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ESE 601: Hybrid Systems

ESE 601: Hybrid Systems. Review material on continuous systems I. References. Kwakernaak, H. and Sivan, R. “ Modern signal and systems ”, Prentice Hall, 1991. Brogan, W., “ Modern control theory ”, Prentice Hall Int’l, 1991. Textbooks or lecture notes on linear systems or systems theory.

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ESE 601: Hybrid Systems

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  1. ESE 601: Hybrid Systems Review material on continuous systems I

  2. References • Kwakernaak, H. and Sivan, R. “Modern signal and systems”, Prentice Hall, 1991. • Brogan, W., “Modern control theory”, Prentice Hall Int’l, 1991. • Textbooks or lecture notes on linear systems or systems theory.

  3. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability

  4. Physical systems Capacitor Inductor Resistor Damper Spring Mass

  5. Electric circuit I(t) I(t) 1 + L V t V(t) L 0 t

  6. More electric circuit L R C + V I(t)

  7. A pendulum r Mg

  8. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability

  9. Linear vs nonlinear • Linear systems: if the set of solutions is closed under linear operation, i.e. scaling and addition. • All the examples are linear systems, except for the pendulum.

  10. Time invariant vs time varying • Time invariant: the set of solutions is closed under time shifting. • Time varying: the set of solutions is not closed under time shifting.

  11. Autonomous vs non-autonomous • Autonomous systems: given the past of the signals, the future is already fixed. • Non-autonomous systems: there is possibility for input, non-determinism.

  12. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability

  13. Techniques for autonomous systems

  14. Techniques for non-autonomous systems

  15. Techniques for non-autonomous systems • Example: u(t) y(t) 1 1 t t

  16. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability

  17. Solution concepts

  18. Example of weak solution

  19. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability

  20. Simulation methods x[1] x[2] x(t) x[3]

  21. Simulation methods

  22. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability

  23. State space representation • One of the most important representations of linear time invariant systems.

  24. State space representation

  25. Solution to state space rep. Solution:

  26. Exact discretization of autonomous systems x[3] x(t) x[1] x[2] t

  27. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • Simulation and numerical methods • State space representation • Stability • Reachability • Discrete time systems

  28. Stability of LTI systems

  29. Stability of nonlinear systems p p stable

  30. Stability of nonlinear systems p Asymptotically stable

  31. Lyapunov functions

  32. Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability

  33. Reachability

  34. Reachability of linear systems

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