Questions: Root Locus Method and Pole Placement Design

Question 1

A designer wants closed-loop poles with ζ = 0.7 (≤10% overshoot) and settling time under 1 second. After drawing the root locus, the designer finds the locus passes only through regions with ζ < 0.4. The correct next step is:

• A. Continue increasing gain K; the locus will eventually bend into the desired region at sufficiently high gain
• B. Accept the nearest achievable poles since ζ = 0.4 is close enough to the specification
• C. Add a compensator — a lead controller or PD element — to introduce a zero that attracts the locus branches toward the desired region
• D. Reduce gain K toward zero, because K=0 places poles at the open-loop locations which may be closer to the target

Answer: (C)

The root locus is determined by the plant's open-loop poles and zeros, plus any compensator poles and zeros. Once the locus is drawn, K only moves the closed-loop poles along it — you cannot bend the locus by changing K. If the locus misses the target region entirely, no value of K will work. Adding a zero (via a lead compensator or PD controller) attracts locus branches toward it, reshaping the locus so it passes through the desired performance region. This is the fundamental design cycle: specify target region, draw locus, check intersection, add compensation to reshape if needed.

Question 2

A second-order system's dominant poles are at −2 ± 4j. Two additional poles are at −3 ± j. Compared to a system with the same dominant poles but additional poles at −25 ± j, the time-domain response of the first system will:

• A. Be identical, because dominant poles fully determine the transient response in all cases
• B. Show noticeably different behavior because the additional poles at −3 are only 1.5× further left than the dominant poles and contribute significantly to the transient
• C. Settle faster because more poles contribute energy to the decay
• D. Have lower overshoot because the additional poles add damping

Answer: (B)

The dominant-pole approximation is valid only when non-dominant poles are at least 3–5 times further left than the dominant poles. With dominant poles at −2 ± 4j (real part −2), non-dominant poles at −3 are only 1.5× further left — well within the range where their contributions to the transient decay are still significant. By contrast, poles at −25 are more than 12× further left, decaying so rapidly that they contribute negligibly. This check — verifying the dominant-pole assumption by simulation — is the closing step of the design cycle.

Question 3

For a second-order closed-loop system, percent overshoot is determined solely by the damping ratio ζ, which corresponds to the angle of the closed-loop poles measured from the negative real axis.

• T. True
• F. False

Answer: True

This is the geometric translation of time-domain specs into the s-plane that makes root locus design possible. The damping ratio ζ = cos(θ) where θ is the angle from the negative real axis (0° = pure real, 90° = purely imaginary). Overshoot = exp(−πζ/√(1−ζ²))×100%. A spec of ≤10% overshoot maps to ζ ≥ 0.59, which means poles must lie within a cone defined by θ ≤ 54° from the negative real axis. This allows you to draw a forbidden angular region in the s-plane and check whether the root locus avoids it.

Question 4

If the root locus does not pass through the desired performance region in the s-plane, continuously increasing the controller gain K will eventually move the closed-loop poles into that region.

• T. True
• F. False

Answer: False

The root locus is a fixed geometric path determined entirely by the positions of the open-loop poles and zeros (and any compensator poles/zeros). Gain K only moves the closed-loop poles along that path — it cannot bend or redirect the locus. If the locus passes through a region of high overshoot and never enters the desired ζ ≥ 0.7 cone, no gain value will place the poles there. The solution is to reshape the locus by modifying the open-loop transfer function — adding a zero pulls the locus toward it, adding a pole pushes the locus away from it.

Question 5

How do time-domain performance specifications (rise time, percent overshoot, settling time) translate into a target region in the s-plane, and what determines whether proportional gain alone can achieve those specifications?

Model answer: Each specification maps to a geometric constraint on the closed-loop pole locations. Overshoot determines a minimum damping ratio ζ, which maps to an angular cone from the negative real axis (ζ = cos θ). Settling time determines a minimum real part σ (settling time ≈ 4/σ for a 2% criterion). Rise time is inversely proportional to the imaginary part ωd. Together these constraints define a feasible region in the s-plane. Proportional gain can achieve the specifications if and only if the root locus passes through that feasible region — because K only moves poles along the existing locus. If the locus misses the region, a compensator must be added to reshape the locus first.

This translation from time-domain specs to s-plane geometry is the core skill of root locus design. The root locus is the tool that shows which poles are achievable with a given plant and controller structure. Understanding that K moves poles along a fixed path — and that the path itself must be reshaped when specs aren't achievable — is what separates a designer who understands root locus from one who just cranks through rules.