Digital control systems theory hardware software constantine h houpis gary b lamont

xii CONTENTS 2.4.5 Correlation of the Control Ratio with Frequency and Time 2.7 Synthesizing a Desired Tracking Control Ratio with a Unit-Step Input 43 3.4 Sampling Process Frequency D

DIGITAL

CONTROL SYSTEMS THEORY, HARDWARE, SOFTWARE

SECOND EDITION

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Constantine H Houpis, Ph.D Gary B Lamont, Ph.D Professors of Electrical Engineering

School of Engineering Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio

Theory, Hardware, Software

INTERNATIONAL EDITION 1992

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Library of Congress Cataloging-in-Publication Data

Houpis, Constantine H

Digital control systems: theory, hardware, software / Constantine

H Houpis, Gary B Lamont.-2nd ed

p cm.-(McGraw-Hill series in electrical engineering

ABOUT THE AUTHORS

Constantine H Houpis is a professor of electrical engineering at the Air Force Institute of Technology, where he has taught since 1952 He also supervises the doctoral program in electrical engineering at the Institute and is a consultant to the Air Force Flight Control Directorate at Wright-Patterson Air Force Base, Ohio Previously,

he had taught at Wayne State University Professor Houpis received his Ph.D in electrical engineering at the University of Wyoming

Dr Houpis is the author of numerous control theory technical articles and textbooks and is a member of Tau Beta Pi, Eta Kappa Nu, Sigma Xi, and ASEE, and is a Fellow of IEEE

Gary B Lamont is a professor of electrical and computer engineering at the Air Force Institute of Technology In addition to teaching graduate courses in digital control theory, computer engineering, and computer science, he does consulting in these areas Prior to joining the faculty of the Institute in 1970, he was a systems analyst and engineer at the Aerospace Division of Honeywell Professor Lamont recieved his Bachelor of Physics, MSEE, and Ph.D degrees from the University of Minnesota

Dr Lamont has authored numerous papers on automatic control (conventional, modem, and digital), expert systems, parallel processing algorithms, computer-aided design, and educational techniques He is a member of Eta Kappa Nu, Tau Beta Pi, ASEE, ACM, and a member of IEEE

1 4 Control-System Analysis and Synthesis 1 2

1 5 The Interdisciplinary Field of Digital Control 13

1 7 Nature of the Engineering Control Problem 1 6

2.4.3 Time-Response Characteristics of a Sixth-Order Plant 3 1 2.4.4 Correlation between Frequency and Time Domains 32

xii CONTENTS

2.4.5 Correlation of the Control Ratio with Frequency and Time

2.7 Synthesizing a Desired Tracking Control Ratio with a Unit-Step Input 43

3.4 Sampling Process (Frequency Domain Analysis) 73

4.2 Definition and Determination of the z-Plane Transform Z 100

4.3.2 Mapping of the Constant Frequency Loci 1 14 4.3.3 Mapping of the Constant Damping-Coefficient Loci 1 14 4.3.4 Mapping of the Constant Damping-Ratio Loci 1 15

4.7 1 Open-Loop Hybrid Sampled-Data Control System 1 3 1 4.7.2 Open-Loop Discrete-Input-Data Control System 1 36 4.7.3 Closed-Loop Sampled-Data Control System 137 4.7.4 Signal Flow Graphs for Hybrid Systems (HSFG) 140

5 Discrete Control Analysis 1 5 1

5.2.4 Extended z-Domain Stability Analysis: Jury's Stability Test 157 5.3 Steady-State Error Analysis for Stable Systems 160 5.3 1 Steady-State Error-Coefficient Formulation 162 5.3.2 Evaluation of Steady-State Error Coefficients 163 5.3.3 Use of Steady-State Error Coefficients 164

7 1 1 Integrated AID and D/ A Interfaces 248

8.5 Software Engineering in Digital Control Systems 278

8.10 Structured Programming and Implementation 293

8.13 Real-Time Operating Systems for Digital Control 299 8.1 3 1 Real-Time Operating Systems Requirements 301 8.13.2 Simple Real-Time Operating System 302

9.5 Linear System Response to Random Signals 319

9.5.3 Spectral Density of Linear Discrete Systems 324 9.6 Vector-Matrix Representation of Random Processes 326

9.7 Summary 329

10 Finite Word Length and Compensator Structure 330

10.3 Compensator Structure and Arithmetic Errors 341

1 0.4 Compensator Coefficient Representation 348

10.7 Limit-Cycle Phenomenon Due to Quantization 358

11 Cascade Compensation-Digitization

1 1 3 Guillemin-Truxal (GT) Compensation Method 374

1 1 7.3 Pseudo-Continuous Time (PeT) Approach 389

12 Cascade Compensation-Direct

12.7 Proportional Integral Derivative (PID) Controller 404

14.4.5 State-Variable Representation Summary 464 14.5 State-Variable Representation in the z Domain 464

15.3 State-Variable Feedback: Parameter Insensitivity 479

15.5 State-Feedback H -Equivalent Digital Control System 483

15.7 Frequency- Domain Compensation Design Using Mean-Square Error

15.9 Direct s-Plane to w-Plane Transformation 499

1 5 9.3 Accuracy Considerations of the w Transformation 503

1 5.9.4 Model Relationship as T � Zero 505

16 Discrete Quantitative Feedback Technique 508

16.2 Continuous MISO and MIMO QFf Approach 509

16.3.1 Analog QFI' Design Procedure for an nmp Plant 5 1 4 16.4 Discrete MISO Model with Plant Uncertainty 5 1 7

16.5.1 Closed-Loop System Specifications 520

18.4 1 General Second-Order System Optimal Controllers 574 18.4.2 Discrete Riccati Equation Solution 575

xviii CONTENTS

19 Discrete Estimation and Stochastic Control 583

19.3.4 Solution to Optimal Filter Equations 597

There is a need for a fundamental textbook on sampled-data control theory and applications that emphasizes the use of the small digital computer as a controller The implementation of a digital controller allows design flexibility and system extendability

in an efficient and effective manner as compared to analog components The real­time adaptability of mode changes due to plant parameter variations, environmental changes, and requirements modifications usually require a digital computer

