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 exponenti
MA270A Homework Problem Set 2
Prof. MCloskey
Due Date: October 24, 2014
1. A 1-state single-input/single-output system is given by the equations
x = ax + bu
y = cx,
where cfw_a, b, c are scalars. The
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
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 eigenval
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 oth
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
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
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.
MAE270A Linear Dynamic Systems
Lecture Notes #5
Prof. MCloskey
nonhomogeneous ODE
finite dimensional linear systems, impulse response, frequency response, transfer
function
Bellman-Gronwall lemma
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 d
MAE270A Linear Dynamic Systems
Lecture Notes #2
Prof. MCloskey
vector norms, matrix norms, induced norms
Schurs theorem
Range space, null space, rank
A-invariant subspaces
Jordan Canonical Form (
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 Singul
MAE270A Homework #6
Prof. MCloskey
Due: Nov. 17, 2016
1. Let A be a Jordan block of dimension
0
.
.
A = .
.
.
.
0
n n,
1
.
.
0
1
.
.
.
.
0
0
.
.
.
.
.
.
0
0
0
.
.
0
1
cn .
.
.
.
.
and let
C = c1 c
MA270A Homework Set 5
Prof. MCloskey
Due: Thursday, Nov 12, 2016
1. Consider the time-invariant differential equation x = Ax where
0
1
A=
.
1 0.5
This differential equation is asymptotically stable. T
MA270A Solution
Prof. MCloskey
1. Here is my Matlab code for generating all of the figures. Just uncomment the Q you want to
use.
A = [0 1;-1 -0.5];
Q = eye(2,2);
% Q = [2,1;1,1];
% Q = [1,1;1,10];
P
MAE270A Linear Dynamic Systems
Lecture Notes #6
Prof. MCloskey
time-varying linear differential equations
time-periodic linear differential equations
time-varying linear systems
asymptotic results
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 equatio
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 th
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
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]
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) triangula
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
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
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 Pla
MAE270A Linear Dynamic Systems
Lecture Notes #1
Prof. MCloskey
Some notation (not comprehensive )
Eigenvalue/vector review
A theorem on matrices with distinct e-vals
Definitions of hermitian and r
MAE270A Linear Dynamic Systems
Lecture Notes #5
Prof. MCloskey
nonhomogeneous ODE
finite dimensional linear systems, impulse response, frequency response, transfer
function
Bellman-Gronwall lemma
MAE270A Linear Dynamic Systems
Lecture Notes #11
Prof. MCloskey
equivalent characterizations of observability
Observability Matrix
Consider the linear ODE with scalar output y:
x = Ax
y = Cx,
(1)
whe
MAE 294A
Compliant Mechanism Design
Lecture 9: Serial Elements and Hybrid Systems
Jonathan B. Hopkins, Ph.D.
Assistant Professor
Mechanical and Aerospace Engineering
University of California Los Angel
Complete Table of Shapes Displaced to Infinity
This table provides a complete list of
freedom spaces that mimic other
freedom spaces when displaced to infinity
in every direction.