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

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

3. 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

4. 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

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

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

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

8. 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

9. 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

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

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

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

13. 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

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

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

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

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

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

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

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

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

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

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

24. 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