This text accomplishes the objective of providing sampled-data control system topics mainly for upper-level undergraduate and first-year graduate students It effec­tively merges and interrelates the two general areas which are vital to a practicing digital control engineer: discrete and sampled-data control theory and computer en­gineering This textbook provides a clear, understandable, logical development and motivated account that spans sampled-data control theory with computer engineering

as an integrated entity

The minimum background required for this book is a fundamental course in continuous-time control systems Some higher-order language programming expe­rience would be useful but not necessary in appreciating the software engineering sections The textbook has been developed to achieve a minimal satisfactory under­standing of discrete-data and sampled-data systems This development is based on the reader having a mathematical background in differential equations, integral calculus, and Laplace transforms

The authors have tried to exert meticuluous care with explanations, diagrams, calculations, tables, and symbols The text provides a strong, comprehensive, and illuminating account of those elements of conventional control theory which have relevance in the design and analysis of sampled-data control systems The variety of different design techniques presented contributes to the development of the student's working understanding of what A T Fuller has called "the enigmatic control system."

To provide a coherent development of the subject, formal proofs and lemmas have

been eschewed with an organization that draws the perceptive student steadily and surely into the demanding theory of multirate, multivariable control systems

The text, which is summarized in the following paragraphs, introduces the con­cepts of sampled-data theory, relates it to continuous analysis methods as appropriate, integrates computer engineering concepts with digital control implementation, and presents both scalar and modem control design and synthesis techniques

Chapter 1 introduces the sampling process model while the development of ideal impulse sampling is detailed in Chapter 3 The use of linear difference equations to model the performance of sampled-data control systems is also presented in Chapter 3

Many of the analysis and design techniques of sampled-data control systems emphasized are extensions of continuous control-theory methods that are reviewed in Chapter 2, including the modeling of a desired system control ratio for tracking and disturbance rejection

Chapter 4 introduces the Z-transform (zee transform) as a method for the analysis and design of sampled-data control systems in the z-plane The correlation between the pole-zero pattern in the s-and z-planes is presented with respect to time­response characteristics The properties and mathematical representations of open-loop and closed-loop sampled-data control systems are developed, and their corresponding block diagram and signal-flow representations are given

Stability analysis as presented in Chapter 5 is accomplished by applying Jury's stability test or Routh's stability criterion in the z- and w-domains respectively The w' -transformation is developed as an approximation to the Laplace transform Chapter 5 binds together the s-, z-, and w'-domain analyses with respect to time­response characteristics by means of the root-locus and frequency-response methods

In Chapter 6 a general sampled-data control-system design technique is devel­oped The Pade approximation, the Tustin transformation, and the pseudo-continuous­time (PCT) control system model are involved in this technique

The basic organization of analog-to-digital (AID) and digital-to-analog (D/A) converters and I/O programming is presented in Chapter 7 The objective is to develop

a detailed understanding of conversion processes in a digital control system General control transducers are also presented, providing insight into their construction, accu­racy, and utilization in control systems

A concise but integrated presentation of the fundamentals of computer engineer­ing as related to digital control system is set forth in Chapter 8 This chapter provides

an introduction to logical operations, hardware architecture, software engineering, and real-time operating systems as used in the detailed design of digital controllers and their implementations

Chapter 9 develops the foundation for the statistical analysis of finite word length discussed in Chapter 1 0 and for optimal estimation considered in Chapter 19, and presents the fundamentals of continuous and discrete random processes Using this sta­tistical background, Chapter 1 0 focuses on modeling the effects of finite word length

Chapters 1 1 and 12 discuss two approaches of analyzing and designing a sampled­data, cascade-compensated control system: in the direct (DIR) techniques analysis and synthesis are done in the z-plane and for the digitization (DIG) technique all work is

accomplished in either the s- or Wi -plane The interrelations and comparison of the results obtained by using these design methods and the effect of T on these results are thoroughly discussed with the PID (proportional integral derivative) controller used •

as an example

Feedback-compensated digital systems are discussed in Chapter 13 The pre­sentation deals with the tracking problem requiring that the system output follow the system input and the disturbance rejection model

Chapter 14 introduces the state-variable methods of representing a sampled­data control system and includes the analysis of system performance by state-space representation Chapter 1 5 develops a design method for minimizing the effect of plant parameter variations on the system output

Chapter 1 6 presents for the first time in any textbook the quantitative feedback theory (QFf) design technique for sampled-data control systems This is a power­ful technique in designing a robust control system for plants with plant parameter uncertainty and disturbances

Chapter 17 develops the concepts of controllability and observability with re­duced state controllers and observers Chapter 1 8 presents the development of discrete optimal controllers along with their digital-computer implementation Chapter 19 discusses the design of estimators for multi-inputs, multi-output systems with emphasis

on identification, and also addresses the concepts of digital adaptive control and stochastic control systems Various appendices support the understanding of various models and design techniques

The two important and major features of Digital Control Systems : Theory, Hard­ware, Software are uniqueness and flexibility of use The unique features are integration

of discrete and sampled-data control theory with computer engineering; degree of accuracy needed as T becomes smaller because of its impact on calculations and digital implementation, with emphasis on computer-aided design (CAD); discussion

of text examples by integrating control theory and digital computer implementation; extensive development of system analysis and design by root-locus techniques in various domains; development of the pseudo-continuous-time (PCT) control system design and analysis; development of discrete optimal control and discrete optimal estimation methods; and emphasis of control law (algorithm) implementation in a digital computer Various design approaches are presented and compared in terms of advantages and disadvantages in order to provide the digital control engineer with appropriate criteria for selecting an applicable method

