Question on @mtkbuild model structure❓
Consider a DAE model described by @mtkmodel in the form:
$$
\frac{dx}{dt} = f(x,z,u)
$$
$$
0 = g(x,z,u)
$$
Here, $x$ is the "differential variable", $z$ the "algebraic variable", and $u$ an input which is set equal to a register_symbolic function. Assume that the user is inexperienced, and may have created some differential equations by differentiating simple algebraic equations, e.g., differentiating $x_1 = x_2 + 1$.
Next, macro @mtkbuild is applied, leading to a transformed model, say, in the form:
$$
\frac{d \xi}{dt} = \phi (\xi,\zeta, u)
$$
$$
0 = \gamma(\xi, \zeta, u)
$$
Question 1: Is this transformed model in Index 1 DAE form? In other words, is $\partial \gamma/ \partial \zeta$ non-singular?
Question 2: Will in, in general, $\dim \xi \le \dim x$, or is that uncertain? [For sure, if "fake" differential equations from differentiating $x_1 = x_2 + 1$ are removed and made into algebraic equations, $\dim \xi$ should be smaller than $\dim x$. But perhaps new differential equations are introduced in the index reduction process?]
In addition to the above model, some observed relations are included. These are of form:
$$
y = h(\xi, \zeta, u)
$$
Question 3: Will the set of variables ${\mathcal S}(x,z,u) \subseteq {\mathcal S}(\xi, \zeta, y, u)$?
