Asymptotes for the phase

Dear Ian,

The mathematical asymptote is definitely a jump in phase.
For example, consider a term 1/(s+1). This has asymptotes 1 (for w<1) and
1/s (for w>1), and
the phase of 1/s is -90. But as noted this asymptote is not very precise as
the actual phase
is -arctan(w).

Alr.1: Tangent
We could improve the approximation by using the tangent at the break
frequency (w=1 in the above example)
where the exact phase is -arctan(1)=-45. This tangent has a slope of
ln(10)/2 rad/decade = 66 degrees/decade and thus crosses the asymptotes of
0 and -90 at 0.68 (=45/66) decades below and above the break frequency,
respectively.

Alt.2: "1 decade on each side rule"
You suggest to let phase change from 1 decade below to 1 decade above the
break frequency.
Is there a mathematical justification for this (like minimizing the ISE
deviation) or is it
a "rule of thumb" (which is simple and gives a reasonably good approximation).

In any case, it may be good to some comment about this in our revised edition.
If you think we should then could you please suggest a text.

I am going through the book these days and making the final changes.
I have to finish up before going to Mallorca in the beginning of August!

Best regards,
Sigurd

At 05:27 PM 7/18/01 +0100, you wrote:
>Dear Michael
>
>Yes, I know what you mean. This is a better approximation and what I would
>usually teach. Still what is in the book is also an approximation, and with
>Matlab who needs an approximation annyway.
>
>Cheers
>
>Ian
>
>PS I will copy this to Sigurd.
>
>
>
> > -----Original Message-----
> > From: Michael Tombs
> > [mailto:michael.tombs@engineering-science.oxford.ac.uk]
> > Sent: 18 July 2001 16:54
> > To: Postlethwaite, Prof I.
> > Subject: Re: Your Book
> >
> >
> > Ian,
> >
> > Thought I should mention that I've been making extensive use
> > of your great
> > book to recap what I used to know and where it went after I
> > left Oxford.
> > One little thing I thought I should tell you which
> > surprisingly doesn't seem
> > to be noted in the corrigenda on your website (perhaps no-one
> > reads this
> > bit!)
> >
> > In your introduction to classical control on page 20/21 you
> > refer to the
> > straight line asymptotes for bode plots and note that the
> > phase doesn't
> > follow the asymptotes closely. The usual straight-line
> > approximation to the
> > phase change is a slope from a decade down to a decade above
> > - which would
> > appear to be a good approximation!
> >
> > I'm sure this is of no consequence for later in the book!
> >
> > Regards
> >
> > Michael
> >
> > Dr M S Tombs
> > Programme Manager
> > Invensys UTC for Advanced Instrumentation
> > University of Oxford
> > Department of Engineering Science
> > Parks Road
> > Oxford
> > OX1 3PJ
> > Tel: +44 1865 283317, Fax: +44 1865 273906
> > michael.tombs@eng.ox.ac.uk
> >
> >

******************************************************************************************
Sigurd Skogestad, Professor and Head of Department Phone: +47-7359-4154
Department of Chemical
Engineering Home: +47-7393-6855
Norwegian Univ. of Science and Technology (NTNU) Fax: +47-7359-4080
N-7491 Trondheim email:
sigurd.skogestad@chembio.ntnu.no
Norway
www.chembio.ntnu.no/users/skoge
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Received on Thu Jul 19 11:53:41 2001