FirstOrder(lowpass=false): documented transfer function differs from implementation for k != 1

Describe the bug 🐞

The documented transfer function for FirstOrder(lowpass=false) differs from the implementation when k ≠
1.

The docstring gives (sT + 1 - k)/(sT + 1), but the component equations imply k*sT/(sT + 1).

These agree at the default k=1. For k=2, their DC gains are −1 and 0, respectively.

Expected behavior

The documentation and implementation should describe the same transfer function. I’m unsure which
behavior is intended.

Minimal Reproducible Example 👇

using ModelingToolkit, Symbolics
  using ModelingToolkit: t_nounits as t, D_nounits as D
  using ModelingToolkitStandardLibrary.Blocks: FirstOrder, Constant

  @named input = Constant()
  @named filter = FirstOrder(lowpass=false)
  @named model = System(
      [connect(input.output, filter.input)], t;
      systems=[input, filter],
  )

  eqs = equations(expand_connections(model))
  rate = only(eq.rhs for eq in eqs if isequal(eq.lhs, D(filter.x)))
  output = only(
      eq.rhs for eq in eqs
      if isequal(eq.lhs, filter.y) &&
         any(v -> isequal(v, filter.x), Symbolics.get_variables(eq.rhs))
  )

  values = Dict(filter.u => 1, filter.x => 2, filter.k => 2, filter.T => 1)
  evaluate(expr) = Symbolics.unwrap_const(Symbolics.unwrap(
      Symbolics.simplify(Symbolics.substitute(expr, values))
  ))

  @show evaluate(rate)    # 0: the state is stationary
  @show evaluate(output)  # 0: actual output
  @show (1 - 2) * 1       # -1: documented DC gain times input

Error & Stacktrace ⚠️

No exception occurs. The discrepancy is in the output:

evaluate(rate) = 0
evaluate(output) = 0
(1 - 2) * 1 = -1

Environment (please complete the following information):

Julia 1.11.9. Relevant entries from using Pkg; Pkg.status():

⌃ [961ee093] ModelingToolkit v11.20.0
⌅ [16a59e39] ModelingToolkitStandardLibrary v2.29.4
⌃ [0c5d862f] Symbolics v7.18.1

Additional context

Found using our formal property prover system.

Start with the FirstOrder docstring at src/Blocks/continuous.jl#L116-L122 and its component equations at #L157-L160. Run the provided Julia reproducer to confirm the differing transfer functions, then determine which behavior is intended and make the documentation and implementation agree; verify that the k=1 and k≠1 cases match.