Systematization of approaches to adaptive boundary stabilization of PDEs

作者：Krstic Miroslav^*

来源：International Journal of Robust and Nonlinear Control, 2006, 16(16): 801-818.

DOI：10.1002/rnc.1098

• backstepping
• Lyapunov stabilization
• boundary control
• distributed parameter systems
• adaptive control

While adaptive control of finite dimensional systems is an advanced field that has produced adaptive control methods for a very general class of LTI systems, adaptive control techniques have been developed for only a few of the classes of PDEs for which non-adaptive controllers exist. We present a catalog of approaches for the design of adaptive controllers for PDEs controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. We differentiate between two major classes of schemes: Lyapunov schemes and certainty equivalence schemes. Within the certainty equivalence class, two types of identifier designs are pursued: passivity-based and swapping designs. Each of those designs is applicable to two types of parametrizations: the plant model in its original form (which we refer to as the 'u-model') and a transformed model to which a backstepping transformation has been applied (which we refer to as the 'w-model'). Hence, a large number of control algorithms result from combining different design tools-Lyapunov schemes, w-passive schemes, u-swapping schemes, etc.
Our method builds upon the explicitly parametrized control formulae that we introduced in our earlier work on non-adaptive backstepping control for PDEs. These formulae allow us to develop tunable controllers that avoid solving Riccati or Bezout equations at each time step.
This paper is primarily a tutorial. Its purpose is to provide structure that helps the future reader of five other papers currently under review which contain the detailed proofs for the designs presented here. Additionally, the paper can serve as an entry point for a non-expert reader interested in an introduction to adaptive boundary control of PDEs. For this reason, our presentation proceeds through a series of examples, which are generalized in the companion papers.

• 出版日期2006-11-10