Linear Systems Theory 線 性系統理論 ( 239014) 2011 Fall, 4bcd

1. Linear Systems Theory線性系統理論 (239014)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

2. 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)

3. Time Line of Systems Theory in Control Engineering 1950’s Linear Systems Theory Ziegler-Nicholas LQR Kalman filter LQG 1960’s Transfermatrix LyapunovTheory Matrix-fraction description 1970’s AdaptiveControl LQG/LTR Sliding Mode 1980’s DynamicInversion H∞ -Synthesis MPC 1990’s Adaptive Back-Stepping 2000’s State-spacetechniques Nonlineartechniques Classical & frequencydomain techniques

4. Lesson Plan • Introduction • Mathemetical Description of Dynamical Systems • Review of Linear Algebra -- Matrix Theory • State-Space Solution • Controllability & Observability, Stability • Transfer Matrix Description and Realization • State Feedback and State Estimators • Introduction to Linear Sampled-Data Systems

5. 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).

6. 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

7. 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.

8. 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.

9. 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)

10. What You Should Expect to Learn • State Feedback and State Estimators • These are immediate applications of controllability and observability concepts. • More later …

11. 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.

12. Lesson Plan

13. 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

14. Course Map

15. 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, Prentice-Hall, 2000. • 鄭大鐘 , 《線形系統理論》，第二版，北京：清華大學出版社, 2002。 • http://staffweb.ncnu.edu.tw/kylum

16. Analytical Softwares (4th generation programming languages) • MATLAB (1984-) By MathWorksCommercial LINPACK (1970-1980) BLAS (1979-) Basic Linear Algebra Subprogram • GNU Octave (1992-) Open source, public license • Scilab (1990-) Open source Developed by INRIA, France LAPACK (1980-)Linear AlgeraPACKage Common Tools Fortran/C++ LibrariesFree!