APPROXIMATE GREATEST COMMON DIVISOR OF POLYNOMIALS AND THE STRUCTURED SINGULAR VALUE

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1 PPROXIMTE GRETEST OMMON DIVISOR OF POLYNOMILS ND THE STRUTURED SINGULR VLUE G Halikias, S Fatouros N Karcanias ontrol Engineering Research entre, School of Engineering Mathematical Sciences, ity University, Northampton Square, London EV H, UK addresses: ghalikiascityacuk, sfatouroscityacuk, nkarcaniascityacuk Keywords: pproximate GD, Sylvester matrix, structured singular value bstract In this note the following problem is considered: Given two monic coprime polynomials a(s) b(s) with real coefficients, find the smallest (in magnitude) perturbation in their coefficients so that the perturbed polynomials have a common root It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix, which can be performed using efficient existing techniques of robust control simple numerical example illustrates the effectiveness of the method The generalisation of the method to calculate the approximate greatest common divisor (GD) of polynomials is finally discussed Introduction The computation of the greatest common divisor (GD) of two polynomials a(s) b(s) is a non-generic problem, in the sense that a generic pair of polynomials has greatest common divisor one Thus, the notion of approximate common factors GD s has to be introduced for the purpose of effective numerical calculations In the context of Systems ontrol applications, the main motivation for developing non-generic techniques for calculating the GD arises from the study of almost zeros [3] The numerical computation of the GD of polynomials in a robust way has been considered using a variety of methodologies, such as ERES [5], extended ERES [5] matrix pencil methods [4] The main characteristic of all these techniques is that they transform exact procedures to their numerical versions In this work, we formalise the notion of approximate coprimeness approximate GD of two polynomials, by considering the minimum-magnitude perturbation in the polynomial s coefficients, such that the perturbed polynomials have a common root The calculation of the minimal perturbation is shown to correspond to the distance of a structured matrix from singularity, or, equivalently to the calculation of the structured singular value of a matrix [6, 7] The later problem has been studied extensively in the context of robust-stability robustperformance design of control systems, efficient methods have been developed for its numerical solution Generalising the definition of the structured singular value by imposing additional rank constraints, leads naturally to the formulation of a problem involving the calculation of the approximate GD of some minimal degree k 2 Main results The problem considered in this paper is the following: Problem : Let a(s) b(s) be two monic coprime polynomials with real coefficients, such that a(s) = m b(s) = n where m n What is the smallest real perturbation (in magnitude) in the coefficients of a(s) b(s) so that the perturbed polynomials have a common root? Formally write: a (s) = s m + α m s m + α m 2 s m α () b (s) = s m + β n s n + β n 2 s n β (2) assume that the coefficients {α i, i =,,, m } {β i, i =,,, n } are subjected to real perturbations δ, δ,, δ m ɛ, ɛ,, ɛ n, respectively, ie the perturbed polynomials are: a(s; δ,, δ m ) = s m + (α m + δ m )s m + + (α + δ ) b(s; ɛ,, ɛ n ) = s n + (β n + ɛ n )s n respectively Let also: + + (β + ɛ ) γ = max{ δ, δ,, δ m, ɛ, ɛ,, ɛ n } (3) Then Problem is equivalent to: min γ such that a(s; δ,, δ m ) b(s; ɛ,, ɛ n ) have a common root In the sequel, it is shown that Problem is equivalent to the calculation of the (real) structured singular value of an appropriate matrix which can be performed efficiently using existing algorithms

