We present two MATLAB toolboxes, provided as open-source code, that expand the capabilities of the Control System Toolbox to large-scale models.

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We develop here a computationally effective approach for producing high-quality H ∞ -approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for H ∞ model reduction introduced by Flagg, Beattie, and Gugercin [1] for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory H 2 -optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many largescale settings. We are able to avoid computationally demanding H ∞ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better H ∞ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation.

In this contribution, a new framework for H 2 -optimal reduction of multiple-input, multipleoutput linear dynamical systems by tangential interpolation is presented. The framework is motivated by the local nature of both tangential interpolation and H 2 -optimal approximations. The main advantage is given by a decoupling of the cost of optimization from the cost of reduction, resulting in a significant speedup in H 2 -optimal reduction. In addition, a middle-sized surrogate model is produced at no additional cost and can be used e.g. for error estimation. Numerical examples illustrate the new framework, showing its effectiveness in producing H 2 -optimal reduced models at a far lower cost than conventional algorithms. The paper ends with a brief discussion on how the idea behind the framework can be extended to approximate further system classes, thus showing that this truly is a general framework for interpolatory H 2 reduction rather than just an additional reduction algorithm.