Читать книгу PID Passivity-Based Control of Nonlinear Systems with Applications - Romeo Ortega - Страница 27

To mathematically define the dissipation obstacle of a passive system with storage function , let us compute its derivative

(2.2)

As a corollary of Hill–Moylan's theorem, see Theorem A.1, we see that the only passive output of relative degree one is the so‐called natural output, that we identify with the subindex , and is given by

(2.3)

Substituting the definition above in 2.2, we can give to it the interpretation of power‐balance equation, where is the energy stored by the system, is the supplied power and is the system's dissipation. In passivity theory, it is said that the system does not suffer from the dissipation obstacle – at an assignable equilibrium – if

(2.4)

Notice that for pH systems, see Definition D.1, the dissipation obstacle translates into

(2.5)

where is the dissipation matrix and is a bona fide energy function – yielding a clear physical interpretation.

The dissipation obstacle is a phenomenon whose origin is the existence of pervasive dissipation, that is, dissipation that is present even at the equilibrium state. It is a multifaceted phenomenon that has been discussed at length in the PBC literature, where it is shown that the key energy shaping step of PBC (Ortega et al., 2008, Proposition 1), the generation of Casimir functions for CbI (van der Schaft, 2016, Remark 7.1.9) and the assignment of a minimum at the desired point to the shaped energy function (Zhang et al., 2015, Proposition 2) are all stymied by the dissipation obstacle.