EE 581 Optimal Control
Final Exam
Tuesday, May 6
Department of Electrical Engineering
Pennsylvania State University
Spring 2008
1. Linear Quadratic Gaussian (LQG) Control. Consider the linear stochastic system
x(t + 1) = A(t)x(t) + B(t)u(t) + w(t),
y (t)
EE 581 Optimal Control
Homework 9
Due on Tuesday, April 28
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. A Variable-Terminal-Time Control Problem. Consider the problem of minimizing
tf
J (tf , u) = tf +
|u(t)| dt
0
fuel
EE 581 Optimal Control
Homework 8
Due on Tuesday, April 21
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Quadratic Optimization via Semidenite Programming. Let A0 Rnn , A Rnn ,
b0 Rn , b Rn , c0 R, and c R be given. Con
EE 581 Optimal Control
Homework 7
Due on Tuesday, April 7
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Innite-Horizon LQR Revisited. Consider the problem of driving the linear system
x1 (t) = x1 (t) + x2 (t) + 2u1 (t)
EE 581 Optimal Control
Homework 6
Due on Thursday, March 19
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Integral Operator. Given a C n [t0 , t1 ], let g(t, s) = (t)T (s) for t, s [t0 , t1 ]. Dene
L : L2 [t0 , t1 ] L2
EE 581 Optimal Control
Homework 5
Due on Thursday, March 5
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Jensens Inequality. Let X be a vector space over the real eld. Suppose f : X R is convex
and continuously Frchet-d
EE 581 Optimal Control
Homework 4
Due on Thursday, February 26
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Frchet Dierentiation on Function Space. Let t0 , t1 R be such that t0 < t1 . Let
e
g : [t0 , t1 ] Rn Rm be suc
EE 581 Optimal Control
Homework 3
Due on Tuesday, February 10
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Dierential Riccati Equation.
(a) Let A Rnn , B Rnm , L Rnn , Q Rnn , R Rmm , and S Rmn . Assume
Q 0, R > 0, and
EE 581 Optimal Control
Homework 2
Due on Thursday, January 29
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Some optimization problems have special structures that make them easy to solve. The orthogonal projection prob
EE 581 Optimal Control
Homework 1
Due on Thursday, January 22
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Let A Rnn , B Rnm, x0 Rn , xf Rn , and t0 R be given.
(a) Minimum Time Control. Consider the problem of minimiz
EE 581 Optimal Control
Final Exam
Tuesday, May 5, 2:30 PM 4:20 PM
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Consider the problem of determining the shortest path between two points (t0 , x0 ) and (t1 , x1 )
in R2 ,
EE 581 Optimal Control
Exam 2
Due on Tuesday, March 31 at 11:15 AM
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Minimum Terminal Cost Problems. Let A(t) Rnn and B(t) Rnn be piecewise
continuous in t [t0 , t1 ]. Let Rm
EE 581 Optimal Control
Exam 1
Due on Thursday, February 19 at 11:15 AM
Department of Electrical Engineering
Pennsylvania State University
Spring 2009
1. Linear Quadratic Tracking [Geering, Exercise 2.9.4 (with typos xed)]. Let A(t) Rnn ,
B(t) Rnn , L(t) R
EE 581 Optimal Control
Exam 2
Due on Tuesday, April 1 by 11:15 AM
Department of Electrical Engineering
Pennsylvania State University
Spring 2008
1. Convex Optimization. Prove the following facts:
(a) 4 pts. If f : Rn R is strongly convex, then there exist
EE 581 Optimal Control
Exam 1
Due on Thursday, February 21 by 11:15 AM
Department of Electrical Engineering
Pennsylvania State University
Spring 2008
1. Discrete-Time LQR. Consider the discrete-time linear system
x(t + 1) = A(t)x(t) + B(t)u(t),
t = t0 , t