Some applications to rational matrices

rmkstruct(N, D; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (KRInfo, iz, nfp, ip)

Determine the Kronecker-structure and infinite pole-zero structure information of the rational matrix R(λ) := N(λ)./D(λ) and return an KRInfo object, the multiplicities of infinite zeros in iz, the number of finite poles in nfp and the multiplicities of infinite poles in ip.

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The information on the Kronecker-structure of the rational matrix R(λ) consists of the right Kronecker indices rki, left Kronecker indices lki, the number of finite zeros nf and the normal rank nrank and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo. Additionally, KRInfo.id contains the infinite elementary divisors of the underlying system matrix pencil S(λ) used as linearization.

The Kronecker-structure information of the rational matrix R(λ) is obtained from the Kronecker-structure information of the underlying linearization using the results of [1] and [2]. The determination of the Kronecker-structure information is performed by building an irreducible descriptor system realization (A-λE,B,C,D) with A-λE a regular pencil of order n, satisfying R(λ) = C*inv(λE-A)*B+D and reducing the structured system matrix pencil S(λ) = [A-λE B; C D] to an appropriate Kronecker-like form (KLF) which exhibits the number of its finite eigenvalues (also the finite zeros of R(λ)), the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite zeros of R(λ)), the left and right Kronecker indices (also of R(λ)) and the normal rank (in excess with n to the normal rank of R(λ)). The Kronecker-structure information and pole-zero stucture information on R(λ) are recovered from the Kronecker-structure information of S(λ) and A-λE, respectively, using the results of [1] and [2].

The right Kronecker indices are provided in the integer vector rki. The number of elements of rki is the dimension of the right nullspace of the rational matrix R(λ) and their sum is the least degree of a right polynomial nullspace basis or the least McMillan degree of a right rational nullspace basis.

The left Kronecker indices are provided in the integer vector lki. The number of elements of lki is the dimension of the left nullspace of the rational matrix R(λ) and their sum is the least degree of a left polynomial nullspace basis or the least McMillan degree of a left rational nullspace basis.

The multiplicities of infinite eigenvalues of S(λ) are provided in the integer vector id, where each i-th element id[i] is the order of an infinite elementary divisor (i.e., the multiplicity of an infinite eigenvalue). To each id[i] > 1 corresponds an infinite zero of R(λ) of multiplicity id[i]-1. The multiplicities of the infinite zeros of R(λ) are returned in iz.

The poles of R(λ) are the zeros of the regular pencil A-λE. The finite eigenvalues of A-λE are the finite poles of R(λ) and their number is returned in nfp. The infinite zeros of A-λE are the infinite poles of R(λ) and their multiplicities are returned in ip.

The irreducible descriptor system based linearization is built using the methods described in [3] in conjunction with pencil manipulation algorithms of [4] and [5] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reductions to appropriate KLFs of S(λ) and A-λE are performed using orthogonal similarity transformations [5] and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false.

The keyword arguments atol and rtol specify the absolute and relative tolerances, respectively, for the nonzero coefficients of N(λ) and D(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

[3] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[4] P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control, vol. AC-26, pp. 111-129, 1981.

[5] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.

rmzeros(N, D; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, iz, KRInfo)

Return the finite and infinite zeros of the rational matrix R(λ) := N(λ)./D(λ) in val, the multiplicities of infinite zeros in iz and the information on the Kronecker-structure of the rational matrix R(λ) in the KRInfo object.

The information on the Kronecker-structure of the rational matrix R(λ) consists of the right Kronecker indices rki, left Kronecker indices lki, the number of finite zeros nf and the normal rank nrank and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo. Additionally, KRInfo.id contains the infinite elementary divisors of the underlying linearization as a structured system matrix pencil.

The Kronecker-structure information of the rational matrix R(λ) is obtained from the Kronecker-structure information of the underlying linearization using the results of [1] and [2]. The determination of the Kronecker-structure information is performed by building an irreducible descriptor system realization (A-λE,B,C,D) with A-λE a regular pencil of order n, satisfying R(λ) = C*inv(λE-A)*B+D and reducing the structured system matrix pencil S(λ) = [A-λE B; C D] to an appropriate Kronecker-like form (KLF) which exhibits the number of its finite eigenvalues (also the finite zeros of R(λ)), the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite zeros of R(λ)), the left and right Kronecker indices (also of R(λ)) and the normal rank (in excess with n to the normal rank of R(λ)).

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The irreducible descriptor system based linearization is built using the methods described in [3] in conjunction with pencil manipulation algorithms of [4] and [5] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reduction to the appropriate KLF of S(λ) is performed using orthogonal similarity transformations [5] and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false.

The keyword arguments atol and rtol specify the absolute and relative tolerances, respectively, for the nonzero coefficients of N(λ) and D(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

[3] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[4] P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control, vol. AC-26, pp. 111-129, 1981.

[5] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.

rmzeros1(N, D; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, iz, KRInfo)

Return the finite and infinite zeros of the rational matrix R(λ) := N(λ)./D(λ) in val, the multiplicities of infinite zeros in iz and the information on the Kronecker-structure of the rational matrix R(λ) in the KRInfo object.

The information on the Kronecker-structure of the rational matrix R(λ) consists of the right Kronecker indices rki, left Kronecker indices lki, the number of finite zeros nf and the normal rank nrank and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo. Additionally, KRInfo.id contains the infinite elementary divisors of the underlying pencil based linearization.

The Kronecker-structure information of the rational matrix R(λ) is obtained from the Kronecker-structure information of the underlying pencil based linearization using the results of [1] and [2]. The determination of the Kronecker-structure information is performed by building a strongly minimal pencil based linearization (A-λE,B-λF,C-λG,D-λH) with A-λE a regular pencil of order n, satisfying R(λ) = (C-λG)*inv(λE-A)*(B-λF)+D-λH and reducing the structured system matrix pencil S(λ) = [A-λE B-λF; C-λG D-λH] to an appropriate Kronecker-like form (KLF) which exhibits the number of its finite eigenvalues (also the finite zeros of R(λ)), the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite zeros of R(λ)), the left and right Kronecker indices (also of R(λ)) and the normal rank (in excess with n to the normal rank of R(λ)).

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The strongly minimal pencil based linearization is built using the methods described in [3] in conjunction with pencil manipulation algorithms of [4] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reduction to the appropriate KLF of S(λ) is performed using orthogonal similarity transformations [5] and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false.

The keyword arguments atol and rtol specify the absolute and relative tolerances, respectively, for the nonzero coefficients of N(λ) and D(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

[3] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[4] F.M. Dopico, M.C. Quintana and P. Van Dooren, Linear system matrices of rational transfer functions, to appear in "Realization and Model Reduction of Dynamical Systems", A Festschrift to honor the 70th birthday of Thanos Antoulas", Springer-Verlag. arXiv:1903.05016

[5] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.

rmpoles(N, D; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, ip, id)

Return the finite and infinite poles of the rational matrix R(λ) := N(λ)./D(λ) in val, the multiplicities of infinite poles in ip and the infinite elementary divisors of the pole pencil of the underlying irreducible descriptor system based linearization of R(λ) in id.

The information on the finite and infinite poles of the rational matrix R(λ) is obtained from the Kronecker-structure information of the underlying linearization using the results of [1] and [2]. The determination of the Kronecker-structure information is performed by building an irreducible descriptor system realization (A-λE,B,C,D) with A-λE a regular pencil of order n, satisfying R(λ) = C*inv(λE-A)*B+D and reducing the regular pencil A-λE to an appropriate Kronecker-like form (KLF) which exhibits the its finite eigenvalues (also the finite poles of R(λ)) and the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite poles of R(λ)). The multiplicities of the infinite zeros of A-λE and of infinite poles of R(λ) are the same [2] and are returned in ip. The number of infinite poles in val is equal to the sum of multiplicites in ip. The multiplicities of the infinite eigenvalues of A-λE are returned in id.

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The irreducible descriptor system based linearization is built using the methods described in [3] in conjunction with pencil manipulation algorithms of [4] and [5] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reduction of A-λE to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of N(λ) and D(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

[3] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[4] P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control, vol. AC-26, pp. 111-129, 1981.

