Algorithms for polynomial multidimensional spectral factorization and sum of squares

Transcription

1 Algorithms for polynomial multidimensional spectral factorization and sum of squares H.L. Trentelman University of Groningen, The Netherlands. Japan November 2007 Algorithms for polynomial multidimensional spectral factorization and sum of squares p.1

2 Outline Polynomial spectral factorization of nd polynomial matrices Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

3 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

4 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Spectral factorization: lifting to 2nD polynomial matrix context Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

5 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Spectral factorization: lifting to 2nD polynomial matrix context Reduction to LMI s Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

6 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Spectral factorization: lifting to 2nD polynomial matrix context Reduction to LMI s Connection with dissipativity of nd system behaviors Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

7 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Spectral factorization: lifting to 2nD polynomial matrix context Reduction to LMI s Connection with dissipativity of nd system behaviors SOS: lifting to 2nD polynomial matrix context Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

8 Outline Polynomial spectral factorization of nd polynomial matrices Sum of squares decomposition of nd polynomial matrices Spectral factorization: lifting to 2nD polynomial matrix context Reduction to LMI s Connection with dissipativity of nd system behaviors SOS: lifting to 2nD polynomial matrix context Example of SOS Algorithms for polynomial multidimensional spectral factorization and sum of squares p.2

9 Multidimensional spectral factorization Notation: n-dimensional indeterminate ξ = (ξ 1,ξ 2,...,ξ n ). Given: a para-hermitian real q q n-variable polynomial matrix Z(ξ), i.e. Z T ( ξ) = Z(ξ) Polynomial spectral factorization problem: compute a real n-variable polynomial matrix F(ξ) such that Z(ξ) = F T ( ξ)f(ξ) Any such F is called a spectral factor Well-known: for n = 1: such F(ξ) exists if and only if Z(iω) 0 for all ω R. F can even be chosen square Algorithms for polynomial multidimensional spectral factorization and sum of squares p.3

10 For n > 1: Z(iω) 0 for all ω R n if and only if there exists an n-variable rational matrix such that Z(ξ) = F T ( ξ)f(ξ) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.4

11 For n > 1: Z(iω) 0 for all ω R n if and only if there exists an n-variable rational matrix such that Z(ξ) = F T ( ξ)f(ξ) In our work: necessary and sufficient conditions for existence of polynomial spectral factors algorithms for their computation in terms of LMI s Algorithms for polynomial multidimensional spectral factorization and sum of squares p.4

12 Sum of squares problem Algorithms for polynomial multidimensional spectral factorization and sum of squares p.5

13 Sum of squares problem Given: a real symmetric q q n-variable polynomial matrix Z(ξ), i.e. Z(ξ) T = Z(ξ) Sum of squares problem: compute an n-variable polynomial matrix F(ξ) such that Z(ξ) = F T (ξ)f(ξ) Note: for q = 1 this amounts to the problem of decomposing the given real polynomial z(ξ) into a sum of squares: with f i (ξ) real polynomials. z(ξ) = f 1 (ξ) 2 + f 2 (ξ) f N (ξ) 2 Well-known: for n = 1: such F(ξ) exists if and only if Z(ξ) 0 for all ξ R. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.6

14 for n > 1: Z(ξ) 0 for all ξ R n if and only if there exists an n-variable rational matrix such that Z(ξ) = F T (ξ)f(ξ) In this presentation: necessary and sufficient conditions for existence of polynomial sum of squares decomposition algorithm for computation of polynomial sum of squares decompositions For the sake of notational simplicity: we will outline the ideas assuming n = 2. However, the results hold for general n. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.7

15 Monomial ordering General 2-variable polynomial matrix: Z(ξ 1,ξ 2 ) = M 1 k=0 M 2 l=0 Z k,l ξ k 1ξ l 2. Associated with Z(ξ 1,ξ 2 ) we have a coefficient matrix mat(z): is determined after fixing an ordering on the (degrees of the) monomials ξ1 kξl 2. We take the ("anti-lexicographic") ordering (0, 0) < (1, 0) < (2, 0) <... < (M 1, 0) < (0, 1) < (1, 1) < (2, 1) <... < (M 1, 1) < (0,M 2 ) < (1,M 2 ) < (2,M 2 ) <... < (M 1,M 2 ).. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.8

