Problem 1. [easy] Describe in detail the relative merits of control design using Nyquist
methods, Bode analysis, and ad hoc simulation. Are any of these complementary? Do any
of these methods give you information and tools that the others don’t?
Problem 2. [easy] Suppose you’re given the Bode plots for each of the following uncompensated systems. Point out what the problems are likely to be with these systems, and what
you would like to do as a control designer to improve performance. Assume the goals are
the “typical” things such as good phase and gain margin, good noise rejection, low error to
reference inputs, etc.
Bode Diagram
0
Magnitude (dB)
−20
−40
−60
−80
0
Phase (deg)
−45
−90
−135
−180
10
−1
10
0
10
Frequency (rad/s)
Figure 1: Part (a)
1
10
2
Bode Diagram
0
−20
Magnitude (dB)
−40
−60
−80
−100
−120
−180
Phase (deg)
−225
−270
−315
−360
−405
−450
10
0
10
1
10
2
10
3
Frequency (rad/s)
Figure 2: Part (b)
Bode Diagram
0
Magnitude (dB)
−10
−20
−30
−40
−50
90
Phase (deg)
45
0
−45
−90
10
−4
10
−3
10
−2
−1
10
Frequency (rad/s)
Figure 3: Part (c)
Page 2
10
0
10
1
10
2
Problem 3. [easy] Derive a description for the sampling error of a sinusoid with frequency
f sampled at frequency fs . How does the sampling error change as fs → ∞? Based on your
results, what do you think a good relationship would be between f and fs ? For the last
question, speak in terms of a theoretical limit on fs and a practical limit for fs , including
some practical considerations.
Problem
[easy]
Explain the
theory
behind
criteria?
B–6–12. Plot root-locus
diagrams4.
for the
nonminimum-phase
B–6–15.
Determine
the valuesthe
of K, TNyquist
system
, and T of thestability
systems shown in Figures 6–102(a) and (b), respectively.
in Figure 6–105 so that the dominant closed-loop
Problem 5. [easy] Considershown
the
unity-feedback
control
system
whose
open-loop transfer
poles have the damping ratio z=0.5 and the undamped
K(s
–
1)
natural frequency v =3 rad兾sec.
function
is
+
(s + 2) (s + 4)
–
K
R
C
10
T s+1
.
G1(s)
+ G(s) =
K
2
1)
T ss(s
+1
–
+ s s(s
++4)
1
2
n
1
2
(a)
Figure 6–102
Determine the value of K such that the phase margin is 50◦ . What is the gain margin with
K(1 – s)
this
gain
K?
+
Figure 6–105 Control system.
(s + 2) (s + 4)
–
ProblemG2(s)
6. [easy]
B–6–16.
Consider
control system
shown in Figure 6–106.
Consider
the control system
shown
intheFigure
4. Determine
the gain K and time constant
(b)
Determine the gain K and time constant T of the controller
T and
of (b)
the
controller systems.
Gc (S) such
that
poles
located
at s = −2 ± 2i.
G (s)
suchthe
that closed-loop
the closed-loop poles
are are
located
at
(a)
Nonminimum-phase
c
s = -2 ; j2.
B–6–13. Consider the mechanical system shown in Figure
6–103. It consists of a spring and two dashpots. Obtain the
transfer function of the system. The displacement xi is the
input and displacement xo is the output. Is this system a
mechanical lead network or lag network?
+
–
K(Ts + 1)
1
s(s + 2)
Gc(s)
G(s)
Figure 6–106 Control system.
xi
Figure 4: Closed-loop control scheme.
B–6–17. Consider the system shown in Figure 6–107. Design a lead compensator such that the dominant closed-loop
poles are located at s = -2 ; j2 13 . Plot the unit-step response curve of the designed system with MATLAB.
k
b2
Problem 7. [hard] Consider the closed-loop system whose open-loop transfer function is
+
b1
–
xo
5
−2s
Kes(0.5s
+ 1)
.
G(s) =
s
Gc(s)
Determine the maximum value of K for which the system is stable.
Problem 8. [easy] Given Figure 6–107 Control system.
Figure 6–103
Mechanical system.
B–6–14. Consider the system shown in Figure 6–104. Plot
the root loci for the system. Determine the value of K such
that the damping ratio z of the dominant closed-loop poles
is 0.5. Then determine all closed-loop poles. Plot the unitstep response curve with MATLAB.
show that
+
–
K
s(s2 + 4s + 5)
B–6–18. Consider the system shown
2 in Figure 6–108. Design a compensator such that the dominant
closed-loop poles
n
are located at s = -1 ;2 j1.
2
G(s) =
+
–
ω
s + 2ζωn s + ωn
Gc(s)
1
s2
Lead
compensator
Space
1
vehicle
and ∠G(jωn ) = −π/2.
|G(jωn )| =
2ζ
(1)
(2)
Figure 6–108
Control system. for the damping ratio in terms of the phase
Problem 9. [hard] A well-known
approximation
P.M.
marginProblems
is ζ ≈ 100 . State the assumptions that go into this approximation
and then derive
395
the approximate relationship.
Problem 10. [easy] Demonstrate that a lead-lag controller can be designed to approximate
a PID controller. If this PID controller has gains kp , ki , and kd , write the transfer function
of the approximating lead-lag controller.
Problem 11. [hard] In Homework 6 you designed a feedback controller for joint 1 of a
serial positioning robot. This feedback controller was designed given an average model of
the transfer function for joint 1 (black line in Figs. 8 and 9 of Bukkems paper). Also given
Figure 6–104 Control system.
Page 3
in Fig. 8 of Bukkems is the experimental frequency response when tested at 16 different
static postures for joints 2 and 3 (gray lines in Fig. 8, see Fig. 3 for diagram of all the
joints). Now consider the design of a feedforward controller to be designed as an addon to your existing feedback controller. What complications should you consider when
implementing a feedforward controller based on this average model? Note: I’m not saying
that a feedforward controller would not be effective for this system, it would probably help
performance some, but there are some potential complications that should be considered
before adding a feedforward controller.
Problem 12.[hard] 6.41 of FPEN 7th edition. See attached document.
Problem 13. [hard] Demonstrate and/or explain why the following rule of thumb is advisable: if you have a plant with no poles or zeros in the right half plane the slope of your
magnitude bode plot should be approximately -20 db/dec at the crossover frequency.
Problem 14. [easy] Draw the open-loop transfer function, K(s)P (s), exclusion regions on
a typical bode plot graph paper if you want your system to track reference signal frequency
components up to 10 rad/sec with 5% attenuation and have 95% sensor noise rejection of
noise signals with a frequency of 150 rad/sec or greater.
Problem 15. [easy] What do you want the DC gain of your typical closed loop transfer
function to be, and why? Describe the idea of closed-loop bandwidth?
Page 4