Adaptive boundary control for unstable parabolic PDEs - Part II: Estimation-based designs

The certainty equivalence approach to adaptive control is commonly used with two types of identifiers: passivity-based identifiers and swapping identifiers. The ''passive'' (also known as ''observer-based'') approach is the prevalent identification technique in existing results on adaptive control for PDEs but has so far not been used in boundary control problems. The swapping approach, prevalent in finite-dimensional adaptive control is employed here for the first time in adaptive control of PDEs. For a class of unstable parabolic PDEs we prove a separation principle result for both the passive and swapping identifiers combined with the backstepping boundary controllers. The result is applicable in any dimension. For physical reasons we restrict our attention to dimensions no higher than three. The results of the paper are illustrated by simulations.

[1]  M. Krstić,et al.  Adaptive control of Burgers' equation with unknown viscosity , 2001 .

[2]  Miroslav Krstic,et al.  Stabilization of a solid propellant rocket instability by state feedback , 2003 .

[3]  Toshihiro Kobayashi,et al.  Adaptive stabilization of the Kuramoto-Sivashinsky equation , 2002, Int. J. Syst. Sci..

[4]  Yury Orlov.  Sliding Mode Observer-Based Synthesis of State Derivative-Free Model Reference Adaptive Control of Distributed Parameter Systems , 2000 .

[5]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[6]  Alan S. I. Zinober,et al.  Nonlinear and Adaptive Control , 2003 .

[7]  K. Ito,et al.  Optimal online parameter estimation for a class of infinite dimensional systems using Kalman filters , 2003, Proceedings of the 2003 American Control Conference, 2003..

[8]  Keum-Shik Hong,et al.  Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis , 1994, IEEE Trans. Autom. Control..

[9]  Bassam Bamieh,et al.  Adaptive distributed control of a parabolic system with spatially varying parameters , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[10]  Yury Orlov,et al.  Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced order spatial differentiation , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  Michael A. Demetriou,et al.  Model Reference Adaptive Control of Distributed Parameter Systems , 1998 .

[12]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[13]  Joseph Bentsman,et al.  Reduced spatial order model reference adaptive control of spatially varying distributed parameter systems of parabolic and hyperbolic types , 2001 .

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

[15]  Toshihiro Kobayashi.  Adaptive regulator design of a viscous Burgers' system by boundary control , 2001 .

[16]  John F. MacGregor,et al.  Proceedings of the American Control Conference , 1985 .

[17]  Tyrone E. Duncan,et al.  Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems , 1994 .

[18]  A. Prudnikov,et al.  Integrals and series of special functions , 1983 .

[19]  Miroslav Krstic,et al.  Adaptive boundary control for unstable parabolic PDEs - Part III: Output feedback examples with swapping identifiers , 2007, Autom..

[20]  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.

[21]  Stuart Townley,et al.  Adaptive control of infinite-dimensional systems without parameter estimation: an overview , 1997 .

[22]  Miroslav Krstic,et al.  Adaptive Boundary Control for Unstable Parabolic PDEs—Part I: Lyapunov Design , 2008, IEEE Transactions on Automatic Control.

[23]  Fumin Zhang,et al.  Adaptive nonlinear boundary control of a flexible link robot arm , 1999, IEEE Trans. Robotics Autom..

[24]  Michael Athans,et al.  Nonlinear and Adaptive Control , 1989 .

[25]  Mark J. Balas,et al.  Robust Adaptive Control In Hilbert Space , 1989 .

[26]  Mihailo R. Jovanovic,et al.  Lyapunov-based distributed control of systems on lattices , 2005, IEEE Transactions on Automatic Control.