Stability analysis of time domain systems. S-Plane: Poles and Zeros . A linear system can be represented by the following transfer function: Zeros: z i (the roots of the numerator) Poles: p i (the roots of the denominator or of the system or characterestic equation).
i) a > 0, y(∞) = 0 → stable
ii) a = 0, y(∞) = 1 → neutral (marginally stable)
iii) a < 0, y(∞) = ∞ → unstable
A system is stable if ALL the closed-loop poles are in the LEFT HALF of the s-plane and have negative real parts, i.e. all the roots of the characteristic equation are in the left-hand s-plane.
Exercise: Use the R-H criterion to determine if the closed-loop system described by the following characteristic equation is stable:
Exercise: Consider closed-loop system shown below. Using the R-H criterion, determine the range of K over which the system is stable.