16 This yields Z(ξ 1,ξ 2 ) = mat(z)v (ξ 1,ξ 2 ) T with V (ξ 1,ξ 2 ) = (I, ξ 1 I,..., ξ M 1 1 I. ξ 2 I,ξ 1 ξ 2 I,..,ξ M 1 1 ξ 2 I ξ M 2 2 I,ξ 1 ξ M 2 2 I,...,ξ M 1 1 ξ M 2 2 I) Note: respects the ordering of the monomials! Algorithms for polynomial multidimensional spectral factorization and sum of squares p.9

17 Spectral factorization: lifting from n to 2n For the given n variable polynomial matrix Z(ξ) we will compute a 2n variable polynomial matrix Φ(ζ, η) such that Z(ξ) = Φ( ξ,ξ) and mat(φ) 0 Here, mat(φ) is the coefficient matrix of Φ(ζ,η), defined by: n = 2: general form of 4-variable polynomial matrix: Φ(ζ 1,ζ 2,η 1,η 2 ) = N 1 k 1,l 1 =0 N 2 k 2,l 2 =0 Φ (k1,k 2 )(l 1,l 2 )ζ k 1 1 ζk 2 2 ηl 1 1 η l 2 1. Again let V (ξ 1,ξ 2 ) = (I, ξ 1 I,..., ξ M 1 1 I. ξ 2 I,ξ 1 ξ 2 I,..,ξ M 1 1 ξ 2 I ξ M 2 2 I,ξ 1 ξ M 2 2 I,...,ξ M 1 1 ξ M 2 2 I) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.10

18 Then we can write: Φ(ζ 1,ζ 2,η 1,η 2 ) = V (ζ 1,ζ 2 )mat(φ)v (η 1,η 2 ) T Now, if Z(ξ 1,ξ 2 ) = Φ( ξ 1, ξ 2,ξ 1,ξ 2 ) and mat(φ) 0 then factorize mat(φ) = F T F and define F(ξ1,ξ 2 ) := FV (ξ 1,ξ 2 ) T. This then yields a polynomial spectral factorization: Z(ξ 1,ξ 2 ) = V ( ξ 1, ξ 2 )mat(φ)v (ξ 1,ξ 2 ) T = F T ( ξ 1,ξ 2 )F(ξ 1,ξ 2 ) The above also holds for arbitrary n, (more difficult notation) Question: Is it possible to find, for a given Z(ξ), a Φ(ζ,η) such that Z(ξ) = Φ( ξ,ξ) and mat(φ) 0? Algorithms for polynomial multidimensional spectral factorization and sum of squares p.11

19 Theorem: Let Z(ξ) be a parahermitian n-variable q q polynomial matrix. Then the following statements are equivalent: (i) Z(ξ) has a polynomial spectral factorization, i.e. there exists a positive integer r and an r q polynomial matrix F(ξ) such that Z(ξ) = F T ( ξ)f(ξ) (ii) there exists a symmetric 2n-variable polynomial matrix ˆΦ(ζ,η) such that ˆΦ( ξ,ξ) = Z(ξ) and mat(ˆφ) 0. Symmetric means: Φ(ζ,η) T = Φ(η,ζ). Algorithms for polynomial multidimensional spectral factorization and sum of squares p.12

20 Two steps But: how to check whether such ˆΦ exists, and if it exists, how to compute it? In two steps: First step: compute any Φ(ζ,η) such that Φ( ξ,ξ) = Z(ξ). Is always possible: take e.g.: Φ(ζ,η) := 1 2 (Z( ζ)t + Z(η)) Complete characterization of all such Φ(ζ, η) can be given in terms of Z(ξ) Second step: change the Φ(ζ,η) that we found to one with mat(φ) 0. The property Φ( ξ,ξ) = Z(ξ) should remain! Look at n-tuples of 2n-variable polynomial matrices Ψ(ζ, η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)). Define: Ψ (ζ,η) := (ζ 1 +η 1 )Ψ 1 (ζ,η)+(ζ 2 +η 2 )Ψ 2 (ζ,η)+...+(ζ n +η n )Ψ n (ζ,η) We have: if Φ( ξ,ξ) = Z(ξ), then for any n tuple Ψ(ζ,η) we have (Φ Ψ)( ξ,ξ) = Z(ξ). Algorithms for polynomial multidimensional spectral factorization and sum of squares p.13

