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 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 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 e-vals, 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 #5
Prof. MCloskey
nonhomogeneous ODE
finite dimensional linear systems, impulse response, frequency response, transfer
function
Bellman-Gronwall lemma
stability
The non-homogeneous case x = Ax + f
Suppose f
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 #2
Prof. MCloskey
vector norms, matrix norms, induced norms
Schurs theorem
Range space, null space, rank
A-invariant subspaces
Jordan Canonical Form (JCF)
Cayley-Hamilton Theorem
Textbook reading: see cor
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 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 c2
Give necessary and sufficient conditions involving t
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. Thus, for any positive definite Q R22 ,
there exists a u
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 = lyap(A,Q);
cir = exp(1i*[0:360]*pi/180);
N = length(c
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 using the Bellman-Gronwall lemma
Time-varying 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
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
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
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];
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
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
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
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
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 real symmetric matrices, unitary matrices, and skewsymme
MAE270A Linear Dynamic Systems
Lecture Notes #5
Prof. MCloskey
nonhomogeneous ODE
finite dimensional linear systems, impulse response, frequency response, transfer
function
Bellman-Gronwall lemma
stability
The non-homogeneous case x = Ax + f
Suppose f
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)
where A Cnn and C C1n (the single output case is covered h
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 Angeles
www.flexible.seas.ucla.edu
Serial Flexure Elements
S
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.