Basic operations on system models

sysinv = inv(sys; atol = 0, atol1 = atol, atol2 = atol, rtol, checkinv = true)

Compute for a descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ), a descriptor realization of its inverse system sysinv = (Ai-λEi,Bi,Ci,Di), such that the transfer function matrix Ginv(λ) of sysinv is the inverse of G(λ) (i.e., G(λ)*Ginv(λ) = I). The realization of sysinv is determined using inversion-free formulas and the invertibility condition is checked, unless checkinv = false.

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

sysldiv = ldiv(sys1, sys2; atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ, checkinv = true)
sysldiv = sys1 \ sys2

Compute for the descriptor systems sys1 = (A1-λE1,B1,C1,D1) with the transfer function matrix G1(λ) and sys2 = (A2-λE2,B2,C2,D2) with the transfer function matrix G2(λ), a descriptor realization sysldiv = (Ai-λEi,Bi,Ci,Di) of sysldiv = inv(sys1)*sys2, whose transfer-function matrix Gli(λ) represents the result of the left division Gli(λ) = inv(G1(λ))*G2(λ). The realization of sysldiv is determined using inversion-free formulas and the invertibility condition for sys1 is checked, unless checkinv = false.

The keyword arguments atol1, atol2 and rtol specify, respectively, the absolute tolerance for the nonzero elements of A1, B1, C1, D1, A2, B2, C2, D2, the absolute tolerance for the nonzero elements of E1 and E2, and the relative tolerance for the nonzero elements of A1, B1, C1, D1, A2, B2, C2, D2, E1 and E2. The default relative tolerance is n*ϵ, where n is the maximum of orders of the square matrices A1 and A2, and ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol and atol2 = atol.

sysrdiv = rdiv(sys1, sys2; atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ, checkinv = true)  
sysrdiv = sys1 / sys2

Compute for the descriptor systems sys1 = (A1-λE1,B1,C1,D1) with the transfer function matrix G1(λ) and sys2 = (A2-λE2,B2,C2,D2) with the transfer function matrix G2(λ), a descriptor realization sysrdiv = (Ai-λEi,Bi,Ci,Di) of sysrdiv = sys1*inv(sys2), whose transfer-function matrix Gri(λ) represents the result of the right division Gri(λ) = G1(λ)*inv(G2(λ)). The realization of sysrdiv is determined using inversion-free formulas and the invertibility condition for sys2 is checked, unless checkinv = false.

The keyword arguments atol1, atol2 and rtol specify, respectively, the absolute tolerance for the nonzero elements of A1, B1, C1, D1, A2, B2, C2, D2, the absolute tolerance for the nonzero elements of E1 and E2, and the relative tolerance for the nonzero elements of A1, B1, C1, D1, A2, B2, C2, D2, E1 and E2. The default relative tolerance is n*ϵ, where n is the maximum of orders of the square matrices A1 and A2, and ϵ is the working machine epsilon. The keyword argument atol can be used to simultaneously set atol1 = atol and atol2 = atol.

sysdual = gdual(sys, rev = false) 
sysdual = transpose(sys, rev = false)

Compute for a descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ), the descriptor system realization of its dual system sysdual = (Ad-λEd,Bd,Cd,Dd), where Ad = transpose(A), Ed = transpose(E), Bd = transpose(C), Cd = transpose(B) and Dd = transpose(D), such that the transfer function matrix Gdual(λ) of sysdual is the transpose of G(λ) (i.e., Gdual(λ) = transpose(G(λ))).

If rev = true, the tranposition is combined with the reverse permutation of the state variables, such that sysdual = (P*Ad*P-λP*Ed*P,P*Bd,Cd*P,Dd), where P is the permutation matrix with ones down the second diagonal.

sysconj = ctranspose(sys) 
sysconj = sys'

Compute the conjugate transpose (or adjoint) of a descriptor system (see adjoint).

 rt = adjoint(r)

Compute the adjoint rt(λ) of the rational transfer function r(λ) such that for r(λ) = num(λ)/den(λ) we have:

(1) `rt(λ) = conj(num(-λ))/conj(num(-λ))`, if `r.Ts = 0`; 

(2) `rt(λ) = conj(num(1/λ))/conj(num(1/λ))`, if `r.Ts = -1` or `r.Ts > 0`.

sysconj = adjoint(sys) 
sysconj = sys'

Compute for a descriptor system sys = (A-λE,B,C,D) with the transfer function matrix G(λ), the descriptor system realization of its adjoint (also called conjugate transpose) system sysconj = (Ac-λEc,Bc,Cc,Dc), such that the transfer function matrix Gconj(λ) of sysconj is the appropriate conjugate transpose of G(λ), as follows: for a continuous-time system with λ = s, Gconj(s) := transpose(G(-s)), while for a discrete-time system with λ = z, Gconj(z) := transpose(G(1/z)).