ESE 601: Hybrid Systems
Review material on continuous
systems I
Spring semester 2006
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
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•
•
•
•
•
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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
Resistor
Damper
Inductor
Mass
Capacitor
Spring
Electric circuit
I(t)
I(t)
1
+
L
V
t
V(t)
L
0
t
More electric circuit
L
R
+
I(t)
V
C
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 autonomous systems
Techniques for non-autonomous systems
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
Solution concepts
Example of weak solution
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]
Simulation methods
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.
State space representation
Solution to state space rep.
Solution:
Exact discretization of autonomous
systems
x[3]
x[1]
x(t)
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 LTI systems
Stability of nonlinear systems
p
stable
p
Stability of nonlinear systems
p
Asymptotically stable
Lyapunov functions
Contents
•
•
•
•
•
•
•
•
Modeling with differential equations
Taxonomy of systems
Solution to linear ODEs
General solution concept
Simulation and numerical methods
State space representation
Stability
Reachability
Reachability
Reachability of linear systems