IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999 319
Model Predictive Satisficing Fuzzy Logic Control
Michael A. Goodrich, Wynn C. Stirling, and Richard L. Frost
Abstract— Model- predictive control, which is an alternative
to conventional optimal control, provides controller solutions to
many constrained and nonlinear control problems. However, even
when a good model is available, it may be necessary for an
expert to specify the relationship between local model predictions
and global system performance. We present a satisficing fuzzy
logic controller that is based on a receding control horizon,
but which employs a fuzzy description of system consequences
via model predictions. This controller considers the gains and
losses associated with each control action, is compatible with
robust design objectives, and permits flexible defuzzifier design.
We demonstrate the controller’s application to representative
problems from the control of uncertain nonlinear systems.
Index Terms— Decision- making, intelligent control, predictive
control, satisficing.
I. INTRODUCTION
A LTHOUGH many useful optimality- based controller de-signs
exist, it is sometimes difficult to define and find op-timal
solutions to highly nonlinear highly complex problems.
This places controller design for such systems in the class
of ill- formed problems wherein there is a lack of sufficient
information, time, or resources to define or to find the optimal
solution [ 1]. Ill- formed problems motivate the search for
intelligent solutions, the success of which rests, to some
degree, upon the belief that finding the optimal decision is
not necessary for making justifiable decisions [ 2]–[ 4]. The
search for intelligent solutions necessarily addresses: 1) the
definition and computation of acceptable solutions; 2) the
identification of models ( whether implicit in expert rules or
explicit in differential equations); and 3) the robust synthesis
of information from multiple sources. To generate intelligent
controllers, each issue in this noninclusive list demands a
formal and justifiable treatment. In this paper, we address these
issues from a perspective that employs strongly satisficing
decision theory and fuzzy logic.
A. Solution Motivation and Background
In conventional optimal control, explicit models describe
possible system consequences and these possible consequences
are ordered using a cost function. For well- formed problems,
minimizing this cost function determines the optimal global
solution with respect to the specified cost function and the
Manuscript received March 27, 1998; revised July 31, 1998. This work was
supported in part by Nissan Cambridge Basic Research, Nissan Research and
Development, Inc.
M. A. Goodrich is with Nissan Cambridge Basic Research, Cambridge, MA
02142 USA.
W. C. Stirling and R. L. Frost are with the Electrical and Computer
Engineering Department, Brigham Young University, Provo, UT 84602 USA.
Publisher Item Identifier S 1063- 6706( 99) 04934- 6.
Fig. 1. Steps of inference: observation to action.
given explicit model. Specifying the cost function is left to
the designer ( an implicit expert); from experience, quadratic
costs are often used because they yield computable solutions
and, in the case of positive definite cost matrices, produce
unique minimizing solutions.
In conventional fuzzy logic control, the predicted system
consequences are implicit in the rules, where the model is
implicit in the rules too. Rules are obtained explicitly from an
expert and presumably have been compiled from the following
sequence of inferences ( see Fig. 1):
1) if control is applied given observation
then the plant consequence is ;
2) given observation , plant consequence
is most desirable ( i. e., maximizing);
3) given observation , control
should be used;
4) given observation , do .
Since expert rules are not easily obtained for all , fuzzy logic
methods are used to interpolate among a grid of selected
points.
Conventional optimal control assumes not only that an
explicit model of the plant exists, but also that an implicit
expert is available to prescribe a cost function that can be
solved using optimization methods. Though frequently effec-tive
for controller design, some problems are not appropriately
addressed by optimality- based methods [ 5]. In contrast to
optimal control, conventional fuzzy logic control assumes that
an explicit expert exists who can construct a rule base that
transforms observations into plant controls via an implicit
model of the system. Again, though frequently effective for
controller design, useful information may be unnecessarily
discarded by compiling steps 1)– 3) into 4).
There exist problems that can benefit from the best of
both optimal and fuzzy approaches to controller design. For
example, the use of local model predictions may free the expert
from some of the burden of performing mental simulations
of the plant [ 6] and instead allow the expert to identify
consequences as beneficial or costly. This paper is interested
in a subset of such problems where an explicit model of
1063– 6706/ 99$ 10.00 ã 1999 IEEE
320 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
the system exists and an explicit expert is used to transform
local model predictions into global evaluations of gains and
losses. Such an approach is necessary when complexity and
uncertainty prevent precise predictions about global plant
behavior, but when useful information is available from local
plant predictions.
Evaluating the gains and losses of a control action using
model predictions is based on the comparative “ cost/ benefit”
structure of strongly satisficing decision theory ( SSDT) [ 4],
[ 7], [ 8]. SSDT modifies the objective of finding the optimal
control ( with respect to a given model structure and cost
function) by including the less ambitious ( and perhaps more
robust) objective of avoiding error [ 4]. This design paradigm
employs the comparative rationality that is suggested by
Simon’s satisficing principle [ 9], [ 10], the domination principle
from multi- attribute utility theory and the mathematics of
Levi’s error avoidance principle [ 11]. Employing fuzzy logic
in the synthesis of SSDT- based model predictive controllers
produces a method for systematically designing fuzzy logic
controllers that avoid error.
A useful property of employing the satisficing principle
in fuzzy logic controller design is the effect upon defuzzi-fier
specification. Usually, the rationality behind defuzzifica-tion—
a process which has been described as “ an art rather
than a science” [ 12]— is that of finding the best decision
or control. This is typified not only by efforts to formulate
the defuzzification problem as “ the problem of optimal selec-tion”
[ 13, p. 38], but also by interpretations of fuzzy logic
systems as universal function approximators [ 12]. For the
cost/ benefit- based satisficing fuzzy logic controller specified
herein, defuzzifier design is motivated by the acceptability of
multiple controls and, hence, enjoys a degree of flexibility that
is useful for the design of robust controllers.
B. Related Literature
In this section, we briefly review related literature. A more
extensive review can be found in the companion technical
report [ 14]. Model predictive control ( MPC), also known as
moving horizon and receding horizon control, is a method for
designing controllers that operate in nonlinear, constrained,
and uncertain environments. Successes in application are sup-ported
by theoretical advances, such as the characterization and
specification of sufficient conditions for stability [ 15]–[ 18],
and by algorithms that are computationally efficient and ensure
disturbance rejection through state feedback [ 19], [ 20]. In this
paper, we significantly extend the results from [ 4], [ 7] to
include considerations and contributions from fuzzy logic and
to include the ability to deal with conflicting information or
multiple experts.
