MATHEMATICAL TRIPOS: PART II
Lent Term 2012
OPTIMIZATION AND CONTROL
Richard Weber
Example Sheet 1
1. Suppose that the matrix Mk is of dimension nk nk+1 , k cfw_1, . . . , h. We wish to compute
the product M1 M2 Mh . Notice that the order of multiplicatio
Exam August 25, 2010 in SF2852 Optimal Control.
Examiner: Ulf J
onsson, tel. 790 84 50.
Allowed books: The formula sheet and mathematics handbook.
Solution methods: All solutions and conclusions should be carefully motivated.
Note! Your personal number mu
Prof. L. Guzzella
Prof. R. DAndrea
Midterm Examination
November 12th, 2008
Dynamic Programming & Optimal Control
(151-0563-00)
Prof. R. DAndrea
Solutions
Exam Duration:
150 minutes
Number of Problems:
4 (25% each)
Permitted aids:
Textbook Dynamic Programm
Tentamen i SF2872 Optimal Control
Thursday August 25 2011 14.0019.00
Answers and solution sketches
1.
(a) Since, tf is fixed the problem simplifies to
(
Z
x = u, x(0) = x0
1 tf 2
u dt subj. to
min
2
u()
x(tf ) = 0
0
Let H(t, x, u, ) = 12 u2 + u. Pointwise
ECE 553, Spring 2008
Problem Set #3
Issued: February 28th, 2008
Due: March 10th, 2008
Primary Reading: Lecture notes and Handout #1 & #2 (Chapter 3, 4 of Sage and Chelsea).
Homework is due at the beginning of class on the due day!
Problems:
1. A different
Exam August 25 2011 in SF2852 Optimal Control.
Examiner: Per Enqvist, tel. 790 62 98.
Allowed books: The formula sheet and mathematics handbook.
Solution methods: All conclusions should be properly motivated.
Note! Your personal number must be stated on t
Quiz 1
October 24th, 2012
Dynamic Programming & Optimal Control
(151-0563-00)
Angela Schoellig
Solutions
Duration:
45 minutes
Number of Problems:
2
Permitted Aids:
None.
Use only the prepared sheets for your solutions.
Quiz 1 Dynamic Programming & Optimal
Exam in SF2852 Optimal Control
May 27, 2010
Answers and solution sketches
1.
(a) The optimal control problem is defined in terms of the following matrices
1 0
0 1
0
x
Q=
, R = 1, A =
, B=
, x(0) = 10
x20
0 0
0 0
1
(b) We have Q = C T C with C = 1 0 .
Formula Sheet for Optimal Control
Division of Optimization and Systems Theory
Royal Institute of Technology
10044 Stockholm, Sweden
2003
April 20, 2011
1
Dynamic Programming
1.1
Discrete Dynamic Programming
General multistage decision problem
N 1
min (xN
ECE 553, Spring 2008
Problem Set #2
Issued: February 16th, 2008
Due: February 25th, 2008
Primary Reading: Handout 1 & 2 and lecture notes.
Homework is due at the beginning of class on the due day!
Problems:
1. A simple target-set control problem (10 point
151-0563-01
Dynamic Programming and Optimal Control
(Fall 2012)
Programming Exercise #1 Topic: Deterministic Systems and the Shortest Path Problem
Issued: Oct 24, 2012
Due: Nov 07, 2012
Nico Huebel (nhuebel@ethz.ch) , 24. Oktober 2012
Find a shortest path
Exam August 22 2012 in SF2852 Optimal Control.
Examiner: Per Enqvist, tel. 790 62 98.
Allowed books: The formula sheet and mathematics handbook.
Solution methods: All conclusions should be properly motivated.
Note! Your personal number must be stated on t
Exam in SF2852 Optimal Control
Wednesday May 30 2012 14.0019.00
Answers and solution sketches
1.
J (t, x)
.
x
(b) FALSE. The optimal control exists only for reachable xf from x0 .
(a) TRUE. (t) =
(c) FALSE. Assume x and u is an optimal solution to the inf
Final Recitation Class
Dec 12th, 2012
Dynamic Programming & Optimal Control
(151-0563-00)
Examples
Number of Examples:
7
Mohanarajah G.
Final Recitation Class Dynamic Programming & Optimal Control
Page 1
Example 1
A burglar broke into a house and found N
Exam May 27, 2010 in SF2852 Optimal Control.
