The below model fleshes out a pair of comments made to Henry here as well as one made to Bill here. If it looks fuzzy, zoom in with your browser (CTRL+) and then refresh the page (it worked for me). It's essentially an update of SIM3.
To do: I'm positive this will match G&L's SIM at Ts = 1 with dg/dt = a constant = G per period. And I'm sure that it will be time invariant if we calculate A and B as shown, while we continue to satisfy ΔH = G - T over any pair of sample times. (I know, because I can force T = G - ΔH). What I don't know is if the other accounting formulas for Y, YD and C will continue to hold at different sample periods (given the way they are all expressed now as a linear combination of H and G with coefficients which are only functions of theta, alpha1 and alpha2) [Update: see new post here]. See here for a listing of what these coefficients are at Ts=1. I suspect I may need to change theta, alpha1 alpha2, or the whole expressions to match different sample periods. This would be akin to interpreting their current values as the result of continuous compounding over Ts=1.
Note that incorporating steps in g(t) (equivalent to scaled and shifted Dirac deltas in g'(t)) in the discrete time version is a bit of a pain in the ass because you have to keep track of when those steps are in time, and constantly check to see if the last sample period (i*Ts to (i+1)*Ts) included any... and then you have to adjust the state transition function (exp(a*((i+1)*Ts-tm)) for each one in the interval (assuming they can happen anywhere (t=tm) withing the sample period). The corresponding element of row vector B for the mth step in g (called gm) would be b*exp(a*((i+1)*Ts-tm)). So if there were a finite number (M) of such steps, you could augment B and Γ each with M more entries. The M extra entries in Γ would always be 0 except when a step occurred over the sample period, in which case the entries corresponding to the steps happening in that sample period would take a value of gm, as per the expressions above (m being the index to gm and tm, ranging from 0 to M-1). So it's not impossible, but messy. At least it's a linear system and the effects from all that stuff just adds together. Allowing a single step at t=0 might be a good compromise (for simulations restricted to t >= 0): it'd be easy to accommodate, and it would allow for easily maintaining sample period invariance. (You could fairly easily allow them at each sample time, but then it wouldn't be so fun if you changed to a new arbitrary sample period).
Also note that I've used a confusing array of notation for variables related to G&L's G (and T) over the past few posts. I think the above document is my most consistent take on it. But just to be explicit:
G[i] = integral of g'(t) over one sample period = g((i+1)*Ts) - g(i*Ts)
T will have a similar definition, and unfortunately lower case t is going to be a mess. So instead I'll define (ΔH)[i] = H[i] - H[i-1], and from that set T[i] = G[i] - (ΔH)[i]. I didn't do any of that above, but based on SIM3, I know that doing so will satisfy ΔH = G - T (obviously) for all sample periods, but also match G&L's T at Ts=1. I have everything I need above and in the expressions for Y, T, YD and C here to find an expression for T and see if theta, alpha1 and alpha2 need adjustment when Ts changes. I'll leave that as an exercise for another day (SIM5?). Ultimately I want to be able to say what G&L should do to preserve sample period invariance while still satisfying all their equations. If that means the parameters have to be adjusted somehow, then so be it.
If you spot an error, do me a favor and let me know in the comments. Or better yet, download the document (in the bar at the bottom), correct it and email it back to me at brown.tom5@gmail.com. Or if you see a way to improve my notation, let me know. I struggled with allowing step functions in g(t) while also doing a Taylor expansion of g'(t)... thus I introduced big and little gamma. Perhaps there's a much cleaner way to state the whole thing, while preserving G as G&L define it.
UPDATE 1:
Jason Smith put a post up with an electrical circuit analogy. I solve it with particular values for R and L. Also Brian Romanchuk has another post specifically on G&L's SIM model.
END UPDATE 1.
Note to self: The embedded document above has height set to 1370 and width at 1060 (in case accidentally erase it again!)
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