2. Textbook and Syllabus Textbook: Syllabus:
“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. Multivariable Calculus
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. 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.

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

5. 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.

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

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

8. 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.

9. State Space Approach State variables: Voltage across C
Chapter 1
Modeling Approach
State Space Approach
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

10. Chapter 1
Modeling Approach
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:

11. Chapter 1
Modeling Approach
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:

12. 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.

13. Classification of Systems
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. 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.

14. 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
Then, it can also be implied that

15. Linear Time-Invariant (LTI) System
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.

16. 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

17. 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

18. 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

19. 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

20. 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:

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

22. 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)

23. 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)