Factorization of descriptor system transfer function matrices

grcf(sys; smarg, sdeg, evals, mindeg = false, mininf = false, fast = true, 
     atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol = n*ϵ) -> (sysn, sysm)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysn = (An-λEn,Bn,Cn,Dn) and sysm = (Am-λEm,Bm,Cm,Dm) of its stable and proper right coprime factorization. If sys, sysn and sysm have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = N(λ)*inv(M(λ)), with N(λ) and M(λ) proper and stable transfer function matrices. The resulting matrix pairs (An,En) and (Am,Em) are in (generalized) Schur form. The stability domain Cs of poles is defined by the keyword argument smarg for the stability margin, as follows: for a continuous-time system sys, Cs is the set of complex numbers with real parts at most smarg < 0, while for a discrete-time system sys, Cs is the set of complex numbers with moduli at most smarg < 1 (i.e., the interior of a disc of radius smarg centered in the origin). If smarg is missing, then the employed default values are smarg = -sqrt(eps) for a continuous-time system and smarg = 1-sqrt(eps) for a discrete-time system.

The keyword argument sdeg specifies the prescribed stability degree for the assigned eigenvalues of the factors. If both sdeg and smarg are missing, then the employed default values are sdeg = -0.05 for a continuous-time system and sdeg = 0.95 for a discrete-time system, while if smarg is specified, then sdeg = smarg is used.

The keyword argument evals is a real or complex vector, which contains a set of finite desired eigenvalues for the factors. For a system with real data, evals must be a self-conjugated complex set to ensure that the resulting factors are also real.

If mindeg = false, both factors sysn and sysm have descriptor realizations with the same order and with An = Am, En = Em and Bn = Bm. If mindeg = true, the realization of sysm is minimal. The number of (finite) poles of sysm is equal to the number of unstable finite poles of sys.

If mininf = false, then An-λEn and Am-λEm may have simple infinite eigenvalues. If mininf = true, then An-λEn and Am-λEm have no simple infinite eigenvalues. Note that the removing of simple infinite eigenvalues involves matrix inversions.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of B, and the relative tolerance for the nonzero elements of A, E and B. The default relative tolerance is n*ϵ, where ϵ is the machine epsilon of the element type of A and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

The preliminary separation of finite and infinite eigenvalues of A-λE is performed using rank decisions based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: The Procedure GRCF from [2] is implemented, which represents an extension of the recursive factorization approach of [1] to cope with infinite eigenvalues. All infinite poles are assigned to finite real values. If evals is missing or does not contain a sufficient number of real values, then a part or all of infinite eigenvalues of A-λE are assigned to the value specified by sdeg. The pairs (An,En) and (Am,Em) result in generalized Schur form with both An and Am quasi-upper triangular and En and Em either both upper triangular or both UniformScalings.

References:

[1] A. Varga. Computation of coprime factorizations of rational matrices. Linear Algebra and Its Applications, vol. 271, pp.88-115, 1998.

[2] A. Varga. On recursive computation of coprime factorizations of rational matrices. arXiv:1703.07307, https://arxiv.org/abs/1703.07307, 2020. (to appear in Linear Algebra and Its Applications)

glcf(sys; smarg, sdeg, evals, mindeg = false, mininf = false, fast = true, 
     atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol = n*ϵ) -> (sysn, sysm)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysn = (An-λEn,Bn,Cn,Dn) and sysm = (Am-λEm,Bm,Cm,Dm) of its stable and proper left coprime factorization. If sys, sysn and sysm have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = inv(M(λ))*N(λ), with N(λ) and M(λ) proper and stable transfer function matrices. The resulting matrix pairs (An,En) and (Am,Em) are in (generalized) Schur form. The stability domain Cs of poles is defined by the keyword argument smarg for the stability margin, as follows: for a continuous-time system sys, Cs is the set of complex numbers with real parts at most smarg, while for a discrete-time system sys, Cs is the set of complex numbers with moduli at most smarg < 1 (i.e., the interior of a disc of radius smarg centered in the origin). If smarg is missing, then the employed default values are smarg = -sqrt(eps) for a continuous-time system and smarg = 1-sqrt(eps) for a discrete-time system.

