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

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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 upper K Subscript upper P Baseline comma upper K Subscript upper I Baseline comma upper K Subscript upper D Baseline element-of double-struck upper R Superscript m times m with upper K Subscript upper P Baseline comma upper K Subscript upper I Baseline greater-than 0, and upper K Subscript upper D Baseline greater-than-or-equal-to 0 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 normal upper Sigma Subscript c Baseline colon y right-arrow from bar left-parenthesis negative u right-parenthesis. This well‐known property (Ortega and García‐Canseco, 2004; van der Schaft, 2016) is summarized in the lemma below.

      Lemma 2.1

integral Subscript 0 Superscript t Baseline y left-parenthesis s right-parenthesis left-parenthesis minus u left-parenthesis s right-parenthesis right-parenthesis d s greater-than-or-equal-to lamda Subscript min Baseline left-parenthesis upper K Subscript upper P Baseline right-parenthesis integral Subscript 0 Superscript t Baseline StartAbsoluteValue y left-parenthesis s right-parenthesis EndAbsoluteValue squared d s plus beta comma for-all t greater-than-or-equal-to 0 comma

      Proof. To prove the lemma, we compute

StartLayout 1st Row 1st Column y Superscript down-tack Baseline left-parenthesis negative u right-parenthesis 2nd Column equals double-vertical-bar y double-vertical-bar Subscript upper K Sub Subscript upper P Superscript 2 Baseline plus y Superscript down-tack Baseline upper K Subscript upper I Baseline x Subscript c Baseline plus y Superscript down-tack Baseline upper K Subscript upper D Baseline ModifyingAbove y With dot 2nd Row 1st Column Blank 2nd Column greater-than-or-equal-to lamda Subscript min Baseline left-parenthesis upper K Subscript upper P Baseline right-parenthesis StartAbsoluteValue y EndAbsoluteValue squared plus ModifyingAbove x With dot Subscript c Superscript down-tack Baseline upper K Subscript upper I Baseline x Subscript c Baseline plus y Superscript down-tack Baseline upper K Subscript upper D Baseline ModifyingAbove y With dot period EndLayout

      Integrating the expression above, we get

StartLayout 1st Row 1st Column Blank 2nd Column integral Subscript 0 Superscript t Baseline y left-parenthesis s right-parenthesis left-parenthesis minus u left-parenthesis s right-parenthesis right-parenthesis d s 2nd Row 1st Column Blank 2nd Column greater-than-or-equal-to lamda Subscript min Baseline left-parenthesis upper K Subscript upper P Baseline right-parenthesis integral Subscript 0 Superscript t Baseline StartAbsoluteValue y left-parenthesis s right-parenthesis EndAbsoluteValue squared d s 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 Baseline comma for-all t greater-than-or-equal-to 0 period EndLayout

      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.

       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.

      

      Remark 2.1: