Including observed ( exogeneous predictor) matrix in the state equation of a state space model

I'm trying to estimate a state space model of the form

x(t) = Ax(t-1) + Gz(t) + Bu(t)

y(t) = Cx(t) + De(t)

where z(t) is a vector of exogenous (predictor and observed one) variables.

I'm using ssm function in Matlab which allow to input only matrix A, B, C, D. Can someone guide me how to include the matrix G into the system? Or I can work in this following way.

[x(t);z(t)] = [A G;0 1] *[x(t-1);z(t-1)] + [B;0]*u(t)

z(t) is other observed variable that I want to include in the state equation to control the result.

How to edit the state vector [x(t)] to include and observed variable or how to include the matrix of exogenous variable?

Thank you.