Second-Order System Performance Specifications

A second-order system is one of the most critical models in control engineering, describing phenomena from robotic arm motion to electronic filter response. Its performance is quantified by a set of clear, measurable specifications in the time domain—percent overshoot, settling time, rise time, and peak time. These metrics don't exist in a vacuum; they are directly determined by the location of the system's poles in the s-plane. Mastering this map from pole location to time response is the essence of predictive design, allowing you to specify desired behavior and then calculate or place the poles to achieve it.

Relating Pole Location to Damping and Natural Frequency

Every second-order system with no finite zeros can be described by its characteristic equation: $s^2+2\zeta {\omega }_{n}s+{\omega }_n^2=0$. The roots of this equation, the system poles, are $s=-\zeta {\omega }_n\pm j{\omega }_n\sqrt{1-{\zeta }^2}$. These two parameters, $\zeta$ (zeta) and ${\omega }_n$, are the keys to the kingdom.

The damping ratio, $\zeta$, determines the shape of the response. It tells you how oscillatory the system will be.

• $\zeta =0$: Undamped, pure oscillation.
• $0<\zeta <1$: Underdamped, oscillatory decay (the most common design region).
• $\zeta =1$: Critically damped, fastest return without oscillation.
• $\zeta >1$: Overdamped, slow, non-oscillatory return.

The natural frequency, ${\omega }_n$, determines the speed or scale of the response. For an undamped system, it is literally the frequency of oscillation. In the s-plane, ${\omega }_n$ is the radial distance from the origin to the poles.

A pole located at $s=-\sigma \pm j{\omega }_d$ has a real part $\sigma =\zeta {\omega }_n$ and an imaginary part ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$, known as the damped natural frequency. The angle $\theta$ this pole makes with the negative real axis satisfies $\cos \theta =\zeta$. This geometric relationship is fundamental: lines of constant damping ratio radiate from the origin at a fixed angle $\theta ={\cos }^{-1}(\zeta )$.

Time-Domain Specifications and Their s-Plane Correlates

Percent Overshoot and the Damping Ratio

Percent overshoot (PO) is the maximum amount the system output exceeds its final steady-state value after a step input, expressed as a percentage. It is a function only of the damping ratio $\zeta$ for a pure second-order system. The relationship is given by:

$$\%OS=100\times e^{\frac{-\zeta \pi }{\sqrt{1-{\zeta }^2}}}$$

This equation shows that overshoot decreases as $\zeta$ increases. There is no overshoot for $\zeta \ge 1$. In the s-plane, this means percent overshoot is controlled strictly by the angle $\theta$ of the poles. To meet a maximum overshoot requirement (e.g., PO < 10%), you must place your poles within a wedge defined by lines of constant $\zeta$. For PO < 10%, $\zeta$ must be > 0.59, meaning the pole angle $\theta$ must be less than ${\cos }^{-1}(0.59)\approx 54^{\circ }$.

Settling Time and the Real Part of Poles

Settling time ($T_s$) is the time required for the system's step response to reach and stay within a certain percentage (typically 2%) of its final value. For a canonical second-order system, the 2% settling time is approximated with remarkable accuracy by:

$$T_s\approx \frac{4}{\sigma }=\frac{4}{\zeta {\omega }_n}$$

Here, $\sigma$ is the magnitude of the real part of the complex poles. This is a crucial insight: settling time depends only on the real part of the poles. To achieve a faster settling time (e.g., $T_s<2$ seconds), you need $\sigma >2$. In the s-plane, this translates to a vertical boundary: all poles must be located to the left of the line $s=-\sigma =-2$. This is a region of the s-plane defined by $\text{Re}(s)\le -2$.

Rise Time and Peak Time

Rise time ($T_r$) is typically defined as the time for the response to go from 10% to 90% of its final value. For underdamped systems, it is inversely related to the natural frequency ${\omega }_n$. An accurate approximation is:

$$T_r\approx \frac{1.8}{{\omega }_n}$$

Since ${\omega }_n$ is the radial distance to the poles, a larger ${\omega }_n$ (poles further from the origin) means a faster rise time. However, moving poles radially outward also increases the imaginary part ${\omega }_d$, which affects the next specification.

