Modeling the Behavior of Hair Follicle Receptors
as Technical Sensors using Adaptive Control
Carsten Behn
Department of Technical Mechanics, Ilmenau University of Technology, Max-Planck-Ring 12, 98693 Ilmenau, Germany
Keywords:
Adaptive Control, Bio-inspired Sensor System, Modeling, Uncertain System.
Abstract:
Based on the paradigm biological receptor and its fundamental feature to filter signals and transduce them, we
set up a mechanical sensor system to find hints to establish a measurement or monitoring system. These tech-
nical systems have to offer high sensitivity to signals from the environment. To mimic the complex behavior
of the biological system, adaptive controllers have to be applied to a mechanical sensor system to compensate
and filter unknown ground excitations (uncertainties of the system). Before doing this we summarize previous
work on controlling such mechanical systems. We expose the need of improvements of already existing strate-
gies from literature, the corresponding problems are formulated. Improved adaptive controllers are presented.
Their working principle is illustrated in various numerical simulations and experiments.
1 INTRODUCTION
In nature there are various senses that allow animals
to perceive their environment. Depending on the dis-
tance of objects to be sensed from the system bound-
aries of the animal (usually the skin), sensors are
distinguished between “far field” (e.g., vision) and
“near field”, of which the sense of vibrations is a spe-
cial case. Here, we focus on the sensing of vibra-
tions for purposes of exteroception(outside the body),
not ignoring the phylogeneticrelation to interoception
mechanisms like proprioception. Vibrations are an
important piece of environmental information that in-
sects rely on, especially arachnids, such as spiders and
scorpions. To perceive vibrations, they have different
types of sensilla (or tactile hairs, (Barth, 2004)). Ver-
tebrates, such as cats, rats and sea lions, also possess
the sense of vibration. They can perceive vibrations
with the help of their vibrissae (whiskers).
Although these biological vibration receptors have
a different physiology – (Iwasaki et al., 1999) or
(Smith, 2008) for classifications, they share common
properties: When in touch with an oscillating ob-
ject, they are moved and stimulate various (pressure-
sensitive) receptors which have to analyze the stimu-
lus and to transduce their gained adequate information
to the central nervous system (CNS).
Mechanoreceptors of this kind are present throughout
the integument of insects (cuticula) and mammals (fur
on skin).
2 HAIR FOLLICLE RECEPTORS
Let us focus on mammalian receptors. The vibrissae
serve mainly as levers for force transmission. If the
hair is deflected due to some excitations, e.g., wind,
this mechanical (oscillation) energy is then transmit-
ted to the various receptors, which respond to any
movement of the hairs, see Fig. 1. A receptor has only
one function: to transduce a (mechanical) stimulus to
neural impulses (Soderquist, 2002). However, a re-
ceptor never continues to respond to a non-changing
stimulus in transducing information to the CNS as
long as the stimulus is present. It rather depends on
the type of stimulus. If some impulses stimulate a re-
ceptor then there is a rapid and brief response of the
receptor to it. This response declines if the stimu-
lus is unchanging. Due to permanently changing en-
vironments the receptors have to be in a permanent
state of adaptation to adjust their behavior. The rate
or time needed to adapt or stop responding to an un-
changing stimulus is the main characteristic to dis-
tinguish two different types of tactile receptors, see
Fig. 2. The classification is, (Soderquist, 2002) and
(Smith, 2008):
•
fast adapting (FA) receptors
encompass hair fol-
licle sense endings: as mentioned previously, FA
receptors react to applied movements or pressures
with a fast (rapid) response of activity, which is
succeeded by a decrease of it even though the
stimulus is still present. This means, that, if a
336
Behn C..
Modeling the Behavior of Hair Follicle Receptors as Technical Sensors using Adaptive Control.
DOI: 10.5220/0004488003360345
In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 336-345
ISBN: 978-989-8565-70-9
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
nerve to CNS
blood
sinus
Merkel cell
Merkel cell
Lancet
nerve ending
Paciniform
corpuscle
cirumferentially
oriented spiny
ending
vibrissal
shaft
Figure 1: Follicle-sinus complex (FSC) of a vibrissa with
various types of receptors (blue); adapted from (Ebara et al.,
2002) and (Rice et al., 1986).
10000
SA - Process
FA - Process
2000200
Time [ms]
Activity
Figure 2: Adaptation processes of FA and SA receptors,
modified from (Soderquist, 2002).
mechanical pressure via an unchanging force is
applied the FA receptor responds quickly with a
steep increase of activity and then decreases this
activity and waits for a further stimulus, it adapts
its activity in order to notice changes in the stim-
ulus;
• the counterpart are
slow adapting (SA) receptors
,
e.g., Merkel cells — these receptors work in a
similar way as FA receptors but offer two ways of
operation: first, as usual to receptors, a rapid re-
sponse is followed by a decrease of activity. But,
there is also a long duration of time of activity
of these receptor cells from the beginning of the
stimulus. This is in contrast to FA receptors.
The described behavior is shown in Fig. 3.
Stimulus 1 Stimulus 2
SA Receptor Behavior
Time
FA Receptor Behavior
on off
Impulses
on
Figure 3: Activity behavior of FA and SA receptors, modi-
fied from (Soderquist, 2002).
Here, we want to focus on the fast adapting re-
ceptors. The sensibility of these cells is continuously
adjusted so that the receptor system converges to the
rest position despite the continued excitation (Dudel
et al., 1996). Hence, the perception of the continu-
ous unchanging excitation is damped. Therefore, the
excitation is considered irrelevant, once it has been
perceived. If however a different excitation, such as
a sudden deviation of the vibrissa sensor, occurs, this
information is relevant and the sensor has to be sensi-
tive to perceive it. If, for example, a cat is exposed to
wind, the recognition of the resulting excitation of the
whiskers will be damped and ignored. If the cat en-
counters an obstacle, the receptors should still be sen-
sitive enough to perceive the sudden deviation of the
whiskers while the wind excitation persists. There-
fore, the adaption process has to ensure enduring sen-
sitivity.
In the following, we set up a simple mechanical
model to map all important features of fast adapting
receptors via adaptive control strategies applied to the
mechanical system.
3 MECHANICAL MODELING
Motivated by the biological observations in the fore-
going section we consider a simple model of a recep-
tor in form of a spring-mass-damper-system within
a rigid frame, which is forced by an unknown time-
dependent displacement a(·). Moreover, the mass
is under the action of an internal control force u(·)
to compensate the unknown ground excitations, see
Fig. 4, where x is the absolute coordinate. The param-
eters of this sensory system are m (the forced seismic
point mass), the damping factor d and the spring stiff-
ness c.
We derivethe differential equation of motion by using
Newton’s second law:
m ¨x(t) = −d
˙x(t) − ˙a(t)
−c
x(t) − a(t)
+ u(t) ,
x(0) = x
0
, ˙x(0) = x
1
.
(1)
ModelingtheBehaviorofHairFollicleReceptorsasTechnicalSensorsusingAdaptiveControl
337
m
d
u(t)
y(t)
x(t)
a(t)
c
-u(t)
Figure 4: Mechanical model of a sensor system (receptor
model) (Behn and Steigenberger, 2010).
With y = x − a as the relative coordinate of the point
mass, we arrive at the following differential equation
of the relative motion with respect to the frame
m ¨y(t) + d ˙y(t) + cy(t) = −m ¨a(t) + u(t),
y(0) = x
0
− a(0), ˙y(0) = x
1
− ˙a(0).
)
(2)
If y(·) is the measured output of the system, (2) is
presented in normalized form
y(t)
˙y(t)
•
=
0 1
−
c
m
−
d
m
y(t)
˙y(t)
+
0
1
m
u(t) +
0
− ¨a(t)
y(0) = x
0
− a(0), ˙y(0) = x
1
− ˙a(0).
(3)
4 SCOPE, PROBLEM AND GOAL
Scope: The goal is to achieve a predefined movement
of the receptor mass m of the sensor system in Fig. 4
such as stabilization of the sensor system or tracking
of a reference trajectory. It is obvious that the sole
possibility of influencing this system lies in the (con-
trol) force u(·). Hence, we have to design and im-
plement a controller which ensures a desired system
output behavior. Therefore, the scope/object is to find
a suitable control strategy that reproduces the special-
ities of the biological system receptor.
This system is similar to seismic sensor systems to de-
tect (unknown)ground excitations due to the principle
of passive oscillation perception.
Problem: In general, one cannot expect to have
complete information about a mechanical or biologi-
cal system, but instead only structural properties are
known. It is important to point out that all system pa-
rameters are supposed to be unknown because of the
sophisticated nature of the biological system. The ex-
ternal excitation a(·) (biological disturbance to the re-
ceptor) is unknown and the mass, spring and damping
factors are uncertain, e.g., vary in time due to thermic
influences. Here, uncertainty of the factors mean that
they have a positive value, but they are not known ex-
actly, only a valid range, e.g., c ∈ [c
, c]. Summarizing,
we have to deal with a highly uncertain (control)
system of known structure. This is why traditional
control methods fail, as they rely on the knowledge of
those parameters. The consideration of uncertain sys-
tems leads us to the use of adaptive control. By the
above mentioned adjustment of the receptor we are
given the task to adaptively compensate the unknown
ground excitation: we have to design an adaptive con-
troller, which learns from the behavior of the system,
so automatically adjusts its parameters in such a way
that the (seismic) point mass tends to the rest position
in spite of the continuing excitation.
Goal: We choose the λ-tracking control objective
(Behn and Zimmermann, 2006) due to the high-gain
property of the sensor system presented in (3). λ-
tracking allows for simple feedback laws and does
not focus on exact tracking since we deal with an un-
certain system. Therefore, the goal is to act on the
system in such a way that the system output y(·) is λ-
tracked, i.e., the system output is forced into an error
neighborhood λ around a set point trajectory y
ref
(·).
If y
ref
(·) ≡ 0, the problem is known as λ-stabilization.
In this case, the receptor is supposed to remain in its
equilibrium state (rest position).
