Obstructions to Genericity in Study of Parametric Problems in Control Theory

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as well as more complicated ones, which act trivially on the parameters. Such a system can be identified algebraically with a certain left module over a non-commutative algebra, where the operators commute with the parameters. We develop, implement and use in practice the algorithm for revealing all the expressions in parameters, for which e.g. homological properties of a system differ from the generic properties. We use Groebner bases and Groebner basics in rings of solvable type as main tools. In particular, we demonstrate an optimized algorithm for computing the left inverse of a matrix over a ring of solvable type. We illustrate the article with interesting examples. In particular, we provide a complete solution to the "two pendula, mounted on a cart" problem from the classical book of Polderman and Willems, including the case, where the friction at the joints is essential . To the best of our knowledge, the latter example has not been solved before in a complete way.Comment: 20 page

Nonparametric likelihood based estimation of linear filters for point processes

We consider models for multivariate point processes where the intensity is given nonparametrically in terms of functions in a reproducing kernel Hilbert space. The likelihood function involves a time integral and is consequently not given in terms of a finite number of kernel evaluations. The main result is a representation of the gradient of the log-likelihood, which we use to derive computable approximations of the log-likelihood and the gradient by time discretization. These approximations are then used to minimize the approximate penalized log-likelihood. For time and memory efficiency the implementation relies crucially on the use of sparse matrices. As an illustration we consider neuron network modeling, and we use this example to investigate how the computational costs of the approximations depend on the resolution of the time discretization. The implementation is available in the R package ppstat.Comment: 10 pages, 3 figure