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

$\begin{array}{l} \dot{x} = A x + B u \\ y = C x + D u \end{array}$

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

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

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.