This design of controllers depends tremendously
on the system properties. The adaptive control strate-
gies should meet the following requirements:
• ability to apply the controllers without any knowl-
edge about system parameters;
• simple feedback structure;
• optimal control performance regarding
- short settling time: a desired quality of adap-
tive controllers is finite time behavior. Since all
system parameters are unknown and the level
of the necessary control gain cannot be antici-
pated, it is also unknown at which point in time
the control objective will be achieved. Con-
trollers with finite time behavior enable the user
to specify a time at which the control objective
will be achieved at the latest.
- simple structure of controller equations;
- small level of gain parameters, level of error in-
side the λ-tube;
- ability to quickly adapt to parameter changes. It
is imperative to keep the sensitivity of the sys-
tem high. If, for example, a recurring excitation
signal a(·) acts on the system, it is supposed
that its influence is to be damped by the con-
troller. Once the sensor/receptor has noticed
this excitation it has to fade it out to wait for
ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics
338
further new information. It has to adaptively
adjust its parameters. If however the excita-
tion subsides or is replaced be one with a much
lower amplitude, the system is supposed to re-
main sensitive and quickly adjust the control
parameters.
5 SYSTEM CLASSES
The equations of motion (3) fall into the cate-
gory of quadratic, finite-dimensional, nonlinearly per-
turbed, m-input u(·), m-output y(·) systems (MIMO-
systems) of relative degree two, for short S
2,nonlin1
, of
the form
¨y(t) = A
2
˙y(t) + f
1
s
1
(t),y(t), z(t)
+ Gu(t),
˙z(t) = A
5
z(t)+ A
0
˙y(t) + f
2
s
2
(t),y(t)
,
y(t
0
) = y
0
, ˙y(t
0
) = y
1
, z(t
0
) = z
0
,
(4)
with y(t), y
0
, y
1
, u(t) ∈ R
m
, z(t), z
0
∈ R
n−2m
, A
2
,
G ∈ R
m×m
, A
5
∈ R
(n−2m)×(n−2m)
, A
0
∈ R
(n−2m)×m
,
n ≥ 2m, and, for natural number q
1
and q
2
it holds
(i) spec(G) ⊂ C
+
, i.e., the spectrum of the “high-
frequency gain” lies in the open right-half com-
plex plane;
(ii) s
1
(·) ∈ L
∞
R
≥0
;R
q
1
, s
2
(·) ∈ L
∞
R
≥0
;R
q
2
may
be thought of as (bounded) disturbance terms,
where s
i
(t) = ψ
i
t , y(t), ˙y(t), z(t)
is also pos-
sible with ψ
i
(·, ·, · , ·) ∈ L
∞
R
≥0
× R
m
× R
m
×
R
n−2m
;R
q
i
;
(iii) the functions f
1
: R
q
1
× R
m
× R
n−2m
→ R
m
and
f
2
: R
q
2
× R
m
→ R
n−2m
are continuous functions
and, for compact sets C
1
⊂ R
q
1
and C
2
⊂ R
q
2
,
there exist c
1
, c
2
≥ 0 such that for all (s, y, z) ∈
C
1
× R
m
× R
n−2m
f
1
(s, y,z)
≤ c
1
1+ kyk + kzk
,
and for all (s, y) ∈ C
2
× R
m
f
2
(s, y)
≤ c
2
1+ kyk
.
(iv) spec(A
5
) ⊂ C
−
, i.e., the system is minimum
phase, provided f
1
= 0, f
2
= 0.
It is easy to prove that every system of this system
class has strict relative degree two. Therefore, rela-
tive degree two means that the control u(·) directly in-
fluences the second derivative of each output compo-
nent. The term A
0
˙y appears in connection with under-
actuated systems (Behn and Zimmermann, 2006).
If we inspect system (3) in more detail and take
the physical meaning of the parameters into account,
i.e., the mass of the forced (seismic) point mass m, the
damping factor d and the spring stiffness c represent
positivereal values, then we can simplify system class
(4) to a very special one:
• we restrict to single-input u(·), single-output y(·)
systems (SISO-system m = 1),
• then, we claim A
2
< 0,
• and we neglect the coupling term, A
0
:= 0.
Hence, we arrive at a subclass of finite-dimensional,
nonlinearly perturbed SISO-system with strict rela-
tive degree two, S
2,nonlin2
for short, of the form
¨y(t) = A
2
˙y(t) + f
1
s
1
(t),y(t), z(t)
+ Gu(t) ,
˙z(t) = A
5
z(t)+ f
2
s
2
(t),y(t)
,
y(t
0
) = y
0
, ˙y(t
0
) = y
1
, z(t
0
) = z
0
,
(5)
with y(t) , y
0
, y
1
, u(t), G, A
2
∈ R, z(t),z
0
∈ R
n−2
,
A
5
∈ R
(n−2)×(n− 2)
, n ≥ 2, and for q
1
, q
2
∈ N it holds
(i) G > 0, i.e., a positive input gain (“high-frequency
gain”);
(ii) s
1
(·) ∈ L
∞
R
≥0
;R
q
1
and s
2
(·) ∈ L
∞
R
≥0
;R
q
2
,
i.e., they may be thought of as (bounded) distur-
bance terms;
(iii) the functions f
1
: R
q
1
× R × R
n−2
→ R and f
2
:
R
q
2
× R → R
n−2
are continuous ones and, for
compact sets C
1
⊂ R
q
1
and C
2
⊂ R
q
2
, there exist
c
1
, c
2
≥ 0 such that for all (s, y, z) ∈ C
1
×R ×R
n−2
f
1
(s, y,z)
≤ c
1
1+ |y| + kzk
,
and for all (s, y) ∈ C
2
× R
f
2
(s, y)
≤ c
2
1+ |y|
;
(iv) spec(A
5
) ⊂ C
−
, i.e., the system is minimum
phase, provided f
1
= 0, f
2
= 0.
