ME 132
Solutions # 5
1 Complex Arithmetic
(a) Notice that for all real ,
2 + a2
|j a|
j a
=1
=
=
j + a
|j + a|
2 + a2
(b) In all four cases the magnitude plots are identical and are shown in the top panel. The
phase plots are shown in the bottom panel.
ME 132 Fall 2009 Solutions to Homework 1
1. There are several examples from each eld that are valid descriptions of feedback systems. As long as you showed the dependence of the control signal on some measured variable, your example should be sucient. 2.
ME 132
Homework # 1
Issued: September 02, 2011
Due: September 09, 2011
Suggested Readings:
1 My lecture notes, Introduction, posted on bspace.
2 Chapter 1, Astrom & Murray.
Problems to be turned in:
1 Structure of Control Systems
Draw the block diagram fo
ME 132
Solutions # 1
1 Structure of Control Systems
Shown below is a block diagram for the temperature control system in a refrigerator. A relay
is an on-o switch, and is often the most elementary actuation in control systems. Not shown
are humidity eects
ME 132
Issued: Monday, November 23, 2009 1 Observers Consider the plant y= 1 u+n s2
Homework # 12
Due: Monday, November 30, 2009
Here n represents measurement noise. (a) Design an observer to estimate the state of the plant using measurements of the outpu
ME 132
Homework # 1
Issued: January 22, 2016
Due: January 29, 2016
Suggested Readings:
1 My lecture notes, Introduction, posted on bspace.
2 Chapter 1, Astrom & Murray. This is a great book, and is posted on b-courses.
Problems to be turned in:
1 Structur
University of California at Berkeley Department of Mechanical Engineering ME 132: Dynamic Systems and Feedback Instructor: Kameshwar Poolla Office: 5105 or 5141 Etcheverry Hall Email: [email protected] Office Hours: Th 1-3 pm Teaching Assistants: GSI 1:
ME 132
Solutions # 1
1 Structure of Control Systems
Shown below is a block diagram for the temperature control system in a refrigerator. A relay is an on-off
switch, and is often the most elementary actuation in control systems. Not shown are humidity eff
ME 132
Issued: October 09, 2009
Homework # 7
Due: October 17, 2009
1 Block diagrams Consider the block diagram shown below. Each block is a gain with the letter in the block denoting the numerical value of the gain. Find the matrix M (which is 1 3) such t
ME 132
Homework # 7
Issued: October 17, 2011
Due: October 24, 2011
1 PI position control for a DC Motor
In this problem we will investigate position control of a DC motor. Use the following
simplified motor model. You do not have to understand details of
ME 132
Homework # 1
Issued: September 05, 2014
Due: September 12, 2014
Suggested Readings:
1 My lecture notes, Introduction, posted on bspace.
2 Chapter 1, Astrom & Murray.
Problems to be turned in:
1 Structure of Control Systems
Draw the block diagram fo
ME 132
Solutions # 3
1 First Order Systems: Time Domain
(a) Comparing the given model to standard form: y + ay = bu, we can see that a > 0.
Thus, the system is stable.
(b) Recall that the time constant, T, of a first-order ODE is:
T =
1
=1
a
(c) Recall th
ME 132
Issued: December 02, 2009
Homework # 13
Due: December 09, 2009
1 Consider the magnetically-suspended ball treated in class. The dierential equation relating the control input u to the height of the ball y is y=g with the following constants: m 0.15
ME 132
Solutions # 7
1 PI position control for a DC Motor
First observe that the motor dynamics in transfer function notation are given by
1.9
1600
=
u
TL
s(s + 0.97)
s(s + 0.97)
Next, our PI controller has the form
sKP + KI
(r )
u=
s
Combining these equa
University of California at Berkeley Department of Mechanical Engineering ME132 Dynamic Systems and Feedback Homework Set #6 Assigned: April 3 (Th) Due: April 15 (Tu) [1] 18.1-1 [2] 21.3-1 Spring 2008
ME 132
Solutions # 11
1 Equilibrium points, Linearization
(a) If x1 = y , x2 = y , then we have the following state-space description:
x1 = x2
x2 = cos(x2 ) x1 2 + u.
