In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, Hankel singular values, named after
Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on ...
, provide a measure of energy for each state in a system. They are the basis for
balanced model reduction, in which high energy states are retained while low energy states are discarded. The reduced model retains the important features of the original model.
Hankel singular values are calculated as the square roots, , of the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s, , for the product of the
controllability Gramian In control theory, we may need to find out whether or not a system such as
\begin
\dot(t)\boldsymbol(t)+\boldsymbol(t)\\
\boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t)
\end
is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...
, ''W
C'', and the
observability Gramian In control theory, we may need to find out whether or not a system such as
\begin
\dot(t)\boldsymbol(t)+\boldsymbol(t)\\
\boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t)
\end
is observable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol ...
, ''W
O''.
Properties
* The square of the Hilbert-Schmidt norm of the Hankel operator associated with a linear system is the sum of squares of the Hankel singular values of this system. Moreover, the area enclosed by the oriented
Nyquist diagram of an
BIBO stable and strictly proper linear system is equal π times the square of the Hilbert-Schmidt norm of the Hankel operator associated with this system.
*Hankel singular values also provide the optimal range of analog filters.
See also
*
Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...
*
Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on ...
Notes
References
*
*
Further reading
* {{Cite book
, author1-link = Athanasios C. Antoulas
, last = Antoulas
, first = Athanasios C.
, title = Approximation of Large-Scale Dynamical Systems
, doi = 10.1137/1.9780898718713
, year = 2005
, publisher = SIAM
, isbn = 978-0-89871-529-3
, s2cid = 117896525
, url = https://semanticscholar.org/paper/31a9c66730500929663f23fc8350a190b471422e
Control theory