Downloading Adobe Acrobat Reader
Course Objective
The goal of this course is to give graduate students and practicing engineers
a thorough exposure to the stateoftheart in multivariable control system
design methodolgies. Emphasis will be placed on design/analysis tools and
their use in solving realworld control problems. CAD homeworks involoving
high performance aircraft, helicopters, submarines, jet engines, chemical
processes, robotics and other physical systems will be the key vehicle for
conveying the main ideas. Students will be exposed to analytical tools needed
to read the literature and pursue advanced research in the areas of systems
and controls. The course is highly recommended for control engineers in the
local aerospace and process industries. It will be based on design courses
which have been taught at M.I.T., Honeywell SRC in Minnesota, and Raytheon
Missiles Systems in Massachussetts.
Selected Topics
 Modelling of Dynamical Systems, Nonlinear Dynamical Systems, Linearization
 Linear Dynamical Systems, Modal Analysis, Eigenstructures Assignment
 Linear Quadratic Regulator (LQR), Kalman Filters, Duality, Model Based Compensators
 Multivariable PoleZero Cancellations, MIMO Feedback Systems, Function Spaces
 Input/Output and Lyapunov Stability, MIMO Nyquist Stability Test
 Performance and Robust Performance Specifications, Internal Model Principle, Small Gain Theorem
 Structured Singular Values, LQG/LTR Design Methodology and Loop Shaping, Hinfinty, muSynthesis
 Youla Parameterization, Feedback linerization, Gain Scheduling, Control Saturation
Roadmap, Topics, Terms, and Assignments
The purpose of the following "roadmap" is to provide you with a detailed list of terms (terminology,
jargon) which you should use to guide you in your studying. You are responsible for all terms. The
schedule is also suppose to let you know what topics we will cover and the general order in which we
will cover them. Topics listed below may be rearranged as the semester progresses...depending on class needs.
The Big Picture, Reasons for Feedback, MIMO Unity Feedback System, Block Diagram Algebra, Sensitivity Transfer
Function, Complimentary Sensitivity Transfer Function, SISO Control  Frequency Domain Loop Shaping Ideas, Bode
Plots, Command Following, Disturbance Rejection, Noise Attenuation, Robustness to Plant Uncertainty, Stability
Robustness Margins, Nyquist Plots
Frequency Domain Performance Specification, Sinusoidal Analysis, Method of the Transfer Function, Low and High
Frequency Performance Barriers, Behavior Near Gain Crossover, Feedback Performance Limitations: Basic Inversion Ideas
Quantitative Measures of Size, Vector Norms, Matrix Norms, Induced Norms, Properties, Triangle Inequality,
Submultiplicative Property, Introduction to Singular Values
Modeling of MIMO LTI Dynamical System: An Introduction to State Space, Laplace Transforms, Realization of SISO
Transfer Functions, Zero Input Response (ZIR), Zero State Response (ZSR), Poles, Eigenvalues, Stability, Transfer
Function Matrices, Transmission Zeros, Natural Modes, Modal Analysis, Interpreting EigenvalueEigenvector Directionality
Information
Internal Model Principle: Command Following, Disturbance Rejection
Concepts from Linear Algebra, Existence, Uniqueness, Four (4) Fundamental Subspaces, Range Space (Column Space), Null
Space (Kernel), Row Space, Basis, Projection, Projection Theorem, Least Square Error Solution, Minimum Norm Solution
The Singular Value Decomposition (SVD), MIMO LTI System Frequency Responses, Interpreting Singular ValueSingular Vector
Directionality Information
Controllability, An Existence Problem, Controllability Matrix, Rank Test, EigenvalueLeft Eigenvector Test, Stabilizability
Observability, An Uniqueness Problem, Observability Matrix, Rank Test, EigenvalueRight Eigenvector Test, Detectability,
Duality
MIMO PoleZero Cancellations, Loss of Controllability and/or Observability
State Feedback, Linear Quadratic Regulator (LQR) Problem, Matrix Riccati Equation, Control Algebraic Riccati Equation
(CARE), Infinite Planning Horizon Properties
Kalman Bucy Filter (KBF) Problem, Filter Algebraic Riccati Equation (FARE), Infinite Observation Horizon Properties
Model Based Compensators, Linear Quadratic Guassian (LQG) Problem, Loop
Transfer Recovery (LTR) Ideas, Limitations, LQG/LTR Design Methodology
Modeling Uncertainty, Additive Modeling Error, Multiplicative Modeling
Error, Small Gain Theorem (SGT)  A Satbility Robustness Result,
Conservatism, Necessity of SGT, Structured Uncertainty, The Structured
Singular Value, Robust Performance
Functions Spaces,
General Control System Design Paradigm, Optimization Methods, Youla Parameterization
MUCH MORE WILL BE COMING FOLKS!