(b) We are given that u = 0. To nd equilibrium points, we solve 0 = f (x, u) as
0 = x2
0
ME 132
Solutions # 9
1 Matrix Exponentials
(a) Since v1 and v2 are linearly independent, we can write the matrix A as
A = T T 1
where T = [ v1 v2 ] and
1
A = [ v1 v2 ]
0
1
1 2
=
0
0 1
1 2
=
0 2
= diagcfw_1 , 2 . For this problem,
0
1
[ v1 v2 ]
2
1 2
0
0
ME 132
1 The state-variable equations for the system are x1 = x2 x2 = g y = x1 c u2 m x2 1
Solutions # 13
(a) The control u needed to maintain y = is found from evaluating the system at equilibrium. This gives the equilibrium x1 x2 , u, y = 0 , mg , c
The
ME 132
Solutions # 10
1 PID Controller Design
(a) The characteristic equation of the ODE describing the system is s 1. The root of the
equation is s1 = 1 > 0 and thus the system is unstable.
(b) We start by taking the time derivative of the ODE y (t) = y
ME 132
Spring 2016
Midterm 1
NAME:
ID # :
#1
#2
#3
#4
#5
#6
#7
TOTAL
8
10
6
8
8
8
10
58
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 Write your solution clearly. Please, please please .
4 This exam has 7 questio
ME 132
Formulas you should know
LTI
x = Ax + Bu
y = Cx + Du
LTI general solution:
Z t
x(t) = eat x(0) +
ea(t ) bu( )d
0
Sensitivity of a quantity A() to changes in the parameter , at an operating point 0 is,
dA 0
d A0
where A0 = A(0 ).
First order step re
ME 132
Homework # 2
Issued: January 28, 2017
Due: February 06, 2017
1 Sensitivity Analysis
Consider the block diagram shown below. Compute the sensitivity Sa of the output y with
respect to changes in the parameter A. Also find the sensitivity Sc of the o
ME 132
Homework # 3
Issued: February 02, 2017
Due: February 10, 2017
Suggested Readings:
1 Chapter 2, Astrom & Murray.
Problems to be turned in:
1 First order systems
Suppose a system is described by the differential equation
1
x = x + 2u
2
where u is giv
ME 132
Homework # 4
Issued: February 09, 2017
Due: February 17, 2017
1 Modeling of First-order Systems
We want to build a model for some physical system. We believe that the system can be modeled
as first-order system
y + ay = bu
We need to find the const
ME 132
Spring 2016
Midterm 2
NAME:
ID # :
#1
#2
#3
#4
#5
#6
#7
TOTAL
10
10
8
10
10
8
4
60
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 This exam has 7 questions worth 60 points.
4 Please write your solution clea
ME 132
Fall 2014
Midterm 1
NAME:
ID # :
#1
#2
#3
#4
#5
#6
TOTAL
7
11
8
12
10
12
60
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 Write your solution clearly. Please, please please .
4 This exam has 6 questions wo
ME 132
Dynamic Systems and Feedback
Fall 2016
List of equations from the course
This is the list of equations and concepts for ME132. It is not exhaustive. Please tell me if Ive left
out any important equations, or made any mistakes.
The numbered equation
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME132 Dynamic Systems and Feedback
Midterm I
Spring 2013
Closed Book and Closed Notes. One 8.5 11 sheet (double-sided) of handwritten notes
allowed. Scientific calculator with no gr
ME 132
Fall 2014
Midterm 2
NAME:
ID # :
#1
#2
#3
#4
#5
#6
TOTAL
14
12
10
8
8
8
60
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 Write your solution clearly. Please, please please .
4 This exam has 6 questions wor
Problem # 4 (1+4+4 = 9 points)
A second order system has transfer function
G(s) =
Let's instead solve this for a first
order system:
n2
s2 + 2n s + n2
The steady state response for the input
u(t) = cos(t)
is plotted below.
2.5
Output
Input
2
1.5
1
0.5
0
0
ME 132
Fall 2011
Midterm 3
NAME:
ID # :
#1
#2
#3
#4
#5
TOTAL
26
18
14
9
3
70
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 This exam has 5 questions worth 70 points.
4 There is plenty of time, so please write you
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