Transient Response Specifications
Rise time, overshoot, and settling time quantify how a system responds to a step, translating design goals into measurable targets.
Describing the Step Response
The response of a system to a sudden step input is the standard way to characterize its transient behavior. Several specifications measure different aspects of that response, converting vague goals like fast and well-behaved into concrete numbers a design must meet.
The key measures
- Rise time: how long the output takes to go from a low to a high fraction of its final value, measuring speed.
- Peak time: the time at which the output reaches its maximum.
- Overshoot: how far the output exceeds its final value, expressed as a percentage, measuring how oscillatory the response is.
- Settling time: how long until the output stays within a small band, often two or five percent, of its final value.
- Steady-state error: the remaining difference between output and reference after settling.
The second-order model
Most transient specs are derived from the standard second-order system, characterized by a damping ratio and a natural frequency. The damping ratio sets overshoot: heavily damped systems barely overshoot but rise slowly, while lightly damped systems rise fast but ring. The natural frequency sets the overall speed, scaling rise and settling times together.
The core trade-off
Rise time and overshoot pull against each other. Making the response faster generally increases overshoot and ringing; suppressing overshoot generally slows the response. A common target is a damping ratio around 0.7, which gives a fast rise with modest overshoot near five percent, a widely used compromise between speed and smoothness.
Mapping specs to pole locations
These time-domain specifications translate into regions of the s-plane where the dominant poles must lie. A settling-time requirement bounds how far left the poles must be; an overshoot requirement bounds their angle from the real axis; a speed requirement bounds their distance from the origin. Design methods such as root locus and pole placement then position the poles inside this allowed region.
For higher-order systems these second-order formulas are approximations valid when a dominant pole pair governs the response, so the final design is always verified against the actual simulated step response, not just the formulas.