12
Lipschitz Continuity
Calculus required continuity, and continuity was supposed to require
the innitely little, but nobody could discover what the innitely
little might be. (Russell)
12.1 Introducti
c
Erik I. Verriest 2016
4.5
ECE 6550 - Chapter 4: State Space Realizations
69
Eigen Problem and Diagonalization
In this section, the eigen problem, i.e., the problem relating to eigenvalues and eigenv
Linear System Theory
Notes to accompany ECE 6550
c
Erik
I. Verriest
School of ECE
Georgia Institute of Technology
Atlanta, GA 30332-0250
August 20, 2016
Chapter 1
Introduction and Motivation
1.1
Overv
Chapter 7
Solutions of State Space Equations
In this chapter explicit solutions to the discrete and continuous time state equations are obtained for time invariant systems. As a direct application, th
Chapter 8
Quadratic Forms and Applications
In this chapter, we digress to introduce some more concepts from matrix theory, which have
plenty of applications in systems theory, optimization, and numeri
Chapter 4
State Space Realizations
This chapter discusses first the notion of a state space representation of a linear time invariant
system. Although this is a bit premature, as we will not precisely
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 9
Solution Problem 9.1:
i) The stationary solution (if it exists) satisfies
X = AX A + X + Q
or
X =
1
1
AX A +
Q.
1
1
(1)
1
A is Schur-Cohn
Con
Chapter 2
Mathematical Structures
2.1
Semigroups, Groups, Rings, Fields
I hope that everyone will agree on the premises that linear algebra is central to linear system
theory (ample motivation for thi
Solution T1.1
From the mod-4 multiplication table we have that 2 2 = 0. Hence 2 cannot have a multiplicative inverse, and therefore the set cfw_0, 1, 2, 3 with addition and multiplication mod 4
cannot
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Take home re-test (20 points to add to test 1)
Date Assigned: October 31, 2016.
Due (for section A) : Wed., November 2, 2016
Distance learning students return
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Solution Take Home Re-test.
Solution Problem TH1.1
i) In the absence of predators,
N1
.
N1 = rN1 1
K
The equilibria are N1 = 0 and N1 = K. In the first case t
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Solutions Test #2
Solution Problem 1:
i) Adapt the proof given for the time-invariant case as follows: If |u(t)| < B for all t > 0,
then
Z t
|y(t)| = h(t, )u(
Chapter 3
Linear Systems and Transforms
The notions of homogeneity, finite additivity, infinite additivity, continuity, and linearity are
introduced. For continuous time signals, we define the Laplace
Chapter 6
Minimal Realizations and the
Fundamental Realization Theorem
This chapter discusses another property of realizations: minimality, and derives algebraic
conditions for it. The main results ar
Chapter 9
Gramians and Energy Principles
From the explicit solution of the state space equations, one can obtain nice conditions for
observability and reachability of a system in any nonzero interval.
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 4
Date assigned: September 19, 2016
Reading Assignment:
Sections 4.3-4 of my Class Notes.
You may also look at Kailath: Chapter 2.
Problem 4.1:
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 10
Date Assigned:
November 21, 2016. Due:
(exceptionally) December 2, 2016
Problem 10.1: Reachability Paradox
Consider a linear time-invariant
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 8
Solution Problem 8.1
The eigen-decomposition of Q yields
0.0295
0.7360 0.6763
0.3774
U
5.3479
Q = 0.8243 0.4006 0.4001
0.5654 0.5457 0.6185
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 1
Date Assigned:
August 26, 2016
Problem 1.1
1
followed by a compensator of
Show that the pole zero cancelation design of the plant s1
s1
the f
GEORGIA INSTITUTE OF TECHNOLOGY
EE 6550
Solutions Problem Set No. 3
Solution Problem 3.1
Let iC (t) denote the current through the capacitor (from + to )
diL
+ R(iL + iC )
u1 = L
dt
u2 = R(iL iC ) vC
GEORGIA INSTITUTE OF TECHNOLOGY
EE 6550
Solutions Problem Set No. 2
Solution Problem 2.1:
This can be solved either over R or C, as long as it is done consistently (same field throughout).
One knows f
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
Problem Set No. 4
Date assigned: September 19, 2016
Solution Problem 4.1
N (D2 + 1) = cfw_y|
y + y = 0 = cfw_ cos t + sin t|, R.
Obviously, since there are 2 d
GEORGIA INSTITUTE OF TECHNOLOGY
ECE 6550
SOLUTIONS Problem Set No. 1
Problem 1.1
See the gure for the internal description. The state equations are
u
+
R
+
y
x1
R
x2
+
-2
-1
Figure 1: Internal descrip
Chapter 5
Properties of Realizations
This chapter is far from complete. On the other hand, this material is standard, and may
be found in most textbooks.
5.1
Observability Problem in Continuous Time
T
Table 15.1. Controllability and Observability Tests for LTI Systems
Controllability Stabilizability Observability Detectability
Denition Every initial state can be Asymptotically stable uncon- Every
Homework 2 Solutions ECE6550, Fall 2014
1
(k+1)
xk+1
=
eA(k+1) ) Bu( )d
eA(k+1)k) xk +
k
= eA xk +
eA(r) Bdruk
0
= eA xk +
eAs Bdsuk .
0
As such, we have
A = eA , B =
eAs Bds.
0
2
a
xk+1 = xk + (Axk +
Midterm: ECE6550
Magnus Egerstedt
Georgia Institute of Technology
Oct. 11, 2011. 3:05-4:25
Closed books, closed notes, closed calculator exam.
This exam gives a maximum of 60 points. (10 points/questi
Solutions to Midterm: ECE6550
Magnus Egerstedt
Georgia Institute of Technology
1
The characteristic polynomial is
A () = det(I A) = ( + 2) 1 = 0
i.e.
2 + 2 1 = 0
= 2 + 1.
A is a stability matrix i Re
164
LECTURE 17
(c) Can you nd a SISO minimal realization for which the matrix A is diagonalizable
with repeated eigenvalues? Justify your answer.
(d) Can you nd a SISO minimal realization for which th
263
INDEX
state transition matrix
continuous time, 4143
discrete time, 44
inverse, 43, 45
semigroup property, 42, 45
state estimation, 153154
state estimation error, 154
state feedback, 135, 154
state
Table 8.1. Lyapunov Stability Tests for LTI Systems
Continuous time Discrete time
Denition Eigenvalue test Lyapunov test Eigenvalue test Lyapunov test
For some A; [A], For some A; [A].
udA] 22- Oor
LECTURE 2
Linearization
CONTENTS
This lecture addresses how state-space linear systems arise in control.
1. State-Space Nonlinear Systems
2. Local Linearization around an Equilibrium Point
3. Local Li