PSEUDOSPECTRA o Application of eigenvalue o Pseudospectra definition - - PowerPoint PPT Presentation

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PSEUDOSPECTRA

D E F I N I T I O N S A N D A P P L I C A T I O N S

S U P E R V I S O R D R . N O B A K H T I

H A M I D Z A R G A R A N

OVERVI EW

• Application of eigenvalue
• Pseudospectra definition
• Examples of pseudospectra
• Normal matrices
• Ill-condition problems
• Fragility of controllers

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W HY EI GEN VALUES?

Diagonalization and separation of variables Resonance: heightened response to selected inputs. Asymptotics and stability: dominant response to general inputs They give a matrix a personality

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QUI Z

1 1 1 , 1 5 2

Which curve is which?

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PSEUDOSPECTRA TOOL

Better question is … Is large? Is z an eigenvalue of A? Is singular?

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• First def.

∈

PSEUDOSPECTRA DEFI NI TI ONS

• Example:

1

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• Second def.

∈ ∈ ∈ .

PSEUDOSPECTRA DEFI NI TI ONS

( CONTI NUED)

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• Third def.

∈ ∈ 1.

The three definitions are equivalent

PSEUDOSPECTRA DEFI NI TI ONS

( CONTI NUED)

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EXAM PLES

1 1 1 1 5 2

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NORM AL M ATRI CES

• Normal matrix

↔ ∗ ∗

• Some other equivalent conditions:
• ∑
• ,

∑

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NORM AL M ATRI CES

• In general we have :

| , ⊆ If A is normal then = | , And for 2-norm ∆

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EXAM PLE

1 1

• B is a normal matrix

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APPLI CATI ON I N TRANSI ENT RESPONSE ANALYSI S

• Theorem.

For all 0 we have sup

• sup
• and

sup lim

→

1 log

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I LL-CONDI TI ON PROBLEM S

• consider a polynomial with zeros

2, 2, … , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Zeros of 2

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I LL-CONDI TI ON PROBLEM S

1 20 ⋯ 2 20 ⋯ ⋮ 3 ⋯ ⋮ ⋮ ⋱ 20

• ⋯

20

• Characteristic equation:

20

• 20

20 ! 1 !

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FRAGI LI TY OF CONTROLLERS

 Any controller should be able to tolerate some uncertainty

in its coefficients

 Inherent imprecision in analog-digital and digital-analog conversion  Finite resolution measuring instruments  Round off errors in numerical computations

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FRAGI LI TY OF CONTROLLERS

EXAM PLE

• Electromagnetic suspension system
• µ-synthesis technique

Design controller

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FRAGI LI TY OF CONTROLLERS

EXAM PLE( CONTI NUED)

• The poles of closed-loop system
• The normalized ratio of required change in controller coefficients to

destabilize the closed loop

• ‖0‖ 1.455 352 715 525 003 1015

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W HY PSEUDOSPECTRA COULD BE A SOLUTI ON?

Σ Perturbed MIMO system Designed MIMO controller

∆

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REFRENCES

• L. N. Trefethen, M. Embree, Spectra and pseudospectra: the behavior of nonnormal matrices

and operators, New Jersey, Princeton University Press, 2005, pp. 12-21

• J. H. Wilkinson, Rounding Errors in Algebraic Processes, New York, Dover Publications, pp

44-45 & 138-139.

• L. H. Keel, P. Bhattacharyya, Robust, fragile or optimal?, American Control Conference, 1997.

Proceedings of the 1997. Vol. 2. IEEE, 1997.

• A. Jensen, Lecture Notes on Spectra and Pseudospectra of Matrices and Operators,

Department of Mathematical Sciences, Aalborg University, 2009.

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THANKS FOR YOUR ATTENTI ON ANY QUESTI ON?

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