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Non-Differentially Flat Systems
Published in Hebertt Sira-Ramírez, Sunil K. Agrawal, Differentially Flat Systems, 2018
Hebertt Sira-Ramírez, Sunil K. Agrawal
An interesting and rather straightforward technique to solve feedback stabilization problems for linear time-varying systems consists in direct instantaneous eigenvalue placement, at constant locations of the complex plane, for the closed loop matrix of the linearized system. This is to be achieved by means of linear time-varying incremental state feedback. It should be pointed out, that this technique is valid as long as the coefficients of the linear system are known to belong to a Hardy field, one in which the largest comparability class is constituted by exponential functions. For details, the reader is referred to the work of Fliess and Rudolph [32] where this technique is applied in the construction of observers for linear time-varying systems arising from linearization of flat systems around nominal trajectories.
A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages
Published in Dynamical Systems, 2023
We now define Hardy field functions. Let B denote the set of germs at infinity of real valued functions defined on a half-line . Then, is a ring. A sub-field of B that is closed under differentiation is called a Hardy field. An example of a Hardy field is the field of logarithmico-exponential functions. These are the functions defined on some half line of by a finite combination of the operations and composition of functions acting on a real variable t and real constants. The set contains functions such as the polynomials , for all real c>0, and . A function is a Hardy field function if , for some Hardy field containing . The assumption that is a necessary in order for us to make use of Theorems 3.3 and 3.4 later on. We will also assume for convenience that our Hardy fields are translation invariant, i.e. if , then for any . We refer the reader to [10, 11] and the references therein for more information about Hardy fields.