AUTOMATYKA • 2006 • Tom 10 • Zeszyt 1
Ewelina Zalot*
Criterions for the stability of the convex combination
of polynomials**
1. Introduction
Let p1 and p2 be two complex and of the same degree polynomials
p1(s) = αnsn + αn–1sn–1+ ... + α1s + α0 = αn(s – x1)(s – x2) ... (s – xn),
p2(s) = βnsn + βn–1sn–1+ ... + β1s + β0 = βn(s – y1)(s – y2) ... (s – yn),
(1)
where αn ≠ 0, βn ≠ 0, αk, βl ∈ +, k = 0, 1, ..., n, l = 0, 1, ..., n and let C(p1, p2) be the convex
combination of these polynomials
C(p1, p2) = {δp1(s) + (1 – δ) p2(s) : δ∈ [0, 1]}.
We say that polynomial p1(s) is Hurwitz (resp. Schur) stable if Re(xk) < 0 (k = 1, 2, ..., n)
(resp. |xk| < 1 (k = 1, 2, ..., n)).
We call the convex combination C(p1, p2) Hurwitz (resp. Schur) stable if every polynomial p(s) ∈ C(p1, p2) is Hurwitz (resp. Schur) stable.
According to paper [2], in 1985 a necessary and sufficient condition for Hurwitz stability of the set C(p1, p2) was proved, for real polynomials p1(s), p2(s). In 1989 in paper [3],
a necessary and sufficient condition for Hurwitz and Schur stability of the set C(p1, p2) was
given, for complex polynomials p1(s), p2(s).
In this work we introduce a new necessary and sufficient condition for Hurwitz and
Schur stability of the set C(p1, p2), where polynomials p1(s), p2(s) are complex and of the
same degree. We then generalize some results from [2] and complement some of [3].
* Faculty of Applied Mathematics, AGH University of Science and Technology, Cra-
cow, Poland; e-mail: [email protected]
** 2000 Mathematics Subject Classification: 93D09, 15A63.
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Ewelina Zalot
2. A necessary and sufficient condition
for the stability of convex combination of polynomials
To polynomials (1), we assign the matrix
⎡α n
⎢0
⎢
⎢...
⎢
...
R( p1 , p2 ) = ⎢
⎢...
⎢
⎢...
⎢0
⎢
⎢⎣βn
α n −1
αn
...
...
...
...
α n− 2
α n −1
...
...
...
...
βn
βn −1
βn −1
βn − 2
...
...
...
αn
βn
...
...
...
α0 0
α1 α0
... ...
... ...
... ...
... ...
... ...
... ...
...
...
...
...
...
...
⎤
⎥
⎥
⎥
⎥
α0 ⎥
β0 ⎥
⎥
... ⎥
0 ⎥
⎥
0 ⎦⎥
α1
β1
...
...
0
⎫
⎪
⎪
⎬n
⎪
⎪⎭
⎫
⎪
⎪
⎬n
⎪
⎪⎭
rows
(2)
rows
It is obvious that R(p1, p2) ∈ +(2n)×(2n) and the elements of the matrix R(p1, p2) do not
depend on s. It has been proved in [1] that:
n
n
det R( p1 , p2 ) = (−1)n (n −1) / 2 α nnβnn ∏∏ ( xk − yl ).
(3)
k =1 l =1
We will use the following notation:
Pn = {p(s} : p(s) = αnsn + αn–1sn–1 + ...
(
(
)
(1)
(1)
+ α1s
+ α0, αn ≠ 0, αk ∈ +∀ k = 0, 1, ..., n},
)
(
(1)
(1)
f1 (s ) = an(1) + ibn(1) s n + an −1 + ibn −1 s n −1 + ... + a1 + ib1
(
)
(
(2)
(2)
)
(
(2)
f 2 (s ) = an(2) + ibn(2) s n + an −1 + ibn −1 s n −1 + ... + a1
) s + (a
(2)
+ ib1
(1)
0
) s + (a
(1)
+ ib0
(2)
0
)
(2)
+ ib0
(4)
)
where:
(k )
an(1) + ibn(1) ≠ 0, an(2) + ibn(2) ≠ 0, a j
(k )
+ bj
∈ + ( k = 1, 2,
j = 0, 1, ..., n),
H n = { p( s) ∈ Pn : p( s) = α n ( s − x1 )( s − x2 ) ... (s − xn ), Re( xk ) < 0 (k = 1, 2, ..., n)} ,
Sn = { p( s) ∈ Pn : p(s ) = α n ( s − x1 )( s − x2 ) ... (s − xn ), | xk |< 1 (k = 1, 2, ..., n)} ,
C ( f1 , f 2 ) = {δf1 (s ) + (1 − δ) f 2 (s ) : δ ∈ [0, 1]}.
