Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 782057, 11 pages
doi:10.1155/2011/782057
Research Article
Annular Bounds for Polynomial Zeros and Schur
Stability of Difference Equations
Ke Li1 and Jin Liang2
1
Shandong Key Laboratory of Automotive Electronic Technology, Institute of Automation, Shandong
Academy of Sciences, Jinan, Shandong 250014, China
2
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Correspondence should be addressed to Jin Liang, jinliang@sjtu.edu.cn
Received 7 October 2010; Accepted 30 October 2010
Academic Editor: Toka Diagana
Copyright q 2011 K. Li and J. Liang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We investigate the monic complex-coefficient polynomial of degree n, f z : zn an−1 zn−1 · · · a0
in the complex variable z and obtain a new annular bound for the zeros of f z, which is sharper
than the previous results and has clear advantages in judging the Schur stability of difference
equations. In addition, examples are given to illustrate the theoretical result.
1. Introduction
It is well known that many discrete-time systems in engineering are described in terms of a
difference equation, and the characteristic equation for the difference equation plays a key
role in the study of the behaviors of the solutions, especially the stability of the solutions,
to the discrete-time systems. Since the characteristic equations for difference equations are
closely related to some complex polynomials, the estimates of the bound for the moduli of
various complex polynomial zeros have been investigated by many researchers cf. e.g., 1–
8 and references therein. In the study on this issue, one of meaningful research ideas is to
indicate such a common property of a lot of polynomials by a few very special polynomials.
Using this idea, a good annular bound by estimating the largest nonnegative zeros of four
specific polynomials is given in 8 recently. As a continuation of this work and our paper
4 , in this paper we investigate further the location of the zeros of complex-coefficient
polynomials on the basis of such a research idea and establish a new annular bound theorem
Theorem 3.1, which improves the previous corresponding result and has clear advantages
in judging the Schur stability of difference equations. Examples are given to illustrate the
advantages of the new result.
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2. Preliminaries
Throughout this paper, we let
2.1
fz : zn an−1 zn−1 an−2 zn−2 · · · a1 z a0
with ai ∈ C, i ∈ {0, 1, 2, . . . , n − 1}, and
gz : −1n fzf−z z2n b2n−2 z2n−2 b2n−4 z2n−4 · · · b2 z2 b0 .
2.2
≡ 0, or, equivalently, b0 /
≡ 0.
Without losing the generality, we assume that a0 /
Basic notations are as follows.
R : {x ∈ R | x < 0},
|z|: the modulus of a complex number z,
Zfz : the set of all zeros of fz,
Ar, R : {z ∈ C | r ≤ |z| ≤ R} with 0 ≤ r ≤ R,
l: the smallest positive integer such that al /
0 in fz,
k: the largest positive integer such that ak /
0 in fz,
q: the smallest positive integer such that b2q /
0 in gz,
p: the largest positive integer such that b2p /
0 in gz,
m : the integer part of a real number m.
In order to simplify the expressions in our study, we define specially that
s
2.3
yi : 0
it
for any positive integers s, t s < t, and sequence {yi ∈ C : s ≤ i ≤ t}. This notation is logical
and useful in the note.
Moreover, we write
c1 x : xnl nl−1
|ai−l |xi in1
n 2l−1
|a0 ai |
|a0 |2
i
ai−l − a0 ai xi −
x
al |al |
|al |
i2l
il1
2.4
with an 1 and 1 ≤ l ≤ n/2 ,
c2 x : x2n−k −
n−1
|aik−n |xi −
ik1
k
in−k
|aik−n − ai ak |xi −
n−k−1
i0
|ai ak |xi
2.5
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3
with n 1/2 ≤ k ≤ n − 1,
d1 x : x
2n2q
2q−1
n 2i b0 b2i 2i |b0 b2i | 2i
x −
b2i−2q x b2i−2q −
x b2q i2q
in1
iq1 b2q
nq−1
|b |2
0 b2q 2.6
with b2n 1 and 1 ≤ q ≤ n/2 ,
d2 x : x4n−2p −
p
n−p−1
n−1 b2ip−n x2i −
b2ip−n − b2i b2p x2i −
b2i b2p x2i
ip1
in−p
2.7
i0
with n 1/2 ≤ p ≤ n − 1,
f1 x : xn n−1
|ai |xi − |a0 |,
f2 x : xn −
k
|ai |xi ,
i0
il
2.8
g1 x : x2n n−1
|b2i |x2i − |b0 |,
g2 x : x2n −
iq
p
|b2i |x2i .
i0
Remark 2.1. By Descartes’ rule of signs, it is easy to see that for each i ∈ {1, 2}, the polynomial
ci xdi x, fi x, gi x has a unique positive zero.
