sol03.pdf - Homework 03 2021-02-06 00:53:46-08:00 EECS...

Homework 03 @ 2021-02-06 00:53:46-08:00
EECS 16B Designing Information Devices and Systems II
Spring 2021 UC Berkeley
Homework 03
This homework is due on Friday, February 5, 2021, at 11:00PM. Self-grades
and HW Resubmission are due on Tuesday, February 9, 2021, at 11:00PM.
1. Reading Lecture Notes
Staying up to date with lectures is an important part of the learning process in this course. Here are links to
the notes that you need to read for this week: Note 2 and Note 3A
(a) How do we deal with piecewise constant inputs as introduced in the notes?
(b) What conclusions do we get after approximating any function u ( t ) as being piecewise constant over
fixed interval widths Δ ?
2. Simple scalar differential equations driven by an input
In class, you learned that the solution for t ≥ 0 to the simple scalar first-order differential equation
d
dt
x ( t ) = λ x ( t ) (1)
with initial condition
x ( t = 0 ) = x
0
(2)
is given for t ≥ 0 by
x ( t ) = x
0
e
λ t
. (3)
In an earlier homework, you proved that these solutions are unique – that is, that x ( t ) of the form in ( 3 ) are
the only possible solutions to the equation ( 1 ) with the specified initial condition ( 2 ).
In this question, we will analyze differential equations with inputs and prove that their solutions are unique.
In particular, we consider the scalar differential equation
d
dt
x ( t ) = λ x ( t )+ u ( t ) (4)
where u ( t ) is a known function of time from t = 0 onwards.
(a) Suppose that you are given an x
g
( t ) that satisfies both ( 2 ) and ( 4 ) for t ≥ 0.
Show that if y ( t ) also satisfies ( 2 ) and ( 4 ) for t ≥ 0 , then it must be that y ( t ) = x
g
( t ) for all t ≥ 0 .
(HINT: You already used ratios in an earlier HW to prove that two things were necessarily equal. This
time, you might want to use differences. Be sure to leverage what you already proved earlier instead
of having to redo all that work.)
EECS 16B, Spring 2021, Homework 03 1

Homework 03 @ 2021-02-06 00:53:46-08:00