Increasingly, fuzzy logic researchers are addressing the
implicit compilation of steps 1)– 3) into 4). For example, in
both [ 21] and [ 22], explicit models are used to determine
system behavior, which is then used to generate fuzzy controls
via conventional fuzzy and classical control methods, respec-tively.
An alternative to these approaches generates a fuzzy
model and then employs stability as the sole performance
criterion [ 23], [ 24]. In terms of Fig. 1, such approaches infer
from and and then use to determine action .
These applications are primarily motivated by the desire
to establish the provable stability of fuzzy logic controllers
and to formulate systematic methods for controller design.
Such approaches seek to combine objective mathematical
knowledge with subjective fuzzy knowledge [ 25], [ 26]. In
this paper, we combine the objective knowledge produced by
mathematical models of nonlinear systems with a subjective
interpretation of how local model predictions imply global
system consequences. The use of local model predictions
may free the expert from some of the burden of performing
mental simulations of the plant [ 6] and instead allow the
expert to identify consequences as beneficial or costly, even
for unintuitive plants [ 21].
Satisficing presents a decision- making paradigm that differs
from the de facto paradigm of optimality [ 27]–[ 30]. Many
cognitive scientists recognize that insistence on optimality is
a misplaced requirement in situations of limited resources
and information and that optimality inadequately describes
observed behavior in naturalistic settings [ 10], [ 31]–[ 33].
Additionally, the definition of and reliance upon an optimal
solution has been questioned by Zadeh [ 5], [ 34] and other
philosophers, scientists, and researchers concerned with prag-matic
decision- making [ 2], [ 3], [ 35], [ 36]. The relationship
between fuzzy logic- based satisficing and set- valued maxi-mization
is further explored in the companion paper [ 14].
Our treatment of uncertainty relies on higher order un-certainty
[ 37] and, specifically, set- based Bayesianism [ 38].
Set- based Bayesianism permits a set of probabilities to de-scribe
uncertainty and under certain conditions subsumes
Dempster– Shafer theory as a special case [ 39]. Rather than
adopting a risk- averse stance such as minimax or a risk- neutral
stance such as expectation, set- based Bayesianism allows an
intermediate stance to be taken. This intermediate stance takes
expectations with respect to a set of probabilities; a control is
justified only if it is acceptable for each expected consequence.
This is similar to requirements for robustness in set- theoretic
estimation, including other developments of the satisficing
concept [ 27], [ 28], [ 40]–[ 43].
II. STRONGLY SATISFICING DECISION THEORY
As discussed in the review of relevant literature, many
cognitive scientists recognize that insistence on optimality is
a misplaced requirement in situations of limited resources
and information. Simon [ 9] addressed the issue of limited or
bounded rationality by defining an aspiration level such that
once this level is met the corresponding solution is deemed ad-equate
or satisficing. 1 An important characteristic of Simon’s
satisficing principle is that decisions are deemed adequate on
the basis of a comparison: any decision which exceeds the
aspiration level is admissible. We employ this characteristic
by constructing and comparing two set membership functions
similar to the way benefit and cost are compared in economics
literature. This comparison leads naturally to a constructive
procedure for identifying satisficing decisions in nonlinear
1A term employed by Simon. A convenient mnemonic is satisfice = satisfy
+ suffice.
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 321
system controller design contexts using receding planning
horizons. The key to this development lies in partitioning
the consequences into a generalized type of benefit called
accuracy and a generalized type of cost called liability. These
two decision attributes may be operationally characterized as
follows.
Accuracy: A natural characterization of the benefit of a
decision is accuracy, meaning conformity to a standard. In
practical contexts, the standard corresponds to whatever goal
or objective is relevant to the problem and accuracy corre-sponds
to the degree of success in achieving that goal. In the
context of fuzzy logic, the term accuracy refers to the set
membership function associated with the linguistic variable
ACCURATE. 2
Liability: Actions may also be evaluated strictly in terms
of their liability, meaning susceptibility or exposure to unde-sirable
consequences. Typically, these consequences may be
manifest in the form of costs or other penalties that would
accrue simply as a result of taking the action, regardless of its
accuracy. Liability corresponds to the degree to which actions
accrue costs or penalties. In the context of fuzzy logic, the
term liability refers to the set membership function associated
with the linguistic variable LIABLE.
For example, in regulator design, the fundamental objective
is to drive the system to and maintain the system at a desired
operating point. Thus, accuracy refers to the degree to which
the possible controlled states satisfy this objective. Indepen-dent
of the desire to regulate the system is the desire to prevent
excessive control authority and oscillatory state transitions.
Thus, liability refers to the cost of possible controlled states
with respect to these undesirable consequences.
Given these two evaluations of consequences, two indepen-dent
principles can be applied: satisficing and domination. The
satisficing principle ( as we have used it) provides a mechanism
for determining what action can be done given the observed
evidence; the domination principle provides a mechanism
for determine what actions should not be done given the
alternative actions. For some problems such as constrained
decision- making and task- based behavior [ 44], the satisficing
principle can be applied without applying domination; and for
other problems such as conventional multi- attribute decision
analysis, the domination3 principle can be applied without
satisficing.
A. Satisficing Decisions
Using Levi’s error avoidance principle [ 11], SSDT provides
a method by which the accuracy and liability set member-ship
functions can be merged: to avoid error, a decision
maker accepts those decisions which are ACCURATE and
not LIABLE. Formally, let denote the set of possible
decisions or actions and let denote the states of nature. The
states of nature represent those conditions, which affect the
consequence of a decision but which cannot be controlled.
2 For the remainder of the paper, we use capital letters and a separate font
when we refer to linguistic variables, but will make no such distinction for
membership functions.
3 Because domination is an extension of the de facto optimality presumption,
this principle is much more frequently encountered in decision making.
For each decision and for each state of nature ,
a consequence results4 that is the effect of making decision
when nature is in state . The accuracy
and liability set membership functions are
defined for each consequence ( i. e., action/ state- of- nature pair).
In SSDT, the set of all decisions which cannot be justifiably
eliminated is called the satisficing set and is linguistically
defined as
For the problems addressed herein, we wish to include mul-tiplicative
hedges , which allow the fuzziness
inherent in the consequences of an action to be parameterized.
Thus, we form the satisficing set membership function as
( 1)
where represents a design parameter that is related to and
, where represents a - norm, and where represents
the complement of the hedged set . When represents the
- norm ( see, for example, [ 46]),
the satisficing set membership becomes
( 2)
where is called the rejectivity and parameterizes
the relative weight5 between accuracy and liability.
The comparative nature of ( 2) is best illustrated by consid-ering
the region of support ( area of nonzero set membership)
for the satisficing set, which is given by
( 3)
From ( 3) we see that decisions are satisficing if and only
if the accuracy membership is large relative to the liability
membership. This comparison provides a set- based mathe-matical
formalism necessary to use the notion of satisficing
in controller design. Through this comparison, each potential
action may be evaluated on its own merits without comparing
it to other actions. It is easily shown that for any membership
functions defined such that
or , a sufficient condition
for is that . In practice, this condition is overly
conservative and is replaced by the operational restriction
. Note that since we restrict attention
to membership functions such that the maximum of and
is unity, the supremum exists and is finite.
4A decision u is often treated as a mapping from into the set of
consequences [ 45].
5The subjective selection of this relative weighting is analogous to the
tradeoff between the size and power of a statistical hypothesis test using
Neyman– Pearson decision theory. Similar to the way in which subjectively
selecting a test’s size determines the test’s power in Neyman– Pearson hy-pothesis
testing, subjectively selecting b determines the relative importance of
accuracy and liability.
322 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
B. Strongly Satisficing Decisions
Although the set contains all possible actions that are
legitimate candidates for adoption, they generally will not be
equal in overall quality. For example, two satisficing actions
may have similar accuracy membership but have significantly
different liability membership and implementing the one with
the lower liability will yield essentially the same fuzzy benefit
with lower fuzzy cost. Thus, we are motivated to refine the
set of satisficing actions. For every let
( 4)
and define the set of actions that are strictly better than ( i. e.,
set of actions that dominate )
( 5)
that is, consists of all possible actions that have lower
liability but not lower accuracy than or have higher accuracy
but not higher liability than . If , then no actions
can be preferred to in both accuracy and liability and is
a ( weakly) nondominated action with respect to . The ( crisp)
nondominated set
( 6)
contains all nondominated actions.
The intersection of the nondominated set with the satisficing
set yields the strongly satisficing set
( 7)
otherwise
( 8)
where represents the crisp support set given . When
is nonempty, it has been shown that is also nonempty
[ 7]. Intuitively, contains only those controls which can
be done given the evidence and does not contain any controls
which should not be done given the alternatives. Elements of
exhibit both properties where, as we shall demonstrate, the
strongly satisficing set can facilitate flexible defuzzifier design.
C. Robustness
The accuracy and liability set membership functions are
defined such that both and are measurable
functions defined over for fixed and both and
are set membership functions for fixed . These
membership functions represent the value of a decision when
the state of nature is or stated simply, represent the statement
if then is . By contrast inferring a
decision directly from the observations ( if then do ), we
instead infer the value of a decision from the observations
( if then is ( not) valuable). When the state of nature is
instead of or when is nonsingleton with membership
, then the accuracy/ liability of is obtained by forming
the composition of the linguistic statements if then
is—. For the applications in this paper, the composition is
defined by taking the expected accuracy/ liability membership
with respect to the state of nature described by the membership
function ( which is constrained to be a probability density
function). Note that other compositions can also be considered.
For some problems, there may exist multiple descriptions
of and it is desirable to include each of these multiple
descriptions in the definition of . For example, there may
be two experts who specify accuracy and liability membership
functions; i. e., there exist and and corresponding
membership functions , , and ,
. For the applications presented herein, 6 we restrict
attention to expert/ system descriptions that share a common
state of nature and common evaluations of consequences
, , but differ in their descriptions of nature
. Formally, let
( 9)
denote a closed set of subjective probability densities that
represent a designer’s understanding of the state of nature,
where the set represents an index of this set ( for example,
). For simplicity, attention is restricted to
countable . Corresponding to each is an expected
accuracy and an expected liability given by
When contains only one element, the definitions for the
satisficing set and nondominated set in ( 2)–( 3) and ( 6),
respectively, are modified by using and instead
of and . By contrast, when contains more
than one element ( i. e., there are more than one measurement
source or expert opinion), a decision is satisficing if
and only if it is satisficing for all . This is similar
to requirements for robustness in set- theoretic estimation,
including other developments of the satisficing concept [ 27],
[ 28], [ 40]. The resulting set is called the robust satisficing set
and is defined as
( 10)
One criticism against using the intersection operator to fuse
results generated by multiple sources is that the resulting
set may be empty. However, the design parameter can be
selected such that the robust satisficing set is always nonempty.
This is achieved if and only if is bounded by ,
where . Loosely speaking, is
bounded by the minimax value over all nonvacuous
expert opinions.
6 Using the more general framework of multiple states of nature we
establish a foundation for designing controllers which fuse multiple sources
of sensor information.
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 323
The robust nondominated set can also be defined by elim-inating
controls for which obviously better ( dominating) con-trols
exist. Let
Then
( 11)
In general, this set can be difficult to compute. Since, as the
following theorem shows, , we can use the
following approximation for the robust equilibrium set:
( 12)
Theorem 1: .
Proof: Suppose . Then there exists a
such that where . Since it
follows that where .
Since each is always nonempty, is also always
nonempty. For the applications considered in this paper,
so we use the more simple notation.
The robust strongly satisficing set can now be defined by
restricting the robust satisficing set to the region of support
defined by the robust equilibrium set . The robust satisficing
set can be expressed as
( 13)
otherwise
( 14)
from which we see that ( 7) and ( 8) are special cases with a
singleton and a delta function placing all belief mass on the
single value . We now have a general definition for a robust
strongly satisficing control. Broadly speaking, a decision is
robustly strongly satisficing if and only if it is satisficing for
every belief held by the designer and nondominated for any
such belief. Thus, if a designer is unsure of precisely how
nature should be fuzzified7 but can restrict the description to
within a set, then the designer can choose a control which is
justifiable for all descriptions.
In the following theorem, we present a sufficient ( but not
necessary) condition to guarantee that the robust strongly
satisficing set is nonempty. The interpretation of this theorem
is as follows: if there is a decision such that
has nonzero support, then the robust
strongly satisficing set has nonzero support. In words, if is
for fuzzified state and not for
fuzzified state, then is strongly satisficing.
Theorem 2: If there exists a such that
then .
7 In our usage, fuzzification is used in the sense of [ 47].
Proof: We will prove that is nonempty for these
conditions by constructing a specific element of this set. Let
( 15)
denote the ( robustly) most discriminating control. We
first show that Since
( otherwise it is not most discriminating
given the hypothesis), then for any ,
implies that
, where .
