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 =
B AB A2 B
1 1 a1 a2 a2 + (a2 1)(a1 + a2 )
1
a2
a2
= 1
EL625
Solution for Problem 8.9
8.9 The controllability and observability matrices are
4 =
=
4
1.5 0.5 3.5 4.5
0.5 3.5 4.5 11.5
1.5 0.5 3.5 4.5
1
3
1
7
2 2 2 2
1 0 2 6
1 2 4 8
2 2 2 2
(1)
(2)
Noting that the rst row of 4 is equal to the third row of 4
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 ) e
2
(t) =
and
H (s) = C [sI A]1 B + D =
5.3 We apply L
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
+
ty
=
+ 2u.
dt3
dt2
dt
dt3
dt
2. Consider the system give
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 between internal
and external stability will be studied.
B
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
discussed
d iinclude
l d
Properties of state transition m
Questions for Homework 1
1-1 An analog system has the zero-state response:
Z
1
y(t) = Scfw_u(t) =
2 u(t + ) d
1
Determine whether the system is or is not a) causal, b) fixed c) zero-state linear. Give an
answer and a reason for each of a), b) and c).
1-2
Lecture 13:
13: Linear Observers and
Output Feedback Control
Lecture Objectives:
This lecture introduces to the students how to
reconstruct the unmeasured states when only
the input and output information is available.
A linear system which gives an asymp
Lecture 10: Observabilityy
G l off this
Goals
thi Lecture:
L t
Observability for continuous-time systems
Tests of observability for LTI systems
Tests of observability for LTV systems
Application examples: observable
canonical form (OCF)
(OCF), duality,
du
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 poles by state feedback?
Constructive design methods
Ex
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 qualitative behavior of
Continuous-time linear systems;
Discret
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 internal (state-space) and
external descriptions for line
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 from some other established farm and the output is the n
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.
Various controllabilityy notions and tests will be
presented.
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 the internal behavior
of a dynamic system.
The other app
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, in
particular linear systems. Moreover, we
study the
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 linear discrete-time systems.
Various tests for controllabil
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. Throughout, continuous-time and discrete-time
systems are tr
Homework
1. Consider the dynamics of a unicycle given as:
x 1 = u1 cos(x3 )
x 2 = u1 sin(x3 )
x 3 = u2 .
This system has two control inputs, u1 and u2 .
(a) Given the initial condition [x1 , x2 , x3 ]T = [1, 1, 4 ]T , design a controller to move the state
Online EL6253
Linear Systems
Zh
Zhong-Ping
Pi Jiang
Ji
ECE Dept, Polytechnic School of Engineering, NYU
E-mail: [email protected]
Web: eeweb.poly.edu/faculty/jiang
Prof. Jiang
NYU Polytechnic School of Engineering
1
Course Goals
Equip
q p students with the b
EL6253 Homework
1. Consider the following system:
0
x = 1
0
1
0
0
0
0
1 x + 0 u.
0
1
Rewrite the above dynamics in the controllable canonical form through
an appropriate change of coordinates. Also, design a control law to place
the poles of the closed-lo
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
=
since ( )T has only two linearly independent columns.
EL625
Solutions to Problem Set 10
9.2
a) To show suciency (if), (i.e., |xi | < for all i = x < ) let x be a vector with
|xi | < for all i and let xn be a sequence of vectors such that as n , xn x (i.e.,
limn |xin xi | = 0 for all i). Consider the vector s
Solutions to Problem Set 11
EL625
9.9 These are all discrete systems.
1.5 1
and the eigenvalues were 1 = 1
1
1
and 2 = 0.5. Thus all eigenvalues have | 1 and the eigenvalue at = 1 is simple so
the equilibrium state at 0 is stable but not AS.
a) In Example
1
Homework Pole placement and observer design 1. The linearized dynamics of an inverted pendulum are as follows:
0 1 0 0
0 1 0 2
x=
0 0 1 0
x+
00 0 1 00 5 0 1000
y=
u
(1)
(a) Using state feedback, place the closed loop poles at -1,-2 and -1j . (b) De
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 =
B AB A2 B
1 1 a1 a2 a2 + (a2 1)(a1 + a2 )
1
a2
a2
= 1
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
=
since ( )T has only two linearly independent columns.
EL625
Solution for Problem 8.9
8.9 The controllability and observability matrices are
4 =
=
4
1.5 0.5 3.5 4.5
0.5 3.5 4.5 11.5
1.5 0.5 3.5 4.5
1
3
1
7
2 2 2 2
1 0 2 6
1 2 4 8
2 2 2 2
(1)
(2)
Noting that the rst row of 4 is equal to the third row of 4
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 the inversion formula. First, the two roots of
1
z 2 z +
Lecture 3
S l
I
Solutions
off LLinear T
Time-Invariant
Time(LTI) Systems:
y
Theoryy
Lecture Goals
This lecture aims to introduce various methods
to compute a matrix exponential, which plays a
crucial
i l role
l iin solving
l i lilinear titime-invariant
i