Adaptive Control: Fooled by False Assumptions?

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Adaptive Control: Fooled by False Assumptions?
Michael G. Safonov
University of Southern California
Los Angeles, CA 90089-2563
Email: msafonov@usc.edu
Abstract—A mismatch between prior modeling assumptions
and reality can fool an adaptive controller design into per-
sistently preferring a destabilizing controller even which the
instability is patently obvious to the eyes of the most casual
observer. To eliminate all possibility of being thusly fooled, the
assumption-free unfalsified control concept was introduced two
decades ago and has since been developed to the point where it
now provides a unifying overarching theoretical framework for
understanding the relationships, benefits and weakness of various
adaptive control methods. We review how the theory allows one
to parsimoniously sift the individual elements of accumulating
evidence and experimental data to determine precisely which of
elements falsify a given performance level and very briefly discuss
recent research on cost-detectable fading-memory cost-functions
for time-varying plants.
I. INTRODUCTION
“Nothing is what it seems. False assumptions
take hold through constant repetition, blowing away
the more complicated evidence in front of our eyes.
Steve Richards [28], 2012
Assumptions and models have always played an important
role in decision and control theory. Modern optimal control
and Bayesian estimation theory revolve around plant and
process models. The models are characterized by assumed
structures and, in the case of uncertain models, the uncer-
tainty itself is typically characterized by further assumptions
about its behavior and, perhaps, its statistical properties. In
statistical learning theory and as well as in some approaches
to adaptive control theory, the powerful mathematical tools
of Bayesian probability theory and optimal control theory
have been brought to bear to produce a priori guarantees
of convergence and stability — provided that the true plant
conforms to the modeling assumptions.
Of course, there is a problem with assumptions: When
assumptions fail to hold, so do conclusions that rest on
those assumptions. If the only theoretical tools one has for
evaluating the implications of raw data rest on assumptions,
then one may be too easily tempted to ignore the possibility
that assumptions may be wrong. When this happens we can
sometimes be fooled into accepting a design or even an entire
design methodology that is fatally flawed from the outset —
like the adaptive controller which is reputed to have caused
the fatal 1967 crash [10] of the experimental X-15 aircraft
(Fig. 1) or the more generic failures of certain ‘standard
assumptions’ of adaptive control theory illustrated by the
Rohr’s counterexamples [29].
Fig. 1. An X-15 aircraft with an adaptive controller crashed in 1967.
Despite the ever present danger of being fooled by wrong
assumptions, there is a consensus that assumptions are un-
avoidable. As the late MIT professor Fred Schweppe (cf. [34])
was fond of telling his estimation theory students, “You have
to assume something to know something.” This premise, which
it turns out is actually false, seems nevertheless to be tacitly
accepted by many researchers in learning theory, estimation
and control. Assumptions can certainly be productive to form
hypotheses about what classes of learning algorithms might
work, but it is quite another matter to design a learning system
that fails to respond intelligently when future experimental
data is inconsistent with prior assumptions. Yet, this seems to
be an unfortunate characteristic of most (but not all) textbook
algorithms for learning, estimation and adaptive control.
II. ASSUMPTION-F RE E LEARNING
“It ain’t so much the things we don’t know that
get us into trouble. It’s the things we know that just
ain’t so.”
Josh Billings [20], c. 1870
Engineers continue to be confronted with situations where
assumptions required by control and estimation theory either
do not hold or are not possible to verify without invoking
further assumptions. Yet, it remains the general consensus
that some assumptions will always be necessary. Everyone
‘knows’ that modern control theory is built on models and
modeling assumptions, so it seems impossible to imagine
that this could change. After all, the oft-quoted maxim ‘all
Sixteenth Yale Workshop on Adaptive and Learning Systems, New Haven, CT, June 5-7, 2013, pp 16-21.

