t right-parenthesis comma u Subscript c Baseline left-parenthesis t right-parenthesis comma y Subscript c Baseline left-parenthesis t right-parenthesis element-of double-struck upper R Superscript m"/>, 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
