Backstepping output-feedback control of moving boundary parabolic PDEs

Abstract This paper extends the backstepping-based observer design to the state estimation of parabolic PDEs with time-dependent spatial domain. The design is developed for the stabilization of a collocated boundary measurement and actuation of an unstable 1D heat equation with the application to the temperature distribution regulation in Czochralski crystal growth process. The PDE system that describes the estimation error dynamics is transformed to an exponentially stable target system through invertible transformations to obtain the time-varying kernel PDE defined on the time-varying triangular-shape domain. The exponential stability of the closed-loop system with an observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system.

[1]  Stevan Dubljevic,et al.  Backstepping control of PDEs with time-varying domain , 2014, 2014 American Control Conference.

[2]  Peter E. Kloeden,et al.  Pullback attractors for a semilinear heat equation in a non-cylindrical domain , 2008 .

[3]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

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

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

[6]  R. Gressang,et al.  Observers for systems characterized by semigroups , 1975 .

[7]  Masataro Nishimura,et al.  Observer for distributed-parameter diffusion systems , 1972 .

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

[9]  Andreas Kugi,et al.  State estimation for parabolic PDEs with reactive-convective non-linearities , 2013, 2013 European Control Conference (ECC).

[10]  Tu Duc Nguyen,et al.  Second-order observers for second-order distributed parameter systems in R2 , 2008, Syst. Control. Lett..

[11]  Michael A. Demetriou,et al.  Natural second-order observers for second-order distributed parameter systems , 2004, Syst. Control. Lett..

[12]  Toshihiro Kobayashi,et al.  Observers and parameter determination for distributed parameter systems , 1981 .

[13]  Miroslav Krstic,et al.  On control design for PDEs with space-dependent diffusivity or time-dependent reactivity , 2005, Autom..

[14]  Stevan Dubljevic,et al.  Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions , 2013 .

[15]  Thomas Meurer,et al.  On the Extended Luenberger-Type Observer for Semilinear Distributed-Parameter Systems , 2013, IEEE Transactions on Automatic Control.

[16]  Jeffrey J. Derby,et al.  Finite-element methods for analysis of the dynamics and control of Czochralski crystal growth , 1987 .

[17]  Andreas Kugi,et al.  An Efficient Implementation of Backstepping Observers for Time-Varying Parabolic PDEs , 2012 .

[18]  M. Balas.  Active control of flexible systems , 1978 .

[19]  M. Amouroux,et al.  Sensors and observers in distributed parameter systems , 1988 .

[20]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

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

[22]  Y. A. LlU,et al.  Observer theory for distributed-parameter systems , 1976 .

[23]  P. A. Orner,et al.  A Design Procedure for a Class of Distributed Parameter Control Systems , 1971 .

[24]  Yoshiyuki Sakawa,et al.  Feedback stabilization of a class of distributed systems and construction of a state estimator , 1975 .

[25]  Miroslav Krstic,et al.  Output-feedback stabilization of an unstable wave equation , 2008, Autom..

[26]  Stevan Dubljevic,et al.  Temperature distribution reconstruction in Czochralski crystal growth process , 2014 .

[27]  Takao Nambu,et al.  On the stabilization of diffusion equations: Boundary observation and feedback , 1984 .