System Identification in Frequency Domain The Bode plot of a dynamic system is shown in Figure 10.73, in which the asymptotes are also given. Following the rules of sketching Bode plots, find the transfer function of the system.

Question

System Identification in Frequency Domain

The Bode plot of a dynamic system is shown in Figure 10.73, in which the asymptotes are also given. Following the rules of sketching Bode plots, find the transfer function of the system.

Accepted Answer

As shown in Figure 10.73, the Bode magnitude plot starts from [latex]−20 ext{dB}[/latex], the slope changes at [latex]10 ext{rad/s}[/latex] from [latex]0 ext{dB/decade}[/latex] to [latex]−40 ext{dB/decade}[/latex], and there is a lightly damped resonant peak at [latex]10 ext{rad/s}[/latex]. This implies that the frequency response function of the system consists of one constant term and one second-order term in the denominator,

[latex]G( ext{j}ω) = \frac{K}{( ext{j}ω/ω_n)^2 + 2 ζ( ext{j}ω/ω_n) + 1}[/latex]

Note that the Bode phase plot starts from [latex]0°[/latex] and changes at [latex]10 ext{rad/s}[/latex] to [latex]−180°[/latex]. This also indicates a second-order term in the denominator.
The initial magnitude [latex]−20 ext{dB} = 20 ext{log}_{10}K[/latex], which gives

[latex]K= 10^{\frac{-20}{20}} = 0.1[/latex]

The slope changes at [latex]10 ext{rad/s}[/latex], indicating the corner frequency, also the natural frequency, of the second-order term is [latex]ω_n = 10 ext{rad/s}[/latex]. The value of the peak is [latex]−12 ext{dB}[/latex], or [latex]8 ext{dB}[/latex] above the asymptotes, indicating

[latex]20 ext{log}_{10}\frac{1}{2ζ} = 8 ext{dB}[/latex]

which gives

[latex]ζ = \frac{1}{2\left( 10 ^{\frac{8}{20}} ight) } = 0.20[/latex]

Thus, the frequency response function of the system is

[latex]G( ext{j}ω) = \frac{0.1}{( ext{j}ω/10)^2 + 2 (0.2)( ext{j}ω/10) + 1}[/latex]

and replacing [latex] ext{j}ω[/latex] with [latex]s[/latex] gives the transfer function of the system as

[latex]G(s) = \frac{10}{s^2 +4s + 100}[/latex]

Question 10.18: System Identification in Frequency Domain The Bode plot of a...

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