Signals and Systems

8.1 EFFECT OF LOCATION OF POLES ON STABILITY

Location of poles has a direct effect on stability. The entire s-plane is taken into account and it is divided as follows:

1. Left half-plane (LHP)
2. jω-axis
3. Right half-plane (RHP)

Let us consider the following cases:

1. LHP Poles
   1. On real axis and simple
   2. On real axis and multiple
   3. Complex conjugates and simple

      The system is stable for all of the above conditions.

2. jω-axis
   1. On jω-axis and simple
   2. On jω-axis and multiple
   3. At origin
   4. At origin and multiple

      In this case, if the pole has zero real part and not repeated, the system is called stable.

3. RHP Poles
   1. On real axis and simple
   2. On real axis and multiple
   3. Complex conjugates and simple

From definition of stability, the impulse response must tend to zero as ...