k w x n ) Waves unaffected by passage of ship Encounter frequency approximated by ω e = d ) (ω 0 t k w u n dt ω 0 k w cos(ψ)u dt ζ : Wave elevation ω 0 : Modal wave frequency k w : Wave number x n : Position x n = u n dt ψ: Heading angle u: Forward speed Roll coupled to heave and pitch m 44 φ + d 44 φ + k 44 φ + k φ 3 +k zφ zφ + k φθ φθ Quasi-steady [ approach ] [ yields explicit solutions for heave/pitch z(t) āz ζ 0 cos ( t ) ] t = 0 ω e (τ)dτ + ᾱ z θ (t) ā θ ζ 0 cos ( t ) t 0 ω e (τ)dτ + ᾱ θ Breu et al.: Ship Model for Parametric Roll Incorporating the Effects of Time-Varying Speed, Parametric Resonance in Dynamical Systems, Springer 2012

11 12 1-DOF Roll Model (con t) [ ( t )] ( t ) m 44 φ + d 44 φ + κ 1 + κ 2 cos ω e dt + κ 3 φ + κ 4 φ 3 = κ 5 sin ω e dt + κ 6 t 0 t 0 Hydrostatic Coefficients Hydrodynamical Coefficients Linear restoring moment: κ 1 Cubic restoring moment: κ 4 Coupling roll heave/pitch: κ 2, κ 3 External wave forcing: κ 5, κ 6 Assumed to be constant Identified by free decay test in calm water (simulations of 6-DOF model without waves) Least square curve fitting κ i = κ i (u,ψ), i {2,3,4,5} κ 5 0 for head seas Identified from simulations of the 6-DOF model Least square curve fitting

12 13 Model Verification 6-DOF Ship Model η = J (Θ)ν M ν + D(ν)ν + C(ν)ν + k(η,t) = τ c + τ e 1-DOF Roll Model m 44 φ + d 44 φ + [ ( t κ 1 + κ 2 cos ω e dt + κ 3 t 0 )] ( t ) φ + κ 4 φ 3 = κ 5 sin ω e dt + κ 6 t 0 Valid for constant and time-varying speed and heading angle Identical Mathieu-type equation for constant ω e Functional expressions for the hydrodynamical coefficients Analytical model Suitable for control purposes

13 14 Maximum Roll Angle 6-DOF Model PSfrag 1-DOF Roll Model PSfrag 1-DOF/Functional Expr. maximum roll angle ( ) u (m/s) 0 0 ψ ( ) maximum roll angle ( ) u (m/s) 0 0 ψ ( ) maximum roll angle ( ) u (m/s) 0 0 ψ ( ) u (m/s) ψ ( ) ψ ( ) u (m/s) u (m/s) ψ ( )

14 15 Time-Varying Speed Holden et al.: Frequency Detuning Control by Doppler Shift, Parametric Resonance in Dynamical Systems, Springer, 2012

15 16 Frequency Detuning Control Frequency Detuning Control Change encounter frequency: ω e (u,ψ,ω 0,βw) n = ω 0 kucos(βw n ψ) Change heading and/or speed ω e : Encounter frequency u: Speed ψ: Heading angle ω 0 : Modal wave freq. k : Wave number β n w: Encounter angle a [t2, ) RI RII RIII [t2, ) t1 t0 σ Holden et al.: 2012 Frequency Detuning Control by Doppler Shift, Parametric Resonance in Dynamical Systems, Springer,

16 Holden et al.: Frequency Detuning Control by Doppler Shift, Parametric Resonance in Dynamical Systems, Springer, 2012

17 18 Frequency Detuning Control Approaches Model Predictive Control Model-based control method Prediction to find an optimal control Extremum Seeking Control Non-model-based adaptive control method Iterative control loop to find optimal control L 1 Adaptive Control Guaranteed robustness with fast adaptation (decoupling) Low-pass filtered parametric estimate Camacho and Bordons: Model predictive control, Springer Verlag London Limited, 2007 Ariyur and Krstić: Real-time optimization by extremum-seeking control, Wiley-Interscience, 2003 Hovakimyan and Cao: L 1 Adaptive Control Theory, SIAM, 2010

18 19 Model Predictive Control Future inputs Past inputs and outputs 3-DOF Ship Model Predicted outputs + Reference trajectory Optimizer Future errors Objective function Constraints Breu et al.: Optimal Speed and Heading Control for Stabilization of Parametric Oscillations in Ships, Parametric Resonance in Dynamical Systems, Springer 2012

19 20 Results MPC Breu et al.: Optimal Speed and Heading Control for Stabilization of Parametric Oscillations in Ships, Parametric Resonance in Dynamical Systems, Springer 2012

20 21 Extremum Seeking & Param. Resonance Control Control Allocation ω e,d + u d Speed n d Surge u Controller System Freq. Coupling Heading Yaw ψ ψ d Controller δd System k s ω l s+ω l s s+ω h ω e Roll System Ship φ Limit Cycle Ampl. Detector asin(ωt) asin(ωt) Extremum seeking loop Breu and Fossen: L 1 Adaptive and Extremum Seeking Control Applied to Roll Parametric Resonance in Ships, Proc. 9th IEEE International Conference on Control and Automation, 2011

21 22 L 1 Adaptive Control Overview ˆη = ˆωu + ˆθ x + ˆσ System ( ) ẋ = A m x + b ωu + θ x + σ r u x k g + kd(s) y = c x θ m x L λ min (P)Γ x ref x L γ 1 Γ u ref u L γ 2 Γ BIBS stable reference ( system: ) ẋ r = A m x r + b ωu r + θ x r + σ u r = C (s)ω 1 (k g r (s) η r (s)) y r = c x r ˆω, ˆθ, ˆσ State Predictor ( ) ˆx = A m ˆx + b ˆωu + ˆθ x + ˆσ ŷ = c ˆx Hovakimyan and Cao: L 1 Adaptive Control Theory, SIAM, 2010 Adaptation Laws ( ˆθ = ΓProj ˆθ, x Pbx) ( ) ˆσ = ΓProj ˆσ, x Pb ( ) ˆω = ΓProj ˆω, x Pbu ˆx x +

22 23 Results Extremum Seeking & L 1 Adaptive Control Breu and Fossen: L 1 Adaptive and Extremum Seeking Control Applied to Roll Parametric Resonance in Ships, Proc. 9th IEEE International Conference on Control and Automation, 2011

23 24 Estimation Challenge: Frequency detuning control assumes knowledge of wave encounter frequency and modal wave frequency Extended Kalman Filter Model-based estimation Optimal in the sense of minimum variance Linear Frequency Estimator Signal-based estimation Globally convergent Frequency Estimation for Irregular Seas Empirical mode decomposition, intrinsic mode function Ongoing research

24 25 Conclusions Nonlinear resonance phenomenon Small parametric excitation may lead to large roll angles Dangerous for container ships, cruise ships, and fishing vessels The steps to solve the problem Understand the phenomenon Mathematical modelling Control approaches Parameter estimation

25 Thank you for your attention

26 27 Further Reading I Ariyur, K. B. and M. Krstić (2003). Real-time optimization by extremum-seeking control. Wiley-Interscience, Hoboken, New Jersey, USA. ISBN: Belleter, D. J. W. (2013). Signal-Based Observer Design for Estimation of the Wave Encounter Frequency. D&C MA thesis. University of Technology Eindhoven. Belleter, D. J. W., D. A. Breu, T. I. Fossen, and H. Nijmeijer (2012). Nonlinear Observer Design for Parametric Roll Resonance. In: Proceedings of the 11th International Conference on the Stability of Ships and Ocean Vehicles, pp ISBN: (2013). A K -Exponential Stable Nonlinear Observer for the Wave Encounter Frequency. In: Submitted to the 9th IFAC Conference on Control Applications in Marine Systems.

27 28 Further Reading II Breu, D. A., L. Feng, and T. I. Fossen (2012). Optimal Speed and Heading Control for Stabilization of Parametric Oscillations in Ships. In: Parametric Resonance in Dynamical Systems. Ed. by T. I. Fossen and H. Nijmeijer. Springer New York. Chap. 11, pp ISBN: DOI: / _11. Breu, D. A. and T. I. Fossen (2010). Extremum Seeking Speed and Heading Control Applied to Parametric Resonance. In: Proceedings of the 8th IFAC Conference on Control Applications in Marine Systems, pp ISBN: DOI: / DE Breu, D. A., C. Holden, and T. I. Fossen (2012a). Ship Model for Parametric Roll Incorporating the Effects of Time-Varying Speed. In: Parametric Resonance in Dynamical Systems. Ed. by T. I. Fossen and H. Nijmeijer. Springer New York. Chap. 9, pp ISBN: DOI: / _9.

28 29 Further Reading III Breu, D. A., C. Holden, and T. I. Fossen (2012b). Stability of Ships in Parametric Roll Resonance Under Time-Varying Heading and Speed. In: Proceedings of the 11th International Conference on the Stability of Ships and Ocean Vehicles, pp ISBN: Breu, D. A. and T. I. Fossen (2011). L 1 Adaptive and Extremum Seeking Control Applied to Roll Parametric Resonance in Ships. In: Proceedings of the 9th IEEE International Conference on Control and Automation, pp ISBN: DOI: /ICCA Camacho, E. F. and C. Bordons (2007). Model predictive control. 2nd. Advanced textbooks in control and signal processing. Springer Verlag Ltd. Fossen, T. I. and H. Nijmeijer, eds. (2012). Parametric Resonance in Dynamical Systems. Springer New York. ISBN: DOI: / France, W. N. et al. (Jan. 2003). An Investigation on Head-Sea Parametric Rolling and Its Influence on Container Lashing Systems. In: Marine Technology, Presented at the SNAME Annual Meeting 2001, pp ISSN:

29 30 Further Reading IV Galeazzi, R. (2009). Autonomous Supervision and Control of Parametric Roll Resonance. PhD thesis. Technical University of Denmark. ISBN: Holden, C. (2011). Modeling and Control of Parametric Roll Resonance. PhD thesis. Norwegian University of Science and Technology. Holden, C., D. A. Breu, and T. I. Fossen (2012). Frequency Detuning Control by Doppler Shift. In: Parametric Resonance in Dynamical Systems. Ed. by T. I. Fossen and H. Nijmeijer. Springer New York. Chap. 10, pp ISBN: DOI: / _10. Hovakimyan, N. and C. Cao (2010). L 1 Adaptive Control Theory. Guaranteed Robustness with Fast Adaptation. Advances in Design and Control. Society for Industrial and Applied Mathematics. ISBN: Nayfeh, A. H. and D. T. Mook (1995). Nonlinear Oscillations. Wiley-VCH Verlag GMbH & Co. KGaA. ISBN:

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