**Funding agency**: Deutsche Forschungsgemeinschaft (DFG)

**Budget**: 161 900 EUR

**Duration**: 09/2014-06/2018

**Summary:** A dynamical system comprises a mathematical model of an underlying physical phenomenon. It has two basic components: external signals which interconnect the system with its environment and the internal state that evolves according to the model description. The external signals can usually further be split into inputs and outputs. One of the basic problem associated with any dynamical system is that of constructing an observer which uses the available information of the external signals to estimate the internal state. The purpose of this project is to develop observers for dynamical systems modeled as switched differential-algebraic equations (DAEs). The motivation to study this particular system class is twofold: 1) In contrast to ordinary differential equations (ODEs), DAEs include differential as well as algebraic equations. Practically every system’s model contains algebraic equations in the first place so it is natural to use DAEs (instead of the simplified ODEs) as a starting point. 2) Possible structural changes (like switches in electrical circuits or component faults in general physical system) can be modeled within the framework of switched systems. As an application of the proposed project, consider for example (national) electrical grids, which are large electrical circuits modeled as DAEs. An observer would then be used to monitor the energy flows through the transmission lines and could prevent overloading. Sudden structural changes in electrical grids are common and have to be taken into account; examples are: tripping of power lines due to harsh weather conditions, or a sudden drop in the energy production by wind turbines when whole wind parks are switched off in the presence of too strong winds. Hence a possible application of the theoretical results obtained by the proposed project could be improved monitoring tools for electrical grids.

**Researchers involved:**

– Stephan Tenn (PI, not paid by project, financial support for travel)

– Aneel Tanwani (Postdoc, paid by project, 09/2014-12/2015, financial support for travel)

– Deepak Patil (Postdoc, paid by project, 03/2016-08/2016, financial support for travel)

– Ferdinand Küsters (PhD, not paid by project, financial support for travel, link to PhD-dissertation)

– Pietro Tesi (financially supported research visit)

– Timo Reis (financially supported research visit)

– Hyungbo Shim (financially supported research visit)

– Kanat Camlibel (financially supported research visit)

– Yashar Kouhi Anbaran (financially supported research visit)

– Andreas Wirsen

– Mihaly Petreczky

– Thomas Berger

– Francesco Vasca

**Research results obtained during the project:**

Unimodular transformations for DAE initial trajectory problems Inproceedings PAMM · Proc. Appl. Math. Mech., pp. e202000322, Wiley-VCH GmbH, 2021, (Open Access.). |

Indiscernible topological variations in DAE networks Journal Article Automatica, 101 , pp. 280-289, 2019. |

Detectability and observer design for switched differential algebraic equations Journal Article Automatica, 99 , pp. 289-300, 2019. |

Switch observability for switched linear systems Journal Article Automatica, 87 , pp. 121-127, 2018. |

Switch observability for a class of inhomogeneous switched DAEs Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 3175 - 3180, Melbourne, Australia, 2017. |

Switch-observer for switched linear systems Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 1749 - 1754, Melbourne, Australia, 2017. |

Switch observability for homogeneous switched DAEs Inproceedings Proc. 20th IFAC World Congress 2017, pp. 9355 - 9360, Toulouse, France, 2017, ISSN: 2405-8963. |

Indiscernible topological variations in DAE networks with applications to power grids Inproceedings Proc. 20th IFAC World Congress 2017, pp. 7333 - 7338, Toulouse, France, 2017, ISSN: 2405-8963. |

Observer design for detectable switched differential-algebraic equations Inproceedings Proc. 20th IFAC World Congress 2017, pp. 2953 - 2958, Toulouse, France, 2017, ISSN: 2405-8963. |

Determinability and state estimation for switched differential–algebraic equations Journal Article Automatica, 76 , pp. 17–31, 2017, ISSN: 0005-1098. |

Observability of linear differential-algebraic systems: A survey Incollection Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations IV, pp. 161–219, Springer-Verlag, Berlin-Heidelberg, 2017. |

Observer design based on constant-input observability for DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 813–814, WILEY-VCH Verlag, 2016, ISSN: 1617-7061. |

On detectability of switched linear differential-algebraic equations Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2957–2962, 2015. |

Observability of switched linear systems Incollection Djemai, Mohamed; Defoort, Michael (Ed.): Hybrid Dynamical Systems, 457 , pp. 205–240, 2015. |