I want to include a condition for the transfer function G(s) which is equivalent to the statement mentioned above inside inverted commas but I am not able to describe it mathematically. I was wondering if I could use the L2 norm in this case .
Please advise.
Thanks
1. If one wants to write in symbols the fact that "|f(x)|
Have a look to the notation used here http://www.maths.bath.ac.uk/~masjde/ODEs/LectureNotes/220ControlNotes.pdf
Dr Joachim, I think the point you made in no. 4 is quite relevant to me. I want to describe a condition where the magnitude of transfer function at all frequencies shouldn't exceed 1. Can the bode plot be helpful under such circumstances ?
@ Biswajit Debnath
The last question
Can the bode plot be helpful under such circumstances ?
requires adding: helful for what? Let me guess that you want to state that the values are all less than 1.
1o: If by writing in symbols - this I have answered already.
2o If graphically, then indeed a readable bode plot should be sufficient, obviously, the limit value 1 should be drawn clearly as the (upper) boundary of the graph.
Special technics of drawing bode plots are not known to me, but mathematically I can list few key points of graphs of functions. Some of them are:
case a: if the function is increasing, then the bound is a horisontal asymptote at infinity; at the origin the minimal left value should be greater than -1.
case b: if the function oscilates, then the upper and lower bounds should be indicated by two horisontal lines touching the graph at the maximum and the minimum of the graph.
Other cases are to be treated correspondingly.
Regards, Joachim
Thank You Dr. Joachim, your ideas have been very helpful.
I have now included "|| f ||\infty < 1" along with the peak value of the norm to show that it has a value much less than unity. Further I have plotted a bode plot to reinforce the fact that it doesn't exceed 1 (0 db) at any frequency.
Joachim has already provided a very good answer. I will only comment about the requirements (and possibly conflicts between) of precison, compactness, and readability.
In your question title, you introduce a symbol s, which may later be interpreted to mean frequency; but since that is not spelled out explicitly the statement becomes unacceptably vague. Replacing frequency by s would mathematically be more presise, if that is what you mean. Writing frequency s is even better (although longer) because it adds context to your statement.
The phrase for all is also unaccaptably vague; you must specify the exact region for which your statement should be valid. I think Joachim has covered that aspect already.
The introduction of precise and compact mathematical notation can be valuable, but this should not at first be used to replace a description which can be understood by a much larger group of readers. You should not use notation which you did not understand long before you started to write your article (it is permissible to introduce new notation that you like or invented). Obfuscation is not an acceptable way to present science.
Dr. Kare, for my case s=jw, where w is the frequency. Basically I am trying to describe a transfer function whose behavior is like that of a low pass filter with pass gain less than 1. This means that the gain shouldn't exceed 1 at any frequency, right?
Further, I wanted to make it mathematically precise in order to avoid any ambiguity associated with it.
Just as it's written. What's wrong with |G(s)| < 1 for all s? The statement that G(s) is a transfer function implies all sorts of consequences, that are ways of restating this fact, among others.
As usual, what's missing is what are the assumptions, that are supposed to imply that |G(s)| < 1 for all s.
Stam Nicolis, that is what I want to figure out. I want to know the difference between writing "|G(s)|
If they are, of course; if they aren't, of course not. Just look up the definition of the infinity norm. But then it would have been better, had you asked that question.
This is, once more, a homework question.
Dear Biswajit,
your problem is related to the values of the Laplace transform (transfer function) in the unitary circle.
sqrt([Re G(s)]2 + [Im G(s)]2) < 1 is just the same thing and you can find the s such that the transfer function stays within the unitary circle in the complex plane.
I read Kåre's remark rather as a general warning/wish/call for just avoiding unnecessary usage of math symbols. I wouldn't say that the question is over saturated by symbols. However, it is unnecessary involving interpretation of the domain of the function. Quite sufficient would be to ask: for a comlex function f are the properties "|f(x)
@ Stefano,
You are extending Martin's question of finding such "frequencies" s for which the transfer function stays within the unit circle, which has an interpretation in terms of filters: which frequencies are quenched? Obviously, answer to such qustions is not expressible in terms of norms since norms say about the total size of the function with taking into account all values, without, however, localizing them. Thus, i particular two tranfer functions, one of low-pass filter and the other of high-pass filter with exactly the same norm:
If G1(s) := \sqrt{2}/(1+s); G2(s) =\sqrt{2}s/(1+s) for s=jw, w \in R, then
||G1||\infty= ||G2||\infty= \sqrt{2}, |G1(s)|
Is your question related with the Nyquist Criterion ? It is related with feedback problems and poles on the imaginary axis
You can see it simply in
https://en.wikipedia.org/wiki/Nyquist_stability_criterion#Nyquist_plot
A possible notation can be
|G(j \omega )| < 1 \ \forall \omega \in {\mathbb R}
and an alternative can be
|G(s)| < \ \forall s \in j{\mathbb R}.
I assume you are working in electrical and/or control engineering since you are using "j" for the imaginary unit.
See also: Bode plot.
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