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Problem 1 - 20 pts
a) Find the range on K for which the following closed-loop system is stable
w(t) +^ K(2s+4) y(t)
s + 6s + 3s
_ 3 2
b) Sketch the root-locus plots for the following open-loop pole/zero configurations:
j ω
σ
i)
j ω
σ
(3 poles)
ii)
j ω
σ (2 poles)
iii)
j ω
σ
iv)
c) What is the type number or each system in part b)?
i) Type ________ ii) Type ________ iii) Type ________ iv) Type ________
d) Draw a point very close to the 3 poles at s=-2 and determine the angle of departure (hint: there may be more than one)
j ω
σ
-6 -4 -2 +
-j
+j
Problem 2 - 20 pts.
a) When should we use the following compensators:
i) PD -
ii) Lag -
iii) PID -
b) Given the following open-loop pole/zero configuration, find the angle which a lead compensator must supply so that the root
locus passes through the desired closed loop dominant pole (s 1
) (use your cheat sheet)
j ω
σ
s1=-4+j
-6 (^) -
c) Suppose our lead compensator in part b) is of the form G c
(s)= K c
(s+ z c
)/(s+p c
). Why can't we pick z c
= 20? What's the
problem with choosing z c
d) Suppose we choose z c
= 5 for our lead compensator in part b). Find an equation for p c
which involves the arctan() function (Do
Not Solve This Equation!).
p c
=__________________________________________________________ (equation)
e) Suppose that when we solve this equation, p c
= 10. Also, suppose |G(s 1
)| = |G(-4+j2)|=4. Find the value of K c
Problem 3 - 15 pts.
Given the following unity-feedback system
w(t)
y(t) 8
s(s + 8)
_
a) Find: the type number, K p
, K
v
, K
a
, and ess| step
, ess| ramp
, ess| parabola
Type # =_____ K p
= ______ K
v
= ______ K
a
= _______ ess| step
= _____ ess| ramp
= _____ ess| parabola
= _______
b) Sketch the root locus
c) Find the settling time and the damping coefficient (ζ) for the uncompensated system.
d) Design the simplest compensator possible to meet the following specs: : t s
≤ 1 second and ζ ≥ 0.
e) Now design a compensator to meet the following specs: t s
≤ 1 second and ζ ≥ 0.707 and ess|parabola =1/10 (find values for your
compensator!
f) Sketch the compensated root locus and give approximate values for all closed-loop poles of the lag compensated system.
Problem 4 - 15 pts.
a) Sketch the magnitude and phase Bode plots for the following open-loop transfer functions:
ω r/s
|G(jw)| dB
40
. (^1 10 100 )
ωr/s
G(jw) deg
0
. 000 1 10 100
i) G(s)=
10(s+1)
s(s/10 +1 )
ω r/s
|G(jw)| dB
40
. (^1 10 100 )
ωr/s
G(jw) deg
0
. (^1 10 100 )
ii) G(s)=
10(s-1)
s(s/10 +1 )
