  In the context of data-driven control of nonlinear systems, many approaches
lack of rigorous guarantees, call for nonconvex optimization, or require
knowledge of a function basis containing the system dynamics. To tackle these
drawbacks, we establish a polynomial representation of nonlinear functions
based on a polynomial sector by Taylor's theorem and a set-membership for
Taylor polynomials. The latter is obtained from finite noisy samples. By
incorporating the measurement noise, the error of polynomial approximation, and
potentially given prior knowledge on the structure of the system dynamics, we
achieve computationally tractable conditions by sum of squares relaxation to
verify dissipativity and incremental dissipativity of nonlinear dynamical
systems with rigorous guarantees. The framework is extended by combining
multiple Taylor polynomial approximations which yields a less conservative
piecewise polynomial system representation. The proposed approach is applied
for numerical and experimental examples. There it is compared to a
least-squares-error model including knowledge from first principle.
