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I R
INSTITUTE FOR SYSTEMS RESEARCH
TECHNICAL RESEARCH REPORT
Border Collision Bifurcation Control of Cardiac Alternans
by Munther A. Hassouneh, Eyad H. Abed
TR 2003-41
Border Collision Bifurcation Control of Cardiac
Alternans
Munther A. Hassouneh and Eyad H. Abed
Department of Electrical and Computer Engineering
and the Institute for Systems Research
University of Maryland
College Park, MD 20742 USA
munther@umd.edu, abed@umd.edu
Revised: October 6, 2003
Abstract
The quenching of alternans is considered using a nonlinear cardiac conduction
model. The model consists of a nonlinear discrete-time piecewise smooth system.
Several authors have hypothesized that alternans arise in the model through a period
doubling bifurcation. In this work, it is first shown that the alternans exhibited by
the model actually arise through a period doubling border collision bifurcation. No
smooth period doubling bifurcation occurs in the parameter region of interest. Next,
recent results of the authors on feedback control of border collision bifurcation are
applied to the model, resulting in control laws that quench the bifurcation and hence
result in alternan suppression.
1 Introduction
In this paper, we give a bifurcation analysis of the cardiac conduction model proposed by
Sun, Amellal, Glass and Billette [Sun et al., 1995] and, based on the conclusions of the
analysis, we investigate control design for suppression of predicted cardiac alternans. Sun
et al. formulated their model as a two-dimensional piecewise smooth map. The model
incorporates physiological concepts of recovery, facilitation and fatigue. It predicts a variety
of experimentally observed complex rhythms of nodal conduction [Sun et al., 1995]. In
particular, alternans, in which there is an alternation in conduction time from beat to
beat, were associated in [Sun et al., 1995] to period doubling bifurcation in the model.
In the present study, we first show that the instability mechanism giving rise to cardiac
alternans in the model is in fact not a smooth period doubling bifurcation as has been
earlier hypothesized, but rather a related bifurcation that occurs in nonsmooth systems, the
1
period doubling border collision bifurcation. We then proceed to apply our recent work on
control of border collision bifurcations [Hassouneh, 2003; Hassouneh & Abed, 2003] to the
design of control laws for alternan quenching.
A border collision bifurcation (BCB) is a bifurcation that occurs when a fixed point (or a
periodic orbit) of a piecewise smooth system crosses or collides with the border between two
regions of smooth operation [Nusse & Yorke, 1992; di Bernardo et al., 1999]. Border collision
bifurcations include bifurcations that are reminiscent of the classical bifurcations in smooth
systems such as the fold and period doubling bifurcations. Despite such resemblances,
the classification of border collision bifurcations is far from complete, and certainly very
preliminary in comparison to the results available in the smooth case. In smooth maps, a
bifurcation occurs from a one-parameter family of fixed points when a real eigenvalue or a
complex conjugate pair of eigenvalues crosses the unit circle. In piecewise smooth (PWS)
maps, on the other hand, a border collision bifurcation can occur when a fixed point (or a
periodic orbit) crosses or collides with the border between two regions of smooth behavior.
This involves a discontinuous change in the eigenvalues of the Jacobian matrix evaluated at
the fixed point (or at a periodic point) when the fixed point hits the border. As a result,
border collision bifurcations for piecewise smooth systems in which the one-sided derivatives
on the border are finite are classified based on the linearizations of the system on both sides
of the border at criticality.
Several researchers have studied the model of [Sun et al., 1995] and developed control
techniques to eliminate the period-2 rhythm and stabilize the underlying period-1 rhythm
(e.g., [Christini & Collins, 1996; Brandt et al., 1997; Chen et al., 1998]). With the exception
of [Chen et al., 1998], all the studies of this model reported in the literature viewed the
border collision period doubling bifurcation in this system as if it were an ordinary period
doubling bifurcation in a smooth dynamical system. Chen, Wang and Chin [1998] identified
the bifurcation in the cardiac model as a border collision bifurcation based on numerical
evidence. However, they didn?t provide analysis to prove this claim. The authors of [Chen
et al., 1998] also investigated the feedback control of the BCB detected in the alternan
model, but the feedback design was largely based on trial and error, and did not involve a
detailed consideration of the border collision bifurcation. In [Brandt et al., 1997], the authors
propose the use of delayed linear feedback to suppress the period doubling bifurcation.