We denote by αi , βi , γi , and δi the unique positive zero of ci x, di x, fi x, and gi x,
respectively.
3. Main Result
The following result is established in 8 .
Theorem A see 8 . Zfz ⊂ Au, v , with : max{γ1 , δ1 } and v : min{γ2 , δ2 }.
Theorem 3.1. Let 1 ≤ l, q ≤ n/2 , and n 1/2 ≤ k, p ≤ n − 1. Then
i
Z fz ⊂ Ar, R ,
3.1
where r : max{α1 , β1 } and R : min{α2 , β2 }
ii
Ar, R ⊆ Au, v ,
3.2
where u, v are constants as in Theorem A;
iii the annular bound of original polynomial fz can be further improved by iterative
procedure.
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Proof. Define
a0
l
cz : fz z −
al
znl −
znl n−1
n−1
a2
a0 ai i
a0 n z ai zil −
z a0 zl − 0
al
al
al
il
il
nl−1
ai−l zi −
i2l
znl nl−1
ai−l zi in1
znl nl−1
ai−l zi in1
cz : fz zn−k − ak
z2n−k k−1
n
a0 ai i a20
z −
al
al
il1
n
n
a0 ai
a2
a0 ai i 2l−1
ai−l zi −
z −
zi − 0
al
al
al
i2l
i2l
il1
n z
ai zin−k −
k
n−1
i0
aik−n zi ik1
k
aik−n − ai ak zi −
n−k−1
ai ak zi ,
i0
in−k
b0
b2q
dz : gz z2q −
z2n2q 3.4
k
aik−n z − ai ak zi
i
in−k
z2n−k ai ak zi
i0
n−1
a0 ai
a2
a0 ai i 2l−1
z −
zi − 0 ,
al
al
al
il1
ai−l −
i2l
i0
2n−k
3.3
nq−1
b2i−2q z2i n
b2i−2q −
i2q
in1
b0 b2i
b2q
z2i −
2q−1
b0 b2i 2i b02
z −
,
b
b2q
iq1 2q
dz : gz z2n−2p − b2p
z4n−2p n−1
ip1
b2ip−n z2i p
b2ip−n − b2i b2p z2i −
in−p
n−p−1
3.5
b2i b2p z2i ,
i0
where an b2n 1. Then it is not difficult to see that
Z fz ⊆ Z cz ∩ Zcz ∩ Z dz ∩ Z dz .
3.6
This implies that for every w ∈ Zfz we have
cw cw dw dw 0,
3.7
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that is,
wnl nl−1
n in1
i2l
ai−l wi n−1
w2n−k aik−n wi ik1
w2n2q nq−1
w4n−2p n−1
n
aik−n − ai ak wi −
b2ip−n w2i p
n−k−1
ai ak wi 0,
i0
b0 b2i
b2i−2q −
b2q
i2q
ip1
k
2l−1
a0 ai
a2
a0 ai
wi −
wi − 0 0,
al
al
al
il1
in−k
b2i−2q w2i in1
ai−l −
2q−1
b0 b2i 2i b02
w −
0,
w2i −
b
b2q
iq1 2q
b2ip−n − b2i b2p w2i −
n−p−1
in−p
3.8
b2i b2p w2i 0.
i0
Hence, by 3.8, one has
nl−1
n |a0 ai | i
a0 ai i 2l−1
|a0 |2
nl
i
≤ |w| |w| ,
|ai−l ||w| ai−l −
|w| al
|al |
|al |
in1
i2l
il1
|w|2n−k ≤
n−1
|aik−n ||w|i ik1
k
|aik−n − ai ak ||w|i n−k−1
|ai ak ||w|i ,
i0
in−k
nq−1
2q−1
n 2i b0 b2i 2i |b0 b2i | 2i
|b0 |2
2n2q
≤ |w|
|w| ,
b2i−2q |w| b2i−2q −
|w| b2q b2q in1
i2q
iq1 b2q
3.9
p
n−p−1
n−1 b2ip−n |w|2i b2ip−n − b2i b2p |w|2i b2i b2p |w|2i ,
|w|4n−2p ≤
ip1
in−p
i0
which imply that
Z fz ⊂ {z ∈ C : c1 |z| ≥ 0, d1 |z| ≥ 0, c2 |z| ≤ 0, d2 |z| ≤ 0}.
