1. Asymptotic Techniques
Introduction
Real world complex for a direct analysis.
Solution simplification.
Basis for simplification small parameter with subsequent expansions.
Often first order term of the expansion gives already good results.
For differential equations, the source of difficulties is two-fold.
 time varying
 Large dim. of x is another
Two asymptotic techniques are
Averaging takes care of ; Singular perturbation takes care of 
Remark:

2. Averaging
Averaging
Motivation
Ex: Consider the system

3. Averaging (Continued)
source of difficulties
Averaging will get rid of this !

4. Averaging (Example) Then the averaged equation is Ex: (1)
Asymptotically stable
Ex:
(1)

5. Example (Continued)
To get rid of it, we use the idea of generating equation
Solve it.
Introduce the substitution.

6. Example (Continued) Then, from (1) or Thus
Thus again, what we obtained is
So if we will be able to analyze such systems in a simple basis,
we will solve the system

7. Example (Continued) The average equations are In the original time or
Equation point

8. Stability analysis
Stability analysis
(i)
(ii)

9. Theory
Theory
slow variable
fast variable

10. Theorem 1
Theorem 1

11. Theorem 2
Theorem 2

12. Theorem 3
Theorem 3
Ex: Van der Pol eq.

13. Example (Continued)

14. Example (Continued)

15. Example
Ex:

16. Singular Perturbations
Idea

17. Singular Perturbations (Continued)~ ~

18. Example
Ex:

19. Example
Ex:

20. Theory
Theory
Conditions

21. Theorem
Theorem:

22. Example
Ex:

23. Linear Systems
Linear Systems

24. NonLinear Systems Nonlinear Systems Theorem
Proof : See Nonlinear Systems Analysis.

25. Example
Ex:

26. Nonlinear Control Indeed, Why do we use nonlinear control :
Modify the number and the location of the steady states.
Ensure the desired stability properties
Ensure the appropriate transients
Reduce the sensitivity to plant parameters
Remark: Consider the following problem :
find
state feedback
dynamic output feedback

27. Nonlinear Control Vs. Linear Control
Why not always use a linear controller ?
It just may not work.
Ex:
Choose
Then
We see that the system can’t be made asymptotically
stable at
On the other hand, a nonlinear feedback does exist :
Then
Asymptotically stable if

28. Example Even if a linear feedback exists, nonlinear one may be better. + _  + _

29. Example (Continued)
Let us use a nonlinear controller : To design it, consider again  If If

30. Example (Continued)
Switch from to appropriately and obtain a variable structure system.
sliding line
Created a new trajectory: the system is insensitive to disturbance in the
sliding regime Variable structure control