Control of a reaction-diffusion PDE cascaded with a heat equation

We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the other boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H-1(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H1(0, 1) rather than the usual L2(0, 1).

[1]  Hartmut Logemann,et al.  Conditions for Robustness and Nonrobustness of theStability of Feedback Systems with Respect to Small Delays inthe Feedback Loop , 1996 .

[2]  Bao-Zhu Guo,et al.  Regularity of a Schrödinger equation with Dirichlet control and colocated observation , 2005, Syst. Control. Lett..

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

[4]  Wim Michiels,et al.  On the delay sensitivity of Smith Predictors , 2003, Int. J. Syst. Sci..

[5]  R. Mahamud,et al.  Reciprocating air flow for Li-ion battery thermal management to improve temperature uniformity , 2011 .

[6]  Bao-Zhu Guo,et al.  Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation , 2009, Autom..

[7]  Miroslav Krstic,et al.  Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time , 2011, SIAM J. Control. Optim..

[8]  Miroslav Krstic,et al.  Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations , 2004, IEEE Transactions on Automatic Control.

[9]  Junmin Wang,et al.  Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks , 2011, Int. J. Control.

[10]  Z. Artstein.  Linear systems with delayed controls: A reduction , 1982 .

[11]  Hassan Hammouri,et al.  Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation , 2012 .

[12]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[13]  Miroslav Krstic.  Control of an unstable reaction-diffusion PDE with long input delay , 2009, CDC.

[14]  Michael P. Polis,et al.  An example on the effect of time delays in boundary feedback stabilization of wave equations , 1986 .

[15]  Miroslav Krstic,et al.  Boundary Controllers and Observers for the Linearized Schrödinger Equation , 2011, SIAM J. Control. Optim..

[16]  R. Datko.  Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks , 1988 .

[17]  Xiaohui Wang,et al.  Three-dimensional thermal modeling of a lithium-ion battery pack , 2012 .

[18]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[19]  Miroslav Krstic,et al.  Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays , 2007, CDC.

[20]  R. Datko,et al.  Two examples of ill-posedness with respect to time delays revisited , 1997, IEEE Trans. Autom. Control..

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

[22]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .

[23]  Qing-Chang Zhong,et al.  Robust Control of Time-delay Systems , 2006 .

[24]  Genqi Xu,et al.  Stabilization of wave systems with input delay in the boundary control , 2006 .

[25]  M. Krstić.  Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch , 2008 .