Servo Motors and Industrial Control Theory

By Riazollah Firoozian

Servo Motors and Industrial Control Theory presents the fundamentals of servo motors and control theory in a manner that is accessible to undergraduate students, as well as practitioners who may need updated information on the subject. Graphical methods for classical control theory have been replaced with examples using mathematical software, such as MathCad and MatLab, to solve real-life engineering control problems. State variable feedback control theory, which is generally not introduced until the Masters level, is introduced clearly and simply for students to approach complicated problems and examples.

• Language: English
• Publisher: Springer
• Release date: Dec 4, 2008
• ISBN: 9780387854601

Book preview

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Riazollah FiroozianMechanical Engineering SeriesServo Motors and Industrial Control Theory10.1007/978-0-387-85460-1© Springer-Verlag US 2009

Mechanical Engineering Series

Editor-in-ChiefFrederick F. Ling

Riazollah Firoozian

Servo Motors and Industrial Control Theory

A978-0-387-85460-1_BookFrontmatter_Figa_HTML.png

Riazollah Firoozian

Firoozian Electronics and Electro-Technique Co, Tehran, Iran

[e-mail]

ISSN 0941-5122

ISBN 978-0-387-85458-8e-ISBN 978-0-387-85460-1

Library of Congress Control Number: 2008933316

© Springer-Verlag US 2009

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

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Series Preface

Mechanical engineering, and engineering discipline born of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series is a series featuring graduate texts and research monographs intended to address the need for information in contemporary areas of mechanical engineering.

The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of series editors, each an expert in one of the areas of concentration. The names of the series editors are listed on page v of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology.

Preface

In recent years, there has been an increased activity in the field of automation, as manufacturers demand increased performance requirements from servo feed drive control systems and acceleration input. The selection of a servo system has become a technology of its own right. This book investigates the performance, characteristics, and design of various types of servo control systems, including electrical, DC, AC, Stepping, and Electrohydraulic servo motors. It is hoped that the topics discussed in this book will help designers achieve optimum performance for the required applications.

When designing servo control systems, several key parameters must be considered, including the effect of input signal (such as steady state error), the following error, and the effect of disturbance (such as external load). Several parameters will be introduced for each servo control system discussed in the book, so that the designer can easily compare several servo control systems.

A chapter is dedicated to Electro-rheological fluid-based servo control systems. These fluids have become commercially available and several parameters have been established for designing such systems, which are discussed in this book. Electro-rheological fluids are fluids that become solids when a voltage is placed across two plates with the gap of 0.5 to 1 mm. In this form, they can be used as valves, clutches, and catches. The design parameters for clutches and torque control are discussed.

The types of servo motors available in the market for high performance applications can be classified in two versions: electric and hydraulic. In the hydraulic category, the performance of the axial motor is considered. These motors are controlled with electrohydraulic servo valve. In the electric category, stepping, AC, and DC motors are discussed. For the stepping motors, the hybrid design is most suitable for high performance applications. These motors are controlled by transistor controlled switching devices.

In the DC motors' category, normal design, brushless, and moving coil motors will be discussed, as well as the four versions of controllers: Thyristor controlled with current frequencies of 50, 100, and 150 Hz, and the Pulse Width Modulated drive system, which has superior performance.

There has been an increase in the development of high power transistors and thyristors, thereby offering a method of controlling high current. Electric motors now can compete with electrohydraulic servo motors, which still have a wide range of applications, specifically in mining and heavy equipments, due to their high power to mass ratios. AC motors are still the first choice for constant speeds. DC motors, now at small to medium power requirements, are available for speed and position control. The invention of invertors allow AC motors to compete with DC motors.

As it was discussed earlier, the choice of a servo motor feed-drive for a specific application requires careful investigation at the design stage. This book analyzes the performance of each type of servo motor and establishes models of different complexity. For each type of velocity, feedback, and lead-lag network will be studied with the view of practical difficulties.

The mathematical model for all servo motors is similar and certain parameters will influence the design. This is discussed in detail for each type of servo motor. The mathematical model for a simple system will be considered first, however, as the required performance increases, the complexity of the mathematical model also increases, and computer programs are, therefore, developed for each type of servo system.

