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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 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.
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability
Physical systems Capacitor Inductor Resistor Damper Spring Mass
Electric circuit I(t) I(t) 1 + L V t V(t) L 0 t
More electric circuit L R C + V I(t)
A pendulum r Mg
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability
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.
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.
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.
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability
Techniques for non-autonomous systems • Example: u(t) y(t) 1 1 t t
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability
Simulation methods x[1] x[2] x(t) x[3]
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concepts • Simulation and numerical methods • State space representation • Stability • Reachability
State space representation • One of the most important representations of linear time invariant systems.
Solution to state space rep. Solution:
Exact discretization of autonomous systems x[3] x(t) x[1] x[2] t
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • Simulation and numerical methods • State space representation • Stability • Reachability • Discrete time systems
Stability of nonlinear systems p p stable
Stability of nonlinear systems p Asymptotically stable
Contents • Modeling with differential equations • Taxonomy of systems • Solution to linear ODEs • General solution concept • Simulation and numerical methods • State space representation • Stability • Reachability