2 Problem 2: (real structured singular value) Let M R n n define the structured set: D = {diag(δ I r, δ 2 I r2,, δ s I rs ) : δ i R, i =, 2,, s} where the r i are positive integers such that s i= r i = n The structured singular value of M (relative to structure D) is defined as: µ D (M) = min{ : D, det(i n M ) = } (4) unless no D makes I n M singular, in which case µ D (M) = The problem here is to calculate µ D (M), provided that µ D (M), to find a D of minimal norm such that det(i n M ) = efore stating the equivalence between Problem Problem 2, we need the following result which relates the existence of common factors of two polynomials to the rank of their corresponding Sylvester resultant matrix: Theorem : onsider the monic polynomials a(s) b(s) with a(s) = m b(s) = n where m n, let be their Sylvester resultant matrix, α m α m 2 α α m α α = α m α α β n β n 2 β β n β β β n β β Then, if φ(s) denotes the GD of a(s) b(s) the following properties hold true: Rank() = n + m φ(s) 2 The polynomials a(s) b(s) are coprime if only if Rank() = n + m 3 The GD φ(s) is invariant under elementary row operations on Furthermore, if we reduce to its row echelon form, the last non-vanishing row defines the coefficients of φ(s) Proof: See [, 5, 2] Using Theorem, the equivalence of Problem Problem 2 can now be established: Theorem 2: Problem is equivalent to Problem 2 by defining: The structured set D as: D = {diag(δ m I n, δ m 2 I n,, δ I n, ɛ n I m, ɛ n 2 I m,, ɛ I m ) : δ i R, ɛ i R} ie s = m + n, r i = n for i m r i = m for m + i m + n 2 M = Z Θ where: ( ) In I Θ = n O n,m O n,m O m,n O m,n I m I m Z = ( (Znm) (Znm m ) (Zmn) (Zmn n ) ) with Θ R n+m,2nm Z R n+m,2nm, where: Z k nm = ( O n,k+ I n O n,m k ) for k =,,, m denotes the (non-singular) Sylvester s resultant matrix of polynomials a(s) b(s) defined above 3 With these definitions: γ = µ D (M) where γ denotes the minimum-magnitude real perturbation in the coefficients of a(s) b(s) such that the perturbed polynomials have a common root Proof: Since a(s) b(s) are assumed coprime, their Sylvester resultant matrix is nonsingular The Sylvester resultant matrix  of the perturbed polynomials a(s; δ,, δ m ) b(s; ɛ,, ɛ n ) can be decomposed as  = + E where E denotes the perturbation matrix : δ m δ m 2 δ δ m δ δ E = δ m δ δ ɛ n ɛ n 2 ɛ ɛ n ɛ ɛ ɛ n ɛ ɛ (5) Matrix E can now be factored as E = Θ Z where Θ Z are defined in Theorem 2 part 2 above, = diag(δ m I n, δ m 2 I n,, δ I n, ɛ n I m, ɛ m 2 I m,, ɛ I m ) learly D, ie it has a block-diagonal structure s = m + n, r i = n for i m r i = m for m + i m + n Note also that: γ = max{ δ, δ,, δ m, ɛ, ɛ,, ɛ n } = Since the resultant Sylvester matrix  loses rank if only if there is a common factor between a(s; δ,, δ m ) b(s; ɛ,, ɛ m ), Problem is equivalent to: min such that det( + Θ Z) = D (6)

3 Using the matrix identity det(i + ) = det(i + ) (7) which holds for any two matrices of compatible dimensions, the fact that is non-singular, we have that: det( + Θ Z) = det(i + Z Θ ) = Hence Problem becomes: det(i M ) = min{ : det(i M ) =, D } (8) which is equivalent to Problem 2, the minimum being µ D (M) Example : Let a(s) = s 3 + α 2 s 2 + α s + α (m = 3) b(s) = s 2 + β s + β (n = 2) The Sylvester resultant matrix of the perturbed polynomials is: α 2 + δ 2 α + δ α + δ α 2 + δ 2 α + δ α + δ Â = β + ɛ β + ɛ β + ɛ β + ɛ (9) β + ɛ β + ɛ which can be written as: α 2 α α δ 2 δ δ α 2 α α Â = β β β β + δ 2 δ δ ɛ ɛ ɛ ɛ β β ɛ ɛ Let E denote the first second matrices, respectively, in the RHS of this equation The perturbation matrix E can be factored as: E = δ 2 I 2 δ I 2 δ I 2 ɛ I 3 ɛ I 3 which is of the required form E = Θ Z with D The minimum coefficient perturbation is the inverse of the structured singular value of the matrix: M = α 2 α α α 2 α α β β β β β β can be computed numerically using efficient existing techniques [6, 7] Example 2: Here the effectiveness of the method is tested with a numerical example onsider the polynomials a (s) = s 3 6s 2 + s 6 b (s) = s 2 6s + 8 with roots {3, 2, } {4, 2} respectively Since there is a common root (s = 2) the resultant Sylvester matrix is singular The polynomials were perturbed to: which have roots a(s) = s 3 65s 2 + s 595 b(s) = s 2 66s + 8 {352, 2454, 9534} {43, 299} respectively The singular values of the corresponding Sylvester resultant matrix, = were obtained as: {227997, 23247, 547, 2264, 7} indicating a numerical rank of 4, hence, an approximate GD of degree one (Theorem ), as expected Next the results of Theorem 2 were applied to a(s) b(s) Note that since the maximum perturbation of the coefficients of

