Stability Analysis for Nyquist Criteria Homework Question

[jaus tail]

Homework Statement

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Homework Equations

Number of encirclements = Number of open loop poles - Number of Close loop poles on Right side of S plane.

The Attempt at a Solution

There is 1 open loop pole on RHS
For Close loop poles I used Routh Herwitz method and got 1 pole on RHS. 1 sign change.
So I get N = 0.
Where am I wrong?

 

[rude man]

Homework Statement

View attachment 216387[/B]

Homework Equations

Number of encirclements = Number of open loop poles - Number of Close loop poles on Right side of S plane.

Why are you trying to deal with closed-loop poles? Nyquist is strictly an open-loop stability criterion. G(s) is the open-loop transfer function.

 

[jaus tail]

Why are you trying to deal with closed-loop poles? Nyquist is strictly an open-loop stability criterion. G(s) is the open-loop transfer function.

Nyquist criteria says encirclement of -1 + j0 is number of open loop poles - series of characteristic equation.
Characteristic equation is 1 + G(s)

 

[rude man]

Nyquist criteria says encirclement of -1 + j0 is number of open loop poles - series of characteristic equation.
Characteristic equation is 1 + G(s)

What do you mean by "series of characteristic equation"?
The Nyquist method does not involve closed-loop transfer functions.
Nyquist determines whether the closed-loop transfer function is stable but its methodology does not involve any closed-loop transfer functions.

 

[rude man]

I see from your attachment that in some cases they do consider closed-loop RHS poles, in others they stick to open-loop only.
I have to admit I never heard of doing Nyquist analysis with anything other than open-loop transfer functions. Seems to me undesirable to have to compute 1 + G(s).
So the only way I know to do this is
(1) determine the Re and Im parts of G
(2) draw polar plot of G
(3) follow rules of Nyquist stability determination.
Sorry that's all I can tell you.

 

[jaus tail]

Sorry for the typo. It was 'zeroes' of characteristic equation and not 'series'. But yeah you're right. I read the question wrong. It says encircle the origin and not encircle -1

 

[rude man]

Sorry for the typo. It was 'zeroes' of characteristic equation and not 'series'. But yeah you're right. I read the question wrong. It says encircle the origin and not encircle -1

OK. I have to admit I don't know on what basis the solution to ex. 39 is given.
If G(s) is an open-loop transfer function then the thing that matters for determining stability of G(s)+1 is encirclement of G(s) of (-1,0), not (0,0). In other words, I guess I really don't understand their reasoning.