29/09/1438 OVERVI EW PSEUDOSPECTRA o Application of eigenvalue o Pseudospectra definition D E F I N I T I O N S A N D o Examples of pseudospectra A P P L I C A T I O N S o Normal matrices o Ill-condition problems S U P E R V I S O R D R . N O B A K H T I o Fragility of controllers H A M I D Z A R G A R A N 2/21 W HY EI GEN VALUES? QUI Z � � � �1 �1 , � � � �1 1 5 0 0 �2 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 Which curve is which? 3/21 4/21 1
29/09/1438 PSEUDOSPECTRA DEFI NI TI ONS PSEUDOSPECTRA TOOL Is z an eigenvalue of A? • First def. Is �� � � singular? � � �������������� � � � �� � �� ��� ��� �� � ∈ � ���� ���� �� � �� �� � � �� Better question is … Is ��� � �� �� Example: large? � � 0 1 0 0 5/21 6/21 PSEUDOSPECTRA DEFI NI TI ONS PSEUDOSPECTRA DEFI NI TI ONS ( CONTI NUED) ( CONTI NUED) • Second def. • Third def. � � � �� �� ��� ��� �� � ∈ � ���� ���� � � � �� �� ��� ��� �� � ∈ � ���� ���� � ∈ � � � � � � � � � � ��� ���� � ∈ � ��� ���� � ��� ���� � ∈ � � ���� � � �. � 1. The three definitions are equivalent 7/21 8/21 2
29/09/1438 EXAM PLES NORM AL M ATRI CES � � � �1 1 �1 � � � �1 5 0 0 �2 • Normal matrix � �� ������ ↔ � ∗ � � �� ∗ • Some other equivalent conditions: � �� ������������ �� � ������� ������ � � � � � � ∑ � �� � ∑ �,��� ��� ����� ����� �� ���������� ��� �� ����� ������� �� � 9/21 10/21 NORM AL M ATRI CES EXAM PLE • In general we have : 1 0 0 • � � 0 �1 0 �|���� �, � � � � ⊆ � � � 0 0 � If A is normal then � � � = �|���� �, � � � � B is a normal matrix And for 2-norm � � � � � � � ∆ � 11/21 12/21 3
29/09/1438 APPLI CATI ON I N TRANSI ENT RESPONSE ANALYSI S I LL-CONDI TI ON PROBLEM S consider a polynomial with zeros � � � 2 ��� � �� Zeros of 2 �� , 2 �� , … , 2 ��� • Theorem. For all � � 0 we have ���� � � �� � � �� � 2 �� � �� � 2 �� � �� � sup� ���� � ����� � �� sup � 2 �� � �� � 2 ��� � �� � 2 ��� � �� � ��� � 2 ��� � �� � 2 ��� � �� � 2 ��� � �� and � 2 ��� � �� � 2 ��� � � � 2 ��� � � 1 � 2 ��� � �� � 2 ��� � � � 2 ���� � � � log � �� sup ��� � � � lim �→� � 2 ���� � � � 2 ���� � � � 2 ���� � � � 2 ���� � � � 2 ���� � � � 2 ���� 13/21 14/21 I LL-CONDI TI ON PROBLEM S FRAGI LI TY OF CONTROLLERS 1 20 0 ⋯ 0 Any controller should be able to tolerate some uncertainty 0 2 20 ⋯ 0 � � ⋮ 0 3 ⋯ ⋮ in its coefficients 0 ⋮ 0 ⋱ 20 Inherent imprecision in analog-digital and digital-analog conversion � 0 ⋯ 0 20 Characteristic equation: Finite resolution measuring instruments �� 20 �� Round off errors in numerical computations �� � � � � � � 20 �� � �� � �� 20 � � ! � � 1 !� ��� 15/21 16/21 4
29/09/1438 FRAGI LI TY OF CONTROLLERS FRAGI LI TY OF CONTROLLERS EXAM PLE( CONTI NUED) EXAM PLE • The poles of closed-loop system • Electromagnetic suspension system • µ-synthesis technique Design controller • The normalized ratio of required change in controller coefficients to destabilize the closed loop � ‖� 0 ‖ � 1.455 352 715 525 003 � 10 �15 17/21 18/21 W HY PSEUDOSPECTRA COULD BE A SOLUTI ON? 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 Designed Perturbed • J. H. Wilkinson, Rounding Errors in Algebraic Processes, New York, Dover Publications, pp MIMO Σ MIMO system 44-45 & 138-139. controller • 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. 19/21 20/21 5
29/09/1438 THANKS FOR YOUR ATTENTI ON ANY QUESTI ON? 21/21 6
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