Stability

The stability of a cone.

Stability in the s-plane.

Tacoma Narrows Bridge (a) as oscillation begins and (b) at catastrophic failure.

Tacoma Narrows Bridge (a) as oscillation begins and (b) at catastrophic failure.

The Routh-Hurwitz Stability Criterion

Case 1: No element in the first column is zero.

Example 5.1 Second-order system

Example 5.2 Thrid-order system

Case 2: Zero in the first column

ExampIe 5.3 Unstable system

Case 3: Zeros in one row

Example 5.4 Robot control

Welding head position control.

Relative Stability of Feedback Control Systems

Root locations in the s-plane.

(a) Turning control system for a two-track vehicle. (b) Block diagram.

The stable region.

Closed-loop control system with T(s) = Y(s)/R(s) = 1/(s3 + s2+ 2s+24).

Routh array for the closed-loop control system with T(s) = Y(s)/R(s) = 1/(s3 s2 2s 24)

The closed-loop disk drive head system with an optional velocity feedback.

Equivalent system with the velocity feedback switch closed.

Response of the system with velocity feedback. (a) MATLAB script. (b) Response with Ka = 100 and K1 = 0.05.