compilation.txt 26 May 1997
In Spring 1997 at MSRI, I did a search to attempt to determine how
common it is for a `transfer function to be not controllable by real data'
For this, I generated 8 groups of 50 transfer functions, and 4 groups of 25.
For each transfer function in the first set of 8, I tested 25 different
polynomials with distinct real roots (for the sets of 25, I tested 50).
Within each group, the transfer functions were generated randomly in
Singular, where I specified the range of the (integral) coefficients for the
polynomial entries, and a scheme to choose the roots of the polynomials.
Below, these choices are detailed. In each group, I summarize the results
of the search for that group in an 8-tuple of integers, where the (i+1)st
entry is the number of times I found 2*i real feedback laws.
Here is the result of these 15,000 trials:
Total Frequency
[23, 804, 3370, 5345, 3854, 1325, 272, 7]
Frequency, (%)
[.15, 5.4, 22.5, 35.6, 25.7, 8.8, 1.8, .047]
This search began in January 1997, and finished in May 1997.
I estimate that it took about 700,000 seconds of CPU
for the Gr\"obner basis calculations in Singular, and about
60,000 seconds for the real root determination in Maple.
All of this was on a HP 9000 D250, 800 series computer.
Frank Sottile,
MSRI
#########################################################################
First set:
50 systems represented via (2x6)-matrices of degree 4 polynomials whose x^4
terms look like:
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
and whose other coefficients were integers chosen randomly as follows:
-8 <= Cubic Terms <= 8
-14 <= Quadratic Terms <= 14
-35 <= Linear Terms <= 35
-70 <= Constant Terms <= 70
The closed loop polynomials \phi were chosen by generating 25
random 8-tuples X1,...,X8 in the following manner:
X1 = -7 and the successive differences X2-X1, X3-X2, ...
were random numbers between 1 and 3.
For each matrix of polynomials and 8 points, we computed the
number of real feedback laws which place the poles at those 8 points.
[ 0, 51, 265, 470, 328, 115, 18, 3 ]
whose jth component is the number of times there were 2j-2 real roots.
###################################################################
Second set:
50 similar systems, but where the poles were chosen with
X1=0 and the same successive differences
[0, 41, 313, 438, 319, 112, 24, 3]
###################################################################
Third set:
50 transfer functions with
-15 <= Cubic Terms <= 15
-25 <= Quadratic Terms <= 25
-35 <= Linear Terms <= 35
-45 <= Constant Terms <= 45
and poles chosen with X1 = -12 and successive differences between 1 and 5.
[0, 58, 240, 482, 339, 115, 16, 0]
###################################################################
Fourth and Fifth sets:
100 more transfer functions, which were not in `standard' form:
There were 100 (2x6)-matrices of random degree 4 polynomials with:
-8 <= Quartic Terms <= 8
-10 <= Cubic Terms <= 10
-12 <= Quadratic Terms <= 12
-14 <= Linear Terms <= 14
-16 <= Constant Terms <= 16
and poles with X1 = -7 and successive differences between 1 and 3.
[15, 101, 348, 466, 253, 54, 13, 0]
[4, 100, 352, 419, 306, 66, 3, 0]
###################################################################
Sixth set:
50 transfer functions, where:
-6 <= Quartic Terms <= 6
-6 <= Cubic Terms <= 6
-6 <= Quadratic Terms <= 6
-6 <= Linear Terms <= 6
-6 <= Constant Terms <= 6
and poles with X1 = -7 and successive differences between 1 and 3.
[2, 63, 331, 483, 279, 77, 15, 0]
###################################################################
Seventh set:
50 transfer functions consisting of (2x6)-matrices of degree 4
polynomials with x^4 terms: and constant terms:
| 1 0 0 0 0 0 | | 0 0 0 0 0 1 |
| 0 1 0 0 0 0 | | 0 0 0 0 1 0 |
and
-5 <= Cubic Terms <= 5
-4 <= Quadratic Terms <= 4
-3 <= Linear Terms <= 3
and poles with X1 = -7 and successive differences between 1 and 3.
[0, 102, 304, 388, 325, 129, 2, 0]
###################################################################
Eighth set:
50 transfer functions chose with the same statistics,
and poles with X1 = -12 and successive differences between 1 and 5.
[2, 91, 351, 367, 307, 113, 19, 0]
###################################################################
Ninth set:
25 similar transfer functions, and for each 50 8-tuples of poles
similarly chosen
[0, 44, 379, 500, 231, 79, 17, 0]
###################################################################
Tenth set:
25 similarly chosen transfer functions, but with poles
with X1=-14 and successive differences between 1 and 7.
[0, 20, 146, 524, 361, 147, 51, 1]
#########################################################################
Eleventh set:
25 similar transfer functions and poles with
X1=-1 and successive differences between 1 and 5.
#########################################################################
Twelfth and final set:
25 similar transfer functions with these statistics:
-8 <= Cubic Terms <= 8
-8 <= Quadratic Terms <= 8
-8 <= Linear Terms <= 8
This time, we tested the same set of poles (listed below) for each:
[0, 17, 177, 439, 350, 223, 44, 0]
##################################################################
-10,-9,-4,-3,-2,5,6,30,
-10,-1,3,-4,-7,7,8,10,
-7,0,3,4,5,7,8,9,
-9,-4,4,5,7,-8,8,10,
-9,-1,1,-3,3,-7,9,10,
-6,0,-1,1,-4,4,9,10,
-10,-2,4,-5,6,7,-8,10,
-9,-1,2,-4,6,-8,9,10,
-19,-17,-16,0,1,16,17,18,
-12,-10,-8,-5,-2,2,10,13,
-15,-11,-8,-2,8,12,15,19,
-17,-5,0,2,3,10,11,20,
-20,-16,-15,1,2,8,19,20,
-30,-13,-12,-11,-3,2,3,20,
-15,-14,-12,-10,-5,9,10,16,
-17,-16,-15,-11,-8,-6,-5,8,
-88,-87,-20,-6,23,24,56,75,
-71,-70,-69,-15,2,5,15,33,
-48,-43,-41,-39,23,38,40,41,
1,2,3,4,5,6,7,8,
-31,-30,-29,-28,-27,2,3,5,
2,3,5,7,11,13,17,19,
23,29,31,37,41,43,47,53,
50,60,80,101,121,138,141,143,
-12,-11,-10,-9,9,10,11,12