The keyword argument sdeg specifies the prescribed stability degree for the assigned eigenvalues of the factors. If both sdeg and smarg are missing, then the employed default values are sdeg = -0.05 for a continuous-time system and sdeg = 0.95 for a discrete-time system, while if smarg is specified, then sdeg = smarg is used.

The keyword argument evals is a real or complex vector, which contains a set of finite desired eigenvalues for the factors. For a system with real data, evals must be a self-conjugated complex set to ensure that the resulting factors are also real.

If mindeg = false, both factors sysn and sysm have descriptor realizations with the same order and with An = Am, En = Em and Cn = Cm. If mindeg = true, the realization of sysm is minimal. The number of (finite) poles of sysm is equal to the number of unstable finite poles of sys.

If mininf = false, then An-λEn and Am-λEm may have simple infinite eigenvalues. If mininf = true, then An-λEn and Am-λEm have no simple infinite eigenvalues. Note that the removing of simple infinite eigenvalues involves matrix inversions.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of C, and the relative tolerance for the nonzero elements of A, E and C. The default relative tolerance is n*ϵ, where ϵ is the machine epsilon of the element type of A and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

The preliminary separation of finite and infinite eigenvalues of A-λE is performed using rank decisions based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: The dual of Procedure GRCF from [2] is used, which represents an extension of the recursive factorization approach of [1] to cope with infinite poles. All infinite eigenvalues are assigned to finite real values. If evals is missing or does not contain a sufficient number of real values, then a part or all of infinite eigenvalues of A-λE are assigned to the value specified by sdeg. The pairs (An,En) and (Am,Em) result in generalized Schur form with both An and Am quasi-upper triangular and En and Em either both upper triangular or both UniformScalings.

References:

[1] A. Varga. Computation of coprime factorizations of rational matrices. Linear Algebra and Its Applications, vol. 271, pp.88-115, 1998.

[2] A. Varga. On recursive computation of coprime factorizations of rational matrices. arXiv:1703.07307, https://arxiv.org/abs/1703.07307, 2020. (to appear in Linear Algebra and Its Applications)

grcfid(sys; mindeg = false, mininf = false, fast = true, offset = sqrt(ϵ), 
       atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol = n*ϵ) -> (sysni, sysmi)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysni = (Ani-λEni,Bni,Cni,Dni) and sysmi = (Ami-λEmi,Bmi,Cmi,Dmi) of its right coprime factorization with inner denominator. If sys, sysni and sysmi have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = N(λ)*inv(M(λ)), with N(λ) and M(λ) proper and stable transfer function matrices and the denominator factor M(λ) inner. The resulting matrix pairs (Ani,Eni) and (Ami,Emi) are in (generalized) Schur form. The system sys must not have poles on the boundary of the stability domain Cs. In terms of eigenvalues, this requires for a continuous-time system, that A-λE must not have controllable eigenvalues on the imaginary axis (excepting simple infinite eigenvalues), while for a discrete-time system, A-λE must not have controllable eigenvalues on the unit circle centered in the origin.

To assess the presence of poles on the boundary of Cs, a boundary offset β can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, then the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

If mindeg = false, both factors sysni and sysmi have descriptor realizations with the same order and with Ani = Ami, Eni = Emi and Bni = Bmi. If mindeg = true, the realization of sysmi is minimal. The number of (finite) poles of sysmi is equal to the number of unstable finite poles of sys.

If mininf = false, then Ani-λEni and Ami-λEmi may have simple infinite eigenvalues. If mininf = true, then Ani-λEni and Ami-λEmi have no simple infinite eigenvalues. Note that the removing of simple infinite eigenvalues involves matrix inversions.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of B, and the relative tolerance for the nonzero elements of A, E and B. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

The preliminary separation of finite and infinite eigenvalues of A-λEis performed using rank decisions based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: An extension of the recursive factorization approach of [1] is used (see [2] for details). The pairs (Ani,Eni) and (Ami,Emi) result in generalized Schur form with both Ani and Ami quasi-upper triangular and Eni and Emi either both upper triangular or both UniformScalings.

References:

[1] A. Varga. Computation of coprime factorizations of rational matrices. Linear Algebra and Its Applications, vol. 271, pp.88-115, 1998.

