Hello Control Friends,

PURPOSE OF NOTE
The purpose of this note is to remind you about a fundamental sinusoidal
analysis result for single-input single-output (SISO) linear time invariant
(LTI) systems.

METHOD OF THE TRANSFER FUNCTION: SINUSOIDAL ANALYSIS FOR SISO LTI SYSTEMS
Let H(s) denote the transfer function of a SISO LTI stable system. Suppose
that the input
              u(t) = A sin( wt + theta )

is applied to the system H. Given, this, it follows that the steady state
output yss is given by

              yss = A |H(jw)| sin( wt + theta + Angle(H(jw)) )
where

              H(jw) = |H(jw)| e^{j Angle(H(jw)) }.

A similar expression holds when a "cosine input" is applied; i.e. if the input

              u(t) = A cos( wt + theta )

is applied to H, then the steady state output yss is given by

              yss = A |H(jw)| cos( wt + theta + Angle(H(jw)) ).


COMMENT
The above result is fundamental for the study of LTI systems.
The idea extends to multiple-input multiple-output (MIMO) LTI systems.
This result is arguably one of the most important ideas which are taught to
undergraduate engineering students.

EXAMPLE: SINUSOIDAL ANALYSIS FOR A SIMPLE SYSTEM
If the input 
             u(t) = 1 + cos t

is applied to the LTI system 

              H(s) = 1/(s+1),

 then the steady state output yss is given by:

           yss = H(0) + |H(j1)| cos( t  + Angle(H(jw)) )

               =  1   +  0.7071 cos( t - pi/4 )

Hope this note was helpful.

Thank you.
AAR