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@PHDTHESIS{Schilli:670544,
      author       = {Schilli, Christian},
      othercontributors = {Zerz, Eva and Walcher, Sebastian},
      title        = {{C}ontrolled and conditioned invariant varieties for
                      polynomial control systems},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      reportid     = {RWTH-2016-07596},
      pages        = {1 Online-Ressource (117 Seiten)},
      year         = {2016},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; Dissertation, RWTH Aachen University, 2016},
      abstract     = {The main goal of this thesis is the generalisation of the
                      notion of “controlled and conditioned invariant subspaces
                      for linear control systems”, introduced by G. Basile and
                      G. Marro in the late sixties. In view of this, we mostly
                      treat input-affine control systems with output, which are
                      defined over a commutative, multivariate, polynomial ring
                      with real or complex ground field. A given variety is
                      called “controlled invariant” for such a system if we
                      can find a feedback law that causes the closed loop system
                      to have this variety as an invariant set, i.e. all
                      trajectories that start on the variety remain there for all
                      time. Several approaches for the feedback law are made,
                      namely polynomial and rational state feedback as well as
                      polynomial and rational output feedback. If it is indeed
                      possible to find an output feedback which makes the variety
                      invariant for the closed loop system, then we call the
                      variety “controlled and conditioned invariant”.The
                      present work begins by giving some mathematical foundation,
                      introducing basic definitions and results of ordinary
                      differential equations, algebraic geometry and the theory of
                      Gröbner bases. We develop computer algebraic methods, for
                      instance for the determination of the intersection of an
                      affine module over a polynomial ring with a free module
                      over a subalgebra of this ring or of a fractional module
                      with a vector space, which help us to decide the properties
                      described above for a given control system and a variety.One
                      crucial object in the context of this thesis is the set of
                      polynomial vector fields which leaves a variety invariant.
                      The elements of this set are called polynomial vector
                      fields on the variety. In fact, this set has a module
                      structure over the considered polynomial ring as a
                      characterisation of the invariance of a variety for a
                      polynomial vector shows, which also gives rise to an
                      algorithmic approach for finding a finite generating
                      system of this module. Furthermore, we investigate the
                      structure of this module: Some submodules will be derived,
                      relations between these submodules as well as conditions on
                      which they already coincide with the whole set of polynomial
                      vector fields on the variety. Moreover, the module of
                      polynomial vector fields on a variety helps us to compare
                      our notion with one made by A. Isidori in the nineties,
                      called distributional invariance, and to characterise the
                      invariance of a variety even for rational vector
                      fields.From this point on, it is easy to find an
                      equivalent condition for a variety being controlled
                      invariant for a polynomial control system with polynomial
                      state feedback, in terms of the given control matrix and the
                      module of polynomial vector fields. We may use techniques
                      from the theory of Gröbner bases to check this condition
                      and, in the affirmative case, to derive the set of all
                      polynomial state feedback making the variety invariant. It
                      turns out that this set is an affine module over the
                      polynomial ring. In view of controlled and conditioned
                      invariance, one has to decide if the intersection of this
                      set with a free module over the subalgebra generated by the
                      individual components of the output of the system is
                      non-empty. The above mentioned algorithms will do this task.
                      Similar considerations can be done to find methods to
                      decide the controlled (and conditioned) invariance of a
                      variety even for rational systems with rational state/output
                      feedback.},
      cin          = {114710 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114710_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-rwth-2016-075965},
      url          = {https://publications.rwth-aachen.de/record/670544},
}