Utilizing SIMULINK and MATLAB in a Graduate Nonlinear Systems
Analysis Course
Asad Azemi1 and Edwin Engin Yaz2
Department of Electrical Engineering
1
Penn State University
2
University of Arkansas
Abstract
Control system design packages like MATLAB,
MATRIXX, Control C, SIMNON, etc. have become
essential
ingredients of
both undergraduate and
graduate courses in the systems and controls area. The
use of MATLAB and its companion toolboxes in
teaching graduate and undergraduate courses have
reported by many educators in the past. This work
describes our experience, at the Pennsylvania State
University the University of Arkansas, with the use
SIMULINK and
MATLAB in
Nonlinear Systems
Analysis and Stability course which is offered to
advanced undergraduate and graduate students. This
paper will emphasize on the use of SIMULINK for
simulation of nonlinear systems and study of their
behavior. We also present the MATLAB features that
are found useful in this course. A discussion of how
SIMULINK and
MATLAB help in enhancing the
students' understanding of the material and reducing the
amount of time spent in performing computational
homework assignments will follow. Examples illustrating
the use of these packages will be included and the
actual code will be posted at a site on the World Wide
Web. Finally, the general positive student reaction to
incorporating these software packages into this course
will be reported.
Introduction
The Electrical and Engineering Departments at
Penn State University and the University of Arkansas are
incorporating computer aided engineering (CAE) and
computer aided design (CAD) packages into their
curricula. The intent of augmenting the curriculum with
these packages is to enhance the students theoretical
understanding of the material with hands on analysis and
design experience. The benefits of CAE and CAD
packages in the classroom have been realized by the
authors and their co-workers before [1-6]. The benefits
of using these packages in a university setting is also
confirmed by the number of new textbooks, and revisions
of previously printed textbooks incorporating new
exercises and problems based on these packages, such as
[7-14]. A summary of the advantages and disadvantages
of incorporating these packages into our graduate
curricula are presented below. The summary is followed
by sections outlining the use of each package in specific
classes. The use of SIMULINK [15] in this course is
fairly new. In this work we will concentrate on those
features that are useful in a graduate level nonlinear
systems analysis and stability course. A summary of the
advantages and disadvantages of incorporating these
packages into our curriculum are presented below.
General Advantages
The main advantages of using these tools are:
the reinforcement of student understanding of theoretical
principles by means of enhanced graphical aids and
interactive simulations,
analysis of more complex
systems that can be treated by pencil and paper, and the
instructors ability to assign fairly complex design
problems that otherwise would have be unrealistic
without the help of such software.
Student response concerning the use of these
packages is generally favorable.
One interesting
response received from students is an increased interest
in the subject material. It is also worth mentioning that
the use of many CAE packages, such as MATLAB and
SIMULINK, are no longer limited to a a specific filed.
Early exposure to these packages will benefit the
students. For a more detailed discussion of this topic
readers can refer to our previous works [2-4].
General Disadvantages
Three of the disadvantages of using these
packages are the maintenance and operation of these
packages on an accessible computer system, the extra
work required by students (and instructors) to learn how
to use CAE packages, and assuring that the packages are
included in the baseline curriculum as part of the
required course material. A more detailed discussion of
this topic can be found in our previous works [2-4].
MATLAB and SIMULINK in a Nonlinear
Systems Analysis
Nonlinear Systems Analysis at the University
of Arkansas
Although, a lot of work has been done in
incorporation of MATLAB [15] into undergraduate
control system textbooks and courses [7-14], there is no
reference textbook for a graduate level nonlinear system
analysis course. It should be mentioned that some of the
work done in an undergraduate control systems textbook,
using MATLAB, could also be applied to a graduate level
course. In this section we will first present the way that
this course is structured at Penn State University Great
Valley Campus and the University of Arkansas. Next, we
present the topics that are covered in this course.
This course is offered to advanced
undergraduate and graduate students. The goal of this
course is to expose the students to the basic methods in
the modeling, analysis, and design of nonlinear control
system. To accomplish this goal, the following topics are
typically covered in a semester:
Nonlinear Systems Analysis at Penn State
Great Valley
Penn State Great Valley Campus, one of the
eighteen campuses of the Penn State University, is a
graduate center designed to address the educational need
of the working engineers in Philadelphia area. Almost
all of our students are working engineers, with a wide
variety of backgrounds using simulation packages. The
nonlinear analysis course deals with the analysis and the
design of nonlinear control systems. In the analysis, a
nonlinear closed-loop system is assumed to have been
designed, and we wish to determine the characteristics of
the system’s behavior. In the design section a nonlinear
plant is given and our task is to construct a controller so
that the closed loop system meets the desired
characteristics. The nonlinear systems course covers the
following topics:
1.
