- Email: [email protected]
A Necessary and Sufficient Condition of Global Positivity on Shift-realizable Multivariate Polynomial by LMI
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
A Necessary and Sufficient Condition of Global Positivity on Shift-realizable Multivariate Polynomial by LMI Y. Kuroiwa ∗ ∗
(e-mail: [email protected]).
Abstract: We introduce a class of multivariate polynomials, which are represented by shift realizations. A necessary and sufficient condition of the global positivity on the shift-realizable polynomial is derived by an LMI. Via the generalization to the multivariable case, a parametrization of positive multivariate polynomials is obtained. Keywords: Global positivity, shift realization, linear matrix inequality. 1. INTRODUCTION The positivity is essential in control and systems theory for the applications to analysis and synthesis of stability and energy-based passivity of systems in Schwarzchild (1916); Krstic et al. (1995); Lindquist (1997); Isidori (1997); Helton (2008); Kuroiwa (2009). The positivity of multivariate polynomial is associated with the multidimensional systems theory in Bose (1982); Kuroiwa (2011a), and the positive realness of multivariate function has applications to the image processing in Jain (1989). For a given multivariate polynomial, we want to know whether or not the polynomial is greater than or equal to zero on the real axes. This is also a fundamental problem not only in engineering but also in mathematics, and quite many efforts must be devoted to answer the question, see Minkowski (1885); Hilbert (1888). For polynomial of one variable on the real axis, the positivity is equivalent that the polynomial has a quadratic form. For rational function of one variable on the complex plane, the positivity is equivalent that the function has a spectral factor in Anderson and Vongpanitlerd (2006). However, the situation is completely different for the multivariate case. There exists a positive polynomial of three variables, p(x1 , x2 , x3 ) = x41 x22 + x21 x42 + x63 − 3x21 x22 x23 ,
The existence of the representation by a quadratic form or a sum-of-squares is a simple sufficient condition for multivariate polynomial to be positive everywhere, see Bose (1982). Consider a polynomial p(x1 , x2 , x3 ) = x41 − (2x2 x3 + 1)x21 + x22 x23 + 2x2 x3 + 2. p(x1 , x2 , x3 ) = 1 +
x21
+ (1 −
x21
+ x2 x3 ) ,
which implies that p(x1 , x2 , x3 ) is positive. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
2
p(x1 , x2 , x3 ) = 2x41 + 2x31 x2 − x21 x22 x23 + 5x42 , is represented by
(2)
T " # 2 x21 x1 20 1 0 5 0 x22 p(x1 , x2 , x3 ) = x22 1 0 −1 x1 x2 x1 x2 2 T " # x1 x21 2 −λ 1 x2 . −λ 5 0 = x22 2 1 0 −1 + 2λ x1 x2 x1 x2
In the firs line, the matrix is not positive semidefinite and we may think that p(x1 , x2 , x3 ) is not positive. However, in the second line, the matrix is positive semidefinite for λ = 3. In this case, T " # 2 x1 2 −3 1 x21 −3 5 0 x22 p(x1 , x2 , x3 ) = x22 1 0 5 x1 x2 x1 x2 2 T 2 x1 x1 = x22 LT L x22 , x1 x2 x1 x2
(1)
which is not represented by a quadratic form in Motzkin (1965).
This polynomial is given by the sum-of-squares
One property is that the representation of a polynomial by a quadratic form might not be unique in Parrilo (2003). The polynomial,
where
1 2 −3 1 . L= √ 2 0 1 3 Therefore, we have the sum-of-squares decomposition 1 [(2x21 − 3x22 + x1 x2 )2 + (x22 + 3x1 x2 )2 ], 2 and we conclude that p(x1 , x2 , x3 ) is positive. p(x1 , x2 , x3 ) =
In this paper, we give a necessary and sufficient condition of the positivity on a class of polynomials by a linear matrix inequality (LMI). The class of polynomials is called shif t-realizable since the polynomial is represented by the 7346
10.3182/20110828-6-IT-1002.00425
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
shift realization. We use a conformal map to transform the system of the shift realization on the real axes to a system on the unit circles. We derive the LMI condition of the positivity of the system on the unit circles. It is based on the positive real lemma of the multidimensional systems without stability constraint Kuroiwa (2011a), which is the generalization of the positive real lemma of one-dimensional case Iwasaki (2007); Kuroiwa (2011b). It turns out that the LMI is linear with respect to the coefficients of the polynomial. We generalize the result to the multivariable case, for which positive is replaced by positive semidef inite. The motivation is as follows. Suppose that a matrix polynomial M (x1 , . . . , xm ) is given, which is positive semidefinite on real axes. Then, for a vector of polynomials y,
We verify P (X ) = c0 +
c11 c12 c21 c22 + 2 + + 2 x1 x1 x2 x2
and x21 x22 P1 (x1 ) = x21 x22
c12 c11 + 2 x1 x1
= c11 x1 x22 + c12 x22 c22 c21 + 2 x21 x22 P2 (x2 ) = x21 x22 x2 x2 = c21 x21 x2 + c22 x21 .
If we consider x1 and x2 as complex variables, the conjugate system P (X )∗ is given by
y T M (x1 , . . . , xm )y ≥ 0 holds. Thus, we can obtain the set of polynomials by the free parameter y, which is positive on real axes.