Criterions for the stability of the convex combination of polynomials
89
Any polynomial fδ(s) ∈ C( f1, f2) is of the form:
f δ (s ) =
n
∑ (ak (δ) + ibk (δ))sk ,
k =0
where
ak (δ) + ibk (δ) = (δak(1) + (1 − δ) ak(2) ) + i (δbk(1) + (1 − δ)bk(2) )
for δ ∈ [0, 1] and
f δ (s ) δ=0 = f 2 (s ),
f δ (s ) δ=1 = f1 (s ).
Let (an + ibn)sn + (an–1 + ibn–1)sn–1 + ... + (a1 + ib1)s + (a0 + ib0) = p(s) ∈ Pn.
We will use the following notation:
s∗ = s
for
s ∈ +,
p∗ (−s∗ ) = p (−s ),
M (s , p) =
1⎡
p (s ) + p∗ (−s ∗ ) ⎤ = a0 + ib1 s + a2 s 2 + ib3 s 3 + ...,
⎦
2⎣
N ( s, p) =
1⎡
p( s) − p∗ ( − s∗ ) ⎤ = ib0 + a1 s + ib2 s 2 + a3 s3 + ...
⎦
2⎣
There, obviously, is p(s) = M(s, p) + N(s, p). The following theorem is true.
Theorem 1 ([3])
The polynomial (an + ibn)sn + (an–1 + ibn–1)sn–1 + ... + (a1 + ib1)s + (a0 + ib0) = p(s) ∈ Pn
is Hurwitz stable (p(s) ∈ Hn) if and only if:
1. anan–1 + bnbn–1 > 0,
2. the zeros of M(s, p) and N(s, p) are simple, lie on the imaginary axis of the s–plane and
the zeros of M(s, p) alternate with the zeros of N(s, p).
The polynomial fδ(s) ∈ C(f1, f2), that is:
where:
fδ(s) = M(s, fδ) + N(s, fδ)
M (s , f δ ) =
1⎡
f (s ) + f δ∗ (− s∗ ) ⎤ = δM ( s, f1 ) + (1 − δ)M ( s, f 2 )
⎦
2⎣ δ
N ( s, f δ ) =
1⎡
f ( s) − fδ∗ (− s∗ ) ⎤ = δN ( s, f1 ) + (1 − δ) N ( s, f2 )
⎦
2⎣ δ
∀δ ∈ [0, 1].
(5)
(6)
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Ewelina Zalot
Matrix (2) formed with the coefficients of M(s, fδ), N(s, fδ) will be denoted by R( f1, f2, δ).
It is obvious that R( f1, f2, δ) ∈ +(2n)×(2n) and every element of this matrix is a linear combination of coefficients of polynomials f1, f2. Let us denote:
R( f1 ) = R( f1 , f 2 , δ) δ=1 , R( f 2 ) = R( f1 , f 2 , δ) δ=0 ,
and:
Δ(δ) = det R( f1, f 2 , δ), Δ(1) = det R( f1 ), Δ(0) = det R( f 2 ).
The following theorem was proved for the family C( f1, f2).
Theorem 2 ([3])
If f1(s), f2(s) ∈ Hn and
an ( δ) an −1 ( δ) + bn ( δ)bn −1 ( δ) > 0 ∀δ ∈ (0, 1)
(7)
then the convex combination C( f1, f2) is Hurwitz stable (C( f1, f2) ⊂ Hn) if and only if Δ(δ) ≠ 0
for all δ ∈ [0, 1].
Now we shall prove a theorem which is complementary to Theorem 2.
