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 $\mathbf {A} \in M_{n}(\Re )$ and a $\mathbf {B} \in M_{n\times m}(\Re )$ the following are equivalent:

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

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