Future Updates
This roadmap will be updated as the semester progresses. All updates will be publicized via email.
Textbook
Kemin Zhou with John C. Doyle, "Essentials of Robust Control," Prentice Hall, 1998.
Homework Assignments
1. Computing the Singular Value Decomposition of a Matrix (Ref #911018)
2. Additive Matrix Perturbations (Ref #911022)
3. Matrix Mapping of a Circle
4. Four Fundamental Vector Spaces Associated With A Matrix (note)
5. A Note On Least Square Error Problems and Solutions (note)
6. A Note On Minimum Norm Problems (note)
7. Introduction to State Space, Eigenvalues, Eigenvectors (note)
8. State Space Realizations for SISO LTI Systems (note)
9. Trasmission Zero and Zero Direction Calculation (note)
EEE582 Spring 1992, Exam #1, Problems 12.
10. Problem #1  Basis for 4 Fundamental Subspaces, General Solution
11. Problem #2  Invariance Under Elementary Column Operations
A Note on Sinusoidal Analysis for LTI Systems
A NOTE ON SVD's AND STEADY STATE SINUSOIDAL ANALYSIS OF MIMO LTI SYSTEMS
NOTE UPDATE: WHAT CAN CAUSE A CLS TO BE UNSTABLE?
EEE582 Spring 1993, Exam #1, Problems 13.
12. Problem #1  SVD, Fundamental Spaces, MoorePenrose, Projections (Scan over MoorePenrose, focus on ideas)
13. Problem #2  General Solution, Fundamental Spaces, Minimum Norm Solutions
14. Problem #3  Least Square & Minimum Norm Problems
15. Introduction to Controllability (note)
16. Normal Equations for Finding Minimum Norm Solution (assigned via email)
17. F404 Engine Singular Value Plots and SVD Analysis
18. F8 Aircraft Singular Value Plots and SVD Analysis
19. Complete F404 Engine Analysis: Including Modal Analysis
20. Complete F8 Aircraft Analysis: Including Modal Analysis
21. State Feedback (LQ Servo) Design for F8 Aircraft
22. Model Based Compensator: Pole Placement and Internal Model Principle
23. LQG/LTR Design for F8 Aircraft
24. LQG/LTR Design for SpringMassDashpot (SMD) System
25. LTR Design for CH47 Tandem Rotor Helicopter
26. LQG/LTR Design for F404 Engine
EEE588 Project: Multivariable Control System Design (Spring 2000)
Please turn in all solutions within two (2) weeks of the assignment date.
Previous Exams
MATLAB MFiles
 F404 Engine
Includes state space representation (ssr), eigenvalues (poles), eigenvectors, modal analysis,
transmission zeros, transmission zero directions, transfer function analysis, controllability test,
observability test, frequency response analysis, singular values, dc analysis, singular value
decomposition (svd), step responses, variable scaling, scaled frequency response (singular values),
linear quadratic gaussian with loop transfer recovery at plant output (LQG/LTRO) design methodology,
dynamic augmentation, integral control, target loop singular value matching at all frequencies, target
loop design, kalman filter (KF) design methodology, target feedback system frequency responses, recovery
of target loop, cheap control problem, resulting feedback system frequency responses, postmultiplicative
stability robustness test.
