Module3 p2 - ECE 514 Nonlinear & Adaptive Control p...

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CE 514 ECE 514 Nonlinear & Adaptive Control Module 3-Part 2 Mathematical Preliminaries 1
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Topics In This Module 1. Inner Products 2. Norms 3. Hilbert & Banach Spaces This material is in Appendix B & Chapter 3 of h lil Khalil 2
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Inner Product Definition: An inner product defined over a vector space V is a function <.,.> defined from V to F where F is either R or C such that x,y,z V: 1. <x,y> = <y,x> * where <.,.> * denotes the complex conjugate. 2. <x,y + z> = <x,y> + <x,z> 3. <x, α y> = <x,y> , ∀α ∈ F x x> where the 0 occurs only for = 4. <x,x> 0 where the 0 occurs only for x = 0 V Note : he usual dot product in n an inner product. . n i xy x y = 3 The usual dot product in R is an inner product. 1 ii i =
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Norms Let |a|: absolute value of a if a is real or magnitude of a if a is complex. efinition: norm (metric) || f a vector a real Definition: A norm (metric) || . || of a vector x is a real- valued function defined on the vector space X such that 1. for all x X with || x || = 0 if and only if x = 0. 2. || α x || = | | || x || for all x X and any scalar α . 0 x 3. || x + y || || x || + || y || for all x , y X ( Triangle inequality ) 4 We say that ( X , ||.||) or ( X, d ) is a normed or metric space
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Common Norms Common norms in X = R n ;where R n is the set of n x1 vectors with real components. orm: n ± 1- norm: - orm: Euclidean norm = = i i x x 1 1 n 2 2. 2 norm: - orm: = = i i x x 1 2 1 3. P norm: - orm: = = = n i p p i p x x 1 ) ( 5 4. norm: i 1,…, n i x x max =
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Norms (cont’d): Example then || x || 1 = 5, || x || 2 = 2 and || x || = 2 T [1 -2 2] x = Lemma || x || a and || x || b any 2 norms of a vector x R n there exist finite positive constants k 1 and k 2 such that k 1 || x || a || x || b k 2 || x || a x R n d x || d x || e said to be equivalent and || x || a and || x || b are said to be equivalent he same holds in any finite- imensional vector space. 6 The same holds in any finite dimensional vector space.
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Norms (Cont’d) Example: For x R n x n x x n x x 1 2 1 b d f i d i d d t l f i x n x 2 A norm may be defined independently from an inner product, but an inner product always defines a orm as: 7 norm as: > < = x x x , || ||
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Matrix Induced Norms m n n m × where atrix f orm duced R R A R A : matrix a of norm Induced matrix norm p x Ax p p p x p x p Ax x A p 1 0 sup sup = Δ For instance, = = m i ij j a 1 1 max A = = n j ij i a 1 max A (max column sum) (max row sum) 2 1 max 2 )] ( [ A A A T λ = A A T of eigenvalue maximum 8
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Definition of Induced Norms Note that p x p p x p Ax x Ax A p 1 0 sup sup = = = p x α x u = = where consider this, show To α Then u u u x p p p p p p p p u Au u Au x Au x Ax = = = Thus only unity length vectors need to be considered instead of all 0. x 9
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Continuity Using Norms ) || || , ( X ) || || , ( Y T X Y Y X T : such that ) , ( , if at continuous is 0 0 x X x T ε δ ε∃ ) , ( when ) ( ) ( 0 0 0 x x x x T x T X Y < < all for continuous
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module3 p2 - ECE 514 Nonlinear &amp; Adaptive Control p...

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