The root locus method will be considered for each servo motor with various mathematical complexities. The governing differential equations are then solved for step, velocity, and acceleration input. In this way, the steady state and following error will be calculated. The effect of disturbance such as external torque will also be calculated. In this way, various servo motors can be compared.

The first and second chapters of this book are dedicated to the modern control theory relevant to servo motors, so that prior knowledge of control theory is not required for understanding the book. Only feedback and state variable feedback control theory techniques are discussed. For digital control, the reader is referred to the relevant digital control books. Due to the widespread use of personal computers and software which makes detailed calculations simple, tedious graphical methods have been omitted.

The third chapter is dedicated to multivariable control theory. Servo control systems are inherently two input variable control systems, that is, input signal and external torque/force input. Chapter 4 is dedicated to electrical DC servo motors. The application of state variable feedback control strategy to high performance servo motors are discussed in detail in this chapter.

In Chapter 5, the characteristic behavior of stepping motors are investigated. In Chapter 6, the properties of AC motors controlled by variable frequency converters are discussed. The dynamic and static behavior of electrohydraulic servo motors are discussed in Chapter 7. The applications of Electrorheological fluids to servo motor applications are discussed in Chapter 8, while Chapter 9 gives a comparison of various types of servo motors and a design strategy for selecting a servo motor is discussed.

This book is the result of PhD work undertaken by the author at the University of Aston in Birmingham and several years of research and teaching work at the University of Liverpool and The University of Sheffield, UK, and Sharif University of Technology, Iran. This book is meant to serve as a practical guide for students or engineers, and includes an appendix with appropriate problems.

Riazollah Firoozian

Tehran

Contents

1 Feedback Control Theory1

1.1 Linear System1

1.2 Nonlinear Systems2

1.3 Linearization Technique3

1.4 Laplace Transform4

1.5 Transfer Function7

1.6 First Order Transfer Function8

1.7 Frequency Response10

1.8 Second Order Transfer Function11

1.9 Block Diagram Representation13

1.10 Frequency Response15

1.11 Conclusion16

2 Feedback Control Theory Continued17

2.1 Introduction17

2.2 Routh-Hurwitz Stability Criteria17

2.3 Root Locus Method19

2.4 Nyquist Plot23

2.5 Bode Diagram28

2.6 Steady State Error39

3 State Variable Feedback Control Theory43

3.1 Introduction43

3.2 State Variables44

3.3 Eigenvalues, Eigenvectors, and Characteristic Equation47

3.4 State Variable Feedback Control Theory50

3.5 Dynamic Observer55

3.6 Controllability and Observability56

3.7 Conclusion58

4 Electrical DC Servo Motors59

4.1 Types of DC Servo Motors59

4.2 Types of Power Unit60

4.3 Speed Torque Characteristic of DC Servo Motors62

4.4 DC Servo Motors in Open and Closed Loop Velocity Control63

4.5 DC Servo Motors in Closed Loop Position Control66

4.6 DC Servo Motors for Very High Performance Requirements68

4.7 Properties of Power Unit77

5 Stepping Servo Motors81

5.1 Principal Operation81

5.2 Stepping Motors with Small Step Angle83

5.3 Torque-Displacement Characteristic of a Stepping Motor84

5.4 Dynamic Response Characteristic over One Step Movement86

5.5 Speed-Torque Characteristic Behavior of Stepping Motors87

5.6 Stepping Motors for Position Control Applications88

6 AC Servo Motors91

6.1 Principle of Operation91

6.2 Variable Speed AC Motors91

6.3 Mathematical Model92

6.4 Frequency Converter97

6.5 Conclusion102

7 Electrohydraulic Servo Motors103

7.1 Introduction103

7.2 A Simple Mechanically Controlled Servo System104

7.3 Electrohydraulic Servo Valves107

7.4 Hydraulic Servo Motors109

7.5 A Numerical Investigation of the Transient Behavior of an Electrohydraulic Servo Motor Under Different Conditions112

7.6 Conclusion117

8 Actuators Based on Electro-Rheological Fluid119

8.1 Introduction119

8.2 Some Possible Applications of ER Fluid120

8.2.1 Valves120

8.2.2 ER Clutch and Catch Type Actuators120

8.2.3 Variable Dampers Based on ER Fluid121

8.3 Properties of ER Fluid in Flow Mode122

8.4 Properties of ER Fluid in Shear Mode127

8.5 Conclusion129

9 The Choice and Comparison of Servo Motors131

9.1 Introduction131

9.2 Theory and Performance Criteria132

9.3 Comparison of Results and Design Procedure135

9.4 An Example on Choosing a Servo Motor145

9.5 Conclusion146

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chapters 1 and 2)149

Appendix B: Exercise Problems on State Variable Feedback Control Theory (Chapter 3)173

