# Module3 p3 - ECE 514 Nonlinear Adaptive Control No ea dap...

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ECE 514 Nonlinear & Adaptive Control No ea & dap ve Co o Module 3-Part 3 Mathematical Preliminaries 1

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Topics In This Module 1. Topological Concepts 2. Contraction Mapping Theorems 3. Existence & Uniqueness of Solutions. This material is in Appendices A, B & Chapter 3 f Kh lil of Khalil 2
Topological Concepts t space linear a be Let X , x of od neighborho - an such that 0 , S x any for if open is X S ε > ε S } z x : X z { ) , x ( N ε < Δ ε is closed if its complement in denoted X/S is open. SX X S x r x r X S < > , such that 0 if bounded is bounded. and closed is it if compact is X S 3

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Continuity Using Topology A continuous function is defined as a function where the pre-image of every open set in Y is open in X : f XY More concretely, a function f ( x ) in a single variable x is said to be continuous at point a if 1. f(a) is defined, so that a is in the domain of f . 2. f(x) exists for x in the domain of f . m ( ) ( ) x fa 3. lim ( ) xa fx fa = 4
Continuity in Pictures A function is continuous at a point if its limit is the same as the value of the function at that point. This function has discontinuities at x=1 and x=2. 1 2 It is continuous at x=0 and x=4, because the one-sided limits match the value of the function 1234 5

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Properties of Continuous Functions If two functions f and g are continuous at x , then ± f+g is continuous at x . ± f-g is continuous at x . ± fg is continuous at x . ± f/g is continuous at x if . ± Providing that g(x) is in the domain of f , () 0 gx fg D is continuous at x , where denotes the composition of the functions f and g . () ( ) ( ( ) ) f f g x = D 6
Properties of Continuous Functions So in essence Continuous functions can be added, ubtracted multiplied divided and multiplied by a constant subtracted, multiplied, divided and multiplied by a constant, and the resulting function remains continuous. Moreover, Composites of continuous functions are continuous. 2 n Examples: ( ) sin yx = cos = 7

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Removable Discontinuities (You can fill the hole.) Essential Discontinuities: 8 jump infinite oscillating
Intermediate (Mean) Value Theorem If a function is continuous between a and b , then it takes n every value between and a b on every value between and . ( ) fa ( ) fb ( ) Because the function is continuous, it must take on very alue between a ( ) every y value between and . ( ) ( ) f b 9 a b

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Differentiable Functions A function is said to be differentiable at a point x if the limit : f RR () ( ) ' lim f xh f x f x +− = exists and f’(x) is called the derivative of f at x. A function is said to be continuously 0 h h nm differentiable at a point x if the partial derivatives exist and are continuous at x , : f / ij f x ∂∂ 1, 1 im jn ≤≤ A function is said to be differentiable or continuously differentiable if it is so for all x in its domain of efinition 10 definition.
Monotonic Sequences 1.

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Module3 p3 - ECE 514 Nonlinear Adaptive Control No ea dap...

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