[5] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.

rmpoles1(N, D;  fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, ip, id)

Return the finite and infinite poles of the rational matrix R(λ) := N(λ)./D(λ) in val, the multiplicities of infinite poles in ip and the infinite elementary divisors of the pole pencil of the underlying strongly irreducible pencil based linearization of R(λ) in id.

The information on the finite and infinite poles of the rational matrix R(λ) is obtained from the pole-structure information of the underlying pencil based linearization using the results of [1] and [2]. The determination of the pole-structure information is performed by building a strongly minimal pencil based linearization (A-λE,B-λF,C-λG,D-λH) with A-λE a regular pencil of order n, satisfying R(λ) = (C-λG)*inv(λE-A)*(B-λF)+D-λH and reducing the structured system matrix pencil Sp(λ) = [A-λE -λF; -λG -λH] to an appropriate Kronecker-like form (KLF) which exhibits the number of its finite eigenvalues (also the finite poles of R(λ)) and the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite poles of R(λ)). The multiplicities of the infinite zeros of Sp(λ) and of infinite poles of R(λ) are the same [2] and are returned in ip. The number of infinite poles in val is equal to the sum of multiplicites in ip. The multiplicities of the infinite eigenvalues of Sp(λ) are returned in id.

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The strongly minimal pencil based linearization is built using the methods described in [3] in conjunction with pencil manipulation algorithms of [4] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reduction of Sp(λ) to the KLF is performed using orthonal similarity transformations [5] and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances, respectively, for the nonzero coefficients of N(λ) and D(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

[3] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[4] F.M. Dopico, M.C. Quintana and P. Van Dooren, Linear system matrices of rational transfer functions, to appear in "Realization and Model Reduction of Dynamical Systems", A Festschrift to honor the 70th birthday of Thanos Antoulas", Springer-Verlag. arXiv:1903.05016

[5] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.

   rmrank(N, D; fastrank = true, atol = 0, rtol = atol > 0 ? 0 : n*ϵ)

Determine the normal rank of the rational matrix R(λ) := N(λ)./D(λ).

If fastrank = true, the rank is evaluated by counting how many singular values of R(γ) have magnitude greater than max(atol, rtol*σ₁), where σ₁ is the largest singular value of R(γ) and γ is a randomly generated complex value of magnitude equal to one. If fastrank = false, first a structured linearization of R(λ) is built in the form of a descriptor system matrix S(λ) = [A-λE B; C D] with A-λE an order n regular subpencil, and then its normal rank nr is determined. The normal rank of R(λ) is nr - n.

The numerator N(λ) is a polynomial matrix of the form N(λ) = N_1 + λ N_2 + ... + λ**k N_(k+1), for which the coefficient matrices N_i, i = 1, ..., k+1 are stored in the 3-dimensional matrix N, where N[:,:,i] contains the i-th coefficient matrix N_i (multiplying λ**(i-1)).

The denominator D(λ) is a polynomial matrix of the form D(λ) = D_1 + λ D_2 + ... + λ**l D_(l+1), for which the coefficient matrices D_i, i = 1, ..., l+1, are stored in the 3-dimensional matrix D, where D[:,:,i] contain the i-th coefficient matrix D_i (multiplying λ**(i-1)).

Alternatively, N(λ) and D(λ) can be specified as matrices of elements of the Polynomial type provided by the Polynomials package. In this case, no check is performed that N(λ) and D(λ) have the same indeterminates.

The irreducible descriptor system based linearization is built using the methods described in [1] in conjunction with pencil manipulation algorithms of [2] and [3] to compute reduced order linearizations. These algorithms employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition, if fast = false. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

The reduction to the appropriate KLF of S(λ) is performed using orthogonal similarity transformations [3] and involves rank decisions based on rank revealing SVD-decompositions.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of N(λ) and D(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of N(λ), and ϵ is the machine epsilon of the element type of coefficients of N(λ).

[1] A. Varga, On computing the Kronecker structure of polynomial and rational matrices using Julia, 2020, arXiv:2006.06825.

[2] P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control, vol. AC-26, pp. 111-129, 1981.

[3] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer Verlag, 2017.