5- Stability & Root Locus

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