8. Stability

Stability plays a fundamental role in control theory, because only stable systems are useful. Basically, when a stable system is disturbed, it returns to its equilibrium state, whereas an unstable system amplifies the disturbance. Most closed-loop systems become unstable when we seek to increase gain to achieve high performance. Thus, stability sets an upper bound on looping. Unstable systems are difficult to control, and one possible strategy is pole-zero simplification.

Consider the linear system : $x^{\prime }\left(t\right)=\mathit{Ax}\left(t\right),x\left(0\right)=x_0$