This paper reviews the method of controlled Lagrangians and the interconnection and damping assignment passivity based control (IDA-PBC)method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange system, respectively Hamiltonian system, by searching for a stabilizing structure preserving feedback law. The conditions under which two Euler-Lagrange or Hamiltonian systems are equivalent under feedback are called the matching conditions (consisting of a set of nonlinear PDEs). Both methods are applied to the general class of underactuated mechanical systems and it is shown that the IDA-PBC method contains the controlled Lagrangians method as a special case by choosing an appropriate closed-loop interconnection structure. Moreover, explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms. The $\lambda$-method as introduced in recent papers for the controlled Lagrangians method transforms the matching conditions into a set of linear PDEs. In this paper the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.
|Title of host publication||Nonlinear and Adaptive Control: Tools and Algorithms for the User|
|Place of Publication||Singapore|
|Number of pages||36|
|Publication status||Published - 2006|
Blankenstein, G., Ortega, R., & van der Schaft, A. (2006). Matching in the method of controlled Lagrangians and IDA-passivity based control. In A. Astolfi (Ed.), Nonlinear and Adaptive Control: Tools and Algorithms for the User (pp. 79-114). Singapore: World Scientific.