A behavioral approach to the pole structure of one-dimensional and multidimensional linear systems.

*(English)*Zbl 0979.93019The aim of this paper is to establish a cohesive theory of poles for nD MIMO linear systems in the new and topical behavioral framework. At the same time, it gives a new perspective for the classical results in the theory of poles of 1D systems.

The basic tool of this theory is the duality theory between finitely generated modules over a polynomial ring and nD differential/difference behaviors with constant coefficients.

The authors provide some relevant results concerning autonomous systems, minimal left annihilators and generalized factor primeness, as well as the characterization of controllable and observable behaviors and minimal realizations.

The important concept of exponential trajectory is studied, as well as the concept of characteristic variety of a behavior, which is a geometric object that contains the essential information on the exponential trajectories. On this basis, a new definition of a pole of an nD system is given and the decomposition of the set of poles into controllable and uncontrollable poles is provided. The former are poles of the controllable part of the system and they are associated with the transfer matrix and the latter play the role of input-decoupling zeros. Some characterizations of these types of poles are derived, as well as the dimension of the characterization variety (pole variety, controllable and uncontrollable pole varieties).

In the case of a behavior which is a latent variable representation of some other behavior, the set of poles is decomposed into observable and unobservable ones. This partitioning is combined with the partitioning from the point of view of controllability and the complete classification of poles is derived.

Finally, the polar decomposition is considered, which expresses the zero-input behavior of a system as the finite sum of subbehaviors associated with the distant poles of the system and a refinement of the controllable-autonomous decomposition is derived for the system behavior. Using the integral representation theorem, it is shown that any zero-input trajectory can be written as a sum of integrals of polynomial exponential trajectories.

The basic tool of this theory is the duality theory between finitely generated modules over a polynomial ring and nD differential/difference behaviors with constant coefficients.

The authors provide some relevant results concerning autonomous systems, minimal left annihilators and generalized factor primeness, as well as the characterization of controllable and observable behaviors and minimal realizations.

The important concept of exponential trajectory is studied, as well as the concept of characteristic variety of a behavior, which is a geometric object that contains the essential information on the exponential trajectories. On this basis, a new definition of a pole of an nD system is given and the decomposition of the set of poles into controllable and uncontrollable poles is provided. The former are poles of the controllable part of the system and they are associated with the transfer matrix and the latter play the role of input-decoupling zeros. Some characterizations of these types of poles are derived, as well as the dimension of the characterization variety (pole variety, controllable and uncontrollable pole varieties).

In the case of a behavior which is a latent variable representation of some other behavior, the set of poles is decomposed into observable and unobservable ones. This partitioning is combined with the partitioning from the point of view of controllability and the complete classification of poles is derived.

Finally, the polar decomposition is considered, which expresses the zero-input behavior of a system as the finite sum of subbehaviors associated with the distant poles of the system and a refinement of the controllable-autonomous decomposition is derived for the system behavior. Using the integral representation theorem, it is shown that any zero-input trajectory can be written as a sum of integrals of polynomial exponential trajectories.

Reviewer: Valeriu Prepeliţă (Bucureşti)

##### MSC:

93B25 | Algebraic methods |

93C35 | Multivariable systems, multidimensional control systems |

35B37 | PDE in connection with control problems (MSC2000) |

93B55 | Pole and zero placement problems |

93C05 | Linear systems in control theory |

93B20 | Minimal systems representations |

13C12 | Torsion modules and ideals in commutative rings |