Modern Control
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
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2 0 1 5
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Modern Control
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
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Modern Control
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, one day before the
schedule of the lecture. Late submission will be penalized by point
deduction of –10·n, where n is the total number of lateness
made.
 There will be 3 quizzes. Only the best 2 will be counted. The
average of quiz grades contributes 20% of the final grade.
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Modern Control
Grade Policy
 Midterm and final exams follow the schedule released by AAB
(Academic Administration Bureau).
 Make up for quizzes must be requested within one week after the
date of the respective quizzes.
 Make up for mid exam and final exam must be requested directly
to AAB.
Modern Control
Homework 4
Ronald Andre
009201700008
21 March 2021
No.1. Answer: . . . . . . . .
 Heading of Homework Papers (Required)
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Modern Control
Grade Policy
 To maintain the integrity, the maximum score of a make up quiz
or exam can be set to 90.
 Extra points will be given if you solve a problem in front of the
class. You will earn 1 or 2 points.
 Lecture slides can be copied during class session. It also will be
available on internet around 1 days after class. Please check the
course homepage regularly.
http://zitompul.wordpress.com
 The use of internet for any purpose during class sessions is
strictly forbidden.
 You are expected to write a note along the lectures to record your
own conclusions or materials which are not covered by the lecture
slides.
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Modern Control
Chapter 1
Introduction
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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.
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Chapter 1
Introduction
Signal Classification
 Continuous signal
 Discrete signal
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Chapter 1
Introduction
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.
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Chapter 1
Introduction
Classification of Systems
 Finite-dimensional system (lumped-parameters, described by
differential equations):
 Linear and nonlinear systems
 Continuous- and discrete-time systems
 Time-invariant and time-varying systems
 Infinite-dimensional system (distributed-parameters, described by
partial differential equations):
 Heat conduction
 Power transmission line
 Antenna
 Fiber optics
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Chapter 1
Introduction
Linear Systems
 A system is said to be linear in terms of the system input u(t) and
the system output y(t) if it satisfies the following two properties of
superposition and homogeneity.
 Superposition
u1 (t )
y1 (t )
u1 (t )  u2 (t )
u2 (t )
y1 (t )  y2 (t )
y2 (t )
 Homogeneity
u1 (t )
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 u1 (t )
y1 (t )
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 y1 (t )
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Chapter 1
Introduction
Example: Linear or Nonlinear
Check the linearity of the following system.
u(t )
y(t )  u(t )  u(t  1)
y (t )
Let u(t )  u1 (t ) , then y1 (t )  u1 (t )  u1 (t  1)
Let u(t )   u1 (t ) , then y1 (t )   u1 (t )   u1 (t  1)
  2u1 (t )  u1 (t 1)
Thus f ( u(t ))   y(t )  The system is nonlinear
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Chapter 1
Introduction
Example: Linear or Nonlinear
Check the linearity of the following system (governed by ODE).
u(t )
y(t )  2 y(t )  y(t )  u (t )  3u(t )
y (t )
Let y1(t )  2 y1 (t )  y1 (t )  u1(t )  3u1 (t )
y2(t )  2 y2 (t )  y2 (t )  u2 (t )  3u2 (t )
Then [ u1 (t )   u2 (t )]  3[ u1 (t )   u2 (t )]
  u1(t )   u2 (t )   3u1 (t )   3u2 (t )
 [u1(t )  3u1 (t )]   [u2 (t )  3u2 (t )]
 [ y1(t )  2 y1(t )  y1 (t )]   [ y2(t )  2 y2 (t )  y2 (t )]
 [ y1 (t )   y2 (t )]  2[ y1 (t )   y2 (t )]  [ y1 (t )   y2 (t )]
 The system is linear
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Chapter 1
Introduction
Properties of Linear Systems
 For linear systems, if input is zero then output is zero.
 A linear system is causal if and only if it satisfies the condition of
initial rest:
u(t )  0 for t  t0  y(t )  0 for t  t0
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Chapter 1
Introduction
Time-Invariance
 A system is said to be time-invariant if a time delay or time
advance of the input signal leads to an identical time shift in the
output signal.
x(t )
y (t )
x(t  t0 )
Time-invariant
system
y (t  t0 )
t0
t0
 A system is said to be time-invariant if its parameters do not
change over time.
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Chapter 1
Introduction
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)
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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
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Chapter 1
Introduction
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
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G(s)
I ( s)
Transfer function
Modern Control 1/17
Chapter 1
Introduction
State Space Approach
 In Laplace Transform, the input-output model is assumed to be
time-invariant, linear, and causal.
 Besides, this method is not effective to model time-varying and
non-linear systems.
 The input-output model regards a system as a black box, with
certain inputs and certain outputs. What is happening inside the
box is not clear.
Inputs
Black box
Outputs
 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.
Inputs
State Space
Outputs
States
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Chapter 1
Introduction
State Space Approach
 In state space approach, the knowledge of the states is enough to
express anything inside the model box.
 With all states known, the future outputs can be determined
based on the future inputs that will be given to a system.
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Chapter 1
Introduction
State Space Equations as Linear System
 A system y(t) = f(x(t),u(t)) is said to be linear if it follows the
following conditions:
 If
and
then
 If
then
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 )
f  x1 (t ), u1 (t )   y1 (t ),
f  x(t ),  u1 (t )    y1 (t )
 Then, it can also be implied that
f  x1 (t )   x 2 (t ),  u1 (t )   u2 (t )    y1 (t )   y 2 (t )
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Chapter 1
Introduction
State Space Approach
Let us now consider the same
RLC circuit and try to use
state space to model it.
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
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Chapter 1
Introduction
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 
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Chapter 1
Introduction
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
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1C 
vC 
 0
x 
A

