# Stability and Relation to Poles Placements

Chapter

## Abstract

The stability of the system (or the lack thereof) is extremely important. In almost all cases, systems need to be stable in order to be useful. In the frequency domain the transfer function of the system describes how the system responds in the frequency domain, and is able to reveal if the system is stable or not. This can be done by examining the poles of the transfer function. Simply put of the real part of the pole lies in the left side of the complex plane then the system is stable; else it is unstable. So the process of identifying stability starts with identifying all poles, and ensuring that they all lie in the left half of the complex plane. In the time domain a system is stable if the impulse response does not grow and dies out. Notice the special case of pole lying right at the center of the plane, such as for example $$H(s) = \frac {1}{s}$$ and in this case the system response does not die out, but at the same time does not grow; we simply get the impulse response h(t) = 1!