(v) A
2
< 0, i.e., this system has a zero-center in
the open left-half complex plane (a “stable zero-
center”), see (Ogata, 1997).
It is easy to check that S
2,nonlin2
⊂ S
2,nonlin1
holds.
In order to capture more relevant SISO-systems we
introduce a generalized system class of S
2,nonlin2
in
the following — system class S
2,nonlin3
:
¨y(t) = f
0
s
0
(t),y(t), z(t)
˙y(t)
+ f
1
s
1
(t),y(t), z(t)
+ Gu(t),
˙z(t) = A
5
z(t)+ f
2
s
2
(t),y(t)
,
y(t
0
) = y
0
, ˙y(t
0
) = y
1
, z(t
0
) = z
0
,
(6)
with y(t) , y
0
, y
1
, u(t), G ∈ R, z(t), z
0
∈ R
n−2
, A
5
∈
R
(n−2)×(n− 2)
, n ≥ 2, and for q
0
, q
1
, q
2
∈ N we have
to claim that the following will hold additionally:
(ii) s
0
(·) ∈ L
∞
R
≥0
;R
q
0
;
ModelingtheBehaviorofHairFollicleReceptorsasTechnicalSensorsusingAdaptiveControl
339
(iii) f
0
: R
q
0
×R × R
n−2
→ R is a continuous function,
and for a compact set C
0
⊂ R
q
0
, there exist c
0
,
˜c
0
> 0 with ˜c
0
> c
0
, such that for all (s, y, z) ∈
C
0
× R × R
n−2
− ˜c
0
< f
0
(s, y,z) < −c
0
;
It follows that S
2,nonlin2
⊂ S
2,nonlin3
. For the fol-
lowing control systems, theorems and proofs we then
focus on class S
2,nonlin3
instead of S
2,nonlin2
.
6 CONTROLLERS
Since we deal with uncertain, nonlinearly perturbed
(ground excitation, see the continuous functions
f
i
) MIMO-systems, which are not necessarily au-
tonomous, particular attention is paid to the adaptive
λ-
tracking control
objective(Behn and Zimmermann,
2006). This is to determine an
online control strategy
that achieves approximate tracking of a given, favored
reference signal in the following sense:
(i) every solution of the closed-loop system is de-
fined and bounded for t ≥ 0, and
(ii) the output y(·) tracks y
ref
(·) with asymptotic ac-
curacy λ > 0 in the sense that
max
n
0,
y(t) − y
ref
(t)
− λ
o
t→+∞
→ 0, (7)
i.e., we tolerate a feasible error of prescribed size λ
(accuracy). Visually, this means that the output y(t)
tends to a tube of radius λ around y
ref
(t), see Fig. 5.
l
t
R
m
y( )
y ( )
ref
.
.
Figure 5: Reference signal and λ-tube.
Classical adaptive high-gain lambda-trackers
from literature are:
1. The first one is a modification of the preferred sta-
bilizer from literature, see (Ilchmann, 1991). The
modified control strategy, which is also presented
in (Behn and Zimmermann, 2006), is:
e(t) := y(t)− y
ref
(t),
u(t) = −
k(t)e(t) +
d
dt
k(t)e(t)
,
˙
k(t) = γ
max
n
0,
e(t)
− λ
o
2
,
(8)
with k(0) = k
0
∈ R, λ > 0, y
ref
(·) ∈ R , u(t),
e(t) ∈ R
m
, k(t) ∈ R , and γ ≫ 1.
Due to the presented λ-tracking control objective
(tolerating a tracking error of size λ, no exact
tracking) this controller consists of a very simple
feedback mechanism and adaptation law, and is
only based on the output of the system and its time
derivative - no knowledge about the system pa-
rameters is required. However, the adaptive con-
troller (8) uses the derivative of the output. The
following two feedback controls avoid the usage
of the derivative of the system output.
2. This one includes a dynamic compensator due to
a controller in (Miller and Davison, 1991). This
controller avoids the possible drawback of using
the derivative of the output.
e(t) := y(t) − y
ref
(t),
u(t) = −k(t)θ(t) −
d
dt
k(t)θ(t)
,
˙
θ(t) = −k(t)
2
θ(t)+ k(t)
2
e(t) ,
˙
k(t) = γ max
n
0,
e(t)
− λ
o
2
,
(9)
with θ(t
0
) = θ
0
, k(t
0
) = k
0
> 0, λ > 0, y
ref
(·) ∈
R , u(t), e(t) ∈ R
m
, k(t) ∈ R, γ ≫ 1 and arbitrary
initial data k
0
> 0, θ
0
∈ R
m
.
