## Continuous-Time System Models

The continuous-time system models are representational schemes for analog filters. Many of the discrete-time system models described earlier are also appropriate for the representation of continuous-time systems:

State-space form

Partial fraction expansion

Transfer function

Zero-pole-gain form

It is possible to represent any system of linear time-invariant differential equations as a
set of first-order differential equations. In matrix or *state-space* form,
you can express the equations as

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

where *u* is a vector of *nu* inputs,
*x* is an *nx*-element state vector, and
*y* is a vector of *ny* outputs. In the MATLAB^{®} environment, `A`

, `B`

, `C`

,
and `D`

are stored in separate rectangular arrays.

An equivalent representation of the state-space system is the Laplace transform transfer function description

$$Y(s)=H(s)U(s)$$

where

$$H(s)=C{(sI-A)}^{-1}B+D$$

For single-input, single-output systems, this form is given by

$$H(s)=\frac{b(s)}{a(s)}=\frac{b(1){s}^{n}+b(2){s}^{n-1}+\dots +b(n+1)}{a(1){s}^{m}+a(2){s}^{m-1}+\dots +a(m+1)}$$

Given the coefficients of a Laplace transform transfer function, `residue`

determines
the partial fraction expansion of the system. See the description
of `residue`

for details.

The factored zero-pole-gain form is

$$H(s)=\frac{z(s)}{p(s)}=k\frac{(s-z(1))(s-z(2))\dots (s-z(n))}{(s-p(1))(s-p(2))\dots (s-p(m))}$$

As in the discrete-time case, the MATLAB environment stores
polynomial coefficients in row vectors in descending powers of *s*. It stores polynomial roots, or zeros
and poles, in column vectors.