e to the power of infinity is infinity (∞).
e is raised to the negative infinity then it becomes a very small number and hence
tends to zero.
According to the definition the examples of Systems become UNSTABLE
S1 – for any input other than cos(t) or sin(t) – the output is bounded like for input 1 or input as any
cos for any angular frequency other than 1
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Input is a sinusoid whose angular frequency is equal to the magnitude of the poles on the
imaginary axis – the output is unbounded(magnitude of the poles – 1)
So some call this system – MARGINALLY STABLE or CRITICALLY STABLE
S2 – Pole at the origin – non-repeating – step input – o/p is unbounded
For all other sinusoidal inputs – o/p is bounded
For Repeating Poles
BIBO Stability
Sufficient Condition for BIBO Stability – above
Poles are connected to System Stability
Impact of zeroes on System Response
Dominant Poles – Poles that are closer to the imaginary axis and these poles have a long lingering
effect on y(t) than which are further from the imaginary axis
Example 2 – zeroes are present –
-
Affects the residues
Affects the steady state value
Relatively the dominance of -1 pole reduced
Zeroes affect the residues of partial fractions(coefficients)
Zero in the RHP – the step response will start of in direction and reverse its direction and go to the
next direction
Pink curve is approximation – used in design(tools cannot be used for the actual step response)