A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system

Abstract This work is motivated by an aeronautical issue: the fuel sloshing in the tank coupled with very flexible wings. Vibrations due to these coupled phenomena can lead to problems like reduced passenger comfort and maneuverability, and even unstable behavior. Port-Hamiltonian systems (pHs) provide a unified framework for the description of multi-domain, complex physical systems and a modular approach for the interconnection of subsystems. In this work, pHs models are proposed for the equations of liquid sloshing in moving containers and for the structural equations of beams with piezoelectric actuators. The interconnection ports are used to couple the sloshing dynamics in the moving tank to the motion the beam. This coupling leads to an infinite-dimensional model of the system in the pHs form. A finite-dimensional approximation is obtained by using a geometric pseudo-spectral method that preserves the pHs structure at the discrete level. Experimental tests on a structure made of a beam and a tank were carried out to validate the finite-dimensional model of liquid sloshing in moving containers. Finally, the pHs model proves useful to design an active control law for the reduction of sloshing phenomena.

[1]  Stefano Stramigioli,et al.  Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach , 2014 .

[2]  Jacquelien M. A. Scherpen,et al.  Modeling for control of an inflatable space reflector, the nonlinear 1-D case , 2008, 2008 47th IEEE Conference on Decision and Control.

[3]  Arjan van der Schaft,et al.  On Interconnections of Infinite-dimensional Port-Hamiltonian Systems , 2004 .

[4]  Arjan van der Schaft,et al.  Multi-variable port Hamiltonian model of piezoelectric material , 2004, 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566).

[5]  A. Schaft.  Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems , 2004 .

[6]  Stefano Stramigioli,et al.  Port-Based Modeling and Simulation of Mechanical Systems With Rigid and Flexible Links , 2009, IEEE Transactions on Robotics.

[7]  Charbel Farhat,et al.  Modeling of Fuel Sloshing and its Physical Effects on Flutter , 2013 .

[8]  Malcolm J. Abzug,et al.  Airplane Stability and Control, Second Edition: Modern Canard Configurations , 2002 .

[9]  Hans Zwart,et al.  Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators , 2005, SIAM J. Control. Optim..

[10]  H. N. Abramson,et al.  The Dynamic Behavior of Liquids in Moving Containers, with Applications to Space Vehicle Technology , 1966 .

[11]  A. Schaft,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[12]  Hamid Alemi Ardakani,et al.  A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing , 2016 .

[13]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[14]  Harvey Thomas Banks,et al.  Smart material structures: Modeling, estimation, and control , 1996 .

[15]  A. Schaft,et al.  Port Hamiltonian formulation of infinite dimensional systems I. Modeling , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[16]  A. J. van der Schaft,et al.  Port-Hamiltonian Differential-Algebraic Systems , 2013 .

[17]  J. A. Villegas,et al.  A Port-Hamiltonian Approach to Distributed Parameter Systems , 2007 .

[18]  Laurent Lefèvre,et al.  Control by interconnection and energy shaping methods of port Hamiltonian models - Application to the shallow water equations , 2009, 2009 European Control Conference (ECC).

[19]  Denis Matignon,et al.  Control design for a coupled fluid-structuresystem with piezoelectric actuators , 2015 .

[20]  Goran Golo,et al.  Interconnection structures in port-based modelling: tools for analysis and simulation , 2002 .

[21]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[22]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[23]  Arjan van der Schaft,et al.  Hamiltonian discretization of boundary control systems , 2004, Autom..

[24]  Romeo Ortega,et al.  On stabilization of nonlinear distributed parameter port-controlled Hamiltonian systems via energy shaping , 2001 .

[25]  D. Arzelier,et al.  Simultaneous $H_\infty$ Vibration Control of Fluid/Plate System via Reduced-Order Controller , 2010, IEEE Transactions on Control Systems Technology.

[26]  Alessandro Macchelli,et al.  Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach , 2004, SIAM J. Control. Optim..

[27]  Airbus,et al.  SLOSHING MECHANICAL MODEL FOR STABILITY AND HANDLING QUALITIES EVALUATION OF THE C 295 AIRCRAFT WITH THE OSD SYSTEM , 2014 .

[28]  Alessandro Macchelli,et al.  Control by interconnection of mixed port Hamiltonian systems , 2005, IEEE Transactions on Automatic Control.

[29]  Jacquelien M. A. Scherpen,et al.  Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems , 2012, Autom..

[30]  Dewey H. Hodges,et al.  Introduction to Structural Dynamics and Aeroelasticity , 2002 .

[31]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[32]  Kirsten Morris,et al.  Strong stabilization of piezoelectric beams with magnetic effects , 2013, 52nd IEEE Conference on Decision and Control.

[33]  Peter C. Breedveld.  Port-based modeling of mechatronic systems , 2004, Math. Comput. Simul..

[34]  Hamid Alemi Ardakani,et al.  Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions , 2011, Journal of Fluid Mechanics.

[35]  Hamid Alemi Ardakani,et al.  Dynamic coupling between shallow-water sloshing and horizontal vehicle motion , 2010, European Journal of Applied Mathematics.

[36]  Thomas J. Bridges,et al.  Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions , 2012 .

[37]  Pierre Rouchon,et al.  Dynamics and solutions to some control problems for water-tank systems , 2002, IEEE Trans. Autom. Control..

[38]  Jonathan de Halleux,et al.  Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations , 2004, Syst. Control. Lett..

[39]  Malcolm J. Abzug,et al.  Airplane Stability and Control, Second Edition: Index , 2002 .

[40]  van der Arjan Schaft,et al.  Port-Hamiltonian Systems Theory , 2014 .

[41]  Bogdan Robu,et al.  Simultaneous H∞ vibration control of fluid/plate system via reduced-order controller , 2010, 49th IEEE Conference on Decision and Control (CDC).

[42]  Henry Martyn Paynter,et al.  Analysis and design of engineering systems : class notes for M.I.T. course 2.751 , 1961 .

[43]  E W Graham,et al.  The Characteristics of Fuel Motion which Affect Airplane Dynamics , 1951 .

[44]  Denis Matignon,et al.  Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation. , 2016 .

[45]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems Theory: An Introductory Overview , 2014, Found. Trends Syst. Control..

[46]  Laurent Lefèvre,et al.  Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws , 2012, J. Comput. Phys..

[47]  Jacquelien M. A. Scherpen,et al.  Port-Hamiltonian Modeling of a Nonlinear Timoshenko Beam with Piezo Actuation , 2014, SIAM J. Control. Optim..

[48]  Denis Matignon,et al.  Modeling by interconnection and control by damping injection of a fluid-structure system with non-collocated actuators and sensors , 2016 .

[49]  Laurent Lefèvre,et al.  Port-based modelling and geometric reduction for open channel irrigation systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[50]  Denis Arzelier,et al.  Modeling of a coupled fluid-structure system excited by piezoelectric actuators , 2014, 2014 IEEE/ASME International Conference on Advanced Intelligent Mechatronics.

[51]  Marcel J. Sidi,et al.  Spacecraft Dynamics and Control: A Practical Engineering Approach , 1997 .