Abstract/Details

## Measuring the distance from a system to the set of all uncontrollable systems

Gao, Mei.   University of Connecticut ProQuest Dissertations Publishing,  1993. 9412441.

### Abstract (summary)

Let $A\in\IR\sp{n,n}$ and $B\in\IR\sp{n,m}$. We suggest a new search algorithm for estimating the distance $\mu(A,B)$ of a controllable pair (A,B) to the set of uncontrollable pairs by estimating the global minimum of the function $\sigma\sb{\rm min}(\lbrack A-\lambda I,B\rbrack )$, $\lambda\in\doubc$, where $\sigma\sb{\rm min}(\cdot)$ denotes the smallest singular value of a matrix. Using simple properties of this function due to Ralph Byers one first observes that, provided $rank(B)<n$, the minimization problem can be transformed to a minimization problem in the bounded region $\{(x,z)\Vert x\vert\le\Vert A\Vert\sb2$, $\vert z\vert\le\Vert A\Vert\sb2\}$ in the two dimensional real plane. The algorithm then progressively partitions this region into simplexes and by determining whether their vertices ($x\sb{j},z\sb{j})$ satisfy that $z\sb{j}>\min\sb{y\in\IR}\sigma\sb{\rm min}(\lbrack A-(x\sb{j}+iy)I,B\rbrack )$, it computes after a finite number of steps upper and lower bounds for $\mu(A,B)$. The difference between the two is small if $\mu(A,B)$ is small, while the lower bound is large if $\mu(A,B)$ is large thus ensuring safe decisions. An error analysis together with numerical examples and an operation count are all presented. Only simple modifications of the search region are necessary to extend the applicability of the algorithm to the case when $rank(B)\le n$.

### Indexing (details)

Subject
Mathematics;
Computer science
Classification
0405: Mathematics
0984: Computer science
Identifier / keyword
Applied sciences; Pure sciences; controllable; minimization problem
Title
Measuring the distance from a system to the set of all uncontrollable systems
Author
Gao, Mei
Number of pages
84
Degree date
1993
School code
0056
Source
DAI-B 54/12, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-208-73851-1
University/institution
University of Connecticut
University location
United States -- Connecticut
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9412441
ProQuest document ID
304054682