Fix input-derivative column pairing in causal simplification call sites

MTK's linearize with allow_input_derivatives = true returns B with 2nu
columns where column nu + i is the derivative of input i. The pairing passed
to causal_simplification was derived by broadcasting 1:nu against the set of
NONZERO derivative columns, which silently pairs every input to the same
derivative channel when exactly one column is nonzero (and errors for other
partial subsets). This produced wrong linearizations whenever inputs enter
retained algebraic equations, e.g. through the numerical (non-inlined)
linear-SCC reassembly: on a planar two-joint robot the input complementary
sensitivity collapsed to a rank-1 projector instead of I at low frequency.
The pairing is now positional over all inputs; all-zero derivative columns
contribute nothing.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01W4JdEEqr6ifdSbp6MvbrU8

Author: baggepinnen

src/ode_system.jl

@@ -114,8 +114,12 @@ function RobustAndOptimalControl.named_ss(
     unames = symstr.(inputs)
     if nu > 0 && size(matrices.B, 2) == 2nu
         # This indicates that input derivatives are present
-        duinds = findall(any(!iszero, eachcol(matrices.B[:, nu+1:end]))) .+ nu
-        u2du = (1:nu) .=> duinds # This maps inputs to their derivatives
+        # Derivative column nu + i is the derivative of input i (see the MTK docs for
+        # `linearize` with `allow_input_derivatives = true`), so the pairing is positional.
+        # All-zero derivative columns contribute nothing and are included for simplicity;
+        # a pairing derived from the set of nonzero columns mispairs inputs when only a
+        # subset of the derivative columns is nonzero.
+        u2du = [i => i + nu for i in 1:nu] # This maps inputs to their derivatives
         lsys = causal_simplification(matrices, u2du; descriptor, simple_infeigs, big, balance)
     else
         lsys = ss(matrices...)
@@ -154,8 +158,12 @@ function RobustAndOptimalControl.named_ss(
     unames = symstr.(inputs)
     if nu > 0 && size(matrices.B, 2) == 2nu
         # This indicates that input derivatives are present
-        duinds = findall(any(!iszero, eachcol(matrices.B[:, nu+1:end]))) .+ nu
-        u2du = (1:nu) .=> duinds # This maps inputs to their derivatives
+        # Derivative column nu + i is the derivative of input i (see the MTK docs for
+        # `linearize` with `allow_input_derivatives = true`), so the pairing is positional.
+        # All-zero derivative columns contribute nothing and are included for simplicity;
+        # a pairing derived from the set of nonzero columns mispairs inputs when only a
+        # subset of the derivative columns is nonzero.
+        u2du = [i => i + nu for i in 1:nu] # This maps inputs to their derivatives
         lsys = causal_simplification(matrices, u2du; descriptor, simple_infeigs, big, balance, verbose=false)
     else
         lsys = ss(matrices...)
@@ -307,8 +315,12 @@ function named_sensitivity_function(
     fm(x) = convert(Matrix{Float64}, x)
     if nu > 0 && size(matrices.B, 2) == 2nu
         # This indicates that input derivatives are present
-        duinds = findall(any(!iszero, eachcol(matrices.B[:, nu+1:end]))) .+ nu
-        u2du = (1:nu) .=> duinds # This maps inputs to their derivatives
+        # Derivative column nu + i is the derivative of input i (see the MTK docs for
+        # `linearize` with `allow_input_derivatives = true`), so the pairing is positional.
+        # All-zero derivative columns contribute nothing and are included for simplicity;
+        # a pairing derived from the set of nonzero columns mispairs inputs when only a
+        # subset of the derivative columns is nonzero.
+        u2du = [i => i + nu for i in 1:nu] # This maps inputs to their derivatives
         lsys = causal_simplification(matrices, u2du; descriptor, simple_infeigs, big, balance)
     else
         lsys = ss(matrices...)