LINEAR SYSTEMS

Department of Mechanical Engineering

Carnegie Mellon University

Spring 1996

Schedule

Personnel

Lecturer:
Professor William Messner
Scaife Hall 303
phone: 268-2510
email: bmessner

Secretary:
Mildred Gibb
Scaife Hall 217
phone: 268-2488
email: mg1n

Texts:

Linear System Theory and Design, C-T Chen, Holt, Rhineha rd and Winston, 1984.

References on Reserve:

Systems
Brocket, Finite Dimensional Linear Systems
Delchamps, State Space and Input-Output Linear Systems
Harris and McClamroch, State Models of Dynamic Systems
Zadeh and Desoer, Linear System Theory: The State Space Approach

Controls
Nise, Control Systems Engineering

Linear Algebra
Golub and Van Loan, Matrix Computations
Strang, Linear Algebra and Its Applications

Analysis
Rudin, Principles of Mathematical Analysis

Matlab
Leonard and Levine, Using Matlab to Analyze and Design Control S ystems
Sigmon, Matlab Primer

Grading:

Mid-term Exam 25%
Final 25%
Homework 50%

Homework Policy

Homework will be assigned approximately every two weeks.

Introduction:

Linear systems are systems in which the input/output relationship satisfy the principle of superpostion. They are found in physics, mathematics, engineering, an d many other fields. In control engineering, there are very few systems which are truly linear, yet the approximation of linearity is often very good.

The different approaches to linear systems theory as they relate to controls can be roughly classified into frequency domain and state-space. The frequency doma in approach is associated with the technology of the 1930's function generators and oscilloscopes. Using this simple equipment input-output gain and phase relat ionships of linear time invariant systems could be determined. This information could be used to derive transfer functions and differential equation models. Con trollers could be designed based on frequency response information alone, or on the derived models. A big advantage of the frequency domain approach is that only i nput/output relationships are needed to design controllers--a physical model is not necessary.

In the 1950's the state space approach of modern control theory emerged with the development of digital computers. The state variable approach uses linear algeb ra, and with computers it became feasible to solve practical control problems using algebraic techniques. The state space methods are particularly well suited to multi-input/multi-output (MIMO) systems and nonlinear systems. The state space approach is now widely us ed because of its power, but frequency domain methods remain very popular for single-input/single-output (SISO) systems. Furthermore, in recent years there has been renewed interest in frequency domain methods as as theory for MIMO systems has been developed. Both of these methods make extensive use of concepts from linea r algebra.

Purpose and Approach

This course is intended to introduce students to important concepts in the study of linear systems, particularly those which are relevant to control. Th ese concepts are not only useful in the analysis and design of linear systems, but also are an important background for more advanced courses in control such a s multivariale control, optimal control, and nonlinear control.

Both frequency domain and state space approaches will be considered, but the state space method will be emphasized. The treatment in this course will be quite mathematical. Some of the material covered in class will not be in the textbook.

Objectives

By the end of the course, students should be able to do the following:

  1. Be familiar with the concepts of vectors and linear spaces. Understand the meaning of basis, span, orthogonality, subspace, and adjoint.
  2. Be familiar with the fundamentals of matrix algebra including rank, determinants, matrix inversion, transpose and adjoint, and elementary row and column oper ations. Also be familiar with functions of square matrices and the Cayley-Hamilton theorem.
  3. Understand the concept and use of eigenvalues and eigenvectors, how they are determined, and the concept and use of singular value decomposition.
  4. Have a basic knowledge of the properties of discrete-time systems.
  5. Write down the solution of linear differential and difference equations using the state transition matrix. Know the properties of the state transition matrix .
  6. Understand the concepts of controllability and observability and the related concepts of stabilizability and detectability.
  7. Use state feedback to arbitrarily place system poles. Use observers to reconstruct the initial state. Use output feedback for partial pole placement.
  8. Be familiar with the concept of Linear Quadratic Optimal Control.

Plan of the Course

The plan is to follow the order in which material is covered in the text. The approximate timetable of the course is listed below.

Reading

Topic

Lecture Number

Introduction to the course. Rings and Fields

1

Handy Relations, Parameterizing Controllers

2

2.1-2.5, 2.8

Vector Spaces and Linear Maps

3

1

First Representation Theorem, norms® 4

Norms and Induced Norms

5

4.1-4.2

State Transition Matrix

6

Adjoints

7

2.6

Eigenvalues and eigenvectors

8

Appendix E

Hermitian Matrices

9

Singular Value Decomposition

10

2.7

Second Representation TheoremÝ

11

2.7

Minimal PolynomialÝ

12

Functions of a Square Matrix

13

Appendix H

Zeros of Transmission

14

5

Controllability and Observability

15-17

Duality

18

Kalman Canonical Form

19

7.1-7.4

State Feedback and Observers

20

Linear Quadratic Regulator

21-23

7.1-7.4

To be determined

24-26

Revised: 23August 1996