### Linear Systems Theory (239085) ******

```Linear Systems Theory

2011 Fall, 4bcd
Kai-Yew Lum 林繼耀
Associate Professor
Dept. of Electrical Engineering
BST-1 #421, ext. 4725
http://staffweb.ncnu.edu.tw/kylum
Objectives
• Motivation
– Linear Systems Theory is the foundation of systems, control and
signal processing.
– Past development of this discipline has produced a mature and
fairly complete set of concepts and methods
– These are fundamental knowledge in electrical engineering,
communications, mechanical engineering, medical engineering,
etc.
• Course Objectives
– Explore the basic theory of linear systems and its applications.
– Provide the necessary tools for engineering problems:
• mathematical description
• analysis (especially numerical analysis)
Time Line of Systems Theory
in Control Engineering
1950’s
Linear Systems Theory
Ziegler-Nicholas
LQR
LQG
1960’s
1970’s
Kalman filter
Transfer
matrix
Lyapunov
Theory
Matrixfraction
description
Control
LQG/LTR
Sliding
Mode
1980’s
H∞
-Synthesis
1990’s
2000’s
Dynamic
Inversion
Back-Stepping
Classical & frequency
domain techniques
State-space
techniques
Nonlinear
techniques
MPC
Lesson Plan
I.
Introduction
II.
Mathemetical Description of Dynamical Systems
III. Review of Linear Algebra -- Matrix Theory
IV. State-Space Solution
V.
Controllability & Observability, Stability
VI. Transfer Matrix Description and Realization
VII. State Feedback and State Estimators
VIII. Introduction to Linear Sampled-Data Systems
What You Should Expect to Learn
• Mathematical Description of Dynamical Systems
– When we study a dynamical system, i.e. a system that evolves in
time with memory effects, we need to describe (represent) its
behavior in equations in order to conduct meaningful analysis
and computation.
– You should also learn the key characteristics that make a system
“linear”, the concept of “state”, and the correspondence between
the state-space representation and what you already know in
frequency domain description (transfer functions).
What You Should Expect to Learn
• Review of Linear Algebra
–
–
–
–
Matrix notations
Properties: determinant, rank, eigenvalues
Characteristic polynomial; Cayley-Hamilton theorem
Special matrices:
• Definite matrices
• Orthogonal matrices
• Singular values & SV decomposition (SVD)
• Transformation & diagonalization
– Generalized eigenvalues & Jordan blocks
What You Should Expect to Learn
• State-Space Solution
– The solution of a dynamical system is its “trajectory” from an
initial state, either on its own or under influence of an external
input.
– The solution of a linear system is structured and easy to
understand if you think of it as linear combination of some
“template” solutions: a basis of solutions.
– Though there is an infinite number of solutions, the dimension of
this basis is finite.
What You Should Expect to Learn
• Controllability, Observability, Stability
– By now you should know that a linear dynamical system has
internal states, which are described in the state-space
representation but not the input-output (transfer) description.
– However, whether the states can be driven by any input, and
observed at the output, is not obvious.
– Also not obvious is whether the internal states are stable, even if
the output is well-behaved.
What You Should Expect to Learn
• Transfer Matrix & Realization
– Here, we go in the reverse direction: given an input-output
transfer description, can we find a state-space representation
that describes the system’s behavior?
– There is in fact an infinite number of representations for the
same system, so we look for some “good” qualities:
• Minimal representation
• Canonical (controllable or observable) forms
• Jordan form (spectral description)
What You Should Expect to Learn
• State Feedback and State Estimators
– These are immediate applications of controllability and
observability concepts.
– More later …
What You Should Expect to Learn
• Introduction to Linear Sampled-Data Systems
– The basic theory of linear systems is discussed in continuous
time.
– However, in engineering problems and especially using digital
computers for control and measurement, we deal with sampled
data and therefore discrete-time systems.
– A quick overview of the discrete theory should equip you for
future learning & practice.
Lesson Plan
I
II
III
IV
Week No.
Topics
1
2
3
9/15
9/22
9/29
4
5
6
7
8
9
Introduction
Mathemetical Description of
Dynamical Systems
Review of Linear Algebra -Matrix Theory
State-Space Solution
Controllability &
Observability, Stability
Transfer Matrix Description
VI
and Realization
State Feedback and State
VII
Estimators
Introduction to Linear
VIII
Sampled-Data Systems
V
Homeworks (due date)
Literature Survey, Report
10
11
12
10/6 10/13 10/20 10/27 11/3 11/10 11/17 11/24 12/1
Mid-Term Exam
S/No.
13
14
15
16
12/8 12/15 12/22 12/29
17
18
1/5
1/12
Core Competency 核心能力
• 具備電機工程專業領域及背景知識
EE domain & background knowledge
• 具備探索新知與解決問題的能力
Continued learning and problem solving
• 具備獨立研究、撰寫論文與研發創新之能力
Independent research and development
• 掌握國際趨勢具全球化競爭挑戰能力
Global competitiveness
• 具備專業倫理道德及社會責任認知
Social ethics and moral duties
Course Map
Text & References
• C.T. Chen, Linear Systems Theory and Design,
3rd ed. Oxford University Press, 1999.
• T. Kailath, Linear Systems, Prentice-Hall, 1998.
• Franklin, Powell and Workman, Digital Control of
Dynamic Systems, 3rd ed. Addison Wesley, 1998.
• Kailath, Sayed and Hassibi, Linear Estimation, PrenticeHall, 2000.
• 鄭大鐘 , 《線形系統理論》，第二版，北京：清華大學

• http://staffweb.ncnu.edu.tw/kylum
Common Tools
Fortran/C++ Libraries
Free!
LINPACK (1970-1980)
BLAS (1979-)
Basic Linear Algebra
Subprogram
LAPACK (1980-)
Linear Algera
PACKage
Analytical Softwares
(4th generation
programming
languages)
• MATLAB (1984-)
By MathWorks
Commercial
• GNU Octave (1992-)