Time Response of First and Second Order Systems Guide.

Time response is how a system's output evolves after an input like a sudden step. First-order systems (one energy store: RC circuit, heated tank) respond with a smooth exponential, never overshooting. Second-order systems (two stores: mass-spring-damper, RLC) can overshoot and oscillate, governed by just two numbers: natural frequency omega_n and damping ratio zeta.

Time response is what users experience: motor spin-up speed, robot-arm overshoot, room warm-up time. Rise time, peak overshoot and settling time are the universal language of control requirements, and most real systems are approximated as first or second order for design.

Key Concepts

• Time constant tau: first-order K/(tau x s + 1) reaches 63.2% at t = tau, within 2% after 4 x tau.
• omega_n: undamped oscillation frequency, rad/s.
• zeta: 0 = oscillates forever; 0-1 = underdamped (overshoot); 1 = critically damped (fastest, no overshoot); >1 = overdamped (sluggish).
• omega_d = omega_n x sqrt(1 - zeta^2): actual oscillation frequency when underdamped.
• Peak overshoot M_p, peak time t_p, settling time t_s: the standard step-response specs.
• Standard form: omega_n^2 / (s^2 + 2 x zeta x omega_n x s + omega_n^2).

Formulas

• First-order step: c(t) = K x (1 - e^(-t/tau)).
• Overshoot: M_p = e^(-pi x zeta / sqrt(1 - zeta^2)), 0 < zeta < 1.
• Peak time: t_p = pi / omega_d. Settling (2%): t_s = 4/(zeta x omega_n).
• Assumptions: linear, unit step, zero initial conditions, dominant complex pole pair.

Worked Example

Servo T(s) = 25/(s^2 + 6s + 25). Step specs:

1. omega_n^2 = 25 → omega_n = 5 rad/s; 2 x zeta x 5 = 6 → zeta = 0.6 (underdamped).
2. omega_d = 5 x 0.8 = 4 rad/s.
3. M_p = e^(-pi x 0.6/0.8) = e^(-2.3562) = 0.0948 → 9.5% overshoot.
4. t_p = pi/4 = 0.785 s; t_s = 4/(0.6 x 5) = 1.333 s.
5. The servo peaks 9.5% high at 0.79 s and settles within 2% by ~1.33 s. More damping would cut overshoot but slow settling.

Common Misconceptions

• "Faster rise is always better." Chasing speed lowers damping: bigger overshoot, longer settling. Balance all three specs.
• "zeta = 1 is fastest." Critically damped is fastest without overshoot; zeta ~0.7 rises faster and still settles quickly.
• "Settling time depends on step size." Linear systems scale: percentages and times are identical for 1 V or 100 V steps.

Real-World Applications

• Motor drives: speed loops tuned to overshoot/settling specs.
• Suspension: damping chosen for comfort vs grip.
• Temperature: first-order models sized from measured tau.
• Robot arms: overshoot limits protect fixtures.

First-order systems creep exponentially with time constant tau; second-order systems are fixed by omega_n and zeta, giving overshoot, peak and settling times from simple formulas. The example (zeta = 0.6): 9.5% overshoot at 0.79 s, settled by 1.33 s. Specify responses in these numbers and every control engineer understands.