MAE 171A: Solutions to
Homework Set 1
Due on April 8, 2015
1
Find the Laplace Transforms
(a) f (t) = e0.4t cos(12t)
The laplace transform of f1 (t) = cos(wt) is given as
L cfw_f1 (t) = F1 (s)
w
L cfw_cos(wt) = 2
s + w2
Using one of the properties of the
MAE 171A:
Due on April 29, 2015
1
Draw the root locus for the following transfer
functions as K goes from 0 to .
(a) G(s)H(s) =
3K(s+1)
s(s+3)
The steps of drawing the root locus (RL) are following
1
P ole : p = 0, 3
(n = 2)
zero : z = 1
(m = 1)
nm=1
2 re
MAE 171A: Homework set 5
Due on May 6, 2015
For the block diagrams in Figure 1 the eect of
dierent compensators are studied.
First, determine the closed-loop transfer function for a) and b).
a)
C = Gp (W + Gc (R C)
Gp
Gp Gc
C=
W+
R
1 + Gp Gc
1 + Gp Gc
b)
MAE 171 A:
Feedback Control Systems
Chapter 9
Gain and Phase Margins and the Nyquist
Criterion
May 6, 2015
Gain and Phase Margin
A margin of safety in ensuring stability in the presence of
uncertainty in the system model is measured by the change in gain
MAE 171 A:
Feedback Control Systems
Chapter 9
Gain and Phase Margins and the Nyquist
Criterion (Continued)
May 15, 2015
Second-Order System: K G =
Positive encirclements are CW.
Nyquist Plot
Bode Plot
40
K
2s
s2
2 + +1
n
=0
< 0.5
slope = -40db/dec
Im(
Intro
MAE 171 A:
Feedback Control Systems
An Introduction to Modern Control Theory
Jason L. Speyer
Mechanical and Aerospace Engineering
University of California, Los Angeles
Spring Quarter
Chapter 12
April 30, 2015
Intro
Outline
Linear-Quadratic Optimal C
Design 3 problem - Solution
May 25, 2011
1. The attitude system of the rocket consists of a gimballed engine acting through a leaver-arm to
generate a torque about the center of gravity of the rocket.
A functional block diagram
The transfer function (TF)
MAE 171 A:
Feedback Control Systems
Chapter 9
Frequency Response
May 3, 2015
Background
The system frequency response can be found experimentally by a
frequency sweep, illustrated below
A sint
G(s)
B sin(t+)
1
0
B
A
-90
or by measuring the system impulse
MAE 171A: Homework set 3
Due on April 22, 2015
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)
The step response of G(s) can is
C(s) = G(s)
MAE 171 A:
Feedback Control Systems
Chapter 8
Root Locus Compensator Design
April 29, 2015
Aircraft Pitch-Rate Autopilot
To illustrate root locus compensator design, an aircraft
pitch-rate command system is to be developed for a Lear Jet in
steady level
Homework set 1
MAE 171A
Due: April 8, 2015
1. Find the Laplace transforms
(a) f (t) = e0.4t cos(12t)
(b) f (t) = sin(4t + )
3
(c) f (t) = t2 eat
(d) f (t) = cos(2t)cos(3t)
(e) f (t) = tu1 (t) (t T )u1 (t T )
2. Find the inverse Laplace Transform
(a) F (s)
Intro
MAE 171 A:
Feedback Control Systems
Jason L. Speyer
Mechanical and Aerospace Engineering
University of California, Los Angeles
Spring Quarter
Lecture 1
March 30, 2015
Intro
Feedback and Control Systems: Introduction
A system is a collection of compo
Intro
MAE 171 A:
Feedback Control Systems
Jason L. Speyer
Mechanical and Aerospace Engineering
University of California, Los Angeles
Spring Quarter
Lectures 2 & 3
April 1, 2015
Intro
Models for Control
Denition: Linear System
A function f (x) is a linear
Homework set 2
MAE 171A
Due: April 15, 2015
1. Find the value of the output c(t) at t =
(a) Let the input be a unit step, u(t), and the plant transfer function be
s+1
G(s) = (s+.5)(s+3)
(b) Let the input be an impulse, (t), and the plant transfer functio
MAE 171 A:
Feedback Control Systems
Chapter 3
Dynamic System Response
April 12, 2015
System Time Response Characteristics
Time response characteristics fall into three categories:
Transient response soon after a command.
Steady-state response long after a
Design Problem 2
MAE 171A
Due: April 29, 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 an inertia I
Homework set 3
MAE 171A
Due: April 22, 2015
1. A second order system has a damping ratio of 0.5, a natural frequency
of 100 rad/s and a dc gain of 1. Find the step response of the system to
a unit step input.
2. For each second order systems below, nd , T
MAE 171 A:
Feedback Control Systems
Chapter 6
Feedback Control
April 12, 2015
Objectives of Feedback Control Systems
Generally, the objectives of a feedback controller are to
improve system performance in the sense of
Speed as measured by the rise-time.
Homework set 4
MAE 171A
Due: April 29, 2015
1. Draw the root locus for the following transfer functions as K goes from 0 to .
(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) =
(d) G(s)H(s) =
K(s+1)
s2
14K
s(s+3.5)(s2 +2s+1)
(e) G(
MAE 171A: Solution : Homework set 2
Due on April 15, 2015
1
Find the value of the output c(t) at t=
(a) Let the input be a unit step, and the plant transfer
function be
G(s) =
s+1
(s + .5)(s + 3)
(1)
The nal value theorem can be applied since the TF G(s)
MAE 171 A:
Feedback Control Systems
Chapter 7
Root Locus
April 19, 2015
Root Locus
Consider the the feedback loop
Y (s)
K(s)G(s)
= GCL (s) =
R(s)
1 + K(s)G(s)H(s)
Let the denominator (which represents the closed-loop poles) be
1 + K(s)G(s)H(s) = 0 K(s)G
Design Problem 1
MAE 171A
Due: April 15, 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 system for th