Stability & Root Locus

Stability• The stability of a system depends on the locations of the poles and zeros

within the system.

• A continuous system is stable if all poles are on the left half of the complex plane.

• A discrete system is stable if all poles are within a unit circle centered at the origin of the complex plane.

• Additionally, both types of systems are stable if they do not contain any poles.

• Additionally, both types of systems are unstable if they contain more than one pole at the origin.

Stability• In terms of the dynamic response, a pole is stable if

the response of the pole decays over time.

• If the response becomes larger over time, the pole is unstable.

• If the response remains unchanged over time, the pole is marginally stable.

• To describe a system as stable, all the closed-loop poles of a system must be stable.

Stability

• Use the CD Pole-Zero Map VI to obtain all the poles and zeros of a system and plot their corresponding locations in the complex plane.

• Use the CD Stability VI to determine if a system is stable, unstable, or marginally stable.

Using the Root Locus Method• The root locus method provides the closed-loop pole

positions for all possible changes in the loop gain K.

• Root locus plots provide an important indication of what gain ranges you can use to keep the closed-loop system stable.

• The root locus is a plot on the real-imaginary axis showing the values of s that correspond to pole locations for all gains, starting at the open-loop poles, K = 0 and ending at K = ∞.

Using the Root Locus Method

• Use the CD Root Locus VI to compute and draw root locus plots for continuous and discrete SISO models of any form.

• You also can use this VI to synthesize a controller.

Root Locus Method Example 1

Root Locus Method Example 1