ECE 486 Assignment # 1
Issued: January 23 Due: January 30, 2009
Reading Assignment:
FPE 5th ed., Chapters 1–3 (ignoring subsections marked “N”).
See also Belanger, Chapters 1–3, and Brogan, Chapters 3 & 5, or Chen Chapters 1–3.
Problems:
1(i) Compute by hand the Laplace transforms Fi=L(fi) with,
f1(t) = e−2tf2(t) = cos(2t)f3(t) = 5e−2t+ cos(2t).
In the second case you should use Euler’s formula for the cosine function.
(ii) Use the Final Value Theorem to compute limt→∞ fi(t) in each case based on the
transform Fi(s). In which cases is the theorem valid?
(iii) Compute by hand the step response for G(s) = 10/(s+ 1). Is the steady state
response to the unit step function equal to the DC gain (G(jω) evaluated at
ω= 0)?
(iv) Explain why Gwill attenuate high frequency sinusoidal inputs.
(v) Repeat (iii) with G(s) = 10/(s−1).
2 Consider again our cruise control problem, but let’s explicitly use the knowledge
that high-frequency throttle inputs will be attenuated. A reasonable model uses the
transfer function G(s) = 10/(s+ 1) as follows:
Y(s) = G(s)[U(s)−1
2D(s)] ,
where Yis the speed of the auto (mph), Uis the throttle angle (degrees), and Dis
the road grade (degrees).
(i) Compute the steady state response to an input u(t) = 1◦,t≥0, when d(t)≡0.
(ii) For a given real number k, consider the control law u=u0+ke where e=r−y,
and u0is a fixed constant. Draw the corresponding block diagram, and compute
the resulting Laplace transform Y /R.
(iii) For what range of kis the closed-loop system BIBO stable? For what values of
u0is the resulting steady state error zero when d≡0?
(iv) Find a gain ksuch that the control in (ii) will bring the automobile from 35 mph
to 55 mph in approximately thirty seconds. Use the value of u0obtained in (iii).