A linear time-invariant continuous (LTIC) system is often represented as linear differential equations like the following:

where all the coefficients and are constants and .

Using operator notaiton to represent and simplify with the summation notation, we can express this equation as

As the system is linear, we can represent it as a sum of zero-input response and zero-state response (decomposition property). Therefore,

total response = zero-input response + zero-state response

In other word, we can decompose the total response into the zero-input response (homogeneous solution) and zero-state response (particular solution) .

The Zero-input Response

The zero-input response is the solution of the previous equation where the input :

This is a homogeneous differential equation, and we can solve it with characteristic equations (see that note for more information).

The Zero-State Response

The zero-state response is a particular solution to our differential equation:

In a LTIC system, the zero-state response can be modeled as the convolution of the unit impulse response and the input :

It is usually easier to compute the zero-state response in the frequency domain using transforms such as the Laplace transform or Fourier transform, because by the convolution theorem, convolution in time domain becomes multiplication in the frequency domain.

Stability

see system stability

Reference

  • Linear Systems and Signals, 3rd Edition, Chapter 2