Backstepping-based observer design for parabolic PDEs with varying parameters

Project focus

• State estimation for scalar linear PDEs with spatially and time-varying parameters defined on a one-dimensional spatial domain and time-varying boundary conditions
• State estimation to systems governed by parabolic PDEs defined on a higher-dimensional spatial domain
• State estimation for quasi-linear parabolic PDEs with locally Lipschitz nonlinearities
• Efficient numerical implementation in view of real-time applications

Description

Model-based control and advanced process monitoring usually require full state information. This is in particular evident if control strategies are utilized by means of state feedback or if algorithms based on full state knowledge are applied to improve the insight into the system behavior, e.g., for diagnostic or error detection purposes. In general, the complete state information cannot be directly measured. This is especially true for distributed-parameter systems, i.e., systems governed by partial differential equations (PDEs). In this case, the measurement information is typically limited to a finite number of spatially discrete sensors, or for spatially higher-dimensional systems by the fact that measurements are usually available only at the boundary of the spatial domain. Therefore, an observer is required to estimate the state variables from the knowledge of measured input and output variables as well as the mathematical model of the system.

This project is concerned with the state observer design for systems governed by parabolic PDEs with boundary sensing. For this, a distributed-parameter Luenberger-type state observer is applied with observer corrections entering both the PDE and the boundary conditions. Thereby, the backstepping method is utilized and extended to determine the output injection weights to ensure the exponential de­cay of the resulting observer error dynamics and hence an accurate state es­ti­ma­tion.

Selected publications

• L. Jadachowski, T. Meurer, and A. Kugi, Backstepping Observers for linear PDEs on Higher-Dimensional Spatial Domains, Automatica, vol. 51, p. 85–97, 2015.
  [BibTex]

  @Article{Jadachowski14a,
  Title = {{Backstepping Observers for linear PDEs on Higher-Dimensional Spatial Domains}},
  Author = {Jadachowski, L. and Meurer, T. and Kugi, A.},
  Journal = {Automatica},
  Pages = {85--97},
  Volume = {51},
  Year = {2015},
  Doi = {10.1016/j.automatica.2014.10.108},
  }

• L. Jadachowski, T. Meurer, and A. Kugi, Backstepping Observers for Periodic Quasi-Linear Parabolic PDEs, in Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, 2014, p. 7761–7766.
  [BibTex]

  @InProceedings{Jadachowski14,
  author = {Jadachowski, L. and Meurer, T. and Kugi, A.},
  title = {Backstepping Observers for Periodic Quasi-Linear Parabolic {PDE}s},
  booktitle = {Proceedings of the 19th IFAC World Congress},
  year = {2014},
  month = {8},
  pages = {7761--7766},
  doi = {10.3182/20140824-6-ZA-1003.01246},
  address = {Cape Town, South Africa},
  }

• L. Jadachowski, T. Meurer, and A. Kugi, State Estimation for Parabolic PDEs with Reactive-Convective Non-Linearities, in Proceedings of European Control Conference ECC 2013, Zürich, Switzerland, 2013, p. 1603 – 1608.
  [BibTex] [Download]

  @InProceedings{Jadachowski13,
  author = {L. Jadachowski and T. Meurer and A. Kugi},
  title = {State Estimation for Parabolic PDEs with Reactive-Convective Non-Linearities},
  booktitle = {Proceedings of European Control Conference ECC 2013},
  year = {2013},
  month = {7},
  pages = {1603 -- 1608},
  url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6669588},
  address = {Z\"{u}rich, Switzerland},
  }

• L. Jadachowski, T. Meurer, and A. Kugi, An Efficient Implementation of Backstepping Observers for Time-Varying Parabolic PDEs, in Proceedings MATHMOD 2012 Vienna, Wien, Austria, 2012, p. 798–803.
  [BibTex]

  @InProceedings{Jadachowski12,
  author = {L. Jadachowski and T. Meurer and A. Kugi},
  title = {{An Efficient Implementation of Backstepping Observers for Time-Varying Parabolic PDEs}},
  booktitle = {Proceedings MATHMOD 2012 Vienna},
  year = {2012},
  editor = {Troch, I. and Breitenecker, F.},
  month = {2},
  pages = {798--803},
  doi = {10.3182/20120215-3-AT-3016.00141},
  address = {Wien, Austria},
  }

• L. Jadachowski, T. Meurer, and A. Kugi, State Estimation for Parabolic PDEs with Varying Parameters on 3-Dimensional Spatial Domains, in Proceedings of the 18th IFAC World Congress, Milano, Italia, 2011, pp. 13338-13343.
  [BibTex]

  @InProceedings{Jadachowski11,
  author = {L. Jadachowski and T. Meurer and A. Kugi},
  title = {{S}tate {E}stimation for {P}arabolic {PDE}s with {V}arying {P}arameters on 3-{D}imensional {S}patial {D}omains},
  booktitle = {Proceedings of the 18th IFAC World Congress},
  year = {2011},
  month = {8},
  pages = {13338-13343},
  doi = {10.3182/20110828-6-IT-1002.02964},
  address = {Milano, Italia},
  }

• L. Jadachowski, T. Meurer, and A. Kugi, State Reconstruction in Higher Dimensional PDEs with Spatially Varying Parameters, PAMM, Proceedings in Applied Mathematics and Mechanics, vol. 11, iss. 1, p. 813–814, 2011.
  [BibTex]

  @Article{Jadachowski11a,
  Title = {{State Reconstruction in Higher Dimensional PDEs with Spatially Varying Parameters}},
  Author = {L. Jadachowski and T. Meurer and A. Kugi},
  Journal = {PAMM, Proceedings in Applied Mathematics and Mechanics},
  Pages = {813--814},
  Volume = {11},
  Year = {2011},
  Number = {1},
  Doi = {10.1002/pamm.201110395},
  }

Applications

• State estimation for heating-up and cooling-down processes in steel industry
• System diagnostics and error detection in fixed-bed tubular reactors in chemical engineering

Funding

TU Wien Doktorandenkolleg PDETech

Partners

Prof. Dr.-Ing. habil. Thomas Meurer