Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the ...Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc. Keywords Routh stablity criterion - Orthonormal basis - Root-mean-squared value - Hankel operater - Nehari problem - Model reduction This work was supported by the Hong Kong Research Grants Council.展开更多
Root-mean-square distance Drms with characteristic of weighted-average is introduced in this article firstly. Drms can be used to capture the general proximity of a site to a dipping fault plane comparing with the rup...Root-mean-square distance Drms with characteristic of weighted-average is introduced in this article firstly. Drms can be used to capture the general proximity of a site to a dipping fault plane comparing with the rupture distance Drup and the seismogenic distance Dseis. Then, using Drup, Dseis and Drms, the hanging wall/footwall effects on the peak ground acceleration (PGA) during the 1999 Chi-Chi earthquake are evaluated by regression analysis. The logarithm residual shows that the PGA on hanging wall is much greater than that on footwall at the same Drup or Dseis when the Drup or Dseis is used as site-to-source distance measure. In contrast, there is no significant difference between the PGA on hanging wall and that on footwall at the same Drms when Drms is used. This result confirms that the hanging wall/footwall effect is mainly a geometric effect caused by the asymmetry of dipping fault. Therefore, the hanging wall/footwall effect on the near-fault ground motions can be ignored in the future attenuation analysis if the root-mean-square distance Drms is used as the site-to-source distance measure.展开更多
文摘Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc. Keywords Routh stablity criterion - Orthonormal basis - Root-mean-squared value - Hankel operater - Nehari problem - Model reduction This work was supported by the Hong Kong Research Grants Council.
基金Basic Science Research Foundation of Institute of Engineering Mechanics, China Earthquake Administration (2006B07) Natural Science Foundation of Heilongjiang Province (E2007-13)Joint Seismological Science Foundation of China (C07025)
文摘Root-mean-square distance Drms with characteristic of weighted-average is introduced in this article firstly. Drms can be used to capture the general proximity of a site to a dipping fault plane comparing with the rupture distance Drup and the seismogenic distance Dseis. Then, using Drup, Dseis and Drms, the hanging wall/footwall effects on the peak ground acceleration (PGA) during the 1999 Chi-Chi earthquake are evaluated by regression analysis. The logarithm residual shows that the PGA on hanging wall is much greater than that on footwall at the same Drup or Dseis when the Drup or Dseis is used as site-to-source distance measure. In contrast, there is no significant difference between the PGA on hanging wall and that on footwall at the same Drms when Drms is used. This result confirms that the hanging wall/footwall effect is mainly a geometric effect caused by the asymmetry of dipping fault. Therefore, the hanging wall/footwall effect on the near-fault ground motions can be ignored in the future attenuation analysis if the root-mean-square distance Drms is used as the site-to-source distance measure.
