State and observer-based feedback control of normal flow equations

Abstract We investigate wellposedness and stability of normal flow equations with two different control schemes. The first one considers the velocity field as a control action and relies on a proportional regulator that is proved to be stable. In the second one, control acts on the source term and is based on an observer providing an estimate of the gradient of the normal flow solution; convergence properties are demonstrated in this case. Numerical results are presented to show the effectiveness of the proposed approaches.

[1]  M. Krstić,et al.  Backstepping observers for a class of parabolic PDEs , 2005, Syst. Control. Lett..

[2]  Stevan Dubljevic,et al.  Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs , 2013, Autom..

[3]  Miroslav Krstic,et al.  An Adaptive Observer Design for $n+1$ Coupled Linear Hyperbolic PDEs Based on Swapping , 2016, IEEE Transactions on Automatic Control.

[4]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[5]  Alexandre M. Bayen,et al.  Nonlinear Local Stabilization of a Viscous Hamilton-Jacobi PDE , 2014, IEEE Transactions on Automatic Control.

[6]  Joachim Deutscher,et al.  Finite-time output regulation for linear 2×2 hyperbolic systems using backstepping , 2017, Autom..

[7]  F. Mazenc,et al.  Strict Lyapunov functions for semilinear parabolic partial differential equations , 2011 .

[8]  Christian A. Ringhofer,et al.  Control of Continuum Models of Production Systems , 2010, IEEE Transactions on Automatic Control.

[9]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[10]  Angelo Alessandri,et al.  Optimal Control of Propagating Fronts by Using Level Set Methods and Neural Approximations , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[11]  Michael Hinze,et al.  Optimal control of the free boundary in a two-phase Stefan problem , 2007, J. Comput. Phys..

[12]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[13]  Ron Kimmel,et al.  Numerical Geometry of Images , 2003, Springer New York.

[14]  Michael Herty,et al.  Feedback boundary control of linear hyperbolic systems with relaxation , 2016, Autom..

[15]  Mohamadreza Ahmadi,et al.  Dissipation inequalities for the analysis of a class of PDEs , 2016, Autom..

[16]  Ian M. Mitchell.  The Flexible, Extensible and Efficient Toolbox of Level Set Methods , 2008, J. Sci. Comput..

[17]  Hans Zwart,et al.  Luenberger boundary observer synthesis for Sturm–Liouville systems , 2010, Int. J. Control.

[18]  G. Barles.  Existence results for first order Hamilton Jacobi equations , 1984 .

[19]  Maxime Theillard,et al.  A coupled level-set and reference map method for interface representation with applications to two-phase flows simulation , 2019, J. Comput. Phys..

[20]  Xiao-Dong Li,et al.  Infinite-dimensional Luenberger-like observers for a rotating body-beam system , 2011, Syst. Control. Lett..

[21]  G. Barles.  An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications , 2013 .

[22]  J. Sethian,et al.  LEVEL SET METHODS FOR FLUID INTERFACES , 2003 .

[23]  Shuxia Tang,et al.  State and output feedback boundary control for a coupled PDE-ODE system , 2011, Syst. Control. Lett..

[24]  Zhiqiang Wang,et al.  Feedback Stabilization for the Mass Balance Equations of an Extrusion Process , 2016, IEEE Transactions on Automatic Control.

[25]  Miroslav Krstic,et al.  Observer design for a class of nonlinear ODE-PDE cascade systems , 2015, Syst. Control. Lett..

[26]  Roland Herzog,et al.  Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation , 2011, SIAM J. Sci. Comput..

[27]  Jordan M. Berg,et al.  On Parameter Estimation Using Level Sets , 1999 .

[28]  Miroslav Krstic,et al.  Backstepping-Forwarding Control and Observation for Hyperbolic PDEs With Fredholm Integrals , 2015, IEEE Transactions on Automatic Control.

[29]  Henrik Anfinsen,et al.  Disturbance Rejection in the Interior Domain of Linear 2 $\times$ 2 Hyperbolic Systems , 2015, IEEE Transactions on Automatic Control.

[30]  Rafael Vazquez,et al.  Nonlinear bilateral output-feedback control for a class of viscous Hamilton-Jacobi PDEs , 2018, Autom..

[31]  Ilyasse Aksikas,et al.  Asymptotic behaviour of contraction non-autonomous semi-flows in a Banach space: Application to first-order hyperbolic PDEs , 2016, Autom..

[32]  Driss Boutat,et al.  Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions , 2016, Syst. Control. Lett..

[33]  Miroslav Krstic,et al.  Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs , 2015, IEEE Transactions on Automatic Control.

[34]  M. Samimy,et al.  Infinite dimensional and reduced order observers for Burgers equation , 2005 .

[35]  Ole Morten Aamo,et al.  Disturbance rejection in 2 x 2 linear hyperbolic systems , 2013, IEEE Transactions on Automatic Control.