P (X )∗ = c0 +
c12 c21 c22 c11 + 2 + + 2. x ¯1 x ¯1 x ¯2 x ¯2
On the real axes,
2. NOTATIONS Real numbers are represented by R and complex numbers are represented by C. c¯ denotes the conjugate of c ∈ C. Denote by Rj×k j × k real matrices and by Cj×k j × k complex matrices. Im denotes m× m identity matrix. 0j×k denotes j×k zero matrix and 0m denotes m×m square zero matrix. They are simply represented by I and 0 if their dimensions are clear in the context. We use the notation A ≥ 0 to denote that a matrix A is positive semidefinite. Denote by AT the transpose of a matrix A and by A∗ the complex conjugate transpose of a matrix A. Denote by Sn the set of n × n Hermitian matrices. Denote by T = {z ∈ C : |z| = 1} the unit circle. The Cartesian product of m real axes R × · · · × R is denoted by Rm . Similarly, the Cartesian product of m unit circles T × · · · × T is denoted by Tm . 3. EXAMPLES OF SHIFT-REALIZABLE POLYNOMIAL We present the results by using example. We use a shift realization to represent a polynomial. We call the polynomial shif t-realizable since it is represented by the state-space realization with the forward shift matrix. The polynomial
P (X )∗ = P (X ) holds. Thus, (4) is written by p(x1 , x2 ) = x21 x22 P (X ) x2 x2 = 1 2 [P (X ) + P (X )∗ ]. (5) 2 The global positivity of p(x1 , x2 ) on R2 is equivalent to P (X ) + P (X )∗ ≥ 0
on R2 since x21 x22 > 0 holds on R2 . It turns out that the representation (5) by the shift realization is fundamental to show the global positivity of p(x1 , x2 ) on R2 . We derive a LMI condition to characterize global positivity of p(x1 , x2 ) on R2 . It is the positive real lemma of multidimensional system, for which the independent variable is not necessarily positive definite. The state-space realization of
is given by
p(x1 , x2 ) = c0 x21 x22 + c11 x1 x22 + c12 x22 + c21 x21 x2 + c22 x21 is represented by the shift realization, p(x1 , x2 ) = x21 x22 P (X ),
P (X ) := C(X − Ap )−1 Bp + c0 X := diag [ x1 I2 x2 I2 ] .
(3)
Ap := diag [ S S ] e Bp := 1 e1
(4)
C := [ C1 C2 ] .
where
We apply the conformal maps,
P (X ) = P1 (x1 ) + P2 (x2 ) + c0 Pk (xk ) := Ck (xk I2 − S)−1 e1 , 00 S := 10 1 e1 := 0 Ck := [ ck1 ck2 ] .
k = 1, 2,
zk =
xk − i , xk + i
k = 1, 2,
to P (X ), which map the upper-half planes to the unit disks, i,e., the real axes R2 is conformally mapped to the unit circles T2 , and we use the formula (19) to derive the state-space realization of zk . Denote by G(Z) the system of zk , which is conformally mapped from P (X ), where 7347
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
H(X ) = H1 (x1 ) + H2 (x2 ) + c0
G(Z) := C(Z − A)−1 B + D
Hk (xk ) := Ck (xk I − S)−1 e1 0 0 00 1 0 0 0 S4 := 0 1 0 0 0 0 10 1 0 e1 := 0 0
Z := diag [ z1 I2 z2 I2 ] .
A := diag [ Sz Sz ] −1 0 Sz := 2i −1 −2i b , b := B := 4 b
D := c0 + CF i f . , f := F := −1 f
(6)
Then, we use the positive real lemma of multidimensional system to check whether or not G(Z) + G(Z)∗ ≥ 0
Ck := [ ck1 ck2 ck3 ck4 ] .
The polynomial h(x1 , x2 ) = c0 x41 x42 + c11 x31 x42 + c12 x21 x42 + c13 x1 x42 x2 + c14 x42
(7)
holds on T2 . The positive real lemma of multidimensional system without stability constraint gives a necessary and sufficient condition of the positivity due to the diagonal structure of the Hermitian matrix of the independent variables. Theorem 1. Let us define
+c21 x41 x32 + c22 x41 x22 + c23 x41 x2 x2 + c24 x41 is given by h(x1 , x2 ) = x41 x42 H(X ) since x41 x42 H1 (x1 ) c12 c13 c14 4 4 c11 + 2 + 3 + 4 = x1 x2 x1 x1 x1 x1
P := diag [ P1 P2 ] , where Pk ∈ S2 . Then, (3) is positive on (x1 , x2 ) ∈ R2 , which is equivalent that (7) holds on (z1 , z2 ) ∈ T2 , if and only if there exists (c0 , c11 , c12 , c21 , c22 ) and P such that M2 (P ) ≥ 0,
(10)
= c11 x31 x42 + c12 x21 x42 + c13 x1 x42 x2 + c14 x42 . x41 x42 H2 (x2 ) c22 c23 c24 4 4 c21 + 2 + 3 + 4 = x1 x2 x2 x2 x2 x2
(8)
= c21 x41 x32 + c22 x41 x22 + c23 x41 x2 x2 + c24 x41 .
where M2 (P ) :=
P − A∗ P A C ∗ − A∗ P B . C − B ∗ P A D + D∗ − B ∗ P B
Moreover, (10) is given by (9) h(x1 , x2 ) = since
Proof. It is in the proof of T heorem 2. It should be noted that (9) is linear with respect to (c0 , c11 , c12 , c21 , c22 ) and P , i.e., the global positivity of p(x1 , x2 ) is characterized by the LMI of (8).
H(X ) = H(X )∗ holds on R2 . If we replace x41 x42 by x41 x22 , then, a polynomial x41 x22 H1 (x1 ) c12 c13 c14 4 2 c11 = x1 x2 + 2 + 3 + 4 x1 x1 x1 x1
Our basic idea is to represent the polynomial by the shift realization. We introduce a class of polynomials, which are called shift-realizable. Clearly, any polynomial of one variable is shit-realizable since c11 c1k c1n n x1 c0 + + ···+ k ···+ n x1 x1 x1
= c11 x31 x22 + c12 x21 x22 + c13 x1 x22 + c14 x22 . is generated. Similarly, a polynomial of three variables
= c0 xn1 + c11 x1n−1 + · · · + c1k x1n−k · · · + c1n
g(x1 , x2 , x3 ) = c0 x21 x22 x23 + c11 x1 x22 x23 + c12 x22 x23
holds. The essence in T heorem 1 is the same for the general case. An obvious necessary condition of positivity of polynomial is that the degree of the polynomial is even. We consider the degree four case. Let us define
x41 x42 [H(X ) + H(X )∗ ]. 2
+c21 x21 x2 x23 + c22 x21 x23 + c31 x21 x22 x3 + c32 x21 x22 can be represented by the shift realization. In fact, g(x1 , x2 , x3 ) = where
7348
x21 x22 x23 [P (X ) + P (X )∗ ], 2
(11)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
is positive on (x1 , . . . , xk , . . . , xm ) ∈ Rm if and only if there exist (c0 , c11 , . . . , cmnm ) and P such that
P (X ) = P1 (x1 ) + P2 (x2 ) + P3 (x3 ) + c0 Pk (xk ) := Ck (xk I2 − S)−1 e1 , 0 0 S := 1 0 1 B := e1 , e1 := 0
k = 1, 2, 3
M (P ) ≥ 0,
where M (P ) :=
Ck := [ ck1 ck2 ] .
