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Digital Control Engineering Analysis and Design.pdf
Cover Page
Copyright page
Copyright page
Preface
Preface
Chapter 1 Introduction to Digital Control
Chapter 1 Introduction to Digital Control
1.1 Why Digital Control?
1.2 The Structure of a Digital Control System
1.3 Examples of Digital Control Systems
1.3.1 Closed-Loop Drug Delivery System
1.3.2 Computer Control of an Aircraft Turbojet Engine
1.3.3 Control of a Robotic Manipulator
Resources
Chapter 2 Discrete-Time Systems
Chapter 2 Discrete-Time Systems
2.1 Analog Systems with Piecewise Constant Inputs
2.2 Difference Equations
2.3 The z-Transform
2.3.1 z-Transforms of Standard Discrete-Time Signals
2.3.2 Properties of the z-Transform
Linearity
Time Delay
Time Advance
Multiplication by Exponential
Complex Differentiation
2.3.3 Inversion of the z-Transform
Long Division
Partial Fraction Expansion
2.3.4 The Final Value Theorem
2.4 Computer-Aided Design
2.5 z-Transform Solution of Difference Equations
2.6 The Time Response of a Discrete-Time System
2.6.1 Convolution Summation
2.6.2 The Convolution Theorem
2.7 The Modified z-Transform
2.8 Frequency Response of Discrete-Time Systems
2.8.1 Properties of the Frequency Response of Discrete-Time Systems
2.8.2 MATLAB Commands for the Discrete-Time Frequency Response
2.9 The Sampling Theorem
2.9.1 Selection of the Sampling Frequency
Resources
Chapter 3 Modeling of Digital Control Systems
Chapter 3 Modeling of Digital Control Systems
3.1 ADC Model
3.2 DAC Model
3.3 The Transfer Function of the ZOH
3.4 Effect of the Sampler on the Transfer Function of a Cascade
3.5 DAC, Analog Subsystem, and ADC Combination Transfer Function
3.6 Systems with Transport Lag
3.7 The Closed-Loop Transfer Function
3.8 Analog Disturbances in a Digital System
3.9 Steady-State Error and Error Constants
3.9.1 Sampled Step Input
3.9.2 Sampled Ramp Input
3.10 MATLAB Commands
3.10.1 MATLAB
Resources
Chapter 4 Stability of Digital Control Systems
Chapter 4 Stability of Digital Control Systems
4.1 Definitions of Stability
4.2 Stable z-Domain Pole Locations
4.3 Stability Conditions
4.3.1 Asymptotic Stability
4.3.2 BIBO Stability
4.3.3 Internal Stability
4.4 Stability Determination
4.4.1 MATLAB
4.4.2 Routh-Hurwitz Criterion
4.5 Jury Test
4.6 Nyquist Criterion
4.6.1 Phase Margin and Gain Margin
Resources
Chapter 5 Analog Control System Design
Chapter 5 Analog Control System Design
5.1 Root Locus
5.2 Root Locus Using MATLAB
5.3 Design Specifications and the Effect of Gain Variation
5.4 Root Locus Design
5.4.1 Proportional Control
5.4.2 PD Control
5.4.3 PI Control
5.4.4 PID Control
5.5 Empirical Tuning of PID Controllers
Resources
Chapter 6 Digital Control System Design
Chapter 6 Digital Control System Design
6.1 z-Domain Root Locus
6.2 z-Domain Digital Control System Design
Observation
6.2.1 z-Domain Contours
6.2.2 Proportional Control Design in the z-Domain
6.3 Digital Implementation of Analog Controller Design
6.3.1 Differencing Methods
Forward Differencing
Backward Differencing
6.3.2 Bilinear Transformation
6.3.3 Empirical Digital PID Controller Tuning
6.4 Direct z-Domain Digital Controller Design
6.5 Frequency Response Design
6.6 Direct Control Design
6.7 Finite Settling Time Design
Resources
Chapter 7 State–Space Representation
Chapter 7 State–Space Representation
7.1 State Variables
7.2 State–Space Representation
7.2.1 State–Space Representation in MATLAB
