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Hankel Singular Values

In control theory, eigenvalues define a system stability, whereas Hankel singular values define the “energy” of each state in the system. Keeping larger energy states of a system preserves most of its characteristics in terms of stability, frequency, and time responses. Model reduction techniques presented here are all based on the Hankel singular values of a system. They can achieve a reduced-order model that preserves the majority of the system characteristics.

Mathematically, given a stable state-space system (A,B,C,D), its Hankel singular values are defined as [1]


where P and Q are controllability and observability grammians satisfying


For example, generate a random 30-state system and plot its Hankel singular values.

G = rss(30,4,3);

Figure contains an axes object. The axes object with title Hankel Singular Values, xlabel Order, ylabel abs contains 2 objects of type bar.

The plot shows that system G has most of its “energy” stored in states 1 through 15 or so. Later, you will see how to use model reduction routines to keep a 15-state reduced model that preserves most of its dynamic response.

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