The proof is completed setting ![beta colon equals minus double-vertical-bar x Subscript c Baseline left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper K Sub Subscript upper I Subscript Superscript 2 Baseline minus double-vertical-bar y left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper K Sub Subscript upper D Subscript Superscript 2]()
.
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 ![normal upper Sigma]()
to ensure ![script upper L 2]()
‐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 ![normal upper Sigma colon u right-arrow from bar y]()
and output strict passivity of the mapping defined by the PID‐PBC.
Proposition 2.1:
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
![integral Subscript 0 Superscript t Baseline StartAbsoluteValue y left-parenthesis s right-parenthesis EndAbsoluteValue squared d s less-than-or-equal-to StartFraction 1 Over lamda Subscript min Baseline left-parenthesis upper K Subscript upper P Baseline right-parenthesis EndFraction integral Subscript 0 Superscript t Baseline StartAbsoluteValue d left-parenthesis s right-parenthesis EndAbsoluteValue squared d s plus beta comma for-all t greater-than-or-equal-to 0 period]()
![Schematic illustration of a block diagram of the closed-loop system of Proposition 2.1.]()
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 ![d left-parenthesis t right-parenthesis element-of script upper L 2]()
, the output ![y left-parenthesis t right-parenthesis element-of script upper L 2]()
. Under some additional assumptions, it also follows that ![limit Underscript t right-arrow infinity Endscripts y left-parenthesis t right-parenthesis equals 0]()
. For instance, the latter property holds true, with ![d left-parenthesis t right-parenthesis equals 0]()
, under the very weak assumption that there exists a steady‐state with ![u]()
and ![]()
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