[2] A. Varga. On recursive computation of coprime factorizations of rational matrices. arXiv:1703.07307, https://arxiv.org/abs/1703.07307, 2020. (to appear in Linear Algebra and Its Applications)

glcfid(sys; mindeg = false, mininf = false, fast = true, offset = sqrt(ϵ), 
       atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol = n*ϵ) -> (sysni, sysmi)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysni = (Ani-λEni,Bni,Cni,Dni) and sysmi = (Ami-λEmi,Bmi,Cmi,Dmi) of its left coprime factorization with inner denominator. If sys, sysni and sysmi have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = inv(M(λ))*N(λ), with N(λ) and M(λ) proper and stable transfer function matrices and the denominator factor M(λ) inner. The resulting matrix pairs (Ani,Eni) and (Ami,Emi) are in Schur forms. The system sys must not have poles on the boundary of the stability domain Cs. In terms of eigenvalues, this requires for a continuous-time system, that A-λE must not have controllable eigenvalues on the imaginary axis (excepting simple infinite eigenvalues), while for a discrete-time system, A-λE must not have controllable eigenvalues on the unit circle centered in the origin.

To assess the presence of poles on the boundary of Cs, a boundary offset β can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, then the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

If mindeg = false, both factors sysni and sysmi have descriptor realizations with the same order and with Ani = Ami, Eni = Emi and Cni = Cmi. If mindeg = true, the realization of sysmi is minimal. The number of (finite) poles of sysmi is equal to the number of unstable finite poles of sys.

If mininf = false, then Ani-λEni and Ami-λEmi may have simple infinite eigenvalues. If mininf = true, then Ani-λEni and Ami-λEmi have no simple infinite eigenvalues. Note that the removing of simple infinite eigenvalues involves matrix inversions.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of C, and the relative tolerance for the nonzero elements of A, E and C. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

The preliminary separation of finite and infinite eigenvalues of A-λEis performed using rank decisions based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: An extension of the recursive factorization approach of [1] is used to the dual system (see [2] for details). The pairs (Ani,Eni) and (Ami,Emi) result in generalized Schur form with both Ani and Ami quasi-upper triangular and Eni and Emi either both upper triangular or both UniformScalings.

References:

[1] A. Varga. Computation of coprime factorizations of rational matrices. Linear Algebra and Its Applications, vol. 271, pp.88-115, 1998.

[2] A. Varga. On recursive computation of coprime factorizations of rational matrices. arXiv:1703.07307, https://arxiv.org/abs/1703.07307, 2020. (to appear in Linear Algebra and Its Applications)

gnrcf(sys; fast = true, ss = false, 
     atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ) -> (sysn, sysm)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysn = (An-λEn,Bn,Cn,Dn) and sysm = (An-λEn,Bn,Cm,Dm) of its normalized right coprime factorization. If sys, sysn and sysm have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = N(λ)*inv(M(λ)), with N(λ) and M(λ) proper and stable transfer function matrices and [N(λ);M(λ)] inner. The resulting En = I if ss = true.

Pencil reduction algorithms are employed which perform 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 atol1, atol2 and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, B, C and D, the absolute tolerance for the nonzero elements of E, and the relative tolerance for the nonzero elements of A, E, B, C and D. The default relative tolerance is n*ϵ, where ϵ is the machine epsilon of the element type of A and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol.

Method: Pencil reduction algorithms are employed to determine the inner range space R(λ) of the transfer function matrix [G(λ); I] using the method described in [1], which is based on the reduction algorithm of [2]. Then the factors N(λ) and M(λ) result from the partitioning of R(λ) as R(λ) = [N(λ);M(λ)].

References:

[1] Varga, A. A note on computing the range of rational matrices. arXiv:1707.0048, https://arxiv.org/abs/1707.0048, 2017.

[2] C. Oara. Constructive solutions to spectral and inner–outer factorizations respect to the disk. Automatica, 41, pp. 1855–1866, 2005.

gnlcf(sys; fast = true, ss = false, 
     atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ) -> (sysn, sysm)

Compute for the descriptor system sys = (A-λE,B,C,D), the factors sysn = (An-λEn,Bn,Cn,Dn) and sysm = (An-λEn,Bm,Cn,Dm) of its normalized right coprime factorization. If sys, sysn and sysm have the transfer function matrices G(λ), N(λ) and M(λ), respectively, then G(λ) = inv(M(λ))*N(λ), with N(λ) and M(λ) proper and stable transfer function matrices and [N(λ) M(λ)] coinner. The resulting En = I if ss = true.