21 Idea: try to compute an n tuple Ψ(ζ,η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)) in such a way that mat(φ Ψ) 0 Since mat(φ Ψ) = mat(φ) mat( Ψ) the latter can be formulated as a linear matrix inequality, with the coefficient matrices of the components Ψ 1 (ζ,η),...ψ n (ζ,η) as the unknowns!!! Indeed: mat((ζ i + η i )Ψ i (ζ,η)) = (σ i,d + σ i,r )(mat(ψ i )) for matrix shift operators σ i,d and σ i,r. Hence mat(φ) mat( Ψ) 0 can be expressed as mat(φ) [(σ 1,D + σ 1,R )(mat(ψ 1 )) + (σ 2,D + σ 2,R )(mat(ψ 2 )) + + (σ n,d + σ n,r )(mat(ψ n ))] 0 (LMI!) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.14

22 Spectral factorization: reducability to LMI Theorem: Let Z(ξ) be a parahermitian n-variable q q polynomial matrix. Let Φ(ζ,η) be a symmetric q q 2n variable polynomial matrix such that Φ( ξ, ξ) = Z(ξ). Then the following statements are equivalent: (i) Z(ξ) has a polynomial spectral factorization, i.e. there exists a positive integer r and an r q polynomial matrix F(ξ) such that Z(ξ) = F T ( ξ)f(ξ). (ii) There exist an n-tuple of symmetric 2n-variable q q polynomial matrices Ψ(ζ,η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)) such that mat(φ) mat( Ψ) 0 LMI! Algorithms for polynomial multidimensional spectral factorization and sum of squares p.15

23 Interpretation: dissipative nd systems Symmetric 2n-variable polynomial matrix: Φ(ζ,η) = k,l Φ k,lζ k η l. Multi-indices k := (k 1,...,k n ), l := (l 1,...,l n ), indeterminates ζ := (ζ 1,...,ζ n ), η := (η 1,...,η n ), ζ k := ζ k 1 1 ζk 2 2 ζk n n, η k := η k 1 1 ηk 2 2 ηk n n. Any such Φ induces a quadratic differential form (QDF) Q Φ : C (R n, R q ) C (R n, R) Q Φ (w) := k,l ( dk w dx k )T Φ k,l d l w dx l. where dk dx k is defined as dk dx k := k 1 x k k n x k n n. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.16

24 System behavior: B = C (R n, R q ). Interpret x 1 as time, (x 2,x 3,...,x n ) as space. Interpretation: Q Φ (w)(x 1,x 2,...,x n ) is the rate of supply into an infinitesimal space volume if the trajectory w occurs. Storage functions: with the n tuple Ψ(ζ,η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)) we associate the vector of quadratic differential forms (VQDF) Q Ψ = (Q Ψ1,Q Ψ2,...,Q Ψn ). Divergence: (div Q Ψ )(w) := x 1 Q Ψ1 (w) x n Q Ψn (w). If (div Q Ψ )(w) Q Φ (w) for all w C (R n, R q ), then Q Ψ is called a storage function for B with respect Q Φ. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.17

25 Recall: Ψ (ζ,η) := (ζ 1 +η 1 )Ψ 1 (ζ,η)+(ζ 2 +η 2 )Ψ 2 (ζ,η)+...+(ζ n +η n )Ψ n (ζ,η). Fact: div Q Ψ is the QDF asociated with Ψ (ζ,η). It can be shown: LMI: mat(φ) mat( Ψ) 0 if and only if (div Q Ψ )(w) Q Φ (w) for all w. Thus: solving the LMI is equivalent to computing a storage function Q Ψ for B = C (R n, R q ) with respect to the supply rate Q Φ. In fact, if mat(φ) mat( Ψ) = F T F and F(ξ) = FV (ξ) T (spectral factor), then Q Φ (w) (div Q Ψ )(w) = F( d dx )w 2. (dissipation rate!) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.18

26 SOS: lifting from n to 2n Given: a real symmetric q q n-variable polynomial matrix Z(ξ), i.e. Z(ξ) T = Z(ξ). For the given n variable polynomial matrix Z(ξ) we will compute a 2n variable polynomial matrix Φ(ζ, η) such that Z(ξ) = Φ(ξ,ξ) and mat(φ) 0 n = 2: next, if Z(ξ 1,ξ 2 ) = Φ(ξ 1,ξ 2,ξ 1,ξ 2 ) and mat(φ) 0 then factorize mat(φ) = F T F and define F(ξ1,ξ 2 ) := FV (ξ 1,ξ 2 ) T. This then yields a SOS decomposition: Z(ξ 1,ξ 2 ) = V (ξ 1,ξ 2 )mat(φ)v (ξ 1,ξ 2 ) T = F T (ξ 1,ξ 2 )F(ξ 1,ξ 2 ) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.19

