PID Passivity-Based Control of Nonlinear Systems with Applications. Romeo Ortega
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where
Lemma 2.1
Consider the PID controller 2.1. The operator
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
Consider the feedback system depicted in Figure 2.1, where is the nonlinear system (1), is the PID controller of 2.1 and is an external signal. Assume the interconnection is well defined.1 If the mapping is passive, the operator is ‐stable. More precisely, there exists such that
Figure 2.1 Block diagram representation of the closed‐loop system of Proposition 2.1.
Remark 2.1:
From Proposition 2.1, we have that, for all signals