Time Domain Specifications

When a control system receives a step input, the output does not instantly reach the desired value. The manner in which the output evolves over time is characterized by a set of standard parameters collectively called time domain specifications. These parameters are universally used in GATE, industry, and design to compare, evaluate, and optimize second-order and higher-order control systems.

Core Concept Explanation

Time domain specifications apply primarily to underdamped second-order systems (0 < zeta < 1) because these systems show a distinct transient with overshoot and oscillations. For overdamped and critically damped systems, overshoot and peak time do not exist, but rise time and settling time still apply.

The delay time (td) is the time for the output to reach 50% of its final value for the first time. It gives an early measure of how quickly the system begins to respond. For a standard underdamped second-order system, td is approximately (1 + 0.7*zeta) / omega_n.

The rise time (tr) is the time taken for the output to rise from 10% to 90% of the final value. For underdamped systems it is often measured from 0% to 100%. The formula is tr = (pi - theta) / omega_d, where theta = cos^(-1)(zeta) and omega_d = omega_n * sqrt(1 - zeta^2) is the damped natural frequency. Smaller zeta gives smaller tr (faster rise) but larger overshoot.

The peak time (tp) is the time at which the response first reaches its maximum value (peak overshoot). It is given by tp = pi / omega_d. Peak time is inversely proportional to damped natural frequency, so increasing omega_n or decreasing zeta both reduce peak time.

Mathematical Expression

The percentage peak overshoot (Mp) is the most commonly asked specification in GATE. It depends only on the damping ratio and is given by Mp = exp(-pi*zeta / sqrt(1-zeta^2)) * 100%. This is a critical result to memorize. As zeta approaches 0, Mp approaches 100%. At zeta = 0.707, Mp is approximately 4.3%.

The settling time (ts) is the time for the output to enter and remain within a tolerance band (typically 2% or 5%) around the final value. For the 2% criterion, ts is approximately 4 / (zeta * omega_n). For the 5% criterion, ts is approximately 3 / (zeta * omega_n). This result comes from the time constant of the exponential envelope e^(-zeta*omega_n*t).

All five specifications are interconnected through zeta and omega_n. Increasing omega_n speeds up the response (reduces tr, tp, ts) without changing overshoot. Increasing zeta reduces overshoot but increases rise time. This fundamental conflict is the core challenge of control system design.

Practical Understanding

In industrial servo drives, specifications such as settling time and overshoot are directly linked to throughput and safety. A robotic arm with 30% overshoot would physically overshoot its target position, potentially causing collisions. A missile guidance system requires fast rise time with near-zero overshoot.

The 2% settling time criterion means the output must stay within plus or minus 2% of the final value permanently. This is harder to satisfy than just reaching the band once, because oscillating systems may enter and exit the band multiple times before finally settling.

Given:
omega_n = 5 rad/s, zeta = 0.5
omega_d = omega_n * sqrt(1 - zeta^2) = 5 * sqrt(1 - 0.25) = 5 * 0.866 = 4.33 rad/s

Why this formula applies:
Underdamped system (0 < zeta < 1) so all specs are defined.

Formula:
tp = pi / omega_d
Mp = exp(-pi*zeta / sqrt(1-zeta^2)) * 100%
ts = 4 / (zeta * omega_n)  [2% criterion]

Substitution:
tp = 3.14159 / 4.33
Mp = exp(-3.14159 * 0.5 / sqrt(0.75)) * 100
ts = 4 / (0.5 * 5)

Calculation:
tp = 0.725 s
Mp = exp(-1.5708 / 0.866) * 100 = exp(-1.814) * 100 = 0.1629 * 100
ts = 4 / 2.5

Final Answer with units:
tp = 0.725 s,  Mp = 16.3%,  ts = 1.6 s

Exam Tip: Peak overshoot formula Mp = exp(-pi*zeta/sqrt(1-zeta^2)) depends only on zeta, not on omega_n. GATE often asks you to find zeta from a given overshoot percentage. Take natural log of (Mp/100) and solve. Also remember: ts = 4/(zeta*omega_n) for 2% criterion and ts = 3/(zeta*omega_n) for 5% criterion.

Mechanism: How Specifications Relate

• Rise time tr = (pi - theta)/omega_d decreases with decreasing zeta and increasing omega_n, meaning faster rise comes at the cost of more overshoot.
• Peak time tp = pi/omega_d depends only on damped natural frequency. Higher omega_d means earlier peak.
• Peak overshoot depends only on zeta. A system with zeta = 0.5 always has 16.3% overshoot regardless of omega_n value.
• Settling time ts = 4/(zeta*omega_n) for 2% band can be reduced by increasing either zeta or omega_n. Increasing omega_n is preferred because it does not affect overshoot.
• Damping ratio zeta = 0.707 is considered optimal as it gives a good balance between rise time and overshoot, with Mp approximately 4.3% and nearly minimum settling time.

Quick Revision

• Delay time: time to reach 50% of final value, td approximately (1+0.7*zeta)/omega_n.
• Rise time: tr = (pi - theta)/omega_d where theta = cos^(-1)(zeta), omega_d = omega_n*sqrt(1-zeta^2).
• Peak time: tp = pi/omega_d, occurs at first peak of response.
• Percentage overshoot: Mp = exp(-pi*zeta/sqrt(1-zeta^2))*100, depends only on zeta.
• Settling time 2%: ts = 4/(zeta*omega_n), settling time 5%: ts = 3/(zeta*omega_n).
• Exam trap: overshoot depends only on zeta, NOT on omega_n. Doubling omega_n does not change overshoot.
• Optimal damping ratio zeta = 0.707 gives Mp approximately 4.3% and fastest practical settling.