We stress that the feedback in (9) does not invoke
any derivatives of observables.
3. If S
2,nonlin1
is restricted to single-input, single-
output systems of class S
2,nonlin3
, then the fol-
lowing simple feedback control is considered,
which reduces in dimension (the number of used
variables calculated by internal differential equa-
tions):
u(t) = −k(t)
y(t) − y
ref
(t)
,
˙
k(t) = γ max
0,
y(t)
− λ
2
.
(10)
with k(0) = k
0
∈ R. Therefore, we have a con-
troller of order 1 whereas (9) is a controller of or-
der 2. We stress that this feedback in (10) does not
invoke any derivatives, too.
The feedback law has a P-structure, a D-term is
not necessary for controlling systems of the class
S
2,nonlin3
. Naturally we need a P- and D-term to
control systems with strict relative degree two, see
(Sontag, 1998).
To summarize, these controllers are simple in their
design, rely only on structural properties of the system
(and not on the system’s data) and do not invoke any
estimation or identification mechanism. They only
consist of a feedback strategy and a simple parameter
ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics
340
adaptation law, and, moreover, do not have to depend
on the derivative of the output of the system.
All three controllers achieve λ-tracking (and, of
course, λ-stabilization using y
ref
(·) ≡ 0, as well) in
applying all three controllers (8), (9) and (10) to the
system classes:
• Theorem I: controller (8) applied to systems of
class S
2,nonlin1
— Proof in (Behn and Zimmer-
mann, 2006);
• Theorem II: controller (9) applied to systems of
class S
2,nonlin1
— Proof in (Behn, 2011);
• Theorem III: controller (10) applied to systems of
class S
2,nonlin3
— proven and submitted.
The parameter γ strongly determines the growth of
the gain parameter k(·). In (Behn and Zimmermann,
2006) the case γ = 1 was dealt with. With small γ
(e.g., γ = 1 as formerly) k(·) often grows too slowly
as to achieve a good tracking behavior. Therefore, a
sufficiently large γ ≫ 1 should be used. But, if we
choose γ too large, we arrive at high feedback values.
Furthermore, these high values keep the sensor not
really to be sensitive to extraordinary impulses, gen-
erally speaking, the receptor is “blind” if the signal is
forced once into the tube, because it cannot detect the
peak in observing the output. The last requirement to
the controllers is not fulfilled: the closed-loop sensor
system has to be sensitive to recurring excitation sig-
nals — fade it out and wait for further new informa-
tion. This is not realized yet. We are able to dominate
the system, but we are not able to get information on
the environment in observing the output. This is ad-
dressed in the next section – design of new adapta-
tion laws – to identify the (whole) ground excitation
or only some basic characteristics of it.
7 ADAPTORS
The drawback of the ‘Classical’ Adaptor:
˙
k(t) = γ
max
n
0,
e(t)
− λ
o
2
,
is
˙
k(t) ≥ 0, ∀t ≥ 0, i.e.,
t 7→ k(t) = k(0)+
t
Z
0
γ
max
n
0,
e(τ)
−λ
o
2
dτ ≥ 0
thus implies monotonic increase of k(·). Typically,
the classical high-gain adaptive controllers (feedback
law including adaptation law) yield a non-decreasing
gain, which is usual. Now we propose some new
adaptation laws, which let k(·) decrease when e is in
the tube.
A very simple modification of the adaptation law
is the so-called σ-modification, σ > 0. For λ-tracking
control including the gain coefficient γ (Georgieva
and Ilchmann, 2001), and revisited in (Behn and
Steigenberger, 2009) in simplified form we have
Adaptor 1:
˙
k(t) = −σk(t) + γ
max
n
0,
e(t)
− λ
o
2
,
with σ > 0, γ ≫ 1. The term −σk(t) decreases
k(·) exponentially, while the second term ensures a
quadratic increase of k(·) when ky(t)k is outside the
λ-strip. Therefore, Adaptor 1 offers two terms which
are active simultaneously and counteract each other.
Depending on the situation, one of the terms over-
comes the effect of the other and results in a global
decrease or increase of k(·). This law often leads
to oscillatory behavior (maybe limit cycles) and even
chaotic one of the system. Hence, this adaptor has to
be treated carefully, because the dynamical behavior
depends crucially on the parameters σ > 0. Therefore,
we will not focus on this adaptor type in sequel.
The idea is now to split the part of increase and
decrease of the gain as follows in Adaptor 2 (Behn
and Steigenberger, 2009):
˙
k(t) =
(
γ
e(t)
− λ
2
,
e(t)
≥ λ,
−σk(t),
e(t)
< λ,
with σ > 0, γ ≫ 1. This adaptor shows alternating
increase and exponential decrease of k(·).
