Romeo Ortega - PID Passivity-Based Control of Nonlinear Systems with Applications

(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.

The proposition below shows that the application of PID‐PBC with the natural output is severely stymied by the dissipation obstacle. Actually, we will prove a much more general result that contains, as a particular case, the PID‐PBC scenario.

Consider the system with passive output 2.3 and a dynamic extension

where . Define the overall system dynamics as , where . A necessary condition for the existence of a constant solution to the equilibrium equation

is that the system does not suffer from the dissipation obstacle, i.e., 2.4 holds.

Proof . The proof is established as follows:

Now

On the other hand,

The proof is completed substituting the last identity in the one above.

An immediate corollary of Proposition 2.2is that the dissipation obstacle hampers the application of PID‐PBC for nonzero equilibrium with the natural passive output.

As shown in Ortega et al. (2008), van der Schaft (2016), Venkatraman and van der Schaft (2010), and Zhang et al. (2015), one way to overcome the dissipation obstacle is to generate relative degree zero outputs. However, it is then not possible to add a derivative term to the controller that, due to its “prediction‐like” feature, is useful in some applications. In Chapter 6, we propose a new construction of PID‐PBC, where it is possible to add a derivative action to systems where the dissipation obstacle is present. We will also show that the integral and derivative terms perform the energy‐shaping process, while the proportional term completes the PBC design by injecting damping into the closed‐loop system.

In this section, we give an interpretation of proportional‐integral (PI) PBC with the natural output as a particular case of CbI, which is a physically (and conceptually) appealing method to stabilize equilibria of nonlinear systems widely studied in the literature, cf, Duindam et al. (2009), Ortega et al. (2008), and van der Schaft (2016).

CbI has been mainly studied for pH systems, where the physical properties can be fully exploited to give a nice interpretation to the control action, viewed not with the standard signal‐processing viewpoint, but as an energy exchange process. Here, we present CbI in the more general case of the ‐system , which we assume passive with storage function , and the controller