On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized Sta

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On the Realization Theory of Polynomial Matrices
and the Algebraic Structure of Pure Generalized
State Space Systems

• A.I.G. Vardulakis, N.P. Karampetakis and E.N.
  Antoniou
• Department of Mathematics
• Faculty of Sciences
• Aristotle University of Thessaloniki
• Thessaloniki 54 006, Greece

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Introduction

• Review of the Realization Theory of Polynomial
  Transfer Function Matrices via Pure" Generalized
  State Space Models
• Study of associated concepts and features
• Comparison to results from the classical State
  Space realization theory of Proper Rational
  Transfer Function matrices.
• Key topics
• Generalized order of GSS realizations
• Cancellations of decoupling zeros at 8
• Irreducibility at infinity Minimality
• Dynamic Non-dynamic variables
• Isomorphism of spectral structures at 8

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Introduction
Given a polynomial transfer function matrix
One may obtain its generalized state space
realization of the form
where
and nilpotent,
Such that
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Introduction
Remark. A GSS realization of a polynomial matrix
A(s) can be obtained from a state space
realization of the strictly proper rational
matrix
Example. Given
where
then
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Generalized Order of a GSS Realization
Example. Notice that the generalized order of the
example in the previous slide is fg3.
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Irreducibility at infinity
Let and let
be a GSS realization of A(s)
Definition. The input decoupling zeros (i.d.z.)
at s 8 of Sg are defined as the zeros at s 8 of
the pencil
Respectively, the output decoupling zeros
(o.d.z.) at s 8 of Sg are defined as the zeros
at s 8 of the pencil
Finally, the input-output decoupling zeros
(i.o.d.z.) at s 8 of Sg are the common zeros at
s 8 of the pencils
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Irreducibility at infinity
Let
where
Then the Smith- McMillan form of the matrix
pencil at s8 is
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Irreducibility at infinity
Remark. Candidates for (i.d.z.) and (o.d.z) at s
8 of a GSS realization of
of a polynomial matrix A(s) are the zeros
at s8 of
Example. Let
i.e. A(s) has one pole at s8 of order q12 and
no zeros at s 8
Continued
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Irreducibility at infinity
Continued
A GSS realization is
with
The generalized order of Sg is
Continued
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Irreducibility at infinity
Continued
It can be seen that both
have a zero of order 1 at s8, and thus Sg has an
i.d.z., an o.d.z. and i.o.d.z.at s8.
Now since we may
easily obtain a smaller GSS realization of
A(s), by simply eliminating the middle blocks
from Sg, , with
Continued
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Irreducibility at infinity
Continued
It can be seen that both have no zeros at
s8, which leads to the following definition
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Irreducibility at infinity
Definition. A GSS realization
of a polynomial matrix A(s) is called
irreducible at s8, iff has no input and no
output decoupling zeros at s 8.
Corollary.
(i) has no zeros at s8, iff
(ii) has no zeros at s8,
iff
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Minimal GSS realizations of a polynomial matrix
Definition. A GSS realization of a polynomial
matrix A(s) is called minimal, iff it has the
least number of generalized states.
Theorem. Let be the Smith McMillan form of
A(s) at s8. A GSS realization of A(s), with
J8 in Jordan canonical form is minimal iff
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Minimal GSS realizations of a polynomial matrix

• Remark.
• A necessary condition for the minimality of a GSS
  realization is
• must be irreducible at s8
• must not have non-dynamic variables
• The least dimension of a GSS realization is
• Where qi are the non-zero orders of the poles at
  s8, of A(s).

Corollary. A GSS realization of a polynomial
matrix A(s) is a minimal GSS realization iff it
is an irreducible at s8 GSS realization and has
no non-dynamic variables.
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Minimal GSS realizations of a polynomial matrix
Example. Continuing the previous example, an
irreducible at infinity GSS realization of A(s),
was
Obviously this realization is not minimal, since
the expected least dimension of a GSS realization
of A(s) should be
(Recall that )
Continued
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Minimal GSS realizations of a polynomial matrix
Continued
This realization Sg can be further reduced by
moving its non-dynamic variables, from the state
vector to the D8 matrix. The resulting minimal
GSS realization is then given by
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Minimal GSS realizations of a polynomial matrix
Proposition. Let A(s) be a column proper
polynomial matrix with column degrees
where are the columns of
A(s). Then an irreducible at s8 GSS realization
of A(s) can be obtained by
inspection and is given by
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Conclusions