BUSINESS ASSIGNMENT

Nyquist Criterion

Stability of mechanical systems

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Nyquist stability criterion • By using polar plot, one can say whether the system is

stable, unstable or maginally stable. • However we cannot determine the absolute stability (

No of poles in the RHP) • N=P-Z where N=number of encirclements about -1+j0 (clockwise (-), anticlockwise(+)) P=number of poles in the RHP for OLTF Z= number of poles in the RHP for the CLTF Number of encirclements about the critical point (-1+j0) must be equal to the poles of characteristic equation

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Rules for drawing Nyquist plots

i. Locate the poles and zeros of open loop transfer function G(s)H(s) in ‘s’ plane. ii. Draw the polar plot by varying ω from zero to infinity. If pole or zero present at s = 0, then varying ω from 0+ to infinity for drawing polar plot. iii. Draw the mirror image of above polar plot for values of ω ranging from −∞ to zero (0− if any pole or zero present at s=0). iv. The number of infinite radius half circles will be equal to the number of poles or zeros at origin. The infinite radius half circle will start at the point where the mirror image of the polar plot ends. And this infinite radius half circle will end at the point where the polar plot starts.

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Nyquist Diagram

The Nyquist diagram is the line joining the series of points plotted on a polar graph when each point represents the magnitude and phase of the open-loop frequency response corresponding to a particular frequency.

To plot the Nyquist diagram from the open-loop transfer function of a system we need to determine the magnitude and the phase as functions of frequency.

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Polar plot

• The polar plot of the frequency response of a system is the line traced out as the frequency is changed from 0 to infinity by the tips of the phasors whose lengths represent the magnitude, i.e. amplitude gain, of the system and which are drawn at angles corresponding to their phase

• i.e if z =x + jy phase= Tan-1 ( 𝑦

𝑥 )

• Magnitude= 𝑥2 + 𝑦2 •

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A polar plot of a sinusoidal transfer function G (jw) is a plot of f magnitude of G(jw) Vs phase angle as w is varied from 0 to ↑ ∞

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Example

• A unity feedback system has a loop TF

• 𝐺 𝑠 = 50

𝑠+1 𝑠+2

• Use Nyquist criterion to determine the system stability in closed loop configuration

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Solution

• Replace s with jw

• 𝐺 𝑠 = 50

𝑗𝑤+1 𝑗𝑤+2

• Magnitude ( |G(jω)|)= 50

1+𝑤2 𝑤2+4

• Phase 𝜑 = − tan−1 𝑤, − tan−1 𝑤

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S.N W M 𝝋

1 0 25 0

2 1 15.8 -45-26.56=-72

3 2 8 -108

4 10 0.5 -163

5 20 0.1 -171

6 100 0.005 -178

7 ∞ 0 -180

M 𝟓𝟎

𝟏+𝒘𝟐 𝒘𝟐+𝟒 𝝋 = − 𝒕𝒂𝒏−𝟏 𝒘, − 𝒕𝒂𝒏−𝟏

𝒘

𝟐

S.N W M 𝝋

1 0 25 0

2 1 15.8 -45-26.56=-72

3 2 8 -108

4 10 0.5 -163

5 20 0.1 -171

6 100 0.005 -178

7 ∞ 0 -180

If P=0, the open loop transfer function is stable

Next we get the magnitude and phase i.e G(jω)=|G(jω)|∠G(jω)

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SN w M 𝝋

1 0 0 -90

1 6.3 -162

2

3.33

10

100

180

∞

0 0 -90

1 6.3 -162

3.33 -180

2 1.58 -198.4

5 0.15 -236.89

10 0.0195 -253

100 0.0002 -268

∞ 0 -270

Checking for stability N=P-Z N=1 P=0 N=-Z (OLTF is stable) For CLTF Z=0 Hence N=P (Stable )

We know that the closed loop control system is stable if there is no closed loop pole in the right half of the ‘s’ plane.

Quizzes

Determine the stability of the above system using Nyquist Criterion Assuming unity feedback

Using Matlab

The -1+j0 point is not encircled so

N=0. There are no poles of L(s) in the right

half plane so P=0. Since N=Z-P, Z=0. This

means that the characteristic equation of

the closed loop transfer function has no

zeros in the right half plane (the closed

loop transfer function has no poles

there). The system is stable.

We can check this by finding the location of

the zeros of the characteristic equation:

Nyquist plot is symmetric along the real- axis

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Critical point in Nyquist plot -1+j0

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Consider the equation of an open loop transfer function

For stability all the zeros of F(s) are in the left-half of the s plane. No need to calculate the poles of closed loop transfer function!!

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Stability Analysis using Nyquist Plots

• From the Nyquist plots, we can identify whether the control system is stable, marginally stable or unstable based on the values of these parameters.

• Gain cross over frequency and phase cross over frequency

• Gain margin and phase margin

Phase Cross over Frequency

• The frequency at which the Nyquist plot intersects the negative real axis (phase angle is 1800) is known as the phase cross over frequency. It is denoted by 𝑤𝑝𝑐

Gain Cross over Frequency

• The frequency at which the Nyquist plot is having the magnitude of one is known as the gain cross over frequency. It is denoted by .

𝑤𝑔𝑐

• A MATLAB program that can be used to plot a Nyquist diagram for, say, the open loop transfer function 4/(s2 + 5s + 2) is:

• num = 4;

• den = [1 5 2];

• subplot (211), nyquist(num, den)

• Note: subplot(211) divides the screen into two equal halves and puts the current plot into the top half of the screen.

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