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

x˙=Ax+Buy=Cx+Du

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(sIA)1B+D

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

H(s)=b(s)a(s)=b(1)sn+b(2)sn1++b(n+1)a(1)sm+a(2)sm1++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)=z(s)p(s)=k(sz(1))(sz(2))(sz(n))(sp(1))(sp(2))(sp(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.