observation is theory laden’ seems to deny even the existence
of uninterpreted raw data.1
In fact, it turns out to be fairly simple to formulate and solve
estimation and control problems without prior assumptions via
a fairly straightforward application of the scientific ‘curve-
fitting’ methods to raw uninterpreted data, whenever such
data is available — e.g., Gauss-Newton curve-fitting methods
(cf. Bertsekas [6, Sect. 1.5]) which select a parameter vector
xof function h(x, y)to approximately fit input-output data
pairs (yi, zi)i= 1, . . . , n by computing the optimal value x∗
minimizing a cost function of the form
f(x)∆
=
n
X
i=1
k(zi−h(x, yi)k2.(1)
The computation of the optimal parameter vector x∗us-
ing such curve-fitting methods requires no assumption about
whether the ‘true’ system is or is not exactly modeled by
some value of the parameter vector x, and (unlike Bayesian
estimation) it requires no noise models and no assumed
prior probabilities. The optimal value of the cost f(x∗)is
the accuracy of the model fit in the sense that the model
z=g(x∗, y)is unfalsified by the data at accuracy level f(x∗).
Here we may find fertile ground for a shift towards a data-
driven paradigm of estimation and control. Such a shift might
occur as engineers are increasingly confronted with problems
in which either priors do not hold or there is a need to adapt
in real-time, as when systems change in unforeseen ways due
to wear, damage, or evolving operating conditions. To do this
reliably, we must have methods that respond correctly when
new data falsifies our prior assumptions; that is, we need
scientific methods.
A. Popper’s Logic of Science
“There can be no ultimate statements in science:
there can be no statements in science which cannot
be tested, and therefore none which cannot in princi-
ple be refuted, by falsifying some of the conclusions
which can be deduced from them.”
Karl Popper [26], 1934
What is science, and how is it distinguished from pseudo-
science and quackery? Philosopher Karl Popper [26] studied
these questions in depth, examining in detail the processes
and practices employed by the best scientists like Newton and
Einstein. In the end, Popper concluded that an empirical theory
(i.e., a theory concerning the material world, as contrasted with
a mathematical theorem) may be properly called a scientific
theory only if it passes three tests, which we may loosely
summarize as follows:
1) Falsifiability. Data from some conceivable future ex-
perimental outcome must be in principle capable of
falsifying (i.e., invalidating) the theory.
1The maxim ‘all observation is theory laden’ can be traced to philosopher
Hanson [13], who was referring to situations similar to human telescopic
observation where the distant thing said to be ‘observed’ is in fact a human re-
interpretation of the unprocessed raw data. Hanson did not deny the existence
or potential accessibility of uninterpreted raw data.
2) Simplicity. The theory must be parsimonious, which is
to say it must not assert extra conditions beyond those
that are necessary to explain observable data.