In [Christini & Collins, 1996], the authors apply a technique for control of chaos to suppress
the alternation resulting from the period doubling bifurcation. In [Hall et al., 1997], a
smooth one dimensional map was used as a model for cardiac conduction. A form of linear
dynamic feedback where the unstable fixed point corresponding to the unstable rhythm
is estimated as the average value of two consecutive beats was used to achieve alternan
quenching [Hall et al., 1997]. The control gain was determined by trial and error.
Another approach to the quenching of alternans by feedback is given in the paper of
Christini et al. [2001], which takes an experimental approach, and determines a stabilizing
controller in a fashion similar to that used in the OGY method [Ott, Grebogi & Yorke, 1990]
for chaos control. The work of Christini et al. [2001] did not use a model, and certainly did
not use any results on border collision bifurcations. So one might ask how this is possible,
and if this implies the role of understanding border collision bifurcations in this context
2
is diminished. We make several comments with regard to the first question. The authors
of [Christini et al., 2001] determine a control law for a given set of system parameters that
does not include the actual point of bifurcation. Thus, the controller could conceivably
fail closer to the bifurcation point. Also, since no model is available, the desired unstable
fixed point is determined by an approximation based on the stable period-2 orbit that is
experienced in the experiment. The extreme values of the period-2 orbit are averaged to
obtain an approximation for the unstable fixed point. This discussion implies that the
control law of [Christini et al., 2001] takes action when an alternan is experienced, whereas
a successful model-based controller would not allow the alternan to occur in the first place.
The second question was on the need for understanding the border collision bifurcation.
Clearly, a better understanding of and appreciation for the system dynamics will lead to
more confidence in the control design. It is of course possible that designs obtained by other
means, such as trial and error with some engineering and medical basis, will also do the job
in some cases. However, there is no substitute for a design based on a correct understanding
of the dynamics. Moreover, in piecewise smooth systems where a border collision bifurcation
exists, the bifurcation is generic. That is, it is maintained under small perturbations of the
model. Using a smooth system approximation does not result in any benefit, because the
di?cult calculations of what bifurcation actually occurs simply will need to be done in a
less local region than if they were done at the fixed point on the border of the nonsmooth
model.
In this paper, we use recent results of the authors on feedback control of border collision
bifurcations [Hassouneh, 2003; Hassouneh & Abed, 2003] to quench the period doubling
border collision bifurcation which consequently suppresses the alternans. The feedback can
be either linear or piecewise linear. Both static and washout filter-aided feedbacks are
considered. Washout filter-aided feedback has certain advantages over static feedback: it
maintains the fixed points of the open-loop system even in the presence of model uncertainty,
and it provides automatic following of the fixed point to be stabilized which alleviates
the need for providing an estimate of the unstable fixed point to the controller. This is
particularly useful in situations where the system model is uncertain and/or cases where
there is parameter drift.
It is important to realize that, since border collision bifurcations arise at the border
separating regions of smooth operation, a linear feedback that seems to ?delay? a border
collision bifurcation to occur away from the border actually does no such thing. If a BCB
seems to have been delayed by feedback, what actually is happening is that the feedback has
changed the BCB to a type that replaces the nominal fixed point by a new one (fixed point
to fixed point BCB), and a new smooth bifurcation has been created elsewhere (away from
the border). Thus, concepts and methods developed in the control of smooth bifurcations
cannot be carried over in a direct way to the nonsmooth case.
The paper proceeds as follows. In Sec. 2, needed results on border collision bifurcations
in PWS maps are recalled. In Sec. 3, results on control of BCBs in two-dimensional and
n-dimensional PWS maps are given. The results of Sec. 2 and Sec. 3 are applied to the
cardiac conduction model in Sec. 4.
3
2 Results on Border Collision Bifurcations
In this section, needed results on border collision bifurcation (BCB) in two-dimensional
and n-dimensional PWS maps are collected from recent work of the authors [Hassouneh,
2003; Hassouneh & Abed, 2003]. Further background on BCBs in PWS maps can be found
in [Nusse & Yorke, 1992; Benerjee & Grebogi, 1999; di Bernardo et al., 1999].
