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Nonlinear Systems

Nonlinear Systems. Prof. Dragan Nesic. Recommended text: H. K. Khalil, “Nonlinear Systems”, Prentice Hall. Second or third editions. Note: I am using the second edition and chapter numbers I refer to are consistent with this edition. Assessment:. Seminar (10%) after each lecture Exam (60%)

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Nonlinear Systems

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  1. Nonlinear Systems Prof. Dragan Nesic

  2. Recommended text:H. K. Khalil, “Nonlinear Systems”, Prentice Hall. Second or third editions.Note: I am using the second edition and chapter numbers I refer to are consistent with this edition.

  3. Assessment: • Seminar (10%) after each lecture • Exam (60%) • Two one hour assessments (15% each): • First assessment after the 8th lecture • Second assessment after the 16th lecture • Exam

  4. Consultations • Each Monday after the lectures. • Any other time, if we manage to arrange it. • My office is 5.14 on the fifth floor of DEEE. • My phone is 8344 5357. • My email is dnesic@unimelb.edu.au

  5. Lecture 1 Introduction to Nonlinear Systems

  6. Recommended reading • Khalil Chapter 1 (2nd edition)

  7. Outline: • Motivation • Examples • General linear and nonlinear system models • Summary

  8. Motivation: • Nonlinear systems: • how they arise in engineering; • why they are important; • their special features; • how to deal with them. • Understand capabilities/limitations of linear tools. • Master various analysis tools (you will be able to understand many other topics in this area). Bonus:We study one of the best books on this topic!

  9. Motivation • All systems in nature are nonlinear! • Global behavior (large variations) can only be studied using nonlinear models. • Linear system can not be used to model some phenomena that occur in physical systems (e.g. limit cycles). • In control engineering, there may be some advantage in using nonlinear feedbacks over linear.

  10. Motivation example (cell dynamics):

  11. Examples of nonlinear systems • Pendulum equations (Mech. Engineering); • Population dynamics (Ecology); • Van der Pol oscillator (Elec. Engineering; biology); and so on. Homework: find another nonlinear model in your immediate area of research.

  12. Swing equation (mech. engineering) Nonlinear: Newton’s law of motion l  Linearization: Small angles  sin  mg

  13. State space equations • Introduce (phase) state variables: Then, we can write the nonlinear state space model of the pendulum equation:

  14. Oscillator equations • If we assume that there is no friction, i.e. k=0, then we obtain the (nonlinear) oscillator: • Or in state-space form:

  15. Forced pendulum Nonlinear: T l  Linearized: mg The torque T is an “input”

  16. Example: forced linear pendulum • The linearized model of a forced pendulum: Using the phase state variables as before: and we can take that the “output” is y=x1=.

  17. Population modelling (ecology): • Population model of two species competing for the same food source: x1 and x2 are numbers of two species respectively.

  18. Negative resistance oscillator (Khalil) • Van der Pol equation (v = voltage): Define

  19. General nonlinear models • We consider general models of this form: Introduce:

  20. Terminology: • x is the state of the system; it is the minimum info (besides the input) required to integrate the equations. E.g. for pendulum the angle and angular velocity qualify as a state vector. • The same system can be represented using infinitely many different state vectors. • u is the input to the system. It is a variable that affects the behaviour of the system. It can either be a control input (i.e. we can change/affect it) or a disturbance input (i.e. we can not affect it).

  21. We can rewrite the nonlinear model as: x is a “state”; u is the “input”. Suppose u is control input and we choose then we have an unforced state equation: Time-invariant or autonomous system:

  22. Controlled forced pendulum • Suppose that we measure x1= and we let Then, the forced linear model takes the form:

  23. Sometimes we also consider “output equations” • Outputs are either the physical variables that we measure using some sensors or they are variables that we are interested in (or both). • E.g. in the case of pendulum, if we are interested in how  changes and/or we measure it, then this can be an output. • The choice of outputs depends on the particular application.

  24. Linear systems Time-varying linear system with inputs and outputs: Time-invariant linear system without inputs/outputs:

  25. Summary: • Physical systems from all branches of science and engineering can be modeled as nonlinear systems. • Some essentially nonlinear phenomena can not be modeled by linear systems (strong motivation to study nonlinear systems). • Analysis of nonlinear dynamical systems is an active area of research: e.g. understanding the dynamics of cell growth requires further research in this area.

  26. Next lecture: • Examples of nonlinear phenomena; • Linearization principle.

  27. Thank you for your attention!

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