Shift unscalarized array variables with their scalarized elements

[baggepinnen]

Fixes SciML/ModelingToolkit.jl#5169.

Problem

shift_discrete_system shifts every discrete variable of a partition forward by one tick
before the partition is compiled. The variables to shift are those that search_variables!
returns for the equations and that are present in fullvars:

for k in discvars
    k in fullvars_set || continue
    MTKBase.isoperator(k, Union{Sample, Hold, Pre}) && continue
    discmap[k] = MTKBase.simplify_shifts(Shift(iv, 1)(k))
end

fullvars only ever contains the scalarized elements of an array variable. The
TearingState constructor searches with SU.Operator atomic, so an occurrence
Shift(t, -1)(x) of an array x is returned whole and scalarized into
Shift(t, -1)(x[i]) entries in fullvars, while shift_discrete_system searches with
only Sample, Hold and Pre atomic and gets back the bare array x, which is not in
fullvars_set.

An array variable used without scalarization, as in f(x(k - 1)) where f is a registered
function of a whole vector and the elements of x are aliased away, is therefore never
entered into the substitution map: its elements are shifted forward while the unscalarized
use is left in place, and the two forms end up one tick apart. For the model in the issue,
the partition equation

0 ~ -Shift(t, 1)(ud(t)) + combine(Shift(t, -1)(xd(t)))*kp

keeps Shift(t, -1)(xd(t)) where combine(xd(t)) was meant, so the compiled system has
unknowns (xdₜ₋₁(t))[1:2] but an observed equation ud(t) ~ combine(xdₜ₋₂(t))*kp.
xdₜ₋₂ is neither an unknown nor a parameter of the system and code generation fails with
UndefVarError: xdₜ₋₂ not defined in ModelingToolkitBase.

This is not specific to the hybrid clock-partition path the issue takes: mtkcompile of a
purely discrete system goes through shift_discrete_system as well and produces the same
result.

Fix

Add the arrays that the scalarized elements of fullvars belong to (with any Shifts
stripped) to fullvars_set, so the unscalarized form is shifted along with the elements.
Entries that are a Sample, Hold or Pre of an array are discarded by the existing
isoperator check.

Tests

Two regression tests, both failing on main and passing here:

• mtkcompile of a purely discrete system whose array variable is used one tick back as a
  whole: the observed equation must refer to the same shifted variable the unknowns are
  lowered to, and the resulting DiscreteProblem must agree with the one built from the
  element-by-element formulation. On main the first assertion fails and the problem
  construction throws UndefVarError: xdₜ₋₂.
• The clock-partition path from the issue: after split_system, the unscalarized xd must
  appear at the same tick as its elements.

The existing ModelingToolkitTearing suite and ModelingToolkit's test/clock.jl pass
unchanged.

🤖 Generated with Claude Code

https://claude.ai/code/session_016KRw9L2c45f9MuTtWcfj8S

[baggepinnen]

Verified against the original repro of SciML/ModelingToolkit.jl#5169. With this branch developed into the environment, the compiled partition's observed equation refers to the same shifted variable as its unknowns and the DiscreteProblem builds:

unknowns   = [(xdₜ₋₁(t))[2], (xdₜ₋₁(t))[1]]
observed:
    ud(t) ~ combine(xdₜ₋₁(t))*kp
    ...
DiscreteProblem: ok

SciML/ModelingToolkit.jl#5168 now carries a regression test for the same pattern through linearize_clocked, asserting that the whole-array and the element-by-element spellings of one control law give identical matrices, and raises its ModelingToolkitTearing bound to 1.20.7.