A linear time-invariant continuous (LTIC) system is often represented as linear differential equations like the following:
where all the coefficients
Using operator notaiton
As the system is linear, we can represent it as a sum of zero-input response and zero-state response (decomposition property). Therefore,
In other word, we can decompose the total response
The Zero-input Response
The zero-input response
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
In a LTIC system, the zero-state response can be modeled as the convolution of the unit impulse response
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