Appendix C: Exercise Problems on Servo Motors (Chapters 4–9)197

C.1 Electrical DC Servo Motors197

C.2 Electrical Stepping Servo Motors209

C.3 Electrical AC Servo Motors213

C.4 Electrohydraulic Servo Motors216

C.5 Actuators based on Electro-Rheological Fluid (ERF)220

C.6 Selection of a Servo Motor for Particular Applications223

Index225

Riazollah FiroozianMechanical Engineering SeriesServo Motors and Industrial Control Theory10.1007/978-0-387-85460-1_1© Springer Science+Business Media, LLC 2009

1. Feedback Control Theory

Riazollah Firoozian¹  

(1)

Firoozian Electronics and Electro-Technique Co, Tehran, Iran

Riazollah Firoozian

Email: [e-mail]

In any system, if there exists a linear relationship between two variables, then it is said that it is a linear system.

1 Linear System

In any system, if there exists a linear relationship between two variables, then it is said that it is a linear system.

For example, the equation

$${\rm{y = K}}{\rm{x}}$$

(1.1)

represents a linear system. It means that if K is constant then the relationship (1.1) represents a linear relationship between two variables y and x. In general, any governing differential equations between two variables x and y in the form of

$$a_n \frac{{d^{\;n} }}{{dt^{\;n} }}y + a_{n - 1} \frac{{d^{\;n - 1} }}{{dt^{\;n - 1} }}y + ............ay = b_{\;m} \frac{{d^{\;m }}}{{dt^{\;m} }}x + ...............bx$$

(1.2)

is linear, where n and m represent the order of differential equations, and a n , b m are constants. For real system n>m, any other form of equations that is not similar to equation (1.2) is called nonlinear system.

There are extensive theories that deal with linear systems, but the theories on nonlinear systems are very complex and little.

Example 1

The circuit diagram of equivalent DC servo motors is shown in Fig. 1.1.

A978-0-387-85460-1_1_Fig1_HTML.jpg

Fig. 1.1

Equivalent circuit diagram of a DC servo motor

The governing differential equation may be written as

$$V_i = RI + L\frac{{dI}}{{dt}} + C_m \omega _m$$

(1.2)

where V i ,I,ω m are the input voltage, current, and angular speed. R and L are the resistance and inductance, respectively. This represents a linear system, where ω m is the output variable and V i represents the input voltage.

For DC servo motor, we can write

$$T = K_t I$$

(1.3)

$$T = J\frac{{d\omega _m }}{{dt}}$$

(1.4)

where K i ,J are the torque constant and rotor moment of inertia.

Eliminating T, from equations (1.3) and (1.4) and substituting for I in equation (1.2) yields

$$V_i = \frac{{RJ}}{{K_t^{} }}\frac{{d\omega _m }}{{dt}} + \frac{{LJ}}{{K_t }}\frac{{d^{\,2} \omega _m }}{{dt^{\,2} }} + C_m \omega _m$$

(1.5)

Equation (1.5) now represents a linear differential system, and in control terminology, V i is called the input variable and ω m is called the output variable. The equation (1.5) can be solved for ω m in terms of the input variable. In deriving equation (1.5), we ignore the external torque acting on the motor. If we consider the external torque, the governing differential equation would have two input variables and one output variable.

For linear systems, the principle of superposition holds. It means that if input x1 causes output y1 and input x2 causes output y2, then input x1+x2 causes output y1+y2. This is a powerful principle, and we will use it throughout this book.

2 Nonlinear Systems

There are different kinds of nonlinearities. For example, on–off control systems are inherently nonlinear. Transport lag, saturation, and transport lag are other kinds of nonlinearities. These kinds of nonlinearities cannot be solved with linear control theory. This is shown in Fig. 1.2 . There is complicated theory that covers discontinuous nonlinearities, but they are beyond the scope of this book. Most nonlinearities that exist in servo control systems are shown in Fig. 1.2.

