Basic Modelling and Design

The MBuct Series in

Applied Control Engineering

- 1 -

Basic Modelling and Design

Martin Braae

Smashwords Edition

Copyright 2016 Martin Braae

First published: 2016

Solely for use in education

License Notes

This ebook is licensed for your personal enjoyment only. This ebook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each recipient. If you’re reading this book and did not purchase it, or it was not purchased for your enjoyment only, then please return to Smashwords.com or your favorite retailer and purchase your own copy. Thank you for respecting the hard work of this author.

MBuct(c)2016. All rights reserved.

Table of Contents

Preface

1. A Pragmatic Mathematical Abstraction

1.1 An Electric Motor System

1.2 A Simple Mathematical Model

1.3 Generalization

2. Design of a First Control System

2.1 Design of the Control System

2.2 Implementation of the Control Law

2.3 Validation of the Design by Simulation

3. An Improved Controller

3.1 A More Practical Design

3.2 The Effect of Loop Gain on Performance

3.3 Some Observations

3.4 The Effect of Changes to the Motor

3.5 A Crucial Observation

3.6 The Effect of the Operating Point on Performance

4. Abstraction for a General Control System

4.1 Problem Formulation

4.2 Summary

4.3 Sensors and Sensor Noise

4.4 Observations Focused on the Feedback Element

4.5 The Pre-Filter

5. Relevant Application and Mini Project

5.1 Experimental Estimation of Plant Models

5.2 Robustness

5.3 A Mini Project

6. Modelling System Transients

6.1 Dynamics

6.2 Time Signals

6.3 Experimental Estimation of a Steady State Curve

6.4 Systems and Signals

6.5 Dynamic Model for the Motor

6.6 Include a Field Coil Model

6.7 Comment on Zero Initial Conditions

6.8 Generalization

6.9 Procedural Summary

7. Standard Models

7.1 Electrical Models

7.2 Mechanical Models

7.3 Electro-Mechanical Models

7.4 Other Models

8. Block Diagram Algebra

8.1 The DC Motor

8.2 Summary of Block Diagram Rules

8.3 A Realistic Application

9. Signals and Systems

9.1 Commonly Encountered Signals

9.2 Common Properties of Signal Transforms

9.3 Use in Engineering

9.4 Prediction of System Response

9.5 A First Order Model

9.6 Cascaded Systems

9.7 A Second Order Model

9.8 A Differentiated Second Order Model

9.9 Dead-Time

9.10 Concept of System Poles

9.11 Pole Dominance

9.12 The Concept of System Zeros

9.13 Pole Zero Cancellation

9.14 Observation and Discussion

10. Basic System Identification

10.1 An Industrial Plant

10.2 Formulating the Control Problem

10.3 The Experimental Task

10.4 The Modelling Task

10.5 Final Results

Appendix A. Laplace Transformation

A.1 A Simple Analogy Based on Logarithms

A.2 The Laplace Transform

A.3 Laplace Properties for Differentials

A.4 Transfer Function Models

A.5 Laplace Property of Integrals

A.6 Laplace Property of Linearity

Appendix B. Application of Laplace Transforms

B.1 Cascade Combination of Two Differential Equations

B.2 Cascade Combination of Three Transfer Functions

Appendix C. Partial Fraction Expansion

C.1 The Method of Residuals

C2. An Example

C3. General Case

About the Author

The MBuct Series in Applied Control Engineering

Support Software

Preface

There can be little doubt that Control Engineering is a magic discipline, especially when it successfully solves tough practical problems found in large industrial systems. Take for instance the crucial contribution that it makes to the design of avionic equipment for automatically piloting aircraft --- a highly skilled task that requires years of training for pilots to master, let alone perfect. Yet the skill can be mimicked by a digital control system using code that is downloadable onto its microchip in a few microseconds. And what traveller alighting from a plane that has landed on a fog-bound airfield under autopilot control can fail to be awed by a system that control engineering methods have helped to create?

The subject of control engineering is very mathematical and over a number of decades has seemingly been infiltrated by mathematicians who may or may not have a background in engineering with its unique modus operandi. Consequently practicing control engineers are often faced with perfectly precise, internally consistent and somewhat incomprehensible solutions to control problems that appear in professional literature from time to time; some may have been demonstrated on laboratory installations though digital simulations are safer, faster and less costly, and hence more widespread. And so seasoned control engineers who have experienced the unexpected, perplexing surprises that industrial applications often dish up tend to be sceptical of such speculative, clinical solutions that may not be viable in practice, let alone optimal.

So you may well ask, "Why the fuss?"