3) Validation. The theory must be thoroughly tested by
surviving aggressive experimental efforts designed to
seek falsifying empirical evidence (i.e., data).
The falsifiability test was Popper’s hallmark contribution
to the definition of science. It addresses the empirical fact
that scientific knowledge does not always advance. It explains
that scientific theories are always tentative, remaining forever
subject to possible falsification by new information. The sim-
plicity test is basically the classical Occam’s razor principle,
but it also implies Newton’s [23] famous hypotheses non fingo
principle (‘make no feigned hypotheses’) prohibiting experi-
mentally unverifiable prior assumptions about the ‘true’ inter-
nal workings and/or motivations of the process that generates
the observed data.2The validation test is what distinguishes a
scientific theory from a mere scientific hypothesis.
B. The Role of Probability Theory
“It’s turtles all the way down!”
Stephen Hawking [14], 1988
As noted by Popper [26], Finetti [8], Kalman [18], [19] and
Taleb [40], [41], Bayesian probability theory is on inherently
shaky ground from a scientific perspective. There is an ‘infinite
regress’ problem associated with the application of Bayes’
rule,
p(X|D) = P(D|X)P(X)
P(D)=P(D, X )
P(D).
To estimate the posterior probability p(X|D)of Xgiven
data Dusing Bayes’ rule, we evidently must first have an
estimate of the prior probability P(X, D). So, any attempt
to estimate a probability rests on having an estimate of a
prior probability, thus leading to an infinite regress — like
the infinite stack of turtles of a primitive cosmology (cf.
Hawking [14]). Unless we terminate the regress at some
level either by simply assuming a prior probability or by
actively intervening to ‘shuffle the deck’ as is done in RCT
experiments,3the probability P(X|D)cannot be computed.
Because of this infinite regress problem, probability statements
are generally experimentally unfalsifiable, and therefore by
Popper’s criterion Bayesian estimation theory may not be a
scientific theory. Unlike RCT methods, Bayesian estimation
2An interesting albeit, to some, possibly disturbing consequence of this is
that any theory for system identification or adaptive control that begins with
prior assumptions about the structure or form of either the ‘true plant’ (LTI,
parameters, order, etc.) or the ‘true noise’ (Gaussian, unknown-but-bounded,
etc.) may not be a scientific theory by Popper’s definition.
3In the randomized controlled trial (RCT) experimental method of
Fisher [11], the experimenter intervenes by randomly assigning members of
the sample set to test and control groups prior to performing an experiment
in order to create a uniform prior probability distribution of potentially
confounding factors, even when these factors are unknown. RCT should
not be confused with the assumption-based Monte Carlo “randomization”
methods like those described in Tempo et al. [42], in which one assumes a
uniform prior probability on an assumed sample space. According to Pearl [25,
pp. 410–418], RCT is the only known statistical method for reliably testing
causal hypotheses.

