Control of a Nonlinear Model of a Robotic Arm with a
Flexible Joint via the Linearization Technique
Consider a single-link robotic manipulator with a flexible joint whose nonlinear mathematical
model is given in Chapter 1, Example 1.3 of the book GL96, page 29. It is desired to control the
obot arm to hold any reference (nominal-, operating-, trim-, steady-state- set-point) angle in the
entire range 0 2. Design a full-state feedback control law for this system using linearization
about a set point. As a controller use the “pole placement” (eigenvalue assignment) controller
(regulator). To find the feedback gain F via MATLAB use the function “place” as: F=place (A, B,
eigenvalues_desired). The project tasks should be achieved in two steps.
Step 1. (3 points). Present the results of linearization at steady state points and find the
co
esponding matrices A and B. Assume that matrix C is identity and matrix D a zero matrix of
appropriate dimensions. Use the Simulink state space block and simulate dynamics of the
linearized system (see Hint). The linearized system initial conditions that should be set in the
neighborhood of the system initial conditions (note x(t0) = x(t0) − xss). The full-state feedback
controller u(t) = −Fx(t) should be obtained using the MATLAB function place. Monitor the
system response and choose the desired eigenvalues such a good transient response is obtained
(small overshoot, short settling and rise times). At the same time observe the magnitudes of the
signals in the closed-loop system.
Step 2. (7 points). Build the Simulink block diagram of the considered nonlinear robot arm
model.
Design a full-state space controller that keeps at steady state 1ss =
4
.Simulate the performance
of the system response using SIMULINK and plot 1(t) as a function of time until it settles down to
its steady state value. Observe also the linearized and actual values of the input signal. Plot all required
signals from the MATLAB window using the MATLAB plot function as: plot (out.t, out. signal), see the
Simulink block diagram in Figure 9 of the Simulink Short Manual. Consider the time interval of several
seconds.
Report Format: For this project, every student must submit a typed technical report that should include:
Project description, project formulation; basic formulas used and the design steps; SIMULINK block
diagrams; graphs of all relevant signals as defined in the project tasks; comments on results obtained;
summary; references; appendix with MATLAB codes.
PubTeX output XXXXXXXXXX:2253
Chapter One
Introduction
This book represents a modern treatment of classical control theory of continuous-
and discrete-time linear systems. Classical control theory originated in the fifties
and attained maturity in the sixties and seventies. During that time control theory
and its applications were among the most challenging and interesting scientific
and engineering areas. The success of the space program and the aircraft industry
was heavily based on the power of classical control theory.
The rapid scientific development between 1960 and 1990
ought a tremen-
dous number of new scientific results. Just within electrical engineering, we
have witnessed the real explosion of the computer industry in the middle of the
eighties, and the rapid development of signal processing, parallel computing,
neural networks, and wireless communication theory and practice at the begin-
ning of the nineties. In the years to come many scientific areas will evolve
around vastly enhanced computers with the ability to solve by virtually
ute
force very complex problems, and many new scientific areas will open in that
direction. The already established “information superhighway” is maybe just
a synonym for the numerous possibilities for “informational
eakthrough” in
almost all scientific and engineering areas with the use of modern computers.
Neural networks—dynamic systems able to process information through large
number of inputs and outputs—will become specialized “dynamic” computers
for solving specialized problems.
Where is the place of classical (and modern) control theory in contemporary
scientific, industrial, and educational life? First of all, classical control theory
values have to be preserved, properly placed, and incorporated into modern scien-
tific knowledge of the nineties. Control theory will not get as much attention and
1
2 INTRODUCTION
ecognition as it used to enjoy in the past. However, control theory is concerned
with dynamic systems, and dynamics is present, and will be increasingly present,
in almost all scientific and engineering disciplines. Even computers connected
into networks can be studied as dynamic systems. Communication networks have
long been recognized as dynamic systems, but their models are too complex to
e studied without the use of powerful computers. Traffic highways of the fu-
ture are already the subject of
oad scale research as dynamic systems and an
intensive search for the best optimal control of networks of highways is under-
way. Robotics, aerospace, chemical, and automotive industries are producing
every day new and challenging models of dynamic systems which have to be
optimized and controlled. Thus, there is plenty of room for further development
of control theory applications, both behind or together with the “informational
power” of modern computers.
Control theory must preserve its old values and incorporate them into modern
scientific trends, which will be based on the already developed fast and reliable
packages for scientific numerical computations, symbolic computations, and
computer graphics. One of them, MATLAB, is already gaining
oad recognition
from the scientific community and academia. It represents an expert system
for many control/system oriented problems and it is widely used in industry
and academia either to solve new problems or to demonstrate the present state
of scientific knowledge in control theory and its applications. The MATLAB
package will be extensively used throughout of this book to solve many control
theory problems and allow deeper understanding and analysis of problems that
would not otherwise be solvable using only pen and paper.