By using the same procedures in T heorem 1, the positivities of (10) and (11) are characterized by LMIs.
P − A∗ P A C ∗ − A∗ P B . C − B ∗ P A D + D∗ − B ∗ P B
(13)
Proof. On the real axes Rm , the polynomial is given by
4. MAIN RESULT
p(x1 , . . . , xk , . . . , xm )
We state the general case of the results in the previous section. Let us define
=
1 m · · · xk2nk · · · x2n x2n m 1 [P (X ) + P (X )∗ ], 2
where Z := diag [ z1 In1 . . . zk Ink . . . zm Inm ]
P (X ) = C(X − A)−1 B + c0
A := diag [ A1 . . . Ak . . . Am ] −1 0 0 0 ··· 0 −2i −1 0 0 · · · 0 1 −2i −1 0 · · · 0 nk ×nk Ak := 0 1 −2i −1 · · · 0 ∈C . .. .. .. .. .. . . . . 0 0 0 0 · · · −1 −2i b1 4 .. 2i . ∈ Cnk ×1 B := bk , bk := 0 . . ... . bm 0
X := diag [ x1 In1 . . . xk Ink . . . xm Inm ]
S := diag [ S1 . . . Sk . . . Sm ] 0 0 ··· 0 1 0 ··· 0 0 1 ··· 0 ∈ Rnk ×nk Sk := . . .. .. .. . 0 0 ··· 0 k = 1, . . . , m e1 1 .. . 0 nk ×1 B := ek , ek = . ... ∈ R . . . 0 em
C := [ C1 C2 · · · Cm ]
Ck := [ ck1 ck2 · · · cknk ] D := c0 + CF f1 .. . F := fk , . . . fm
We apply conformal maps,
i −1 0 nk ×1 . fk := . ∈C ..
zk = (12)
0
They are used to represent a polynomial by the shift realization. Theorem 2. Let us define
+···+ +···+
nk X i=1
nm X i=1
n1 X i=1
(14)
P (X ) =⇒ G(Z) = C(Z − A)−1 B + D,
where the matrices are defined by (12).
The positivity of the polynomial is equivalent to P (X ) + P (X )∗ ≥ 0 m
1 x2n 1
(15) k · · · x2n k
m · · · x2n m
on (x1 , . . . , xk , . . . , xm ) ∈ R since ≥ 0 on Rm . Conformally, the positivity of (15) is equivalent to
p(x1 , . . . , xk , . . . , xm ) = c0 xn1 1 · · · xnk k · · · xnmm +
k = 1, . . . , m,
to P (X ), which map Rm to Tm . We use the formula (19) to derive the state-space realization of zk ,
P := diag [ P1 . . . Pk . . . Pm ] , where Pk ∈ Snk . The polynomial
xk − i , xk + i
c1i xi1 · · · xnk k · · · xnmm
cki xn1 1 · · · xik · · · xnmm cmi xn1 1 · · · xnk k · · · xim .
G(Z) + G(Z)∗ ≥ 0 m
(16)
on (z1 , . . . , zk , . . . , zm ) ∈ T , where the conjugate of G(Z) is given by G(Z)∗ = B ∗ (Z −1 − A∗ )−1 C ∗ + D∗
since (eiθk )∗ = (eiθk )−1 , k = 1, . . . , m, i.e., 7349
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Z ∗ = Z −1 holds on (z1 , . . . , zk , . . . , zm ) ∈ Tm . We derive the LMI condition of (16). We use G(Z) + G(Z)∗ 0 C∗ = B ∗ (Z −1 − A∗ )−1 I C D + D∗ (Z − A)−1 B × I
and
y T M (x1 , . . . , xm )y ≥ 0 holds. Thus, we can obtain the set of polynomials by the free parameter y, which is positive on Rm . For real matrices C0 and Cjk , an ℓ × ℓ matrix polynomial p(x1 , . . . , xk , . . . , xm ) = (C0 + C0T )xn1 1 · · · xnk k · · · xnmm n1 X T (C1i + C1i )xi1 · · · xnk k · · · xnmm + i=1
+···+
P − A∗ P A −A∗ P B I ∗ ∗ −B P A −B P B
B ∗ (Z −1 − A∗ )−1 (Z − A)−1 B ≡ 0, × I
+···+
which is derived by
+A∗ P (Z − A) + (Z −1 − A∗ )P A ⇐⇒ (Z
= P + (Z
∗ −1
−A )
−1
i=1
T (Cmi + Cmi )xn1 1 · · · xnk k · · · xim
(17)
2nm 1 x2n · · · xk2nk · · · xm 1 [P (X ) + P (X )∗ ] 2 on the real axes (x1 , . . . , xk , . . . , xm ) ∈ Rm . The statespace realization is given by
=
−1
{P − A P A}(Z − A)
∗ −1
−A )
∗
p(x1 , . . . , xk , . . . , xm )
A∗ P + P A(Z − A)−1
=⇒ B ∗ (Z −1 − A∗ )−1 {P − A∗ P A}(Z − A)−1 B
= B ∗ P B + B ∗ (Z −1 − A∗ )−1 A∗ P B
+B ∗ P A(Z − A)−1 B since P and Z are commutative due to the diagonal structure. The sum of them yields
P (X ) = C(X − A)−1 B + C0 + C0T , where X :=
G(Z) + G(Z)∗ ∗ −1 (Z − A)−1 B ∗ −1 . = B (Z − A ) I M (P ) I
diag [ x1 Iℓn1 . . . xk Iℓnk . . . xm Iℓnm ] S := diag [ S1 0ℓ 0ℓ Iℓ 0ℓ 0 I ℓ ℓ Sk := . . .. .. 0ℓ 0ℓ k = 1, . . . , m
Clearly, G(Z)+G(Z)∗ ≥ 0 on Tm if and only if M (P ) ≥ 0. Conformally,
on Rm
i=1
nm X
T )xn1 1 · · · xik · · · xnmm (Cki + Cki
is represented by the shift realization,
P − A∗ P A = (Z −1 − A∗ )P (Z − A) −1
nk X
P (X ) + P (X )∗ ≥ 0 if and only if M (P ) ≥ 0. On the real axes Rm ,
P (X ) = P (X )∗ holds, and we obtain P (X ) ≥ 0.