and system identification problem formulations usually assume
both a ‘true system’ satisfying prior assumptions about its
form and internal structure, and noises with assumed prior
probabilities.
The problem of infinite regress is not isolated to probabilis-
tic problem formulations. Infinite regress issues can also arise
with unknown-but-bounded estimation methods favored by
robust control theorists that begin with prior assumption that
the ‘true plant’ lies in a predetermined set with a given a priori
known internal structure and, perhaps, noises or other uncer-
tain elements that satisfy a priori uncertainty bounds (e.g.,
[24], [12]). Any prior assumptions about the internal structure
of the ‘true plant’ violate the hypothesis non fingo principle
of Isaac Newton [23]. To correct this, we need data-driven
black-box methods; i.e., methods in which only observed data
is regarded as known, without any prior assumptions about the
properties or internal workings of the so-called ‘true system’
or measurement noises.
III. THE U NFALSIFICATI ON PAR AD IG M
Heavier-than-air flying machines are impossible.
Lord Kelvin [43], President, British Royal Society, 1895
Paradigm shifts are controversial, even risky. To completely
eliminate prior assumptions in estimation and control is to
develop a assumption-free methodology, and perhaps thereby
even to risk being tarred with the label of simpleton or even
charlatan. Let us tentatively and cautiously take this risk and
see where it leads.
What we are seeking is a data-driven theory, i.e., a theory
in which no prior assumptions corrupt our interpretation of
results. One implication of this is that the performance cri-
terion by which we judge a candidate controller or estimator
must be expressed directly in terms of the raw noise-corrupted
past input-output data collected from the outputs of sensors
attached to the real plant. And, since with a finite amount of
past plant input-output data we cannot know for sure without
further assumptions how plant would respond to other as yet
untried input signals, or even that it would respond exactly the
same way in the future, the best we can hope to conclude from
raw data analysis is that the particular past observed behavior
is, or is not consistent with our performance goals. In other
words, the best data-driven theory of control and estimation
can hope to conclude that a given candidate controller or
estimator is as yet unfalsified by the past data.
So to have proper scientific data-driven theory, we must
limit our aims to judging performance goals expressed as
functions of raw data. And, after we examine any candidate
estimator or controller, we must be careful to limit the claims
to unfalsification of performance, and resign ourselves to the
fact that traditional a priori guarantees of asymptotic stability
are unattainable without gratuitously introducing additional
assumptions to constrain future data in ways that nature may
not. That is, the strongest guarantee that a scientific data-
driven theory of estimation or control can logically offer is
that given controller’s or estimator’s performance is as yet
unfalsified by past data. But, this may be enough when the
data is information rich (e.g., persistently exciting), so that
bad controllers and estimators will be quickly and efficiently
falsified and discarded.
It turns out that all this is not hard, and has in fact been
done without much fanfare for a very long time.
A. Unfalsified estimation theory
“Probability does not exist.”
Bruno de Finetti [8], 1974
For the case of data-driven estimator design, a simple exam-
ple of a data driven ‘curve-fitting’ solution is provided by the
classical weighted least-squares algorithms that select model
parameters by minimizing a cost function similar to (1)(e.g.,
[34], [6]). Without unneeded the presumption of a ‘true’ plant
or noise model, weighted least-squares estimators solve the
problem of computing optimal model parameter estimates such
that the input-output behavior of the models minimizes an
arbitrary quadratic function of the difference between the raw
experimental input-output data and the model output [34].
For some problems, the solution even takes the exact form
of a Kalman filter. But of course weighted-least-squared is
not Kalman filtering theory, as it does not require access to
the ‘true plant’ or to probabilistic noise models, nor does it
claim to be able to a priori predict future error-covariance.
More significantly unlike the Kalman filtering theory, the data-
driven least square curve fitting solutions parsimoniously make
no use of prior assumptions and, consequently, make none of
the purely assumption-driven claims of the Kalman filtering
theory to have precisely pre-computable Gaussian statistics.
Such unfalsifiable claims of access to certain knowledge
about future statistics goes beyond limits of what is logically
knowable from data alone, violating Popper’s simplicity re-
quirement for a scientific theory as well as hypotheses non
fingo principle of Newton [23]. In fact, this seems to be one
of the reasons why Kalman [18], [19] now questions both
Bayesian probability and the probabilistic interpretation of his
Kalman filter equations.
Willems [48] and Smith [37] provide further examples of
assumption-free curve-fitting-type formulations of estimation
problems. Explicitly citing Popper [27] as motivation, Willems
defined an unfalsified model as one whose graph contains all
currently available plant data, and he introduced the term most
powerful unfalsified model (MPUM) to describe an unfalsified
model that offers the best (in a particular sense) fit to the
data. Smith proposed optimally curve-fitting past plant input-
output data to minimize a robust-control-oriented ‘LFT’ error
criterion that admits a curve-fitting cost function with terms
penalizing multiplicative errors among others.
Essentially, these data-driven methods involve nothing more
than curve-fitting plant and controller models to raw data so
as to optimize ‘unfalsified’ performance levels by minimizing
a curve-fit-error performance criterion. Because the process of
curve-fitting raw data to models requires no prior assumptions
about plant or noise, such methods are wholly incapable of
being confounded by prior assumptions and are therefore
inherently robust.