7.2.2 Linear versus Nonlinear State–Space Equations
7.3 Linearization of Nonlinear State Equations
7.4 The Solution of Linear State–Space Equations
7.4.1 The Leverrier Algorithm
Leverrier Algorithm
7.4.2 Sylvester’s Expansion
7.4.3 The State-Transition Matrix for a Diagonal State Matrix
Properties of Constituent Matrices
7.5 The Transfer Function Matrix
7.5.1 MATLAB Commands
7.6 Discrete-Time State–Space Equations
7.6.1 MATLAB Commands for Discrete-Time State–Space Equations
7.7 Solution of Discrete-Time State–Space Equations
7.7.1 z-Transform Solution of Discrete-Time State Equations
7.8 Z-Transfer Function from State–Space Equations
7.8.1 z-Transfer Function in MATLAB
7.9 Similarity Transformation
7.9.1 Invariance of Transfer Functions and Characteristic Equations
Resources
Problems
Computer Exercises
Chapter 8 Properties of State–Space Models
Chapter 8 Properties of State–Space Models
8.1 Stability of State–Space Realizations
8.1.1 Asymptotic Stability
Remark
8.1.2 BIBO Stability
8.2 Controllability and Stabilizability
8.2.1 MATLAB Commands for Controllability Testing
8.2.2 Controllability of Systems in Normal Form
8.2.3 Stabilizability
8.3 Observability and Detectability
8.3.1 MATLAB Commands
8.3.2 Observability of Systems in Normal Form
8.3.3 Detectability
8.4 Poles and Zeros of Multivariable Systems
8.4.1 Poles and Zeros from the Transfer Function Matrix
8.4.2 Zeros from State–Space Models
8.5 State–Space Realizations
8.5.1 Controllable Canonical Realization
Systems with No Input Differencing
Systems with Input Differencing
8.5.2 Controllable Form in MATLAB
8.5.3 Parallel Realization
Parallel Realization for MIMO Systems
8.5.4 Observable Form
8.6 Duality
Resources
Chapter 9 State Feedback Control
Chapter 9 State Feedback Control
9.1 State and Output Feedback
9.2 Pole Placement
9.2.1 Pole Placement by Transformation to Controllable Form
9.2.2 Pole Placement Using a Matrix Polynomial
9.2.3 Choice of the Closed-Loop Eigenvalues
9.2.4 MATLAB Commands for Pole Placement
9.2.5 Pole Placement by Output Feedback
9.3 Servo Problem
9.4 Invariance of System Zeros
9.5 State Estimation
9.5.1 Full-Order Observer
9.5.2 Reduced-Order Observer
9.6 Observer State Feedback
9.6.1 Choice of Observer Eigenvalues
9.7 Pole Assignment Using Transfer Functions
Resources
Chapter 10 Optimal Control
Chapter 10 Optimal Control
10.1 Optimization
10.1.1 Unconstrained Optimization
10.1.2 Constrained Optimization
10.2 Optimal Control
10.3 The Linear Quadratic Regulator
10.3.1 Free Final State
10.4 Steady-State Quadratic Regulator
10.4.1 Output Quadratic Regulator
10.4.2 MATLAB Solution of the Steady-State Regulator Problem
10.4.3 Linear Quadratic Tracking Controller
10.5 Hamiltonian System
Resources
Chapter 11 Elements of Nonlinear Digital Control Systems
Chapter 11 Elements of Nonlinear Digital Control Systems
11.1 Discretization of Nonlinear Systems
11.1.1 Extended Linearization by Input Redefinition
11.1.2 Extended Linearization by Input and State Redefinition
11.1.3 Extended Linearization by Output Differentiation
11.1.4 Extended Linearization Using Matching Conditions
11.2 Nonlinear Difference Equations
11.2.1 Logarithmic Transformation
11.3 Equilibrium Of Nonlinear Discrete-Time Systems
11.4 Lyapunov Stability Theory
11.4.1 Lyapunov Functions
11.4.2 Stability Theorems
11.4.3 Rate of Convergence
11.4.4 Lyapunov Stability of Linear Systems
11.4.5 MATLAB
11.4.6 Lyapunov’s Linearization Method
11.4.7 Instability Theorems
