# Control Systems/System Representations

## Contents

## System Representations[edit]

This is a table of times when it is appropriate to use each different type of system representation:

Properties | State-Space Equations |
Transfer Function |
Transfer Matrix |
---|---|---|---|

Linear, Distributed | no | no | no |

Linear, Lumped | yes | no | no |

Linear, Time-Invariant, Distributed | no | yes | no |

Linear, Time-Invariant, Lumped | yes | yes | yes |

## General Description[edit]

These are the general external system descriptions. *y* is the system output, *h* is the system response characteristic, and *x* is the system input. In the time-variant cases, the general description is also known as the convolution description.

General Description | |
---|---|

Time-Invariant, Non-causal | |

Time-Invariant, Causal | |

Time-Variant, Non-Causal | |

Time-Variant, Causal |

## State-Space Equations[edit]

These are the state-space representations for a system. *y* is the system output, *x* is the internal system state, and *u* is the system input. The matrices A, B, C, and D are coefficient matrices.

[Analog State Equations]

State-Space Equations | |
---|---|

Time-Invariant | |

Time-Variant |

These are the digital versions of the equations listed above. All the variables have the same meanings, except that the systems are digital.

[Digital State Equations]

State-Space Equations | |
---|---|

Time-Invariant | |

Time-Variant |

## Transfer Functions[edit]

These are the transfer function descriptions, obtained by using the Laplace Transform or the Z-Transform on the general system descriptions listed above. *Y* is the system output, *H* is the system transfer function, and *X* is the system input.

[Analog Transfer Function]

Transfer Function | |
---|---|

[Digital Transfer Function]

Transfer Function | |
---|---|

## Transfer Matrix[edit]

This is the transfer matrix system description. This representation can be obtained by taking the Laplace or Z transforms of the state-space equations. In the SISO case, these equations reduce to the transfer function representations listed above. In the MIMO case, * Y* is the vector of system outputs,

*is the vector of system inputs, and*

**X***is the transfer matrix that relates each input X to each output Y.*

**H**

[Analog Transfer Matrix]

Transfer Matrix | |
---|---|

[Digital Transfer Matrix]

Transfer Matrix | |
---|---|