Nyquist and Bode Criterion

Question 1

What is the Nyquist Diagram?

Answer 1

It is an undirect method to analyze the stability of the system. It is based on the open loop transfer function

Question 2

What are the two components on which the Nyquist Diagram is based?

Answer 2

Mapping and Cauchy principle of argument

Question 3

What is the Cauchy principle of argument?

Answer 3

The Cauchy principle of argument says: The map of an s function in the F plane (The curve described in the F plane by F as s is varied in the path Cs) encircles the origin of the F plane as many times N as the number of zeros minus the number of poles of the function F which are included in the path Cs (N=Z-P)

Question 4

How is the direction of the path defined?

Answer 4

If N (which is equal to the number of zeros minus the number of poles) is positive (Z>P) then the encirclements are clockwise. If N is negative (Z<P) the encirclements are counterclockwise

Question 5

What is the important condition for the Cauchy principle of argument?

Answer 5

We must count only the poles and zeros inside the path Cs

Question 6

How is the function F(s) for the Nyquist diagram defined?

Answer 6

-In order to determine the stability of the system, the function F is defined as the denominator of the closed loop transfer function L (den(L) = 1+GH) so the zeros of F(s) are equals to the poles of L (Zeros F = Poles L) and the poles of F(s) would be equal to the poles of the function GH (Poles F = Poles GH)

Question 7

How is the path Cs for the Nyquist diagram defined?

Answer 7

-For the system to be stable we need no Poles of L (zeros of F) in the right part of the Re-Im plane (all poles must be negative for the system to be stable) so the Cs plane will have an infinite radius covering the whole positive real axis of the plane

Question 8

How to determine the stability of the system according to the Cauchy principle of argument?

Answer 8

For the system to be stable, the number of loops around the origin done by the maps of the path Cs must be equal to the number of zeros of F inside Cs minus poles of F inside path Cs (N=-P). In sinthesis, for L to be stable no poles of L must lie inside Cs

Question 9

What is the Nyquist criterion for stability?

Answer 9

The closed loop transfer function L is stable if and only if the number of anticlockwise encirclements of point (-1,0) done by the Nyquist diagram of the open loop transfer function GH is equal to the number of unstable poles of GH

Question 10

Why we consider the point -1,0 for the encirclements?

Answer 10

We “shift” the origin based on the selection of the function F(s), which is equal to the denominator of the closed loop transfer function L and has the form 1+GH. Adjusting the “origin” to the -1 we would be able to evaluate the stability simply based on the open loop transfer function GH

Question 11

Which are the steps for drawing the Nyquist Diagram?

Answer 11

1. Look at the OLTF GH and count the number of unstable poles in order to calculate P
2. Draw the Nyquist of GH
3. Count the encirclements in the anticlockwise direction
4. Compare N with P

Question 12

What are the posible cases for stability using Nyquist?

Answer 12

1. N = 0, then L is stable and P=0 (no unstable poles of GH)
2. N<0, then L is stable and GH must have -P unstable poles
3. N>0, the L is unstable

Question 13

What is the procedure when theres a pole in the origin when studying GH for the Nyquist diagram?

Answer 13

We must close the map GH with an encirclement of infinite radius from pi/2 to -pi/2 (clockwise) to close the path. The number of clockwise encirclements must be equal to the number of poles in the origin (zero)

Question 14

What are the two hypothesis needed to be validated to use the Bode criterion?

Answer 14

1. The open loop transfer function GH does not have unstable poles
2. Amplitude of |GH|in dB crosses once the 0 (zero) dB axis

Question 15

What is the crossover frequency?

Answer 15

It is the frequency at which the phase of GH is equal to -pi

Question 16

What is the sufficient condition from Bode Criterion for stability?

Answer 16

If the amplitude of GH when its phase is equal to -pi is larger than one, then the transfer function L is unstable (sufficient condition only)

Question 17

What are the indexes of robust stability?

Answer 17

The phase margin and gain margin

Question 18

What are the conditions to determine stability using Bode?

Answer 18

1. Minimum phase. Which means that the transfer function GH has no unstable poles or zeros.
2. Both gain margin and phase margin are positive

Question 19

What are the values of robust stability?

Answer 19

The gain margin being bigger than 6 dB and phase margin bigger than 30° - 60°