2.1 Border collision bifurcation in 2-D PWS maps
Consider a general two-dimensional PWS map of the form
parenleftbigg
x
k+1
y
k+1
parenrightbigg
= f(x
k
,y
k
,?)
where
f(x, y, ?)=
braceleftbigg
f
A
(x, y, ?),x? 0
f
B
(x, y, ?),x>0
(1)
Here ? is the bifurcation parameter and R
A
:= {(x, y):x ? 0}, R
B
:= {(x, y):x>0} are
two regions of smooth behavior separated by the border x =0.Themapf(?,?,?) is assumed
to be PWS: f
A
(x, y, ?)issmoothonR
A
, f
B
(x, y, ?)issmoothonR
B
and f is continuous
in (x, y) but not di?erentiable at the border and depends smoothly on ? everywhere. Let
(x
0
(?),y
0
(?)) be a path of fixed points of f; this path depends continuously on ?. Suppose
also that the fixed point hits the border at a critical parameter value ?
b
. Assume without
loss of generality that ?
b
=0. Thus,(x
0
(0),y
0
(0)) = (x
b
,y
b
). Suppose that the coordinate
system is chosen such that (x
b
,y
b
)=(0, 0).
Expanding (1) in a Taylor series near the fixed point (0, 0, 0) and ignoring the higher
order terms gives
f(x, y, ?)=
?
?
?
?
?
?
?
A
parenleftbigg
x
y
parenrightbigg
+ b?, (x, y) ? R
A
B
parenleftbigg
x
y
parenrightbigg
+ b?, (x, y) ? R
B
(2)
where A :=
parenleftbigg
a
11
a
12
a
13
a
14
parenrightbigg
is the limiting Jacobian of f at (x, y, ?) close to (0, 0, 0) with
(x, y) ? R
A
, B :=
parenleftbigg
a
21
a
22
a
23
a
24
parenrightbigg
is the limiting Jacobian of f at (x, y, ?) close to (0, 0, 0)
with (x, y) ? R
B
and b =
parenleftbigg
b
1
b
2
parenrightbigg
is the derivative of f with respect to ?. We assume that
the elements of A and B are finite. Since the map f is not di?erentiable at the border x =0,
A negationslash= B. The continuity of f at the border implies that the second column of A equals the
second column of B, i.e., a
12
= a
22
=: a
2
and a
14
= a
24
=: a
4
.Let?
A
:=trace(A)=a
11
+a
4
,
?
A
:=det(A)=a
11
a
4
? a
2
a
13
, ?
B
:=trace(B)=a
21
+ a
4
and ?
B
:=det(B)=a
21
a
4
? a
2
a
23
.
4
Border collision bifurcations are classified based on the eigenvalues or, equivalently, by
the trace and the determinant of the Jacobian matrices on both sides of the border [Nusse &
Yorke, 1992; Benerjee & Grebogi, 1999]. Next, we give two propositions on border collision
bifurcations in two-dimensional PWS maps.
Denote the eigenvalues of A by ?
+
A
, ?
?
A
and the eigenvalues of B by ?
+
B
, ?
?
B
.The
following proposition gives su?cient conditions for nonbifurcation with persistent stability
in two-dimensional PWS maps.
Proposition 1 [Hassouneh, 2003] (Su?cient Conditions for Nonbifurcation with Persistent
Stability in 2-D PWS Maps) If the eigenvalues of the Jacobian matrices A and B on either
side of the border of the two-dimensional PWS map (2) are real and satisfy any of the fol-
lowing conditions:
(i) 0 <?
?
A
<?
+
A
< 1and0<?
?
B
<?
+
B
< 1
(ii) ?1 <?
?
A
< 0 <?
+
A
< 1and? 1 <?
?
B
< 0 <?
+
B
< 1
(iii) 0 <?
?
A
<?
+
A
< 1and ? 1 <?
?
B
< 0 <?
+
B
< 1with?
+
B
+ ?
?
B
> 0
or 0 <?
?
B
<?
+
B
< 1and ? 1 <?
?
A
< 0 <?
+
A
< 1with?
+
A
+ ?
?
A
> 0
then a locally unique and stable fixed point on one side of the border leads to a locally unique
and stable fixed point on the other side of the border as ? is varied through zero.
For PWS maps of dimension two or higher, having the eigenvalues of the Jacobian
matrices on both sides of the border within the unit circle does not imply that the fixed
points are the only attractors as ? is increased (decreased) through its critical value. For
example, if the eigenvalues of one of the Jacobian matrices or both Jacobian matrices are
within the unit circle but nonreal, then higher period periodic attractors may exist on one
side or both sides of the border in addition to the stable fixed points [Benerjee & Grebogi,
1999; Hassouneh, 2003].
The following proposition gives a su?cient condition for the occurrence of a supercritical
period doubling BCB in two-dimensional PWS maps.
Proposition 2 Suppose that the fixed point of (2) to the left of the border is stable for ?<0
(i.e., |?
A
| < 1 and ?(1 + ?
A
) <?
A
< (1 + ?
A
)) and that it crosses the border and becomes
unstable as ? is increased through zero. If
|?
A
?
B
| < 1,
?(1 ? ?
B
)(1 ? ?
A
) <?
B
?
A
< (1 + ?
B
)(1 + ?
A
).
then a supercritical period doubling border collision bifurcation occurs as ? is increased
through zero. That is, a stable fixed point to the left of the border for ?<0 crosses the
border and becomes unstable and a period two attractor is born as ? is increased through
zero.
5
For a proof and further details, see [Hassouneh, 2003]. Regarding the stability condition in
Prop. 2, it can be understood by noting that the stability of the bifurcated period-2 orbit
is determined by the eigenvalues of the Jacobian matrix of the second iterate map with
one point in R
A
and the other point in R
B
, i.e., by the matrix product AB. The stability
condition follows by a straightforward application of the Jury Test for second order systems
to AB.
2.2 Border collision bifurcations in n-dimensional PWS maps
Next, a su?cient condition for nonbifurcation with persistent stability in n-dimensional
PWS maps is given. This result, which is based on using a quadratic Lyapunov function
and linear matrix inequalities, is derived in [Hassouneh, 2003; Hassouneh & Abed, 2003].
Consider the one-parameter family of piecewise smooth maps
f(x, ?)=
braceleftbigg
f
A
(x, ?),x
1
? 0
f
B
(x, ?),x
1
> 0
(3)
where f : R
n+1
? R
n
is piecewise smooth in x (f is smooth everywhere except on the border
{x ? R
n
: x
1
=0} whereitisonlycontinuous),f is smooth in ? and R
A
:= {x ? R
n
: x
1
?
0}, R
B
:= {x ? R
n
: x
1
> 0} are two regions of smooth behavior. The notation x
1
denotes
the first component of the vector x. Suppose that at ? = ?
b
,afixedpointoff is at the
border separating R
A
and R
B
. Assume without loss of generality that ?
b
=0andx
0
(0) = 0.
Border collision bifurcations occurring in (3) can be studied using the piecewise-linearized
representation [di Bernardo et al., 1999]
x(k +1) = f
1
(x(k),?)(4)
where
f
1
(x(k),?)=
braceleftbigg
Ax(k)+b?, if x
1
(k) ? 0
Bx(k)+b?, if x
1
(k) > 0
Here, A is the linearization of the PWS map f in R
A
at a fixed point on the border
approached from points in R
A
near the border and B is the linearization of f at a fixed
point on the border approached from points in R
B
and b is the derivative of the map f with
respect to ?.
Proposition 3 [Hassouneh, 2003] (Su?cient Condition for Persistent Stability in n-Dimensional
PWS Maps) Consider the system (4). If there is a P = P
T
> 0 such that
A
T
PA? P<0, (5)
B
T
PB? P<0, (6)
then system (4) has a globally stable fixed point for all ? ? R.
6
3 Feedback Control of Border Collision Bifurcation
The normal form for BCBs contains only linear terms in the state [Nusse & Yorke, 1992; di
Bernardo et al., 1999]. This leads us to seek linear or piecewise linear feedback controllers to
modify the system?s bifurcation characteristics. The feedback can either be applied on only
one side of the border or on both sides. However, using switching feedback with switching
depending on the location of the state requires accurate knowledge of where the border lies,
which is not necessarily available in practice. To achieve robustness to uncertainties in the
border itself, stabilization is pursued using the same stabilizing feedback acting on both
sides of the border. We call this the ?simultaneous stabilization? problem for BCBs. Below,
the design of simultaneous static feedback is considered followed by design of simultaneous
washout filter-aided feedback.
3.1 Simultaneous static feedback
In this method, the same linear state feedback control is applied additively on both the left
and right sides of the border. This leads to the closed-loop system
parenleftbigg
x
k+1
y
k+1
parenrightbigg
=
?
?
?
?
?
?
?
A
parenleftbigg
x
k
y
k
parenrightbigg
+ b? + bu
k
,x
k
? 0
B
parenleftbigg
x
k
y
k
parenrightbigg
+ b? + bu
k
,x
k
> 0
(7)
u
k
=(?
1
?
2
)
parenleftbigg
x
k
y
k
parenrightbigg
= ?
1
x
k
+ ?
2
y
k
(8)
where ?
1
, ?
2
are the control gains to be chosen. The following proposition gives su?cient
conditions for the existence of a stabilizing control policy as in (7)-(8) above. The conditions
are given in terms of linear inequalities. The existence of a solution can be easily checked
numerically.
Proposition 4 [Hassouneh, 2003] Suppose that the fixed point in R
A
for ?<0 is stable?
that is, assume |?
A
| < 1 and ?(1+?
A
) <?
A
< (1+?
A
). Suppose also that a border collision
bifurcation occurs as ? is increased through zero. A simultaneous control that renders the
BCB to be from a locally unique stable fixed point to a locally unique stable fixed point exists
if there is a (?
1
,?
2
)and0 <epsilon1<1 such that the following inequalities are satisfied:
(b
1
a
4
? a
2
b
2
)?
1
< ?(a
11
b
2
? a
13
b
1
)?
2
? ?
A
+ epsilon1 (9)
(b
1
a
4
? a
2
b
2
)?
1
> ?(a
11
b
2
? a
13
b
1
)?
2
? ?
A
(10)
b
1
?
1
> ?b
2
?
2
+2
?
epsilon1 ? ?
A
(11)
(b
1
a
4
? a
2
b
2
? b
1
)?
1
> ?(?b
2
+ a
11
b
2
? a
13
b
1
)?
2
? (1 ? ?
A
+ ?
A
) (12)
and
(b
1
a
4
? a
2
b
2
)?
1
< ?(a
21
b
2
? a
23
b
1
)?
2
? ?
B
(13)
7
(b
1
a
4
? a
2
b
2
)?
1
> ?(a
21
b
2
? a
23
b
1
)?
2
? ?
B
? 1 (14)
b
1
?
1
> ?b
2
?
2
? ?
B
(15)
(b
1
a
4
? a
2
b
2
? b
1
)?
1
> ?(?b
2
+ a
21
b
2
? a
23
b
1
)?
2
? (1 ? ?
B
+ ?
B
) (16)
Any (?
1
,?
2
) satisfying these inequalities is stabilizing.
The assertion of Prop. 4 follows by choosing the control gains so that the eigenvalues of
the closed-loop system satisfy Prop. 1 (iii).
Let g := (?
1
?
2
). The following proposition gives a su?cient condition for stabilization
of border collision bifurcation in terms of linear matrix inequalities (LMIs) [Hassouneh,
2003; Hassouneh & Abed, 2003]. If the LMIs are feasible, a stabilizing control gain can be
calculated using any LMI solver such as the LMI toolbox in MATLAB.
Proposition 5 If there exist a P = P
T
> 0 and a feedback gain (row) vector g such that
P ? (A + bg)
T
P(A + bg) > 0 (17)
P ? (B + bg)
T
P(B + bg) > 0, (18)
then any border collision bifurcation that occurs in the open-loop system (u ? 0)of(7)can
be eliminated using simultaneous feedback (8). Equivalently, if there exist a Q and y such
that
parenleftbigg
QAQ+ by
(AQ + by)
T
Q
parenrightbigg
> 0, (19)
parenleftbigg
QBQ+ by
(BQ + by)
T
Q
parenrightbigg
> 0, (20)
then any border collision bifurcation that occurs in (7) can be eliminated using simultaneous
feedback (8). Here Q = P
?1
and the feedback gain is given by g = yP.
3.2 Simultaneous washout filter-aided feedback
System (7) augmented with a washout filter-aided simultaneous feedback control is given
by the closed-loop system
parenleftbigg
x
k+1
y
k+1
parenrightbigg
=
?
