Modern Control
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
President University
Erwin Sitompul
Modern Control 1/1
Textbook and Syllabus
Textbook:
“Linear System Theory and Design”,
Third Edition, Chi-Tsong Chen,
Oxford University Press, 1999.
Syllabus:
Chapter 1: Introduction
Chapter 2: Mathematical Descriptions
of Systems
Chapter 3: Linear Algebra
Chapter 4: State-Space Solutions and Realizations
Chapter 5: Stability
Chapter 6: Controllability and Observability
Chapter 7: Minimal Realizations and Coprime Fractions
Chapter 8: State Feedback and State Estimators
Additional: Optimal Control
President University
Erwin Sitompul
Modern Control 1/2
Multivariable Calculus
Grade Policy
Final Grade =
10% Homework + 20% Quizzes +
30% Midterm Exam + 40% Final Exam +
Extra Points
 Homeworks will be given in fairly regular basis. The average of
homework grades contributes 10% of final grade.
 Homeworks are to be submitted on A4 papers, otherwise they
will not be graded.
 Homeworks must be submitted on time. If you submit late,
< 10 min.
 No penalty
10 – 60 min.  –20 points
> 60 min.
 –40 points
 There will be 3 quizzes. Only the best 2 will be counted. The
average of quiz grades contributes 20% of the final grade.
 Midterm and final exam schedule will be announced in time.
 Make up of quizzes and exams will be held one week after the
schedule of the respective quizzes and exams.
President University
Erwin Sitompul
Modern Control 1/3
Multivariable Calculus
Grade Policy
 The score of a make up quiz or exam, upon discretion, can be
multiplied by 0.9 (i.e., the maximum score for a make up is then
90).
 Extra points will be given if you solve a problem in front of the
class. You will earn 1, 2, or 3 points.
 You are responsible to read and understand the lecture slides. I
am responsible to answer your questions.
Modern Control
Homework 4
Ronald Andre
009201700008
21 March 2021
No.1. Answer: . . . . . . . .
 Heading of Homework Papers (Required)
President University
Erwin Sitompul
Modern Control 1/4
Chapter 1
Introduction
Classical Control and Modern Control
Classical Control
• SISO
(Single Input Single Output)
• Low order ODEs
• Time-invariant
• Fixed parameters
• Linear
• Time-response approach
• Continuous, analog
• Before 80s
Modern Control
• MIMO
(Multiple Input Multiple Output)
• High order ODEs, PDEs
• Time-invariant and time variant
• Changing parameters
• Linear and non-linear
• Time- and frequency response
approach
• Tends to be discrete, digital
• 80s and after
 The difference between classical control and modern control
originates from the different modeling approach used by each
control.
 The modeling approach used by modern control enables it to have
new features not available for classical control.
President University
Erwin Sitompul
Modern Control 1/5
Chapter 1
Modeling Approach
Laplace Transform Approach
R
L
i(t )

C
u(t )
v  L di
dt
v  Ri

RLC Circuit
Input variables:
• Input voltage u(t)
Output variables:
• Current i(t)
President University
V  s   R  I  s  V  sL  I  s 
Resistor
Inductor
i  C dv
dt
I  s   sC V  s 
Capacitor
t
di (t ) 1
  i ( )d  v0  u (t )
dt
C0
v0
1
RI ( s)  L( sI ( s )  i0 ) 
I (s)   U (s)
Cs
s
Ri (t )  L
Erwin Sitompul
Modern Control 1/6
Chapter 1
Modeling Approach
Laplace Transform Approach
v0
1
RI ( s)  L( sI ( s )  i0 ) 
I (s)   U (s)
Cs
s
v0
1 

