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
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
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 act
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
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
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 po
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]
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 val
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 a
ME 132
Homework # 6
Issued: October 10, 2011
Due: October 17, 2011
1 Second-Order Systems
In this problem we will study the effect of damping and natural frequency on step responses
of second-order sy
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
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
= dia
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
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 constan
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
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 st
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 controll
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
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. T
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 n
ME 132
Homework # 1
Issued: January 20, 2017
Due: January 27, 2017
Suggested Readings:
1 Chapter 1, Astrom & Murray. This is a great book, and is posted on b-courses.
Problems to be turned in:
1 Struc
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
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
ME 132
Homework # 5
Issued: February 16, 2017
Due: February 24, 2017
1 Frequency Response
Consider a linear time-invariant system modeled by the input-output differential equation
y(t) + a1 y(t)
+ a2
ME 132
State-space Methods for Control of LTI Plants
1 Introduction
2 Controllable Canonical Form
3 Observable Canonical Form
4 State Feedback
5 Pole Placement by State Feedback
6 Observers
7 Observer
ME 132
General LTI Systems Transfer Function Methods
1 General LTI ODE Models
2 Transfer Functions
3 Poles and Zeros
4 The Algebra of Transfer Functions
5 The Total Response
6 The Free Response
7 Stab
ME 132
Classical Control System Design: Time Domain Methods
1 Preliminaries
2 Design Objectives
3 Proportional Control
4 Integral Control
5 Actuator Saturation
6 Anti-windup Strategies to be done
7 Ca
ME 132
Linear Algebra
1 Matrices and Vectors
2 Linear Simultaneous Equations
3 Transposes and Adjoints
4 Determinants and Inverses
5 Eigenvalues and Eigenvectors
6 Simple Case of the Jordan Form
7 The
ME 132
Static Systems
1 Static Systems
2 A Simple Cruise Control System for a Car
3 Static Model
4 Open-loop Control
5 Closed-loop Control
6 Sensitivity Analysis
7 Problems
page 0 version January 27,