Finite block-Hankel MIMO model reduction
========================================

.. admonition:: Tutorial goal

   Generalize the SISO finite-Hankel reducer to MIMO Markov matrices and return a reduced state-space model.

.. note::

   New to the terminology? See the :doc:`lattice DSP concept map <../../algorithms/concept_map>` and the :doc:`causality/data-use guide <../../theory/causality_and_data_use>` for how online, offline, block, and MIMO examples should be read.

Context
-------

The SISO reducer works with one impulse response.  A MIMO system has a sequence of Markov
matrices instead, where each matrix maps input channels to output channels at a particular
lag.  The natural finite-Hankel baseline is therefore a block-Hankel matrix.

This tutorial is deliberately a baseline: it gives a reference MIMO Ho--Kalman/ERA-style
reduction as a finite block-Hankel baseline separate from matrix Nehari or AAK optimality claims.

Key idea and equations
----------------------

For Markov matrices ``M_k`` with shape ``outputs x inputs``, the finite block-Hankel matrix is

.. math::

   \mathcal{H}_0 =
   \begin{bmatrix}
     M_1 & M_2 & M_3 & \cdots \\
     M_2 & M_3 & M_4 & \cdots \\
     M_3 & M_4 & M_5 & \cdots \\
     \vdots & \vdots & \vdots & \ddots
   \end{bmatrix}.

The reduced model is returned as state-space matrices ``A, B, C, D`` rather than as scalar
numerator/denominator coefficients.

How to read the result
----------------------

Look for block-Hankel singular-value decay, retained energy, Markov-response error, and stable reduced state matrices.

Run command
-----------

.. code-block:: bash

   python examples/mimo_finite_hankel_model_reduction.py

Source code
-----------

.. literalinclude:: ../../../examples/mimo_finite_hankel_model_reduction.py
   :language: python
   :linenos: