MAE 171A Homework #1 Solutions
Summer 2014
Solution 1:
(a) Use frequency shift transform
s
s2 + 122
=
s + 0.4
L e0.4t cos(12t) =
(s + 0.4)2 + 122
L(cos(12t) =
=
s2
s + 0.4
+ 0.8s + 144.16
(b) Use trig. identity sin( + ) = sin()cos() + cos()sin() to get
si
MAE 171A Homework #3 Solutions
Summer 2014
Solution 1:
(a) First, reduce the block diagram as shown:
G2
+
r
+
y
G1
1+G1
The transfer function for R to Y in the gure above is given by
G1
Y
= G2 +
R
1 + G1
(b) Reduce the block diagram
G7
G6
r
+
G1
G2
G3
+ +
171A Project Solutions
Summer 2014
Solution 1:
The functional block diagram is given by
+
e
c
Compensator
v
Amp
i
M
Motor
Wheel Inertia
Satellite Inertia
The dynamic equations are given by
v(t) = Ka (c (t) (t)
v(t)
i(t) =
R
M (t) = K1 i(t) K2 (t) + (t)
(1
Use MATLAB for any of the following questions.
EmblﬂmlConsider the plant transfer function
G[s] = [bs + k]/ [52[lHMS2 + [M + m]bs + [M + m}k]
to be put in the unity feedback loop [see figure]. This is the transfer function
relating the input force u[t] an
MAE 171A Homework #1 Solutions
Summer 2014
Solution 1:
In each of these problems, if the poles are stable, we may apply the Final Value Theorem.
(a) The output transfer function is given by
Y (s) = G(s) U (s) =
s+1
.
s s + 1 (s + 3)
2
Since all the poles
MAE 171A
Winter 2015
Design Project 2
Due February 20/2015
A control system has the job of driving a controlled shaft so that its angle of
rotation duplicates the angle of another command shaft, which is positioned
manually. The controlled shaft, carrying
Name: _
UID: _
MAE 171A
Final Exam August 13, 2014
Problem 1) (40%)
Consider the feedback control system of figure with a compensator
K(s) = K(1 + 1/s ) and a plant G(s) = 1/(s-1) .
a) Draw the root locus for the closed-loop poles.
b) Choose a gain K that
MAE 171A
Winter 2015
Design Project 1
Due January 30/2015
Certain devices require a constant temperature environment for proper operation.
Examples are gyroscopes, accelerometers and crystals used as frequency standards.
A particular temperature control s
MAE 171A
Winter 2015
Homework 1
Due January 16, 2015
Problem 1. Find the Laplace transforms
0.4t
(a) f ( t )=e cos ( 12 t)
(b) f (t)=sin ( 4 t+ /3)
2 at
(c) f ( t )=t e
(d) f ( t )=cos ( 2 t )cos ( 3 t )
(e) f ( t )=tu (t )(tT )u ( tT )
u(t) is the unit s
MAE 171A
Winter 2015
Homework 2
Due January 23/2015
Find the value of the output y(t) at t big
(a)
Let the input be a unit step, u(t), and the plant transfer function be
G(s)=
(b)
Let the input be an impulse, (t), and the plant transfer function be
G(s)=
MAE 171A
Winter 2105
Homework 3
Due January 30, 2015
Problem 1 Sketch the output time response of a step input into the following
transfer functions. Use partial fractions to decompose the response.
a)
b)
c)
d)
e)
f)
G(s) = 10/(s+1)(s+10)
G(s) = 10(s+.9)/
MAE 171A
Winter 2015
Homework 4
Due February 6, 2015
Problem 1 Sketch the step response of the systems
a) G(s) = (s+2)/(s2+3s+36)
b) G(s) = (-s+2)/(s2+3s+36)
Hint: write G(s) = zsG(s) + G(s) where z = +1;-1.
Problem 2 Determine the input/output transfer f
MAE 171A
Winter 2015
Homework 7
Due March 13/2015
Use MATLAB for any of the following questions.
Problem 1) Sketch the Nyquist Plots for the following transfer functions.
Indicate when the Nyquist plot indicates stability. Comment your results.
