ECE 515 / ME 540 Solution 1
1. Problem 1
Current through a capacitor is the time derivative of charge CVc . Since the circuit connected
serially,
d
i = (CVc )
dt
Voltage drop at an inductor is the tim
ECE 515 / ME 540 Solution 2
1. Problem 1
Let our candidate inverse be
X Y
A =
Z W
0
then
X + BZ Y + BW
AA =
Z
W
0
For AA to be identity, Z = 0 and W = I can be directly concluded, then the upper row
b
ECE515: Control System Theory and Design
Fall 2012
Homework 7 Solution
Solutions.
Author: Navid Aghasadeghi
Solutions:
1. (a)
u pT Bu
ij
=
X
ui pTjk Bkl ul =
X
k,l
pTjk Bkl il =
X
k,l
ptjk Bki =
k
X
T
ECE 515 / ME 540 Solution 3
1. Problem 1
(a) The characteristic polynomial of At is (s at)2 + (bt)2 = s2 2ats + (at)2 + (bt)2 = 0,
hence complex eigenvalues are:
p
= at (at)2 (at)2 + (bt)2 ) = at bti
ECE 515/ME 540, Spring 17
PROBLEM SET 3
Due Thursday, Feb 16
Reading: Class Notes, Sections 3.13.5.
Problems:
1. Consider the following matrix, whose exponential was computed in class: A =
a b
, a, b
ECE 515/ME 540, Spring 17
PROBLEM SET 1
Due Thursday, Feb 2
Reading: Class Notes, Sections 1.1, 1.4, 1.5, 2.12.6.
Problems:
1. Consider the electrical circuit discussed in class, but now suppose that
ECE 515
Assignment # 1
Issued: January 18
Due: January 28, 2010
Reading Assignment:
Lecture Notes Ch. 1 and 2. See also Brogan, 3.13.4, 5.15.8, or Chen 2.12.8, 3.33.5.
Problems:
1. Based on Sections 1
ECE 515 homework 5 solution
Author: Xiaobin Gao
Problem 1; Solution:
1 = 2
cfw_ 2 = 31 22 +
= 1 + 2
Taking Laplace Transform on both sides and assume zero initial conditions (since we only want to
ECE 515 / ME 540 Solution 4
1. Problem 1
(a) Writing down the differential equation,
(
x 1 = (1 + cos t)x1
x 2 = (2 + cos t)x2
These are two disjoint first order differential equations, which we can s
ECE 515: Control System Theory and Design
Fall 2015
Problem Set 6
Author: Xiaobin Gao
Due Date: 11/19
1. Problem 1
Since the cost function is given by
Z
L(x, x)
=
1
(x2 + x 2 )dt
0
The Lagrangian L(x
Problem Set # 3
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Tue Oct 6 2015
Reading
Problem Set # 4
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Thu Oct 15 2015
Readin
Problem Set # 6
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Thu Nov 19 2015
Readin
ECE 515, Fall 2012
PROBLEM SET 7
Due Thursday, Nov 29
Reading: Class Notes, Chapter 10.
Note: Feel free to use Matlab for numerical computations.
Problems:
1. This problem is a quick (and easy) refres
ECE 515/ME 540, Spring 17
PROBLEM SET 5
Due Thursday, Mar 2
Reading: Class Notes, Sections 1.21.4 and 2.82.10
Problems:
1. Let X be an inner product space with elements x, y X and norm k k induced by
ECE 515/ME 540, Spring 17
PROBLEM SET 4
Due Thursday, Feb 23
Reading: Class Notes, Sections 3.63.8
Problems:
1. Compute the state transition matrix (t, ) of the linear time-varying systems x = A(t)x w
ECE 515 / ME 540 Solution 5
1. Problem 1
(a) If x is orthogonal to y, hx, yi = hy, xi = 0, so
kx + yk2 = hx + y, x + yi
= hx, x + yi + hy, x + yi
= hx, xi + hx, yi + hy, xi + hy, yi
= hx, xi + 0 + 0 +
Problem Set # 1
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Tue Sep 15 2015
Readin
Problem Set # 5
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Tue Nov 10 2015
Readin
ECE 515, Fall 2007
Problem Set #6 Solution
Solutions:
1. (a) Read off the second order realization in controllable canonical form (this concept is from
HW1/Chapter 1):
0
1
0
x =
x+
u
2 3
1
y= 1 1 x
Linear Systems Notes
1
1
Lecture 1
Introduction Consider the following linear system
x = Ax + Bx
(1)
where x Rn , which describes the dynamics of x, given by two components: Ax the position of
x (A L(
Controllable subspace decomposition
M.-A. Belabbas
1. We deal here with linear time-invariant systems
x = Ax + Bu
where A Rnn and B Rn . The restriction that B be a column vector is not an essential
o
ECE 515, Spring 2012
PROBLEM SET 10
Due Thursday, Apr 19
Solutions:
1.
a) System state space representation is
0 1
0
x =
x+
u.
0 0
1
Using HJB,
h
@V
@V x2 i
4
2
= min x1 + u +
u
u
@t
@x
h
@V
@V i
= mi
ECE 515, Spring 2012
PROBLEM SET 10
Due Thursday, Apr 19
Reading: Class Notes, Sections 10.1, 10.2; my optimal control lecture notes (posted on the class website),
Section 5.1.
Problems:
1. Consider t
Problem Set # 5
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, [email protected], CSL 166
Teaching Assistant:Xiaobin Gao, [email protected]
Due date: Tue Nov 10 2015
Readin
FA16-ECE515:Homework 7 Solutions
TA: Ivan Abraham
November 14, 2016
Not all steps are worked out. Effort has been made to sure there are no errors; but if you find any, please
e-mail me at [email protected]
FA16-ECE515:Homework 5-6 Solutions
TA: Ivan Abraham
November 6, 2016
Not all steps are worked out. Effort has been made to sure there are no errors; but if you find any, please
e-mail me at [email protected]
FA16-ECE515:Homework 2 Solutions
TA: Ivan Abraham
September 15, 2016
Not all steps are worked out. Effort has been made to sure there are no errors; but if you find any, please
e-mail me at [email protected]
FA16-ECE515:Homework 3 Solutions
TA: Ivan Abraham
September 22, 2016
Not all steps are worked out. Effort has been made to sure there are no errors; but if you find any, please
e-mail me at [email protected]
FA16-ECE515:Homework 1 Solutions
TA: Ivan Abraham
September 7, 2016
1
Ivan Abraham
1
1.1
Homework 1 Solutions
Problem 1
Part (a) - 10 points
We will work with the nodal analysis method and generalized
FA16-ECE515:Homework 4 Solutions
TA: Ivan Abraham
October 3, 2016
Not all steps are worked out. Effort has been made to sure there are no errors; but if you find any, please
e-mail me at [email protected]