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]

$${\sigma}_{H}=\sqrt{{\lambda}_{i}\left(PQ\right)}$$

where *P* and *Q* are *controllability* and *observability
grammians* satisfying

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

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

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

The plot shows 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.

- Approximate Plant Model by Additive Error Methods
- Approximate Plant Model by Multiplicative Error Method

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