Pencil reduction algorithms are employed which perform 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 atol1, atol2 and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, B, C and D, the absolute tolerance for the nonzero elements of E, and the relative tolerance for the nonzero elements of A, E, B, C and D. The default relative tolerance is n*ϵ, where ϵ is the machine epsilon of the element type of A and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol.

Method: Pencil reduction algorithms are employed to determine the coinner coimage space R(λ) of the transfer function matrix [G(λ) I] using the dual of method described in [1], which is based on the reduction algorithm of [2]. Then the factors N(λ) and M(λ) result from the partitioning of R(λ) as R(λ) = [N(λ) M(λ)].

References:

[1] Varga, A. A note on computing the range of rational matrices. arXiv:1707.0048, https://arxiv.org/abs/1707.0048, 2017.

[2] C. Oara. Constructive solutions to spectral and inner–outer factorizations respect to the disk. Automatica, 41, pp. 1855–1866, 2005.

giofac(sys; atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol, 
       fast = true, minphase = true, offset = sqrt(ϵ)) -> (sysi, syso, info)

Compute for the descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ), the square inner factor sysi = (Ai-λEi,Bi,Ci,Di) with the transfer function matrix Gi(λ) and the minimum-phase quasi-outer factor or the full row rank factor syso = (Ao-λEo,Bo,Co,Do) with the transfer function matrix Go(λ) such that

 G(λ) = Gi[:,1:r](λ)*Go(λ)    (*),

where r is the normal rank of G(λ). The resulting proper and stable inner factor satisfies Gi'(λ)*Gi(λ) = I. If sys is stable (proper), then the resulting syso is stable (proper). The resulting factor Go(λ) has full row rank r. Depending on the selected factorization option, if minphase = true, then Go(λ) is minimum phase, excepting possibly zeros on the boundary of the appropriate stability domain Cs, or if minphase = false, then Go(λ) contains all zeros of G(λ), in which case (*) is the extended QR-like factorization of G(λ). For a continuous-time system sys, the stability domain Cs is defined as the set of complex numbers with real parts at most -β, while for a discrete-time system sys, Cs is the set of complex numbers with moduli at most 1-β (i.e., the interior of a disc of radius 1-β centered in the origin). The boundary offset β to be used to assess the stability of zeros and their number on the boundary of Cs can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

The resulting named triple info contains (nrank, nfuz, niuz), where info.nrank = r, the normal rank of G(λ), info.nfuz is the number of finite zeros of syso on the boundary of Cs, and info.niuz is the number of infinite zeros of syso. info.nfuz is set to missing if minphase = false.

Note: syso may generally contain a free inner factor, which can be eliminated by removing the finite unobservable eigenvalues.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A and B, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of C and D, and the relative tolerance for the nonzero elements of A, E, B, C and D. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

For the assessment of zeros, the system pencil [A-λE B; C D] is reduced to a special Kronecker-like form (see [2]). In this reduction, the performed rank decisions are based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: For a continuous-time system, the factorization algorithm of [1] is used, while for a discrete-time system, the factorization algorithm of [1] is used.

References:

[1] C. Oara and A. Varga. Computation of the general inner-outer and spectral factorizations. IEEE Trans. Autom. Control, vol. 45, pp. 2307-2325, 2000.

[2] C. Oara. Constructive solutions to spectral and inner–outer factorizations respect to the disk. Automatica, 41, pp. 1855–1866, 2005.

goifac(sys; atol = 0, atol1 = atol, atol2 = atol, atol3 = atol, rtol, 
       fast = true, minphase = true, offset = sqrt(ϵ)) -> (sysi, syso, info)

Compute for the descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ), the square inner factor sysi = (Ai-λEi,Bi,Ci,Di) with the transfer function matrix Gi(λ) and the minimum-phase quasi-outer factor or the full column rank factor syso = (Ao-λEo,Bo,Co,Do) with the transfer function matrix Go(λ) such that