k 1 m By multiplying x2n · · · x2n · · · x2n m , which is positive on 1 k m R , we conclude
B :=
1 k m x2n · · · x2n · · · x2n m P (X ) = p(x1 , . . . , xk , . . . , xm ) 1 k
≥ 0.
e1 .. . ek .. . em
,
Sk . . . Sm ] 0ℓ 0ℓ 0ℓ ∈ Rℓnk ×ℓnk .. . · · · 0ℓ ... ··· ··· ···
Iℓ 0ℓ ℓnk ×ℓ ek = ... ∈ R
0ℓ
C := [ C1 C2 · · · Cm ]
5. GENERALIZATION FOR PARAMETERIZATION We first consider the multivariable generalization of T heorem 2, for which positive is replaced by positive semidef inite. The motivation is as follows. Suppose that an ℓ × ℓ matrix polynomial M (x1 , . . . , xm ) is given, which is positive semidefinite on real axes Rm . Then, for an ℓdimensional vector of polynomials y,
Ck = T T T . Ck1 + Ck1 Ck2 + Ck2 · · · Cknk + Ckn k
By (14), we obtain a system of zk ,
where
7350
P (X ) =⇒ G(Z) = C(Z − A)−1 B + D,
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Z := diag [ z1 Iℓn1 . . . zk Iℓnk . . . zm Iℓnm ]
A := diag [ A1 . . . Ak . . . Am ] −Iℓ 0ℓ 0ℓ 0ℓ · · · −2iIℓ −Iℓ 0ℓ 0ℓ · · · Iℓ −2iIℓ −Iℓ 0ℓ · · · Ak := Iℓ −2i −Ie ll · · · 0ℓ . .. .. .. .. . . . 0ℓ 0ℓ 0ℓ 0ℓ · · ·
0ℓ 0ℓ 0ℓ 0ℓ .. . −Iℓ
Thus, we obtain
G(z) = (iI + A) (A − iI) 2i(iI + A) B . C −C(iI + A)−1 B + D −1
−2
(19)
REFERENCES
∈ Cℓnk ×ℓnk −2iIℓ B1 4Iℓ .. 2i . ∈ Cℓnk ×ℓ B := Bk , Bk := 0 ℓ . . . .. . Bm 0ℓ
D := C0 + C0T + CF F1 iIℓ . . −Iℓ . 0 ℓnk ×ℓ ℓ . (18) F := Fk , Fk := . ∈C . .. . . 0ℓ Fm Similar to T heorem 2, P (X ) is positive semidefinite on Rm if and only if G(Z) is positive semidefinite on Tm . Theorem 3. Let us define P := diag [ P1 . . . Pk . . . Pm ] , where Pk ∈ Sℓnk . The matrix polynomial (17) is positive definite on (x1 , . . . , xk , . . . , xm ) ∈ Rm if and only if there exist (C0 , C11 , . . . , Cmnm ) and P such that M (P ) ≥ 0, where M (P ) is given by (13), and the matrices are given by (18). Proof. It is similar to the proof of T heorem 2. APPENDIX We present the formula of the state-space realization. For a given P (x) = C(xI − A)−1 B + D A B = , C D
we transform it to a system of G(z) by using a conformal map ax + b cx + d for ad−bc 6= 0. The state-space realization of G(z) is given by z=
(a, b, c, d) = (1, −i, 1, i).
Anderson, B.D.O. and Vongpanitlerd, S. (2006). Network Analysis and Synthesis-A Modern Systems Approach. Dover. Bose, N.K. (1982). Applied multidimensional systems theory. Van Nostrand Reinhold. Helton, J.W. (2008). Sets which lift to a set with an LMI representation. In Proceedings of 18th International Symposium of Mathematical Theory of Networks and Systems. Blacksburg Virginia, USA. ¨ Hilbert, D. (1888). Uber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann., 32, 342– 350. Isidori, A. (1997). Nonlinear Control Systems. SpringerVerlag, Berlin, 3rd edition. Iwasaki, T. (2007). Multivariable Control. Lecture Notes, University of Virginia. Jain, A.K. (1989). Fundamentals of Digital Image Processing. Prentice Hall, Englewood Cliffs, New Jersey. Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V. (1995). Robust and Adaptive Control Design. John Wiley and Sons. Kuroiwa, Y. (2009). A Parameterization of Positive Real Residue Interpolants with McMillan Degree Constraint. Ph.D. thesis, Royal Institute of Technology, Stockholm. Kuroiwa, Y.S.H. (2011a). Positive real lemma of multidimensional systems with application to rational spectral factorization. submitted. Kuroiwa, Y.S.H. (2011b). Positive real lemma with positive semidefinite solution and a difference in continuoustime and discrete-time systems theories. submitted. Lindquist, A. (1997). Recent progress in the partial stochastic realization problem. In U. Helmke, D. Pratzel-Wolters, and E. Zerz (eds.), Operators, Systems, and Linear Algebra. Minkowski, H. (1885). Untersuchungen u ¨ber quadratische Formen. Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enth¨ alt. K¨onigsberg. Motzkin, T.S. (1965). The arithmetic-geometric inequality. In Proc. of Sympos. at Wright-Patterson AFB, 19– 27. Parrilo, P.A. (2003). Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming Ser. B, 96(2), 293–320. Schwarzchild, K. (1916). On the gravitational field of a mass point according to Einstein’s theory. arXiv:physics/9905030.
