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# A Pertubatttion Result for Linear Control Problems

IP.com Disclosure Number: IPCOM000128568D
Original Publication Date: 1981-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 8 page(s) / 26K

## Publishing Venue

Software Patent Institute

## Related People

Daniel Boley: AUTHOR [+3]

## Abstract

In this paper we will discuss some problems in com-puting the controllable (reachable) space for a linear system and give some perturbation analysis results that are significant for a popular algorithm used to compute that space, herein called the Staircase Algorithm.

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This is the abbreviated version, containing approximately 20% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

A Pertubatttion Result for Linear Control Problems

By

Daniel Boley

Computer Science Department Institute of Technology

136 Lind Hall University of Minnesota Minneapolis, Minnesota 55455 A Perturbation Result for Linear Control Problems

Technical Report 83-10

May A Perturbation Result for Linear Control Problems

Daniel Bode University of Minnesota

ABSTRACT

In this paper we will discuss some problems in com-puting the controllable (reachable) space for a linear system and give some perturbation analysis results that are significant for a popular algorithm used to compute that space, herein called the Staircase Algorithm.

Introduction

In this paper we will discuss some numerical problems in com- puting the controllable (reachable) space for a linear system

(Equation Omitted)

In this pager we confine our attention to the case where A,B are constant matrices, and x,u are vector function of time. Under these conditions controllable and reachable are equivalent. One classic algebraic definition of the controllable space

(Equation Omitted)

[1]. But numerical methods based on computing this matrix are very unstable. In [2] and [4] it is pointed 'out that using (2) directly to compute the space Sc can be much more unstable than using a method based on orthogonal similarity transformations of A. Con-sider the system of the form (1) with

(Equation Omitted)

and B equal to the vector:

University of Minnesota Page 1 Dec 31, 1981

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A Pertubatttion Result for Linear Control Problems

The author grateftfly acknowledges the support of the National Science Foundation grant ECS- 8204468 and U.S. Army Research office grant DAHC04-?15-G-0185.

(Equation Omitted)

We see that dim S, = 8, but if we compute the singular values of the matrix in (2) we obtain:

(Equation Omitted)

which, depending on the choice of relative zero tolerance, would imply that dim Sc could have any value in the range 2 to 9 In this paper we will indicate a more robust approach (see 2) and give a limited sensitivity analysis of the problem (sets 3-4,. We then show how to apply the sensitivity analysis to this approach (see 5), give some bounds for the numerical errors committed by the algorithm (see F), and indicate where the limitations of the analysis may lie (see 7). We finally give results of some numerical experiments (see 8) and some concluding remarks (see 9).

2. The Method

A more stable method than using (2) is the so-called Staircase Algorithm described in detail in [2J [4J. Briefly, this method con-sists of applying a series of orthogonal similarity= transformations Q to (1) to obtain a system

(Equation Omitted)

(3) where we have

(Equation Omitted)

and the matrices

(Equation Omitted)

are in a special reduced form:

(Equation Omitted)

with A'I1 block upper Hessenberg [2] [4]. The transformation Q=CQI R2J can be partitioned as in (3a), where Q1 is an orthogonal basis for the controllable space Sc. In the s...