Jacquelien Scherpen

Delft University of Technology

**On Minimality and Similarity Invariance and its Relation with
Input-Output Notions for Nonlinear Systems. **

For linear systems the understanding of the relations between input-output systems, Hankel operators and minimal state-space realizations is well-developed and has turned out to be very important over the last decades. However, for nonlinear systems the relations are far less clear and well-developed. A set of sufficient conditions in terms of controllability and observability functions will be given, under which a given state space realization of a formal power series is minimal. Specifically, it will be shown that positivity of these functions, plus a few technical conditions, implies minimality. In doing so, connections are established between Hamilton-Jacobi type optimal control theory and the well known necessary and sufficient conditions for minimality in terms of Kalman type rank conditions. Furthermore, a definition of a system Hankel operator is developed for causal L2-stable input-output systems. If a generating series representation of the input-output system is given then an explicit representation of the corresponding Hankel operator is possible. If, in addition, an affine state space model is available with certain stability properties then a unique factorization of the Hankel operator can be constructed with direct connections to well known and new nonlinear Gramian extensions. Finally, preliminary results on singular value type of considerations in combination with the Hankel operator will be briefly discussed.

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