Chapter 2 Review of Linear Algebra
Notation: The n-dimensional Euclidean space is denoted by R n . Hence,
x R n refers to a n-dimensional vector of real numbers and A R nm refers
to a n m matrix of real numbers.
2.1 Preliminaries
The following identities

Chapter 10 Servo Control
Tutorial 10
10.1
Servo Control
Consider the second order system
x1 2 1 x1 0
1 0 w2
u
x 0 1 x 1
0 1 w ,
3
2
2
where w2 and w3 are unknown constant disturbance signals. It is desired that the state
x1 follows a reference in

Tutorial 12
Tutorial 12 Collection of Problems
12.1 Consider the following plant:
1 0
1
x
x
u,
0 1
1
and
y 1 1x .
(a) Find the state feedback u Kx v , which assigns the closed-loop poles at 1 j .
(b) Sketch the resultant feedback system.
(c) Obtain the

Solution to Tutorial 2
by C.J. Ong
September 9, 2016
Q1
Assuming that I = 0, we have
I + b + k = H cos
Let x1 = , x2 = then
x 1 = x2
b
H
k
x 2 = x1 x2 + cosx1
I
I
I
Note that this system is nonlinear. To linearize the system, consider
x = x0 + x and = 0 +

Solution to Tutorial 1
by C.J. Ong
September 29, 2016
Q1
(a) and (b) are straight forward.
(c) Not possible because det(A) = 0.
Q2
eAt =
k
t
k=0
k!
Ak = I + At +
1 2 2
1
A t + A3 t3 +
2!
3!
deAt
1
= A + A2 t + A3 t2 +
dt
2!
1
1
= A(I + At + A2 t2 + A3

Chapter Eleven
State Estimation
Tutorial 11 State Estimation
11.1 Design an observer for instrument servo:
0 1
0
x
x u ,
0
y 1 0x .
Solution:
Consider an observor in the form:
x (t ) Ax (t ) Bu (t ) l y c T x ,
where x (t ) denotes the estimate of x

Chapter Seven
Pole Placement
Tutorial 7 Pole Placement
7.1
Consider a plant with transfer function
G ( s)
10
.
s( s 1)
Describe the system in state space representation. Choose suitable closed-loop poles
and calculate the corresponding feedback gain.
Sol

Chapter Eight
Quadratic Optimal Control
Tutorial 8 Quadratic Optimal Control
8.1 Consider a first-order system
x 0 x 0 ,
x t ax t u t ,
y t x t ,
with
J
y t ru t dt .
2
2
0
find the optimal control.
Solution:
Let us assume a feedback control of the form

National University of Singapore
Department of Mechanical Engineering
ME5401/MCH5201/EE5101 Linear System 2016/2017
Tutorial 3
1. Assume that a SISO system cfw_A,B,C,D is expressed in controllable canonical
form. Show that the controllability matrix has a

National University of Singapore
Department of Mechanical Engineering
ME5401/MCH5201/EE5101 Linear System 2016/2017
Tutorial 2
1.
Consider the system defined by the differential equation
I b k H cos .
Write the state space representation of the system if

National University of Singapore
Department of Mechanical Engineering
ME5401/EE5101 Linear System 2016/2017
Tutorial 1
Note that Questions 8-10 are optional.
1.
Find the inverse of the following matrices, if they exist.
2 5
(a) A=
10 1
3 0 1
(b) A= 2 1

EE5101R/ME5401:
Linear Systems: Part II
Xiang Cheng
Associate Professor
Department of Electrical & Computer Engineering
The National University of Singapore
Phone: 65166210 Office: Block E4-08-07
Email: elexc@nus.edu.sg
1
Teaching Arrangement
Part I:
Syst

EE5101R/ME5401:
Linear Systems: Part II
Xiang Cheng
Associate Professor
Department of Electrical & Computer Engineering
The National University of Singapore
Phone: 65166210 Office: Block E4-08-07
Email: elexc@nus.edu.sg
1
Chapter Nine
Decoupling Control
F

EE5101R/ME5401:
Linear Systems: Part II
Xiang Cheng
Associate Professor
Department of Electrical & Computer Engineering
The National University of Singapore
Phone: 65166210 Office: Block E4-08-07
Email: elexc@nus.edu.sg
Where are we now?
INPUT to
Desired

EE5101R/ME5401:
Linear Systems: Part II
Xiang Cheng
Associate Professor
Department of Electrical & Computer Engineering
The National University of Singapore
Phone: 65166210 Office: Block E4-08-07
Email: elexc@nus.edu.sg
1
Chapter 10 Servo Control
What has

EE5101R/ME5401:
Linear Systems: Part II
Xiang Cheng
Associate Professor
Department of Electrical & Computer Engineering
The National University of Singapore
Phone: 65166210 Office: Block E4-08-07
Email: elexc@nus.edu.sg
1
INPUT to
Desired
performance:
REF

Chapter Nine
Decoupling Control
Tutorial 9 Decoupling Control
9.1 Consider the 2-input/2-output system described by
0
0
1
1 0
x 0 2 0 x 2
3 u,
3 3
0
0 3
1 0 0
y
x.
1 1 1
Find the state-feedback decoupler.
Solution:
It is readily checked that
c1T A 0 B