The Art of Controller Design

The MBuct Series in

Applied Control Engineering

- 2 -

The Art of Controller Design

Martin Braae

Smashwords Edition

Copyright 2017 Martin Braae

First published: 2017

Solely for use in education

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Table of Contents

Preface

1. A Design Method

1.1 Procedural Design

1.2. The Educational Approach

1.3. Software and Hardware

1.4. Design Brief for the Project

2. An Interesting Dynamic System

2.1. The Helicopter and Problem Formulation

2.2. The Model Form

2.3. Parameter Estimation

2.3.1. Open Loop Step Tests

2.3.2. First Method for Parameter Estimation

2.3.3. Tweaking the Parameter Values

2.3.4. Model Validation

2.3.5. Another Method for Parameter Estimation

2.4. Calibration of the Altitude Sensor

2.5. Observations

3. Controller Design

3.1. Design for Steady State Performance

3.1.1. Developing Design Equations

3.1.2. Decision Variable, Cost Function and Visualization --- Steady State

3.2. Design of Closed Loop Dynamics

3.2.1. Closed Loop Stability

3.2.2. Closed Loop Speed of Response

3.2.3. Fastest Attainable Speed of Response

3.2.4. Closed Loop Damping Factor

3.2.5. The Engineering Result

3.2.6. Brief Summary

3.3. Trimming the Design

3.4. Observation

4. Controller Implementation and First Improvement

4.1. Commissioning the Proposed Controller

4.2. Evaluation of Controller Performance

4.3. Addressing the Slow Dynamics

4.3.1. Controller Realization

4.3.2. Design Equations

4.3.3. Tweaking the Design

4.4. Controller Realization

4.5. A Few Observations

5. Design Robustness

5.1. Analysis of Robustness

5.2. Operating Performance

5.3. Analysis of the Loading Operation

5.4. An Improved Controller

5.4.1. The Engineering Outcome

6. A Prototype Controller

6.1. Tweaking the Design

6.1.1. Robustness of the Design

6.1.2. Sensitivity to Model Changes

6.1.3. Commissioning and Evaluation

6.2. Tweaking the Decision Space

7. Handling Ocean Swell

7.1. Response to Ocean Swell

7.2. Proposed Adjustment to the Design

7.3. Tracking of Sinusoidal Setpoints

7.4. Practical Aspects

7.4.1. Initial Conditions

7.4.2. Bumpless Transfer of Control

8. Summary and Conclusion

Tasks

#1. Familiarization

#2. Dynamic Modelling

#3. Design a Value for the Controller Gain

#... An Intellectual Challenge

#4. Design a Type One Controller

#... A Design Challenge

#5. A Realistic Design Project

Appendix A. The Importance of Visualization

A.1 Use of Theory

A.2. Reaching Beyond Current Theory

A.3. An Engineering Approach to Solving the Problem

A.4. Application to an Engineering Project

B. System Type Number

B.1. Constant Position Setpoints

B.2. Constant Velocity Setpoints

B.3. Constant Acceleration Setpoints

B.4. System Type Number

B.5. Examples

B.5.1. Position Errors

B.5.2. Velocity Errors

B.5.3. Acceleration Errors

B.6. Use in Engineering

B.7. Internal Model Principle

B.8. An Important Observation

C. The Routh-Hurwitz Methods

C.1. The Routh-Hurwitz Tests

C.1.1. The Routh-Hurwitz Array

C.1.2. An Example from Control Engineering

C.1.3. An Example Providing Useful Engineering Insight

C.2. A Modified Routh-Hurwitz Method

C.2.1. Fastest Possible Speed of Response

C.2.2. Generalization of the Pre-Test Mapping

D. The Root Locus Design Methods

D.1. Root Locus Plots

D.2. Use in Design

D.3. Modified Root Locus Plots

E. Transfer Functions as Electronic Circuits

E.1. Analog Implementation of Transfer Functions

E.1.1. Two Standard Circuits

E.1.2. General Transfer Function

E.1.2.1. Two Basic OpAmp Building Blocks

E.1.3. Evaluating the Circuit Signals

E.1.4. Optimizing the Electronic Circuit

E.2. Initial Conditions

E.2.1. Bumpless Transfer of Control

About the Author

The MBuct Series in Applied Control Engineering

Support Software

Preface

The addition of appropriate electronics to virtually any physical process (a.k.a. plant or system) can result in a composite system with significantly improved performance, provided it is well designed. As an example from one industrial project, the throughput of a mineral extraction plant was increased by well over ten percent simply by optimizing two constants in its digital controller and including an essential signal conditioning unit on its actuator. Such achievements are not uncommon in industrial projects yet tend to astound even knowledgeable onlookers with limited practical skills that preclude profitable outcomes in applications.

