On control by interconnection of port Hamiltonian systems

In the standard approach of control by interconnection the plant and the controllerare assumed to be passive and coupled via a power–preserving interconnection—generating an overall passive system with storage function the sum of the plant andcontroller storage functions. To achieve stabilization of a desired equilibrium one mustmake this point a minimum of the new storage function. Towards this end, dynamicinvariants—called Casimirs—are first computed. Restricting the dynamics to the levelsets of the Casimirs, the overall storage function becomes a bona fide function of theplant states and the storage function can be shaped. Unfortunately, this procedure isapplicable only if one fixes the initial conditions of the controller to some specfic values.To remove this drawback we propose in this paper to carry out the stability analysis inthe full plant and controller state spaces. The new storage function is then the sum ofthe plant and the controller Hamiltonians and an arbitrary functions of the correspondingCasimir functions. We also provide some examples which illustrate the possibilities andlimitations of the new method.