Chapter 3 Islamic University of Gaza Faculty of Engineering

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Chapter 3 Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Chapter 3 Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

must have a unique solution. Islamic University of Gaza Faculty of Engineering Electrical Engineering Department For mathematical model to predict the future state of the system from its current state at 𝑡 0 , the initial-value problem: must have a unique solution. “Existence and Uniqueness of Solutions is important” Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Continuous function Given a function f : D → R and an element  𝒙 𝟎  of the domain D,  f is said to be continuous at the point  𝒙 𝟎 when the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of f with  𝒙 𝟎  − δ < x <  𝒙 𝟎 + δ, the value of f(x) satisfies Alternatively written, continuity of f : D → R at  𝒙 𝟎  ∈ D means that for every ε > 0 there exists a δ > 0 such that for all x ∈ D : Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Piecewise continuous function Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Piecewise continuous function Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department A differentiable function of one real variable is a function whose derivative exists at each point in its domain. if  𝒙 𝟎  is a point in the domain of a function f, then f is said to be differentiable at  𝒙 𝟎  if the derivative f ′( 𝒙 𝟎 ) exists. This means that the graph of f has a non-vertical tangent line at the point ( 𝒙 𝟎 , f( 𝒙 𝟎 )) and doesn’t contains break, bend, or cusp. A function f is said to be continuously differentiable if the derivative f′(x) exists and is itself a continuous function.  Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

The solution 𝒙(𝒕) will be piecewise continuously differentiable. Islamic University of Gaza Faculty of Engineering Electrical Engineering Department If 𝒇(𝒕,𝒙) is continuous in t and x and 𝒙 is defined, then the solution 𝒙(𝒕) will be continuously differentiable. We will assume that 𝒇(𝒕,𝒙) is continuous in 𝒙 , but only piecewise continuous in t. The solution 𝒙(𝒕) will be piecewise continuously differentiable. Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Local Existence and Uniqueness Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Local Existence and Uniqueness Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5

Deadline : ----- with next H.W Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Take your notes for extra explanation and examples H.W 3.2 Deadline : ----- with next H.W Nonlinear Dynamic Control Systems , Dr. Moayed Almobaied Lecture5