27 The above also holds for arbitrary n, (more difficult notation) Question: Is it possible to find, for a given Z(ξ), a Φ(ζ,η) such that Z(ξ) = Φ(ξ,ξ) and mat(φ) 0? Theorem: Let Z(ξ) be a positive semidefinite n-variable q q polynomial matrix. Then the following statements are equivalent: (i) Z(ξ) is SOS, i.e. there exists a positive integer r and an r q polynomial matrix F(ξ) such that Z(ξ) = F T (ξ)f(ξ) (ii) there exists a symmetric 2n-variable polynomial matrix ˆΦ(ζ,η) such that ˆΦ(ξ,ξ) = Z(ξ) and mat(ˆφ) 0. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.20

28 Two steps But: how to check whether such ˆΦ exists, and if it exists, how to compute it? Again in two steps: First step: compute any Φ(ζ,η) such that Φ(ξ,ξ) = Z(ξ). Is always possible: take e.g.: Φ(ζ,η) := 1 2 (Z(ζ)T + Z(η)) Complete characterization of all such Φ(ζ, η) can be given in terms of Z(ξ) Second step: change the Φ(ζ,η) that we found to one with mat(φ) 0. The property Φ(ξ,ξ) = Z(ξ) should remain! Look at n-tuples of 2n-variable polynomial matrices Ψ(ζ, η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)). Define: Ψ(ζ,η) := (ζ 1 η 1 )Ψ 1 (ζ,η)+(ζ 2 η 2 )Ψ 2 (ζ,η)+...+(ζ n η n )Ψ n (ζ,η) We have: if Φ(ξ,ξ) = Z(ξ), then for any n tuple Ψ(ζ,η) we have (Φ + Ψ)(ξ,ξ) = Z(ξ). Algorithms for polynomial multidimensional spectral factorization and sum of squares p.21

29 Idea: try to compute an n tuple Ψ(ζ,η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)) in such a way that mat(φ + Ψ) 0 Since mat(φ + Ψ) = mat(φ) + mat(ψ) the latter can be formulated as a linear matrix inequality, with the coefficient matrices of the components Ψ 1 (ζ,η),...ψ n (ζ,η) as the unknowns!!! Indeed: mat((ζ i η i )Ψ i (ζ,η)) = (σ i,d σ i,r )(mat(ψ i )) for matrix shift operators σ i,d and σ i,r. Hence mat(φ) + mat(ψ) 0 can be expressed as mat(φ) + (σ 1,D σ 1,R )(mat(ψ 1 )) + (σ 2,D σ 2,R )(mat(ψ 2 )) + + (σ n,d σ n,r )(mat(ψ n )) 0 (LMI!) Algorithms for polynomial multidimensional spectral factorization and sum of squares p.22

30 SOS and reducability to LMI Theorem: Let Z(ξ) be a positive semidefinite n-variable q q polynomial matrix. Let Φ(ζ,η) be a symmetric q q 2n variable polynomial matrix such that Φ(ξ, ξ) = Z(ξ). Then the following statements are equivalent: (i) Z(ξ) is SOS, i.e. there exists a positive integer r and an r q polynomial matrix F(ξ) such that Z(ξ) = F T (ξ)f(ξ). (ii) There exist an n-tuple of skew-symmetric 2n-variable q q polynomial matrices Ψ(ζ,η) = (Ψ 1 (ζ,η),.., Ψ n (ζ,η)) such that mat(φ) + mat(ψ) 0 LMI! Algorithms for polynomial multidimensional spectral factorization and sum of squares p.23

31 Example Z(ξ 1,ξ 2 ) = ( 1 2ξ 1 + ξ1 2 + (ξ2 1 ξ 2) 2 ξ 1 ξ1 2 ξ 1 ξ ξ 1 + 2ξ1 2 + (ξ2 1 ξ 2) 2 ) Find, if these exist, a positive integer r and a r 2 matrix F(ξ 1,ξ 2 ) such that Z(ξ 1,ξ 2 ) = F T (ξ 1,ξ 2 )F(ξ 1,ξ 2 ) First step: construct symmetric Φ(ζ 1,ζ 2,η 1,η 2 ) such that Φ(ξ 1,ξ 2,ξ 1,ξ 2 ) = Z(ξ 1,ξ 2 ) Take Φ(ζ 1,ζ 2,η 1,η 2 ) equal to: Algorithms for polynomial multidimensional spectral factorization and sum of squares p.24