1. G(s) =K
MAE 171A
Winter2015
Homework 5
Due February 23/2015
Problem 1 Draw the root locus for the following transfer functions as K goes
from 0 to1.
(a) G(s)H(s) = 3K(s+1)/s(s+3)
(b) G(s)H(s) = K(s+4)/2s(s+2)
(c) G(s)H(s) = K(s+1)/s2
(d) G(s)H(s) = 14K/s(s+3.5)(s
MAE 171A
Winter 2015
Homework 6
Due March 6/2015
Use MATLAB for any of the following questions.
Problem 1 Consider the plant transfer function
G(s) = (bs + k)/ (s2[mMs2 + (M + m)bs + (M + m)k])
to be put in the unity feedback loop (see figure). This is th
Use MATLAB fer any {if the fellewing questinns.
Prﬂblem 1] Sketch the Nyquist Plots for the following transfer ﬁtnetiens.
Indicate when the Nyquist plot indicates stability. Comment your results.
1. G[s] =Kf[s + 1}[s + 2]; K = 1
Myquist Diagram
g
E"
E
R
MAE 171A
Spring 2013
Homework 3
Due May 1 / 13
Problem 1 Sketch the output time response of a step input into the following
transfer functions. Use partial fractions to decompose the response.
a) G[s) = 10/[s+1)(s+10]
Y(s) = :GCS‘) = 1°
s(s+1)(s+10)
y(t)
1 Draw the root locus for the following transfer
functions as K goes from U to 00.
(a) Grams} =
The steps of drawing the root locus [BL] are following
(9
Polc:p=ﬂ,—3 [11:2]
36.1“013 = —1 (m :1]
11—11:
{'2} real segments: —1 <_Z s <_Z CI, 3 <_Z —3
@ asymp
MAE 171 A:
Feedback Control Systems
Chapter 9
Gain and Phase Margins and the Nyquist
Criterion (Continued)
May 23, 2012
Second-Order System: K G =
K
2s
s2
2 + +1
n
Second-Order System: K G =
Bode Plot
40
=0
< 0.5
slope = -40db/dec
db 0
=1
-40
0.1n
n
log
MAE 171A
Winter 2015
Design Project 3
Due March 13/2015
Consider the design of a satellite attitude control system. The satellite is
assumed to be in drag free space. We are interested in controlling rotation
about its axis of rotational symmetry. We do t
MAE 171A: Dynamic Systems Control
Lecture 6: Stability
Goele Pipeleers
Overview of the Lectures so Far
L1: review of the Laplace transform and transfer function
L2: deriving a transfer function model from a physical model
L3: system poles, zeros and im
MAE 171A: Dynamic Systems Control
Lecture 7: Framework of Control Systems
Goele Pipeleers
Overview
So Far: Mainly Systems Theory
L1: review of the Laplace transform and transfer function
L2: deriving a transfer function model from a physical model
L3:
MAE 171A: Dynamic Systems Control
Lecture 8: Steady-state Error and System Type
Goele Pipeleers
Recall from Lecture 7
Feedback Control
control conguration
r
D (s )
controller
u
w
G (s )
system
y
v
r: reference
w: disturbance
u: control signal
y : output
MAE 171A: Dynamic Systems Control
Lecture 9: PID Controller
Goele Pipeleers
PID Control
Feedback Control
control conguration
r
em
D (s )
u
w
G (s )
y
v
ki
PID controller: D(s) = kp + + kds
s
t
u(t) = kp em(t) + ki
em( )d + kd em(t)
0
1
ki
PID controlle
MAE 171A: Dynamic Systems Control
Lecture 10: Root Locus
Goele Pipeleers
Situation
Feedback Control
r
D (s )
u
w
G (s )
y
v
In the Previous Lecture
P, D, I, PD, PID controllers
analysis of their advantages and disadvantages
1
In the Subsequent Lectures
MAE 171A: Dynamic Systems Control
Lecture 11: Control Design Using Root Locus
Goele Pipeleers
Situation
In Lecture 10
P, D, I, PD, PID controllers
analysis of their advantages and disadvantages
In Lecture 11: Root Locus
problem formulation
basic chara