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Nyquist-criterion :Example

5 Effect of Poles at the Origin Nyquist-criterion :Example

Application of the Nyquist criterion in the Bode diagram G(jω)H(jω) locus traverses the left real axis of the point (-1, j0) in G(jω)H(jω)- plane → L(ω)≥0dB and φ(ω) = － 180 o in Bode diagram We have the Nyquist criterion in the Bode diagram : The sufficient and necessary condition of the stability of the linear closed loop systems is : When ω vary from 0→+ , the number of the net “positive traversing” is P/2. Here: the net “positive traversing” —the difference between the number of the “positive traversing” and the number of the “negative traversing” in all L(ω)≥0dB ranges of the open-loop system’s Bode diagram. “positive traversing” — φ(ω) traverses the “-180 o line” from below to above in the open-loop system’s Bode diagram; “negative traversing” — φ(ω) traverses the “-180 o line” from above to below. 6

Application of the Nyquist criterion in the Bode diagram Example The Bode diagram of a open-loop stable system is shown, determine whether the closed loop system is stable. In terms of the Nyquist criterion in the Bode diagram: Because the open-loop system is stable, P = 0. The number of the net “positive traversing” is 0 ( = P/2 = 0 ). The closed loop system is stable. 0dB, － 180 o － 20 － 40 － 60 － 40 － 20 － 40 － 60 － 270 o － 90 o L(ω) φ(ω) Solution 7

Nyquist-criterion : relative stability

In frequency domain, the relative stability could be described by the “gain margin” and the “phase margin”. 1. Gain margin K g Nyquist criterion and the relative stability (Relative stability of the control systems) 2. Phase margin γ c 3. Geometrical and physical meanings of the K g and γ c 9

Nyquist criterion and the relative stability The geometrical meanings is shown. Re －1－1 Im γ c 1/K g stable Critical stability unstable The physical signification : K g — amount of the open-loop gain in decibels that can be allowed to increase before the closed-loop system reaches to be unstable. For the minimum phase system: K g >1 the closed loop system is stable. γ c —amount of the phase shift of G(jω)H(jω) to be allowed before the closed-loop system reaches to be unstable. For the minimum phase system: γ c >0 the closed loop system is stable. 10

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13 The specification of phase and gain margins requires a compromise between performance and robustness. In general, large values of GM and PM correspond to sluggish closed-loop responses, while smaller values result in less sluggish, more oscillatory responses. Guideline: In general, a well-tuned controller should have a gain margin between 1.7 and 4.0 and a phase margin between 30°and 45°.

Nyquist criterion and the relative stability The changes of the open-loop gain only alter the magnitude of G(jω)H(jω). The changes of the time constants of G(s)H(s) only alter the phase angle of G(jω)H(jω). Attention : For the linear systems: Example: (1) Determine K g and γ c when K =1 and τ =1. The open loop transfer function of a control system is: 16

Nyquist criterion and the relative stability Solution (1) Determine K g and γ c ( K =1 and τ =1)

Nyquist criterion and the relative stability Re Im ( - 1, j0) G(jω)H(jω) Example: (1) Determine K g The G(jω)H(jω) polar plot of a system is shown (2) Determine the stable range of the open loop gain. Solution (1) Determine K g (2) Determine the stable range of the open loop gain. 18

Re Im ( - 1, j0) G(jω)H(jω) 19