Most contemporary control textbooks originated in the sixties or have kept
the structure of the textbooks written in the sixties with a lot of emphasis
on frequency domain techniques and a strong distinction between continuous-
and discrete-time domains. At the present time, all undergraduate students in
electrical engineering are exposed to discrete-time systems in their junior yea
while studying linear systems and signals and digital signal processing courses
so that parallel treatment of continuous- and discrete-time systems saves time
and space. The time domain techniques for system/control analysis and design
are computationally more powerful than the frequency domain techniques. The
time domain techniques are heavily based on differential/difference equations and
linear alge
a, which are very well developed areas of applied mathematics, fo
which efficient numerical methods and computer packages exist. In addition, the
INTRODUCTION 3
state space time domain method, to be presented in Chapter 3, is much more
convenient for describing and studying higher-order systems than the frequency
domain method. Modern scientific problems to be addressed in the future will
very often be of high dimensions.
In this book, the reader will find parallel treatment of continuous- and
discrete-time systems with emphasis on continuous-time control systems and on
time domain techniques (state space method) for analysis and design of linea
control systems. However, all fundamental concepts known from the frequency
domain approach will be presented in the book. Our goal is to present the
essence, the fundamental concepts, of classic control theory—something that
will be valuable and applicable for modern dynamic control systems.
The reader will find that some control concepts and techniques for discrete-
time control systems are not fully explained in this book. The main reason
for this omission is that those “untreated topics” can be simply obtained by
extending the presented concepts and techniques given in detail for continuous-
time control systems. Readers particularly interested in discrete-time control
systems are refe
ed to the specialized books on that topic (e.g. Ogata, 1987;
Franklin et al., 1990; Kuo, 1992; Phillips and Nagle, XXXXXXXXXXInstructors who
are not enthusiastic about the simultaneous presentation of both continuous- and
discrete-time control systems can completely omit the “discrete-time parts” of
this book and give only continuous-time treatment of control systems. This book
contains an introduction to discrete-time systems that naturally follows from thei
continuous-time counterparts, which historically are first considered, and which
physically represent models of real-world systems.
Having in mind that this textbook will be used at a time when control theory
is not at its peak, and is merging with other scientific fields dealing with dynamic
systems, we have divided this book into two independent parts. In Chapters 2–5
we present fundamental control theory methods and concepts: transfer function
method, state space method, system controllability and observability concepts,
and system stability. In the next four chapters, we mostly deal with applications
so that techniques useful for design of control systems are considered. In Chapte
10, an overview of modern control areas is given. A description of the topics
considered in the introductory chapter of this book is given in the next paragraph.
Chapter Objectives
In the first chapter of this book, we introduce continuous- and discrete-time
invariant linear control systems, and indicate the difference between open-loop
4 INTRODUCTION
and closed-loop (feedback) control. The two main techniques in control system
analysis and design, i.e. state space and transfer function methods, are
iefly
discussed. Modeling of dynamic systems and linearization of nonlinear control
systems are presented in detail. A few real-world control systems are given
in order to demonstrate the system modeling and linearization. Several othe
models of real-world dynamic control systems will be considered in the following
chapters of this book. In the concluding sections, we outline the book’s structure
and organization, and indicate the use of MATLAB and its CONTROL and
SIMULINK toolboxes as teaching tools in computer control system analysis and
design.
1.1 Continuous and Discrete Control Systems
Real-world systems are either static or dynamic. Static systems are represented
y alge
aic equations, and since not too many real physical systems are static
they are of no interest to control engineers. Dynamic systems are described eithe
y differential/difference equations (also known as systems with concentrated o
lumped parameters) or by partial differential equations (known as systems with
distributed parameters). Distributed parameter control systems are very hard
to study from the control theory point of view since their analysis is based
on very advanced mathematics, and hence will not be considered in this book.
At some schools distributed parameter control systems are taught as advanced
graduate courses. Thus, we will pay attention to concentrated parameter control
systems, i.e. dynamic systems described by differential/difference equations. It
is important to point out that many real physical systems belong to the category
of concentrated parameter control systems and a large number of them will be
encountered in this book.
Consider, for example, dynamic systems represented by scalar differen-
tial/difference equations
_x(t) = fc(x(t)); x(t0) = x0 (1.1)
x(k +1) = fd(x(k)); x(k0) = x0 (1.2)
where t stands for continuous-time, k represents discrete-time, subscript c in-
dicates continuous-time and subscript d