SLIDE 1
Motivation
In order to reason about complex dynamical systems, we need to write system dynamics in a convenient form. How encode dynamics of an inverted pendulum? How plan walking trajectories? How plan flying trajectories?
Writing Dynamics in State Space Form Robert Platt Northeastern - - PowerPoint PPT Presentation
Writing Dynamics in State Space Form Robert Platt Northeastern - - PowerPoint PPT Presentation
Writing Dynamics in State Space Form Robert Platt Northeastern University Motivation In order to reason about complex dynamical systems, we need to write system dynamics in a convenient form. How encode dynamics of an inverted pendulum? How
SLIDE 2
SLIDE 3
A simple system
k b m
Force exerted by the spring: Force exerted by the damper: Force exerted by the inertia of the mass:
SLIDE 4
A simple system
k b m
Consider the motion of the mass
- there are no other forces acting on the mass
- therefore, the equation of motion is the sum of the forces:
This is called a linear system. Why?
SLIDE 5
A simple system
k b
m
Let's express this in ''state space form'':
SLIDE 6
A simple system
k b
m
Let's express this in ''state space form'':
SLIDE 7
A simple system
k b
m
Let's express this in ''state space form'':
SLIDE 8
A simple system
k b m
Let's express this in ''state space form'':
SLIDE 9
A simple system
k b m
Let's express this in ''state space form'':
where
SLIDE 10
A simple system
k b
m
Your finger
f
Suppose that you apply a force:
SLIDE 11
A simple system
Suppose that you apply a force:
SLIDE 12
A simple system
Suppose that you apply a force:
Canonical form for a linear system
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Continuous time vs discrete time
Continuous time Discrete time
SLIDE 14
Continuous time vs discrete time
Continuous time Discrete time
What are A and B now?
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Continuous time vs discrete time
Continuous time Discrete time
What are A and B now?
SLIDE 16
Simple system in discrete time
We want something in this form:
SLIDE 17
Simple system in discrete time
We want something in this form:
SLIDE 18
Simple system in discrete time
We want something in this form:
SLIDE 19
Simple system in discrete time
We want something in this form:
SLIDE 20
Continuous time vs discrete time
CT DT CT DT
SLIDE 21
Continuous time vs discrete time
CT DT CT DT
In this class, we’re going to focus on discrete time representations...
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Think-pair-share
Viscous damping External force Express DT dynamics of this system in state space form
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Think-pair-share
Express DT dynamics of this system in state space form