System transformation function H(z) for a discrete-time LTI system expressed in state variable form with zero initial conditions is

Explanation:

Transfer function which is the ratio of Laplace output to the Laplace input when the initial conditions are zero in discrete is the same as continuous but in the z-domain.

Transfer function from state model is given as:

\(\dot{X}=AX+BU\) -----(1)

Y = CX + DU -----(2)

Where,

Y is single output 

X is single input

X(0) = 0, as initial condition is zero.

Taking Z transform of equation (1):

zX(z) = AX(z) + BU(z)

[zI - A] X(z) = BU(z)

Taking Z transform of equation (2):

Y(z) = CX(z) + DU(z)

putting value of X(z) from equation (1) in equation (2) we get:

\(T(z)=\frac{Y(z)}{U(z)}\)

T(z) = c[zI - A]^-1b + d

So, option (1) is correct answer.