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Assorted Notes for

MEM 255: Introduction To Controls

&

MEM 355: Performance Enhancement of

Dynamic Systems June 11, 2003

Professor Harry G. Kwatny Office: 3-151A

hkwatny@coe.drexel.edu http://www.pages.drexel.edu/faculty/hgk22

mailto:hkwatny@coe.drexel.edu

Table of Contents

Preliminaries ....................................................................................................5

Control Engineering & Mathematics.........................................................5

A Little History..........................................................................................6

Impact of the Digital Computer .................................................................8

Course Objectives ......................................................................................8

What is a Linear System? ..............................................................................11

Laplace Transform Summary ........................................................................13

Partial Fraction Expansion.......................................................................14

Case 1: distinct roots, 1 2 n ............................................. 14

Case 2: Nondistinct roots. .................................................................. 14

Example 1: ......................................................................................... 15

Example 2: ......................................................................................... 16

State Space & Transfer Function Models ......................................................18

Example: Accelerometer..........................................................................18

Reduction to State Space Form.......................................................... 19

The Transfer Function ....................................................................... 20

Analyzing The 2nd Order System.............................................................21

Time Domain: Response to a Step Input ........................................... 21

Frequency Domain: Response to Sinusoidal Inputs .......................... 24

Factoring the Quadratic: s sn2 2+ + n

2 ......................................... 27

Reduction of an nth Order ODE to State Equations .................................29

General Approach .............................................................................. 29

Example ............................................................................................. 30

MEM 255/355 Notes Professor Kwatny

Example: Inverted Pendulum...................................................................32

Equations via Newtons Law............................................................. 32

Equations via Lagranges Equations.................................................. 33

Reduction to State Space ................................................................... 33

Control Design...............................................................................................35

Preliminary Examples..............................................................................35

Cruise Control.................................................................................... 35

Automobile Directional Stability....................................................... 37

Problem Definition...................................................................................38

Regulation: Ultimate State Tracking Errors ...................................... 40

Transient Dynamics ........................................................................... 40

Root Locus Summary ..............................................................................41

Basic Rules ........................................................................................ 41

Derivation of behavior at infinity.................................................... 42

Additional Rules ................................................................................ 43

Nyquist Method .......................................................................................44

Basics ................................................................................................. 44

First Examples ................................................................................... 45

Gain & Phase Margin ........................................................................ 45

Performance: Sensitivity Peaks & Bandwidth................................... 45

The Sensitivity Functions ................................................................................45

Bode Waterbed Formula ..................................................................................46

Sensitivity Peaks ..............................................................................................46

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MEM 255/355 Notes Professor Kwatny

Bandwidth ........................................................................................................47

Example ............................................................................................. 47

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MEM 255/355 Notes Professor Kwatny

Preliminaries

These notes began with an attempt to bring coherence to what must seem like a

disconnected and abstract assortment of concepts and to students who confront them for

the first time. Unfortunately, they seem to grow each time I teach these courses to the

point where I worry that they might confuse rather than clarify.

Control Engineering & Mathematics

Control engineering is a discipline dealing with the design of devices, called control

systems, that influence the performance of a system through manipulation of control

devices on the basis of observations of system behavior. Mechanisms of this sort have

been employed for centuries but today they are truly ubiquitous. Control systems are an

essential part of chemical and manufacturing processes, communication systems, electric

power plants and systems, ground vehicles, ships, aircraft and spacecraft, robots and

manipulators, computers and so on.

During the last century engineering has been transformed from a craft into a

science. Those interested in profiting from societys thirst for new technology have found

it impossible to rely on time consuming trial and error to develop new products or resolve

problems in existing ones. Modern technologies like automobiles, aircraft,

telecommunications, and computers are too complex to thrive solely on vast compilations

of empirical data and decades of experience. Some intellectual constructs that organize

and explain essential facts and principles are required. So engineering, in general, has

come to adopt the style and methods of the natural sciences.

At the core of this point of view is the distinction between two thought processes:

the physical, and the mathematical. While engineers conceive of problems in the physical

world and construct solutions intended for application in the physical world, the solution

is almost always developed in the mathematical world. Todays engineers must be

comfortable with translating between them. In the mathematical domain we work with

abstractions of the physical. Abstraction is essential because most systems or devices

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MEM 255/355 Notes Professor Kwatny

involve so many irrelevant attributes that their complete characterization would only

obscure practical solutions. On the other hand, abstraction can be dangerous because it is

often easy to overlook important features and consequently to develop designs that fail to

perform adequately in the physical world. Herein lies the challenge and the art of

engineering.

Because of its inter-disciplinary nature and the breadth of its applications, control

engineering is especially reliant on a scientific perspective. Unifying principles that bring

together seemingly diverse subjects within a single inclusive concept are of great

significance. Mathematics, which may be regarded as the ultimate unifying principle in

science and technology, is very much at the heart of control engineering. In some circles

control theory is considered to be a branch of applied mathematics. But while

mathematics is a necessary part of control engineering there is much more to it. A control

system design project begins (with a problem definition) and ends (with a solution

implemented) in the physical world.

A Little History

It is almost certain that feedback controllers in primitive form existed many centuries

ago. But the earliest to receive prominence in the written history is James Watts fly-ball

governor, a device that received a patent in the late 18th century. Governors, or speed

regulators, were important in many systems of the late 18th and 19th centuries during

which time several alternatives were developed to meet increasingly stringent

performance requirements. A paper by the noted physicist James Clerk Maxwell, On

Governors, published in 1868 is considered to be the first paper dealing directly with the

theory of automatic control1. At that time period, the Russian engineer Vyshnegradskii

worked on similar control problems, publishing his results i