The time-domain transformations are done on the discrete-time state-space matrices Ad, Bd, Cd, Dd, and T is the sampling period of the discrete-time system. The matrices Ac, Bc, Cc, Dc represent the continuous-time matrices. I is the identity matrix.
Ac = (Ad - I)/T
Bc = Bd/T
Cc = Cd
Dc = Dd
Ac = (I - Ad^(-1))/T
Bc = Ad(-1).Bd/T
Cc = Cd.Ad(-1)
Dc = Dd - Cd.Bc*T
M = (Ad + I)*T/2
Ac = M^(-1).(Ad - I)
Bc = Bd/sqrt(T) - sqrt(T)/2*Ac.Bd
Cc = Cd/sqrt(T) + sqrt(T)/2*Cd.Ac
Dc = Dd - sqrt(T)/2*Cd.Bc
Ac = ln(Ad)/T
Bc = (exp(Ac*T) - I)^(-1).Ac.Bd
Cc = Cd
Dc = Dd
Ac = ln(Ad)/T
Bc = (exp(Ac*T) - I)^(-2).Ac^2.Bd*T
Cc = Cd
Dc = Dd - Cc.(Ac^(-1).(exp(Ac*T)/T - I) - I)Ac^(-1).Bc