 Ls  R 
 I ( s )  U ( s )  Li0 
s
Cs 

LCsi0  Cv0
Cs
I (s) 
U (s) 
2
LCs  RCs  1
LCs 2  RCs  1
Current due
to input
Current due to
initial condition
 For zero initial conditions (v0 = 0, i0 = 0),
I ( s )  G ( s )U ( s )
where
U ( s)
Cs
G ( s) 
LCs 2  RCs  1
President University
Erwin Sitompul
G(s)
I ( s)
Transfer function
Modern Control 1/7
Chapter 1
Modeling Approach
State Space Approach
 Laplace Transform method is not effective to model time-varying
and non-linear systems.
 The state space approach to be studied in this course will be able
to handle more general systems.
 The state space approach characterizes the properties of a system
without solving for the exact output.
 Let us now consider the same RLC circuit and try to use state
space to model it.
President University
Erwin Sitompul
Modern Control 1/8
Chapter 1
Modeling Approach
State Space Approach
R
L
i(t )

C
u(t )

RLC Circuit
State variables:
• Voltage across C
• Current through L
dvC
dvC 1
iC  C

 iC
dt
dt
C
dvC 1
 iL
dt
C
diL 1
diL

 vL
vL  L
dt L
dt
diL 1
 (u  vR  vC )
dt L
diL 1
 (u  RiL  vC )
dt L
• We now have two first-order ODEs
• Their variables are the state
variables and the input
President University
Erwin Sitompul
Modern Control 1/9
Chapter 1
Modeling Approach
State Space Approach
 The two equations are called state
equations, and can be rewritten in the
form of:
1 C  vC   0 
 dvC dt   0
 di dt    1 L  R L   i   1 L  u
 L   
 L
 
dvC 1
 iL
dt
C
diL 1
 (u  RiL  vC )
dt L
 The output is described by an output
equation:
vC 
iL  0 1  
 iL 
President University
Erwin Sitompul
Modern Control 1/10
Chapter 1
Modeling Approach
State Space Approach
 The state equations and output equation, combined together, form
the state space description of the circuit.
1 C  vC   0 
 dvC dt   0
 di dt    1 L  R L   i   1 L  u
 L   
 L
 
vC 
iL  0 1  
 iL 
 In a more compact form, the
state space can be written as:
x  Ax  Bu
y Cx
President University
1C 
vC 
 0
x 
A

 1 L  R L 
 iL 
 0 
B 
dvC dt 
x
1 L 

di
dt
 L

C  0 1
Erwin Sitompul
Modern Control 1/11
Chapter 1
Modeling Approach
State Space Approach
 The state of a system at t0 is the information at t0 that, together
with the input u for t0 ≤ t < ∞, uniquely determines the behavior
of the system for t ≥ t0.
 The number of state variables = the number of initial conditions
needed to solve the problem.
 As we will learn in the future, there are infinite numbers of state
space that can represent a system.
 The main features of state space approach are:
 It describes the behaviors inside the system.
 Stability and performance can be analyzed without solving for
any differential equations.
 Applicable to more general systems such as non-linear
systems, time-varying system.
 Modern control theory are developed using state space
approach.
President University
Erwin Sitompul
Modern Control 1/12
Chapter 2
Mathematical Descriptions of Systems
Classification of Systems
 Systems are classified based on:
 The number of inputs and outputs: single-input single-output
(SISO), multi-input multi-output (MIMO), MISO, SIMO.
 Existence of memory: if the current output depends on the
current input only, then the system is said to be memoryless,
otherwise it has memory  purely resistive circuit vs. RLCcircuit.
 Causality: a system is called causal or non-anticipatory if the
output depends only on the present and past inputs and
independent of the future unfed inputs.
 Dimensionality: the dimension of system can be finite
(lumped) or infinite (distributed).
 Linearity: superposition of inputs yields the superposition of
outputs.
 Time-Invariance: the characteristics of a system with the
change of time.
President University
Erwin Sitompul
Modern Control 1/13
Chapter 2
Mathematical Descriptions of Systems
Linear System
 A system y(t) = f(x(t),u(t)) is said to be linear if it follows the
following conditions:
 If
then
 If
and
then
f  x1 (t ), u1 (t )   y1 (t ),
f  x(t ),  u1 (t )    y1 (t )
f  x1 (t ), u1 (t )   y1 (t )
f  x 2 (t ), u2 (t )   y 2 (t )
f  x1 (t )  x 2 (t ), u1 (t )  u2 (t )   y1 (t )  y 2 (t )
 Then, it can also be implied that
f  x1 (t )   x 2 (t ),  u1 (t )   u2 (t )    y1 (t )   y 2 (t )
President University
Erwin Sitompul
Modern Control 1/14
Chapter 2
Mathematical Descriptions of Systems
Linear Time-Invariant (LTI) System
 A system is said to be linear time-invariant if it is linear and its
parameters do not change over time.
President University
Erwin Sitompul
Modern Control 1/15
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 The state equations of a system can generally be written as:
 x1 (t )   a11
 x (t )  
a22
 2 

 

 
 xn (t )   an1
a1n   x1 (t )   b11
  x (t )  
b22
 2 

 

 
ann   xn (t )  bn1
x1 (t ), x2 (t ),
, xn (t )
are the state variables
u1 (t ), u2 (t ),
, ur (t )
are the system inputs
b1r   u1 (t ) 
 u (t ) 
 2 




bnr  ur (t ) 
• State equations are built of n linearly coupled first
order ordinary differential equations
President University
Erwin Sitompul
Modern Control 1/16
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 By defining:
 x1 (t ) 
 x (t ) 
x (t )   2  ,




 xn (t ) 
we can write
President University
 u1 (t ) 
u (t ) 
u(t )   2  ,




un (t ) 
x(t )  Ax(t )  Bu(t )
Erwin Sitompul
State Equations
Modern Control 1/17
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 The outputs of the state space are the linear combinations of the
state variables and the inputs:
 y1 (t )   c11
 y (t )  
c22
 2 

 

 
y
(
t
)
 m  cm1
y1 (t ), y2 (t ),
, ym (t )
President University
c1n   x1 (t )   d11
  x (t )  
d 22
 2 

 

 
cmn   xn (t )   d m1
d1r   u1 (t ) 
 u (t ) 
 2 




d mr  ur (t ) 
are the system outputs
Erwin Sitompul
Modern Control 1/18
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 By defining:
 y1 (t ) 
 y (t ) 
y (t )   2  ,




 ym (t ) 
y (t )  C x (t )  Du(t )
we can write
Output Equations
D
u(t )
B
x(t )
+
+

x(t )
y(t )
C
+
+
A
President University
Erwin Sitompul
Modern Control 1/19
Chapter 2
Mathematical Descriptions of Systems
Example: Mechanical System
y0
y(t )
dy (t )
d 2 y (t )
u (t )  ky (t )  b
m
dt
dt 2
k
b
u(t ) State variables:
m
frictionless
Input variables:
• Applied force u(t)
Output variables:
• Displacement y(t)
President University
x1 (t )  y (t )
dy (t )
x2 (t ) 
dt
 x1 (t )  x2 (t )
d 2 y (t )
 x2 (t ) 
dt 2
State equations:
x1 (t )  x2 (t )
k
b
1
x2 (t )   x1 (t )  x2 (t )  u (t )
m
m
m
Erwin Sitompul
Modern Control 1/20
Chapter 2
Mathematical Descriptions of Systems
Example: Mechanical System
 The state space equations can now be constructed as below:
1   x1 (t )   0 
 x1 (t )   0
 x (t )     k m  b m   x (t )   1 m  u (t )
 2   
 2  
 x1 (t ) 
y (t )  1 0 

x
(
t
)
 2 
President University
Erwin Sitompul
Modern Control 1/21
Chapter 2
Mathematical Descriptions of Systems
Homework 1: Electrical System
 Derive the state space representation of the following electic
circuit:
R
C2


u(t )
C1
L
vL (t )


Input variables:
• Input voltage u(t)
Output variables:
• Inductor voltage vL(t)
President University
Erwin Sitompul
Modern Control 1/22