Theorem 3
If f1(s), f2(s) ∈ Hn and coefficients of any polynomial fδ(s) ∈ C( f1, f2) satisfy inequality
(7), then the following conditions are equivalent:
(i) C ( f1 , f 2 ) ⊂ H n ,
(ii) Δ (δ) ≠ 0 ∀ δ ∈ (0, 1),
(iii) λ k (R( f1 )R −1 ( f 2 )) ∉ (−∞, 0) ∀k = 1, 2, ..., 2n,
where λ k (R( f1 )R −1 ( f2 )) k = 1, 2, ..., 2n are eigenvalues of the matrix R( f1 )R −1 ( f 2 ).
Proof
In this proof the reasoning developed in paper [2] will be applied.
(i) ⇒ (ii) It follows from Theorem 2.
(ii) ⇒ (iii) Let Δ(δ) ≠ 0 ∀δ ∈ (0, 1).
From (5) i (6) it follows that:
R( f1, f2, δ) = δR( f1) + (1– δ)R( f2), δ ∈ (0, 1).
The assumption Δ(δ) ≠ 0 ∀δ ∈ (0, 1) and condition (8) imply:
Δ(δ) = det(δR( f1) + (1 – δ)R( f2)) ≠ 0, ∀δ ∈ (0, 1).
(8)
Criterions for the stability of the convex combination of polynomials
91
Since the polynomial f2(s) is stable, R–1( f2) exists because of (3) and Theorem 1, which
says that Δ(0) ≠ 0. Using some of properties of matrix determinants, we conclude that:
det(δR( f1)R–1( f2) + (1 – δ)I) ≠ 0,
and the last condition implies
⎛ δ −1
⎞
det ⎜
I − R( f1 )R −1 (f 2 ) ⎟ ≠ 0, ∀δ ∈ (0, 1).
δ
⎝
⎠
From this relation we infer that the eigenvalues λ k (R( f1 )R −1 ( f 2 )) ∉ (−∞, 0) for k = 1,
2, ..., 2n, which finishes the proof of the implication (ii) ⇒ (iii).
(iii) ⇒ (i) We assume that λ k (R( f1 )R −1 ( f 2 )) ∉ (−∞, 0) k = (1, 2, ..., 2n). Hence
⎛ δ −1
⎞
det ⎜
I − R( f1 )R −1 ( f 2 ) ⎟ ≠ 0, ∀δ ∈ (0, 1),
δ
⎝
⎠
and then
Δ(δ) = det(δR( f1 ) + (1 − δ)R( f 2 )) ≠ 0, ∀δ ∈ (0, 1)
Conditions (7) i (9) imply the stability of the family C( f1, f2) via Theorem 2.
(9)
o
Similar results can be achieved in the case of a convex combination of two complex
Schur stable polynomials.
Let (an + ibn)sn + (an–1 + ibn–1)sn–1 + ... + (a1 + ib1)s + (a0 + ib0) = p(s) ∈ Pn.
We accept further notation:
p∗ ((s∗ )−1 ) = p((s )−1) ),
1
1
D1 ( s, p) = [ p( s) + s n p∗ ((s∗ )−1 )] = [(a0 + an + i (b0 − bn )) +
2
2
+ (a1 + an −1 + i (b1 − bn −1 )) s + ... + (an + a0 + i (bn − b0 )) s 2 ]n ,
1
1
D2 ( s, p) = [ p( s) − s n p∗ (( s∗ )−1 )] = [(a0 − an + i(b0 + bn )) +
2
2
+ (a1 − an −1 + i (b1 + bn −1 )) s + ... + (an − a0 + i (bn + b0 )) s 2 ]n ,
for s ≠ 0. Obviously, p(s) = D1(s, p) + D2(s, p). For the polynomial p(s) ∈ Pn the Theorem 4
is true.
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Ewelina Zalot
Theorem 4 ([3])
The polynomial (an + ibn)sn + (an–1 + ibn–1)sn–1 + ... + (a1 + ib1)s + (a0 + ib0) = p(s) ∈ Pn
is Schur stable (p(s) ∈ Sn) if and only if:
1. |(a0 + ib0)/(an + ibn)| < 1,
2. the zeros of D1(s, p) and D2(s, p) are simple, lie on the circus |s| = 1 and the zeros of
D1(s, p) alternate with the zeros of D2(s, p).
