Multidimensional realisation theory and polynomial system solving
| Autor: | Kim Batselier, Bart De Moor, Philippe Dreesen |
|---|---|
| Přispěvatelé: | Electricity |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
0209 industrial biotechnology
Polynomial Computer science System of polynomial equations Systems and Control (eess.SY) 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Overdetermined system Kernel (linear algebra) Matrix (mathematics) 020901 industrial engineering & automation ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION FOS: Electrical engineering electronic engineering information engineering FOS: Mathematics Observability 0101 mathematics Mathematics - Optimization and Control Eigendecomposition of a matrix SISTA polynomial system solving Univariate Multidimensional systems difference equations 16. Peace & justice Computer Science Applications Algebra Optimization and Control (math.OC) Control and Systems Engineering Linear algebra DYSCO Computer Science - Systems and Control eigenvalue problem Realization (systems) realisation theory |
| Popis: | Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper we relate the realization theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyze and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realization theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems. Comment: 21 pages, accepted for publication in International Journal of Control (in press), 2018 |
| Databáze: | OpenAIRE |
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