We now show8 that . For any , suppose that
. Then there is a where
which means that is most discriminating. This contradicts
our assumption where for all , where .
Since and then , where .
Note that the operator de- emphasizes the good
and the operator overemphasizes the bad yielding
a conservative controller.
D. Defuzzifier Design
Defuzzifiers for typical fuzzy logic systems employ su-perlative
methods such as the maximum defuzzifier or av-eraging
methods such as the centroid defuzzifier [ 12]. In
these methods, a single control is selected from the set of
controls with nonzero set membership because it is superior
or most representative. Selection and design of a defuzzifier
can be a significant obstacle in designing a fuzzy logic
controller [ 12]. We suggest that one reason defuzzifier design
is difficult is that many rule bases are based upon local
rather than than global performance. A well- known result
from optimal control theory is that optimizing over a local
planning horizon does not necessarily yield global optimal
performance. Thus, we suggest that rule bases determined by
local performance considerations may be incompatible with
optimal defuzzifying and may instead require “ procedurally
rational” defuzzifying [ 4].
The satisficing fuzzy logic controller ( SFLC) includes both
the accuracy set membership function as well as the liability
set membership function. Unlike most fuzzy control applica-tions
( but similar to the ART model [ 48, p. 106]), the output
vector of the SFLC does not correspond to a final control
actions; i. e., inference is not made from the observation to
a control solution. Instead, inference from the observation to
the control solution proceeds in three distinct steps: 1) observe
nature and infer consequences of control actions through an
explicit dynamic model; 2) infer values from the consequences
using an explicit expert; and 3) infer the fuzzy set control
solutions from values using both the satisficing and domination
principles. Note that when and when the accuracy
inference is made directly from observations to values then
8 Note that we actually show the stronger result that u D 2 \
2 ?? E
.
324 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
the conventional FLC is obtained and, thus, the SFLC can be
viewed as a generalization of the FLC.
For the SFLC, any defuzzifier which selects a control from
is justifiable according to two criteria: 1) all such controls
are satisficing ( in ), which means that evidence is sufficient
to justify their use and 2) all such controls are maximizing
[ in ( 1), , see ( 16)] for some , which means that
no other control is superior and thereby precludes their use.
Furthermore, when ( the set of beliefs held by the designer)
is a singleton, then the set of strongly satisficing controls can
claim superiority in the following sense and, hence, can be
justified as the defuzzifier output. Let
( 16)
This set consists of those actions which maximize the
difference between the accuracy and liability set membership
functions for some value of . A well- known result from multi-objective
optimization, which is presented in the companion
report [ 14] in this context for completeness and for a specific
case in [ 7], implies that the set . This means that non-dominated
solutions are not only superior to all other solutions
in some sense, they are also mathematically optimizing.
III. MODEL PREDICTIVE SATISFICING
FUZZY LOGIC CONTROLLER SYNTHESIS
It is desirable to employ expert knowledge to determine
how local information translates into global goal achievement.
For example, automobile drivers are ( usually) very effective
at interpreting how local measurements ( such as time head-way
and relative velocity) translate into global consequences
( such as safe but expedient travel) [ 44]. Such a translation
between local measurements and decisions reflecting global
consequences must either implicitly or explicitly address the
inferences diagrammed in Fig. 1. In this section, we develop
methods for implementing such inferences and identify re-strictions
that facilitate computable methods for performing
such inferences. We first briefly review how consequences can
be inferred from observations using receding horizon model
predictions [ 7]. We then discuss how these predictions can be
used to infer valuations of control. Finally, we review how
these valuations can be used to infer justifiable behavior.
A. Consequences: Receding Planning
Horizons and Influence Vectors
Consider a discrete time, time- varying single input nonlinear
plant of the form
( 17)
where represents the system state, is the system input,
and is a disturbance. Given a desired state and
positive definite cost matrices , , and , the conventional
model predictive controller ( see, for example, [ 17] and [ 18])
is obtained by minimizing
( 18)
with respect to the control sequence
subject to the control bounds and dynamics constraint
given by, respectively [ 17], for all and
for all . The first control of the
resulting minimizing sequence is applied and the constrained
minimization is repeated for the next and all subsequent time
steps. Although for some model predictive systems there exist
methods for guaranteeing plant stability ( such as including
a terminal constraint), the cost matrices , , and are
specified by an implicit expert ( the designer) to infer how
predicted states translate into global plant performance.
In [ 7], we developed the notion of an influence vector
defined for the discrete time dynamical system in ( 17) and
then use this influence vector and the standard quadratic cost
function in ( 18) to develop a receding horizon controller. Let
be the substate of that is an explicit
function of , , , , where is the
maximum number of time increments for which
has explicit influence on any component of the state. For a
planning horizon , the influence vectors are the sequences
of substates of the form
By restricting attention to single step receding horizon,
is the only relevant control variable where we can write
. For the plant in ( 17), it can be shown ( see [ 7])
that the influence vector can be written in the form
where , , , and where is the delay
before influences the th element of the vector . A
necessary restriction9 is that each is controllable via ;
that is, each is defined and there exists and such that
for any there exists a sequence of controls , ,
such that . Given these restrictions,
. Using the procedure in [ 7], each is easily
computed subject to the plant in ( 17). As in [ 7], we require
that although all system matrices can be time- varying, the
variations are restricted such that each is constant.
B. Observations, Consequences, and Evaluations
Since modeling is subject to uncertainty , , and in
( 17) may not be precisely known) it is desirable to develop
controllers that work for multiple system models. It is also
desirable to develop controllers that operate effectively in
9An area of current active research is the development of stable satisficing
controllers. A key to this development is the controllability of the plant.
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 325
the presence of external disturbances . This is one of the
objectives of robust control design ( see, for example, [ 41],
[ 42], [ 43]). In this paper, 10 we restrict attention to problems
for which precise measurements of are available, thereby
focusing emphasis on robustness with respect to nonwhite
disturbances as well as with respect to model uncertainty.
Thus, . An estimate and its associated
distribution then gives us information necessary to make
a decision. When disturbances are difficult to predict it is
helpful to consider a set of such predictions in determining
a control. Similarly, when system models are subject to error,
it is helpful to consider a set of such models. Thus, if more than
one disturbance/ system model11 is possible, then the designer
can consider more than one in determining a control .
In the cost function given by ( 18), there are two different
factors: the terminal cost and the “ cost to go”
. Rather than aggregating the terminal cost and “ cost to
go” into a single global performance metric, we can instead
independently identify and compare each factor from a local
perspective [ 7]. Thus, we define the accuracy cost functional
for a single step control horizon as the terminal cost portion
of the receding horizon cost function
( 19)
and a liability cost functional for a single step control hori-zon
as the “ cost- to- go” portion of the receding horizon cost
function
( 20)
The accuracy cost functional is associated with the global goal
of the controller and the liability cost functional is associated
with the proximate design considerations. In [ 7] the matrices
, , and were assumed given by an implicit expert. In this
paper, we specify these matrices using an explicit rule base
obtained from expert knowledge. In words, the plant model
specifies the inference from observations to consequences
and the explicit expert specifies the relation between local
( receding horizon) consequences and global ( plant behavior)
values.
C. Decisions
By normalizing ( 19)–( 20), and may be determined as
( 21)
( 22)
where and are the ( possibly state dependent) normal-izing
constants required to create membership functions. Ob-serve
that since can be directly computed from ( 19)–( 20),
normalization can occur so that all mass is restricted to
10 Following the example of [ 47], we could treat measurement uncertainty
using fuzzified observations ^ x . Thus, the state of nature could be extended to
include , where y = h ( x ( t ) ; t ) + w ( t ) .
11 Although criteria such as Akaike’s Information Theoretic Criterion exist,
it may be difficult for a designer to justify a model class as being the best
class for a problem [ 49].
the appropriate region of support . Furthermore, observe
that this normalization yields set membership functions that
situationally depend on the set of available controls where
fuzzy values are situationally inferred.
It is worth noting that the most discriminating control
defined in ( 15) closely resembles the solution obtained by
minimizing for . As observed in [ 7],
depends on a state- dependent evaluation of predicted
consequences on the basis of what can be done ( the control
) given the circumstances ( the observation . By contrast,
minimizing reduces to a nonlinear state feedback
control law that does not explicitly consider the set of available
alternatives. Additionally, as demonstrated in Section IV- B2
for plants with no known feedback law in closed form, a search
to find need not be performed each sample time whereas
a search for the minimizing solution of must be
performed for each sample time.
IV. EXAMPLES
In this section, we develop rule bases and present results for
the rotational translation actuator ( RTAC) [ 50], [ 51] and the
inverted pendulum problems. These problems are appropriate
because they serve as benchmarks for demonstrating “ proof of
concept.” In the examples, not only is the set- based structure
of SSDT used to provide robustness to nonwhite disturbances
and model uncertainty, but also the resulting flexibility in
defuzzifier design is also illustrated.
In the preceding sections, we have taken care to discriminate
between three types of inference: 1) inferring ( partial) conse-quences
from observations; 2) inferring valuations from ( par-tial)
consequences; and 3) inferring decisions from valuations.
These inferences were resolved in Section III by 1) one- step
predictions of consequences obtained using an explicit plant
model; 2) the terminal and cost- to- go performance metrics
defined by an expert and subject to facilitatory restrictions;
and 3) the strongly satisficing decision theory described in
Section II. In this section, we focus on implementing these
inferences in examples, and illustrate some of the implications.
Special mention needs to be made of the process of inferring
valuations from consequences. As implemented, an expert
must not only specify a rule base structure to determine cost
functions, but also specify numerical parameters for these cost
functions. This rule base structure and numerical specification
are required during design- time but not employed during
execution time. We present rule bases for two problems and
assume that an expert exists who can translate the resulting
structured cost- functions into numerical representations. In
practice, such translation was accomplished by iteratively
adjusting these parameters in simulation. Note that, because
the specified cost functions are normalized to produce set-membership
functions and thereby ensure the comparability
of accuracy and liability, the expert need only specify relative
numbers ( ratios) of the usefulness of each rule. This process
is compatible with a “ tune by simulations” approach.
A. The RTAC
The RTAC system is shown in Fig. 2 and represents a
translational oscillator with an eccentric rotational proof mass
326 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
Fig. 2. Translational oscillator with rotational actuator.
connected by a linear spring of stiffness to a fixed wall.
The cart is constrained to have one- dimensional travel. The
rotational proof mass actuator affixed to the cart has mass
and moment of inertia about its center of mass. Its center of
mass is located a distance from the axis about which it rotates.
It is assumed that motion occurs in a horizontal plane so that
there are no gravitational forces to be considered. The control
denotes a torque applied to the proof mass and denotes
a disturbance force on the cart. The equations of motion are
given by
( 23)
( 24)
which can easily be written in the form of ( 17) with
and satisfies the conditions specified in
Section III. The states and denote the translational position
and velocity of the cart, respectively, and and denote
the angular position and velocity of the rotational proof mass,
respectively. In the simulation results, we use Euler integration
with sample time s, and the following parameter
values: kg; kg; m;
kg m ; and ( nominally) N/ m.
Although the discretized sub- states and
are not explicit functions of , the substates ,
, , and are. Thus, we may identify
the components of the state that are influenced ( due to the
relative order of the system) by the current input as
the velocity vector and
position vector yielding
.
1) Specification of Attributes: We now turn attention to de-termining
a structure for , , and , and selecting and
in ( 19)–( 22) to yield desirable results. We normalize the
membership functions such that they have unit area over the
region of support . The corresponding normalizing constants
are given by
For the simulation results presented herein, the robustly most
discriminating defuzzifier defined in Theorem 2 is used.
The structures of the cost matrices and, hence, the values
inferred from the set of consequences are determined by
associating elements of these matrices with particular rules as
follows. This problem is a regulator problem with the objective
of keeping the cart close to the origin. Since regulation
( a global consequence) cannot be precisely determined12 by
considering only a single- step control horizon , the
set of accurate controls is fuzzy and, consequently, associated
with a set- membership function. The accuracy of a control is
dictated by the following rule: those controls for which is
close to the origin ( a consequence similar to regulation) have
high accuracy. Since each element of is equally important
with respect to this purpose, we let .
Liability is determined by a set of rules obtained from
expert- specified design principles and tuned by simulation. The
design principles dictate the following rules for generating the
liability membership function.
If cart position, proof mass angle, or velocities produced
by a control are relatively large the control is liable.
If the cart position and cart velocity have the same sign
the control is liable. This reflects expert understanding
that when the cart is moving away from the origin the
system is not very close to being in equilibrium.
If a control requires a great deal of energy it is liable.
These rules dictate a block diagonal structure of the matrix .
We identify the relationships between elements of dictated
by the rules by placing the appropriate rule ( A, B, C) in
the matrices below. If no relationship exists, then the matrix
element is zero
Given this rule- base structure, numerical values are tuned
in simulation. Such tuning is tantamount to determining the
relative importance of these local rules in relation to the global
performance of the controller. After tuning, becomes
Since rejecting disturbances by damping unfavorable cart
motion is the most important factor for this system, the second
rule gets the most emphasis ( weight ). Note that
is not positive semi- definite; this formulation is permissible
because we treat the quadratic forms as fuzzy inference
engines rather than as cost functions used to produce the
optimal behavior.
Note that , , and are specified by experts at design time
and are used to infer valuations from a partial understanding of
global consequences. Run- time inferences to action are done
using the resulting valuations via satisficing, domination, and
defuzzifying. Also note that normalizing the resulting cost
functions to give set membership functions yields valuations
12 For example, nonminimum phase systems exhibit unintuitive behaviors
for single- step control horizons [ 21].
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 327
( a) ( b)
Fig. 3. Phase planes for the RTAC controllers. ( a) Rotational- proof mass phase angle ( in degrees). ( b) Translational cart position ( in meters).
that situationally depend on the set of available controls. Thus,
values are situationally inferred.
2) Results: In the simulations, we consider three cases.
The first case has known system parameters ( and from
( 17) are known) and is disturbance free ( from ( 17) is
zero) where is known precisely. The second
case has no disturbances but does have uncertain
spring constant ( uncertain, but known) where
. The third case has known system parameters
and are known) but has nonwhite disturbances generated
by an unknown model structure where . The
time histories of the rotational- proof mass angle and the cart
position obtained using a SFLC for each case are shown in
Fig. 3( a) and ( b), respectively. For qualitative comparison,
simulation results using a discretized version of the stabilizing
controller developed by Bupp et al., in [ 50] are also shown. For
quantitative comparison, we define the observed cost function
based on ( 18) for a 20 s simulation duration as
( 25)
For case one, . The time histories of the
rotational- proof mass angle and the cart position in Fig. 3 are
for a satisficing controller with . The satisficing
controller behaves qualitatively ( as shown in the plots) and
quantitatively and similar
to Bupp’s controller. 13
For case two and the uncertainty in
occurs because we suppose that the spring constant is
known only as an element in the set [ 150, 250] N/ m and can
drift over time. The drift of the spring constant is simulated
by a random walk with reflecting boundaries at 150 and 250
given by where is distributed
13 Since Bupp’s controller does not explicitly consider the cost function
( 25), ~ J c a s e 1
t = 2 0 < ~ J B u p p
t = 2 0 should not be interpreted as implying inferiority
of Bupp’s controller. The result should instead be interpreted that Bupp’s
controller exhibits similar performance as the satisficing controller.
uniformly over . For any there exists a
corresponding , such that all belief is
placed on where and .
Simulation results for are shown in Fig. 3 and
indicate that, despite the drifting spring constant, the controller
regulates the cart about the origin with .
Notice the small angular offset of approximately 0.025 radians
which decreases approximately 0.001 radians every second.
This terminal offset contributes significantly to the resulting
cost and can be decreased by increasing the - weighting for
, but only in exchange for less damping.
For case three, . The disturbance is a
nonwhite random sequence generated by an autoregressive
moving- average ( ARMA) model given by
( 26)
where we have adopted the notation of [ 49] in defining
the transfer operators and and where is an
independently identically distributed zero mean Gaussian noise
sequence with variance equal to two. In practice,
and . Using system identification
methods, we identify from which is
predicted from past observations of the disturbance. We restrict
attention to a set of model structures which uniquely predict
the next value of . Associated with each model structure
is the best model , which yields the prediction14
of the disturbance . Thus,
and the corresponding set of belief functions
is the set .
Continuing case three, models of and were
identified using MATLAB’s System Identification Toolbox
from four model structures by applying a pseudorandom input
signal to the discretized equations ( 23)–( 24)
for a Gaussian white noise sequence and estimating
by: 1) predicting using known ; 2) determining
; and 3) finding from
14 ^ v ( t j t ?? 1 ) denotes an estimate of the disturbance at time t given past
observations
328 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
TABLE I
IDENTIFIED DISTURBANCE MODELS
. Four model structures were chosen: a first- order
( ARMA ), a second- order ( ARMA , a second- order auto-regressive
( AR), and a second- order state space ( SS) model
identified using the prediction error method [ 49]. Two models
were identified, one from each of two noise/ input realizations,
for each of these four model structures. This yields a total
of eight identified models where . Table I
presents the model structures and the corresponding parameter
estimates.
Simulation results using the robust strongly satisficing set
are shown for in Fig. 3 and yield .
Observe that the cart is effectively regulated about the origin
despite disturbances. Also observe that to maintain similar
performance, . This decrease in is a
result of the fact that as grows to include differing models,
must decrease for to be nonempty. For comparison
purposes, it is insightful to omit the two frist order model
classes since they have the same model structure as the actual
disturbance. Doing so yields indicating
that it is not necessary to have the actual disturbance model
structure to produce acceptable performance.
B. The Inverted Pendulum: Robustness and Defuzzification
The inverted pendulum problem can be stated as follows.
Control an inverted pendulum in a vertical plane with full
circular freedom by applying a lateral force to the cart to which
the pendulum is attached, while regulating
the position of the cart to any desired point. This problem
has become a benchmark for nonlinear control design with
examples from conventional control, fuzzy logic control, and
other nonclassical control methods. For example, conventional
controllers linearize the dynamics model of the pendulum in
a small region within say 10 of the vertical. FLC controllers
include a FLC trained by a genetic algorithm, which has
been shown to balance the pendulum 90% of the time if the
pendulum is given a random initial position within 80 of
the vertical and a random initial velocity less than 80 / s [ 52].
An example of a particularly inventive nonclassical controller
uses deterministic rules, fuzzy logic, and model- dependent
information to control the pendulum with 360 of motion on
a constrained track [ 53].
A model- predictive satisficing controller for this problem
has previously been presented [ 7], [ 8], which balances the
pendulum and brings the cart to the origin for any initial cart
position, cart velocity, pendulum angle, and pendulum angular
Fig. 4. Inverted pendulum on a cart. The pendulum has full rotational motion
and must be regulated about the vertical while simultaneously keeping the
cart near the origin.
velocity less than 180 per sample time . The approach
employs predictions obtained from a nonlinear model of the
plant dynamics, uses a single time step receding control
horizon, and determines accuracy and liability using quadratic
cost functions. A criticism of this design is the omission of
either a stability argument or an appeal to expert rules. In
this paper, we extend the previous results by: 1) employing
expert information to generate the appropriate cost functions;
2) designing a controller with explicit attention to robustness;
and 3) illustrating the flexibility of reduced- search defuzzifier
design.
The inverted pendulum apparatus is illustrated in Fig. 4,
where is the mass of the cart, is the length of the
pendulum, is the mass of the pendulum, is the angle
from vertical ( measured counterclockwise), is the horizontal
position of the cart, and , the control input, is a lateral force
applied to the cart. The continuous- time dynamical equations
for this problem are
( 27)
( 28)
which can be easily written in the form of ( 17) with state
variable and which can be
shown to satisfy the conditions in Section III. In the simulation
results, we use Euler integration with a sample time of
s and the following parameter values: kg,
m, , and ( nominally)
kg. For this problem, we consider the disturbance free case
with uncertain pendulum mass where we use the simplified
notation .
It is clear that, although the substates and
are not explicit functions of , the substates ,
, , and are. Thus, we may identify the
components of the state that are influenced by the current input
as the velocity vector
and position vector yielding
.
1) Specification of Attributes: We now turn attention to
choosing , , , , and in ( 19)–( 22) to yield desirable
results. We normalize the membership functions so that the
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 329
maximum membership is unity and the minimum membership
is zero ( over the region of support ). The corresponding
normalizing constants are given by
The values in the cost matrices are determined by associating
elements of these matrices with particular rules. For the first
simulation result, the most discriminating defuzzifier defined
in Theorem 2 is used. The flexibility of defuzzifier design is
demonstrated in the second simulation result.
The inverted pendulum can be thought of as a regulator
problem since the goal of the system is to regulate the
system about a desired operating point. Since regulation is
the purpose of the controller and since regulation cannot
be precisely determined by considering only a single- step
control horizon , the set of accurate controls is fuzzy
and, consequently, associated with a set- membership function.
The accuracy of a control is dictated by the following rule:
those controls for which is close to the origin ( regulating
point) have high accuracy. Since each element of is equally
important with respect to this purpose, we let .
Liability is determined by an expert who not only specifies
the structure of the cost matrix according the observed design
principles but also tunes the numerical values in simulation.
The quadratic cost function can be used because no more
than second order relationships between elements of are
required15 ( as evident from the following rule base). The
following rules specify the design principles:
If cart position, pendulum angle, or velocities produced
by a control are relatively large the control is liable.
If cart position and pendulum angle produced by the
control have the same sign the control is liable. This
reflects the expert understanding that when the pendulum
is pointing to the left/ right and the cart is also to the
left/ right the system is not very close to being balanced.
If cart velocity and pendulum velocity produced by the
control have the same sign the control is liable. This
reflects the expert understanding that when the pendulum
is swinging the same direction that the cart is traveling,
the system is either getting farther from being balanced
or needs to avoid overshoot.
If a control requires a lot of energy then it is liable.
These rules dictate a block diagonal structure of the matrix
. We identify the relationships between elements of
dictated by the rules by placing the appropriate rule ( 1, 2,
3, 4) in the matrices below. If no relationship exists, then the
matrix element is zero.
15 For more complicated relationships, a more sophisticated inference engine
is required.
Obviously, the relative weight of these rules will influence
the system behavior. For example, if minimizing velocities is
more important than minimizing positions, then the pendulum
may never generate enough energy to move above the horizon.
Such issues are resolved by tuning the weights to the nominal
cart/ pendulum system yielding [ 7]
For the cart/ pendulum system used in the simulation, the
values indicate that making the pendulum angle small
( weight 30) is more important than keeping the cart near the
origin ( weight 0.3). They also indicate that keeping the cart
position and pendulum angle opposite signs ( weight 1) is more
important than all other considerations except bringing the
pendulum to vertical.
2) Results: In the simulations we consider two cases. The
first case has no disturbances but does have an uncertain
pendulum mass from ( 17) is uncertain) where .
This case forms an illustration of the application of robust
satisficing receding horizon control. A second case is also
presented when the plant has certain system parameters and no
disturbances, where with known and fixed. This case
illustrates two important aspects of our methodology: 1) that
the defuzzifier design is flexible and 2) that reduced search
alternatives are possible.
For case one and the uncertainty in the system
model occurs because we suppose that the mass at time
is given by where is
distributed uniformly over . Suppose further that
this random walk is restricted so that for
all time. In the absence of other information, it is desirable
to design a control such that for all16
the plant performance is justifiable. For any
there exists a corresponding such
that all belief is placed on where and
.
Applying a robust satisficing fuzzy logic controller to this
problem with yields the rotational and translational
phase- plane performance illustrated in Fig. 5( a) and ( b). The
“ ” symbol represents the initial conditions ( the cart at the
origin with the pendulum in the vertical down position) and the
“ ” symbol represents the terminal conditions ( the cart at the
origin with the pendulum balanced in the vertical up position).
The controller balances the pendulum while regulating cart
position by swinging the pendulum back and forth while
the cart oscillates around the origin. As the cart oscillates,
the pendulum gathers momentum. In the translational and
rotational phase planes, this motion is manifest as growing
spirals. When the amplitude increases sufficiently, the oscilla-tion
ceases and the pendulum then converges to the vertical
upright position. The cart then returns slowly to the origin. For
16The results presented in [ 7] for a randomized mass do not explicitly
account for the unknown value of m ( t ) , but instead employ the known ( albeit
randomly time- varying) value of m ( t ) via to produce acceptable performance
via local model predictions.
330 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999
( a) ( b)
Fig. 5. Phase planes for the inverted pendulum for uncertain system. ( a) Rotational pendulum angle ( in radians per second and radians). ( b) Translational
cart position ( in meters per second and meters).
these simulations, the most discriminating defuzzifier defined
in Theorem 2 is used.
Observe that the perturbations of the mass are
as large as 10%. We can interpret this in two ways. First
and most simply, this means that a large uncertainty is in-cluded
in the discrete time system and that as designed, the
controller is robust with respect to this uncertainty. Second,
the large uncertainty in can be interpreted as evidence
that the discrete time control law would be appropriate ( i. e.,
the controller handles significant errors between model- based
predicted performance and simulated plant performance) for
the continuous time system ( the physical system was not
available for testing).
For case two, is certain and a flexible reduced- search
defuzzifier design is explored. 17 When no closed- form solu-tion
can be found, applying maximizing defuzzifiers requires
a search. Flexible reduced- search defuzzifying is important
since, for example, applying a receding control horizon to
many nonlinear control problems requires a planning horizon
greater than one step ( for example, to guarantee stability)
and the resulting control law may not have a closed form. A
search must therefore by applied and such search should not
be computationally prohibitive. For the satisficing approach, a
defuzzifying algorithm can be constructed that reduces search
by “ staying the course” with the current control when possible
and otherwise selecting any satisficing solution. The algorithm
for such search is as follows. Let represent the current
control. If , then no search needs to be done because
is a satisficing solution. Otherwise, a search
needs to be done because . Using the randomized
defuzzifier, this search randomly selects possible solutions
until one is obtained that is satisficing. As soon as a satisficing
solution is found, this solution is implemented. Because any
element in the strongly satisficing set can be justifiably applied
as a control, any defuzzifier that selects from this set can be
17 For case one, the most discriminating defuzzifier was applied, but this is
not the only choice. For example, applying a centroid defuzzifier produces a
result nearly identical to the maximum defuzzifier.
used. To emphasize this point in case two, a simulation was
performed using a defuzzifier which randomly ( with uniform
probability) selected a control from the strongly satisficing set
for known and fixed. The rotational and translational phase
planes for one such simulation with are presented
in Fig. 6( a) and ( b). Observe that the behavior produced by
the random defuzzifier is qualitatively very similar to that
produced by the most discriminating defuzzifier.
V. CONCLUSIONS
Conventional optimal control employs an explicit system
model and assumes an implicit expert who defines a cost
function and solves the resulting optimization problem. Con-ventional
fuzzy control assumes an implicit model and em-ploys
an explicit expert to define solvable performance criteria.
In this paper, we employ an explicit model to predict local
plant behavior and an explicit expert to transform these local
model predictions into global evaluations of gains and losses.
This formulation implements these characteristics in a three
step process: 1) explicit models are used to infer local con-sequences
from observations; for problems where complexity
and uncertainty dictate that long range computable and precise
plant descriptions are infeasible, local predictions may be
available and applicable to generating a useful solution; 2)
explicit expert rules are used to infer valuations from local
consequences; such inferences are applicable to problems
where mathematically stable control laws are not trivially
constructed, but where local information can be related to
global performance; and 3) finally, satisficing decision theory
infers control actions from valuations by combining indepen-dent
assessments of goal achievement gains with proximate
performance losses.
Since the relation between local plant predictions and global
system performance is nonstatistically uncertain, fuzzy de-scriptions
are required. Given these fuzzy descriptions, it
seems unreasonable to suppose that a single unique cost
function unambiguously specifies a best global performance.
Instead the error- avoiding principle associated with compara-
GOODRICH et al.: MODEL PREDICTIVE SATISFICING FUZZY LOGIC CONTROL 331
( a) ( b)
Fig. 6. Phase planes for the inverted pendulum with constant mass and reduced search/ random defuzzifier. ( a) Rotational pendulum angle ( in radians per
second and radians). ( b) Translational cart position ( in meters per second and meters).
tive rationality can be used to produce the set of satisficing
system controls. This application of the satisficing princi-ple
determines when a control solution is justified given
the observed evidence. When combined with the domination
principle which eliminates control solutions when superior
solutions exist, the resulting set- valued formulation facilitates
both flexible defuzzifier design as well as robustness with
respect to uncertain models and nonwhite disturbances.
In this paper, we have presented a synthesis procedure for
designing fuzzy logic controllers. This procedure is derived
from the SSDT- based model predictive approach, but which
has been extended to: 1) appropriately employ the power
of expert knowledge and 2) account for robust performance
in the presence of uncertainty. We have demonstrated the
synthesis of satisficing fuzzy logic controllers for two discrete
time nonlinear problems. The synthesis procedure is based on
identifying rules and cost functions, which simultaneously use
objective ( albeit possibly uncertain) local one- step predictions
derived from mathematical models and subjective ( expert) in-terpretation
of the global system consequences. These sources
of knowledge are used to identify the accuracy and liability of
a possible control and, though the provable stability of such
systems remains an open research question, expert knowledge
was used to generate feasible controllers. Using the RTAC,
we demonstrated robustness with respect to nonwhite distur-bances.
Using the inverted pendulum, we demonstrated that
successful control of the model- based problem is flexible with
respect to the defuzzification procedure. Additionally, using
each of the problems, we demonstrated robustness properties
with respect to model uncertainty.
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Michael A. Goodrich received the B. S. ( cum
laude), M. S., and Ph. D. degrees in electrical
and computer engineering from Brigham Young
University, Provo, UT, in 1992, 1995, and 1996,
respectively.
From 1996 to 1998, he was a Research Associate
with Nissan Cambridge Basic Research, Nissan
Research and Development, Inc., Cambridge, MA,
where he maintains status as a Visiting Scientist.
Since 1998 he has been with the Computer Science
Department at Brigham Young University where
he is an Assistant Professor. His research interests include modeling and
controlling intelligent systems, decision theory, multiple- agent coordination,
human- centered engineering, fuzzy logic, and estimation theory.
Wynn C. Stirling received the B. A. ( honors magna
cum laude, mathematics) and M. S. ( electrical engi-neering)
degrees from the University of Utah, Salt
Lake City, Utah, and the Ph. D degree ( electrical en-gineering)
from Stanford University, Stanford CA,
in 1969, 1971, and 1983, respectively.
From 1972 to 1975, he was with Rockwell Inter-national
Corporation, Anaheim, CA, and from 1975
to 1984 he was employed by ESL, Inc., Sunnyvale,
CA. Since 1984 he has been with Brigham Young
University, Provo, UT, where he is a Professor in
the Department of Electrical and Computer Engineering. His research interests
include decision theory, control theory, estimation theory, and stochastic
processes.
Dr. Stirling is a member of Phi Beta Kappa.
Richard L. Frost received the B. S. ( physics, magna
cum laude), M. S. E. E. ( electrical engineering), and
Ph. D. ( electrical engineering) degrees, in 1975,
1977, and 1979, respectively, all from the University
of Utah, Salt Lake City.
He was with the Massachusetts Institute of
Technology ( MIT) Lincoln Laboratory, Cambridge,
MA, from 1979 to 1981, on the faculty of the
University of Utah from 1981 to 1984, and with
the Communication Systems Division, Sperry ( now
L3 Communications), Salt Lake City, UT, from
1984 to 1987. In 1987 he joined the faculty at Brigham Young University.
He is currently an Associate Professor in the Department of Electrical and
Computer Engineering, Brigham Young University. His principal research
interests include quantization and source coding and intelligent control.