Solutions to Problem Set 6
EL625
5.1 For this system, we see that
s1
0
2 s + 3
[sI A] =
Since
[sI A]1 =
so
1
s1
2
(s1)(s+3)
0
1
s+3
.
2
1
1
1
=
,
(s 1)(s + 3)
2 s1 s+3
we nd that
et
0
1
t
3t
3t
(e e )
Solutions to Problem Set 7
EL625
8.1
a) The system has the state equation
x1 (k + 1)
a1 1 a2 0
x1 (k )
1
0 x2 (k ) + 1 u(k )
a2
x2 (k + 1) = 0
x3 (k + 1)
0
0
a3
x3 (k )
1
The matrix 3 is given by
3 =
Lecture 5 Solutions of Linear TimeTimeInvariant Systems: Applications
Lecture Goal
This lecture addresses some preliminary
applications
off state transition
li i
i i matrix
i to
analyze the qualitativ
Lecture 11
11: Extensions to Linear
Discrete-Time Systems
Discrete
Lecture Objective:
This lecture is devoted to the generalization of
controllability and observability, and their properties,
to linea
Week 7
Midterm examination
Prof. Jiang
NYU Polytechnic School of Engineering
201
Lecture 7 Internal Stability
Lecture Goals
This lecture introduces the notion of
internal stability for dynamic systems
Lecture 2: State
State-Space Model
L t
Lecture
Goals
G l
This lecture aims to introduce the state-space model ,
also known as the internal description of systems,
to provide detailed descriptions of t
Lecture 9:
9: Controllability
Lecture Goals
This lecture focuses on the concept of controllability
f linear
for
li
systems,
t
which
hi h plays
l
a crucial
i l role
l iin
control systems design.
Variou
EL6253 - Midterm - Fall 2016
Closed Books and Notes one page one sided allowed
1. Write the state differential equation in the matrix form for the following system:
du
d3 u
d3 y d2 y
2 dy
+
+
t
+t
+
t
Lecture 12:
12: Linear State Feedback
Lect re Objecti
Lecture
Objectives
es
What is linear state-feedback?
What is the Pole Placement technique?
When can the poles be placed arbitrarily?
How to place
EL625
Additional Homework 1
You are to model a population dynamics problem. Specifically, we are interested
in predicting the pelt production of a mink farm. The input is the newborn minks
purchased f
Lecture 6 Relations Between
Internal and External Descriptions
Lecture Goal
This lecture introduces external (i/o)
d
description
i i ffor lilinear systems, and
d studies
di the
h
relations between in
State-space methods form the basis of modern control theory. This textbook is
devoted to a description of these methods in the analysis of linear multiple-input,
multiple-output dynamic systems. Throu
Review of Signals and Systems
Part I
Copyright: Dr. A. Safaai-Jazi
February 2014
A.
Signals (continuous-time)
Unit impulse function: (t )
(t ) =0, t 0 ; (t ) = , t = 0 and
f (t ) (t t0 ) = f (t0 ) (t
Lecture 8 External Stability
L t
Lecture
Objective
Obj ti
This lecture is aimed at studying the
stability also called input-output
stability,
input output stability
stability,
systems. Connections bet
Lecture 4
E
Extensions
i
to Ti
Time-Varying
TimeV i and
d
Discrete-Time Systems
Discrete
y
Lecture Outline
This lecture is devoted to time-varying and
discrete-time linear systems. Topics to be
di
dis
Solutions to Problem Set 10
EL625
7.13
a) For
F (s) =
1
(s + a)(s + b)
we get,
1
( + a)( + b)(1 esT eT )
1
1
=
+
sT eT )
( + b)(1 e
( + a)(1 esT eT )
=a
1
1
1
=
b a 1 e(s+a)T
1 e(s+b)T
(s+b)T
(s+a)T
1
Solutions to Problem Set 9
EL625
7.1 a) The Z -transformation of the transition matrix is
Z cfw_(k ) = z (zI A)1 =
1
2z+
z
1
z (z 2 )
z
2
z (z 1 )
2
z
2
1
2
To get the inverse transformation, we use t
Solutions to Problem Set 8
EL625
8.2
a) This system has the matrices
1 0
0
A = 0 1 1
0
0 2
1
B= 0
1
C=
110.
For observability we can test,
C
1
1
0
T
(3 ) = CA = 1 1 1
2
CA
1
1 3
rank( )T = 2 < 3
3