@phdthesis{Lieb2017,
  author    = {Lieb, Julia},
  title     = {Counting Polynomial Matrices over Finite Fields : Matrices with Certain Primeness Properties and Applications to Linear Systems and Coding Theory},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-064-1 (print)},
  doi       = {10.25972/WUP-978-3-95826-065-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-151303},
  school      = {W{\"u}rzburg University Press},
  pages     = {164},
  year      = {2017},
  abstract  = {This dissertation is dealing with three mathematical areas, namely polynomial matrices over finite fields, linear systems and coding theory. Coprimeness properties of polynomial matrices provide criteria for the reachability and observability of interconnected linear systems. Since time-discrete linear systems over finite fields and convolutional codes are basically the same objects, these results could be transfered to criteria for non-catastrophicity of convolutional codes. We calculate the probability that specially structured polynomial matrices are right prime. In particular, formulas for the number of pairwise coprime polynomials and for the number of mutually left coprime polynomial matrices are calculated. This leads to the probability that a parallel connected linear system is reachable and that a parallel connected convolutional codes is non-catastrophic. Moreover, the corresponding probabilities are calculated for other networks of linear systems and convolutional codes, such as series connection. Furthermore, the probabilities that a convolutional codes is MDP and that a clock code is MDS are approximated. Finally, we consider the probability of finding a solution for a linear network coding problem.},
  subject      = {Lineares System},
  language  = {en}
}
@phdthesis{Rueppel2014,
  author    = {R{\"u}ppel, Frederike},
  title     = {Accessibility of Bilinear Interconnected Systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-99250},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {The subject of this thesis is the controllability of interconnected linear systems, where the interconnection parameter are the control variables. The study of accessibility and controllability of bilinear systems is closely related to their system Lie algebra. In 1976, Brockett classified all possible system Lie algebras of linear single-input, single-output (SISO) systems under time-varying output feedback. Here, Brockett's results are generalized to networks of linear systems, where time-varying output feedback is applied according to the interconnection structure of the network. First, networks of linear SISO systems are studied and it is assumed that all interconnections are independently controllable. By calculating the system Lie algebra it is shown that accessibility of the controlled network is equivalent to the strong connectedness of the underlying interconnection graph in case the network has at least three subsystems. Networks with two subsystems are not captured by these proofs. Thus, we give results for this particular case under additional assumption either on the graph structure or on the dynamics of the node systems, which are both not necessary. Additionally, the system Lie algebra is studied in case the interconnection graph is not strongly connected. Then, we show how to adapt the ideas of proof to networks of multi-input, multi-output (MIMO) systems. We generalize results for the system Lie algebra on networks of MIMO systems both under output feedback and under restricted output feedback. Moreover, the case with generalized interconnections is studied, i.e. parallel edges and linear dependencies in the interconnection controls are allowed. The new setting demands to distinguish between homogeneous and heterogeneous networks. With this new setting only sufficient conditions can be found to guarantee accessibility of the controlled network. As an example, networks with Toeplitz interconnection structure are studied.},
  subject      = {Steuerbarkeit},
  language  = {en}
}