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ESE 601: Hybrid Systems

Explore modeling with differential equations, system taxonomy, solution methods, and state space representation for linear and nonlinear systems. Learn stability and reachability concepts along with simulation and numerical methods.

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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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