Let f1(s), f2(s) be polynomials defined in (4) and fδ(s) ∈ C(f1, f2). We can write the
polynomial fδ(s) in the form:
where:
fδ(s) = D1(s, fδ) + D2(s, fδ),
1
D1 (s , f δ ) = [ f δ (s ) + s n f δ∗ ((s∗ )−1 )] = δD1 (s, f1 ) + (1 − δ) D1 (s, f 2 ),
2
1
D2 ( s, f δ ) = [ f δ (s ) − s n f δ∗ ((s∗ )−1 )] = δD2 ( s, f1 ) + (1 − δ) D2 ( s, f 2 ),
2
∀δ ∈ [0, 1].
Let us build matrix (2) with the coefficients of polynomials D1(s, fδ), D2(s, fδ), which
we denote by RS(f1, f2, δ). As it was in Hurwitz’s stability case, this matrix is of order 2n and
its elements depend on δ ∈ [0, 1]. Let us denote:
and:
R S ( f1 ) = R S ( f1 , f 2 , δ) δ=1 , R S ( f 2 ) = R S ( f1 , f 2 , δ) δ=0 ,
Δ S ( δ) = det R S ( f1 , f 2 , δ), δ ∈ [0, 1], Δ S (1) = det R S ( f1 ), Δ S (0) = det R S ( f 2 ).
The Theorem 5 is true.
Theorem 5 ([3])
If polynomials f1(s) and f2(s) are in Sn and
|(a0(δ) + ib0(δ))/(an(δ) + ibn(δ)| < 1, ∀ δ ∈ (0, 1)
(10)
then the family C( f1, f2) is Schur stable (C( f1, f2) ⊂ Sn) if and only if the determinant
ΔS(δ) ≠ 0 for all δ ∈ [0, 1].
Now we shall prove a theorem which is complementary to Theorem 5.
Theorem 6
If f1(s), f2(s) ∈ Sn and coefficients of any polynomial fδ(s) ∈ C( f1, f2) satisfy inequality
(10) then the following conditions are equivalent:
Criterions for the stability of the convex combination of polynomials
93
(i) C ( f1 , f 2 ) ⊂ Sn ,
(ii) Δ S (δ) ≠ 0 ∀δ ∈ (0, 1),
(iii) λk (R S ( f1 )R S−1 ( f 2 )) ∉ ( −∞, 0) ∀k = 1, 2, ..., 2n,
where λ k (R S ( f1 )R S−1 ( f 2 )) k = 1, 2, ..., 2n are eigenvalues of the matrix RS ( f1 )RS−1 ( f 2 ).
Proof of this Theorem is analogous to proof of Theorem 3. It is enough to notice that
RS( f1, f2, δ) = δRS( f1) + (1 – δ)RS( f2), δ ∈ (0, 1)
and to use Theorems 4 and 5.
Example 1
Let us consider two complex Hurwitz polynomials:
f1(s) = (1 – i)s2 + (2 – 2i)s + (2 – 2i) = (1 – i)(s + 1 – i)(s + 1 + i),
f2(s) = (1 + i)s2 + (3 + 3i)s + (2 +2i) = (1 + i)(s + 2)(s + 1),
and the convex combination of these polynomials:
C( f1, f2) = {δf1(s) + (1 – δ)f2(s) : δ ∈ [0, 1]} =
= { f (s) = [1 + i(l – 2δ) ]s2 +
+ [3 – δ + i(3 – 5δ)]s + [2 + i(2 – 4δ)]}.
The matrices R( f1), R( f2) are of the form:
⎡1
⎢0
R( f1 ) = ⎢
⎢0
⎢
⎣ –i
–2i
1
2
–2i
–i
2
2
–2i
0 ⎤
⎡1 3i 2 0 ⎤
⎥
⎢0 1 3i 2 ⎥
2
⎥ , R( f ) = ⎢
⎥
2
⎢0 i 3 2 i ⎥
–2i ⎥
⎥
⎢
⎥
0 ⎦
⎣ i 3 2i 0 ⎦
and the matrix
⎡ −1 0
⎢
1 0 −1
R( f1 )R −1 ( f 2 ) = − ⎢
6 ⎢ 0 5i
⎢
⎣ 5i 0
0
5i
1
0
5i ⎤
0⎥
⎥.
0⎥
⎥
1⎦
94
Ewelina Zalot
Since the characteristic polynomial of the matrix R( f1) R–1( f2) is of the form
2
2⎞
⎛
det(λI − R( f1 )R −1 ( f 2 )) = ⎜ λ 2 + ⎟ ,
3⎠
⎝
then λk(R( f1)R–1( f2)) ∉ (–∞, 0) (k = 1, 2, 3, 4). Hence the family C( f1, f2) is Hurwitz stable
for all δ ∈ [0, 1].
Example 2
Consider the following two complex Schur stable polynomials:
4
⎛ 1 ⎞
f1 ( s) = −8s 2 − is + 2 ⎜1 + i ⎟ ,
3
⎝ 3 ⎠
f 2 ( s) = 3s 2 − i
and the convex combination of these polynomials:
4
⎧
⎫
⎛5
⎞
C ( f1 , f 2 ) = ⎨ f ( s ) = ( −11δ + 3) s 2 − δis + 2δ + i ⎜ δ − 1 ⎟ , δ ∈ [0, 1]⎬
3
⎝3
⎠
⎩
⎭
The matrices RS(f1), RS(f2) are of the form:
1
1
⎡
⎤
0
0 ⎥
−3 + i
⎢ −3 − 3 i
3
⎢
⎥
⎢
1
1 ⎥
0
−3 − i
−3 + i ⎥
⎢ 0
3
3 ⎥
⎢
R S ( f1 ) = ⎢
⎥,
1
4
1 ⎥
⎢ 0
5+ i
−5 + i
− i
⎢
3
3
3 ⎥
⎢
⎥
⎢
⎥
1
4
1
5+ i
0 ⎥
− i
⎢ −5 + 3 i
3
3
⎣
⎦
0
3−i
0 ⎤
⎡3 + i
⎢
⎥
0
3− i ⎥
1 ⎢ 0 3+ i
⎥
R S ( f2 ) = ⎢
2 ⎢ 0 3−i
0
−3 − i ⎥
⎢
⎥
⎢⎣3 − i
0 −3 − i
0 ⎥⎦
Criterions for the stability of the convex combination of polynomials
95
and the matrix
0
0
−3i ⎤
⎡ 13
⎢
⎥
0
13
3
0
⎢
⎥
−
i
1
⎥.
R S ( f1 )R −S 1 ( f 2 ) = − ⎢
6 ⎢i (3 − i )
6i
22
−i (3 + i ) ⎥
⎢
⎥
⎢⎣ 6i
22 ⎥⎦
i (3 + i) i(3 − i)
For λ = –2
det(λI – RS( f1)RS–1( f2)) = 0,
therefore the family C( f1, f2) is not Schur stable. For δ =
−
1
polynomial
3
2⎡ 2
3s + 2is + ( −3 + 2i ) ⎤ = f 1 ( s ) ∈ C ( f1 , f 2 )
⎦
9⎣
3
is not Schur stable.
3. Conclusion
In this work necessary and sufficient conditions for Hurwitz (Schur) stability of convex combination of two complex polynomials of the same degree were proved and examples were given to show the use of those theorems. Theorems 3 and 6, proved in this paper,
give answers for the question of stability of a convex combination of complex polynomials
of the same degree, faster than theorems known from paper [3].
Acknowledgements
The author wishes to thank Prof. S. Bia³as for many helpful comments and the referee
for important remark.
References
[1] Barnett S.: Matrices in Control Theory. London, England, Van Nostrand Reinhold 1971
[2] Bia³as S.: A necessary and sufficient condition for the stability of convex combinations of stable
polynomials or matrices. Bull Polish Academy of Sciences, Tech. Sci., 33(9)–(10), 1985, 474–480
[3] Bose N.K.: Tests for Hurwitz and Schur Properties of Convex Combination of Complex Polynomials. IEEE Trans. on Circuits and Systems, 36(9), 1989, 1245–1247
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Ewelina Zalot