Examiner: Ulf J
onsson, tel. 790 84 50.
Allowed books: The formula sheet and mathematics handbook.
Solution methods: All solutions and conclusions should be carefully motivated.
Note! Your personal number must
ECE 553, Spring 2008
Problem Set #4
Issued: March 24th, 2008
Due: March 31st, 2008
Primary Reading: Lecture notes and Handouts.
Homework is due at the beginning of class on the due day!
Problems:
1. Hamilton-Jacobi and Riccati equations. (10 points)
Consi
Final Exam
January 28th, 2010
Dynamic Programming & Optimal Control
(151-0563-00)
Prof. R. DAndrea
Solutions
Exam Duration:
150 minutes
Number of Problems:
4
Permitted aids:
One A4 sheet of paper.
Use only the provided sheets for your solutions.
Page 2
Fi
Tripos Questions in Optimization and Control
20
00314 In a television game show a contestant is successively asked questions Q1 , . . . , Q9 .
After correctly answering Qi and hearing Qi+1 she has the option of either going home with
2i pounds or attempti
ECE 553, Spring 2008
Problem Set #7
Issued: April 23th, 2008
Due: April 30th, 2008
Primary Reading: Lecture notes LQG and Kalman filters.
Homework is due at the beginning of class on the due day!
Problems:
1. In problem 6 of homework 5, replace the determ
Quiz 1
November 2nd, 2011
Dynamic Programming & Optimal Control
(151-0563-00)
Prof. R. DAndrea
Solutions
Duration:
45 minutes
Number of Problems:
2
Permitted Aids:
None.
Use only the prepared sheets for your solutions.
Quiz 1 Dynamic Programming & Optimal
ECE 553, Spring 2008
Problem Set #7 Solution
Posted: May 2nd, 2008
Solutions:
1. The optimal controller is still the one given in the solution to the Problem 6 in Homework
#5:
u (x, t) = p(t)x k(t), t 0.
The minimum expected cost is:
1
Z
p(t) dt + p(0) +
Exam in SF2852 Optimal Control
Wednesday May 30 2012 14.0019.00
Answers and solution sketches
1.
False The stability of the model predictive control algorithm can be guaranteed
when a few assumptions are made: f0 (0, 0) is a strictly positive definite fu
ECE 553, Spring 2008
Problem Set #1
Issued: February 5, 2008
Due: February 13th, 2008
Primary Reading: Lecture notes, and review analysis, nonlinear optimization, and differential
equations.
Homework is due at the beginning of class on the due day!
Proble
ECE 553, Spring 2008
Problem Set #6
Issued: April 15th, 2008
Due: April 23th, 2008
Primary Reading: Lecture notes on dynamical programming and LQR control.
Homework is due at the beginning of class on the due day!
Problems:
1. Singular arc (10 points)
Giv
Dynamic Programming and Optimal Control
Recitation Session
Mohanarajah Gajamohan
Institute for Dynamic Systems and Control
ETH Zurich, Switzerland
gajan@ethz.ch
December 12, 2012
Plan for Today
1. The Dynamic Programming Algorithm (DPA)
2. Deterministic S
Prof. L. Guzzella
Prof. R. DAndrea
Final Examination
January 28th, 2009
Dynamic Programming & Optimal Control
(151-0563-01)
Prof. R. DAndrea
Solutions
Exam Duration:
150 minutes
Number of Problems:
4 (25% each)
Permitted Aids:
Textbook Dynamic Programming
Quiz 2
December 12th, 2012
Dynamic Programming & Optimal Control
(151-0563-00)
Angela Schoellig
Solutions
Duration:
45 minutes
Number of Problems:
3
Permitted Aids:
Provided cheat sheet on page 1.
Use only the prepared sheets for your solutions.
Quiz 2 Dy
Exam in SF2852 Optimal Control
June 2, 2009
Answers and solution sketches
1.
(a) The optimal control problem can be written on standard form
(
Z
x(t)
= Ax(t) + Bu(t),
T
T
min
(x(t) Qx(t) + u(t) Ru(t)dt subj. to
x(0) = x0 ,
0
with the following matrices
Tripos Questions in Optimization and Control
1
060229 A policy is to be chosen to maximize
"
F (, x) = E
X
t=0
#
r(xt , ut ) x0 = x ,
t
where 0 < 1. Assuming r 0, prove that is optimal if F (, x) satisfies the optimality
equation.
An investor receives at