Deep learning

In control theory and in particular when studying the controllability of a linear time-invariant system in state space form, the Hautus lemma, named after Malo Hautus, can prove to be a powerful tool. This result appeared first in and. Today it can be found in most textbooks on control theory.

The main result

The Hautus lemma says that given a square matrix $\mathbf{A}\in M_n(\mathrm{\Re })$ and a $\mathbf{B}\in M_{n\times m}(\mathrm{\Re })$ the following are equivalent:

1. The pair $(\mathbf{A},\mathbf{B})$ is controllable
2. For all $\lambda \in \mathbb{ℂ}$ it holds that $\mathrm{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n$
3. For all $\lambda \in \mathbb{ℂ}$ that are eigenvalues of $\mathbf{A}$ it holds that $\mathrm{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n$

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534