• Transfer function reduction math

    From bitrex@21:1/5 to All on Tue Jun 11 14:08:23 2024
    The "Lecture Notes in Control and Information Sciences" can be a
    fascinating book to page through, particularly if you can get your hands
    on a hard copy.

    The particular one I'm looking through is from the early 1980s and
    there's a lot of interesting material related to optimization problems
    here, like "Optimal turning strategy for a supercruiser" (aircraft) and "Optimal maintenance policy and sale date for a machine with random deterioration and subject to random catastrophic failure"...

    A fair bit of the mathematics assumes a certain baseline knowledge of
    the field of systems optimization/linear programming/etc and I don't
    easily follow most papers, but there are some papers of interest to
    electrical engineering, e.g. one about reduction of order of transfer
    functions using a minimum-phase approximation, higher order transfer
    functions sometimes contain more information than you need for a
    restricted bandwidth.

    Unfortunately partly due to the pre-Latex typesetting e I'm unclear what
    this one is saying exactly:

    <https://imgur.com/a/9HIKOEN>

    H(s) is just a regular s-domain transfer function with polynomials top
    and bottom, so they decompose it into odd and even parts and set it
    equal to...what's tanh phi(s) supposed to mean? Tanh(s)phi(s)?
    Tanh(phi(s))?

    Seems like they're doing some kind of tanh interpolation but it's not
    entirely obvious to me how they get from equation (3) to the expression
    in (5).

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Don@21:1/5 to bitrex on Tue Jun 11 19:44:32 2024
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not entirely obvious to me how they get from equation (3) to the expression
    in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    A clearer scan may enable me to continue.

    Danke,

    --
    Don, KB7RPU, https://www.qsl.net/kb7rpu
    There was a young lady named Bright Whose speed was far faster than light;
    She set out one day In a relative way And returned on the previous night.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From bitrex@21:1/5 to Don on Tue Jun 11 17:00:07 2024
    On 6/11/2024 3:44 PM, Don wrote:
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not
    entirely obvious to me how they get from equation (3) to the expression
    in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    Right, I see that.

    A clearer scan may enable me to continue.

    Danke,


    Sure, here's the full page in question:

    <https://imgur.com/a/as3jfNo>

    I have a hardcopy from an academic library which is a relatively massive
    (800+) page tome so difficult to get a good scan of...the only full-text
    online I can find is on Springerlink (blech) and despite my having an "institutional login" that should grant access to it. it never seems to
    work with them.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From bitrex@21:1/5 to Don on Tue Jun 11 21:12:03 2024
    On 6/11/2024 3:44 PM, Don wrote:
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not
    entirely obvious to me how they get from equation (3) to the expression
    in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    A clearer scan may enable me to continue.

    Danke,


    (not sure if my response posted as I don't see it on my newsreader,
    apologies if this reply appears twice)

    Sure, here's the full page in question:

    <https://imgur.com/a/as3jfNo>

    I have a hardcopy from an academic library which is a relatively massive
    (800+) page tome so difficult to get a good scan of...the only full-text
    online I can find is on Springerlink (blech) and despite my having an "institutional login" that should grant access to it. it never seems to
    work with them.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Don@21:1/5 to bitrex on Wed Jun 12 03:17:43 2024
    bitrex wrote:
    Don wrote:
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not
    entirely obvious to me how they get from equation (3) to the expression
    in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    A clearer scan may enable me to continue.

    Danke,


    (not sure if my response posted as I don't see it on my newsreader,
    apologies if this reply appears twice)

    Sure, here's the full page in question:

    <https://imgur.com/a/as3jfNo>

    I have a hardcopy from an academic library which is a relatively massive (800+) page tome so difficult to get a good scan of...the only full-text online I can find is on Springerlink (blech) and despite my having an "institutional login" that should grant access to it. it never seems to
    work with them.

    Your first followup was indeed posted.

    The first three steps from (4) to (5) are easy-peasy:

    tanh(s) = (e^s - e^-s) / (e^s + e^-s)
    H(s) = Q(s) / D(s) = (e^s - e^-s) / (e^s + e^-s)
    (e^s + e^-s)Q(s) = (e^s - e^-s)D(s)

    Control Theory must now be reviewed by me in order to continue.

    # # #

    "Lecture Notes in Control and Information Sciences" seems to be a series
    of books, each about three hundred pages long. Where do you find page
    808?