The text is designed so that, depending on the course and the instructor, chapters and/or sections can be presented in a variety of sequences It may be used in ad­vanced undergraduate and first-year graduate courses (two quarters or two semesters

in length); for a short course (40 lecture hours); as a self-study text; and for a single course restricted to only scalar sampled-data control theory, Chapter 1 to 6 and 1 1

to 13 Of course, the appropriate elements of computer engineering and transducer modeling as developed in Chapters 7 through 10 are critical for detailed design and implementation of digital controllers Chapters 14 to 1 9 are included for modem digital control and the estimation theory

xxii PREFACE

Use of a CACSD (computer-aided control system design) package is advan­tageous in providing insight to design and analysis of digital control systems by employing the various examples and exercises in the text Also, a physical laboratory for student implementation of various digital controller designs is of considerable educational benefit (see the Instructor' s Manual for a series of suggested experiments and facilities and for case studies)

We feel that with the mastery of the text material the student should be able to analyze and design sampled-data control systems and implement digital controllers as well as to provide the background for more extensive studies in the area

The goals of achieving an integrated sampled-data theory and computer en­gineering text have been extended to this second edition Additional clarification of design processes is included and applied to real-world design problems Modem digital control techniques have been added that employ quantitative feedback, optimal control, stochastic control, and adaptive control theory methods

COMPUTER-AIDED DIGITAL CONTROL

SYSTEM DESIGN

In the understanding of digital-control system-design methods the use of computer­aided design (CAD) packages can provide a rich educational environment The student can validate textbook examples as well as various textbook exercises There are various commercial CAD packages (CNTLC, MATLAB, MATRIXx, SIMON ) for control that could be used ICECAP-PC (Interactive Control Engineering Computer Aided Package for the Personal Computer) has been developed at the Air Force Institute of Tech­nology and in conjunction with a number of universities to provide a public domain CACSD package The purpose of ICECAP-PC is to provide an educational CAD program for control engineering students to analyze, design, synthesize and simulate control systems (continuous and discrete) The public domain MS-DOS ICECAP-PC executable program is available via the Instructor's Manual Also included is a series of macro files for various examples and problems in this text and a users manual (ASCII file) If you are interested, an ICECAP source code can be provided for enhancements

as part of a university/industry consortium

We express our thanks to the students who have used this book and to the faculty who have reviewed it for their helpful comments and corrections Particular appreciation is expressed to Dr J J D'Azzo, Head of the Electrical Engineering De­partment; Professor Emeritus R B Fontana, and the digital control students of the Air Force Institute of Technology for their inspiration The continual encouragement and review of the text by Dr T 1 Higgins, Professor Emeritus, University of Wisconsin, has been a very important catalyst in the completion of this second edition Special thanks to Dr Donald McLean, Senior Lecturer, University of Southhampton, England, formerly a visiting Professor at the Air Force Institute of Technology, who provided a detailed review of the first edition The following reviewers also gave many valuable comments: Richard K Blandford, Evansville University; Christos G Cassandras, University of Massachusetts-Amherst; John F Dorsey, Georgia Institute of Technol­ogy; Warren J Guy, Jr., Lafayette College; Nairn A Kheir, Oakland University; and

Renjeng Su, University of Colorado-Boulder The perception and insight of all these individuals has contributed extensively to the clarity and rigor of the presentation Our association with them has been a enlightening and refreshing experience Finally, we thank McGraw-Hill, especially Eleanor Castellano for her help and encouragement

Constantine H Houpis Gary B Lamont

Engineers and scientists attempt to design control systems to perfection so that ideal system perfonnance is achieved For a practical control system, physical realiz­ability of components limits the extent to which the ideal system perfonnance can be achieved The advent of the digital computer as a computational device penn its more accurate control in general but also constrains the speed of operation However, this accuracy has proven to be a critical element in the success of modem space explo­ration and intricate process control The advent of the microprocessor and its use as a control element have provided the impetus, not only to enhance the theoretical anal­ysis and synthesis techniques for many systems, but also to motivate control-system designers to progress closer to their goal of "ideal system perfonnance."

The purpose of this text is to present an extensive discussion of digital-control­system tenninology, sampled-data control-system analysis and synthesis, and practical

implementation techniques and considerations from a software and hardware point of view including measurement technology Various aspects of a general digital control system are discussed in this text from various levels of observation (theory, hardware, software)

The three simplest control-system configurations or architectures are shown in the block diagram of Fig 1 1 Figure 1 1 (a) is an open-loop control system representing many industrial process structures The other three closed-loop configurations represent the most commonly used control systems where the performance specifications are more restrictive The process block in Fig 1 1 can represent a dc motor speed or position system (translational or rotational), a thermal system, a hydraulic system, etc These block diagrams represent the fundamental control-system notation: The process

to be controlled is called the plant, and the controller or compensator The process may also contain sensors for measurement of plant dynamics (measured variables) Figure 1 1 (d) represents a general sampled-data feedback control system with a digital controller consisting of three elements The analog-to-digital (AID) converter samples the analog signal and converts the sampled signal to digits for numerical processing

by the digital computer The digital-to-analog (D/A) converter transforms the resulting discrete or digital signal values to an analog control signal

Acceptable system performance requires the control system output c(t) (con­trolled variables) to track an input ret) (manipulated variables) despite various system disturbances If the input is zero, then the system is defined as a regulator Satisfac­tory regulation or tracking with disturbances is associated with a disturbance rejection system If the plant parameters vary but the system retains acceptable regulation or tracking characteristics with disturbances, then the system is defined as robust Var­ious design approaches presented in the following chapters depend upon the desired performance using these general system definitions

1.2 DIGITAL CONTROL·SYSTEM

MODELING

A digital-control-system model can be viewed from many different levels including the control law (algorithm), the computer program, conversion between analog and digital signal domains, and system performance One of the most important aspects leading to the understanding of digital control systems is at the sampling process level, which is introduced in this section The associated system terminology, which

is critical in understanding digital control concepts, is also presented

1.2.1 Sampling Process1,2

In continuous control systems, all system variables are continuous signals as repre­sented by Fig 1 2(a) That is, whether the system is linear or nonlinear, all variables are continuously present and are therefore known at all times (This text deals only with linear or linearized systems.) Another category of control systems is one in which one signal e(t) is sampled at intervals of time T This is depicted in Fig 1 2(c) so that the sampled signal appears as a pulse train of varying amplitudes, as shown in Fig 1 2(d)