 Generic Single Rotor Helicopter
Includes state space representation (ssr), eigenvalues (poles), transmission zeros, transfer functions,
frequency response singular values, LQR design, dynamic augmentation, LQ Servo adaptation for command
following, Open loop frequency response analysis at plant input and at error signal, Gain, phase, and
delay margins at plant input and at error signal, Closed loop step reference command analysis, Closed loop
frequency response analysis at plant input and at error signal, uncertainty analysis at plant input and
output, dependence of design properties on design parameters.
 CH47 Tandem Rotor Helicopter
Includes state space representation (ssr), eigenvalues (poles), eigenvectors, transmission zeros,
transmission zero directions, transfer function analysis, controllability test, observability test,
frequency response analysis, singular values, dc analysis, singular value decomposition (svd), loop transfer
recovery at plant output (LTRO) design methodology, target feedback system frequency responses, recovery of
target loop, cheap control problem, resulting feedback system frequency responses, postmultiplicative stability
robustness test.
 F8 Aircraft with Flaperon
Includes description of F8 model, flight condition, aerodynamic variable definitions, control variables,
state space representation (ssr), variable scaling, eigenvalues (poles), eigenvectors, transmission zeros,
transmission zero directions, transfer function analysis, controllability test, observability test, dc analysis,
singular value decomposition (svd), svd at phugoid and short period frequencies, frequency response, singular
values, step responses, integral dynamic augmentation, linear quadratic regulator (LQR) design methodology, lqr
design parameters, open loop ssr at error signal, closed loop ssr, closed loop step command following, lq open
loop singular values  at plant input and at plant output (error signal), lq closed loop sensitivity and
complementary sensitivity singular values  at plant input and at plant output (error signal).
 Blakelock 3DOF Commercial Transport (Landing Configuration) Macros
 Twin Lift Helicopter System Data and Model Generator
 Matrix Mapping Of A Circle
Specify any 2x2 matrix and see how it (generally) maps a unit circle into an ellipse. The major axis of the ellipse
is associated with the maximum singular value and corresponding left singular vector of the matrix. The minor axis
of the ellipse is associated with the minimum singular value and corresponding left singular vector of the matrix.
The macro draws the unit circle and the resulting ellipse. The right and left singular vector information is also plotted.

Visualizing Robust Performance – Simple Examples
 Impact of BW on Closed Loop Performance for Unstable First Order Plant (PI Controller)

Impact of BW on Closed Loop Performance for Unstable NonMinimum Phase First Order Plant (PI Controller)

Magnitude of S and T vs Loop Phase and Magnitude Variations
 2D Heat Equation PDE Model
 Condition # of Scaled Matrix vs Condition # of Scaling Matrix
 2DOF PUMA 560 Robotic Manipulator
 2DOF PUMA 560 Robotic Manipulator: Hinfinity Control Design
 F8 Aircraft: Hinfinity Control Design
 Impact of Plant Coupling on Decentralized PI Control Law mfile, Slides
Class Notes
Special Problems
Related Materials
Mathtools.net
 Mathtools.net is a technical computing portal for
all scientific and engineering needs. The portal is free and contains over 20,000 useful links to
technical computing programmers, covering MATLAB, Java, Excel, C/C+, Fortran and others.
References
Signals and Systems
 Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab, "Signals and Systems," Prentice Hall, 1997.
Classical Controls and Feedback Systems
 Hendrik W. Bode, "Network Analysis and Feedback Amplifier Design," D. Van Nostrand Company, 1945.
 Jan C. Willems, "The Analysis of Feedback Systems," MIT Press, 1969.
