Hankel Singular Values

In control theory, eigenvalues define a system stability, whereasHankel singular valuesdefine 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 astablestate-space system (A,B,C,D), its Hankel singular values are defined as[1]

$σ_{H} = \sqrt{λ_{i} (P Q)}$

wherePandQarecontrollabilityandobservability grammianssatisfying

$\begin{array}{l} A P + P A^{T} = - B B^{T} \\ A^{T} Q + Q A = - C^{T} C . \end{array}$

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

rng(1234,'twister'); G = rss(30,4,3); hankelsv(G)

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 systemGhas 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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