MMP.9:4.1 - Express the retained result and its change

Name the source state z, its initial possibilities and the admitted inputs u. Specify the answer and horizon: a quantity at time T, a threshold crossing, a response to an input, or a distributional feature. Construct the retained description x=r(z) from those needs. A.3.3.PI:4.1-4.2 tests whether merged source states can still answer that question; use an already sufficient result from that test.

For a discrete source update z_next=F(z,u), substitute it into the retained description:

x_next = r(F(z,u)).

For a differentiable r and a differential law z_dot=F(z,u), the chain rule gives:

x_dot = Dr(z) F(z,u).

Dr is the derivative of r. If r also depends explicitly on time, include its time derivative. A discontinuous readout or event needs its own change relation rather than this differentiable formula. A.3.3.TR supplies composition of the relevant changes.

Rewrite the result using x, the admitted inputs and whatever other information the smaller model proposes to carry. The part still depending on discarded quantities is the unclosed contribution. One convenient decomposition is x_dot=f0(x,u)+c(z,u), where f0 is the part you will compute directly and c is the remaining contribution. Another decomposition can be useful; what matters is the complete retained law and what information obtains each term.

If every term is obtainable from the retained information, construct the closed update through A.3.3.PI and compare it through C.29.1. Continue here when the replacement itself still needs construction.