Digital Control Systems [MEE 4003] - Kckong.info

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2.3. FREQUENCY RESPONSE ANALYSIS 47 0 −10 dB −20 degree −30 −40 10 −1 10 0 10 1 10 2 0 −50 −100 −150 Frequency (rad/sec) −200 10 −1 10 0 10 1 10 2 Frequency (rad/sec) Figure 2.14: Smoothly connected asymptotes. ½ ¾ ¿ÀÁÂÃ ÄÅÆÇÂÄ È ÉÄÅÆÊË Ì ÍÎÀÏÐÏÐÑ Ò ¿ÓÒÐÔÀÎÓ ÀÕÐÖ¿Ï×Ð ØÙÚÛÁ½Ê Ì Ü×ÝÎÔ ×À ½ÁÔÊ ÞÛßÙÁ½Ê Ì àÎÓ×Ô ×À ½ÁÔÊ áÙâÛÁ½ÊË Ì ÍÓÒã Ò ä×åÎ æÝ×¿ Figure 2.15: Matlab code for drawing a Bode plot Digital Control Systems, Sogang University Kyoungchul Kong
2.3. FREQUENCY RESPONSE ANALYSIS 48 [Example 2-17] The Bode plot of a transfer function G(s) = G(jω) = 4 s 2 +s+4 4 (jω) 2 +jω +4 is obtained by: √ 15 [Step 1] Since p 1,2 = −0.5 ± j , they are sorted by the absolute values (i.e., 2). 4 There is only one section boundary at 2. [Step 2-4] There are two sections: 0 < ω < 2, 2 < ω. For each section, the asymptote is • at ω = 0, |G(j0)| = | 4 4 | = 1 (i.e.,0dB) • 0 < ω < 2: G(jω) ≈ 4 4 (slope: 0, phase: 0) • 2 < ω: G(jω) ≈ 4 (jω) 2 (slope: -40, phase: -180) The exact value of|G(jω)| at the section boundary is ∣ |G(2j)| = 4 ∣∣∣ ∣(2j) 2 +2j +4∣ = 4 2j∣ = 2 [Step 5] Finally, the Bode plot is obtained by smoothly connecting the asymptotes as in Fig. 2.16, where the line must pass through the exact value at the section boundary. Graphical representation ofG(s) in the Laplace-domain andG(jω) in the frequency- Domain Remind that the Laplace domain and the frequency domain are related by s = jω. Since s is a complex number, which can be defined as s = σ +jω where σ = Re{s} ∈ R and ω = Im{s} ∈ R, the Bode plot is the line of intersection between G(s) and a plane of σ = 0. Namely, the Bode plot is a part of G(s) plotted on the s-plane, and thus information on the locations of poles and zeros of G(s) is resolved in the Bode plot. Digital Control Systems, Sogang University Kyoungchul Kong
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