Process Dynamics and Control - Dale, Thomas and Duncan

Anyone using this book for study or reference? I need help on Section 7, page 167, example 7.1.
I followed the example and used the equations shown but I ended up with
different values for the model parameter ao and a1.
Your help is very much appreciated.
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On Mon, 16 Jul 2012 11:15:25 -0700, The Romanov wrote:

If it doesn't take too long you can always try posting an outline of the problem here (or the whole thing if it's short).
Speaking as both a reader and an author -- it's not inconceivable that the author messed up his own calculations.
--
Tim Wescott
Control system and signal processing consulting
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On Wednesday, July 18, 2012 6:00:12 PM UTC+3, Tim Wescott wrote:

Hi Tim.
I wish I could post or attach scan copy of the problem outline here but it is not possible due to limited functionality of group discussion. It is for the earlier reasons that I quoted the exact section, page and example no. so that someone using the book can refer to. However, I'll try to explain below where possible though it can be difficult to type legible equations in this group discussion.
The example provided several equations as below;
a0S00 + a1S01 = T0
a0S10 + a1S11 = T1
where,
S00 = sum of square 1 for data i=1 to 5
S01 = sum of 1 time xi for data i=1 to 5
S10 = sum of xi time 1 for data i=1 to 5
S11 = sum of square of xi for data i=1 to 5
T0 = sum of 1 times yi for data i=1 to 5
T1 = sum of xi times yi for data i=1 to 5
Data refers to;
xi 1.0 2.3 2.9 4.0 4.9 yi 2.0 4.4 5.4 7.5 9.1
When I followed the example question, I go the following results;
S00 = 1
S01 = 15.1
S10 = 15.1
S11 = 54.71
T0 = 28.4
T1 = 102.37
On substituting these numbers into first two equations above and solve algebraically, I got different result than stipulated on the book.
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On Sat, 21 Jul 2012 07:57:30 -0700, The Romanov wrote:

So it's some sort of least-mean square minimization problem?
The sum of 1, five times, is 5, not 1 -- perhaps that's your problem?
--
Tim Wescott
Control system and signal processing consulting
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On Sat, 21 Jul 2012 07:57:30 -0700, The Romanov wrote:

Hah. It took me a while to realize what exactly you're doing. (you _could_ give an overview).
They're asking you to do a 1st-order least-squares fit of x and y; this is the optimal result if y = a*x + b + n, where a and b are constant and n is a vector of independent Gaussian variables of zero mean and constant variance.
Taking x and y as column vectors, what you really want to solve for is
[x^0 x] * A = y,
where A = [a b]^T ("^T" meaning "transform of", so A is a two-element column vector)
They don't want you wrapped around the axle with higher-order linear math like singular value decomposition, so they're defining
X = [x^0 x] (note that x^0 is just a column of ones),
then you want to solve
X * A = y
but X is 2 x 5 and y is 1 x 5, so you don't have the tools to do it. However, thanks to some clever mathematicians in the 18th and 19th century, you know that you can multiply through by X^T to get
X^T * X * A = X^T * y
This reduces the problem to a 2x2 times a 1x2 = a 1x2 -- and you can solve that with "ordinary" linear algebra, at the expense of some numerical precision from squaring X. (as X gets bigger, you change the reading to "at the expense of lots of numerical precision..., and you investigate the use of better linear algebra techniques).
--
Tim Wescott
Control system and signal processing consulting
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