Название: PID Passivity-Based Control of Nonlinear Systems with Applications
Автор: Romeo Ortega
Издательство: John Wiley & Sons Limited
Жанр: Отраслевые издания
isbn: 9781119694182
isbn:
2.2 Well‐Posedness Conditions
As discussed earlier, it is necessary to ensure that the control law 2.1 can be computed without differentiation nor singularities. The latter may arise due to the presence of the derivative term
The required well‐posedness assumptions in both cases are stated in the lemma below, whose proof follows immediately computing the expressions of
Lemma 2.2
If the system has relative degree zero, that is , the feedback system of Figure 2.1 with is well‐posed if the matrix
is full‐rank. On the other hand, if the system has relative degree one, that is and , the feedback system is well posed if the matrix
is full‐rank.
Remark 2.3:
As a final comment of this section, we note that in van der Schaft (2016), PID control is viewed from a different perspective. Namely, assuming that
2.3 PID‐PBC and the Dissipation Obstacle
In this section, we reveal a subtle aspect of the practical application of PID‐PBC, namely that for passive systems of relative degree one, there exists a steady state only if the energy extracted from the controller is zero at the equilibrium. The latter condition is known in PBC as dissipation obstacle and is present in many physical systems, for instance, all electrical circuits with leaky energy storing elements operating in nonzero equilibria – i.e. capacitors in parallel, or inductors in series, with resistors. Interestingly, this obstacle is absent in position regulation of mechanical systems since dissipation (due to Coulomb friction) is zero at standstill.
After briefly recalling the nature and mathematical definition of the dissipation obstacle, we prove the claim of inexistence of equilibria stated above in a more general context than just PID‐PBC, namely for all dynamic controllers incorporating an integral action on a passive output of relative degree one.
2.3.1 Passive Systems and the Dissipation Obstacle
To mathematically define the dissipation obstacle of a passive system with storage function
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
Substituting the definition above in 2.2, we can give to it the interpretation of power‐balance equation, where
(2.4)
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