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 
where P and Q are controllability and observability grammians satisfying
For example, generate a random 30-state system and plot its Hankel singular values.
rng(1234,'twister'); G = rss(30,4,3); hankelsv(G)
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
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