Structured singular values

If the uncertainty has a known structure, structured singular values are used. In the book by Morari and Zafiriou (Robust Process Control, 1989, Prentice-Hall) this topic is discussed in detail. 

Packard, A., and J. C. Doyle. 1993. The complex structured singular value. Auto-matica 29 71-109. 

Balas and Doyle [28] formulate the problem of disturbance rejection problem for a prototype space structural system. They used a structured singular value jj, synthesis approach considering uncertainties due to unmodeled dynamics and equivalent uncertainty formulations of the performance requirements on actuator limits, disturbance rejection and sensor noise by choosing appropriate weightings. 

The structured singular value for the RP problem at frequency can then be computed as [47]... 

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

EFA. m is a short Matlab function that computes forward and backward EFA matrices for a given number, ne, of singular values. Its structure is essentially identical to the one discussed for Main EFA2, m. 

In mathematical system theory, the subject of model reduction has been studied for about 30 years. The focus is on model reduction of linear systems, in particular methods based on singular value decomposition. One of the best known of these methods is balanced truncation. It is used extensively for various engineering purposes, such as electronic chip design and the reduction of models of aerospace structures. This method does not require the type of a priori information about the system mentioned above. Only recently has it been tried out on biochemical systems [105, 106]. 

There has been some discussion as to whether CD can distinguish parallel from antiparallel p sheets. As stable, well-defined model compounds are lacking, the spectra available have been derived from secondary structure deconvolutions (see below). Overall, the ability of CD to provide adequate estimates of both parallel and antiparallel p sheet contents is still an ongoing question. Johnson and co-workers were the first to derive basis spectra which corresponded to both parallel and antiparallel p sheet structures in globular proteins using the singular value deconvolution method [11, 12, 51-53], However, the basis spectra were significantly different from spectra reported for model sleet structures. Recently, Perczel et al. [54] employed another approach, convex curve analysis, to obtain improved p sheet baas spectra. The major improvement was to include more p sheet proteins into the data base. 

Various refinements to the basic LPSVD method include using the noise singular values to correct the signal singular values,100, l01,109 restoration of the Hankel structure of X after truncation of the noise singular values,110,111 and regularization of Eq. (103) to determine K.111,112... 

Hull et al. have developed the Latent Semantic Structure Indexing (LaSSI) approach to perform similarity search in low-dimensional chemical space. To reduce the dimension of initial chemical space, the singular value decomposition method is applied for the descriptor-molecule matrix. Ranking molecules by similarity to a query molecule was performed in the reduced space... 

Table 4. Minimum singular value and eondition number for each structure (Ml).

Figure 1. Comparison of SVD and PP PCA for linear structure data with outliers. Scatter point plots in PC1-PC2 plane for data sets LINOl, L1N02 and LINOS (from top to bottom) calculated by SVD (A) and PP PCA (B). The corresponding singular values (X j) and projective indexes ( j) ate given.

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