Stabilization of a coupled PDE-ODE system by boundary control

A coupled system of an ODE and a diffusion PDE is considered in this paper. Special techniques as well as the method of PDE backstepping are employed to construct controllers. Through transforming the system into an exponentially stable PDE-ODE cascade, a state feedback boundary controller is established. Moreover, an observer for anti-collocated setup is proposed, and the observer error is shown to exponentially converge to zero, then an output feedback boundary controller is obtained. For a scalar coupled PDE-ODE system, the boundary controller and observer, as well as the solution of the closed-loop system are given explicitly.

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

[2]  H. B. Veiga.  On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem , 2004 .

[3]  M. Krstić.  Delay Compensation for Nonlinear, Adaptive, and PDE Systems , 2009 .

[4]  Enrique Zuazua,et al.  Large Time Behavior for a Simplified 1D Model of Fluid–Solid Interaction , 2003 .

[5]  Miroslav Krstic,et al.  Compensating actuator and sensor dynamics governed by diffusion PDEs , 2009, Syst. Control. Lett..

[6]  Enrique Zuazua,et al.  Control, observation and polynomial decay for a coupled heat-wave system , 2003 .

[7]  Miroslav Krstic,et al.  Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays , 2007, 2007 46th IEEE Conference on Decision and Control.

[8]  Enrique Zuazua,et al.  Polynomial decay and control of a 1−d hyperbolic–parabolic coupled system , 2004 .

[9]  I. Lasiecka.  Mathematical control theory of couple PDEs , 2002 .

[10]  Enrique Zuazua,et al.  Polynomial decay and control of a 1-d model for fluid-structure interaction , 2003 .

[11]  Enrique Zuazua,et al.  Approximate controllability of a hydro-elastic coupled system , 1996 .

[12]  Enrique Zuazua,et al.  Polynomial decay for a hyperbolic–parabolic coupled system , 2005 .

[13]  E. Zuazua,et al.  Null control of a 1− d model of mixed hyperbolic-parabolic type , 2002 .

[14]  Miroslav Krstic,et al.  Control of PDE-ODE cascades with Neumann interconnections , 2010, J. Frankl. Inst..

[15]  Enrique Zuazua,et al.  Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction , 2006 .

[16]  Miroslav Krstic,et al.  Compensating a string PDE in the actuation or sensing path of an unstable ODE , 2009, 2009 American Control Conference.