Input

Controller (digital computer system)

FIGURE 1.1

General architecture of digital control systems: (a) Feedforward or open-loop system; (b) unity-feedback

system; (e) nonunity-feedback system; (d) general digital control system

with the sampled output e;(t)==e(t)p(t) That is, the pulse train of Fig 1 2(b) is mod­

ulated with the continuous-time signal of Fig 1 2(a) to yield the sampled function

of Fig 1 2(d) Such sampling may be an inherent characteristic of the system For

example, a radar tracking system supplies information on a vehicle 's position at dis­

crete periods of time This information is therefore available at a succession of time

intervals as data levels

an analog system not containing a digital device in which some of the signals were sampled (pulse or amplitude modulated) was referred to as a sampled-data system

In this text the term sampled-data control system is used to describe a system that contains at least one sampled signal With the advent of the digital computer, the term discrete-time system denoted a system in which all its signals were in a digital coded form (digitized) Most practical systems today are of a hybrid nature, i.e., contain both analog and digital components

Digital computers are available for performing the computations necessary in

a complex control system The AID conversion device samples the device's input signal at some sampling frequency Ws or equivalently at sampling instants t = kT

where T is defined as the sampling time The process is completed with the sam­pled analog signal value being converted to a discrete digital value (digital encoding) The D/A converter performs the reverse conversion process generating an analog control signal The details of this conversion process are discussed in later chap­ters A system in which the digital computer is utilized as a control device is re­ferred to as a digital control system If the digital computer interfaces directly with

•

I.2 DIGITAL CONTROL-SYSTEM MODELING 5

the plant (or process), the system is referred to as a direct digital control (DDC) system

In practice, the output of the pulse modulator of Fig 1 2(c) is generally fed into •

a data-hold device (a device that converts a discrete signal into a continuous signal), as shown in Fig 1 3(a) The simplified representation of the sampling and hold devices

is shown in Fig 1 3(b), where the "ideal sampler" represents the unit impulse train

of Fig 1 3(c) and the output of the sampler is the amplitude-varying impulse train

e* (t) of Fig 1 3(d) The * indicates a sampled continuous signal such as e* (t) with b

defined as the kronecker delta function:

bet -kT) == {� t = kT

A train of these delta functions, bT==L::O bet -kT), multiplied times the continuous signal e(t), is a syntactical method of representing a mathematical model of the sam­pled continuous signal, e* (t) as shown in Fig 1.3(d) The utilization of a hold device, which is shown in Fig 1 3(e), simplifies the mathematical analysis of the sampled signal The hold process in this case is defined as a "zero-order hold" since the output

at time kT is held constant over the next sampling interval Thus, the structures of Fig 1 1 , where the controller contains the sampler and the hold device of Fig 1 3(a), are sampled-data control systems An example of a sampled-data control system is that

of a human, which can be illustrated in a setting of observing the environment while performing a function such as driving a vehicle or dialing a phone The human eye samples the appropriate environmental signals; processes the information in the brain; and controls the hands, arms, and legs accordingly Another example is that of an air­port automated aircraft landing system that uses radar information, an inherent sampled process, defining aircraft lateral and vertical positions These data are used as input to the controller (an autopilot) for aircraft pitch, yaw, and roll control These examples represent multiple-input-multiple-output (MIMO) control systems Such systems can

be negatively influenced by unmodeled disturbances such as noise, wind, and light

1.2.3 General Sampled-Data System Variables

The variables of a sampled-data system can be described in terms of their time and am­plitude characteristics grouped in four general categories: discrete amplitude-discrete time (D-D), discrete amplitude-continuous time (D-C), continuous amplitude-discrete time (C-D), and continuous amplitude-continuous time (C-C) The first letter refers

to the amplitude characteristic and the second letter refers to the time characteristic Table 1 1 summarizes these categories

A general example of the classification of sampled-data control systems is shown

in Fig 1 4 Here the continuous (amplitude and time) input signal e(t) is impulse­sampled, generating a continuous-amplitude-discrete-time signal by definition The hold circuit of Fig 1 4 generates a piecewise-continuous step function ("staircase") The sampled data e*(t) can be any value (an infinite number of values) within some predefined range of amplitude values

sampling and data-hold devices;

(b) simplified representation

of (a); (c) ideal sampler representation; (d) output of ideal sampler, (e) output of hold device

Figure 1 5 presents the various signal classifications in a digital-computer control

system Here again the impulse sampler generates a continuous-amplitude-discrete­

time C-D signal The output eD(kT) of the AID quantizes e*(t) such that it can

be only one of a finite number of values within a specific value range This phe­

nomenon is due to the finite word length of the computer The digital computer itself

manipulates the quantized value (a base 2 number) into another discrete-amplitude

discrete-time D-D signal f(kT) Finally the D/A transforms f(kT) into a discrete­

amplitude-continuous-time D-C signal m(t) Observe that the m(t) in this case as

TABLE 1.1

The nature of sampled-data-system

variables

Time (temporal) Amplitude

(spatial) Discrete, D Continuous, C

1.2 DIGITAL CONTROL·SYSTEM MODELING 7

compared to the output of the hold device in Fig 1 4 has a discrete-amplitude char­acteristic because of the quantization of the AID which is not present in Fig 1 4 The D/A, however, does include a hold device and is modeled in Fig 1 6 The "fictitious" sampler in this D/A model is needed to represent the fact that the input value f*(t) is immediately converted (AID) and manipulated by the digital computer, resulting in a value at the hold device input during the same sampling instant That is, there is no computational time delay TD in the computer This instantaneous computation is, of course, impossible If the computational time delay is much less than the sampling time T, then the assumption that TD � 0 is appropriate, resulting in a simplified analytical model This model is used in the theoretical development of later chapters

1.2.4 Systems Modeling

The generation of a plant model for which a controller is realized as a digital in­formation processing device is of primary importance Generally, in an academic environment the plant model is given in the form of a linear system with constant coefficients This type of model is usually easy to analyze with proper techniques In