In addition, it follows from 2.4–2.7 that
c1 x < 0,
c1 x ≥ 0,
d1 x < 0,
d1 x ≥ 0,
∀x ∈ 0, α1 ,
∀x ∈ α1 , ∞,
∀x ∈ 0, β1 ,
∀x ∈ β1 , ∞ ,
3.10
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Advances in Difference Equations
c2 x ≤ 0,
∀x ∈ 0, α2 ,
∀x ∈ α2 , ∞,
d2 x ≤ 0, ∀x ∈ 0, β2 ,
c2 x > 0,
∀x ∈ β2 , ∞ .
d2 x > 0,
3.11
Therefore, for each w ∈ Zfz we have
|w| ≥ β1 ,
|w| ≥ α1 ,
|w| ≤ α2 ,
|w| ≤ β2 ,
3.12
which imply that 3.1 is hold. So i is proved.
Next we prove that ii holds. Actually, we have
c1 γ1 γ1nl nl−1
|ai−l |γ1i in1
|a0 | −
n−1
n 2l−1
|a0 ai | i |a0 |2
ai−l − a0 ai γ i γ −
al 1 il1 |al | 1 |al |
i2l
|ai |γ1i γ1l −
nl−1
|ai−l |γ1i
|a0 |
|al |
γ1n nl−1
|ai−l |γ1i
in1
i2l
−
|ai−l |γ1i in1
il
|a0 |γ1l
nl−1
n 2l−1
|a0 ai |
|a0 |2
i
ai−l − a0 ai γ i −
γ
al 1 il1 |al | 1 |al |
i2l
n |a0 ai |
a0 ai i 2l−1
ai−l −
γi
γ
al 1 il1 |al | 1
i2l
3.13
n−1
|ai |γ1i
il
n
n n
|a0 ai |
a0 ai i 2l−1
|a0 ai | i
i
− |ai−l |γ1i ai−l −
−
γ
γ
γ
1
1
a
|a
|
|al | 1
l
l
i2l
i2l
il1
il1
n a0 ai a0 ai i
−
− ai−l −
γ
|ai−l | al al 1
i2l
≤ 0,
where an 1 and 1 ≤ l ≤ n/2 .
On the other hand, since the polynomial equation c1 x 0 has a unique positive root
α1 and
c1 x ≤ 0,
c1 x > 0,
∀x ∈ 0, α1 ,
∀x ∈ α1 , ∞,
we get α1 ≥ γ1 by combining3.13 and 3.14.
3.14
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In addition, we have
n−1
c2 γ2 γ22n−k −
ik1
k
n
|aik−n − ai ak |γ2i −
n−1
|aik−n |γ2i −
|aik−n |γ2i −
n−1
k
|aik−n |γ2i −
k
|aik−n − ai ak |γ2i −
k
|aik−n |γ2i −
k
k
|ai ak |γ2i
i0
|aik−n − ai ak |γ2i −
n−k−1
|ai ak |γ2i
i0
|aik−n − ai ak |γ2i −
n−k−1
|ai ak |γ2i − |a0 ||ak |
3.15
i1
in−k
|aik−n | − |aik−n − ai ak |γ2i −
n−k−1
|ai ak |γ2i −
γ2n −
i1
in−k
n−k−1
in−k
in−k
|ak |γ2n k
in−k
ik1
|ak |γ2n |ai ak |γ2i
i0
ik1
in−k
n−k−1
in−k
|ai |γ2in−k −
i0
k
|aik−n |γ2i −
k
|ai |γ2i |ak |
i1
|aik−n | |ai ak | − |aik−n − ai ak |γ2i
in−k
≥ 0.
Since
c2 x < 0,
∀x ∈ 0, α2 ,
3.16
∀x ∈ α2 , ∞,
c2 x ≥ 0,
we have α2 ≤ γ2 .
In the same way, we can obtain β1 ≥ δ1 and β2 ≤ δ2 ; therefore,
Ar, R ⊆ Au, v .