G(z) = (dI + cA)−1 (aA + bI) (ad − bc)(dI + cA)−2 B . C −cC(dI + cA)−1 B + D The conformal map (14) is given by setting 7351
A Necessary and Sufficient Condition of Global Positivity on Shift-realizable Multivariate Polynomial by LMI Y. Kuroiwa ∗ ∗
(e-mail: [email protected]).
Abstract: We introduce a class of multivariate polynomials, which are represented by shift realizations. A necessary and sufficient condition of the global positivity on the shift-realizable polynomial is derived by an LMI. Via the generalization to the multivariable case, a parametrization of positive multivariate polynomials is obtained. Keywords: Global positivity, shift realization, linear matrix inequality. 1. INTRODUCTION The positivity is essential in control and systems theory for the applications to analysis and synthesis of stability and energy-based passivity of systems in Schwarzchild (1916); Krstic et al. (1995); Lindquist (1997); Isidori (1997); Helton (2008); Kuroiwa (2009). The positivity of multivariate polynomial is associated with the multidimensional systems theory in Bose (1982); Kuroiwa (2011a), and the positive realness of multivariate function has applications to the image processing in Jain (1989). For a given multivariate polynomial, we want to know whether or not the polynomial is greater than or equal to zero on the real axes. This is also a fundamental problem not only in engineering but also in mathematics, and quite many efforts must be devoted to answer the question, see Minkowski (1885); Hilbert (1888). For polynomial of one variable on the real axis, the positivity is equivalent that the polynomial has a quadratic form. For rational function of one variable on the complex plane, the positivity is equivalent that the function has a spectral factor in Anderson and Vongpanitlerd (2006). However, the situation is completely different for the multivariate case. There exists a positive polynomial of three variables, p(x1 , x2 , x3 ) = x41 x22 + x21 x42 + x63 − 3x21 x22 x23 ,
The existence of the representation by a quadratic form or a sum-of-squares is a simple sufficient condition for multivariate polynomial to be positive everywhere, see Bose (1982). Consider a polynomial p(x1 , x2 , x3 ) = x41 − (2x2 x3 + 1)x21 + x22 x23 + 2x2 x3 + 2. p(x1 , x2 , x3 ) = 1 +
x21
+ (1 −
x21
+ x2 x3 ) ,
which implies that p(x1 , x2 , x3 ) is positive. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
2
p(x1 , x2 , x3 ) = 2x41 + 2x31 x2 − x21 x22 x23 + 5x42 , is represented by
(2)
T " # 2 x21 x1 20 1 0 5 0 x22 p(x1 , x2 , x3 ) = x22 1 0 −1 x1 x2 x1 x2 2 T " # x1 x21 2 −λ 1 x2 . −λ 5 0 = x22 2 1 0 −1 + 2λ x1 x2 x1 x2
In the firs line, the matrix is not positive semidefinite and we may think that p(x1 , x2 , x3 ) is not positive. However, in the second line, the matrix is positive semidefinite for λ = 3. In this case, T " # 2 x1 2 −3 1 x21 −3 5 0 x22 p(x1 , x2 , x3 ) = x22 1 0 5 x1 x2 x1 x2 2 T 2 x1 x1 = x22 LT L x22 , x1 x2 x1 x2
(1)
which is not represented by a quadratic form in Motzkin (1965).
This polynomial is given by the sum-of-squares
One property is that the representation of a polynomial by a quadratic form might not be unique in Parrilo (2003). The polynomial,
where
1 2 −3 1 . L= √ 2 0 1 3 Therefore, we have the sum-of-squares decomposition 1 [(2x21 − 3x22 + x1 x2 )2 + (x22 + 3x1 x2 )2 ], 2 and we conclude that p(x1 , x2 , x3 ) is positive. p(x1 , x2 , x3 ) =
In this paper, we give a necessary and sufficient condition of the positivity on a class of polynomials by a linear matrix inequality (LMI). The class of polynomials is called shif t-realizable since the polynomial is represented by the 7346
10.3182/20110828-6-IT-1002.00425
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
shift realization. We use a conformal map to transform the system of the shift realization on the real axes to a system on the unit circles. We derive the LMI condition of the positivity of the system on the unit circles. It is based on the positive real lemma of the multidimensional systems without stability constraint Kuroiwa (2011a), which is the generalization of the positive real lemma of one-dimensional case Iwasaki (2007); Kuroiwa (2011b). It turns out that the LMI is linear with respect to the coefficients of the polynomial. We generalize the result to the multivariable case, for which positive is replaced by positive semidef inite. The motivation is as follows. Suppose that a matrix polynomial M (x1 , . . . , xm ) is given, which is positive semidefinite on real axes. Then, for a vector of polynomials y,
We verify P (X ) = c0 +
c11 c12 c21 c22 + 2 + + 2 x1 x1 x2 x2
and x21 x22 P1 (x1 ) = x21 x22
c12 c11 + 2 x1 x1
= c11 x1 x22 + c12 x22 c22 c21 + 2 x21 x22 P2 (x2 ) = x21 x22 x2 x2 = c21 x21 x2 + c22 x21 .
If we consider x1 and x2 as complex variables, the conjugate system P (X )∗ is given by
y T M (x1 , . . . , xm )y ≥ 0 holds. Thus, we can obtain the set of polynomials by the free parameter y, which is positive on real axes.