11.4.8 Estimation of the Domain of Attraction
11.5 Stability of Analog Systems with Digital Control
11.6 State Plane Analysis
11.7 Discrete-Time Nonlinear Controller Design
11.7.1 Controller Design Using Extended Linearization
11.7.2 Controller Design Based on Lyapunov Stability Theory
Resources
Chapter 12 Practical Issues
Chapter 12 Practical Issues
12.1 Design of the hardware and software architecture
12.1.1 Software Requirements
12.1.2 Selection of ADC and DAC
12.2 Choice of the Sampling Period
12.2.1 Antialiasing Filters
12.2.2 Effects of Quantization Errors
12.2.3 Phase Delay Introduced by the ZOH
12.3 Controller Structure
12.4 PID Control
12.4.1 Filtering the Derivative Action
12.4.2 Integrator Windup
12.4.3 Bumpless Transfer between Manual and Automatic Mode
12.4.4 Incremental Form
12.5 Sampling Period Switching
12.5.1 Matlab Commands
12.5.2 Dual-Rate Control
Resources
Appendix I Table of Laplace and z-Transforms
Appendix I Table of Laplace and z-Transforms
Appendix II Properties of the z-Transform
Appendix II Properties of the z-Transform
Appendix III Review of Linear Algebra
Appendix III Review of Linear Algebra
A.1 Matrices
A.2 Equality of Matrices
A.3 Matrix Arithmetic
A.3.1 Addition and Subtraction
A.3.2 Transposition
A.3.3 Matrix Multiplication
A.3.3.1 Multiplication by a Scalar
A.3.3.2 Multiplication by a Matrix
A.4 Determinant of a Matrix
Determinant
Properties of Determinants
A.5 Inverse of a Matrix
A.6 Eigenvalues
Upper triangular matrix
Lower triangular matrix
A.7 Eigenvectors
A.8 Norm of a Vector
Norm Axioms
lp Norms
Equivalent Norms
A.9 Matrix Norms
Frobenius Norm
Induced Matrix Norms
Submultiplicative Property
A.10 Quadratic Forms
A.11 Matrix Differentiation/Integration
A.12 Kronecker Product
Resources
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Preface
Approach
Control systems are an integral part of everyday life in today’s society. They
control our appliances, our entertainment centers, our cars, and our office envi-
ronments; they control our industrial processes and our transportation systems;
they control our exploration of land, sea, air, and space. Almost all of these appli-
cation use digital controllers implemented with computers, microprocessors, or
digital electronics. Every electrical, chemical, or mechanical engineering senior
or graduate student should therefore be familiar with the basic theory of digital
controllers.
This text is designed for a senior or combined senior/graduate-level course in
digital controls in departments of mechanical, electrical, or chemical engineering.
Although other texts are available on digital controls, most do not provide a sat-
isfactory format for a senior/graduate-level class. Some texts have very few exam-
ples to support the theory, and some were written before the wide availability of
computer-aided-design (CAD) packages. Others make some use of CAD packages
but do not fully exploit their capabilities. Most available texts are based on the
assumption that students must complete several courses in systems and control
theory before they can be exposed to digital control. We disagree with this
assumption, and we firmly believe that students can learn digital control after a
one-semester course covering the basics of analog control. As with other topics
that started at the graduate level—linear algebra and Fourier analysis to name
a few—the time has come for digital control to become an integral part of the
undergraduate curriculum.