2.
3.
4.
5.
6.
7.
Phase Plane Analysis
Fundamentals of Lyapunov Theory
Advanced Stability Theory
Describing Function Analysis
Feedback Linearization
Sliding Control and Estimation
Adaptive Control
[16] and [17] are used in teaching the course. Students
are given weekly assignments that also include computer
simulation/usage. Exams also include a take home part
that have computer simulations. Student have access to
student versions of MATLAB and SIMULINK.
1.
2.
3.
Examples of nonlinear control systems (4 classes).
Second order systems (3 classes).
Fundamental properties of nonlinear differential
equations (4 classes).
4. Lyaponov stability (4 classes).
5. Invariance principle (2 classes).
6. Input-output stability (1 class).
7. Control and observer design based on local
linearization (1 class).
8. Exact linearization (4 classes).
9. Analysis of perturbed systems by the Lyapunov
method and Bellman-Gronwall lemma.
10. Ultimate boundedness and its achievement by minmax design of controller and observers (2 classes).
One class is normally 80 minutes of lecture. Usually two
in class exam is given and the rest of the grading is based
on weekly homework assignments and the student
portfolio that includes the class notes, corrected
homework,
etc.
The
major
use
of
MATLAB/SIMULINK is in homework assignments.
Use of MATLAB/SIMULINK in a Nonlinear
Systems Analysis Course
In this section we will present those features of
the MATLAB/SIMULINK that we have found most
useful in our classes, based on the aforementioned course
outlines. Due to the nature of the SIMULINK, a
graphical user interface program, that is basically using
block diagram approach for simulating different system,
it can be effectively used in simulation of different
nonlinear systems. The block diagram nature of the
program will enable us to study the behavior of the
system under different nonlinearities. This will enable us
to use the software in investigating finite escape time,
multiple equilibria, limit cycles and their nature (stable
or unstable), etc. For example students are given 2dimensional tunnel-diode model and are asked to
investigate the behavior of the system by plotting the
phase portrait diagram. In another example students are
given the chaotic Lorenz attraction equation , with proper
ranges of parameters and initial conditions. They are
asked to plot the three phase plane portraits for this 3dimensional system for various parameter values. Then,
they are asked to change the initial conditions to observe
the sensitivity of the system to the initial conditions.
Individual state trajectories are plotted to see the
seemingly random (chaotic) behavior. SIMULINK also
contains Sources library that allows one to choose the
input (forcing functions) necessary for simulation. The
Linear library contains transfer functions and summers
where as the Nonlinear library has many common
nonlinearities like dead zone, saturation, etc. The Sinks
library has the Scope, for example, that displays any
desired signal. Blocks are activated by double clicking,
and holding the mouse which the object is being dragged
to the proper location. Connections are also made by the
same “dragging” procedure. Parameters are selected
from the simulation menu after the block diagram is
complete and Start is used to carry out the simulation.
There are many other facilities available in SIMULINK
which is beyond the scope of this paper. In addition to
the simulation capability of the SIMULINK, MATLAB’s
Control System Toolbox can be used e.g. to investigate
the stability of the perturbed linear systems. EIG can be
used for finding the eigenvalues of the system matrix of a
linear system to check whether they are all in the left half
complex plane or inside the unit circle for continuousand discrete-time systems, respectively. When these
systems are perturbed, there are many stability robustness
bounds that can be calculated via the use of LYAP or
DLYAP for continuous- and discrete-time algebraic
matrix Lyapunov equations [17]-[18]. Controllers and
observers can be designed using linear techniques e.g.
PLACE to assign the poles of a linear system by state
feedback or LQR to design an optimal state feedback
control used on local linearizations of nonlinear systems.
Observer design s can be realized by duality. Finally, the
performance of nonlinear controller and observer
schemes like feedback linearization and min-max designs
can be assessed using SIMULINK.
Conclusion
In this paper, we have presented the use of
SIMULINK/MATLAB software packages in our graduate
curriculum, at Penn State University and the university of
Arkansas. Several of the advantages provided by
computer simulation packages, such as SIMULINK,
include a reinforcement of student understanding of
theoretical principles, allowing assignment of larger, and
more complex designs, increased student attentiveness,
and enhanced professional development. The main
disadvantages of using computer simulation packages are
the extra work required of students and instructors, the
maintenance and operation of these packages on an
accessible computer system, and assuring that the
packages are inserted in the baseline curriculum as part
of the required course material. The general student
reaction to the use of SIMULINK and MATLAB has
been very positive.
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