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 Linearity. Show that any linear map between nite dimensional ve
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. This is clearly true for the subspaces of the state sp
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 solution to the above equation is
1 et1 + et+1
X (t)
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 e1
0
e1 sin e1
cos e1
cos e1
cos e2
0
0
e2 sin e2
cos e2
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. The output space Y can be as chosen to be the same as t
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 . Therefore it is continuous.
Problem 2. (a) Call the rst s
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 basis cfw_u1 , , uk
of N (A), the set cfw_u1 + x , , uk +
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 x = (x1 , x2 , x3 ) . Then f (x) = (x1 + x3 , 0, x2 +
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 form.
Consider the linear time invariant system with s
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 Internal (State Space) Stability for LTI systems.
(a) Sh
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 Regulator.
Consider the scalar control system (x(t) R):
x(t)
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 polynomial (s 1 )5 (s 2 )3 , it has four linearly indepe
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, time invariance.
(a) Suppose that the output of a system i
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.
Problem 1: Solutions to linear equations (this was part of P
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 : R3 R3 , dened
1
f (x) = Ax, A = 0
0
as
0
0
1
1
0 , x R
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 dynamical system admits a (global) unique solution and so does
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 2012
Lecture Information
Lectures: TuTh 9.30-11, 240 B
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 alternative derivation of the optimal controller for the linear
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 analysis, proof, or counterexample.
There are 6 questio
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 analysis, proof, or counterexample.
There are 5 questio