Features
To meet the needs of the typical senior/graduate-level course, this text includes
the following features:
Numerous examples. The book includes a large number of examples. Typically,
only one or two examples can be covered in the classroom because of time
Preface
limitations. The student can use the remaining examples for self-study. The
experience of the authors is that students need more examples to experiment
with so as to gain a better understanding of the theory. The examples are varied
to bring out subtleties of the theory that students may overlook.
Extensive use of CAD packages. The book makes extensive use of CAD packages.
It goes beyond the occasional reference to specific commands to the integration
of these commands into the modeling, design, and analysis of digital control
systems. For example, root locus design procedures given in most digital
control texts are not CAD procedures and instead emphasize paper-and-pencil
design. The use of CAD packages, such as MATLAB®, frees students from the
drudgery of mundane calculations and allows them to ponder more subtle
aspects of control system analysis and design. The availability of a simulation
tool like Simulink® allows the student to simulate closed-loop control systems
including aspects neglected in design such as nonlinearities and disturbances.
Coverage of background material. The book itself contains review material from
linear systems and classical control. Some background material is included in
appendices that could either be reviewed in class or consulted by the student
as necessary. The review material, which is often neglected in digital control
texts, is essential for the understanding of digital control system analysis and
design. For example, the behavior of discrete-time systems in the time domain
and in the frequency domain is a standard topic in linear systems texts but
often receives brief coverage. Root locus design is almost identical for analog
systems in the s-domain and digital systems in the z-domain. The topic is covered
much more extensively in classical control texts and inadequately in digital
control texts. The digital control student is expected to recall this material or
rely on other sources. Often, instructors are obliged to compile their own
review material, and the continuity of the course is adversely affected.
Inclusion of advanced topics. In addition to the basic topics required for a one-
semester senior/graduate class, the text includes some advanced material to
make it suitable for an introductory graduate-level class or for two quarters
at the senior/graduate level. We would also hope that the students in a single-
semester course would acquire enough background and interest to read the
additional chapters on their own. Examples of optional topics are state-space
methods, which may receive brief coverage in a one-semester course, and
nonlinear discrete-time systems, which may not be covered.
Standard mathematics prerequisites. The mathematics background required
for understanding most of the book does not exceed what can be reasonably
expected from the average electrical, chemical, or mechanical engineering
senior. This background includes three semesters of calculus, differential
equations, and basic linear algebra. Some texts on digital control require more
mathematical maturity and are therefore beyond the reach of the typical senior.
Preface
i
On the other hand, the text does include optional topics for the more advanced
student. The rest of the text does not require knowledge of this optional
material so that it can be easily skipped if necessary.
Senior system theory prerequisites. The control and system theory background
required for understanding the book does not exceed material typically covered
in one semester of linear systems and one semester of control systems. Thus,
the students should be familiar with Laplace transforms, the frequency domain,
and the root locus. They need not be familiar with the behavior of discrete-time
systems in the frequency and time domain or have extensive experience with
compensator design in the s-domain. For an audience with an extensive
background in these topics, some topics can be skipped and the material can
be covered at a faster rate.
Coverage of theory and applications. The book has two authors: the first is
primarily interested in control theory and the second is primarily interested
in practical applications and hardware implementation. Even though some
control theorists have sufficient familiarity with practical issues such as
hardware implementation and industrial applications to touch on the subject
in their texts, the material included is often deficient because of the rapid
advances in the area and the limited knowledge that theorists have of the
subject.
It became clear to the first author that to have a suitable text for his course
and similar courses, he needed to find a partner to satisfactorily complete the text.
He gradually collected material for the text and started looking for a qualified and
interested partner. Finally, he found a co-author who shared his interest in digital
control and the belief that it can be presented at a level amenable to the average
undergraduate engineering student.
For about 10 years, Dr. Antonio Visioli has been teaching an introductory and
a laboratory course on automatic control, as well as a course on control systems
technology. Further, his research interests are in the fields of industrial regulators
and robotics. Although he contributed to the material presented throughout the
text, his major contribution was adding material related to the practical design
and implementation of digital control systems. This material is rarely covered in
control systems texts but is an essential prerequisite for applying digital control
theory in practice.