    Danke,

    --
    Don, KB7RPU, https://www.qsl.net/kb7rpu
    There was a young lady named Bright Whose speed was far faster than light;
    She set out one day In a relative way And returned on the previous night.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From bitrex@21:1/5 to Don on Wed Jun 12 00:05:52 2024
    On 6/11/2024 11:17 PM, Don wrote:
    bitrex wrote:
    Don wrote:
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not
    entirely obvious to me how they get from equation (3) to the expression >>>> in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    A clearer scan may enable me to continue.

    Danke,


    (not sure if my response posted as I don't see it on my newsreader,
    apologies if this reply appears twice)

    Sure, here's the full page in question:

    <https://imgur.com/a/as3jfNo>

    I have a hardcopy from an academic library which is a relatively massive
    (800+) page tome so difficult to get a good scan of...the only full-text
    online I can find is on Springerlink (blech) and despite my having an
    "institutional login" that should grant access to it. it never seems to
    work with them.

    Your first followup was indeed posted.

    The first three steps from (4) to (5) are easy-peasy:

    tanh(s) = (e^s - e^-s) / (e^s + e^-s)
    H(s) = Q(s) / D(s) = (e^s - e^-s) / (e^s + e^-s)
    (e^s + e^-s)Q(s) = (e^s - e^-s)D(s)

    Control Theory must now be reviewed by me in order to continue.

    Thanks, I think I see sorta see how (5) is derived now. I believe they
    mean by their notation tanh(phi(s)) and phi(s) = arctan(Q(s)/D(s)).

    The denominator of (4) will be real, and the portions of the numerator
    that are an even function times an even function will be real and the
    parts that are anything else will be imaginary, cuz in e^ix = cos(x) + i
    sin(x) the sin is imaginary and sin is an odd function, when each term
    in the expanded fraction is expressed as a magnitude and phase angle.

    Then there's a logarithmic form of the arctangent, arctan(z) =
    -i/2*ln[(1 + iz)/(1 - iz)] and for z(s) = Q(s)/D(s) as decomposed into
    even and odd parts in (4), I think plugging that form into
    tanh(arctan(z)) = (e^z - e^-z) / (e^z + e^-z) should then give (5),
    though I haven't grunged it all out to check.


    # # #

    "Lecture Notes in Control and Information Sciences" seems to be a series
    of books, each about three hundred pages long. Where do you find page
    808?

    Danke,


    This one:

    <https://link.springer.com/book/10.1007/BFb0006119?page=6>

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From bitrex@21:1/5 to bitrex on Wed Jun 12 00:20:54 2024
    On 6/12/2024 12:05 AM, bitrex wrote:

    even and odd parts in (4), I think plugging that form into
    tanh(arctan(z)) = (e^z - e^-z) / (e^z + e^-z)

    tanh(arctan(z)) = [e^(arctan(z)) - e^(-arctan(z))]/[e^(arctan(z)) + e^(-arctan(z))], rather.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From bitrex@21:1/5 to bitrex on Wed Jun 12 00:16:48 2024
    On 6/12/2024 12:05 AM, bitrex wrote:
    On 6/11/2024 11:17 PM, Don wrote:
    bitrex wrote:
    Don wrote:
    bitrex wrote:

    <snip>

    Seems like they're doing some kind of tanh interpolation but it's not >>>>> entirely obvious to me how they get from equation (3) to the
    expression
    in (5).

    Unless the fuzzy form of your scan deceives my eyes, it appears the
    numerator and denominator are multiplied by the conjugate to obtain
    (4) from (3).

    A clearer scan may enable me to continue.

    Danke,


    (not sure if my response posted as I don't see it on my newsreader,
    apologies if this reply appears twice)

    Sure, here's the full page in question:

    <https://imgur.com/a/as3jfNo>

    I have a hardcopy from an academic library which is a relatively massive >>> (800+) page tome so difficult to get a good scan of...the only full-text >>> online I can find is on Springerlink (blech) and despite my having an
    "institutional login" that should grant access to it. it never seems to
    work with them.

    Your first followup was indeed posted.

    The first three steps from (4) to (5) are easy-peasy:

         tanh(s) = (e^s - e^-s) / (e^s + e^-s)
         H(s) = Q(s) / D(s) = (e^s - e^-s) / (e^s + e^-s)
         (e^s + e^-s)Q(s) = (e^s - e^-s)D(s)

    Control Theory must now be reviewed by me in order to continue.

    Thanks, I think I see sorta see how (5) is derived now. I believe they
    mean by their notation tanh(phi(s)) and phi(s) = arctan(Q(s)/D(s)).

    The denominator of (4) will be real, and the portions of the numerator
    that are an even function times an even function will be real and the
    parts that are anything else will be imaginary

    oops, odd*odd will also give an even function.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)