A978-0-387-85460-1_1_Fig2_HTML.jpg

Fig. 1.2

Some discontinuous nonlinearities

For linearized equation, it is better to use Laplace Transform. In this way, the differential equations become algebraic equation in s. Throughout this book, the lower case s represents Laplace Transform.

Some nonlinearity is continuous, and they can be solved by the linearization technique. One example of this kind of nonlinearity is

$$y = Kx^{\,2}$$

(1.6)

This is shown in Fig. 1.3

A978-0-387-85460-1_1_Fig3_HTML.jpg

Fig. 1.3

A continuous nonlinearity

3 Linearization Technique

If there is a continuous nonlinearity in the form of

$$Y = F(X)$$

(1.7)

Assuming small perturbation from the equilibrium point, equation (1.7) can be linearized as

$$Y + y = F(X) + \frac{{dY}}{{dx}}dx$$

(1.8)

or it can be written as

$$y = \frac{{dY}}{{dx}}x$$

(1.9)

In equations (1.9) and (1.8) x,y represent small perturbation from the equilibrium point. Equation (1.9) can be written as

$$y = Kx$$

(1.10)

where

$$K = \frac{{dY}}{{dX}}$$

(1.11)

K is constant at an operating point. Throughout this book, the lower case variable represents small perturbation from equilibrium point. This is shown in Fig. 1.3.

Equation (1.7) represents one variable system. For a multivariable system, similar linearized equation can be obtained.

The solution of the governing equation simplifies if Laplace Transform is used.

4 Laplace Transform

By the definition, the Laplace Transform is defined as

$$F(s) = L\left[\; {f(t)} \right] = \int\limits_0^\infty {f(t)e^{ - st} dt}$$

(1.12)

By taking the Laplace Transform, the variable t is eliminated and the result is only function of s.

Equation (1.12) appears to be very complicated, and indeed for complicated transformation, the integral becomes very complex. Fortunately, for control systems only a few functions are needed.

Example 2

Constant A.

$$L(A) = \int\limits_0^\infty {Ae^{ - st} dt}$$

(1.13)

This is a simple integration, and the integral becomes

$$L(A) = \frac{A}{s}$$

(1.14)

The transformation of some common functions that are used in control are shown in Table 1.1 . There are a few important Laplace Transform that are often used in defining performance of servo control systems. These are constant values which represent step input; the ramp and acceleration inputs are other parameters that are often used in analysis to determine the performance.

Table 1.1

Laplace Transform of some common functions

The above table gives the Laplace Transform of the most useful function. Also note that the Laplace Transform is the only function of s only. This enables us to treat differential equation like algebraic function. Some examples will clarify this point, and they show how differential equation is converted to s domain and how the solution can be obtained from the Laplace Transform.

Example 3

Speed of DC motors assuming negligible inductance.

In this example, only a brief discussion on modeling of DC motors will be given. Full details of analysis will be discussed in a different chapter.

The voltage equation can be written as

$$v_i = Ri + C_m \omega _m$$

(1.15)

the torque current relation and the equation of motion become

$$T_m = J\frac{{d\omega _m }}{{dt}}$$

(1.16)

$$T_m = K_t i$$

(1.17)

The parameters K i ,C m are the torque and voltage constants of the motor respectively. Eliminating T m ,i from the three equations gives

$$v_i = \frac{{RJ}}{{K_t }}\frac{{d\omega _m }}{{dt}} + C_m \omega _m$$

(1.18)

Taking Laplace Transform from both sides of equation (1.18) and referring to Table 1.1 yields

$$V_i (s) = \frac{{RJ}}{{K_t }}s\omega _m (s) + C_m \omega _m (s)$$

(1.19)

rearranging gives

$$\omega _m (s) = \frac{1}{{\frac{{RJ}}{{K_t }}s + C_m }}V_i (s)$$

(1.20)

now equation (1.20) can be solved for any input function, for a step input of V, where L(V)= $$\frac{V}{s}$$ , using partial fraction method, we have

$$\omega _m (s) = \frac{1}{{C_m }}\left[ {\frac{{ - \tau }}{{\tau s + 1}} + \frac{1}{s}} \right]V$$

(1.21)

Taking the inverse of Laplace Transform of equation (1.21) gives

$$\omega _m (t) = \frac{V}{{C_m }}(1 - e^{ - \tau t} )$$

(1.22)

The above analysis shows that by taking Laplace Transform from the differential equations, they become algebraic equation. After re-arrangement using partial fraction method and referring to the Table 1.1 the solution can be obtained. Fortunately, there is no need to solve complicated Laplace Transforms. There are certain parameters that the performance of a system for various input signal can be obtained.

