Mathematical modeling for nonlinear control: a Hamiltonian approach

Modern model-based nonlinear control requires a good mathematical description of the system we want to control, for both, the system analysis and the controller design. Obviously, the term nonlinear system is too broad, and one is interested in subclasses of nonlinear systems with at least two properties. These classes should cover real world problems, and there should exist controller design methods, powerful enough to admit a systematic design of the closed loop with certain properties. Now, classical Hamiltonian systems have a rich mathematical structure, which has been extended such that dissipative effects and inputs, outputs, or better ports, are included in this class. This contribution starts with a Hamiltonian description of linear time invariant lumped parameter systems to motivate the introduction of certain mathematical ideas, which will be exploited in the nonlinear case afterwards. After that the approach will be extended to the distributed parameter case. Finally, the applicability of the presented methods is shown with the help of a piezoelectric elastic structure.