Hankel singular value
In the article we present today we want to address the topic of Hankel singular value from a broad and varied perspective. Hankel singular value is a topic that has generated great interest and debate over the years, covering different aspects and triggering multiple reflections. In this sense, we propose to analyze in depth the various aspects that Hankel singular value presents, as well as its implications in today's society. To do this, we will examine different approaches and opinions of experts on the subject, in order to offer a complete and truthful vision of this very relevant topic. Through an exhaustive analysis, we aim to provide our readers with a broad and updated vision of Hankel singular value, with the aim of promoting critical and enriching reflection.
In control theory, Hankel singular values, named after Hermann Hankel, 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, {σi ≥ 0, i = 1,…,n}, of the eigenvalues, {λi ≥ 0, i = 1,…,n}, for the product of the controllability Gramian, WC, and the observability Gramian, WO.
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.[1]
- Hankel singular values also provide the optimal range of analog filters.[2]
See also
Notes
- ^ Hanzon, B. (1992). "The area enclosed by the (oriented) Nyquist diagram and the Hilbert-Schmidt-Hankel norm of a linear system". IEEE Transactions on Automatic Control. 37 (6): 835–839. doi:10.1109/9.256345. hdl:1871/12152. ISSN 0018-9286.
- ^ Groenewold, G. (1991). "The design of high dynamic range continuous-time integratable bandpass filters". IEEE Transactions on Circuits and Systems. 38 (8): 838–852. doi:10.1109/31.85626. ISSN 0098-4094.
References
- Kenney, C.; Hewer, G. (Feb 1987). "Necessary and sufficient conditions for balancing unstable systems". IEEE Transactions on Automatic Control. 32 (2): 157. doi:10.1109/TAC.1987.1104553.
Further reading
- Antoulas, Athanasios C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM. doi:10.1137/1.9780898718713. ISBN 978-0-89871-529-3. S2CID 117896525.