P (X )∗ = c0 +
c12 c21 c22 c11 + 2 + + 2. x ¯1 x ¯1 x ¯2 x ¯2
On the real axes,
2. NOTATIONS Real numbers are represented by R and complex numbers are represented by C. c¯ denotes the conjugate of c ∈ C. Denote by Rj×k j × k real matrices and by Cj×k j × k complex matrices. Im denotes m× m identity matrix. 0j×k denotes j×k zero matrix and 0m denotes m×m square zero matrix. They are simply represented by I and 0 if their dimensions are clear in the context. We use the notation A ≥ 0 to denote that a matrix A is positive semidefinite. Denote by AT the transpose of a matrix A and by A∗ the complex conjugate transpose of a matrix A. Denote by Sn the set of n × n Hermitian matrices. Denote by T = {z ∈ C : |z| = 1} the unit circle. The Cartesian product of m real axes R × · · · × R is denoted by Rm . Similarly, the Cartesian product of m unit circles T × · · · × T is denoted by Tm . 3. EXAMPLES OF SHIFT-REALIZABLE POLYNOMIAL We present the results by using example. We use a shift realization to represent a polynomial. We call the polynomial shif t-realizable since it is represented by the state-space realization with the forward shift matrix. The polynomial
P (X )∗ = P (X ) holds. Thus, (4) is written by p(x1 , x2 ) = x21 x22 P (X ) x2 x2 = 1 2 [P (X ) + P (X )∗ ]. (5) 2 The global positivity of p(x1 , x2 ) on R2 is equivalent to P (X ) + P (X )∗ ≥ 0
on R2 since x21 x22 > 0 holds on R2 . It turns out that the representation (5) by the shift realization is fundamental to show the global positivity of p(x1 , x2 ) on R2 . We derive a LMI condition to characterize global positivity of p(x1 , x2 ) on R2 . It is the positive real lemma of multidimensional system, for which the independent variable is not necessarily positive definite. The state-space realization of
is given by
p(x1 , x2 ) = c0 x21 x22 + c11 x1 x22 + c12 x22 + c21 x21 x2 + c22 x21 is represented by the shift realization, p(x1 , x2 ) = x21 x22 P (X ),
P (X ) := C(X − Ap )−1 Bp + c0 X := diag [ x1 I2 x2 I2 ] .
(3)
Ap := diag [ S S ] e Bp := 1 e1
(4)
C := [ C1 C2 ] .
where
We apply the conformal maps,
P (X ) = P1 (x1 ) + P2 (x2 ) + c0 Pk (xk ) := Ck (xk I2 − S)−1 e1 , 00 S := 10 1 e1 := 0 Ck := [ ck1 ck2 ] .
k = 1, 2,
zk =
xk − i , xk + i
k = 1, 2,
to P (X ), which map the upper-half planes to the unit disks, i,e., the real axes R2 is conformally mapped to the unit circles T2 , and we use the formula (19) to derive the state-space realization of zk . Denote by G(Z) the system of zk , which is conformally mapped from P (X ), where 7347
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
H(X ) = H1 (x1 ) + H2 (x2 ) + c0
G(Z) := C(Z − A)−1 B + D
Hk (xk ) := Ck (xk I − S)−1 e1 0 0 00 1 0 0 0 S4 := 0 1 0 0 0 0 10 1 0 e1 := 0 0
Z := diag [ z1 I2 z2 I2 ] .
A := diag [ Sz Sz ] −1 0 Sz := 2i −1 −2i b , b := B := 4 b
D := c0 + CF i f . , f := F := −1 f
(6)
Then, we use the positive real lemma of multidimensional system to check whether or not G(Z) + G(Z)∗ ≥ 0
Ck := [ ck1 ck2 ck3 ck4 ] .
The polynomial h(x1 , x2 ) = c0 x41 x42 + c11 x31 x42 + c12 x21 x42 + c13 x1 x42 x2 + c14 x42
(7)
holds on T2 . The positive real lemma of multidimensional system without stability constraint gives a necessary and sufficient condition of the positivity due to the diagonal structure of the Hermitian matrix of the independent variables. Theorem 1. Let us define
+c21 x41 x32 + c22 x41 x22 + c23 x41 x2 x2 + c24 x41 is given by h(x1 , x2 ) = x41 x42 H(X ) since x41 x42 H1 (x1 ) c12 c13 c14 4 4 c11 + 2 + 3 + 4 = x1 x2 x1 x1 x1 x1
P := diag [ P1 P2 ] , where Pk ∈ S2 . Then, (3) is positive on (x1 , x2 ) ∈ R2 , which is equivalent that (7) holds on (z1 , z2 ) ∈ T2 , if and only if there exists (c0 , c11 , c12 , c21 , c22 ) and P such that M2 (P ) ≥ 0,
(10)
= c11 x31 x42 + c12 x21 x42 + c13 x1 x42 x2 + c14 x42 . x41 x42 H2 (x2 ) c22 c23 c24 4 4 c21 + 2 + 3 + 4 = x1 x2 x2 x2 x2 x2
(8)
= c21 x41 x32 + c22 x41 x22 + c23 x41 x2 x2 + c24 x41 .
where M2 (P ) :=
P − A∗ P A C ∗ − A∗ P B . C − B ∗ P A D + D∗ − B ∗ P B
Moreover, (10) is given by (9) h(x1 , x2 ) = since
Proof. It is in the proof of T heorem 2. It should be noted that (9) is linear with respect to (c0 , c11 , c12 , c21 , c22 ) and P , i.e., the global positivity of p(x1 , x2 ) is characterized by the LMI of (8).
H(X ) = H(X )∗ holds on R2 . If we replace x41 x42 by x41 x22 , then, a polynomial x41 x22 H1 (x1 ) c12 c13 c14 4 2 c11 = x1 x2 + 2 + 3 + 4 x1 x1 x1 x1
Our basic idea is to represent the polynomial by the shift realization. We introduce a class of polynomials, which are called shift-realizable. Clearly, any polynomial of one variable is shit-realizable since c11 c1k c1n n x1 c0 + + ···+ k ···+ n x1 x1 x1
= c11 x31 x22 + c12 x21 x22 + c13 x1 x22 + c14 x22 . is generated. Similarly, a polynomial of three variables
= c0 xn1 + c11 x1n−1 + · · · + c1k x1n−k · · · + c1n
g(x1 , x2 , x3 ) = c0 x21 x22 x23 + c11 x1 x22 x23 + c12 x22 x23
holds. The essence in T heorem 1 is the same for the general case. An obvious necessary condition of positivity of polynomial is that the degree of the polynomial is even. We consider the degree four case. Let us define
x41 x42 [H(X ) + H(X )∗ ]. 2
+c21 x21 x2 x23 + c22 x21 x23 + c31 x21 x22 x3 + c32 x21 x22 can be represented by the shift realization. In fact, g(x1 , x2 , x3 ) = where
7348
x21 x22 x23 [P (X ) + P (X )∗ ], 2
(11)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
is positive on (x1 , . . . , xk , . . . , xm ) ∈ Rm if and only if there exist (c0 , c11 , . . . , cmnm ) and P such that
P (X ) = P1 (x1 ) + P2 (x2 ) + P3 (x3 ) + c0 Pk (xk ) := Ck (xk I2 − S)−1 e1 , 0 0 S := 1 0 1 B := e1 , e1 := 0
k = 1, 2, 3
M (P ) ≥ 0,
where M (P ) :=
Ck := [ ck1 ck2 ] .
