سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي - دکتر سید مجید اسما عیل زاده

Frequency Response

3 DIRECT POLAR PLOTS We can plot transfer function in polar plan

Transfer function: 1. Proportional element Re Im K Polar plot DIRECT POLAR PLOTS

2. Integrating element Transfer function: Polar plot Re Im DIRECT POLAR PLOTS

3. Inertial element Transfer function: Polar plot Re Im 1 DIRECT POLAR PLOTS

4. Oscillating element Transfer function: maximum value of ： Make: DIRECT POLAR PLOTS

Polar plot Re Im 1 DIRECT POLAR PLOTS

5. Differentiating element Transfer function: Polar plot Re Im Re Im 1 Re Im 1 differential1th-order differential2th-order differential DIRECT POLAR PLOTS

6. Delay element Transfer function: Polar plot Re Im R=1 DIRECT POLAR PLOTS

11 DIRECT POLAR PLOTS To obtain the direct polar plot of a system’s forward transfer function, the following criteria are used to determine the key parts of the curve.

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16 Construction of Nyquist loci –Loop transfer function –By hand Calculate features –Asymptotes: behavior as –Location of axes crossings Example: Example:

17 System loop –Construct Nyquist:

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19 –Quantitative analysis Limits

20 Re L Im L 0.3

21 Asymptotes for large  –Keep dominant terms –For  positive: + imaginary axis –For  negative:  imaginary axis

22 Re L Im L 0.3

23 –Imaginary axis crossing(s) Real part = 0

24 Re L Im L 0.3 

25 –Real axis crossing(s) Imaginary part = 0

26 Re L Im L 0.3  0.8  0.64

27 Matlab: Nyquist  0.8 