# ESE 601: Hybrid Systems - Rensselaer Polytechnic Institute 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 Intl,

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 Resistor Damper Inductor

Mass Capacitor Spring Electric circuit I(t) I(t) 1

+ V L 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 x x(t) x 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 x x(t) x

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