### Abstract

Let B = {w 2 Lloc 1 (R,Rq) | R( d/dt )w = 0}. It is well-know that B is controllable if and only if R(λ) has the same rank for all complex λ. We want to
re-examen the proof of this fundamental result. Denote by B1 the behavior B
intersected with set of smooth functions. For B1 the proof is easy and one would
expect that the fact that B1 is dense in B would provide a quick and easy proof for the controllability test for B. This, unfortunately, is not true. For the smooth case the proof uses a differential transformation of the behavior. This transformation corresponds to a right unimodular transformation V(E) of R(E) yielding its Smith form. Generally, such a transformation cannot be extended to B since B contains non-smooth trajectories. In this contribution we argue that, despite this fact, the unimodular matrix V(E) defines an injection from B into the behavior defined by the Smith form. With this observation, that is interesting in its own right, the controllability test can be proved relatively easy.

Original language | Undefined |
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Pages | 1204-1208 |

Number of pages | 5 |

Publication status | Published - 2005 |

Event | 16th IFAC World Congress 2005 - Prague, Czech Republic Duration: 3 Jul 2005 → 8 Jul 2005 Conference number: 16 http://www.utia.cas.cz/news/608 |

### Conference

Conference | 16th IFAC World Congress 2005 |
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Country | Czech Republic |

City | Prague |

Period | 3/07/05 → 8/07/05 |

Internet address |

### Keywords

- IR-68681
- EWI-16603

## Cite this

Polderman, J. W., & Piztek, P. (Ed.) (2005).

*The controllability test for behaviors revisited*. 1204-1208. Paper presented at 16th IFAC World Congress 2005, Prague, Czech Republic.