# discretization of continuous systems

Consider a continuous-time system with the following state space representation

$$P:\mathit{\hspace{1em}\hspace{1em}}\{\begin{array}{ccc}\hfill \dot{x}(t)\hfill & \hfill =\hfill & Ax(t)+Bu(t),\hfill \\ \hfill y(t)\hfill & \hfill =\hfill & Cx(t)+Du(t),\hfill \end{array}$$ | (1) |

where $x(t)\in {\mathbb{R}}^{n}$, $u(t)\in {\mathbb{R}}^{r}$ and $y(t)\in {\mathbb{R}}^{m}$ are the state vector, input vector and output vector of the system, respectively; $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times r}$, $C\in {\mathbb{R}}^{m\times n}$ and $D\in {\mathbb{R}}^{m\times r}$ are the constant real or complex matrices.

Suppose that the sampling interval is $\tau $. By using the step invariance transform or the zero-order hold (ZOH), i.e., $$, discretizing the system in (1) gives a discrete-time model,

$${P}_{\tau}:\mathit{\hspace{1em}\hspace{1em}}\{\begin{array}{ccc}\hfill x(k\tau +\tau )\hfill & \hfill =\hfill & {G}_{\tau}x(k\tau )+{F}_{\tau}u(k\tau ),\hfill \\ \hfill y(k\tau )\hfill & \hfill =\hfill & Cx(k\tau )+Du(k\tau ),k=0,1,2,\mathrm{\cdots}\hfill \end{array}$$ | (2) |

where $x(k\tau )=x(t)|{}_{t=k\tau}$, $y(k\tau )=y(t)|{}_{t=k\tau}$, and

$${G}_{\tau}:={\mathrm{e}}^{A\tau},{F}_{\tau}:={\int}_{0}^{\tau}{\mathrm{e}}^{At}dtB.$$ | (3) |

Title | discretization of continuous systems |
---|---|

Canonical name | DiscretizationOfContinuousSystems |

Date of creation | 2013-03-22 15:50:45 |

Last modified on | 2013-03-22 15:50:45 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Topic |

Classification | msc 93C55 |

Synonym | Transform |