Well, engineers who have mastered the mathematical tools of control engineering, have fully understood its fundamental concepts and can confidently apply them to produce robust engineering solutions that prove themselves in industrial applications are becoming somewhat of an endangered species. The slide towards extinction is hastened by slick software for digital simulations that can too glibly provide supporting evidence for ill-conceived solutions that have yet to be tested through exacting applications. Unfortunately such simulation results can turn out to be little better than smoke screens that, from a pragmatic perspective, may be hiding shaky proposals for practical application. In addition misguided human-years may be devoted to perfecting mathematical models of physical systems for such simulators; models that will forever remain approximations of reality.

Harsh words no doubt but necessary albeit with considerable regret since novel rigorous mathematical results are vital to the long-term relevance and hence survival of control engineering --- as is good simulation software for rapid feasibility studies. (And in case you had not yet noticed, this book is aimed squarely at aspiring applications engineers; practical engineers who thrive on working with real physical systems that are full of interesting, intellectually challenging phenomena awaiting discovery and intelligent attention. Early exposure to such inconvenient, unforeseen eventualities would engender a healthy suspicion of untested ideas but should also aim to awaken a desire for cautious evaluation of novel proposals rather than blind acceptance or outright rejection.)

One possible reason for the shrinking supply of applications-hardened control engineers could be the style adopted for modern education in control engineering. Senior students enter control courses with open, eager and unbiased minds suitably primed with two years of tertiary-level mathematics and one or more of physics, in which they have encountered complex numbers, linear algebra, differential equations and the all-important Laplace transformation with its Fourier companion that dominate control engineering theory. They can also wield the laws of Newton and Ohm, and have demonstrated skills in electronics; in examination venues at least, if nowhere else.

Unfortunately many control courses launch into veritable tirades of mathematical formulations that would dampen the enthusiasm of all but the toughest applications-orientated audience. And such introductions to control engineering tend to bury its key engineering features prematurely in superfluous mathematical detail so it is not surprising that many students fail to see the wood for the trees. What may be worse however is that it also alarms the very audience that the subject should be appealing to, namely those attendees that have the desire, innate intellectual ability and cerebral stamina to master the material and acquire the necessary engineering skills to apply its powerful methods aptly for solving tough practical problems that will ultimately be encountered in industry.

And so the resulting ignorance of engineering fundamentals and concomitant lack of appropriate capabilities may emerge later in industry under the guise of wannabe applications engineers or be recycled back into the educational system.

Hence it is suggested that the relevance and power of mathematics in control engineering (as well as its limitations) should be clearly understood from the start so that its methods can be utilized properly with confidence; not feared, and never believed until proven in an application or two. Under no circumstances should the mathematical content be allowed to overshadow the engineering characteristics of control engineering.

Consequently an application-centric introduction to the subject is essential; or so it seems. And an alternative approach to control engineering education is advocated; one in which its engineering features are emphasized up front and continually re-enforced by hands-on experiences that are aided by suitable mathematics yet free from excessive mathematical fervour. Thus the introductory approach should answer the question, "We have this engineering problem; what mathematical methods are needed to solve it? rather than With all this mathematics what engineering problem can be solved?" This necessitates a slower educational process in which applications are put squarely in the spotlight to teach the versatile knowledge and skills of control engineering comprehensively. The process cannot be hurried along and may increase initial training costs but these will be recouped many times over once the acquired expertise is confidently and competently exercised in industry to yield tangible and successful deliverables.

Proficient control engineers take time to reach maturation; as with all skills it cannot be overlooked that "practice makes perfect" and this implies making mistakes from which to learn, preferably before graduation. And the process starts by taking the first steps in the right direction rather than by crawling out later from under a pile of mathematical formulae designed to facilitate examination processes.

Martin Braae

Oakridge, March 2016

Chapter 1 A Pragmatic Mathematical Abstraction

Not all knowledge is amenable to flick-skimming; some needs to be read, studied, mulled over, tinkered with, conscientiously digested and tested out in suitable laboratory exercises to attain full comprehension. Then and only then can it be competently applied with confidence that is based on appropriate experience.

Skills still have to be learnt the hard way and perfected by practice.

So if you are too busy, don’t need the skill, or prefer flick-skimming knowledge bases in cyberspace then it is suggested that you read no further.

Professional engineers work with their minds rather than their hands and this does bias what follows although the mathematics is kept as simple as each situation allows. The aim is to develop pragmatic thinking based on practical experience in order to build up intuitive common sense that, with the support of appropriate mathematical skills, generally underpins successful engineering applications --- The forte of practical engineers.

1.1 An Electric Motor System

A simple yet not simplistic mathematical model is deduced for a DC motor. The model is derived from a set of experimental data points that is obtained by recording its input voltage and