Читать книгу PID Passivity-Based Control of Nonlinear Systems with Applications - Romeo Ortega - Страница 19

Consider the PID controller 2.1. The operator is output strictly passive. More precisely, there exists such that the following inequality holds:

Proof. To prove the lemma, we compute

Integrating the expression above, we get

The proof is completed setting .

The main idea of PID‐PBC is to exploit the passivity property of PIDs and, invoking the Passivity Theorem, see Section A.2 of Appendix A, wrap the PID around a passive output of the system to ensure ‐stability of the closed‐loop system. This result is summarized in the proposition below, whose proof follows directly from the Passivity Theorem, passivity of the mapping and output strict passivity of the mapping defined by the PID‐PBC.