Root Locus Gain Design

Core Idea

Root locus gain design selects the controller gain K so that the closed-loop poles lie at desired locations in the s-plane, meeting time-domain performance specifications such as percent overshoot, settling time, and rise time. The design procedure maps performance specs into s-plane regions: a damping ratio ζ requirement defines lines of constant angle θ = cos⁻¹(ζ) from the negative real axis, a natural frequency ωn requirement defines a circle of radius ωn centered at the origin, and a settling time requirement defines a vertical boundary at σ = −4/t_s (for 2% criterion). The designer identifies where the root locus crosses the desired damping line or enters the acceptable region, then computes the corresponding K using the magnitude condition |G(s)H(s)| = 1/K at that point. When the locus does not pass through the desired region, a compensator (adding poles or zeros) must reshape the locus before gain selection — pure gain adjustment alone cannot place poles arbitrarily. The dominant pole approximation assumes that the closed-loop response is primarily governed by the poles nearest the imaginary axis, provided other poles are at least five times farther to the left.

How It's Best Learned

Given a plant transfer function with specified overshoot and settling time requirements, convert the specs to a target region in the s-plane, sketch the root locus, and graphically determine the gain K at the intersection point. Verify by computing the closed-loop step response and checking whether higher-order poles violate the dominant-pole assumption. Repeat for systems where the locus does not intersect the desired region to motivate compensator design.

Common Misconceptions

• Meeting the damping ratio specification by placing dominant poles on the correct ζ line does not guarantee the predicted overshoot if there are nearby zeros or non-dominant poles that are not sufficiently far to the left — the dominant-pole approximation has limits.
• The gain K found from the root locus is the open-loop gain parameter, not the closed-loop DC gain — the steady-state value of the step response depends on the closed-loop transfer function and may require separate steady-state error analysis.
• Increasing K to speed up the response (higher ωn) eventually drives branches into the right half-plane for most systems, so there is a fundamental tradeoff between speed and stability that gain alone cannot resolve.

Explainer

From your prerequisite work on root locus construction rules, you can draw the paths that closed-loop poles trace in the s-plane as the gain K varies from 0 to ∞. At K = 0, the closed-loop poles sit at the open-loop poles; as K → ∞, they migrate toward the open-loop zeros (or to infinity along asymptotes). The root locus tells you *where* the poles can go. Root locus gain design answers the follow-up question: where *should* they go, and what value of K puts them there?

The design procedure starts by translating time-domain performance specifications into geometric regions in the s-plane. A specified percent overshoot maps to a minimum damping ratio ζ via OS% = 100·e^(−πζ/√(1−ζ²)), which in turn defines a pair of lines radiating from the origin at angle θ = cos⁻¹(ζ) from the negative real axis. Poles on the left side of these lines are damped enough; poles to the right are not. A specified settling time t_s ≈ 4/σ (2% criterion) defines a vertical boundary: poles must be at least this far to the left of the imaginary axis. A specified natural frequency ωn defines a circle of radius ωn — poles inside meet the speed requirement. The intersection of all these regions is the target zone where the desired closed-loop poles should land.

Once you identify the target zone, you look at where the root locus passes through it. If the locus intersects the target zone, you apply the magnitude condition to find K: at the desired pole location s* on the locus, the condition |G(s*)H(s*)| = 1/K must hold. Rearranging: K = 1/|G(s*)H(s*)|. Geometrically, K equals the product of distances from s* to all open-loop poles divided by the product of distances to all open-loop zeros. This is the gain you program into the controller.

If the root locus does not pass through the target zone at any finite gain — a common situation — then gain alone cannot achieve the specifications. This is the fundamental limitation that motivates compensator design: adding poles or zeros to the open-loop transfer function reshapes the locus so it does pass through the desired region. The dominant pole approximation simplifies verification: if the closed-loop poles you placed are the leftmost ones and all others are at least five times farther left, the response is well-approximated by just those two poles, and the predicted overshoot and settling time are accurate. Checking this assumption is always the last step — a design that looks correct on the root locus can still produce incorrect behavior if non-dominant poles are not far enough away.