# controllability of LTI systems

Consider the linear time invariant (LTI) system given by:

$$\dot{x}=Ax+Bu$$ |

where $A$ is an $n\times n$ matrix, $B$ is an $n\times m$ matrix, $u$ is an $m\times 1$ vector - called the control or input vector, $x$ is an $n\times 1$ vector - called the state vector, and $\dot{x}$ denotes the time derivative of $x$.

Definition Of Controllability Matrix For LTI Systems: The controllability matrix of the above LTI system is defined by the pair $(A,B)$ as follows:

$$C(A,B)=[B,AB,{A}^{2}B,{A}^{3}B,\mathrm{\dots},{A}^{n-1}B]$$ |

Test for Controllability of LTI Systems: The above LTI system $(A,B)$ is controllable if and only if the controllability matrix $C(A,B)$ has rank $n$;
i.e. has $n$ linearly independent^{} columns.

Title | controllability of LTI systems |
---|---|

Canonical name | ControllabilityOfLTISystems |

Date of creation | 2013-03-22 14:32:50 |

Last modified on | 2013-03-22 14:32:50 |

Owner | GeraW (6138) |

Last modified by | GeraW (6138) |

Numerical id | 5 |

Author | GeraW (6138) |

Entry type | Definition |

Classification | msc 93B05 |

Defines | controllability matrix |