It could happen that e rapidly traverses the λ-tube.
Then it would be inadequate to immediately decrease
k(·) after e entered the tube. Rather we should distin-
guish three cases:
1. increasing k(·) while e is outside the tube,
2. constant k(·) after e entered the tube - no longer
than a pre-specified duration t
d
of stay, and
3. decreasing k(·) after this duration has been ex-
ceeded.
So, a another adaptation law of this kind is Adaptor 3
(Behn and Steigenberger, 2010):
˙
k(t) =
γ
e(t)
− λ
2
,
e(t)
≥ λ,
0,
e(t)
< λ
∧ (t −t
E
< t
d
),
−σk(t),
e(t)
< λ
∧ (t −t
E
≥ t
d
),
with given σ > 0, γ ≫ 1, and t
d
> 0, whereas the entry
time t
E
is an internal time variable.
If the norm of the error value kek is close to the
λ-strip, i.e., the system output y is already close to
ModelingtheBehaviorofHairFollicleReceptorsasTechnicalSensorsusingAdaptiveControl
341
the λ-tube, and 0 < kek − λ < 1 holds, an exponent
of p = 2 leads to an even smaller number. This is
the main disadvantage in such a way, that, if kyk is
already close to the λ-strip around the prescribed ref-
erence signal, the adaption process, i.e., the increase
of k(·), is very slow.
In order to make the attraction of the tube stronger, it
would be advantageous to use different exponents p
with better performance such as a square root. Hence,
a kind of scheduling of
˙
k is introduced, different ex-
ponents for large/small distances from the tube, see
(Behn and Steigenberger, 2010) and Adaptor 4:
˙
k(t) =
γ
e(t)
− λ
2
,
e(t)
≥ λ + 1,
γ
e(t)
− λ
0.5
, λ + 1 >
e(t)
≥ λ,
0,
e(t)
< λ
∧ (t −t
E
< t
d
),
−σk(t),
e(t)
< λ
∧ (t − t
E
≥ t
d
),
with σ, γ,t
d
,t
E
as before.
Let λ > 0 be chosen in regard of certain require-
ments given by the context. To ensure that the system
output y stays within that λ-tube along the reference
signal (e will not leave the λ-strip after entering the
strip) is to track a smaller safety radius ελ < λ, sug-
gestions for adaptation laws are now Adaptor 5:
˙
k(t) =
γ
e(t)
− ελ
2
,
e(t)
≥ ελ + 1,
γ
e(t)
− ελ
1
2
, ελ+ 1 >
e(t)
≥ ελ,
0,
e(t)
< ελ
∧ (t −t
E
< t
d
),
−σk(t),
e(t)
< ελ
∧ (t −t
E
≥ t
d
),
with σ, γ,t
d
,t
E
as before.
8 SIMULATION
We point out, that the adaptive nature of the con-
trollers is expressed by the arbitrary choice of the
system parameters. Obviously numerical simulation
needs fixed (and known) system data, but the con-
trollers adjust their gain parameter to each set of sys-
tem data. The numerical simulations will demonstrate
and illustrate that the adaptive controllers work suc-
cessfully and effectively.
Choosing the parameters from Table 1 (which are
arbitrarily chosen, not measured or identified from
the biological paradigm, just for simulation purposes)
and ε = 0.7 we get the results shown in Figs. 6 and 7
in applying Adaptor 5.
Table 1: Global simulation parameters (dimensionless).
sensor mass m 1
damping coefficient d 5
spring stiffness c 10
initial values
y(0), ˙y(0)
− a(0), − ˙a(0)
tolerance λ 0.2
initial gain value k
0
0
ref. signal t 7→ y
ref
(t) 0 (rest position)
ground excitation t 7→ a(t) = sin(2πt)
0 5 10 15 20 25 30
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
t
y(t)
l-tube
e-tube
Figure 6: Output y(·) and tubes.
0 5 10 15 20 25 30
0
10
20
30
40
50
60
70
t
k(t)
Figure 7: Gain parameter k(·).
In former simulations, the output apparently pe-
riodically leaves the λ-tube. Then ελ-tracking (we
will call this kind of tracking ε-safe λ-tracking) with
ε = 0.7 makes e not to leave the desired λ-tube, see
Fig. 6.
The steep increase of k(·) at the beginning is due to
the “switching on” of the controller and the small ini-
tial value of k(0) = k
0
= 0. This could be prevented
in choosing a larger k
0
. But, it depends on the system
data which is unknown a-priori. The constant high
level is due to the (only) monotonic increase of the
ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics
342
gain. This problem is addressed in the next section.
Comparing the simulation results and the sensor be-
havior of the gain parameter k(·) in Fig. 7 with the im-
pulse sequences in Fig. 3, one clearly recognizes that
we achieved the behavior of the biological paradigm.
9 EXPERIMENT
This section is devoted to the experimental verifica-
tion of the successful implementation of the controller
(feedback including Adaptor 5) developed above. For
this purpose, we built up a demonstrator in form of an
electrical oscillating circuit, the test rig is presented in
Fig. 8.
