Poles, modes and stability

Rule

Real pole p gives . Complex pair σ ± jω gives .

Takeaway

σ sets decay (left = dies out, right = explodes), ω sets how fast it wiggles. Every response is a sum of these modes.

Try ω = 0, then σ > 0, then σ = 0 with ω = 5, then σ = −3.

SS-05 p.3 (poles and zeros) · SS-07 p.3–6 (modes) · p.7 (example shapes)

System

, poles fixed at −1 and −5, final value 1.

Takeaway

Moving the zero never adds or removes a shape; it changes how much of each shape appears. At z = 1 it cancels the −1 mode.

SS-07 p.4 (coefficients depend on the zeros) · p.15 note 5 (a zero close to a pole cancels it)

Model

. Target 90°, wall at 99°.

Takeaway

c sits in the denominator, so it moves the poles. Overdamped (sobreamortiguado): real poles, slow. Critically damped: double pole, fastest without overshoot. Underdamped (subamortiguado): complex poles, overshoot.

SS-07 p.6 (overdamped and underdamped modes) · SS-10 p.3 (second order) · DC-04 (D term adds damping)

Rule

Stable ⇔ every pole has σ < 0. One pole on the right is enough to make it unstable, whatever the input or gain.

Edge cases

A single pole at 0 is limitedly stable (limitadamente estable). A pure pair ±jω is marginally stable (marginalmente estable).

SS-07 p.9–10 (definition, ball example) · p.11 (pole criterion) · p.12 (impulse and step) · *balancing robot is a simplified model

Idea

Modes from poles far left die fast. After a short time only the slowest mode is left, so the pole closest to the imaginary axis decides the shape and the settling time.

Takeaway

Approximate the system by its dominant pole (or pair). Compare systems by their dominant poles, never the leftmost ones.

A race between two modes

Approximating by the dominant pole

Four systems: when is a pole dominant enough?

Same slow pole at −1, second pole moving farther left. Rule of thumb: others ≥ 5× farther left (textbook rule; SS-07 gives no number).

The 4th-order example from the slides

SS-07 p.15 (dominant poles) · p.16–17 (4th-order example) · p.18 (1st/2nd-order approximation) · FR-06 p.7

Takeaway

Far enough left (fast) and inside a narrow cone (small overshoot). A little wiggle (B) beats none (A). Very far left (D) costs actuator effort and lets noise through.

Damping ratio

for a pole pair −σ ± jω_d.

A single real pole (A, D) is a first-order system, so "underdamped" and "overdamped" don't apply to it: those words describe pole pairs. It behaves like an overdamped system in that it never oscillates.

FR-06 p.3 (response vs pole position) · p.7 (placing closed-loop poles)

Setup

The same robot arm, ω_n = 5 rad/s. The only difference is the damping ratio ζ.

Takeaway

A slightly underdamped arm settles almost 40% faster, because its small overshoot never leaves the ±5% band.

Reason 1

Less brake, so it moves faster. The critically damped arm has to slow down on the way so it never passes the target.

Reason 2

The 4.6% overshoot (sobreoscilación) stays inside the ±5% band, so it already counts as settled. You get the speed for free.

Reason 3

Margin for model error. Real damping drifts with friction, temperature and wear. Designed exactly at ζ = 1, a little less damping brings overshoot anyway and a little more makes it slow. Around ζ ≈ 0.7 a small error changes almost nothing.

Use ζ = 1

Only when overshoot is forbidden (M_p = 0%): an arm right next to a wall, a cutter at the workpiece, a needle at a patient, a lift arriving at a floor.

SS-08 p.6 (±5% band) · SS-10 p.3, p.13 (ζ and overshoot) · ζ ≈ 0.7 is a common design rule of thumb, not stated in the slides

Recipe

Set the denominator to 0 and solve (watch the sign). Check every σ. The pole closest to the axis dominates: t_s ≈ 3/|σ|.

SS-05 p.3 · SS-07 p.11, p.15