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

      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.

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Notes

      1 1 Essentially, it is necessary to ensure that the control law 2.1 can be computed without differentiation nor singularities. We will elaborate on this issue in Section 2.2.

      2 2 We recall that a necessary condition for passivity of the system is that the relative degree is smaller or equal to one (van der Schaft, 2016).