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 plan walking trajectories?
How plan flying trajectories?
A simple system
k
b
Force exerted by the spring: Force exerted by the damper:
Force exerted by the inertia of the mass:
m
A simple system
k
b
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:
m
This is called a linear system. Why?
Let’s express this in ”state space form”:
k
b
A simple system
m
A simple system
Let’s express this in ”state space form”:
k
b
m
A simple system
Let’s express this in ”state space form”:
k
b
m
Let’s express this in ”state space form”:
k
b
A simple system
m
m
Let’s express this in ”state space form”:
k
b
A simple system
A simple system
m
Let’s express this in ”state space form”:
k
b
where
k
b
Suppose that you apply a force:
f
Your finger
A simple system
m
A simple system Suppose that you apply a force:
A simple system Suppose that you apply a force:
Canonical form for a linear system
Continuous time vs discrete time
Continuous time
Discrete time
Continuous time vs discrete time
Continuous time
Discrete time
What are A and B now?
Continuous time vs discrete time
Continuous time
Discrete time
What are A and B now?
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Continuous time vs discrete time
CT DT
CT
DT
Continuous time vs discrete time
CT DT
CT
DT
In this class, we’re going to focus on discrete time representations…
Think-pair-share External force
Viscous damping
Express DT dynamics of this system in state space form
Think-pair-share
Express DT dynamics of this system in state space form
Think-pair-share
Consider a point robot in the plane with state (x, y). It can make fixed motions dx and dy in either direction. Express this as a linear system of the form: