rel="nofollow" href="#fb3_img_img_5ed29cdc-1592-5c74-a030-a28aee7db4f8.png" alt="normal upper Sigma Subscript c Baseline Start 2 By 2 Matrix 1st Row 1st Column ModifyingAbove x With dot Subscript c 2nd Column equals y 2nd Row 1st Column u 2nd Column equals minus upper K Subscript upper P Baseline y minus upper K Subscript upper I Baseline x Subscript c Baseline minus upper K Subscript upper D Baseline ModifyingAbove y With dot comma EndMatrix"/>
where with , and are the PID tuning gains. The key property of PID controllers that we exploit in the book is that it defines an output strictly passive map . This well‐known property (Ortega and García‐Canseco, 2004; van der Schaft, 2016) is summarized in the lemma below.
Lemma 2.1
Consider the PID controller2.1. 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.
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 , the output . Under some additional assumptions, it also follows that . For instance, the latter property holds true, with , under the very weak assumption that there exists a steady‐state with and
