EE221A Linear System Theory
Problem Set 3
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 9/20; Due 10/2
Problem 1: Continuity and Linearit
EE221A Linear System Theory
Problem Set 1
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 8/30; Due 9/11
Problem 1: Functions. Consider f :
EE221A Linear System Theory
Problem Set 1
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 8/30; Due 9/11
Problem 1: Functions. Consider f :
EE 221A Problem Set 8 - Solutions
Problem 1. The closed loop system is
0
1
0
0
0
1 x + 0 (v k1 k2 k3 x)
x= 0
3 2 1
1
0
1
0
0
x + 0 v.
0
0
1
=
3 k1 2 k2 1 k3
1
(1)
Its characteristic polynomial is s3
EE 221A Problem Set 7 - Solutions
Problem 1. (a) For internal stability, we simply need the state to be bounded for all t t0 . This implies that
eJt must be bounded, where J is the Jordan matrix of A.
EE 221A Problem Set 6 - Solutions
Problem 1. The optimal control is given by u(t) = B (t) X (t)X (t)1 x(t), where X and X solves
d X (t)
0
=
1
dt X (t)
1
0
X (t)
X (t)
with X (1) = 1 and X (1) = .
The
EE 221A Problem Set 5 - Solutions
Problem 1. We can determine the Jordan matrix to be
1 1
0 1
1 1
0 1
J =
1
2
0
0
1
2
0
,
0
1
2
where J = T AT 1 . Hence,
1
1
cos e
0
e sin e1
cos e1
cos(eJ ) =
cos e
EE 221A Problem Set 4 - Solutions
Problem 1. (a) To show that this is a dynamical system we have to identify all the ingredients: The input space U is
as specied in the problem. The state space is R.
EE 221A Problem Set 3 - Solutions
Problem 1. Let A : U V be a linear function. For any x, y U , we have
Ax Ay = A(x y ) A
i
xy .
(1)
Hence, A is Lipschitz continuous with a Lipschitz constant A i . Th
EE 221A Problem Set 2 - Solutions
Problem 1. If b R(A), then there are no solutions. Hence S = .
/
If b R(A), then there exists at least one (particular) solution x such that Ax = b. Then for any basi
EE 221A Problem Set 1 - Solutions
Problem 1. It is a function; matrix multiplication is well dened.
Not injective; easy to nd a counterexample where f (x1 ) = f (x2 )
x1 = x2 .
Not surjective; suppose
EE221A Linear System Theory
Problem Set 8
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 11/24; Due 12/7
Problem 1: Controllable canonical
EE221A Linear System Theory
Problem Set 7
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 11/13; Due 11/21
Problem 1: Characterization of I
EE221A Linear System Theory
Problem Set 6
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 11/4; Due 11/13
Problem 1: Linear Quadratic Regul
EE221A Linear System Theory
Problem Set 5
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 11/25; Due 11/1
Problem 1.
A has characteristic p
EE221A Linear System Theory
Problem Set 4
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 10/2; Due 10/11
Problem 1: Dynamical systems, tim
EE221A Linear System Theory
Problem Set 2
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2012
Issued 9/12; Due 9/20
All answers must be justied.
Probl
EE 221A Practice Midterm
10/12/12
Problem 1.
Consider the solution (t, t0 , x0 ) at a given t of x = f (x), x(t0 ) = x0 as a mapping : Rn Rn dened by
(x0 ) = (t, t0 , x0 ). We assume that the dynamica
EE221A Linear System Theory
http:/inst.eecs.berkeley.edu/ee221a/
Course Outline
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley
Fall
EE 221A: Linear System Theory
U NIVERSITY OF C ALIFORNIA , B ERKELEY
Fall 2012
Jerry Ding
Alternative Derivation of Linear Quadratic Regulator
The purpose of this short note is to provide an alternati
EE221A Linear System Theory
Midterm Test
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2011
10/13/11, 9.30-11.00am
Your answers MUST BE supported by
EE221A Linear System Theory
Midterm Test
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2010
10/19/10, 9.30-11.00am
Your answers MUST BE supported by