The following result is established in .
Theorem A (see ).
, and .
Let , , and , . Then
where , are constants as in Theorem A;
the annular bound of original polynomial can be further improved by iterative procedure.
where . Then it is not difficult to see that
This implies that for every we have
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
we have .
In the same way, we can obtain and ; therefore,
Finally, we prove (iii). Set
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
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.
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.
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).
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.
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.
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.