Example of analog signal classifications in a sampled-data system (see Table 1 1)

c-c

T 2T

Many control problems tum out to be directly related to the use of an incorrect plant model The development of the original plant model is probably the most im­portant and probably the most difficult aspect of control engineering The application

of the various analysis and design techniques usually proceeds in a straightforward manner after the model is obtained

1.2 DIGITAL CONTROL-SYSTEM MODELING 9

In developing pedagogical models, one can select various examples Physical examples include electrical motors, mechanical translation systems, mechanical rota­tional systems, thermo systems, hydraulic systems, and combinations of these models

Many of these physical systems can be represented by the same type of transfer function or differential equation resulting in the same number of poles and zeros but with different values and different gains The block diagram of Fig 1 l (d) depicts

a general control loop with plant, sensor, actuator, samplers, and compensator In most cases the sensor and actuator dynamics are incorporated into the plant model Although various units (English, mks) can be specified for a given physical system, they are excluded in the following linearized models for emphasis on structure The specified references detail not only the numerical conversion between appropriate units but also the derivation of the model equation Some common Laplace transform and differential equation models follow

FIRST-ORDER MODEL A temperature control system input and output relationship'

SECOND-ORDER MODELS The modell for a servo motor is

=

E(s) s(J Ras + BRa + KTKb) where () is the motor shaft position output, E is the motor (armature) voltage input,

KT is the mo Jr-torque constant defined by motor-torque = KT* armature current,

J is the total moment of inertia connected to the motor shaft, Ra is the armature resistance, B is the total viscous friction, and Kb is the motor back-emf constant defined by motor back-emf = Kb * B() / Bt A normalized version of this model is:

()(s) K

which is used as a second-order model in this text Physical examples of this sim­ple model include individual-joint robot-arm control, motor antenna control, process servo-motor control, radar azimuth antenna control, rotating power amplifiers, engine control, etc Associated gearing is modeled as a gain (gear ratios)

Another second-order example is represented by a satellite attitude control model The model for a single-axis satellite attitude control system is

=

where () is the relative angle (position) of the satellite, Tq is the input torque, and J

is the satellite's moment of inertia about the chosen axis

A further second-order example is a process control model of a mixing tank with two unpure liquids flowing in and the resulting mixture flowing out Assuming small perturbations about a steady-state mixing condition, the dynamics are defined

by the equations:

vet) = fl (t) + 12(t) + 2fo vet)

Vo e(t) = cdl (t) + c212(t) - eov(t) - eofo - foc(t) 2110

where vet) is the volume of liquid in the tank, c(t) is the concentration in the tank, fl

is the flow of liquid I into the tank with concentration CI , and 12 is the flow of liquid

2 into the tank with concentration C2 The steady state concentration, outflow, and volume are given by eo, fo, and Vo, respectively The input flows can be considered constants resulting in a linear time-invariant system Note, the two coupled variables constitute a second-order system

THIRD· ORDER MODELS The third-order model for a single-axis hydraulic pump motor isl

( 1 5) where 8m is the motor angle and X is the hydraulic displacement The gains are appropriate constant values With certain approximations hydraulic systems can be model as second-order transfer functions The second-order motor model becomes a third-order transfer function when the motor inductance, Lm, is not assumed zero although it is usually relatively quite small

FOURTH·ORDER MODEL For the fourth-order model the classical inverted pendu­lum positioning system is chosen In this system the pendulum pivots on a carriage which can move in a horizontal direction in order to balance the pendulum Assuming small perturbations about a norm results in the following equations using trigonometric approximations:

" g 1

¢(t) - I ¢(t) + Li(t) = 0

Mi(t) = u(t) - fret)

where ¢(t) is the perturbed angle of rotation, x(t) is the horizontal displacement, g is the gravitational acceleration, M is the mass of the carriage, u(t) is the control force

on the carriage, and f is the coefficient of friction L is defined as

L == J + mcg mCg

where m is the pendulum mass, cg is the pendulum center of gravity, and J is the moment of inertia with respect to the center of gravity Because the two variables

¢ and x appear with second derivatives, a linear fourth-order time-invariant model results This inverted pendulum system is usually very difficult to control due to its inherent instability!

U WHY USE DIGITAL CONTROL" 1 1

All the preceding models are approximations to the associated continuous or analog physical plant structure The various s-domain models can also be augmented with a multiplicative time delay e - Td S • To measure and design controllers for appro- ' priate system parameters, sensor and actuator equations also have to be incorporated into the system model Examples of sensors are rate/position gyros, accelerometers, potentiometers, Wheatstone bridge circuits, pressure transducers, switches, and signal converters Examples of actuators are control surfaces, relays, switches, and valves

1.3 WHY USE DIGITAL CONTROL?

The choice of designing a continuous-time or a digital control system for a given application may be based upon the knowledge of the following incomplete list of advantages and disadvantages:

ADVANTAGES Many systems inherently contain a sampling process such as human vision and radar systems Digital control permits the use of sensitive control elements with relatively low-energy signals The advantages of using a digital transducer is the relative immunity of its digital signals to distortion by noise and nonlinearities and its high accuracy and resolution as compared to analog transducers A digital transducer is a coupling device that transforms continuous and discrete data into a digital code/binary number The employment of discrete or digital signals provides for the design and development of complex and sophisticated control systems This phenomenon results from the ability to store discrete information for long time inter­vals, to process complicated algorithms, and to transmit discrete information with high accuracy, using inexpensive low-power microcomputers along with data multiplexing For some control-system applications, improved system performance can be achieved

by a sampled-data control-system design over a continuous-data control-system design because of better noise filtering

