2.0 Mathematical Backgroundsummarize continued

2.3 Response Descriptors

I n section 2.2 we characterized the response of the differential equation without solving for the complete time domain solution. As a designer you should be able to sketch an approximate time domain solution give the poles, zeros and gain of the system. The next step is to use a set of quantifiable parameter, or descriptors, to simplify the description of the time domain solution (as opposed to having to sketch the solution every time). On this site we focus on descriptors for a second order system. Second order system is the lowest order system that has complex poles.

Settling Time

Settling time is the time it takes for the transient response to decay. Recall that the transient response for a 2nd order system has the form

xtransient(t)= Aec·t sin(d·t+ϕ)

We define the Settling Time as the time required for the transient response to decay to about 2% of its final value. Knowing that e-4=0.018 it follows that the transient system will decay to 1.8% (about 2%) of its final value when the settling time is

tsettling=-4/c (seconds)
where c is the real component of the root of the characteristic equation. Thus a second order system with a pole that has a real component of -4 will die out in about 1 second. A system with pole that has a real component of -1 will die out in about 4 seconds.

Note that the exact definition of settling time is not unique. Some texts use 5% or 10%. However since we are only using settling time to approximately describe the response of a dynamic system and exact definition is not necessary.

Damping Ratio

Damping ratio is a measure of how much the response oscillates before it dies out. The characteristic equation for a second order system has the form

a2λ2+ a1λ + a0 =0

Eqn 2.3.3 can be rewritten as

A second order system can be written as

λ2+ 2ζ ωnλ + ωn2 = 0
where ζ is the damping ratio and ωn is the natural frequency (discussed below).

Settling time and damping ratio are commonly used descriptors of a systems transient response. Figure 2.3.4 show the relationship between the time domain response, pole location, settling time and damping ratio. The red line is the time domaing response plot. The blue line is envelope defined by the settling time. The complex plane shows the pole locations and lines of constant damping ratio.

Time Domain ResponsePole Locations
ts (settling time) = 1.0 sec     ζ (damping ratio) =
Figure 2.3.2. Effects of Settling Time and Damping Ratio on System Response


Damped Natural Frequency

The damped natural frequency, ωd, is the frequency at which the response oscillates. This corresponds to the magnitude of the imaginary component of a complex pole. For a complex pole the real component determines the settling time and the imaginary component determines the frequency of oscillation.

Natural Frequency

The natural frequency, ωn, is the frequency at which the system would oscillate in the absence of damping. The natural frequency is always faster than the damped natural frequency. You should convince yourself that is true.


The following figure shows the relationship between the pole location on the complex plane,tsettling, ζ ωn and ωd.

image/svg+xmlωnζ ωnpole location (-c+bj)jωd = ωn1- ζ2 b-ctsettling = -4/c = -4ζ ωnXComplex Plane
Figure 2.3.1. Damped frequency

System Response

The following experiment lets see the relationship between response descriptors discussed here and pole location.

img Response Descriptors for a DEQ



  Exercise 2.3.1
Calculate the settling time and damping ratio for the following closed loop system.
Then use the simulation tool to verify your results.


Answer: (show)
The closed loop transfer function is
Gclosed loop(s) =
1.5(s+1)_________s2 + 5.5s + 27.5
This has poles at s=-2.75±4.476i
Tsettling=-4/a = 4/2.74 = 1.45sec
Writing the characteristic equation in standard form
s2 + 5.5s + 27.5 = s2 + 2ζωns + ωn2
matching terms
ζ=0.52, ωn=5.25 r/s
Based on these parameter the system will reach steady state in 1.45sec and have some oscillation.
Verify through simulation