Problem Set #9:
1(4). Is the following state equation controllable? observable?
0 1 1
0
1 1 1 x + 1 u,
x=
y = [1 0 1] x
1 1 0
0
If not controllable, reduce it to a controllable one;
If not observable, reduce it to an observable one.
0 1 1
0
1
Problem Set #11:
1(5). Design a robust tracking strategy for the system
u
g o ( s) =
s2 + s + 1
s 3 s 2 + 2s 1
y
so that the output y follows a step signal asymptotically. Choose design parameters so that the
closed-loop poles are at 2 + j 2, 2 j 2, 4 and
16.513 Control Systems - Final Exam (Spring 2006)
There are 5 problems (including 1 bonus problem, total 100+20 points)
1. (30pts) Given two matrices:
1 1
A 1=
,
1 3
1) (16) Compute
e
A1t
0 1 0
A 2 = 1 1 1
1 0 0
, e A2t .
2) (6) For the differential eq
Problem Set #8:
1(2). Use the first definition of a matrix function to compute e At for
1 2
A=
4 3
Solution: ( ) = I A =
+1
4
2
= ( + 1)( 3) + 8 = ( 1) 2 + 4 = 0
3
1 = 1 + 2 j , 2 = 1 2 j.
Define f ( ) = et , g ( ) = 0 + 1 ,
f (1 ) = g (1 )
e(1+ 2 j
16.513 Control Systems - Final Exam (Spring 2007)
There are 5 problems (Total 100)
1. (15pts) For the differential equation
1 2
1
0
&
x=
x + 1u; y = [1 1]x, with x(0) = 1 , u(t)=0, what is y(t) for t > 0?
5 5
2. (15pts) Consider the following system
16.513 Control Systems
Last Time:
Introduction
Motivation
Course Overview
Project
Math. Descriptions of Systems ~ Review
Classification of Systems
Linear Systems
LTI Systems
The notion of state and state variables were
introduced
1
Today:
Matrix O
Problem Set #3:
1(1.5). Derive state-space description for the circuit:
R
i2
vC
R
u=Vin
i1
V=0
iL
L
C
+
y =Vo
Solution: Denote the current through L as iL and the voltage across C as vC , from KVL and KCL,
u 0 = L
i1 = i2
diL
dt
v
dv
u 0
+ iL = C + C C
R
16.513 Control Systems
Problem set #5 solutions (modified from homework of Jinming Chen)
1. Find:
1) Nullities
2) Bases for the range spaces and
3) Bases for the null spaces
For the following Matrics
A1 = [1 0 1],
0 1 0
A4 = 0 0 0,
0 0 1
1 1 0
A2 =
,
Problem Set #4:
1(2). Compute the inverses for:
1 1 1
1 1 1
0 1 1 , S = 0 1 0
S1 =
2
0 1 1
0 1 1
Use the formula on slide 19 for the inverse of block diagonal matrices.
A1 A1CB 1
A C
1
Solution: For block diagonal matrix S =
,
S =
B 1
0 B
0
1 1
Problem Set #1:
1(3). Give examples for nonlinear systems and infinite dimensional systems respectively. What are the
inputs, outputs and states?
Nonlinear System:
I
Vi
D1
D1 is a silicon junction Diode with the i - v relationship,
I = I s (eVi / nVT 1)
V
Control Systems
Solution for problem set #6 (Modified from Jinming Chens homework)
1. Find Jordan-form representations and transformation matrix Q for the following
matrices:
1 1 0
2 0 0
1 0 0
A1 = 0 0 1 , A 2 = 1 6 4 , A 3 = 0 1 0
0 0 1
1 9 6
4 2
. Name: March 22, 2007
16.513 Midterm Exam (Spring 2007)
There are 5 problems and two bonus problems.
1. (16) For each of the following sets of vectors, determine if it is linearly dependent or independent:
1121121111 82 $23 333111-18411111111
ll
l . .
16.513 Control Systems
Last Time:
Matrix Operations - Fundamental to Linear Algebra
Determinant
Matrix Multiplication
Eigenvalue
Rank
Math. Descriptions of Systems ~ Review
LTI Systems: State Variable Description
Linearization
1
Today:
Modeling of Selec
Problem Set #12:
1(10). The open-loop system
0 1 0
0
0 0 1 x + 0 u,
x=
1 4 3
2
1
x(0) = 1 .
1
1) Assume that x is available for state feedback. Design an LQR control law by letting R = 1 and
0 1 0
y=
x,
0 0 1
choosing Q so that all the elemen
16.513 Control Systems
Lyapunov Stability: An introduction to
Nonlinear/Uncertain Systems
1
Stability for a linear system
The system:
x = Ax + Bu,
y = Cx + Du
Basic requirement for a system:
req irement
s stem:
If a bounded input is applied, a bounded ou
Problem Set #9:
1(4). Is the following state equation controllable? observable?
0
x
1
1
1
1 x
0
1
1
0
1
1 u,
0
y
1 0 1 x
If not controllable, reduce it to a controllable one;
If not observable, reduce it to an observable one.
0
Solution: Given A
1
1
1
1 ,
An. 0 c')
deJ'CAIA) 0 -I A a
-1 g] {V1
o o
4:}
.6 0
4+
{1'2 I
*1 4+ J. "4
[IL 5.5 5+ 1 3" -+ .334" l -] 1 ~
Mzéétig) _| (3],. iieab) 1 {3:35; 4*
C. 3. 3" O I -1 -;_ I a E 2-3:.
CJ +0 GFWV} Qw- CoyxlnlnuoU} 1E5 4'35,me A1: 64+ %: A-lElrllg
At}: [0.2113 £1
16.513 Control Systems
Controllability and Observability
(Chapter 6)
1
A General Framework in State-Space Approach
Given an LTI system:
x Ax Bu; y Cx
(*)
The system might be unstable or doesnt meet the required
performance spec. How can we improve the sit
16.513 Control Systems (Lecture note #7)
Last Time:
Generalized eigenvectors, Jordan form
Polynomial functions of a square matrix, eAt
A big picture: one branch of the course
Vector spaces
matrices
Algebraic
equations
Eigenvalues
Eigenvectors
Solutions
16.513 Control Systems
Summary of Results From Last Lecture:
Consider the system: x Ax Bu; y Cx Du
Given x(0) and u(t) for t 0. The solution is
t
x(t) e At x(0) e A(t ) Bu ( )d ;
0
t
y(t) Ce x(0) Ce A(t ) Bu ( )d Du(t)
At
0
The main problem involved is to
16.513 Control Systems
Today, we are going to cover part of Chapter 6
and part of Chapter 8
Controllability and Observability
State Feedback and State Estimators
Last Time :
Controllability
Observability
Canonical decomposition
Controllable/uncontro
16.513 Control systems -
Last lecture
Last time, we constructed
Full dimensional estimator
SISO case via observable canonical form
MIMO case by solving matrix equation
Today: We conclude the design part
Reduced order observer
Connection of state-feedba
16.513 Control systems
Last time,
Controllability and observability (Chapter 6)
Two approaches to state feedback design (Chapter 8)
Using controllable canonical form
By solving matrix equations
1
Today, we continue to work on feedback design
(Chapter