The closed loop control of velocity and angular position and the effect of external torque will be discussed in different chapters.

5 Transfer Function

Taking Laplace Transform from both sides of equation (1.2) and assuming zero initial conditions yield

$$(a_n \,s^{\; n} + a_{\,n - 1} s^{\;n - 1} ..................\,a)y(s) = (b_m s^{\;m} + b_{\;m - 1} s^{\;m - 1} ........... + b)x(s)$$

(1.23)

which can be written in the form of

$$\frac{{y(s)}}{{x(s)}} = \frac{{b_m s^{\; m} + b_{\;m - 1} s^{\;m - 1} ..............b}}{{a_n s^{\;n} + a_{\;n - 1} s^{\;n - 1} + ..............a}}$$

(1.24)

The right hand side of equation (1.24) is called the transfer function. a m … … . ,b m … … . are constants and y(s),x(s) are called the output and input variables. Equation (1.24) can be of any form but normally for real system n > m and n is called the order of transfer function.

The principle of superposition may be used for simple multivariable systems.

Once the transfer function is obtained, the following performance must be studied.

1.

Stability

2.

Transient response

3.

Steady state error for various standard input

4.

The above analysis should be carried out for various input functions

5.

Frequency response

There are some standard transfer functions that can be solved and exact solution may be obtained. In the following, some standard transfer function is studied.

6 First Order Transfer Function

First order transfer function in standard form may be written as

$$\frac{{y(s)}}{{x(s)}} = \frac{A}{{\tau s + 1}}$$

(1.25)

For unit step input of x(t)=1, the Laplace Transform becomes

$$x(s) = \frac{1}{s}$$ and substituting in equation (1.25) gives

$$y(s) = \frac{A}{{s(\tau s + 1)}}$$

(1.26)

Solving equation (1.26) by partial fraction yields

$$y(s) = \frac{1}{s} - \frac{\tau }{{\tau s + 1}}$$

(1.27)

Taking inverse Laplace Transform using Table 1.1, the solution becomes

$$y(t) = 1 - e^{\frac{t}{\tau }}$$

(1.28)

The solution graphically is shown in Fig. 1.4 . The important points on the graph are

$$t = 0 \quad y(t) = 0$$$$\begin{array}{l} t = \tau \quad {y(\tau ) = 0.632} \\ {t: = 3\tau } \quad {y(3\tau ): = 0.95} \\ {t = 5\tau } \quad {y(5\tau ) = 0.99} \end{array} $$A978-0-387-85460-1_1_Fig4_HTML.gif

Fig. 1.4

Step input response of first order lag

It shows that after t = τ, t = 3τ, t = 5τ, the output variable reaches its 63, 95, and 99% of its final value.

Similarly, the transient response for a ramp input which is a commonly used test signal can be obtained. For a ramp input of

$$x(t) = t$$

(1.29)

The Laplace Transform is

$$x(s) = \frac{1}{{s^2 }}$$

(1.30)

The output then becomes

$$y(s) = \frac{1}{{s^{\;2} (\tau s + 1)}} = \frac{A}{s} + \frac{B}{{s^{\;2} }} + \frac{C}{{\tau s + 1}}$$

(1.31)

Calculating the coefficients A, B, C, by equating the common factors in s, the output transfer equation becomes

$$y(s) = - \frac{\tau }{s} + \frac{1}{{s^2 }} + \frac{{\tau ^2 }}{{\tau s + 1}}$$

(1.32)

Taking the inverse of Laplace Transform by referring to Table 1.1, the solution becomes

$$y(t) = t - (\tau - \tau e^{ - \frac{t}{\tau }} )$$

(1.33)

The solution is shown graphically in Fig. 1.5 .

A978-0-387-85460-1_1_Fig5_HTML.gif

Fig. 1.5

Ramp input response of first order lag

It should be noted that both for step and ramp inputs, a unity gain was used. If different gain is used, the solution will be multiplied by that factor. For a ramp