By using the same procedures in T heorem 1, the positivities of (10) and (11) are characterized by LMIs.
P − A∗ P A C ∗ − A∗ P B . C − B ∗ P A D + D∗ − B ∗ P B
(13)
Proof. On the real axes Rm , the polynomial is given by
4. MAIN RESULT
p(x1 , . . . , xk , . . . , xm )
We state the general case of the results in the previous section. Let us define
=
1 m · · · xk2nk · · · x2n x2n m 1 [P (X ) + P (X )∗ ], 2
where Z := diag [ z1 In1 . . . zk Ink . . . zm Inm ]
P (X ) = C(X − A)−1 B + c0
A := diag [ A1 . . . Ak . . . Am ] −1 0 0 0 ··· 0 −2i −1 0 0 · · · 0 1 −2i −1 0 · · · 0 nk ×nk Ak := 0 1 −2i −1 · · · 0 ∈C . .. .. .. .. .. . . . . 0 0 0 0 · · · −1 −2i b1 4 .. 2i . ∈ Cnk ×1 B := bk , bk := 0 . . ... . bm 0
X := diag [ x1 In1 . . . xk Ink . . . xm Inm ]
S := diag [ S1 . . . Sk . . . Sm ] 0 0 ··· 0 1 0 ··· 0 0 1 ··· 0 ∈ Rnk ×nk Sk := . . .. .. .. . 0 0 ··· 0 k = 1, . . . , m e1 1 .. . 0 nk ×1 B := ek , ek = . ... ∈ R . . . 0 em
C := [ C1 C2 · · · Cm ]
Ck := [ ck1 ck2 · · · cknk ] D := c0 + CF f1 .. . F := fk , . . . fm
We apply conformal maps,
i −1 0 nk ×1 . fk := . ∈C ..
zk = (12)
0
They are used to represent a polynomial by the shift realization. Theorem 2. Let us define
+···+ +···+
nk X i=1
nm X i=1
n1 X i=1
(14)
P (X ) =⇒ G(Z) = C(Z − A)−1 B + D,
where the matrices are defined by (12).
The positivity of the polynomial is equivalent to P (X ) + P (X )∗ ≥ 0 m
1 x2n 1
(15) k · · · x2n k
m · · · x2n m
on (x1 , . . . , xk , . . . , xm ) ∈ R since ≥ 0 on Rm . Conformally, the positivity of (15) is equivalent to
p(x1 , . . . , xk , . . . , xm ) = c0 xn1 1 · · · xnk k · · · xnmm +
k = 1, . . . , m,
to P (X ), which map Rm to Tm . We use the formula (19) to derive the state-space realization of zk ,
P := diag [ P1 . . . Pk . . . Pm ] , where Pk ∈ Snk . The polynomial
xk − i , xk + i
c1i xi1 · · · xnk k · · · xnmm
cki xn1 1 · · · xik · · · xnmm cmi xn1 1 · · · xnk k · · · xim .
G(Z) + G(Z)∗ ≥ 0 m
(16)
on (z1 , . . . , zk , . . . , zm ) ∈ T , where the conjugate of G(Z) is given by G(Z)∗ = B ∗ (Z −1 − A∗ )−1 C ∗ + D∗
since (eiθk )∗ = (eiθk )−1 , k = 1, . . . , m, i.e., 7349
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Z ∗ = Z −1 holds on (z1 , . . . , zk , . . . , zm ) ∈ Tm . We derive the LMI condition of (16). We use G(Z) + G(Z)∗ 0 C∗ = B ∗ (Z −1 − A∗ )−1 I C D + D∗ (Z − A)−1 B × I
and
y T M (x1 , . . . , xm )y ≥ 0 holds. Thus, we can obtain the set of polynomials by the free parameter y, which is positive on Rm . For real matrices C0 and Cjk , an ℓ × ℓ matrix polynomial p(x1 , . . . , xk , . . . , xm ) = (C0 + C0T )xn1 1 · · · xnk k · · · xnmm n1 X T (C1i + C1i )xi1 · · · xnk k · · · xnmm + i=1
+···+
P − A∗ P A −A∗ P B I ∗ ∗ −B P A −B P B
B ∗ (Z −1 − A∗ )−1 (Z − A)−1 B ≡ 0, × I
+···+
which is derived by
+A∗ P (Z − A) + (Z −1 − A∗ )P A ⇐⇒ (Z
= P + (Z
∗ −1
−A )
−1
i=1
T (Cmi + Cmi )xn1 1 · · · xnk k · · · xim
(17)
2nm 1 x2n · · · xk2nk · · · xm 1 [P (X ) + P (X )∗ ] 2 on the real axes (x1 , . . . , xk , . . . , xm ) ∈ Rm . The statespace realization is given by
=
−1
{P − A P A}(Z − A)
∗ −1
−A )
∗
p(x1 , . . . , xk , . . . , xm )
A∗ P + P A(Z − A)−1
=⇒ B ∗ (Z −1 − A∗ )−1 {P − A∗ P A}(Z − A)−1 B
= B ∗ P B + B ∗ (Z −1 − A∗ )−1 A∗ P B
+B ∗ P A(Z − A)−1 B since P and Z are commutative due to the diagonal structure. The sum of them yields
P (X ) = C(X − A)−1 B + C0 + C0T , where X :=
G(Z) + G(Z)∗ ∗ −1 (Z − A)−1 B ∗ −1 . = B (Z − A ) I M (P ) I
diag [ x1 Iℓn1 . . . xk Iℓnk . . . xm Iℓnm ] S := diag [ S1 0ℓ 0ℓ Iℓ 0ℓ 0 I ℓ ℓ Sk := . . .. .. 0ℓ 0ℓ k = 1, . . . , m
Clearly, G(Z)+G(Z)∗ ≥ 0 on Tm if and only if M (P ) ≥ 0. Conformally,
on Rm
i=1
nm X
T )xn1 1 · · · xik · · · xnmm (Cki + Cki
is represented by the shift realization,
P − A∗ P A = (Z −1 − A∗ )P (Z − A) −1
nk X
P (X ) + P (X )∗ ≥ 0 if and only if M (P ) ≥ 0. On the real axes Rm ,
P (X ) = P (X )∗ holds, and we obtain P (X ) ≥ 0.
k 1 m By multiplying x2n · · · x2n · · · x2n m , which is positive on 1 k m R , we conclude
B :=
1 k m x2n · · · x2n · · · x2n m P (X ) = p(x1 , . . . , xk , . . . , xm ) 1 k
≥ 0.
e1 .. . ek .. . em
,
Sk . . . Sm ] 0ℓ 0ℓ 0ℓ ∈ Rℓnk ×ℓnk .. . · · · 0ℓ ... ··· ··· ···
Iℓ 0ℓ ℓnk ×ℓ ek = ... ∈ R
0ℓ
C := [ C1 C2 · · · Cm ]
5. GENERALIZATION FOR PARAMETERIZATION We first consider the multivariable generalization of T heorem 2, for which positive is replaced by positive semidef inite. The motivation is as follows. Suppose that an ℓ × ℓ matrix polynomial M (x1 , . . . , xm ) is given, which is positive semidefinite on real axes Rm . Then, for an ℓdimensional vector of polynomials y,
Ck = T T T . Ck1 + Ck1 Ck2 + Ck2 · · · Cknk + Ckn k
By (14), we obtain a system of zk ,
where
7350
P (X ) =⇒ G(Z) = C(Z − A)−1 B + D,
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Z := diag [ z1 Iℓn1 . . . zk Iℓnk . . . zm Iℓnm ]
A := diag [ A1 . . . Ak . . . Am ] −Iℓ 0ℓ 0ℓ 0ℓ · · · −2iIℓ −Iℓ 0ℓ 0ℓ · · · Iℓ −2iIℓ −Iℓ 0ℓ · · · Ak := Iℓ −2i −Ie ll · · · 0ℓ . .. .. .. .. . . . 0ℓ 0ℓ 0ℓ 0ℓ · · ·
0ℓ 0ℓ 0ℓ 0ℓ .. . −Iℓ
Thus, we obtain
G(z) = (iI + A) (A − iI) 2i(iI + A) B . C −C(iI + A)−1 B + D −1
−2
(19)
REFERENCES
∈ Cℓnk ×ℓnk −2iIℓ B1 4Iℓ .. 2i . ∈ Cℓnk ×ℓ B := Bk , Bk := 0 ℓ . . . .. . Bm 0ℓ
D := C0 + C0T + CF F1 iIℓ . . −Iℓ . 0 ℓnk ×ℓ ℓ . (18) F := Fk , Fk := . ∈C . .. . . 0ℓ Fm Similar to T heorem 2, P (X ) is positive semidefinite on Rm if and only if G(Z) is positive semidefinite on Tm . Theorem 3. Let us define P := diag [ P1 . . . Pk . . . Pm ] , where Pk ∈ Sℓnk . The matrix polynomial (17) is positive definite on (x1 , . . . , xk , . . . , xm ) ∈ Rm if and only if there exist (C0 , C11 , . . . , Cmnm ) and P such that M (P ) ≥ 0, where M (P ) is given by (13), and the matrices are given by (18). Proof. It is similar to the proof of T heorem 2. APPENDIX We present the formula of the state-space realization. For a given P (x) = C(xI − A)−1 B + D A B = , C D
we transform it to a system of G(z) by using a conformal map ax + b cx + d for ad−bc 6= 0. The state-space realization of G(z) is given by z=
(a, b, c, d) = (1, −i, 1, i).
Anderson, B.D.O. and Vongpanitlerd, S. (2006). Network Analysis and Synthesis-A Modern Systems Approach. Dover. Bose, N.K. (1982). Applied multidimensional systems theory. Van Nostrand Reinhold. Helton, J.W. (2008). Sets which lift to a set with an LMI representation. In Proceedings of 18th International Symposium of Mathematical Theory of Networks and Systems. Blacksburg Virginia, USA. ¨ Hilbert, D. (1888). Uber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann., 32, 342– 350. Isidori, A. (1997). Nonlinear Control Systems. SpringerVerlag, Berlin, 3rd edition. Iwasaki, T. (2007). Multivariable Control. Lecture Notes, University of Virginia. Jain, A.K. (1989). Fundamentals of Digital Image Processing. Prentice Hall, Englewood Cliffs, New Jersey. Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V. (1995). Robust and Adaptive Control Design. John Wiley and Sons. Kuroiwa, Y. (2009). A Parameterization of Positive Real Residue Interpolants with McMillan Degree Constraint. Ph.D. thesis, Royal Institute of Technology, Stockholm. Kuroiwa, Y.S.H. (2011a). Positive real lemma of multidimensional systems with application to rational spectral factorization. submitted. Kuroiwa, Y.S.H. (2011b). Positive real lemma with positive semidefinite solution and a difference in continuoustime and discrete-time systems theories. submitted. Lindquist, A. (1997). Recent progress in the partial stochastic realization problem. In U. Helmke, D. Pratzel-Wolters, and E. Zerz (eds.), Operators, Systems, and Linear Algebra. Minkowski, H. (1885). Untersuchungen u ¨ber quadratische Formen. Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enth¨ alt. K¨onigsberg. Motzkin, T.S. (1965). The arithmetic-geometric inequality. In Proc. of Sympos. at Wright-Patterson AFB, 19– 27. Parrilo, P.A. (2003). Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming Ser. B, 96(2), 293–320. Schwarzchild, K. (1916). On the gravitational field of a mass point according to Einstein’s theory. arXiv:physics/9905030.
G(z) = (dI + cA)−1 (aA + bI) (ad − bc)(dI + cA)−2 B . C −cC(dI + cA)−1 B + D The conformal map (14) is given by setting 7351