Speaker
Description
The design of flatness-based open-loop control for various kinds of hyperbolic partial differential equations (PDEs) has been discussed in several publications [1,2], including its application to quasi-linear hyperbolic PDEs [3]. Furthermore, flatness-based tracking control for quasi-linear hyperbolic PDEs has already been analyzed in [4]. However, the study of flatness-based observer design remains pending, despite energy-based and backstepping-based approaches already being considered (see e.g. [5]). This contribution, therefore, focuses on the flatness-based observer design for quasi-linear systems, specifically accounting for non-collocated measurements for error injection. The design approach relies on the so-called hyperbolic observer form [6]. Transforming a given system into this specific state-space representation can be achieved through a flatness-based analysis of the system under consideration. In these new normal form coordinates, the transient response of the observer is governed by a stable difference-differential equation. The design methodology is illustrated using the well-known Saint-Venant equations [7]. These equations are widely employed to model shallow water in open channels, such as irrigation channels with rectangular cross-sections [8, 9], and for modeling snow avalanches [10]. The system discussed in [11] has been analyzed for flatness-based open-loop control, while flatness-based tracking control has been studied in [12]. Additionally, a model transformation using material-fixed coordinates, as presented in [13], leads to an adapted quasi-linear hyperbolic PDE. This transformation significantly simplifies the model structure for observer design. The proposed design utilizes position measurement alongside the non-collocated measurement of the water level and transforms the given system into a hyperbolic observer canonical form. Finally, an efficient implementation strategy based on higher-order approximation methods, as discussed in [14], is implemented in an energy-preserving manner.
[1] J. Rudolph, Int. J. Control 81, 2008.15
[2] F. Woittennek, IFAC Proc. Vol. 46, 2013.
[3] T. Knüppel, IEEE Trans. Autom. Control 60, 2015.
[4] N. Gehring, IFAC-PapersOnLine 51, 2018.
[5] T. Strecker, Automatica, 2022.
[6] F. Woittennek, On the hyperbolic observer canonical form, 2013.
[7] A. J. C. B. d. Saint-Venant, Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit, 1871.
[8] H. Chanson, Hydraulics of Open Channel Flow (Elsevier, May 2004).
[9] W. H. Graf, Fluvial Hydraulics: Flow and Transport Processes in Channels of Simple Geometry, 1998.
[10] A. N. Nazarov, Fluid Dynamics, 1991.
[11] J. Kopp, Automatica, 2020.
[12] J. Wurm, AT.
[13] J. Wurm, IFAC-PapersOnLine, 2022.
[14] L. Mayer, PAMM, 2023.
