Syllabus

1. Syllabus • Phenomena of nonlinear dynamics • Second order systems • Mathematical foundation • Lyapunov stability • Variable gradient method • Advanced stability theory • Input-output stability • Averaging method • Singular perturbations • Absolute stability • Describing function method • Exact feedback linearization • Lyapunov redesign • Back-stepping • Sliding mode control • High gain observers Lecture notes will be available at Website : http://ece.clemson.edu/crb/ece874/main.htm The relative importance of homeworks and tests on determining the final grade will be as follows : Homeworks 10% Mid - term exam 40% Final Exam 50%

2. General Considerations • Nonlinear • not linear (x) • not necessarily linear (o) • Why study nonlinear system ? • All physical systems are nonlinear in nature. • Nonlinearities may be introduced intentionally into a system in order to compensate for the effect of other undesirable nonlinearities, or to obtain better performance (on-off controller). • Linear system  closed form solutionNonlinear system  closed form solution X (Some predictions – qualitative analysis)

3. 1. Phenomena of Nonlinear Dynamics • Linear vs. Nonlinear Input Output System state, Definitions : Linear : when the superposition holds Nonlinear : otherwise

4. Stability & Output of systems • Stability depends on the system’s parameter (linear) • Stability depends on the initial conditions, input signals as well as the system parameters (nonlinear). • Output of a linear system has the same frequency as the input although its amplitude and phase may differ. • Output of a nonlinear system usually contains additional frequency components and may, in fact, not contain the input frequency.

5. Sys. Sys. Sys. Superposition * Superposition  = Is (1) linear ? + So is it linear?  No, under zero initial conditions only.

6. Linearity What is the linearity when ? + A mnemonic rule: All functions in RHS of a differential equation are linear. System is linear atleast at zero input or zero initial condition Ex:

7. Time invariant vs. Time varying • Time invariant vs. Time varying • System (1) is time invariant  parameters are constant - Linear time varying system • System (2) is time invariant  no function has t as its argument. - Nonlinear time varying system

8. Autonomous & Non - Autonomous • Time invariant system are called autonomous and time varying are called non - autonomous. In this course, ‘autonomous’ is reserved for systems with no external input, i.e., • Thus autonomous are time invariant systems with no external input. This course will address nonlinear system, both time invariant and time varying, but mostly autonomous. Ex:

9. If det(A)0,(1)has a unique equilibrium point, (Linear System). Equilibrium Point • Equilibrium Point • We start with an autonomous system. Definition: is an equilibrium point (or a steady state, or a singular point)  Nonlinear system ? × × × × × × multiple equilibrium points

10. Linear Autonomous Systems • What can a linear autonomous system do? where For 1-dim sys. For 2-dim sys.

11. Linear Autonomous Systems (Contd.)

12. Solution of Linear systems • For linear sys, the following facts are true • Solution always exists locally. • Solution always exists globally. • Solution is unique each initial condition produces a different trajectory. • Solution is continuously dependent on initial conditions for every finite t, • Equilibrium point is unique (when det A0).

13. Periodic Solution • If there is one periodic solution, there is an infinite setof periodic solutions. (There is no isolated closed solution.)Ex: ( many periodic solutions, w.r.t. I.C.)

14. Non - linear Autonomous System • What can a nonlinear autonomous system do ? • A solution may not exist, even locally. Basically everything. Here the solution is chattering, because Therefore, no differential function satisfying the equation exists.

15. Solutions • Solution may not exist globally. • Solution may not be unique. Assume finite escape time (= : linear system)

16. Equilibrium point • Equilibrium point doesn’t have to be unique. Ex: Ex:

17. Periodic Solutions • Nonlinear system may have isolated closed (periodic) solutions. Ex:

18. Isolated closed solution • Chaotic regimes  non periodic, bounded behavior Isolated closed solution ( only one periodic solution.)  Isolated attractive periodic solution Ex: ( lightly damped structure with large elastic deflections )

19. 2. Second Order Systems • Isoclines called “vector field”  Set of all trajectories on plane  Phase portrait

20. Isocline(contd.) • Curve c is called an isocline: when a trajectory intersect the isocline, it has slope c, connecting isoclines, we can obtain a solution. Ex:

21. Linearization • Linearization A nonlinear system can be represented as a bunch of linear systems - each valid in a small neighborhood of using linearization. Specifically, assume that is continuously, differentiable, Take one of the equilibrium, say Introduce, where =0

22. Consider a sufficiently small ball around The linearization of at is defined by Linearization(contd.) Ex:

23. Linearization(contd.) Then the two linearizations are

24. Singular Points • Nature of singular points (a)

25. Phase Portraits

26. (b) Phase Portraits(contd.)

27. Phase Portraits(contd.) (c) Let stable focus unstablefocus center

28. Nonlinear system • Nonlinear system Assume that the nature of this singular point in the linear system is What is the nature of the singular point in the nonlinear system ? Ans) Same, except for center.Center for the linear system doesn’t mean center in the nonlinear system. Equilibrium of a nonlinear system such that the linearization has no eigenvalues on the imaginary axis is called hyperbolic. Thus, for hyperbolic equilibria, the nature is the same as the linearization.

29. Nonlinear system(contd.) Ex: