Add slow_fast decomposition and modal_reduction

[baggepinnen]

This PR adds a decomposition of a system into slow and fast parts, and a model-reduction method based on it.

slow_fast(sys, ωc)

Decomposes sys = slow + fast, where slow contains the poles with natural frequency less than ωc and fast contains the remaining poles and the direct feedthrough term D. The natural frequency of a pole p is |p| in continuous time and |log(p)|/Ts in discrete time, consistent with damp. The implementation uses an ordered Schur decomposition followed by a Sylvester equation, as in MatrixPencils.ssblkdiag. ssblkdiag cannot be reused because it only supports a half-plane or a disc centred at the origin as the selection region.

stab_unstab(sys; smarg) splits continuous-time systems by the real part of the poles, so a lightly damped high-frequency mode is assigned to the slow part. For discrete-time systems, the disc centred at the origin separates the poles by decay rate rather than by frequency. slow_fast assigns such modes by natural frequency in both cases.

modal_reduction(sys, ωc; residual = true, coordinates = :modal, states = nothing)

Removes the modes with natural frequency greater than or equal to ωc by residualization (default, preserves the static gain) or truncation. The poles of the reduced model are the slow poles of sys. The returned matrix T maps the original state to the reduced state.

• coordinates = :modal: the reduced state is in the slow modal coordinates. With residual = true, the reduced model equals slow + ss(dcgain(fast)).
• coordinates = :original: the reduced state is a subset x[states] of the original state variables. The slow invariant subspace is represented as x2 = X*x1, where X solves the nonsymmetric algebraic Riccati equation A22*X - X*A11 - X*A12*X + A21 = 0, and the deviation x2 - X*x1 is residualized or truncated. This is the singular perturbation approximation with the exact slow manifold. X is computed from the ordered Schur basis of A. This is the same computation MatrixEquations.nare performs for this equation (added in MatrixEquations.jl v2.7.0), so the MatrixEquations compat bound does not change. If states is not provided, the retained state variables are selected by a QR decomposition with column pivoting of the slow invariant subspace basis. A user-provided states for which the subspace cannot be represented throws an error.

The two choices of coordinates yield reduced models with the same poles and static gain but in general different transfer functions. The method does not account for controllability and observability. The docstring therefore refers to baltrunc for reduction based on input-output importance.

Tests

The tests in test_matrix_comps.jl cover continuous-time, discrete-time, complex-coefficient and BigFloat systems, both coordinate choices, the cases where all poles are slow or all are fast, and invalid arguments. BigFloat systems require GenericSchur for the Schur decomposition, as elsewhere in ControlSystemsBase. The conditioning check for coordinates = :original uses the 1-norm condition number, since the 2-norm condition number requires an SVD that is unavailable for BigFloat without GenericLinearAlgebra. The full ControlSystemsBase test suite passes locally (including Aqua). ExplicitImports.check_no_stale_explicit_imports reports no new stale imports, and ordschur! was added to the explicit LinearAlgebra imports.

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Host: demeter2, session ID: 527acd1d-a7ce-41f9-9f9e-d6aff56e7521, working directory: ~/.julia/dev/ControlSystems/.claude/worktrees/expressive-yawning-squid

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