MA270A Homework Problem Set 4 Solution
Prof. MCloskey
1. Consider
A=
0 1
0 0
= eAt =
1 t
.
0 1
Both eigenvalues are 0, yet the ODE has unbounded solutions because of the t term in the
matrix exponential so Re(A) 0 is not sucient for stability.
2. Suppose
MA270A Homework Problem Set 2
Prof. MCloskey
Due Date: October 24, 2014
1. A 1state singleinput/singleoutput system is given by the equations
x = ax + bu
y = cx,
where cfw_a, b, c are scalars. The impulse response of this scalar system is given by the
MA270A Homework Set 5
Prof. MCloskey
Due: Monday, Nov 24, 2014
1. We have seen in class that the stability of constant linear dierential equation comes down
to computing the real parts of the eigenvalues of the A matrix. Consider the linear timeperiodic s
MA270A Homework #5 Solution
Prof. MCloskey
1. Dening the state variables x1 := x and x2 := x leads to the following rst order dierential
equation
x1
0
1
x1
=
.
x2
(1 + 0.1 cos(2t) 0.01 x2
A(t)
Any other choice of state variables is also valid as long as t
MAE270A Homework Problem Set 1
Prof. MCloskey
Due Friday, Oct 17, 2014
1. Find two orthonormal vectors that span the same space as the following vectors:
2
1
v1 = 3 , v2 = 4 .
1
0
First show that cfw_v1 , v2 are linearly independent (see the note bel
MA270A Homework 2 Solution
Prof. MCloskey
1. Change coordinates in which the new state vector is called z, where x = T z, det T = 0, to
get a dierent representation of the system equations:
z = T 1 AT z + T 1 Bu,
y = CT z.
Since A has distinct evals, it
MAE270A Homework 1 Solution
Prof. MCloskey
1. With v1 and v2 given in the problem statement, dene V = [v1 v2 ] and note
V V =
14 14
.
14 17
Since det(V V ) = 0 then v1 and v2 are linearly independent.
Given two linearly independent vectors v1 and v2 , an
MA270A Homework 7 Solution
Prof. MCloskey
1. (a) System 1:
x = x + u1
y1 = x
Compute the controllability and observability matrices,
C = 1, so Sys 1 is controllable.
O = 1, so Sys 1 is observable.
The transfer function for System 1 is
H1 (s) = 1 (s + )1 1
MAE270A Linear Dynamic Systems
Lecture Notes #2
Prof. MCloskey
vector norms, matrix norms, induced norms
Schurs theorem
Range space, null space, rank
Ainvariant subspaces
Jordan Canonical Form (JCF)
CayleyHamilton Theorem
Textbook reading: see cor
MAE270A Linear Dynamic Systems
Lecture Notes #8
Prof. MCloskey
quadratic forms
positive definite matrices
matrix congruence
level sets of positive definite quadratic forms
Lyapunov Funtions
The definitions of stability and Lyapunov functions are disc
MAE270A Linear Dynamic Systems
Lecture Notes #3
Prof. MCloskey
the Singular Value Decomposition (SVD)
applications of the SVD
Textbook reading: SVD material is in the course text appendix
The Singular Value Decomposition
Theorem: Given A 2 Cmn of rank r
MAE270A Linear Dynamic Systems
Lecture Notes #5
Prof. MCloskey
nonhomogeneous ODE
finite dimensional linear systems, impulse response, frequency response, transfer
function
BellmanGronwall lemma
stability
The nonhomogeneous case x = Ax + f
Suppose f
MAE270A Linear Dynamic Systems
Lecture Notes #6
Prof. MCloskey
timevarying linear differential equations
timeperiodic linear differential equations
timevarying linear systems
asymptotic results using the BellmanGronwall lemma
Timevarying Linear D
MAE270A Linear Dynamic Systems
Lecture Notes #4
Prof. MCloskey
SVD applied to the least squares, least norm problems
pseudo inverses, left and right inverses
homogeneous linear differential equations
properties of the matrix exponential
Textbook readi
MAE270A Linear Dynamic Systems
Lecture Notes #1
Prof. MCloskey
Some notation (not comprehensive )
Eigenvalue/vector review
A theorem on matrices with distinct evals
Definitions of hermitian and real symmetric matrices, unitary matrices, and skewsymme
MAE270A Linear Dynamic Systems
Lecture Notes #7
Prof. MCloskey
phase plane and classification of singular (equilibrium) points
geometric interpretation of eAt eAs = eA(t+s) and det eAt
The Phase Plane
Consider a two state system
x 1
a11 a12 x1
, whe
Solution
Prof. MCloskey
1. Represent A1 as
1 0
0 1
A1 =
+
.
0 1
0 0
 cfw_z  cfw_z
I
M
Since (I)M = M (I) we can compute eA1 t = e(I+M )t = eIt eM t ,
eIt = et I
eM t = I +
=I+
1
=
0
1
=
0
1
1
1
M t + M 2 t2 + M 3 t3 +
1!
2!
3!
1
k
M t since M = 0
MA270A Homework Problem Set 3
Prof. MCloskey
Due Date: October 19, 2016
1. The figure below represents the set of vectors produced when multiplying the set of unit
vectors by the matrix
1 2
A=
1 0
2.5
2
1.5
1
x2
0.5
0
0.5
1
1.5
2
2.5
2.5
2
1.5
1
0.5
0
x1
MA270A Homework 2 Solution
Prof. MCloskey
1. Consider A Cnn . Show
(a) Show that the product of the eigenvalues equals the determinant. First show that the
determinant of an upper (or lower) triangular matrix is the product of its diagonal elements (this
Solution
Prof. MCloskey
1. Here is the Matlab code to produce the ellipsoids. Just uncomment the A matrix you are
interested in (or create your own).
%
clear
close all
%
A
%
%
%
A
=
A
A
A
= [1 2;1 0];
[1 1; 2 1];
= [1 1;1 2];
= [1 1;1 2];
= [1 1;1 1];
MAE270A Homework 1 Solution
Prof. MCloskey
1. To show A is a normal matrix, just compute A A and AA and show that these products are
equal,
2 1 1
A A = AA = 1 2 1 .
1 1 2
Note that
Av = 2v,
so v is an eigenvector corresponding to eigenvalue 2.
The followi
MA270A Homework Problem Set 2
Prof. MCloskey
Due Date: October 12, 2016
1. Consider A Cnn . Show
(a) The product of the eigenvalues equals the determinant (hint: use Schurs theorem).
(b) The sum of the eigenvalues equals the sum of the diagonal elements o
MA270A Homework Problem Set 4
Prof. MCloskey
Oct 27, 2016
1. Compute the matrix exponentials of the following matrices,
1 0
0
1
0 1
A1 =
= I2 +
, A2 = 0 1 = I3 + 0
0
0 0
0 0
0
 cfw_z

M
1 0
0 1 .
0 0
cfw_z
M
Use the fact that each matrix can be re
MAE270A Homework Problem Set 1
Prof. MCloskey
Due Tuesday, Oct 4, 2016
1. Show that the following matrix is a normal matrix
1 1 0
A = 0 1 1 .
1 0 1
Next, construct a unitary matrix (lets call it U ) such that U AU is diagonal. Start the
process by first s