# Control Systems/System Representations

## System Representations

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

+-------------------------------------+--------------+-----------+-----------+
| Properties                          | State-Space\ | Transfer\ | Transfer\ |
|                                     | Equations    | Function  | 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

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

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.

  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.

  State-Space Equations
  -----------------------
  Time-Invariant
  Time-Variant

## Transfer Functions

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.

  Transfer Function
  -------------------
  $Y(s) = H(s)X(s)$

  Transfer Function
  -------------------
  $Y(z) = H(z)X(z)$

## Transfer Matrix

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,***X***is the vector of system inputs,
and***H*** is the transfer matrix that relates each input X to each
output Y.

  Transfer Matrix
  ----------------------------------------------
  $\mathbf{Y}(s) = \mathbf{H}(s)\mathbf{X}(s)$

  Transfer Matrix
  ----------------------------------------------
  $\mathbf{Y}(z) = \mathbf{H}(z)\mathbf{X}(z)$

