## Description

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