d
Disturbance
Noise
Adjustment
Mechanism =(u,y)=Data
K
^
Controlle
r
yu
ξ
θ
r
Reference
A
ctuato
r
Process Senso
r
Disturbance
Noise
Adaptive Controller K
K
^
Signal Output
Plant P
Control
Signal
AdjustmentMechanism:
=arg minV(K,y,u,t)+(1‐
δ
(K,K(t‐))h
K(t)
^
K
where
δ
(
∙
,
∙
)denotes the Kronecker delta function,
^
where
δ
( ,

)

denotes

the

Kronecker

delta

function,
h>0iscalleda‘hysteresisconstant’
Fig. 2. In adaptive control it is desirable that the cost-function V(K, y, u, t)
be cost-detectable; otherwise, the adjustment mechanism can be destabiliz-
ing [47], [38].
B. Unfalsified Control Theory
“Just have lots of ideas and throw away the bad
ones.” Linus Pauling [1], 1935
It is almost as easy to formulate the problem of directly
finding closed-loop feedback controller parameters as a data-
driven curve-fitting problem formulation as it is to do so for
open-loop estimator design problems [32], [30]. The problem
is essentially a controller identification problem, which may
be reliably solved by choosing controllers that minimize any
suitable cost-function expressing ones performance goals as a
causal function of raw plant input-output data, uninterpreted
by plant/noise models or other prior beliefs. Further, a most
basic requirement is that this cost function should be cost-
detectable [47], [38], which is to say the cost must remain
finite if, only if, the data does not falsify a controllers ability
to stabilize the plant.
As first proved by Wang-Paul-Stefanovich-Safonov [46],
[47, Thm. 2], by simply substituting a cost-detectable
cost function together into a now classic Morse-Mayne-
Goodwin [22] adaptive hysteresis switching algorithm, one can
prevent model-mismatch instability.
Theorem 1 (Safe Adaptive Control —[46], [47], [38]):
Consider the adaptive feedback control system in Figure 2.
Suppose that the adaptive control problem is feasible in the
sense that there exists at least one candidate controller K∈K
that stabilizes the plant. If the cost-function V(K, y, u, t)is
monotone in t, the controllers K∈Kare minimum-phase
and the(V, K )is cost-detectable, then the adaptive system
converges to a stabilizing K∈Kafter at most finitely many
controller switches. 2
The safe adaptive control theorem says that merely substi-
tuting a cost-detectable cost function into the classic Morse-
Mayne-Goodwin hysteresis switching algorithm is sufficient
to prevent model-mismatch instabilities, thereby ensuring that
no mismatch between prior modeling assumptions and reality
can fool the adaptive control logic into persistently preferring
candidate
controlle
r
hypotheses goals
data
COMPUTER SIEVE
V
(
K,y,u,
t
)
+ h
≤
V(K,y,u,t)
^
FALSIFIED
Unfalsified K
FALSIFIED
Fig. 3. An unfalsification process used in unfalsified control design is
essentially a data-driven sieve. The sieve requires three types of input: (1)
goals, (2) candidate controllers and (3) data. Controllers are sifted to find those
that are consistent with both performance goals and physical data. No plant
models are required while the process is running, though a plant model can
be useful for prior selections of the candidate controllers and the performance
goal.
a destabilizing candidate controller over a stabilizing one. The
algorithm is essentially a data-driven sieve that rejects/falsifies
controllers K∈Kwhose cost is not at least h > 0lower than
the cost of the currently active controller ˆ
K(t), as depicted in
Figure 3. Using a cost-detectable cost function V(K, y, u, t)in
the unfalsification sieve solves the problem posed by the Rohrs
counterexamples [29] and perhaps even could help to prevent
problems like the fatal 1967 failure of the adaptively controlled
X-15 aircraft [10]. For the unity negative feedback adaptive
control system in Figure 2 with minimum-phase candidate
controllers K(s)∈K, an example of a cost-detectable cost
function is [32], [33], [47], [38]
V(K, y, u, t) = sup
τ≤t
kW1eKk2
τ+kW2uk2
τ
krKk2
τ+(2)
where  > 0is a small constant, kxkτdenotes the L2-norm
of the signal xtruncated at time τ, and eKand rKare K(s)-
dependent ‘fictitious’ signals, defined as the signals eand r
that would have generated the data (y, u)had the controller K
been in the loop when the data was collected. For minimum-
phase K(s), they are computed via the formula eK
∆
=K−1u,
rK
∆
=eK+y.4
The above cost function V(K, y, u, t)is an unfalsified
lower-bound on the standard weighted-sensitivity cost function
of robust control [31], [36], viz.



W1(I+P K)−1
W2K(I+P K)−1


∞
.
The theory of unfalsified control, including its application to
the design of inherently robust assumption-free adaptive con-
trol systems is now developing rapidly, including several recent
4If K−1(s)is unstable, then we may substitute eK=DKuand rK=
NKy+eKwhere K(s) = D−1(s)N(s)is any stable left-coprime matrix
fraction description of K(s)[21].