The text is written to be as self-contained as possible. However, the reader is
expected to have completed a semester of linear systems and classical control.
Throughout the text, extensive use is made of the numerical computation and
computer-aided-design package MATLAB. As with all computational tools, the
enormous capabilities of MATLAB are no substitute for a sound understanding of
the theory presented in the text. As an example of the inappropriate use of sup-
porting technology, we recall the story of the driver who followed the instructions
ii
Preface
of his GPS system and drove into the path of an oncoming train!1 The reader must
use MATLAB as a tool to support the theory without blindly accepting its compu-
tational results.
Organization of Text
The text begins with an introduction to digital control and the reasons for its
popularity. It also provides a few examples of applications of digital control from
the engineering literature.
Chapter 2 considers discrete-time models and their analysis using the z-
transform. We review the z-transform, its properties, and its use to solve differ-
ence equations. The chapter also reviews the properties of the frequency
response of discrete-time systems. After a brief discussion of the sampling
theorem, we are able to provide rules of thumb for selecting the sampling rate
for a given signal or for given system dynamics. This material is often covered in
linear systems courses, and much of it can be skipped or covered quickly in a
digital control course. However, the material is included because it serves as a
foundation for much of the material in the text.
Chapter 3 derives simple mathematical models for linear discrete-time systems.
We derive models for the analog-to-digital converter (ADC), the digital-to-analog
converter (DAC), and for an analog system with a DAC and an ADC. We include
systems with time delays that are not an integer multiple of the sampling period.
These transfer functions are particularly important because many applications
include an analog plant with DAC and ADC. Nevertheless, there are situations
where different configurations are used. We therefore include an analysis of a
variety of configurations with samplers. We also characterize the steady-state
tracking error of discrete-time systems and define error constants for the unity
feedback case. These error constants play an analogous role to the error constants
for analog systems. Using our analysis of more complex configurations, we are
able to obtain the error due to a disturbance input.
In Chapter 4, we present stability tests for input-output systems. We examine
the definitions of input-output stability and internal stability and derive con-
ditions for each. By transforming the characteristic polynomial of a discrete-time
system, we are able to test it using the standard Routh-Hurwitz criterion for
analog systems. We use the Jury criterion, which allows us to directly test the
stability of a discrete-time system. Finally, we present the Nyquist criterion for
the z-domain and use it to determine closed-loop stability of discrete-time
systems.
Chapter 5 introduces analog s-domain design of proportional (P), proportional-
plus-integral (PI), proportional-plus-derivative (PD), and proportional-plus-integral-
1The story was reported in the Chicago Sun-Times, on January 4, 2008. The driver, a computer
consultant, escaped just in time before the train slammed into his car at 60 mph in Bedford Hills,
New York.
Preface
iii
plus-derivative (PID) control using MATLAB. We use MATLAB as an integral part
of the design process, although many steps of the design can be competed using
a scientific calculator. It would seem that a chapter on analog design does not
belong to a text on digital control. This is false. Analog control can be used as a
first step toward obtaining a digital control. In addition, direct digital control
design in the z-domain is similar in many ways to s-domain design.
Digital controller design is topic of Chapter 6. It begins with proportional
control design then examines digital controllers based on analog design. The
direct design of digital controllers is considered next. We consider root locus
design in the z-plane for PI and PID controllers. We also consider a synthesis
approach due to Ragazzini that allows us to specify the desired closed-loop trans-
fer function. As a special case, we consider the design of deadbeat controllers that
allow us to exactly track an input at the sampling points after a few sampling
points. For completeness, we also examine frequency response design in the w-
plane. This approach requires more experience because values of the stability
margins must be significantly larger than in the more familiar analog design. As
with analog design, MATLAB is an integral part of the design process for all digital
control approaches.
