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 1990’s 2000’s 1980’s 1970’s 1960’s Classical & frequency domain techniques State-space techniques H∞H∞  -Synthesis LQR LQG Nonlinear techniques Ziegler-Nicholas Dynamic Inversion Sliding Mode MPC Adaptive Control LQG/LTR Adaptive Back-Stepping 1950’s Kalman filter Lyapunov Theory Linear Systems Theory Transfer matrix Matrix- fraction description

4. 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

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, 3 rd 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. Common Tools LINPACK (1970-1980) LAPACK (1980-) Linear Algera PACKage BLAS (1979-) Basic Linear Algebra Subprogram Fortran/C++ Libraries Free! MATLAB (1984-) By MathWorks Commercial GNU Octave (1992-) Open source, public license Scilab (1990-) Open source Developed by INRIA, France Analytical Softwares (4 th generation programming languages)