application studies, e.g., [39], [44], [35]. Significant recent
progress includes alternative cost functions with windowed
and fading memory suitable for adaptive control of plants
that may vary so greatly with time that the controller must
be switched to maintain stability and performance [3], [4],
[17], [16], [15], [5]. Interestingly, there has also been recent
success in developing cost-detectable cost functions that reflect
plant-model-based goals [2].
IV. DISCUSSION
“It is a capital mistake to theorize before one
has data. Insensibly one begins to twist facts to suit
theories, instead of theories to suit facts.”
A. C. Doyle [9], 1891
“It may be so, there is no arguing against facts
and experiments.” Issaac Newton [7], 1726
Audiences of mathematical system theorists at past talks that
I have given on unfalsified control usually express discomfort,
and occasionally even outrage, at the thought that plant models
and noise assumptions do not seem to play a direct role
in the unfalsified control theory. A common refrain is that
‘science has always relied on models and assumptions and
to throw these away is to throw away valuable information’.
A slight problem with that view of science is that prior
assumptions, mathematical models and even basic physical
laws like f=ma and e=mc2are regarded by scientists
as mere theories, always subject to possible refutation by
future experiments data. Indeed, it is precisely when some
prior assumptions or some models are falsified by new data
that the need for learning and adaptation arises.
The fact is that models can, and usually do, have a pivotal
role in the application of unfalsified control theory just as they
do in most of engineering. In unfalsified control applications,
plant models are normally used in the synthesis of the candi-
date controllers that are subsequently fed to the assumption-
free unfalsification algorithm. In particular, the set of candidate
controllers Kis typically created by applying traditional
model-based and assumption-based control synthesis methods.
Each candidate controller in the set is designed to meet perfor-
mance goals for a member of a set of possible ‘true’ plant and
‘true’ noise models. These model-based candidate controller
designs are then fed to the unfalsified control algorithm, which
separately performs its assumption-free and plant-model-free
checks for falsification of each controller using only raw plant
input-output data.5Each unfalsified controller that emerges
from this process has passed two tests: (1) a purely model-
driven test for robust performance with one or more of the
hypothesized candidate plant and noise models that was used
for the controller’s initial design, and (2) a purely data-driven
test for whether the controller’s performance is falsified by
5Using raw data in formulating the cost-functions used to evaluate per-
formance is essential. It avoids the need to invoke the peculiarly circular
logic of introducing unrealistic Santa Claus assumptions to ‘prove’ stability
of adaptive systems that in fact are not—e.g., as exemplified by the Rohr’s
counterexamples [29].
past input-output data. The sole role of data-driven second
test of unfalsification is to allow adaptive feedback to robustly
correct any performance problems that may arise when some
of the assumed plant or noise models are later falsified by new
data. A key point here is that information in prior models is
not discarded at all, but is instead fully exploited in designing
a set of candidate controllers to pass the model-based first test.
V. CONCLUSION
“The task of science is to stake out the limits of
the knowable and to center the consciousness within
them.” Rudolf Virchow [45], 1849
In the present paper, I have argued that there may be
paradigm shift now in the making as assumption-laden theories
of learning and adaptive control theory face potentially falsi-
fying empirical tests. Of particular concern are assumptions of
prior knowledge of uncertainty structure and bounds, leading
the questions like, “How should controllers and estimators
change when these assumptions are falsified by new data?”
The next revolution in estimation and control could be a
discomforting shift towards assumption-free methods for the
design of learning systems and adaptive control algorithms
based on Popper’s unfalsification paradigm. The unfalsified
control paradigm allows one to clearly and sharply separate
the implications prior assumptions from those of data, which
is a logical necessity if one is to avoid being fooled by prior
assumptions.
ACKNOWLEDGMENT
I wish to express my sincere appreciation to the many
colleagues and students whose contributions have helped shape
the development of the unfalsified control concept that is the
essential element of the plant-assumption-free formulation of
the adaptive control problem presented in this paper. Former
students and postdocs who have made direct contributions to
the theory or its application include Michael W. Chang, Shin-
Young Cheong, Seung-Gyun Cheong, Margareta Stefanovic,
Ayanedu Paul, Crisy Wang, Fabricio Cabral, Tom Brozenec,
Myungsoo Jun, Vincent Fromion, Paul Brugarolas, Claudia
Manuelli, Hardy Siahaan and especially Tom Tsao who helped
lay the initial foundations of the theory. I also thank Robert
Kosut, Brian Anderson, Lennart Ljung, Michel Gevers, Mike
Athans, Charlie Rohrs, Masami Saeki, Petar Kokotovic, Petros
Ioannou, Karl Astrom, Edoardo Mosca, Bob Narendra, Jan
Willems, Jeroen van Helvoort, Martin Steinbuch, Huiyu Jin
whose work and words helped to shape my perspective on the
adaptive control problem and to focus my thoughts on the need
for a plant-assumption-free theory for safe adaptive control.
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