The following result is established in [8].
Theorem A (see [8]).
,
and
.
Theorem 3.1.
Let
,
, and
,
. Then
-
(i)
-
(ii)
where
,
are constants as in Theorem A;
-
(iii)
the annular bound of original polynomial
can be further improved by iterative procedure.
Proof.
Define
where
. Then it is not difficult to see that
This implies that for every
we have
that is,
Hence, by (3.8), one has
which imply that
In addition, it follows from (2.4)–(2.7) that
Therefore, for each
we have
which imply that (3.1) is hold. So (i) is proved.
Next we prove that (ii) holds. Actually, we have
where
and
.
On the other hand, since the polynomial equation
has a unique positive root
and
we get
by combining(3.13) and (3.14).
In addition, we have
Since
we have
.
In the same way, we can obtain
and
; therefore,
Finally, we prove (iii). Set
with
and let
be the smallest positive integer such that
in
. If
, in analogy to (3.3) and (2.4), we can define
and
, respectively. It is not difficult to see that, the unique positive root of polynomial
,
. Similarly, we can define
,
, and
, respectively. Moreover, their respective positive roots
,
, and
satisfy that
Consequently, new annular bound of
, namely,
with
is better than (3.1). This procedure can be applied iteratively.
can be further transformed into
respectively, and
into
until the last iteration brings no practical improvement. Obviously, when
increases,
will approach the smallest and largest modulus of polynomial zero, respectively, where
denotes the unique positive root of
This means that (iii) is true.
Remark 3.2.
-
(a)
When
, it follows from (3.14) and (3.15) that for every
,
, that is,
, that is,
is
-stable. Similarly, we can draw the same conclusion when
, and
when
or
.(b) By the similar arguments in the proof of (iii) of Theorem 3.1, the results in (a) can be improved. This also provides an iterative algorithm to test the r-stability and Schur stability of polynomials. (c) The question "What happens to Theorem 3.1 when
,
, and
" is worth considering further.
Example 3.3.
Let
where
. By Theorem 3.1, we obtain
If we start the iterative procedure given in the proof of (iii) of Theorem 3.1, after five iterations, we obtain
On the other hand, by Theorem A, one only can have
The following examples show the advantages of Theorems 3.1 over Theorem A in analyzing the Schur stability of difference equations (discrete-time systems).
Example 3.4.
Let the characteristic polynomial of a difference equation (discrete-time system) be given by
where
. Then by Theorem 3.1, we get
, which implies that all zeros of
lie in the open unit disk, that is, this system is Schur stable. However, by Theorem A, one has
So Theorem A cannot guarantee the stability of such a system.
Example 3.5.
Suppose the characteristic polynomial of a difference equation (discrete-time system) is given by
where
. Then by Theorem 3.1, we have
, which implies that all zeros of
are outside the open unit disk, namely, such a system is instable. By Theorem A, one has
which cannot determine the instability of this system.
Example 3.6.
Consider the following characteristic polynomial of a difference equation (discrete-time system):
In this example,
Consequently, such a difference equation (discrete-time system) is Schur stable.