ME 506 – Advanced Control Systems
Rose-Hulman Institute of Technology
Homework 1
Due: Tuesday, 10 September 2015
Day 1
1) Given the following non-linear equations for the cart-pendulum system, linearize the
equations about the operating points θ = 180o and θ& = 0 (inverted mode). Write the
resulting equations in state-space form.
(J
)
+ m p lcg2 θ&& + m p &x&lcg cos θ + m p g lcg sin θ = 0
(mc + m p )&x& + m p lcgθ&&cosθ − m pθ&2 lcg sin θ + b1x& = F
p
2) Given the ballistic pendulum schematic below where h = (1 − cos θ )L
Use the Taylor series expansion to show
that h can be approximated as: h ≈
HINTS: Write sin (θ ) =
x2
2L
x
, then use the
L
Pythagorean theorem to get
cos(θ ) = f ( x ) . Subs the result into the
original expression for h. Expand the
Taylor series out to the second order term
and evaluate at x = 0.
3) Nise problem 3.31 b & c.
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