Figure 8: Test rig with electrical oscillating circuit: 1 -
I/O-system (BNC-2110), 2 - DAQ-6036-PCMCIA-card, 3
- demonstrator, 4 - PC with LabView.
The demonstrator, see (3) in Fig. 8, is shown in
Fig. 9.
Figure 9: Circuit: 1 - capacitor (C = 800µF) , 2 - resistor
(R = 100Ω), 3 - one inductor (overall inductance L
ges
=
640mH), 4 - communication to PC.
Then, the equations of motion are, using La-
grange’s equations of the 2nd kind
L ¨q(t)+ R ˙q(t) +
1
C
q(t) = U(t) + u(t). (11)
The system output shall be the charge q(·). The goal
is to adaptively compensate changes ofU(t) by means
of the control input u(t) to λ-track q
ref
(·) ≡ 0. As a
rule, the charge is measured due to the voltage at the
capacitor in form of
q(t) = CU
C
(t).
Due to the small system parameter values the gain k(·)
will increase tremendously and we need high comput-
ing capacity. To avoid this we will directly control the
voltageU
C
(·) which depends linearly on the measured
q(·), see above.
We apply Adaptor 5 to guarantee that the error will
not leave the λ-tube in tracking a tube of smaller ra-
dius ελ. We have
• excitation: t 7→ U(t) = U
0
sin(ωt) with amplitude
U
0
= 5V and frequency f = 0.5Hz;
• Adaptor 5: γ = 1000, λ = 0.03V, ελ = 0.02V
(much smaller tolerance), σ = 0.05, t
d
= 6s.
We perform this experiment in using LabView to
handle and to control the circuit. By means of a pro-
grammed LabView control panel, see Fig. 10, we are
able to switch on/off the excitation U(·) and the con-
trol strategy u(·). Furthermore, several signals are dis-
played via this panel:
• in the top window (actual measured data): the
control input u(·) (orange line), the system output
U
C
(·) (red line), the excitation signal U(·) (blue
line);
• the bottom window (data on time horizon): the
depicted curves are capacity voltage U
C
(·) (i.e.,
the output y(·), red line), the λ-strip (blue lines),
and gain parameter k(·) (green line), in only one
window.
The plots of the exported data-files from LabView
are shown in Fig. 11 and 12.
The capacitor voltage (output) never leaves
the λ-tube, the adaptor works effectively. Further
experiments can be found in (Behn, 2013).
Comparing the simulation results with σ = 0.05 in
Fig. 12 with the impulse sequences in Fig. 3 and the
adaptation behavior in Fig. 2, we conclude that this
technical device offers the behavior of the biological
paradigm.
ModelingtheBehaviorofHairFollicleReceptorsasTechnicalSensorsusingAdaptiveControl
343
Figure 10: LabView front panel on PC screen, using adaptor
(11); depicted curves in bottom window: capacity voltage
U
C
(·), i.e., new output y(·), (red line), λ-strip (blue lines),
and gain parameter k(·) (green line).
l-tube
U (t)
C
t
e-tube
Figure 11: Output U
C
(·) and tubes using Adaptor 5.
k(t)
t
Figure 12: Gain parameter using Adaptor 5.
10 CONCLUSIONS
The development of new control strategies and sen-
sor models was motivated by the open question which
occurred during analysis of the functional morphol-
ogy of vibrissal sensor systems. The vibrissa re-
ceptors are in a permanent state of adaption to fil-
ter the perception of tactile stimuli. This behavior
now may be mimicked by the artificial sensor sys-
tem. The sensor system was modeled as a spring-
mass-damper system with relative degree two and the
system parameters are supposed to be unknown, due
to the complexity of biological systems. Using a sim-
ple linear model of a sensory system, adaptive con-
trollers have been considered which compensate un-
known permanent ground excitations. Classical adap-
tors suffer from a monotonic increase of the control
gain parameter, thereby possibly paralyzing the sen-
sor’s capability to detect future extraordinary excita-
tions. The existing adaptivecontrollers from literature
were improved with respect to performance, sensitiv-
ity and capabilities. Various modifications of existing
controllers are made and new controller designs were
discussed:
- tuning parameters γ and gain exponent p increase
the growth rate of the gain parameter k;
- new adaptors allow for gain parameter decrease
that improves the sensor system’s sensitivity to
further ground excitations;
- a smaller ελ-tube is introduced to prevent the out-
put y from leaving the λ-neighborhood.
These proposed and modified adaptors avoid the
drawbacks from literature and do not invoke any
estimation or identification techniques. The working
principle of the new controller (feedback law includ-
ing Adaptor 5) is shown in a numerical simulation
which proves that this controller in fact works
successfully and effectively. This controller is simple
in its design: its adaptation law is not complex
as current adaptive control strategies in literature.
Moreover, a practical implementation of this con-
troller to a demonstrator in form of an electrical
oscillating circuit results in a successful experiment
which confirms the theoretical results. However, the
preceding simulation results shed some light upon the
behavior of the sensor system under the governance
of various adaptive controllers. It becomes clear that
both the choice of controller type and the tuning of
the chosen controller (optimize controller data) is a
delicate task. Simulation and experiment show that
the developed adaptive control strategy applied to the
mechanical sensor system achieve the fast adapting
behavior of the biological receptor.
