14
15
Solving convex
controller design problems
Book layout.
Figure
1.5
18
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
Notes and References
A history of feedback control is given in Mayr May70] and the book Ben79] and arti-
cle Ben76] by Bennett.
Sensors and Actuators
Commercially available sensors and actuators for control systems are surveyed in the books
by Hordeski Hor87] and DeSilva DeS89] the reader can also consult commercial catalogs
and manuals such as ECC80] and Tra89b].
The technology behind integrated sensors and actuators is discussed in the survey article by
Petersen Pet82]. Commercial implications of integrated sensor technology are discussed
in, e.g., All80] (many of the predictions in this article have come to pass over the last
decade). Research developments in integrated sensors and actuators can be found in
the conference proceedings Tra89a] (this conference occurs every other year), and the
journal Sensors and Actuators, published by Elsevier Sequoia. The journal IEEE Trans.
on Electron Devices occasionally has special issues on integrated sensors and actuators
(e.g., Dec. 1979, Jan. 1982).
Overviews of GPS can be found in the book compiled by Wells Wel87] and the two
volume set of reprints published by the Institute of Navigation GPS84].
Modeling and Identification
Formulation of dynamics equations for physical modeling of mechanical systems is covered
in Kane and Levinson KL85], Crandall et. al. CKK68], and Cannon Can67]. Texts
treating identication include those by Box and Jenkins BJ70], Norton Nor86], and
Ljung Lju87], which has a complete bibliography.
Linear Controller Design
P and PI controllers have been in use for a long time for example, the advantage of a
PI controller over a P controller is discussed in Maxwell's 1868 article Max68], which is
one of the rst articles on controller design and analysis. PID tuning rules that have been
widely used originally appeared in the 1942 article by Ziegler and Nichols ZN42].
The book Theory of Servomechanisms JNP47], edited by James, Nichols, and Philips,
gives a survey of controller design right after World War II. The 1957 book by Newton,
Gould, and Kaiser NGK57] is among the rst to adopt an \analytical" approach to
controller design (see below). Texts covering classical linear controller design include
Bode Bod45], Ogata Oga90], Horowitz Hor63], and Dorf Dor88]. The root locus
method was rst described in Eva50].
Recent books covering classical and state-space methods of linear controller design include
Franklin, Powell, and Emami FPE86] and Chen Che87]. Linear quadratic methods
for LTI controller design are covered in Athans and Falb AF66, ch.9], Kwakernaak and
Sivan KS72], Anderson and Moore AM90], and Bryson and Ho BH75].
Three recent books on LTI controller design deserve special mention: Lunze's Robust Mul-
tivariable Feedback Design Lun89], Maciejowski's Multivariable Feedback Design Mac89],
and Vidyasagar's Control System Synthesis: A Factorization Approach Vid85]. The rst
two, Lun89] and Mac89], cover a broad range of current topics and linear controller
design techniques, although neither covers our central topic, convex closed-loop design.
Compared to this book, these two books address more directly the question of how to
NOTES AND REFERENCES
19
design linear controllers. Vidyasagar's book Vid85] contains the \recent results" that we
referred to at the beginning of section 1.4. Our book can be thought of as an extension or
application of the ideas in Vid85].
Digital Control
Digital control systems are covered in the books by Ogata Oga87], Ackermann Ack85],
and Astrom and Wittenmark AW90]. A recent comprehensive text covering all aspects
of digital control systems is by Franklin, Powell, and Workman FPW90].
Control Processors and Controller Implementation
Programmable logic controllers and other industrial control processors are covered in
Warnock War88]. An example of a commercially available special-purpose chip for con-
trol systems is National Semiconductor's lm628 precision motion controller Pre89], which
implements a PID control law.
The use of DSP chips as control processors is discussed in several articles and manufac-
turers' applications manuals. For example, the implementation of a simple controller on
a Texas Instruments tms32010 is described in SB87], and the implementation of a PID
controller on a Motorola dsp56001 is described in SS89]. Chapter 12 of FPW90] de-
scribes the implementation of a complex disk drive head positioning controller using the
Analog Devices adsp2101. The article Che82] describes the implementation of simple
controllers on an Intel 2920.
The design of custom integrated circuits for control processors is discussed in JTP85]
and TL80].
General issues in controller implementation are discussed in the survey paper by Hansel-
mann Han87].
The book AT90] discusses real-time software used to program general-purpose computers
as control processors. Topics covered include implementing the control law, interface to
actuators and sensors, communication, data logging, and operator display.
Computers and Control Engineering
Examples of computer-based equipment for control systems engineering include Hewlett-
Packard's hp3563a Control Systems Analyzer HeP89], which automates frequency re-
sponse measurements and some simple identication procedures, and Integrated Systems'
ac-100 control processor ac100], which allows rapid implementation of a controller for
prototyping.
A review of various software packages for structural analysis is given in Nik86], in particu-
lar the chapter Mac86]. A widely used nite element code is nastran Cif89]. Computer
software packages (based on Kane's method) that symbolically form the system dynamics
include sd/exact, described in RS86, SR88] and autolev, described in SL88]. Exam-
ples of software for system identication are the system-id toolbox Lju86] for use with
matlab and the system-id package mat88] for use with matrix-x (see below).
Examples of controller design software are matlab MLB87] (and LL87]), matrix-
x SFL85, WSG84], delight-mimo PSW85], and console FWK89]. Some of these
programs were originally based on the linear algebra software packages linpack DMB79]
and eispack SBD76]. A new generation of reliable linear algebra routines is now being
developed in the lapack project Dem89] lapack will take advantage of some of the
advances in computer hardware, e.g., vector processing.
20
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
General discussion of CACSD can be found, for example, in the article Ast83], and the
special issue of the Proceedings of the IEEE PIE84]. See also JH85] and Den84].
Determining Limits of Performance
The value of being able to determine that a set of specications cannot be achieved, and
the failing of many controller design methods in this regard, has been noted before. In
the 1957 book Analytical Design of Linear Feedback Controls, by Newton, Gould, and
Kaiser NGK57, 1.6], we nd:
x
Unfortunately, the trial and error design method is beset with certain fun-
damental diculties, which must be clearly understood and appreciated in
order to employ it properly. From both a practical and theoretical viewpoint
its principal disadvantage is that it cannot recognize an inconsistent set of
specications.
...The analytical design procedure has several advantages over the trial and
error method, the most important of which is the facility to detect immediately
and surely an inconsistent set of specications. The designer obtains a \yes"
or \no" answer to the question of whether it is possible to fulll any given
set of specications he is not left with the haunting thought that if he had
tried this or that form of compensation he might have been able to meet the
specications.
...Even if the reader never employs the analytical procedure directly, the
insight that it gives him into linear system design materially assists him in
employing the trial and error design procedure.
This book is about an analytical design procedure, in the sense in which the phrase is used
in this quote.
There are a few results in classical controller design that can be used to determine
some specications that cannot be achieved. The most famous is Bode's integral theo-
rem Bod45] a more recent result is due to Zames Zam81, ZF83]. These results were
extended to unstable plants by Freudenberg and Looze FL85, FL88], and plants with
multiple sensors and actuators by Boyd and Desoer BD85].
The article by Barratt and Boyd BB89] gives some specic examples of using convex
optimization to numerically determine the limits of performance of a simple control system.
The article by Boyd, Barratt, and Norman BBN90] gives an overview of the closed-loop
convex design method.
About the Example in Section 1.4.1
The plant used is described in section 2.4 the process and sensor noises are described in
chapter 11, and the precise denitions of RMS actuator eort, RMS regulation, and step
response overshoot are given in chapters 3, 5, and 8.
The method used to determine the shaded region in gure 1.2 is explained in section 12.2.1
a similar gure appears in Kwakernaak and Sivan KS72, p205]. The method used to
determine the shaded region in gure 1.3 is explained in detail in chapter 15.
The exact form of the PD controller was
( ) =
+
k
sk
p
d
(1.1)
K
s
p
d
(1 + 20)2
s=
NOTES AND REFERENCES
21
where and are constants, the proportional and derivative gains, respectively.
k
k
p
d
Determining the shaded region shown in gure 1.4 required the solution of many global
optimization problems in the variables
and . We rst used a numerical local op-
k
k
p
d
timization method designed especially for parametrized controller design problems with
RMS specications see, e.g., the survey by Makila and Toivonen MT87]. This produced
a region that was likely, but not certain, to be the whole region of achievable specications.
To verify that we had found the whole region, we used the Routh conditions to determine
analytically the region in the , plane that corresponds to stable closed-loop systems
k
k
p
d
this region was very nely gridded and the RMS actuator eort and regulation checked
over this grid. This exhaustive search revealed that for this example, the local optimiza-
tion method had indeed found the global minima it simply took an enormous amount of
computation to verify that the solutions were global. Of course, in general, local methods
can miss the global minimum. (See the discussion in section 14.6.4.)
A more sophisticated global optimization algorithm, such as branch-and-bound, could
have been used (see, e.g., Pardalos and Rosen PR87]). But all known global optimization
algorithms involve computation that in the worst case grows exponentially with the number
of variables. A similar ve parameter global optimization problem would probably be
computationally intractable.
22
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
