Читать книгу PID Passivity-Based Control of Nonlinear Systems with Applications - Romeo Ortega - Страница 32

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

with , which is also passive with storage function . Clearly, the integral action of the PI‐PBC is a particular case of this controller with the choices , , and .

These systems are coupled via an interconnection that preserves power, that is which satisfies . For instance, the classical negative feedback interconnection

The proportional action of the PI‐PBC may be assimilated as a preliminary damping injection to the plant giving rise to the new process model

In view of the passivity properties, the storage function of the overall system

(2.6)

is nonincreasing, alas, not necessarily positive definite – with respect to the desired equilibrium . To construct a bona‐fide Lyapunov function, it is proposed in CbI to prove the existence of an invariant foliation

with a smooth mapping and . In CbI, a cross‐term of the form , with a free differentiable function, is added to the function given in 2.6 to create the function

that, due to the invariance property of , satisfies , hence, is still nonincreasing. If we manage to prove that is positive definite, the desired equilibrium will be stable. However, the asymptotic stability requirement, and the fact that is invariant, imposes the constraint on the initial conditions

That is, the trajectory should start on the leaf of that contains the desired equilibrium – fixing the initial conditions of the controller. Invoking Sard's theorem (Spivak, 1995), we see that is a nowhere dense set, hence, the asymptotic stability claim is nonrobust (Ortega, 2021). Two solutions to alleviate this problem – estimation of the constant or breaking the invariance of via damping injection – have been reported in Castaños et al. (2009), but this adds significant complications to the scheme.

In Chapter 6, we give a solution to the robustness problem using the PI‐PBC. In this case, instead of adding the cross term, we project the function of 2.6 onto to generate the function:

that we might be able to use as a Lyapunov function (for the projected dynamics). To avoid the difficulty of the constraint on the initial conditions mentioned above, we replace the PI by a static state‐feedback law of the form

which exactly coincides with the control generated with the PI‐PBC evaluated at . Such an implementation does not impose any constraint on and is therefore robust.

The details of this construction are given in Chapter 6, where we also prove that the set of solutions of the partial differential equations (PDEs) that must be solved to generate the invariant foliation in CbI is strictly smaller than the ones needed in the PID‐PBC, yielding a design procedure that is applicable for a broader class of systems.