 G(λ) = Go(λ)*Gi[1:r,:](λ)    (*),

where r is the normal rank of G(λ). The resulting proper and stable inner factor satisfies Gi'(λ)*Gi(λ) = I. If sys is stable (proper), then the resulting syso is stable (proper). The resulting factor Go(λ) has full column rank r. Depending on the selected factorization option, if minphase = true, then Go(λ) is minimum phase, excepting possibly zeros on the boundary of the appropriate stability domain Cs, or if minphase = false, then Go(λ) contains all zeros of G(λ), in which case (*) is the extended RQ-like factorization of G(λ). For a continuous-time system sys, the stability domain Cs is defined as the set of complex numbers with real parts at most -β, while for a discrete-time system sys, Cs is the set of complex numbers with moduli at most 1-β (i.e., the interior of a disc of radius 1-β centered in the origin). The boundary offset β to be used to assess the stability of zeros and their number on the boundary of Cs can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

The resulting named triple info contains (nrank, nfuz, niuz), where info.nrank = r, the normal rank of G(λ), info.nfuz is the number of finite zeros of syso on the boundary of Cs, and info.niuz is the number of infinite zeros of syso. info.nfuz is set to missing if minphase = false.

Note: syso may generally contain a free inner factor, which can be eliminated by removing the finite unobservable eigenvalues.

The keyword arguments atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A and C, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of B and D, and the relative tolerance for the nonzero elements of A, E, B, C and D. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon and n is the order of the system sys. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

For the assessment of zeros, the dual system pencil transpose([A-λE B; C D]) is reduced to a special Kronecker-like form (see [2]). In this reduction, the performed rank decisions are based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: For a continuous-time system, the dual system is formed and the factorization algorithm of [1] is used, while for a discrete-time system, the factorization algorithm of [1] is used.

References:

[1] C. Oara and A. Varga. Computation of the general inner-outer and spectral factorizations. IEEE Trans. Autom. Control, vol. 45, pp. 2307–2325, 2000.

[2] C. Oara. Constructive solutions to spectral and inner–outer factorizations respect to the disk. Automatica, 41, pp. 1855–1866, 2005.

 sysf = grsfg(sys, γ; fast = true, stabilize = true, offset = β, 
              atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ)

Compute for the descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ) and ${\small γ > \|G(λ)\|_∞}$, the minimum-phase right spectral factor sysf = (Af-λEf,Bf,Cf,Df) with the transfer-function matrix F(λ), such that F(λ)'*F(λ) = γ^2*I-G(λ)'*G(λ). If stabilize = true (the default), a preliminary stabilization of sys is performed. In this case, sys must not have poles on the imaginary-axis in the continuous-time case or on the unit circle in the discrete-time case. If stabilize = false, then no preliminary stabilization is performed. In this case, sys must be stable.

To assess the presence of poles on the boundary of the stability domain Cs, a boundary offset β can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, then the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

If stabilize = true, a preliminary separation of finite and infinite eigenvalues of A-λEis performed using 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 atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of C, and the relative tolerance for the nonzero elements of A, E and C. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

Method: Extensions of the factorization approaches of [1] are used.

References:

[1] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, 1996.

 sysf = glsfg(sys, γ; fast = true, stabilize = true, offset = β, 
              atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ)

Compute for the descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ) and ${\small γ > \|G(λ)\|_∞}$, the minimum-phase right spectral factor sysf = (Af-λEf,Bf,Cf,Df) with the transfer-function matrix F(λ), such that F(λ)*F(λ)' = γ^2*I-G(λ)*G(λ)'. If stabilize = true (the default), a preliminary stabilization of sys is performed. In this case, sys must not have poles on the imaginary-axis in the continuous-time case or on the unit circle in the discrete-time case. If stabilize = false, then no preliminary stabilization is performed. In this case, sys must be stable.

To assess the presence of poles on the boundary of the stability domain Cs, a boundary offset β can be specified via the keyword parameter offset = β. Accordingly, for a continuous-time system, the boundary of Cs contains the complex numbers with real parts within the interval [-β,β], while for a discrete-time system, then the boundary of Cs contains the complex numbers with moduli within the interval [1-β,1+β]. The default value used for β is sqrt(ϵ), where ϵ is the working machine precision.

If stabilize = true, a preliminary separation of finite and infinite eigenvalues of A-λEis performed using 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 atol1, atol2, atol3, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, the absolute tolerance for the nonzero elements of E, the absolute tolerance for the nonzero elements of C, and the relative tolerance for the nonzero elements of A, E and C. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol, atol3 = atol.

Method: Extensions of the factorization approaches of [1] are used.

References:

[1] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, 1996.