Department of Electrical & Computer Engineering
University of Tennessee
Course Title: Multivariable Linear Control
System Design
Credit Hours: 3
Course Code: ECE 512
Prerequisite: ECE 511 Linear Systems
Theory
Instructor: Dr. Seddik M. Djouadi
E-Mail: djouadi@ece.utk.edu
Office # Ferris 307
Tel: (865) 974-5447
Office Hours: Wednesday 10:30AM – 12:00 PM, Friday 10:30AM-12:00PM or by
Appointment
Text Book:
Essentials of Robust Control, Kemin Zhou, Prentice-Hall 1998.
References:
Robust and Optimal Control, Kemin Zhou, J.C. Doyle, and K. Glover, Prentice Hall, 1996.
A Course in Robust Control Theory: A convex Approach, G.E. Dullerud, F.G. Paganini,
Springer-Verlag 1999.
Multivariable Feedback Control, Sigur Skogestad, Ian Postlethwaite, John Wiley & Sons 2001.
Feedback Control Theory, J.C. Doyle, B.A. Francis and A.R. Tannenbaum, McMillan, N.Y.,
1990.
Course Objectives:
Most systems encountered in science and engineering, and particularly in economics and
social science have highly uncertain dynamics, and their fundamental structure is very
complex. In the last two decades there has been explosive developments in control theory
dealing with dynamic uncertainty, which culminated in a new field: robust and optimal
control. The objective of this course is to introduce the students to analysis and synthesis
methods to design reliable, robust and efficient control systems.
The following schedule is approximate and subject to change. You will be responsible for
any changes that are announced in class, and for all the material covered in class.
Tentative Schedule
1
Week 1
• Linear spaces
• Eigenvalues and eigenvectors
• Invariant subspaces
• Vector and matrix norms
• Singular valued decomposition
• Semi-definite matrices
Week2
• Linear dynamical systems
• Controllability and observability
• Operation on systems
• State space realization
• Multivariable system poles and zeros
Week 3
• Hilbert spaces
• H2 and H ∞ spaces
• Power and spectral norms
• Computing L2 and H2 norms
Week 4
• Computing L ∞ and H ∞ norms
• Feedback structure
• Well-posedness of feedback loop
• Internal stability
Week 5
• Model Reduction
• Lyapunov equations and inequalities
• Observability operator and grammian
• Controllability operator and grammian
• Balanced realizations
Week 6
• Model uncertainty
• Small gain theorem
• Stability under unstructured uncertainty
Week 7
• Static state feedback stabilization via LMIs
• An LMI characterization of the stabilization problem
• Controller parametrization,
• Existence of stabilizing controllers
2
Week 8
• Parametrizing all stabilizing controller
• Coprime factorization approach
• H2 optimal control: motivation
• Standard LQG problem
• Extended LQR problem
Week 9
• Riccati equation and Hamiltonian matrix
• Synthesis, state feedback H2 via LMIs
• H ∞ control
• Problem formulation
Week 10
• The Kalman-Yakubovich-Popov lemma
• H ∞ synthesis
Week 11
• H ∞ synthesis
• General H ∞ solutions
Week 12-13
•
General H ∞ solutions
Week 14-15
• Review and Final Examination.
Grading (Tentative):
This course requires a major design project, where students must complete a semester-long
project requiring specification, design, and testing. Formal written project proposals and final
reports are required.
Homework
Project/Term Paper
Final Exam.
25% (No late assignment will be accepted unless you
have a legitimate excuse)
35%
40%
Grades
90% ≤ A≤ 100 %, 85% ≤ B+< 89 %, 80% ≤ B< 84 %, 75% ≤ C+< 79 %, 70% ≤ C< 74 %,
60% ≤ D< 69 %, F <60 %
3