## 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

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

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

`$Y\left(s\right)=H\left(s\right)U\left(s\right)$`

where

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

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

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

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

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

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.