The electronic hardware, whether analog, digital or hybrid, is often designed with the aid of control engineering methods that utilize adequate mathematical descriptions of the process dynamics, in the form of transfer function (or state space) models. The control units for such processes emerge from an approach that is known as procedural design and are rarely invented. This means that the design process follows a well-trodden path from formulation of the problem to implementation of its solution. Even so, the approach does require skills in the art of controller design that is guided by suitable visualization. This simple fact is often obscured by layers of well-intentioned mathematical formulae that constitute the knowledge of control engineering.

The fundamental theory that is required for procedural design is usually provided in courses on control engineering that utilize second year levels of mathematics, particularly Complex numbers, Linear algebra, Differential equations and Laplace transforms with their Fourier transform companion. Unfortunately many of these courses fall short of imparting the actual skill of applying their vended knowledge, presumably on the tacit assumption that this blatant omission will be rectified elsewhere, possibly later on the job. In addition many course participants tend to focus on the skill of passing impending examination papers than on understanding the material thoroughly, let alone being able to apply it. Be that as it may, acquiring the skill of designing robust practical control systems with confidence and competence is essential to round off presentations of course material; yet modern production-line education rarely caters for this need even though "on the job experience is a notoriously expensive teacher". (Clearly this educational scenario is not entirely sound in that it is a bit like teaching budding surgeons all about anatomy and surgical instruments but omitting to add much on how these tools are to be wielded on patients.)

This book in the series on Applied Control Engineering builds on the material presented in Basic Modelling and Design and extends its hands-on approach to the design of control systems. There will be references to the first book but these are kept to a minimum to ensure that the present book can stand on its own. Textbook material like the Routh Hurwitz Methods is widely available and assumed to be known so it is not repeated. Instead such design methods are summarized in appendices so that the main text can focus on the central theme of this book, namely the art of designing controllers.

Martin Braae

Oakridge, March 2017

Chapter 1 A Design Method

Control engineering consists largely of a set of procedural design methods with clearly specified steps from modelling plant dynamics mathematically to commissioning the feedback control system.

Many diverse skills are involved in the successful design of useful thingamabobs, and all these skills are rarely found in one individual since they depend somewhat on specialist expertise, personality, past experiences and more. Luckily competent and far-sighted managers, who bring their own unique skills to the task, can assemble teams to focus a suitable mix of talents on any given problem.

The team might include inventors who have the ability to explore phenomena based on an intuitive "gut-feel" understanding that has been built up from years of experience, without giving too much attention or credence to existing theory, constraints and scholarly finesse. Their main strength lies in their courage to venture into territory that set theory might classify as unreachable truths. (i.e. Their approach may unearth thingamajigs that lie outside those that can be reached with current technologies, theories and methods.) On the other hand, scientists in the team are often at the other extreme, giving conscientious attention to scholarship and rigor based on axioms, theorems and fundamental laws of nature that allow exploration of the boundaries of reachable truths thereby avoiding unreachable falsehoods (like perpetual motion mechanisms.) In many ways engineers tend to be pragmatic, fit in somewhere within these two useful extremes, and can move between them as required by the problem.

Thus team members can view the same theory yet see different things; this book tends towards the engineering perspective.

1.1. Procedural Design

Since control engineering is largely characterized by procedural design methods, it is probably closer to the scientific than the inventive approach in that it uses the rigour of appropriate mathematics to design systems that enhance the performance of processes. With reference to a standard configuration for control systems:

Figure 1.1 A Two Degree of Freedom Control Configuration

procedural design systematically cycles through a well-established, highly successful sequence of steps:

* Study the process (existing or planned) to become familiar with its operational objectives and problems.

* Identify a key variable that encapsulates its performance and estimate the potential benefit(s) of controlling this variable to a set value.

* Formulate the problem mathematically as an open loop dynamic system {g(s)} with input and output signals {u(s), y(s)}. (The focus of this book is deliberately restricted to single-variable input-output methods.)

* Find the form of a mathematical model {g(s)} first and then its parameters so that it adequately approximates the steady state and transient behaviour of the process (relative to a set of mobile axes positioned within an operating region).

* Design a controller {k(s)} that will satisfy the steady state specifications for the controlled process while ensuring stable, damped and fast dynamics.

* Improve the initial controller by adding feedback and pre-filter elements {f(s), p(s)} so that the performance of the closed loop system meets the specifications set in the project brief.

* Check the control system {k(s), f(s), p(s)} for optimality and robustness, and tweak its design as necessary.

* Construct the control system in hardware and implement it