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CDSSC · Repaso · Step 01

Static gain and the first-order system

Interactive plots and self-checks for re-learning. Every sheet names the concept, gives the one-line takeaway and points to the lecture slide.

1.1

K and T ganancia estática · constante de tiempo

Model
Takeaway
K sets how high the output ends. T sets how quickly it gets there: 63% at t = T, inside ±5% at t = 3T.
Static gain K2.5
Time constant T [s]0.50
y(t) final value K ±5% band 63% at t = T settled at t = 3T

SS-09 p.3 (standard form) · p.5 (63%) · p.6 (ts = 3T) · SS-08 p.3–4 (static gain)

1.2

Four motors side by side comparar sistemas

Setup
Input: voltage [V]. Output: speed [rad/s]. Each motor is given the way an exam gives it.
Takeaway
Quick or slow is set by T (pole farther left = quicker). High or low is set by K. The two are independent.
Input voltage5 V

Pole positions on the real axis, :

Check yourself

Which motor reaches 95% of its own final speed first?

With 5 V applied, which motor reaches the highest final speed?

SS-09 p.3 (K and T) · p.9 (K as vertical scale, smaller T = quicker)

1.3

Read K and T from measured data identificación

Method
1. Final value = average of the flat part (not the highest spike). 2. Take 63% of it. 3. The time where the curve crosses that level is T.
Why 63%
Check yourself · unknown motor, 1 V step

SS-09 p.5 (T = time to 63% of the final value)

1.4

Stable vs unstable pole estabilidad

Compare
Takeaway
A pole with positive real part makes the response grow without limit. G(0) still gives a number (−2.5), but it means nothing: check the poles first.
pole at −2: settles at 2.5 pole at +2: keeps growing
y(1) ≈ 16y(2) ≈ 134y(5) ≈ 55 000y(10) ≈ 1.2·10⁹

SS-07 p.11 (stability criterion) · SS-04 p.11 (e−at ↔ 1/(s+a)) · SS-04 p.14 (final value theorem needs stable poles)

1.5

The integrator 1/s integrador

Meaning
s means derivative, 1/s means integral (accumulate). Tank: constant inflow, the level rises forever.
Takeaway
A step into 1/s gives a ramp, so : no finite static gain. As a signal, 1/s is the unit step; as a system, it is an integrator.
input: constant flow (step) output of 1/s: ramp first-order system: settles

SS-04 p.9 (step ↔ 1/s) · SS-04 p.13 (derivative ↔ s, integral ↔ 1/s)

1.6

Drill: K, T and settling time ejercicios

Recipe
Make the denominator end in "+1". Then K is on top, T multiplies s, and ts = 3T. For any order, K = G(0).

SS-09 p.3, p.6 · SS-08 p.3