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

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

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

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t right-parenthesis comma u Subscript c Baseline left-parenthesis t right-parenthesis comma y Subscript c Baseline left-parenthesis t right-parenthesis element-of double-struck upper R Superscript m"/>, which is also passive with storage function upper S Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis. Clearly, the integral action of the PI‐PBC is a particular case of this controller with the choices f Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis equals 0, g Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis equals upper I Subscript m, and h Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis equals upper K Subscript upper I Baseline x Subscript c.

      These systems are coupled via an interconnection that preserves power, that is which satisfies u Superscript down-tack Baseline y plus u Subscript c Superscript down-tack Baseline y Subscript c Baseline equals 0. For instance, the classical negative feedback interconnection

StartLayout 1st Row 1st Column u 2nd Column equals minus y Subscript c Baseline 2nd Row 1st Column u Subscript c 2nd Column equals y 0 period EndLayout

      The proportional action of the PI‐PBC may be assimilated as a preliminary damping injection to the plant giving rise to the new process model

StartLayout 1st Row 1st Column ModifyingAbove x With dot 2nd Column equals left-bracket f left-parenthesis x right-parenthesis minus g left-parenthesis x right-parenthesis upper K Subscript upper P Baseline g Superscript upper T Baseline left-parenthesis x right-parenthesis nabla upper S left-parenthesis x right-parenthesis right-bracket plus g left-parenthesis x right-parenthesis u Subscript c Baseline period EndLayout

      In view of the passivity properties, the storage function of the overall system

      is nonincreasing, alas, not necessarily positive definite – with respect to the desired equilibrium left-parenthesis x Superscript star Baseline comma x Subscript c Superscript star Baseline right-parenthesis. To construct a bona‐fide Lyapunov function, it is proposed in CbI to prove the existence of an invariant foliation

script upper M Subscript kappa Baseline colon equals StartSet left-parenthesis x comma x Subscript c Baseline right-parenthesis element-of double-struck upper R Superscript n Baseline times double-struck upper R Superscript m Baseline vertical-bar x Subscript c Baseline equals gamma left-parenthesis x right-parenthesis plus kappa EndSet comma

      with gamma colon double-struck upper R Superscript n Baseline right-arrow double-struck upper R Superscript m a smooth mapping and kappa element-of double-struck upper R Superscript m. In CbI, a cross‐term of the form normal upper Phi left-parenthesis x Subscript c Baseline minus gamma left-parenthesis x right-parenthesis minus kappa right-parenthesis, with normal upper Phi colon double-struck upper R Superscript m Baseline right-arrow double-struck upper R a free differentiable function, is added to the function upper S Subscript c l Baseline left-parenthesis x comma x Subscript c Baseline right-parenthesis given in 2.6 to create the function

upper S Subscript d Baseline left-parenthesis x comma x Subscript c Baseline right-parenthesis colon equals upper S Subscript c l Baseline left-parenthesis x comma x Subscript c Baseline right-parenthesis plus normal upper Phi left-parenthesis x Subscript c Baseline minus gamma left-parenthesis x right-parenthesis minus kappa right-parenthesis comma

      that, due to the invariance property of script upper M Subscript kappa, satisfies ModifyingAbove upper S With dot Subscript d Baseline equals ModifyingAbove upper S With dot Subscript c l, hence, is still nonincreasing. If we manage to prove that upper S Subscript d Baseline left-parenthesis x comma x Subscript c Baseline right-parenthesis is positive definite, the desired equilibrium will be stable. However, the asymptotic stability requirement, and the fact that script upper M Subscript kappa is invariant, imposes the constraint on the initial conditions

script í’Ÿ equals StartSet x element-of double-struck upper R Superscript n Baseline vertical-bar gamma left-parenthesis x right-parenthesis minus x Subscript c Baseline equals gamma left-parenthesis x Superscript star Baseline right-parenthesis minus x Subscript c Superscript star Baseline EndSet period

      That is, the trajectory should start on the leaf of script upper M Subscript kappa that contains the desired equilibrium – fixing the initial conditions of the controller. Invoking Sard's theorem (Spivak, 1995), we see that script í’Ÿ is a nowhere dense set, hence, the asymptotic stability claim is nonrobust (Ortega, 2021). Two solutions to alleviate this problem – estimation of the constant kappa Superscript star Baseline colon equals x Subscript c Superscript star Baseline minus gamma left-parenthesis x Subscript p Superscript star Baseline right-parenthesis or breaking the invariance of script upper M Subscript kappa via damping injection – have been reported in Castaños et al. (2009), but this adds significant complications to the scheme.

upper V left-parenthesis x right-parenthesis colon equals upper S Subscript c l Baseline left-parenthesis x comma gamma left-parenthesis x right-parenthesis plus kappa right-parenthesis comma

      that we

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