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1: 30 points
Consider the simple pendulum system shown below with a motor at the hinge that produces
torque τ . The system dynamics are easy to derive from first principles.
¨θ +
g
l
sin θ =
τ
ml2
(a) Linearize the differential equation about equilibrium point θ =
π
2
. Analyze the stability of the
system in this configuration.
(b) Assume the mass is 1 kg and the length is 4 m. Design a PD controller (maps angle error to
motor torque) that places the closed loop poles such that the closed loop system nominally has a
25% overshoot and a 2 second 2% settling time.
(c) Construct the nonlinear dynamics in Simulink and apply the controller designed in part (b).
Plot the initial condition response starting at rest from θ0 = 45o
. For what range of initial
conditions does the PD controller stabilize the nonlinear system? No need to derive this
analytically – you can use the simulation to answer this. Be sure to include an image of your
Simulink model in the solution.
2: 20 points
A system has an input u(t) and an output y(t) which are related by the information provided
below. Classify each system as linear or nonlinear and time invariant or time varying.
(a) y(t) = a, a 6= 0 ∀t
(b) y(t) = −3u(t) + 2
(c) y(t) = u
3
(t)
1
24-677 (LCS) Homework 1 Due 2/10/2021
(d) y(t) = u(t
3
)
(e) y(t) = e
−tu(t − T)
3: 20 points
The figure below shows a model commonly used for automobile suspension analysis. In the
model, the uneven ground specifies the position of the wheel’s contact point. The wheel itself is
not shown, as its mass is considered to be negligible relative to the mass of the rest of the car.
(a) Write a differential equation and a state variable description for this system, considering the
height of the car x(t) to be the output and the road height y(t) to be the input.
(b) Using the parameters M = 300 kg, K = 20000 N/m, and B = 1000 N·s/m, use Matlab to
plot the response of the system to
• y(t) is a step of size 0.15 m (simulates hitting a curb). HINT: You can use the step
command.
• y(t) is a 1/2 sine wave of amplitude 0.08 m and frequency 50 Hz (simulates hitting a speed
bump at 30 mph). Be sure to pad your y(t) with zeros after the 1/2 sine to let the system
ring out. HINT: You can use the lsim command.
4: 20 points
For the mechanical system shown below, friction between surfaces is modeled as viscous damping
with damping coefficients denoted by Bi
. Use the principles of dynamics to find the equations of
motion, then create state space realizations using the following states.
2
24-677 (LCS) Homework 1 Due 2/10/2021
(a) State variables x =
x1
x˙ 1
x2
x˙ 2
and output variable y = x2.
(b) State variables x =
x1
x˙ 1
x2 − x1
x˙ 2 − x˙ 1
and output variable y = x2 − x1.
5: 10 points
Consider the state space system given by
˙x =
18 9 13
50 23 35
−65 −31 −46
x +
−1
0
1
u(t)
y(t) =
5 −5 5
x.
Write the system equations in terms of the new state variables
ˆx =
−4×1 − 2×2 − 3×3
15×1 + 7×2 + 10×3
−5×1 − 2×2 − 3×3
.
6: 20 points
The robot shown in the figure below has the equations of motion given. Symbols
m1, m2, I1, I2, l1, and g are constant parameters, representing the characteristics of the rigid
body links. Quantities θ1 and d2 are the coordinate variables and are functions of time. The
3
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