32 (I, Iζ 1, Iζ1 2 Iζ 2, Iζ 1 ζ 2, Iζ1 2ζ 2) C A 0 I Iη 1 Iη 2 1 Iη 2 Iη 1 η 2 Iη 2 1 Iη 2 1 C A I is the 2 2 identity matrix Algorithms for polynomial multidimensional spectral factorization and sum of squares p.25

33 Second step: compute Ψ 1 (ζ 1,ζ 2,η 1,η 2 ) and Ψ 2 (ζ 1,ζ 2,η 1,η 2 ), skew-symmetric, with highest monomial degrees (2, 0) and (1, 1) respectively. Their (unknown) coefficient matrices are given by mat(ψ 1 ) = 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 0 a 8 a 9 a 10 a 11 a 12 a 13 a 2 a 8 0 a 14 a 15 a 16 a 17 a 18 a 3 a 9 a 14 0 a 19 a 20 a 21 a 22 a 4 a 10 a 15 a 19 0 a 23 a 24 a 25 a 5 a 11 a 16 a 20 a 23 0 a 26 a 27 a 6 a 12 a 17 a 21 a 24 a 26 0 a 29 a 6 a 13 a 18 a 22 a 25 a 27 a 29 0 Algorithms for polynomial multidimensional spectral factorization and sum of squares p.26

34 mat(ψ 2 ) = 0 b 1 b 2 b 3 b 4 b 5 b 1 0 b 6 b 7 b 8 b 9 b 2 b 6 0 b 10 b 11 b 12 b 3 b 7 b 10 0 b 13 b 14 b 4 b 8 b 11 b 13 0 b 15 b 5 b 9 b 12 b 14 b 15 0 the LMI we want to solve is mat(φ) + (σ 1,D σ 1,R )(mat(ψ 1 )) + (σ 2,D σ 2,R )(mat(ψ 2 )) 0. Algorithms for polynomial multidimensional spectral factorization and sum of squares p.27

35 A solution to this inequality is given by: mat(ψ 1 ) = and mat(ψ 2 ) the zero matrix Algorithms for polynomial multidimensional spectral factorization and sum of squares p.28

36 This yields mat(φ + Ψ) = Algorithms for polynomial multidimensional spectral factorization and sum of squares p.29

37 Rank of this matrix is equal to 4, it can be factorized as F T F, with F given below: F(ξ 1,ξ 2 ) = Hence: Z(ξ 1,ξ 2 ) = F T (ξ 1,ξ 2 )F(ξ 1,ξ 2 ) = ( ) ( 1 ξ 1 0 ξ1 2ξ I Iξ 1 Iξ 2 1 Iξ 2 Iξ 1 ξ 2 Iξ 2 1 Iξ 2 1 ξ 1 0 ξ 2 1 ξ C A ) T. ξ 1 1 ξ 1 0 ξ 2 1 ξ 2 ξ 1 1 ξ 1 0 ξ 2 1 ξ 2 Algorithms for polynomial multidimensional spectral factorization and sum of squares p.30

38 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31

39 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable A symmetric positive semidefinite n variable polynomial matrix has a SOS decomposition if and only if a given LMI is solvable Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31

40 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable A symmetric positive semidefinite n variable polynomial matrix has a SOS decomposition if and only if a given LMI is solvable After solving the LMI, a spectral factorization/sos decomposition can be computed Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31

41 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable A symmetric positive semidefinite n variable polynomial matrix has a SOS decomposition if and only if a given LMI is solvable After solving the LMI, a spectral factorization/sos decomposition can be computed Lifting to 2n variable polynomial context Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31

42 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable A symmetric positive semidefinite n variable polynomial matrix has a SOS decomposition if and only if a given LMI is solvable After solving the LMI, a spectral factorization/sos decomposition can be computed Lifting to 2n variable polynomial context For spectral factorization the inspiration comes from dissipative nd systems: spectral factors correspond to dissipation rates Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31

43 Conclusions A parahermitian n variable polynomial matrix has a polynomial spectral factorization if and only if a given LMI is solvable A symmetric positive semidefinite n variable polynomial matrix has a SOS decomposition if and only if a given LMI is solvable After solving the LMI, a spectral factorization/sos decomposition can be computed Lifting to 2n variable polynomial context For spectral factorization the inspiration comes from dissipative nd systems: spectral factors correspond to dissipation rates Thank you! Algorithms for polynomial multidimensional spectral factorization and sum of squares p.31