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66 IEEE CONTROL SYSTEMS MAGAZINE »  june 2013 1066-033X/13/$31.00©2013ieee

Teaching Mathematics to Control Engineers

How can mathematics courses be taught so as to be

more effective for the training of control engineers?

Without doubt, the range of mathematical techniques

relevant to at least one branch of control engineering is

large and includes multivariable calculus, ordinary and

partial differential equations, real and complex analysis,

linear algebra, multivariable statistics, convex optimiza-

tion, functional analysis, and differential geometry. To beeffective in applications, a control engineer should also be

versed in at least one other engineering discipline (such as

electrical, mechanical, or aeronautical) as an understanding

of a system is required before an effective control system

can be designed. A control engineer should also be trained

in systems and control problems and techniques, such as

model identification, experimental design, fault detection

and diagnosis, dynamic optimization, model predictive

control, robust control, and nonlinear control. Collectively,

the number of courses needed to cover all of these topics

would be too high to fit into one curriculum, or even two

curricula, which means that most control engineers take

courses in only some of these topics.

There is not enough time in a control engineering

curriculum for students to take all of the mathematics

courses that could be potentially relevant, so it would be

 best if the mathematics courses that are taken by students

have content that is most relevant to the training of con-

trol engineers.

Probably the most obvious example of a mismatch

 between the topics that are covered and the needs of con-

trol engineers is in linear algebra. A well-trained engineer

should have some knowledge of linear algebra, and so mostengineering degree programs require its students to take

an introductory course in linear algebra, early in their stud-

ies. The coverage in most of these courses is poorly aligned

with the needs of most engineers, and the alignment is

even poorer for training control engineers. A typical linear

algebra course covers matrix multiplication and addition,

eigenvalues, eigenvectors, matrix inverses, and determi-

nants, and a great deal of time is spent teaching students

how to determine matrix inverses and solve systems of lin-

ear equations for matrices of arbitrary dimensions by hand

using the method of elimination of variables, Gaussian and

Gauss-Jordan elimination, and Cramer’s rule. The concepts

in these methods are important for some students, but

rarely will a control engineer solve a large system of linear

equations by hand.

A different selection of material would increase the use-

fulness of the linear algebra course for control engineers.

For example, instead of spending weeks learning Gaussianelimination, that time would be better spent covering qua-

dratic forms and positive-definite matrices so students can

understand the definition of a Lyapunov function.

Due to the mismatch between the content of an intro-

ductory linear algebra course and the linear algebra most

useful for educating control engineers, the subsequent

introductory control course usually begins with a review

of basic concepts of linear algebra that were not taught in

the earlier linear algebra course, such as matrix identities,

singular value decompositions, and quadratic forms. A

typical student learns much more linear algebra useful to

control engineering from reading the appendix of a con-

trol textbook [1] or skimming through a reference book on

matrix theory [2] than from reading an entire introductory

textbook on linear algebra.

The textbook on convex optimization by Stephen Boyd

and Lieven Vandenberghe [3] includes many examples

from control theory, including an optimal control prob-

lem in which the objective is to minimize the amount of 

fuel consumed to move a linear dynamical system from

one value of the state vector to another, an optimal con-

trol problem to compute smooth and small manipulated

variable signals for tracking a desired target output trajec-tory over a finite time horizon, the design of experimental

inputs to maximize the information content of the result-

ing experimental data, and the estimation of parameters

in the presence of uncertainties. This textbook is written

in a manner that a student interested only in convex opti-

mization can retain that focus, without being distracted by

the control examples as they all flow naturally from the

discussion of topics in convex optimization. Also, many of 

the control examples are in the homework problems, so an

instructor who is not interested in control can just avoid

assigning those problems as homework. For a student

interested in both convex optimization and control theory,the textbook is optimal.

RIChARd d. BRAATZ

Digital Object Identifier 10.1109/MCS.2013.2249471Date of publication: 16 May 2013

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june 2013 « IEEE CONTROL SYSTEMS MAGAZINE 67

The recently published textbook on functional analysis by

Yutaka Yamamoto [4] is tuned to the needs of educating con-

trol engineers. The first five chapters provide the background

in vector spaces, normal linear spaces and Banach spaces,

inner product and Hilbert spaces, normed dual spaces, and

the space of linear operators useful for understanding the

subsequent four chapters on distributions, Fourier series andtransforms, Laplace transforms, and Hardy spaces. These

chapters provide the strong foundation for the last chapter,

which covers linear systems and control, controllability and

observability analysis, state-space realization, and H∞-control.

Most control engineers learn bits of functional analysis

along the way as part of their control courses, whereas [4]

provides a single cohesive treatment of functional analy-

sis that would serve as background for all of the control

courses. It seems likely that some of the mathematically

oriented students learning from the textbook will become

attracted to careers in control engineering and that a con-

trol engineer is more likely to take a course on functionalanalysis if taught from this textbook.

Perhaps I am overly biased by being a control engineer,

 but I feel that mathematics courses in such topics as lin-

ear algebra, convex optimization, and functional analysis

designed for teaching control engineers would provide a

 better education for students regardless of whether they

are in control engineering or in some other mathematically

oriented discipline. I believe that mathematics is taught best when in concert with practically relevant examples,

and what other discipline has a richer variety of mathemat-

ical examples than control engineering?

REfERENCES[1] T. Kailath, Linear Systems. Piscataway, NJ: Prentice Hall, 1980.

[2] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, 2nd ed.Princeton, NJ: Princeton Univ. Press, 2009.

[3] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:

Cambridge Univ. Press, 2004.[4] Y. Yamamoto, From Vector Spaces to Function Spaces: Introduction to Func-

tional Analysis with Applications. Philadelphia, PA: SIAM Press, 2012.

 

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