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There are various aspects of adaptive controllers
that can be improved and modified in current and fu-
ture work on the subject.
• Avoidance of Error Derivative
Since the derivative term is difficult to measure, it
might be possible to implement an observer in or-
der to estimate the system state. If this is also not
possible, it would be best to omit the derivative
term. This would avoid the occurrence of noise
in the feedback loop, see (Behn, 2011). How-
ever, such a term is imperative to achieve stabil-
ity. Thus, methods of not having to differentiate
the system output can be investigated.
One possibility to do so is the control with output
delay feedback. With delay feedback, the output
derivative is approximated by computing a differ-
ence quotient with a fixed time span:
˙y ≈
y(t) − y(t − h)
h
with h > 0.
This method computes a value for ˙y, which is not
exact, but might be sufficiently approximated if h
is chosen sufficiently small. However, there re-
mains an error in the derivative feedback term.
Current investigations on this topic are done.
• Constrained Control Input
In technical realizations of controllers, there usu-
ally exists a limit for the control value that can-
not be exceeded. This is quite obvious, as there
are no actuators that can generate an infinite force,
for example. Therefore, some adaptivecontrollers
may not be implemented in certain applications,
as they rely on the possibility to increase the con-
trol value as high as necessary. In order to cope
with this issue, controllers with constrained input
values might be investigated.
• Intelligent Control
The fuzzy controller is built upon expert knowl-
edge that is used to form the rule set of the con-
troller. This knowledge is not given a priori and is
obtained from experimenting with previous con-
trollers. However, it is possible to generate expert
knowledge by using intelligent control methods,
such as artificial neural nets. The expert knowl-
edge – or “intelligence” – in a neural net is ob-
tained by training the controller with data gener-
ated by the system. However, this training process
takes time and therefore diminishes the adaptive
capabilities of the controller.
REFERENCES
Barth, F. (2004). Spider mechanoreceptors. Current Opin-
ion in Neurobiology, 14:415–422.
Behn, C. (2011). Adaptive control of straight worms with-
out derivative measurement. Multibody System Dy-
namics, 26(3):213–243.
Behn, C. (2013). Mathematical modeling and control of
biologically inspired uncertain motion systems with
adaptive features. Habilitation submitted to TU Ilme-
nau.
Behn, C. and Steigenberger, J. (2009). Improved adap-
tive controllers for sensory systems - first attempts.
In Awrejcewicz, J., editor, Modeling, Simulation and
Control of Nonlinear Engineering Dynamical Sys-
tems, pages 161–178. Springer.
Behn, C. and Steigenberger, J. (2010). Experiments in adap-
tive control of uncertain mechanical systems. Interna-
tional Review of Mechanical Engineering (IREME),
4(7):886–898.
Behn, C. and Zimmermann, K. (2006). Adaptive λ -
tracking for locomotion systems. Robotics and Au-
tonomous Systems, 54:529–545.
Dudel, J., Menzel, R., and Schmidt, R. (1996). Neurowis-
senschaft. Springer.
Ebara, S., Kumamoto, K., Matsuura, T., Mazurkiewicz,
J., and Rice, F. (2002). Similarities and differences
in the innervation of mystacial vibrissal follicle-sinus
complexes in the rat and cat: A confocal micro-
scopic study. The Journal of Comparative Neurology,
449:103–119.
Georgieva, P. and Ilchmann, A. (2001). Adaptive λ-tracking
control of activated sludge processes. International
Journal of Control, 74(12):1247–1259.
Ilchmann, A. (1991). Non-identifier-based adaptive con-
trol of dynamical systems: a survey. IMA Journal of
Mathematical Control and Information, 8:321–366.
Iwasaki, M., Itoh, T., and Tominaga, Y. (1999). Vi
mechano- and phonoreceptors. In Eguchi, E. and
Tominaga, Y., editors, Atlas of Arthropod Sensory Re-
ceptors - Dynamic Morphology in Relation to Func-
tion, pages 177–190. Springer.
Miller, D. and Davison, E. (1991). An adaptive con-
troller which provides an arbitrarily good transient and
steady-state response. IEEE Transactions on Auto-
matic Control, 36:68–81.
Ogata, K. (1997). Modern Control Engineering. Prentice
Hall, 3rd edition.
Rice, F., Mance, A., and Munher, B. (1986). A compar-
ative light microscopic analysis of the sensory inner-
vation of the mystacial pad. i. innervation of vibrissal
follicle-sinus complexes. The Journal of Comparative
Neurology, 252:154–174.
Smith, C. (2008). Biology of Sensory Systems. John Wiley
& Sons Ltd., 2nd edition.
Soderquist, D. (2002). Sensory Processes. Sage Publica-
tions.
Sontag, E. (1998). Mathematical Control Theory. Springer,
2nd edition.
ModelingtheBehaviorofHairFollicleReceptorsasTechnicalSensorsusingAdaptiveControl
345
