[SOLVED] CS Writing Dynamics in State Space Form

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?

Lets express this in state space form:
k
b
A simple system
m

A simple system
Lets express this in state space form:
k
b
m

A simple system
Lets express this in state space form:
k
b
m

Lets express this in state space form:
k
b
A simple system
m

m
Lets express this in state space form:
k
b
A simple system

A simple system
m
Lets 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, were 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: