Some Remarks on Real and Complex Output Feedback
Abstract: We provide some new necessary and sufficient conditions
which guarantee arbitrary pole placement of a particular linear system
over the complex numbers. We exhibit a non-trivial real linear system which
is not controllable by real static output feedback and discuss a conjecture
from algebraic geometry concerning the existence of real linear systems
for which all static feedback laws are real.
Post Script version
A non-trivial real system that cannot be controlled by real output
This non-trivial example of a real system which is not controllable
by real static output feedback, while presented in the paper (pages 5 and
6 of the manuscript above), is not easy to
verify by hand. Below are some files which contain the feedback laws, and
can be used to verify our claims.
- A MAPLE input file which contains the feedback laws for that example,
the system (represented as a matrix [D(s):N(s)]),
and the poles (-8,-6,-4,-2,1,2,3,4).
Running it will verify these are feedback laws for the given system. The
beginning of the file contains some additional documentation.
- This file contains polynomials whose roots are the coordinates of all
the feedback laws.
When run, it computes these roots, and then assembles them into
the matrices which are displayed on page 6 of the manuscript.
Verifying that these are the only feedback laws (equivalently, that
the given system is nondegenerate) is more difficult. Algebraically, it
amounts to showing that the 8 polynomial equations of displayed equation
(7) on page 5 of the manuscript describe 14 distinct points in complex
8-space. We used the Gr\"obner basis package
to check this for us.
- This SINGULAR input file contains the equations (7), will compute a
Gr\"obner basis for the ideal, and then the dimension of the quotient
ring and its degree. Since the degree is 14, and we have given 14 distinct
solutions, this constitutes a proof that the system was nondegenerate.
- This MAPLE input file creates the singular file input.singular,
and may be used to verify that the equations in input.singular
are indeed those of (7), as claimed.
Feedback laws for real m=2, p=4, n=8 systems
As part of this project, we generated a considerable number (15,000)
of real m=2, p=4, n=8 systems with characteristic polynomials and for each,
determined the number of real and complex feedback laws. This was accomplished
by randomly generating linear systems and characteristic polynomials in
SINGULAR. Then, also in SINGULAR, we computed an eliminant for each system
and characteristic polynomial. Finally, we used the realroot routine
of MAPLE to determine the number of real roots of the eliminant, and hence
the number of real feedback laws. The results of this, as well as SINGULAR
input files, are contained in the following files.
- This will create 250 systems and compute an eliminant for each. It
generates a MAPLE input file, which must be edited (documentation contained
in search.singular). When the MAPLE file is run, it prints a matrix
which records the number of real feedback laws for each system.
- This MAPLE input file contains the output matrices from 60 runs of
search.singular, as well as some documentation in the
form of comments.
When run, it prints out
[ 23, 804, 3370, 5345, 3854, 1325, 272, 7 ]
[.1533, 5.360, 22.47, 35.63, 25.69, 8.833, 1.813, .04667]
The first vector records the frequency; its (j+1)th component is the
number of systems generated with 2j real feedback laws, and the second
vector similarly records the percentages.
- This contains further documentation of this part of our project, in
particular, the parameters used in the generation of the systems with SINGULAR.
- This contains some remarks about this study of m=2, p=4 systems and
some questions that arose naturally out of the data we generated.
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