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

Автор: Romeo Ortega

Издательство: John Wiley & Sons Limited

Жанр: Отраслевые издания

Серия:

isbn: 9781119694182

isbn:

СКАЧАТЬ see Definition D.1, the dissipation obstacle translates into

      (2.5)double-vertical-bar left-parenthesis nabla upper H right-parenthesis Superscript star Baseline double-vertical-bar Subscript script upper R Sub Superscript star Subscript Superscript 2 Baseline not-equals 0 comma

      where script upper R left-parenthesis x right-parenthesis is the dissipation matrix and upper H left-parenthesis x right-parenthesis is a bona fide energy function – yielding a clear physical interpretation.

      The dissipation obstacle is a phenomenon whose origin is the existence of pervasive dissipation, that is, dissipation that is present even at the equilibrium state. It is a multifaceted phenomenon that has been discussed at length in the PBC literature, where it is shown that the key energy shaping step of PBC (Ortega et al., 2008, Proposition 1), the generation of Casimir functions for CbI (van der Schaft, 2016, Remark 7.1.9) and the assignment of a minimum at the desired point to the shaped energy function (Zhang et al., 2015, Proposition 2) are all stymied by the dissipation obstacle.

      2.3.2 Steady‐State Operation and the Dissipation Obstacle

      The proposition below shows that the application of PID‐PBC with the natural output is severely stymied by the dissipation obstacle. Actually, we will prove a much more general result that contains, as a particular case, the PID‐PBC scenario.

StartLayout 1st Row 1st Column ModifyingAbove x With dot Subscript c 2nd Column equals 3rd Column g Superscript down-tack Baseline left-parenthesis x right-parenthesis nabla upper S left-parenthesis x right-parenthesis 2nd Row 1st Column ModifyingAbove xi With dot 2nd Column equals 3rd Column f Subscript c Baseline left-parenthesis xi comma x comma x Subscript c Baseline comma u right-parenthesis comma EndLayout

       where . Define the overall system dynamics as , where . A necessary condition for the existence of a constant solution to the equilibrium equation

upper F left-parenthesis chi Superscript star Baseline comma u Superscript star Baseline right-parenthesis equals 0 comma

       is that the system does not suffer from the dissipation obstacle, i.e., 2.4 holds.

      Proof. The proof is established as follows:

upper F left-parenthesis chi Superscript star Baseline comma u Superscript star Baseline right-parenthesis equals 0 right double arrow StartBinomialOrMatrix ModifyingAbove x With dot Choose ModifyingAbove x With dot Subscript c Baseline EndBinomialOrMatrix equals 0 period

      Now

ModifyingAbove x With dot equals 0 right double arrow ModifyingAbove upper S With dot equals 0 left right double arrow left-bracket left-parenthesis nabla upper S right-parenthesis Superscript star Baseline right-bracket Superscript down-tack Baseline f Superscript star Baseline plus left-bracket left-parenthesis nabla upper S right-parenthesis Superscript star Baseline right-bracket Superscript down-tack Baseline g Superscript star Baseline u Superscript star Baseline equals 0 period ModifyingAbove x With dot Subscript c Baseline equals 0 left right double arrow left-bracket left-parenthesis nabla upper S right-parenthesis Superscript star Baseline right-bracket Superscript down-tack Baseline g Superscript star Baseline equals 0 period

      The proof is completed substituting the last identity in the one above.

      Remark 2.4:

      Remark 2.5:

      As shown in Ortega et al. (2008), van der Schaft (2016), Venkatraman and van der Schaft (2010), and Zhang et al. (2015), one way to overcome the dissipation obstacle is to generate relative degree zero outputs. However, it is then not possible to add a derivative term to the controller that, due to its “prediction‐like” feature, is useful in some applications. In Chapter 6, we propose a new construction of PID‐PBC, where it is possible to add a derivative action to systems where the dissipation obstacle is present. We will also show that the integral and derivative terms perform the energy‐shaping process, while the proportional term completes the PBC design by injecting damping into the closed‐loop system.

      

      In this section, we give an interpretation of proportional‐integral (PI) PBC with the natural output y 0 as a particular case of CbI, which is a physically (and conceptually) appealing method to stabilize equilibria of nonlinear systems widely studied in the literature, cf, Duindam et al. (2009), Ortega et al. (2008), and van der Schaft (2016).

      CbI has been mainly studied for pH systems, where the physical properties can be fully exploited to give a nice interpretation to the control action, viewed not with the standard signal‐processing viewpoint, but as an energy exchange process. Here, we present CbI in the more general case of the left-parenthesis f comma g comma h comma j right-parenthesis‐system normal upper Sigma, which we assume passive with storage function upper S left-parenthesis x right-parenthesis, and the controller

StartLayout 1st Row 1st Column ModifyingAbove x With dot Subscript c 2nd Column equals f Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis plus g Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis u Subscript c Baseline 2nd Row 1st Column y Subscript c 2nd Column equals h Subscript c Baseline left-parenthesis x Subscript c Baseline right-parenthesis comma EndLayout