Peak time ($T_p$) is the time at which the maximum overshoot occurs. It is directly related to the damped natural frequency:

$$T_p=\frac{\pi }{{\omega }_d}=\frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}$$

A larger imaginary part ${\omega }_d$ (faster oscillation) results in an earlier peak time. In the s-plane, for a fixed damping ratio (fixed angle), moving poles outward along a constant $\zeta$ line increases ${\omega }_n$, which decreases both rise time and peak time.

Synthesizing Specifications for Pole Placement

Design is the art of managing trade-offs. The s-plane region mapping makes these trade-offs visual. Let's say your design requires:

1. Percent Overshoot ≤ 16% (which requires $\zeta \ge 0.5$, or $\theta \le 60^{\circ }$).
2. Settling Time ≤ 1 second (which requires $\sigma =\zeta {\omega }_n\ge 4$).
3. Rise Time ≤ 0.25 seconds (which requires ${\omega }_n\ge \frac{1.8}{0.25}=7.2$ rad/s).

Your acceptable pole region is the intersection of all these constraints.

• From (1): Poles must lie inside a $60^{\circ }$ wedge from the negative real axis.
• From (2): Poles must lie to the left of the vertical line $s=-4$.
• From (3): Poles must lie outside a circle of radius ${\omega }_n\approx 7.2$.

You would graphically (or computationally) find the overlapping region that satisfies all three. Any pole pair placed within this region will, in theory, meet all your performance specifications. This is the foundation of pole-placement design techniques.

Common Pitfalls

1. Ignoring the Dominant Pole Assumption: The elegant equations for $T_s$, $T_r$, and $\%OS$ are exact only for pure, prototype second-order systems (no zeros, two dominant poles). Adding zeros or additional poles changes the response. A common mistake is to use these formulas for higher-order systems without verifying that the two complex poles are truly dominant—that their real parts are at least 5-6 times closer to the imaginary axis than other poles/zeros.

1. Treating Approximations as Exact Formulas: The settling time formula $T_s=4/(\zeta {\omega }_n)$ is a 2% criterion approximation based on the envelope of the decay. It is remarkably robust but is still an approximation. For very high or low $\zeta$, the accuracy decreases. Similarly, the rise time formula $1.8/{\omega }_n$ is an empirical approximation that works well for common damping ratios (0.5 < $\zeta$ < 0.8) but breaks down near critical damping.

1. Overlooking the Effect of Zeros: A zero in the left-half plane (LHP) can increase overshoot and shorten rise time, making the system feel "faster" but more oscillatory. A zero in the right-half plane (RHP) can cause an initial undershoot and severely degrade performance. The standard second-order specs do not account for zeros. Your pole-placement region may guarantee performance for a pure second-order plant, but a controller that adds zeros could invalidate that guarantee.

1. Chasing Conflicting Specs: The s-plane map reveals inherent conflicts. You cannot have zero overshoot and minimum rise time simultaneously. A critically damped response ($\zeta =1$) has no overshoot but a relatively slow rise. To get a very fast rise (large ${\omega }_n$), you often need to accept some oscillation (lower $\zeta$), which means overshoot. Good design involves finding the optimal compromise within the application's constraints.

Summary

• The damping ratio ($\zeta$) exclusively determines percent overshoot, mapped in the s-plane as the angle of the complex poles relative to the real axis.
• The real part of the poles ($\sigma$) exclusively determines settling time, mapped as a vertical boundary in the left-half plane.
• The natural frequency (${\omega }_n$) primarily determines rise time and peak time, mapped as the radial distance of the poles from the origin.
• Pole placement is the process of choosing pole locations whose $\zeta$ and ${\omega }_n$ values satisfy all desired performance specifications, visualized as finding a region in the s-plane that satisfies all constraints.
• Always validate that your system's dynamics are dominantly second-order and account for the effects of any zeros before relying solely on these classic relationships for design.