4 a(s) b(s) relative to those of a (s) b (s) (which have a common root) is, we expect that γ Two functions from Matlab s µ-optimisation toolbox (mum unwrapm) were used to calculate the (real) structured singular value of matrix M the corresponding minimum-norm singularising perturbation The lower upper bounds of the structured singular value were obtained as: µ D (M) = γ corresponding to γ = It was also checked that the singularising perturbation corresponding to µ D (M) had the right structure (ie D) in fact δ = δ = δ 2 = γ ɛ = ɛ = γ Polynomials with a common factor nearest to a(s) b(s) were obtained with the help of as: algorithm This requires the refinement of the definition of the structured singular value by introducing rank constraints Specifically, we define: Definition: (generalised real structured singular value) Let M R n n define the structured set: D = {diag(δ I r, δ 2 I r2,, δ s I rs ) : δ i R, i =, 2,, s} where the r i are positive integers such that s i= r i = n The generalised structured singular value of M relative to structure D for a non-negative integer k is defined as: ˆµ D,k (M) = min{ : D, null(i n M ) > k} unless there does not exist a D such that null(i n M ) > k, in which case ˆµ D,k (M) = â(s) = s s s ˆb(s) = s s It follows immediately from the definition that ˆµ D, (M) = µ D (M) that ˆµ D,k (M) ˆµ D,k+ (M) for each integer k Further if for some integer k, ˆµ D,k (M) > ˆµ D,k+ (M) =, then for any minimiser of ˆµ D,k (M), null(i n M ) = k + The roots of â(s) ˆb(s) were calculated as: { , , } { , } respectively corresponding to an optimal approximate GD, φ(s) = s pproximate GD of polynomials The technique developed in section 2 may be used to define a conceptual algorithm to calculate the numerical GD of any two polynomials a(s) b(s) This sequentially extracts approximate common factors φ i (s), by calculating the corresponding structured singular value µ D the minimumnorm singularising matrix perturbation fter the extraction of each factor, the quotients a i+ = a i (s)/φ i (s) b i+ (s) = b i (s)/φ i (s) are calculated, ignoring possible (small) remainders of the divisions The procedure is initialised by setting a (s) = a(s), b (s) = b(s), iterates by constructing at each step of the algorithm the reduceddimension Sylvester matrix corresponding to the polynomial pair (a i+ (s), b i+ (s)), followed by calculating the new µ D, which in turn results to the extraction of the new factor φ i+ (s) The whole process is repeated until a tolerance condition is met, at which stage the approximate GD φ(s) can be constructed by accumulating the extracted common factors φ i (s) Special care is needed in the real case, to ensure that any complex roots in φ(s) appear in conjugate pairs ompared to this procedure, a more elegant approach would be to extract the approximate common factor via a non-iterative Theorem 3: Let a (s) b (s) be two monic coprime polynomials of degrees a (s) = m b (s) = n with m n defined in equations () (2) Let a(s) b(s) be two perturbed polynomials, set γ = max{ δ, δ,, δ m, ɛ, ɛ,, ɛ n } () where {δ i } {ɛ i } denote the perturbed coefficients of a (s) b (s) respectively Further, let γ (k) denote the the minimum value of γ such that a(s) b(s) have a common factor φ(s) of degree φ(s) > k (k =,,, n ) Then, γ (k) = ˆµ D,k (M) () where ˆµ D,k (M) denotes the generalised structured singular value of M = Z Θ with respect to the structure D defined in equation (8),, Θ Z are the matrices defined in equations (7), (9) () respectively Further, φ(s) may be constructed from any D which minimises subject to the constraint null(i n M ) > k Proof: This is almost identical to the proof of Theorem 2, on noting that the transformations in equation (8) do not change the nullity of the corresponding matrices Theorem 3 suggests that the GD of two polynomials a(s) b(s) can be obtained by calculating successively ˆµ D,k (M) for k =,, n The procedure terminates when either k = n is reached, or when the generalised structured singular value falls below a pre-specified tolerance level The effective numerical calculation of ˆµ D,k (or at least of tight upper lower bounds) will be the subject of future research further generalisation of our method involves the calculation of

5 the approximate GD for an arbitrary number of polynomials (rather than just two) This problem can also be translated to our framework using some recent results on generalised resultants [2] will also be a topic of future work 4 onclusions In this paper we have proposed a new method for calculating numerically the approximate GD of a set of polynomials It was shown that, for two coprime polynomials, the problem of calculating the smallest l -norm perturbation in the polynomials coefficient vector so that the perturbed polynomials have a common root, is equivalent to the calculation of the structured singular value of an appropriate matrix This is a fundamental problem in the area of robust control various techniques have been successfully developed for its solution The effectiveness of one such method for calculating the GD of lowdegree polynomials has been demonstrated via a numerical example We have further shown that calculating the minimum l -norm perturbation in the coefficient vector of two (or more) coprime polynomials so that the perturbed polynomials have a GD of degree at least k, reduces to a structured singular value-type calculation with additional rank constraints This is a non-stard problem developing effective algorithms for its solution will be the topic of future research work References [] S arnett (983), Polynomials Linear onstrol Systems, Marcel Dekker, Inc, New York [2] S Fatouros N Karcanias (22), The GD of many polynomials the factorisation of generalised resultants, Research Report, ontrol Engineering Research entre, ity University [3] N Karcanias, Giannakopoulos M Hubbard (984), lmost zeros of a set of polynomials, Int J ontrol, 4, [4] N Karcanias M Mitrouli (994), matrix pencilbased numerical method for the computation of the GD of polynomials, IEEE Tran uto ontr, -39, [5] M Mitrouli N Karcanias (993), omputation of the GD of polynomials using Gaussian transformations shifting, Int J ontrol, 58, [6] PM Young J Doyle (996), Properties of the mixed µ problem its bounds, IEEE Tran uto ontr, 4(), [7] K Zhou J Doyle (998), Essentials of Robust ontrol, Prentice Hall, Inc, Upper Saddle River, NJ, US