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

x_{transient}(t)= Ae^{c·t} sin(d·t+ϕ)

2.3.1

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

t_{settling}=-4/c (seconds)

2.3.2

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

a_{2}λ^{2}+ a_{1}λ + a_{0} =0

2.3.3

Eqn 2.3.3 can be rewritten as

A second order system can be written as

λ^{2}+ 2ζ ω_{n}λ + ω_{n}^{2} = 0

where ζ is the damping ratio and ω2.3.4

EX

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

t_{s} (settling time) = 1.0 sec ζ (damping ratio) =

0.707

Figure 2.3.2. Effects of Settling Time and Damping Ratio on System Response

- How do the complex poles move when the settling time is held constant? (answer)Poles with constant settling time move vertically on the complex plane. They have a constant real value.
- How do the complex poles move when the damping ratio is held constant? (answer)Poles with a constant damping ratio move along a line originating at the origin.
- A damping ratio of 0.707 corresponds to what pole location on the complex plane? (answer)When poles lie on a 45° line, or correspondingly when the real and imaginary magnitudes are equal, they have a damping ratio of 0.707.
- How does the shape of the response change as you change only the damping ratio? (answer)The system decays at a different rate but the response curve has a similar shape.
- What happens to the pole location when ζ = 1 (answer)When ζ = 1 the poles of the system are repeated. Verify that in Eqn 2.3.4. When ζ > 1 the system has two real poles and does not oscillate - its over damped. You can experiment with over damped system in the experiment at the end of this page.

The damped natural frequency, ω_{d}, 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.

The natural frequency, ω_{n}, 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,t_{settling}, ζ ω_{n} and ω_{d}.

Figure 2.3.1. Damped frequency

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

Response Descriptors for a DEQ

Answer: (show)

The closed loop transfer function is

Verify through simulation

G_{closed loop}(s) =

This has poles at s=-2.75±4.476i1.5(s+1)_________s^{2} + 5.5s + 27.5

T_{settling}=-4/a = 4/2.74 = 1.45sec

Writing the characteristic equation in standard form s^{2} + 5.5s + 27.5 = s^{2} + 2ζω_{n}s + ω_{n}_{2}

matching terms ζ=0.52, ω_{n}=5.25 r/s

Based on these parameter the system will reach steady state in 1.45sec and have some oscillation.Verify through simulation