Location, location, location… Relating pole/zero positions to step response specifications
Step Response Specifications
Steady-state value Undershoot Overshoot
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Raise time Settling time
Canonical 2nd Order Lowpass System
Assume stability and underdamping ( 0 < ζ<1), i.e. the poles have nonzero imaginary parts and negative real parts.
We will see that overshoot, rise time and settling time specs will impose constraints on the permitted pole locations
Partial Fraction
𝑠𝑠 𝑠𝑠 𝑠𝑠2+2𝜁𝜁𝜔𝜔𝑛𝑛𝑠𝑠+𝜔𝜔𝑛𝑛2 𝑠𝑠 (𝑠𝑠+𝜎𝜎)2+𝜔𝜔2
Denote step response by y(t)
𝐺𝐺 𝑠𝑠 𝜔𝜔 𝑛𝑛2 𝜔𝜔 𝑛𝑛2
=1𝑠𝑠− 2𝜁𝜁𝜔𝜔𝑛𝑛 − 𝑠𝑠𝑑𝑑
𝑦𝑦 𝑡𝑡 𝑠𝑠2 +2𝜁𝜁𝜔𝜔𝑛𝑛𝑠𝑠+𝜔𝜔𝑛𝑛2 𝑠𝑠2 +2𝜁𝜁𝜔𝜔𝑛𝑛𝑠𝑠+𝜔𝜔𝑛𝑛2
Table look-u𝜁𝜁p:
=1 − 1−𝜁𝜁2𝑒𝑒−𝜎𝜎𝜎𝜎sin 𝜔𝜔𝑑𝑑𝑡𝑡 −𝑒𝑒−𝜎𝜎𝜎𝜎cos 𝜔𝜔𝑑𝑑𝑡𝑡
Settling time
Calculated for
Brown colour represents a “forbidden region”.
Complex domain (pole locations)
Brown colour represents a “forbidden region”.
Complex domain (pole locations)
Complex domain (pole locations)
Taken from Franklin, Powel & Emami-Naeini “Feedback control of dynamic systems”.
Given a combined requirement on overshoot, rise time and settling time, we could get:
Complex domain (pole locations)
Brown colour represents a “forbidden region”.
From 2nd to Higher Order
Formulas for canonical 2nd order systems provide a “rule of thumb” for designing higher order lowpass systems
They work well if there are two complex conjugate, stable, “dominant” poles that satisfy the required constraints. (Dominant means all other poles are far to the left, i.e. their modes decay much more rapidly)
The formulas also work for proper stable systems with zeros all much larger than the dominant poles.
Graphical illustration
Dominant poles
Complex domain (pole locations)
Approximating by 1st Order System
By designing a single, real-valued dominant pole, we could even use a 1st order approximation. Even simpler!
However, we then lose some useful degrees of design freedom.
E.g. in a 1st order step response, the rise time is always about the same as the settling time.
But for a 2nd order system, we could potentially place the poles so that the rise time is much shortera controller that’s more responsive, at the expense of overshoot.
Higher order systems also have superior noise rejection properties (more after we introduce freq. response)
Fundamental trade-offs
Sometimes, a step-response specification may not be achievable, due to a clash with another spec.
In other words, there may be fundamental limitations, or trade-offs, in performance.
We look next at a couple of the trade-offs caused by zeros
Example (real unstable zero)
Consider this system with zero at
Overshoot: when zero is negative and
small relative to the dominant pole at -1. Undershoot: when zero is positive. Slow zeros seem to have biggest impact.
Fundamental limitation: (undershoot vs. settling time with unstable, real zero)
Fundamental limitation:
(overshoot vs. settling time with slow, stable real zero)
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