3.17
Finally, we prove iii. Set
1
1
1
c1 z : cz znl anl−1 znl−1 · · · al1 zl1 a0 ,
3.18
with
1
ai
⎧
a ,
⎪
⎪
⎪ i−l
⎪
⎪
⎪
⎪
a0 ai
⎪
⎪
,
a −
⎪
⎪
⎨ i−l
al
a0 ai
⎪
⎪
−
,
⎪
⎪
al
⎪
⎪
⎪
⎪
⎪
⎪ a20
⎪
⎩− ,
al
n 1 ≤ i ≤ n l − 1;
2l ≤ i ≤ n;
l 1 ≤ i ≤ 2l − 1;
i 0,
3.19
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1
and let l1 be the smallest positive integer such that al1 /
0 in c1 z. If l 1 ≤ l1 ≤ n l/2 ,
in analogy to 3.3 and 2.4, we can define
⎛
c2 z : c1 z⎝zl1 −
1
a0
1
⎞
⎠
3.20
al1
2
and c1 x, respectively. It is not difficult to see that, the unique positive root of polynomial
2
2
2
2
2
c1 x, α1 ≥ α1 . Similarly, we can define c2 x, d1 x, and d2 x, respectively. Moreover,
2
2
2
their respective positive roots α2 , β1 , and β2 satisfy that
2
2
α2 ≤ α2 ,
β1 ≥ β1 ,
2
β2 ≤ β2 .
3.21
Consequently, new annular bound of fz, namely, Ar 2 , R2 with
2 2
r 2 : max α1 , β1 ,
2 2
R2 : min α2 , β2 ,
3.22
is better than 3.1. This procedure can be applied iteratively.
2
c1 x,
2
d1 x,
3
d1 x,
3
c2 x,
2
d2 x
2
3.23
3
d2 x,
3
3.24
3
d2 x
3
3.25
4
d2 x,
4
3.26
can be further transformed into
3
c2 x,
3
c2 x,
d1 x,
4
d1 x,
c1 x,
respectively, and
c1 x,
into
4
c1 x,
c2 x,
until the last iteration brings no practical improvement. Obviously, when m increases,
m m
,
r m : max α1 , β1
m m
Rm : min α2 , β2
3.27
will approach the smallest and largest modulus of polynomial zero, respectively, where
m
α1
m
m
m
resp. α2 , β1 , β2
3.28
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denotes the unique positive root of
m
m
m
m
c1 x resp. c2 x, d1 x, d2 x .
3.29
This means that iii is true.
Remark 3.2. a When c2 r > 0, it follows from 3.14 and 3.15 that for every w ∈ Zfz ,
|w| ≤ α2 < r, that is, Zfz ⊂ Br, that is, fz is r-stable.
c
Similarly, we can draw the same conclusion when d2 r > 0, and Zfz ⊂ B r when
c1 r < 0 or d1 r < 0.
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 n − 1 ≥ l, q > n/2 , and 1 ≤ k, p <
n 1/2 ?” is worth considering further.
Example 3.3. Let
fz z3 1 j z2 2jz 1,
where j √
3.30
−1. By Theorem 3.1, we obtain
Z fz ⊂ A0.389, 1.647 .
3.31
If we start the iterative procedure given in the proof of iii of Theorem 3.1, after five
iterations, we obtain
Z fz ⊂ A0.390, 1.644 .
3.32
On the other hand, by Theorem A, one only can have
Z fz ⊂ A0.387, 1.938 .
3.33
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
1 √
fz z 2 j z2 2
3
√
1 √
2
1
j z−
2 − 5j ,
4
2
8
3.34
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√
where j −1. Then by Theorem 3.1, we get c2 1 7/16 > 0, which implies that all zeros
of fz lie in the open unit disk, that is, this system is Schur stable. However, by Theorem A,
one has
Z fz ⊂ A0.638, 1.175 .
3.35
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
fz z3 11 1
1
3
j z2 −
−j z−
j ,
2
4
8
4
3.36
√
where j −1. Then by Theorem 3.1, we have c1 1 −9/16 < 0, which implies that all zeros
of fz are outside the open unit disk, namely, such a system is instable. By Theorem A, one
has
Z fz ⊂ A0.824, 1.517 ,
3.37
which cannot determine the instability of this system.
Example 3.6. Consider the following characteristic polynomial of a difference equation
discrete-time system:
fz z4 2z3 2z2 z √
11 − 3.
3.38
In this example,
1
c2 −6.316,
1
d2 0;
2
c2 −7.584,
2
d2 0.203.
3.39
Consequently, such a difference equation discrete-time system is Schur stable.
Acknowledgments
This work was supported partially by the NSF of China 10771202 and the Specialized
Research Fund for the Doctoral Program of Higher Education of China 2007035805.
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