DISADVANTAGES The mathematical analysis and design of a sampled-data control system is some times more complex and tedious as compared to continuous-data control-system development In general, converting a given continuous-data control system into a sampled-data control system degrades the system stability margin The purpose of the hold device is to reconstruct the continuous signal from the discrete signal The best reconstructed signal that can be achieved is defined as met), which

is an approximation to e ( t ) Thus, a loss of signal information occurs The complexity

of the control process for a multidimensional system is embedded in the software­implemented algorithms These algorithms may contain logical errors resulting in considerable testing in order to find the incorrect software statements Because the AID, D/A, and the digital computer in reality delay the control signal ouput, the performance objectives can be more difficult to achieve since the theoretical design approaches normally do not model this small delay

1.4 CONTROL-SYSTEM ANALYSIS AND

SYNTHESIS

Since digital-control-system development is interdisciplinary in nature, the techniques •

employed cover a wide spectrum, including the technique of classical control theory and its extensions, discrete mathematical procedures, as well as computer-related de­sign programs and simulators and computer engineering All these techniques are used

in approaching a satisfactory design

The analysis of sampled-data control systems relies heavily on the extension of the complex frequency-domain and the time-domain methods developed for continuous­data control systems 1 21 The major approaches in analyzing sampled-data control

systems are:

1 Complex frequency domain s-transform method (Laplace transformation), root­locus method, frequency-response method, Z-transform method and approxima­tions, Wi -transform method (bilinear transformation), and state-variable method

2 Time domain Linear difference equation method, impulse-response method, and state-variable method

The Z and W i domain transformations and difference equations methods are addi­tional approaches to those employed for continuous-time control-system analysis and synthesis

By incorporating probabilistic models of noise into a digital-control-system model, a better understanding of system performance can be achieved This area

of study requires the understanding of random processes

To develop digital control systems in an efficient manner, various aids are sug­gested that perform accurate and repeatable calculations and permit relatively easy modification of system constants/parameters and software development Examples of various general aids are economical simulators, software development systems, and computer-aided-design (CAD) tools Hybrid simulation using an analog computer is employed to represent the continuous plant or process that is to be digitally controlled

In a digital simulation, a digital computer is used for both the process and digital controller A software development facility permits easy, economical, and efficient generation of control programs, simulations, and associated documentation A con­trol engineer must be proficient in the use of available computer-aided control-system design (CACSD) programs that minimize and expedite the tedious and repetitive cal­culations involved in the synthesis and analysis of a control-system design

If the analysis of the basic system reveals that the desired system performance specifications have not been achieved, then a compensator must be inserted into the system The compensator is designed so that the system achieves the desired perfor­mance specifications Two possible approaches for achieving cascade compensation are shown in Fig 1 7 Figure 1 7(a) illustrates the utilization of a continuous-data compensator Ge(s) in cascade with the basic plant Gx(s) An alternate method of

\ 5 THE INTERDISCIPLINARY FIELD OF DIGITAL CONTROL 13

Basic plant

C(I) c-c

<"(I) c-c

compensation can be achieved by utilizing a digital controller (computer), as shown

1.5 THE INTERDISCIPLINARY FIELD OF

DIGITAL CONTROL

As summarized earlier in this chapter, many disciplines are involved with developing

a successful digital control system These elementary disciplines include differential and difference equations, classical control theory, numerical analysis, discrete control theory, computer systems architecture (hardware/software), digital integrated circuits, signal converters, information structures, control-algorithm design, digital signal pro­cessing, software engineering, and test generation

Additional areas providing extended capabilities for analysis and synthesis are matrix algebra,7 1 discrete stochastic processes,3,4 modem control theory,3,4,7, 19,70,85 and estimation theory.3,4

The intent of this text is not to focus on all elementary disciplines as sepa­rate areas but to use appropriate topics within each of them to support the overall theme, digital control systems Thus, some fields such as numerical analysis are only

discussed in terms of discrete control algorithms or difference equation approxima­tions The main areas of concern are discrete control theory, control-algorithm design, and control-system structure at various levels of observation (models) based upon the interdisciplinary nature of digital control systems

Digital signal-processing concepts provide insight into the processing of digital control signals within a computer Many topics associated with digital processing are discussed here because of the large role this field plays in design and synthesis of control systems A comparison of the design methodology used in the two disciplines shows that in the area of digital-filter design the type of specifications used to define the desired filter and the method of approximation used to model the filter reflect the primary differences The specifications for digital control systems are usually given in terms of time constraints or frequency constraints (magnitude and phase) For digital signal processing the constraint is usually in the form of frequency magnitude con­straints The same considerations of word length in the difference equation coefficients and converters also exist

1.6 DIGITAL CONTROL DEVELOPMENT

With the decreasing cost of digital hardware, economical digital control implementa­tion is now feasible Such applications include process control, automatic aircraft sta­bilization and control, guidance and control of aerospace vehicles, robotic control, and numeric control of manufacturing machines An extensive example is state-feedback control systems for which the computer is used to estimate the inaccessible states and

to help minimize parameter variations The development of digital control systems can be illustrated by the following examples, which center on digital flight control systems, process control systems, and robotics

Example 1.1 Aircraft digital control development Modem technology has brought about some numerous changes in aircraft flight control systems A step in the evolution

of flight control systems is the use of a fly-by-wire (FBW) control system In this de­ sign all pilot commands are transmitted to the control-surface actuators through electric wires Individual component reliability was also increased by replacing the older analog circuitry with newer digital hardware This updated system is referred to as a digital flight control system (DFCS) as shown in Fig 1 8 The use of airborne digital proces­ sors further reduced the cost, weight, and maintenance of modem aircraft and spacecraft Other advantages associated with newer digital equipment included greater accuracy, in­ creased modification flexibility through the use of software changes, improved dynamic reconfiguration techniques, and more reliable preflight and postflight maintenance testing

Example 1.2 Process control Analog computers were initially utilized to monitor and control small dimensional processes Specific installations may have used regulation control (set-point), supervisory control (monitor), or on-line instruction In direct digital control (DOC) the digital computer measures directly the set-point values and current outputs in order to generate control signals (value position, vehicle turning rate, etc.) for tracking or regulation With supervisory control, the objective is to optimize a mul­ tiprocess system by measuring system parameters and generating the required controls

