EEE588: Multivariable Control System Design

Arizona State University

EEE 588: Multivariable Control System Design

Table of Contents

Course Objective
Selected Topics
Homework Assignments
Previous Exams
Matlab M-files
Class Notes and Special Problems
Related Materials
Downloading Adobe Acrobat Reader

Course Objective

The goal of this course is to give graduate students and practicing engineers a thorough exposure to the state-of-the-art in multivariable control system design methodolgies. Emphasis will be placed on design/analysis tools and their use in solving real-world 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

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 Eigenvalue-Eigenvector 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 Value-Singular Vector Directionality Information

Controllability, An Existence Problem, Controllability Matrix, Rank Test, Eigenvalue-Left Eigenvector Test, Stabilizability

Observability, An Uniqueness Problem, Observability Matrix, Rank Test, Eigenvalue-Right Eigenvector Test, Detectability, Duality

MIMO Pole-Zero 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


Future Updates

This roadmap will be updated as the semester progresses. All updates will be publicized via email.


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 1-2.
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

EEE582 Spring 1993, Exam #1, Problems 1-3.
12. Problem #1 - SVD, Fundamental Spaces, Moore-Penrose, Projections (Scan over Moore-Penrose, 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 Spring-Mass-Dashpot (SMD) System

25. LTR Design for CH-47 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


Class Notes

Special Problems

Related Materials


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.

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