Spring 2021 ECE 244-Digital Control Homework 1
NAME: SCORE: / 100 points
INSTRUCTIONS: Solve the following problems. Make sure to clearly box your unique final answer after
all of your derivations. All submission should be clearly named as follows:
“Lastname”-HW “homework number”-“additional info(if any)”.pdf
where “...” should be replaced with appropriate information, e.g., McGuire-HW1.pdf or McGuire-HW1-
Q1.pdf
Note that ONLY pdf files are acceptable in Canvas.
SUGGESTED REVIEW SECTIONS: Homework 1 problems involve materials covered in Lecture 1 to
Lecture 3.
SUBMISSION DEADLINE: Full submission is due before the end of the day 04/11/2021 (11:59pm) in
electronic form at Canvas/Assignment
HOMEWORK POLICY: No late homework submissions are allowed. Please check how to submit your
homework via Canvas before the deadline to avoid technical problems with your submission. Collaborations
are encouraged and feel free to consult anyone, particularly me (via office hours) and those in your team.
However, all solutions handed in for credit must reflect your own understanding of the material. If you do
collaborate or receive help from anyone outside of me, you must credit them by placing their name(s) at the
top of your paper.
FORMATTING REQUIREMENTS: Please reduce and consolidate your answers (including any needed
figures, MATLAB codes used and texts, you don’t have to include the problem statements, but make sure
you label your answers and box them.), using a maximum of 10MB for the final PDF submitted to Canvas.
Each page shall be maximum US letter sized (as defined within the PDF headers, such as this homework is).
Failure to follow these formatting requirements impedes my ability to grade your work in a timely manner
and will be penalized at my discretion.
Question 1.
(30 points) Consider a stable continuous-time system described by
x˙(t) = ax(t) + sin(ω0t)
where a < 0, subject to x(0) = x0 = 1.
The backward-difference discretization method approximates the continuous-time system with
x(t) ≈ x(k + 1)
x˙(t) ≈ x(k + 1)− x(k)
T
Conduct a stability analysis of the backward-difference discretization method and determine any stability
requirement placed on the sampling period T . Solve the differential equation exactly. Then, compute and
plot the responses of the forward-difference, backward-difference, and trapezoidal discretized models of the
continuous-time system for a = −5, ω0 = 2pi, and T = 0.1, 0.2, 0.4 sec versus the exact solution using t = kT
with T = 0.001 sec for 10 sec. Compute the RMS (root-mean-square) errors for the three methods for
T = 0.1, 0.2 and 0.4 sec by the formula
e =
√√√√ 1
N
N∑
k=0
[x(t = kT )− x(k)]2
Question 2.
(30 points) A triangular hold is a device that has an output, as sketched in Figure 1 that connects the samples
of an input with straight lines.
Figure 1: Response of a sample and triangle hold
(a) Sketch the impulse response of the triangle hold. Notice that it is noncausal.
(b) Compute the transfer function of the hold.
(c) Use Matlab to plot the frequency response of the triangle hold.
(d) How will the frequency response be changed if the triangle hold is made to be causal by adding a delay
of one sample period?
Question 3.
(40 points) Suppose a continuous-time signal x(t) = a0 sin(ω0t)+ap sin(ωpt) is to be sampled. The signal has
an undesired parasitic frequency ωp due to plant dynamics. To remove the parasitic frequency, the signal is
pre-filtered by a first-order anti-aliasing filter
Haa(jω) =
α
jω + α
The cut-off frequency ωc is defined to be the frequency at the half-power point of the frequency response
Haa(jω) where the magnitude of Haa(jω) is equal to
1√
2
or −3dB. Determine α in terms of ωc. Obtain the
continuous-time filtered signal xf (t) by the inverse Fourier transform.
Let a0 = 1, ω0 = 4pi rad/sec, ap = 0.2, ωp = 40pi rad/sec, and ωc = 10pi rad/sec. Assuming that the
cutoff frequency is the Nyquist frequency, compute the sampling period T . Compute and plot the unfiltered
sampled signal x(kT ), and filtered sampled signal xf (kT ) for t ≤ 2 sec.

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