Questions: Controller Design via Root Locus

Question 1

A lead compensator has been designed so that its zero and pole satisfy the angle condition at the desired closed-loop pole location s*. What additional step is required to complete the design?

• A. The design is complete — satisfying the angle condition places the closed-loop pole at s* for all values of gain K
• B. Gain K must be set separately using the magnitude condition to place the closed-loop pole exactly at s*
• C. A lag compensator must be added to cancel the phase contribution of the lead compensator
• D. The plant transfer function must be re-linearized around the new operating point defined by s*

Answer: (B)

Satisfying the angle condition means s* lies on the root locus — it is a *possible* closed-loop pole location. But the root locus contains infinitely many points, one for each value of K. To place the closed-loop pole exactly at s*, gain K must be set using the magnitude condition: K = 1/|G(s*)H(s*)|. Satisfying the angle condition and setting K are two separate steps; confusing them is the most common design error.

Question 2

A control engineer needs to improve a system's steady-state tracking accuracy (reduce position error) without significantly changing its transient response speed or overshoot. Which compensator is appropriate?

• A. Lead compensator — adds phase near the desired pole to increase response speed
• B. Lag compensator — adds low-frequency gain to reduce steady-state error without substantially reshaping the locus near the desired poles
• C. Both lead and lag simultaneously to address both transient and steady-state performance
• D. Neither — steady-state accuracy can only be improved by increasing plant gain K directly

Answer: (B)

Lead and lag compensators serve fundamentally different purposes. A lead compensator (zero closer to imaginary axis than pole) adds phase near the desired closed-loop poles, improving speed and damping. A lag compensator (pole closer to origin than zero) places its pole and zero very close together near the origin, contributing nearly zero phase near the desired poles but adding significant low-frequency gain — exactly what is needed to reduce steady-state error. Using a lead compensator for steady-state accuracy (or a lag compensator for speed) conflates two distinct design objectives.

Question 3

A lead compensator improves both transient response speed and steady-state accuracy, making it the preferred choice over a lag compensator in most control design scenarios.

• T. True
• F. False

Answer: False

Lead and lag compensators address different objectives and are not substitutes. A lead compensator adds phase at the desired closed-loop pole location, improving speed and damping — transient performance. It does not significantly improve steady-state accuracy. A lag compensator adds low-frequency gain (improving steady-state error) without substantially adding or removing phase near the desired poles. Claiming a lead compensator improves steady-state accuracy conflates the two and ignores the distinct mechanics of each compensator type.

Question 4

The dominant pole assumption is considered valid when non-dominant closed-loop poles are at least 5 times further to the left in the s-plane than the dominant poles.

• T. True
• F. False

Answer: True

The 5× rule-of-thumb ensures that non-dominant poles produce transient terms that decay roughly 5× faster than the dominant pole terms, making their contribution negligible in the step response within a fraction of the dominant settling time. If non-dominant poles are closer to the imaginary axis — especially if they nearly cancel with closed-loop zeros — their contribution can significantly alter the response and the second-order approximation breaks down. Always verify this condition and simulate the full response.

Question 5

Why is the angle condition central to root locus controller design, and what does it mean geometrically?

Model answer: The angle condition states that a point s* lies on the root locus if and only if the total phase angle of the open-loop transfer function evaluated at s* equals ±180° (an odd multiple). Geometrically, you draw vectors from every open-loop pole and zero to s*, measure each vector's angle, and sum them (subtracting pole angles, adding zero angles). If the sum equals 180°, s* is on the locus. If it does not, the angle deficiency tells you exactly how many degrees of phase a compensator must contribute. The compensator's zero and pole are then positioned so that their angular contributions make up the deficiency — reshaping the locus to pass through the desired closed-loop pole location.

This geometric interpretation is what makes root locus design tractable: instead of solving for compensator parameters algebraically, you graphically determine the phase contribution needed and use simple angle geometry (often bisecting angles or using circular arcs) to place the compensator elements.