How to Calculate Rise Time, Overshoot, and Settling Time in Control Systems

 1. Introduction

Dynamic (control) systems in engineering—such as motor controllers, climate control systems, or vehicle suspension systems—are often tested using step functions (sudden jumps). The response to these step inputs provides an indication of how fast, stable, accurate, and reliable the system is. Three important parameters related to time response performance are:

• Rise Time – the time it takes for the output response to rise from a low value to a target value.
• Overshoot – the extent to which the output response exceeds the desired final value.
• Settling Time – the time it takes for the output response to stabilize around a final value within a specified tolerance band.

This material will discuss definitions, formulas, example graphs, analytical and experimental methods, and practical interpretations of each parameter.

Basic Model of Time Response Systems

2.1 First-Order Systems

A first-order system is defined as:

If the input is a unit step, that is, $R(s) = 1/s$, then the output $C(s)$ is:

The reverse transformation produces:

Parameters:

• K: steady-state gain, which is the final value after a sufficient amount of time.
• τ: time constant, describes the speed at which the system approaches steady-state.

For second-order systems, overshoot occurs, and the rise and settling times can be analyzed using the system's natural characteristics (natural frequency ω_n and damping ratio ζ).

3. Response Time Parameters

3.1 Rise Time 

Definition: The time required for the system output to respond from the initial value (usually 10% or 0%) to the final value (90% or 100%) of the step amplitude.

First-order systems:

Because the graph is exponentially rising, the pressure often takes the range 10–90%.

After t = τ, the output reaches ~63.2% of its final value.

Second-order system (underdamped):

Transfer function:

Why use rise time?

Because it measures the speed of the initial response, it's especially important in applications where initial delays impact performance—such as tracking systems, anti-lock braking, or robotic control.

3.2 Overshoot

Definition: The extent to which the response peak exceeds the steady-state value the first time.

In percentage terms:

Second-order system:

Overshoot:

• if ζ → 0 → M is large (large waves)
• If ζ → 1 → M approaches zero (non-oscillatory response)

Overshoot is important because of its impact on performance:

• High overshoot can cause physical damage (motor spin) or process instability.
• Low overshoot means smooth control but may be slow to reach the final value.

3.3 Settling Time

Definition: The time for the output response to remain within a deviation range of ±δ% from steady-state, typically δ = 2% or δ = 5%.

First-order system:

For a deviation of ±2%:

Untuk ±5%:

Settling time reflects the final stability of the system: whether after the initial oscillations, the system can settle down and no longer “vibrate”.

4. Graphs and Interpretation

Let's look at an example of a response graph of an underdamped second-order system (ζ < 1).

^

|                /\           _

|              /   \       _/

|             /     \     /

|    ________/       \___/____________ steady-state = 1

|

+--------------------------------------------> t

The main parameters are clearly visible:

• Rise Time: the duration when the response reaches 0.1 → 0.9
• Overshoot: the peak area above steady-state (peak minus 1)
• Settling Time: the point at which the curve enters and remains within a ±2% band of 1

5. How to Determine Analytically

5.1 Calculate System Parameters

For the Laplace transform of G(s), you can determine parameters such as ω_n and ζ:

• Characteristic Response of a Quadratic Equation

From G(s) we can determine ζ and ω

5.2 Parametric Formula

Use the formulas and approaches mentioned above:

• Rise Time:

First Order: 2.197τ

• Overshoot:
• 
  
  
  

6. Experimental/Practical Method

6.1 Obtain Data

Measure the output c(t) when the system is subjected to a unit step (e.g., apply a voltage of 0→1 V to a plant or motor). Record the data with a sufficiently high sampling rate.

6.2 Data Analysis

• Rise Time: Find the time indexes at which the signal passes 10% and 90%, and calculate the difference.
• Overshoot: Find the first peak signal compared to the final stabilization value (average after a sufficient amount of time).
• Settling Time: Find the time at which the signal enters and remains within ±2%.

6.3 Tools

MatLab, Python, Pandas, and SciPy software can be used. Python examples:

import numpy as np

t, y = load_data()  # misal via CSV

y_ss = np.mean(y[-100:])

t_r_start = t[np.where(y >= 0.1*y_ss)[0][0]]

t_r_end = t[np.where(y >= 0.9*y_ss)[0][0]]

tr = t_r_end - t_r_start

yp = np.max(y)

Mp = (yp-y_ss)/y_ss * 100

tol, ts = 0.02, None

for ti, yi in zip(t, y):

    if abs(yi-y_ss) <= tol*y_ss:

        ts = ti

        break

7. Interpretation of Design Decisions

This parameter helps engineers:

• Fast (short rise time) but can have high overshoot and possibly vibrate ("aggressive").
• Requires trade-offs (see specifications), e.g., overshoot <5%, settling time <0.2 s).
• Gain control, filters, PID, or higher-order system design can be used for optimization.

8. Optimization Methods

8.1 PID Tuning

• P → fast but can overshoot
• D → dampen overshoot
• I → reduce steady-state error, but can be slow

8.2 Pole Placement

Determine the pole position s = –σ ± jω. Select ζ and ω_n according to rise, overshoot, and settling requirements.

8.3 Root Locus / Bode / Lead-Lag

Frequency analysis to ensure the system is robust to real-world dynamics without sacrificing transmission time performance.

9. Case Studies and Examples

Provide real-world examples:

• Robot arm: requires fast response and low overshoot for precision
• HVAC: requires fast settling time for comfortable temperature, but overshoot can be uncomfortable
• Audio: requires smooth response without overshoot for sound quality