 1 L  R L 
 iL 
 0 
B 
dvC dt 
x
1 L 

di
dt
 L

C  0 1
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Chapter 1
Introduction
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.
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Modern Control
Chapter 2
Mathematical Descriptions of Systems
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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 )  
 2 

 

 
 xn (t )   an1
a22
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
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Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 By defining:
 x1 (t ) 
 x (t ) 
x (t )   2  ,




 xn (t ) 
 u1 (t ) 
u (t ) 
u(t )   2  ,




un (t ) 
x(t )  Ax(t )  Bu(t )
we can write
State Equations
• A is called system matrix
• B is called input matrix
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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 

 

 
 ym (t )  cm1
y1 (t ), y2 (t ),
, ym (t )
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c1n   x1 (t )   d11
  x (t )  
 2 

 

 
cmn   xn (t )   d m1
d 22
d1r   u1 (t ) 
 u (t ) 
 2 




d mr  ur (t ) 
are the system outputs
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Chapter 2
Mathematical Descriptions of Systems
State Space Equations
 By defining:
 y1 (t ) 
 y (t ) 
y (t )   2  ,




y
(
t
)
 m 
y (t )  C x (t )  Du(t )
we can write
Output Equations
• C is called output matrix
• D is called feedthrough matrix
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Chapter 2
Mathematical Descriptions of Systems
Block Diagram of a State Space
D
u(t )
B
x(t )
+
+

x(t )
y(t )
C
+
+
A
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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)
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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
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Modern Control 1/31
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 
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Chapter 2
Mathematical Descriptions of Systems
Homework 1: Electrical System
 Derive the state space representation of the following electric
circuit:
R
C2


u(t )
C1
L
vL (t )


Input variables:
• Input voltage u(t)
Output variables:
• Inductor voltage vL(t)
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Chapter 2
Mathematical Descriptions of Systems
Homework 1A: Electrical System
 Derive the state space representation of the following electric
circuit:
C1
C2


u(t )
R
L
vL (t )


Input variables:
• Input voltage u(t)
Output variables:
• Inductor voltage vL(t)
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