Second-Order System Time Response

The behavior of a second-order dynamic system in response to an input is fundamental to engineering design, from the suspension in your car to the positioning of a robotic arm. Mastering this concept allows you to predict and control oscillations, stability, and speed of response. At its core, the step response—how the system output reacts to a sudden, sustained input—is completely characterized by two parameters: the damping ratio and the natural frequency.

The Governing Parameters: $\zeta$ and ${\omega }_n$

Every linear, time-invariant second-order system can be described by its standard transfer function, which leads to a characteristic equation of the form $s^2+2\zeta {\omega }_{n}s+{\omega }_n^2=0$. The two parameters that emerge are the damping ratio, denoted by $\zeta$ (zeta), and the undamped natural frequency, denoted by ${\omega }_n$ (omega_n).

The natural frequency, ${\omega }_n$, represents the system's inherent oscillatory frequency if no damping were present. It is measured in radians per second and fundamentally sets the time scale of the response; a higher ${\omega }_n$ means a faster system. The damping ratio, $\zeta$, is a dimensionless number that describes how oscillations in the system decay after a disturbance. It is the critical factor determining the shape of the transient response. The value of $\zeta$ directly classifies the system's behavior into three distinct regimes.

The Three Damping Regimes

The damping ratio $\zeta$ acts as a switch between fundamentally different response types.

1. Underdamped ($0<\zeta <1$): This is the most common regime for analysis. When $\zeta$ is less than one, the system's response to a step input is oscillatory. The output rises, overshoots the final steady-state value, and then oscillates with a decaying amplitude until it settles. The actual frequency of these decaying oscillations is called the damped natural frequency, given by ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$. The lower the damping ratio, the more pronounced the oscillations and the higher the overshoot.

1. Critically Damped ($\zeta =1$): This represents the threshold between oscillatory and non-oscillatory behavior. A critically damped system returns to the steady-state value in the shortest possible time without oscillating. It is often a design target for systems where any overshoot is unacceptable, such as in elevator or door mechanisms.

1. Overdamped ($\zeta >1$): In this regime, the system's response is sluggish and non-oscillatory. It approaches the final value slower than a critically damped system. Overdamped behavior is like moving through a very thick fluid; the system is so damped that it cannot oscillate but is also slow to respond.

Performance Metrics for Underdamped Systems

Since underdamped systems are so prevalent, engineers quantify their transient performance with specific metrics derived directly from $\zeta$ and ${\omega }_n$. Consider a system subjected to a unit step input.

• Rise Time ($t_r$): The time it takes for the response to first reach the final steady-state value. An approximate formula is $t_r\approx \frac{1.8}{{\omega }_n}$. This shows that rise time is inversely proportional to natural frequency; to make a system faster, increase ${\omega }_n$.

• Percent Overshoot (\%OS): The maximum amount the response exceeds the final value, expressed as a percentage. It is determined exclusively by the damping ratio:

$$\%OS=100\times e^{(-\zeta \pi /\sqrt{1-{\zeta }^2})}$$ For example, a damping ratio of $\zeta =0.6$ yields an overshoot of approximately 9.5%. This relationship is crucial for design: if a specification limits overshoot to 10%, you must design for $\zeta \approx 0.6$ or higher.

• Settling Time ($t_s$): The time required for the response to enter and remain within a certain percentage band (typically 2%) of the final value. It is approximated by:

$$t_s\approx \frac{4}{\zeta {\omega }_n}$$ Settling time is inversely proportional to the product $\zeta {\omega }_n$. To reduce settling time, you can increase either the damping ratio or the natural frequency.

A Practical Worked Example

Let's apply these concepts. Suppose a servo motor system is modeled as a second-order system with ${\omega }_n=10$ rad/s and $\zeta =0.4$.

1. Classify the response: Since $\zeta =0.4$ which is $<1$, the system is underdamped and will exhibit oscillatory step response.
2. Calculate key metrics:

• Damped Natural Frequency: ${\omega }_d=10\ast \sqrt{1-0.4^2}=10\ast \sqrt{0.84}\approx 9.17$ rad/s. This is the frequency of the oscillations you would observe.
• Percent Overshoot: $\%OS=100\times e^{(-0.4\pi /\sqrt{1-0.16})}=100\times e^{(-1.256/0.9165)}\approx 100\times e^{-1.37}\approx 25.4\%$.
• 2% Settling Time: $t_s\approx \frac{4}{0.4\times 10}=\frac{4}{4}=1$ second.
• Approximate Rise Time: $t_r\approx \frac{1.8}{10}=0.18$ seconds.

This tells an engineer that after a command, the motor's position will quickly rise (0.18s) but overshoot the target by about 25%, oscillating around 9 times per second (${\omega }_d/(2\pi )$), and will finally settle within 2% of the target position after about 1 second.

Common Pitfalls

1. Confusing ${\omega }_n$ and ${\omega }_d$: A frequent error is using the natural frequency ${\omega }_n$ to calculate the period of visible oscillations. Remember, the oscillations you see occur at the damped natural frequency, ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$, which is always less than ${\omega }_n$. Using ${\omega }_n$ will predict oscillations that are too fast.
2. Misapplying Settling Time Formula: The formula $t_s\approx 4/(\zeta {\omega }_n)$ is an approximation for the 2% criterion. Using it for a different tolerance band (e.g., 5%) without adjustment is incorrect. The 5% settling time is roughly $3/(\zeta {\omega }_n)$.
3. Assuming Higher Damping is Always Better: While increasing $\zeta$ reduces overshoot and can decrease settling time, it also increases rise time. There is a fundamental trade-off between speed of response (rise time) and oscillation (overshoot). A critically damped system has no overshoot but is slower to initially rise than a lightly underdamped one.
4. Ignoring the Underdamped Assumption: The formulas for percent overshoot, rise time, and settling time provided here are derived for the underdamped case ($\zeta <1$). They do not apply to critically damped or overdamped systems, whose performance is analyzed through different methods, often involving the roots of the characteristic equation.

Summary

• The step response of a linear second-order system is entirely defined by the damping ratio ($\zeta$) and the natural frequency (${\omega }_n$).
• The damping ratio $\zeta$ categorizes the response: underdamped ($\zeta <1$, oscillatory), critically damped ($\zeta =1$, fastest non-oscillatory), or overdamped ($\zeta >1$, slow non-oscillatory).
• For underdamped systems, key performance metrics are derived from $\zeta$ and ${\omega }_n$: percent overshoot depends only on $\zeta$, while rise time is inversely proportional to ${\omega }_n$, and settling time is inversely proportional to the product $\zeta {\omega }_n$.
• Design involves balancing trade-offs: increasing ${\omega }_n$ speeds up the response, while adjusting $\zeta$ controls the oscillation and overshoot.
• Always ensure you are using the correct formulas for the damping regime of your system, and remember that the visible oscillation frequency is the damped natural frequency ${\omega }_d$, not ${\omega }_n$.