EECS221A
Linear System Theory

Course information: UCB On-Line Course Catalog and Schedule of Classes

• Claire Tomlin
  721 Sutardja Dai Hall
  tomlin at eecs.berkeley.edu
  Office hours: Tu 1-2, W 11-12

• Insoon Yang
  iyang at eecs.berkeley.edu
  Office hours: M 5-6, W 5-6 (258 Cory)
  

This course provides a comprehensive introduction to the modeling, analysis, and control of linear dynamical systems. Topics include: A review of linear algebra and matrix theory. The solutions of linear equations. Least-squares approximation and linear programming. Linear ordinary differential equations: existence and uniqueness of solutions, the state-transition matrix and matrix exponential. Input-output and internal stability; the method of Lyapunov. Controllability and observability; basic realization theory. Control and observer design: pole placement, state estimation. Linear quadratic optimal control: Riccati equation and properties of the LQ regulator. Advanced topics such as robust control and hybrid system theory will be presented based on allowable time and interest from the class.

This course provides a solid foundation for students doing research that requires the design and use of dynamic models. Students in control, circuits, signal processing, communications and networking are encouraged to take this course.

• Linear Algebra: Fields, vector spaces, subspaces, bases, dimension, range and Null spaces, linear operators, norms, inner products, adjoints.
• Matrix Theory: Eigenspaces, Jordan form, Hermitian forms, positive definiteness, singular value decomposition, functions of matrices, spectral mapping theorem, computational aspects.
• Optimization: Linear equations, least-squares approximation, linear programming.
• Differential Equations: existence and uniqueness of solutions, Lipschitz continuity, linear ordinary differential equations, the notion of state, the state-transition matrix.
• Stability: Internal stability, input-output stability, the method of Lyapunov.
• Linear Systems - open-loop aspects: controllability and observability, duality, canonical forms, the Kalman decomposition, realization theory, minimal realizations.
• Linear systems - feedback aspects: pole placement, stabilizability and detectability, observers, state estimation, the separation principle.
• Linear quadratic optimal control: least-squares control and estimation, Riccati equations, properties of the LQ regulator.
• Advanced topics: robust control, hybrid systems.

• 8/23 Course Outline
• 8/23 Lecture Notes 1
• 8/23 Lecture Notes 2, Introduction and Functions, Fields and Vector Spaces, Subspaces and Bases
• 8/28 Lecture Notes 3, Linear Maps, Matrix Representation of Linear Maps, Change of Basis
• 8/30 Lecture Notes 4, Norms, Induced Norms, Inner Products, Adjoints
• 9/6 Lecture Notes 5
• 9/7 Addendum to Lecture Notes 4 and 5 (by Anil Aswani)
• 9/6 Lecture Notes 6, Hermitian Matrices, Singular Value Decomposition
• 9/11 Lecture Notes 7, Fundamental Theorem of Ordinary Differential Equations, Bellman-Gronwall Lemma
• 9/19 Lecture Notes 8, Dynamical Systems, Linearity and Time invariance
• 9/19 Lecture Notes 9, Linear Time Varying Systems, State Transition Matrix, Solutions to Linear Time Varying Systems, Jacobian Linearization
• 9/26 Lecture Notes 10, The Matrix Exponetial
• 9/30 Lecture Notes 11
• 10/18 Optimal Control Notes (by Jerry Ding)
• 9/30 Lecture Notes 12, Cayley Hamilton Theorem
• 9/30 Lecture Notes 13
• 10/19 Lecture Notes 14
• 10/19 Lecture Notes 15
• 11/7 Lecture Notes 16
• 11/7 Lecture Notes 17
• 11/13 Lecture Notes 18
• 11/13 Lecture Notes 19
• 11/18 Lecture Notes 20
• 11/18 Lecture Notes 21
• 11/18 Lecture Notes 22

• 8/30 Problem Set 1 (Due 9/11), Solution 1
• 9/12 Problem Set 2 (Due 9/20), Solution 2, Problem 5
• 9/20 Problem Set 3 (Due 10/2), Solution 3
• 10/2 Problem Set 4 (Due 10/11), Solution 4
• 10/10 Sample midterms from previous years: Fall 2010 Fall 2011
• 10/25 Problem Set 5 (Due 11/1), Solution 5
• 11/4 Problem Set 6 (Due 11/13), Solution 6
• 11/13 Problem Set 7 (Due 11/21), Solution 7
• 11/24 Problem Set 8 (Due 12/7), Solution 8
• 11/27 Sample final from 2011

Announcements

Discussion

• 8/31 Discussion 1
• 9/7 Discussion 2
• 9/14 Discussion 3
• 9/21 Discussion 4
• 9/28 Discussion 5
• 10/5 Discussion 6
• 10/12 Discussion 7 (Practice Midterm)
• 10/26 Discussion 8
• 11/2 Discussion 9
• 11/9 Discussion 10
• 11/16 Discussion 11
• 11/30 Discussion 12 (Final Review)

• Online Controls Tutorial in MATLAB
• MATLAB Documentation Help Desk
• ODE Software for MATLAB (by Professor Polking at Rice University)

Please sign the handout sheet on the first day of lectures (Thurs Aug 23), OR email Tomlin, so that your email will be added to the class mailing list.

Homework 40%
Midterm 20%
Final 40%

There is no required text book. I will provide notes throughout the term, however I encourage you to take your own notes during lecture.

References

Systems:

• F. Callier & C. A. Desoer, Linear Systems, Springer-Verlag, 1991.
• C.T. Chen, Linear Systems Theory and Design, Holt, Rinehart & Winston, 1999.
• T. Kailath, Linear Systems Theory, Prentice-Hall.
• R. Brockett, Finite-dimensional Linear Systems, Wiley.
• W. J. Rugh, Linear System Theory, Prentice-Hall, 1996.

• G. Golub and C. Van Loan, Matrix Computations, Johns Hopkins Press.
• M. Gantmacher, Theory of Matrices, Vol 1 & 2, Chelsea.
• G. Strang, Linear Algebra and its Applications, 3rd edition, 1988.

• J. Hale, Ordinary Differential Equations, Wiley.
• W. Rudin, Principles of Mathematical Analysis, Mcgraw-Hill.
• W. Rudin, Real and Complex Analysis, Mcgraw-Hill.