Because of the increasing dimensionality of processes along with accuracy spec­ ifications, minicomputers and microcomputers are used to implement efficiently DOCs and supervisory control systems The added flexibility and general-purpose display ca­ pabilities of a digital-computer system associated with micro-miniaturization [very large scale integrated circuit (VLSI) technology] have evolved contemporary process control systems with considerable distributed processing, These physically small multicomputer systems are used in modem manufacturing as integrated components of automobile pro­ cesses, appliance sequencing, medical operations, and remote environmental monitoring and control systems,

Example 1.3 Robotic arm A specific example of a process control function is the use of assembly robots Figure 1 9 depicts a general-purpose robot arm that can be employed on a production line for assembling parts, mixing combustible chemicals, or inspecting components in an austere environment Although the various arm joints are coupled through the structural links, an initial approach is to control each joint separately Each arm joint can be rotated in a two-dimensional plane by a motor/gear system Other arms may use direct drive motors, A coupled-joint robot-arm-controller design usually requires linearizing a set of nonlinear coupled equations in terms of position and velocity variables At a supervisory level, the trajectories of the arm are defined in a software routine so that the desired movements can occur Because of the power of large robots, the supervisory level also requires measurements of the environment so that destructive activity does not happen Most position control systems historically have used a feedback combination of position, integral and derivative or rate signals (PID controller)

Example of robot arm (PUMA 600 Series)

1.7 NATURE OF THE ENGINEERING

2 Generate a linear model (difference or differential equations) of the plant that describes the basic (or original) physical system The model is usually a linear approximation to the real world This effort is critical in understanding the process model

3 As a result of system analysis or testing to desired performance specifications,

a control problem exists and must be solved Determine the performance of the basic system model by application of the available methods of control-theory analysis in the time domain or frequency domain

4 Augment the system model with appropriate sensor and actuator dynamic models that respectively measure or control the desired plant parameters Proper selection

of measurement and actuator transducers is also of primary importance

I X ITXT Ol 'TLI�E 17

S Use conventional control-theory quantitative feedback theory (QFT) eigenstruc­ture assignment or modem control theory design techniques to design and syn­thesize digital controllers to meet system performance specifications

6 Simulate the overall control system iterating the design until performance speci­fications are achieved

7 Emulate test, and iterate the implemented design Technical knowledge of con­temporary digital-computer hardware and software architectures has a critical in­fluence on cost and real-time operation

Design of the system to obtain the desired performance is the control problem The necessary basic equipment is then assembled into a system to perform the desired control function To a varying extent, most systems are nonlinear In many cases the nonlinearity is small enough to be neglected, or the limits of operation are small enough to allow a linear analysis to be made In this textbook linear systems or those which can be approximated as linear systems are considered Because of the relative simplicity and straightforwardness of this approach, the reader can obtain a thorough understanding of linear systems After mastering the terminology, definitions, and methods of analysis for linear control systems, the engineer will find it easier to undertake a study of nonlinear systems

A basic system has the minimum amount of equipment necessary to accomplish the control function The differential and difference equations that describe the physical system are derived and an analysis of the basic system is made If the analysis indicates that the desired performance has not been achieved with this basic system, additional equipment must be inserted into the system or new control algorithms employed Generally, this analysis also indicates the characteristics for the additional equipment

or algorithms that are necessary to achieve the desired performance After the system

is synthesized to achieve the desired performance, based upon linear analysis, final adjustments can be made on the actual system to take into account the nonlinearities that were neglected For digital control systems it is necessary to use good structured programming techniques and to document on a continuing basis all aspects of the software development

1.8 TEXT OUTLINE

Chapter 2 serves as a review, and for those with no control-theory background it presents the control-theory concepts necessary for them to proceed into the study of sampled-data control systems Chapters 3 and 4 describe the linear system models for the discrete case To represent a discrete linear difference equation in the frequency domain, a mathematical transform called the Z transform (zee transform) is developed Many techniques used in modeling and designing continuous controllers with the Laplace transform can be reused with this new transform

Various analysis and design techniques as presented in Chaps 5 and 6 exist for developing discrete or digital controllers The conventional design methods presented

in this text are based upon the root-locus formulation using the pole and zeros of the transfer function in the s , w , or z domain

As presented in Chap 7, the real-world implementation of a digital controller requires conversion from a continuous or analog signal to a digital signal using an AID converter Also, the digital "control signal" generated within the computer must

be converted into a continuous signal through a D/A process As presented in Chap 8, another aspect of digital-controller implementation requires an appreciation of software development A limited number of skills from software engineering are presented for proper development of control programs The text also introduces general real-time executives for embedding control software

To study the impact of finite word length, a statistical or probabilistic approach

is used The fundamentals of random variables and random processes are developed

in Chap 9, which provides the tools for analyzing the impact of finite word length for specific difference equation implementations in Chap 10

Chapters 1 1 , 12, and 1 3 develop the design techniques associated with discrete cascade compensation and feedback compensation using the direct approach (DIR) and the digitization approach (DIG)

Chapter 14 introduces the state-vector modeling approach and Chap 15 presents some fundamental state-space design techniques for digital controllers The robust frequency-domain quantitative feedback theory (QFf) design technique for continuous­time systems having plant parameter uncertainty is extended to include the design of sampled-data control systems in Chap 16 Chapter 1 7 extends the state-space model

to consider controllability, observability, and observer models Chapter 1 8 discusses the development of optimal controllers which involve the solution to a Riccati equa­tion Digital adaptive control using parameter identification techniques is presented in Chap 19 along with combining the optimal control model with the optimal estimation model resulting in a stochastic controller

1.9 SUMMARY

This chapter has introduced the model of a sampled-data control system Such sys­tems with at least one sampled analog signal and a hold device are used in many applications The development of a discrete real-time digital controller requires the understanding of numerous theoretical and empirical subjects, a number of which are presented in this text

The many and varied methods described in this text result of course in different controller structures and organizations Some may have performance characteristics very similar in nature; others may be quite different across a wide performance spec­trum Some may have the same theoretical performance, but, as implemented in a digital controller, the associated responses can reflect sharp differences

The highly developed analysis and design techniques for cascade and feedback com­pensators for continuous-data control systems form the basis for the analysis and design of sampled-data control systems Thus, knowledge of the desired figures of merit (FOM) of the conventional control time response for a simple second-order system is required These FOM are used in designing an nth-order system so that its time-response characteristics are essentially those of a simple second-order system The frequency response or the root-locus technique yields a closed-loop response which is based upon some of the FOM There are other techniques that first require the modeling of a desired control ratio which satisfies the desired FOM The desired control ratio is used to determine the necessary compensation Formulating the desired control ratio is the main objective of this chapter Figure 2.1 represents a multiple­input-single-output (MISO) control system in which ret) is the input, c(t) is the output which is required to track this input, d(t) is a disturbance input that must have minimal effect on the output, and F(s) is a prefilter to assist in the tracking of ret) by c(t)

Thus, two types of control ratios need to be modeled: a tracking transfer function

,1l,[T(S) and a disturbance transfer function MD(S)

The control systems that are initially considered in this text represent systems for which F(s) = I and d(t) = 0; thus, they are single-input-single-output (SISO) control systems

2.2 BACKGROUND

If the analysis of the basic SISO system reveals that the desired specifications cannot

be satisfied, then one of two common methods (cascade or feedback compensation) may be used to try to achieve these desired specifications

The performance of a tracking control system, based upon a step input, generally falls into one of the following four categories:

1 A given system is stable and its transient response is satisfactory, but its steady­state error is too large Therefore, the gain must be increased to reduce the steady­state error without appreciably reducing the system stability

2 A given system is stable, but its transient response is unsatisfactory

3 A given system is stable, but both its transient response and its steady-state re­sponse are unsatisfactory

4 A given system is unstable for all values of gain

Generally, all four categories require the utilization of additional system components

to achieve the desired system performance

Note that the part of the control system on which control is exerted is usually called the plant or basic system Generally, the basic system is fixed or unalterable except possibly for gain adjustment to improve the system performance The addi­tional equipment that is inserted into the system in order to obtain the desired system performance is referred to as the controller or compensator Compensation of a sys­tem by the introduction of compensator poles and zeros is thus used to improve the system performance However, each additional compensator pole increases the num­ber of roots of the closed-loop characteristic equation If an underdamped response of the form shown in Fig 2.2 is desired, the system gain can be adjusted so that there

is only one pair of dominant complex poles This requires that any other pole be far

Simple second-order system tracking time response

to the left or near a zero so that the magnitude of the transient term due to that pole

is small and therefore has a negligible effect on the total time response The effect

of compensator poles and zeros on the system's FOM performance can be evaluated rapidly by use of computer-aided-design (CAD) techniques (see Appendix E).14,15

Three common basic types of analog compensators 1 that are used are shown

in Fig 2.3 They are called passive compensators since they consist of only resistors and capacitors in conjunction with an amplifier The transfer function of the lag compensator, shown in Fig 2.3(a), is

Ge(s) = A(1 + Ts) = A s + l iT

l + aTs a s + 1 /aT (2 1 ) where 1 < a < 1 0 (nominal for passive networks) It i s referred to as a lag network since for a sinusoidal input the output phase always lags relative to the input phase This type of compensator, a low-pass filter, can be used for systems whose perfor­mance falls into category 1 , where the steady-state error can be reduced approximately

by a factor of a The transfer function of the lead compensator, shown in Fig 2.3(b), is

Ge(s) = Aa( 1 + Ts) = A s + l iT (2.2)

1 + aTs s + 1 /aT where 1 > a 2 0 1 (nominal for passive networks) It is referred to as a lead network since for a sinusoidal input the output phase always leads relative to the input phase The lead network, a high-pass filter, can be used for systems whose performance falls into category 2 or 4

(a)

(b)

(c) FIGURE 2.3

Compensators: (a) Lag network; (b) lead network; (e) lag-lead network

The improvement in system performance that can be obtained by using each network separately can be achieved by a single network [see Fig 2.3(c)], which is referred to as a lag-lead network An improvement to system performance for a system that falls into category 3 can be achieved by using a lag-lead network whose transfer function is

G c(s) = A ( 1 + aT] s)[l + (Tda)s] (1 + T] s)(l + T2s)

= A (s + l /T] )(s + 1 /T2) (s + l /aT] )(s + a/T2) (2.3)

KJ' (b)

FIGURE 2.4

A position, integral, and derivative (PID) controller: (a) block diagram; (b) signal flow graph (SFG)

where 0 > 1 and Tl > T2• The fraction ( 1 + T1s)/(l + oTI S) represents the lag compensator, and the fraction ( l + T2s)/[1 + (Tz/ o)s] represents the lead com­pensator

A compensator that is generally used in the process control industry is the position, integral, and derivative (PID) controller of Fig 2.4 This controller combines the characteristics of the lag and lead compensators to provide proportional, integral, and derivative control action for improving a system's perfonnance Its transfer func-tion is

K

s The analog PID controller of Fig 2.4 is not of a practical fonn that can be readily implemented due to the ideal derivative operation A practical fonn of this controller is

to replace KdS by a lead network of the fonn of Eq (2.2), which is an approximation

to the ideal derivative process

Another procedure for designing the cascade compensator Gc(s) of Fig 2.1

is the Guilleman-Truxel (G-T) method This method, developed later in the book,

is commonly called a pole-zero placement technique In other words, some desired tracking control ratio MT(s) is synthesized for an nth-order system that yields the desired values for the FOM, where d(t) = 0, F(s) = 1 , and ret) = U- l (t)

2.3 SIMPLE SECOND-ORDER SYSTEM

TRACKING-RESPONSE CHARACTERISTICS

The tracking response to a unit-step-function input is usually used as a means of evaluating the response of a system As an illustrative example, consider the simple second-order system described by the differential equation

e(t) 2(

- 2 + - e(t) + e(t) = ret) (2.5)

Wn Wn This is defined as a simple second-order equation because there are no derivatives of