Chapter 7 covers state-space models and state-space realizations. First, we
discuss analog state-space equations and their solutions. We include nonlinear
analog equations and their linearization to obtain linear state-space equations. We
then show that the solution of the analog state equations over a sampling period
yields a discrete-time state-space model. Properties of the solution of the analog
state equation can thus be used to analyze the discrete-time state equation. The
discrete-time state equation is a recursion for which we obtain a solution by induc-
tion. In Chapter 8, we consider important properties of state–space models: stabil-
ity, controllability, and observability. As in Chapter 4, we consider internal
stability and input-output stability, but the treatment is based on the properties of
the state-space model rather than those of the transfer function. Controllability is
a property that characterizes our ability to drive the system from an arbitrary initial
state to an arbitrary final state in finite time. Observability characterizes our ability
to calculate the initial state of the system using its input and output measurements.
Both are structural properties of the system that are independent of its stability.
Next, we consider realizations of discrete-time systems. These are ways of imple-
menting discrete-time systems through their state-space equations using summers
and delays.
Chapter 9 covers the design of controllers for state-space models. We show
that the system dynamics can be arbitrarily chosen using state feedback if the
system is controllable. If the state is not available for feedback, we can design a
state estimator or observer to estimate it from the output measurements. These
are dynamic systems that mimic the system but include corrective feedback to
account for errors that are inevitable in any implementation. We give two types
of observers. The first is a simpler but more computationally costly full-order
observer that estimates the entire state vector. The second is a reduced-order
iv
Preface
observer with the order reduced by virtue of the fact that the measurements are
available and need not be estimated. Either observer can be used to provide an
estimate of the state for feedback control, or for other purposes. Control schemes
based on state estimates are said to use observer state feedback.
Chapter 10 deals with the optimal control of digital control systems. We con-
sider the problem of unconstrained optimization, followed by constrained optimi-
zation, then generalize to dynamic optimization as constrained by the system
dynamics. We are particularly interested in the linear quadratic regulator where
optimization results are easy to interpret and the prerequisite mathematics
background is minimal. We consider both the finite time and steady-state regulator
and discuss conditions for the existence of the steady-state solution. The first 10
chapters are mostly restricted to linear discrete-time systems. Chapter 11 examines
the far more complex behavior of nonlinear discrete-time systems. It begins with
equilibrium points and their stability. It shows how equivalent discrete-time
models can be easily obtained for some forms of nonlinear analog systems
using global or extended linearization. It provides stability theorems and insta-
bility theorems using Lyapunov stability theory. The theory gives sufficient condi-
tions for nonlinear systems, and failure of either the stability or instability tests is
inconclusive. For linear systems, Lyapunov stability yields necessary and sufficient
conditions. Lyapunov stability theory also allows us to design controllers by select-
ing a control that yields a closed-loop system that meets the Lyapunov stability
conditions. For the classes of nonlinear systems for which extended linearization
is straightforward, linear design methodologies can yield nonlinear controllers.
Chapter 12 deals with practical issues that must be addressed for the success-
ful implementation of digital controllers. In particular, the hardware and software
requirements for the correct implementation of a digital control system are ana-
lyzed. We discuss the choice of the sampling frequency in the presence of anti-
aliasing filters and the effects of quantization, rounding, and truncation errors. We
also discuss bumpless switching from automatic to manual control, avoiding
discontinuities in the control input. Our discussion naturally leads to approaches
for the effective implementation of a PID controller. Finally, we consider nonuni-
form sampling, where the sampling frequency is changed during control opera-
tion, and multirate sampling, where samples of the process outputs are available
at a slower rate than the controller sampling rate.
Supporting Material
The following resources are available to instructors adopting this text for use in
their courses. Please visit www.elsevierdirect9780123744982.com to register for
access to these materials:
Instructor solutions manual. Fully typeset solutions to the end-of-chapter
problems in the text.
PowerPoint images. Electronic images of the figures and tables from the
book, useful for creating lectures.
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