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 find
the relationship between and ), we have
1 () = 2
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
Corresponding eigenvectors v are, solving (I A)v = 0,
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 solve in various
ways. Using separation of variables for
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 + hy, yi
= kxk2 + kyk2
(b) Similarly, we can decompose a
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 R.
b a
a) Re-derive the expression for eAt by diagonali
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 with the
following matrices:
1 2
1 + cos t
0
1 et+ 2 t
a
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 the inner product.
a) Let x be orthogonal to y. Prove t
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
becomes X + 0 = I and Y + B = 0 X = I and Y = B.
It can
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 time derivative of flux Li. Using KVL,
u = Vc + iR +
Let x
ECE 515/ME 540, Spring 17
PROBLEM SET 2
Due Thursday, Feb 9
Reading: Class Notes, Sections 2.72.9.
Problems:
I B
1. Find the inverse of the matrix A =
where B is a matrix.
0 I
2. Compute limn An for A =
3. Is the set cfw_I, A,
A2
7/5 1/5
.
1 1/2
linearly
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 its characteristics R, L and C
vary with time. Starting
Controllability and Stabilizability
Point-to-Point Control and Controllability
The Kalman Matrix and its Relevance
Controllability Canonical Form (SI) and Normal Form (MI)
Uncontrollable Modes and the Hautus-Test
Stabilizability
Open-loop and State-
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
Bik
pkj = B T p ij ,
k
where ij = 1 if i = j and is 0
ROBUSTNESS AND SENSITIVITY
Sadegh Bolouki
Lecture slides for ECE 515
University of Illinois, Urbana-Champaign
Fall 2016
S. Bolouki (UIUC)
1 / 17
Sensitivity
Sensitivity
S. Bolouki (UIUC)
2 / 17
Sensitivity
Sensitivity
One degree-of-freedom
control configu
OBSERVABILITY, DUALITY, AND MINIMALITY
Sadegh Bolouki
Lecture slides for ECE 515
University of Illinois, Urbana-Champaign
Fall 2016
S. Bolouki (UIUC)
1 / 19
Overview
1
An Overview
2
Definition
3
LTI Case
4
General LTV Case
5
Duality
6
Kalman Canonical For
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(Rn , Rn ), a linear mapping), and Bu, where u Rm is cal
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
Readings: Class notes Chapters 2 and 3. Linear algebra review
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
Readings: Class notes Chapters 10-11 Do the following problem
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
one, it does make the notation a tad simpler though.
2.
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
= min x41 + u2 +
x2 +
u
u
@x1
@x2
with
V (x(t1 ), t1 ) = m(
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 the optimal control problem
x
= u,
V =
!
t1
t0
"
#
x4 (
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
Readings: Class notes Chapters 6-8 Do the following problems.
ECE 515, Fall 2007
Problem Set #8 Solution
Solutions:
1. Tracking
Let e = y r. Then e = y r = x r = x + u r.
Want e = e,
> 0 x + u r = e u = x + r e = y + r (y r).
Thus the control is u = (1 + )y + r + r for some > 0.
Simulation should show that y(t) app
ECE 515, Fall 2007
Problem Set #5 Solution
Solutions:
1. Stability Margin (10 points)
Let B = A + I. We have:
AT P + P A + 2I = Q
(A + I)T P + P (A + I) = Q B T P + P B = Q.
If is an eigenvalue of A then Av = v (A + I)v = ( + )v ( + ) is an eigenvalue
of
ECE 515, Fall 2007
Problem Set #4 Solution
1. The Transition Matrix
(Hint: Recall that the solution to the linear equation x = Ax + Bu is
Z t
A(tt0 )
eA(t ) Bu( )d.
x(t) = e
x(t0 ) +
(1)
t0
This formula is extremely important for linear systems and should
ECE 515, Fall 2005
Problem Set #3 Solution
1. Linear vs. Nonlinear Maps (10 points)
(a) The map f : X 7 AX + BXC is linear:
f (X + Y ) = A(X + Y ) + B(X + Y )C
= (AX + BXC) + (AY + BY C)
= f (X) + f (Y ).
(b) The map f : X 7 AXA B is not linear if B 6= 0:
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
By the rank test, the realization is controllable but n
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
Readings: Class notes Chapters 6-8 Do the following problems.