2. Textbook and Syllabus Textbook: Syllabus:
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

3. Modern Control
Grade Policy
Final Grade = 10% Homework + 20% Quizzes % 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.

4. 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 March No.1. Answer:
Heading of Homework Papers (Required)

5. 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.
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.

6. Modern Control
Chapter 1
Introduction

7. Classical Control and Modern Control
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.

8. Signal Classification
Chapter 1
Introduction
Signal Classification
Continuous signal
Discrete signal

9. Classification of Systems
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. RLC-circuit.
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.

10. Classification of Systems
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

11. 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
Homogeneity

12. Example: Linear or Nonlinear
Chapter 1
Introduction
Example: Linear or Nonlinear
Check the linearity of the following system.
Let , then
Let , then
Thus  The system is nonlinear

13. Example: Linear or Nonlinear
Chapter 1
Introduction
Example: Linear or Nonlinear
Check the linearity of the following system (governed by ODE).
Let
Then
 The system is linear

14. Properties of Linear Systems
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:

15. 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.
Time-invariant
system
A system is said to be time-invariant if its parameters do not change over time.

16. Laplace Transform Approach
Chapter 1
Introduction
Laplace Transform Approach
RLC Circuit
Resistor
Inductor
Capacitor
Input variables:
Input voltage u(t)
Output variables:
Current i(t)

17. Laplace Transform Approach
Chapter 1
Introduction
Laplace Transform Approach
Current due to input
Current due to initial condition
For zero initial conditions (v0 = 0, i0 = 0),
where
Transfer function

18. 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
Outputs
Black box
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
Outputs
State Space
States

19. 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.

20. State Space Equations as Linear System
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
Then, it can also be implied that

21. Chapter 1
Introduction
State Space Approach
Let us now consider the same RLC circuit and try to use state space to model it.
RLC Circuit
State variables:
Voltage across C
Current through L
We now have two first-order ODEs
Their variables are the state variables and the input

22. Chapter 1
Introduction
State Space Approach
The two equations are called state equations, and can be rewritten in the form of:
The output is described by an output equation:

23. Chapter 1
Introduction
State Space Approach
The state equations and output equation, combined together, form the state space description of the circuit.
In a more compact form, the state space can be written as:

24. 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.

25. Mathematical Descriptions of Systems
Modern Control
Chapter 2
Mathematical Descriptions of Systems

26. Chapter 2
Mathematical Descriptions of Systems
State Space Equations
The state equations of a system can generally be written as:
are the state variables
are the system inputs
State equations are built of n linearly-coupled first-order ordinary differential equations

27. State Space Equations By defining: we can write State Equations
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
By defining:
we can write
State Equations
A is called system matrix
B is called input matrix

28. 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:
are the system outputs

29. State Space Equations By defining: we can write Output Equations
Chapter 2
Mathematical Descriptions of Systems
State Space Equations
By defining:
we can write
Output Equations
C is called output matrix
D is called feedthrough matrix

30. Block Diagram of a State Space
Chapter 2
Mathematical Descriptions of Systems
Block Diagram of a State Space

31. Example: Mechanical System
Chapter 2
Mathematical Descriptions of Systems
Example: Mechanical System
State variables:
Input variables:
Applied force u(t)
Output variables:
Displacement y(t)
State equations:

32. Example: Mechanical System
Chapter 2
Mathematical Descriptions of Systems
Example: Mechanical System
The state space equations can now be constructed as below:

33. Homework 1: Electrical System
Chapter 2
Mathematical Descriptions of Systems
Homework 1: Electrical System
Derive the state space representation of the following electric circuit:
Input variables:
Input voltage u(t)
Output variables:
Inductor voltage vL(t)

34. Homework 1A: Electrical System
Chapter 2
Mathematical Descriptions of Systems
Homework 1A: Electrical System
Derive the state space representation of the following electric circuit:
Input variables:
Input voltage u(t)
Output variables:
Inductor voltage vL(t)