What is the relation between states in the original and reduced model

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Hello, I'm using balred to simplify a state space model. This works fine, but I want to use this to simulate and compare it with the original model results, and this from specific inital conditions. The problem is that I don't know how to convert the initial state in the original model to the states in the reduced model. I would think that there should be a conversion matrix (or maybe 2?).
So:
dX/dt = A.X + B.U
--> reduced: dZ/dt = Ar.Z + Br.U
The question: is there something like a matrix Q?
Z0 = Q.X0 (or another relationship)

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