 C.A. Desoer and M. Vidyasagar, "Feedback Systems: Input/Output Properties," Academic Press, 1975.
 Katsuhiko Ogata, "Modern Control Engineering," Prentice Hall, 3rd Edition, 1997.
Linear Algebra
 Gilbert Strang, "Linear Algebra and Its Applications," Academic Press, 1980.
 Gene H. Golub and Charles F. Van Loan, "Matrix Computations," Johns Hopkins University Press, 1983.
 James M. Ortega, "Matrix Theory: A Second Course," Plenum Press, 1987.
Probability
 Athanasios Papoulis, "Probability, Random Variables, and Stochastic Processes," McGraw Hill, 1965.
 Alvin W. Drake, "Fundamentals of Applied Probability Theory," McGraw Hill, 1967.
 Paul L. Meyer, "Introductory Probability and Statistical Applications," Addison Wesley, 1970.
Linear Systems
 Roger W. Brockett, "Finite Dimensional Linear Systems," John Wiley and Sons, 1969.
 C.T. Chen, Introduction to Linear Systems, Holt, Rinehart, Winston, 1970.
 W. Murray Wonham, "Linear Multivariable Control: A Geometric Approach," SpringerVerlag, 1978.
 David G. Luenberger, Introduction to Dynamic Systems: Theory, Models, & Applications," Wiley, 1979.
 Thomas Kailath, "Linear Systems," Prentice Hall, 1980.
 Wilson J. Rugh, "Linear System Theory," Prentice Hall, 1996.
 Panos J. Antsakalis and Anthony N. Michel, "Linear Systems," McGraw Hill, 1997.
Optimal Control
 Huibert Kwakernaak and Raphael Sivan, "Linear Optimal Control Systems," Wiley, 1972.
Analysis
 Erwin Kreyszig, "Introductory Functional Analysis and Applications," Krieger Publishing, 1989.
Selected Master's Theses
 ChenI Lim's Master's Thesis, "Development of Interactive Modeling, Simulation, Animation, and RealTime
Control (MoSART) Tools for Research and Education" (PDF Version)
(Download Zipped Version)
 Carlos Enrique Rios Sanchez's Master's Thesis, "Development of an Extensible Interactive Modeling,
Simulation, Animation, and RealTIme Control (MoSART) CartPendulumSeesaw Environment: A Tool for Enhancing
Research and Education" (PDF Version)
(Download Zipped Version)
 Jose Ignacio HernandezSanchez's Master's Thesis, "Development of an Extensible Interactive Modeling,
Simulation, Animation, and RealTime Control (MoSART)
Flexible Inverted Pendulum Environment: A Tool for Enhancing Research and Education"
(PDF Version)
(Download Zipped Version)
 SungSik Kwak's Master's Thesis, "Development of an Extensible Interactive Modeling,
Simulation, Animation, and RealTime Control (MoSART)
Submarine Environment: A Tool for Enhancing Research and Education"
(PDF Version)
(Download Zipped Version)
 TaeYoung Kim's Master's Thesis, "Development of an Extensible Interactive Modeling,
Simulation, Animation, and RealTime Control (MoSART)
Aircraft Environment: A Tool for Enhancing Research and Education"
(PDF Version)
(Download Zipped Version)
Selected MoSART Environments
 MoSART ALab Toolbox V1.05: Contains files necessary to use the MoSART ALab
Environments which are listed below, Unzip and follow instructions in readme file
(Download Zip File)
 MoSART Aircraft Toolbox V2.00: Contains files necessary to use MoSART Aircraft
Environment, Unzip and follow instructions in readme file
(Download Zip File)
Many exams, problems, and notes on this page can be read using Adobe's Acrobat Reader. A free copy of this
program for many operating systems (Windows 3.1, 95, NT, Mac, Unix) can be
downloaded from Adobe for your computer.
Return to Dr. Rodriguez's Home Page
Email questions for Dr. Rodriguez