?
?
?
?
?
?
A
parenleftbigg
x
k
y
k
parenrightbigg
+ b? + bu
k
,x
k
? 0
B
parenleftbigg
x
k
y
k
parenrightbigg
+ b? + bu
k
,x
k
> 0
(21)
w
k+1
= x
k
+(1? d)w
k
(22)
z
k
= x
k
? dw
k
(23)
u
k
= ?
1
z
k
(24)
8
Here, w
k
is the state of the washout filter, z
k
is the output of the washout filter and d is the
washout filter constant (d ? (0, 2) for a stable washout filter). Note that only one washout
filter was used in this feedback?this will be found in the next section to be su?cient for
application to the alternan model. The closed-loop system can be rewritten as
?
?
x
k+1
y
k+1
w
k+1
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
A
?
?
x
k
y
k
w
k
?
?
+
?
b?, x
k
? 0
?
B
?
?
x
k
y
k
w
k
?
?
+
?
b?, x
k
> 0
(25)
where
?
A =
?
?
a
11
+ b
1
?
1
a
2
?b
1
d?
1
a
13
+ b
2
?
1
a
4
?b
2
d?
1
101? d
?
?
, (26)
?
B =
?
?
a
21
+ b
1
?
1
a
2
?b
1
d?
1
a
23
+ b
2
?
1
a
4
?b
2
d?
1
101? d
?
?
, (27)
and
?
b =
parenleftbigg
b
0
parenrightbigg
. (28)
The following proposition gives a su?cient condition for the existence of a stabilizing
washout filter-aided feedback.
Proposition 6 If there exist P = P
T
> 0, ?
1
? R and d ? (0, 2) such that
?
A
T
P
?
A ? P<0 (29)
?
B
T
P
?
B ? P<0 (30)
then any border collision bifurcation that occurs in the open-loop system (21) (with u ? 0)
can be eliminated by simultaneous washout filter-aided feedback.
Note that the matrix inequalities (29)-(30) used in Prop. 6 are equivalent, respectively,
to
parenleftbigg
P
?
A
T
P
P
?
AP
parenrightbigg
> 0, (31)
parenleftbigg
P
?
B
T
P
P
?
BP
parenrightbigg
> 0. (32)
Next, the results on border collision bifurcation and its control are applied to the cardiac
conduction model that undergoes a period doubling border collision bifurcation.
9
4 The Cardiac Conduction Model
In this section, we consider the cardiac conduction model of [Sun et al., 1995]. The model
incorporates physiological concepts of recovery, facilitation and fatigue. It is formulated as a
two-dimensional PWS map. Two factors determine the atrioventricular (AV) nodal conduc-
tion time: the time interval from the atrial activation to the activation of the Bundle of His
and the history of activation of the node. The model predicts a variety of experimentally
observed complex rhythms of nodal conduction. In particular, alternans, in which there is
an alternation in conduction time from beat to beat, were associated with period-doubling
bifurcation in the theoretical model.
The authors of [Sun et al., 1995] first define the atrial His interval, A,tobethetime
interval between cardiac impulse excitation of the lower interatrial septum and activation
of the Bundle of His. (See [Sun et al., 1995] for definitions.) The model is
parenleftbigg
A
k+1
R
k+1
parenrightbigg
= f(A
k
,R
k
,H
k
) (33)
where
f(A
k
,R
k
,H
k
)=
?
?
?
?
?
?
?
?
?
?
?
parenleftBigg
A
min
+ R
k+1
+ (201? 0.7A
k
)e
?
H
k
?
rec
R
k
e
?
A
k
+H
k
?
fat
+ ?e
?
H
k
?
fat
parenrightBigg
, for A
k
? 130
parenleftBigg
A
min
+ R
k+1
+ (500? 3.0A
k
)e
?
H
k
?
rec
R
k
e
?
A
k
+H
k
?
fat
+ ?e
?
H
k
?
fat
parenrightBigg
, for A
k
> 130
with R
0
= ?exp(?H
0
/?
fat
). Here H
0
is the initial H interval and the parameters A
min
,
?
fat
, ? and ?
rec
are positive constants. The variable H
k
represents the time interval between
bundle of His activation and the subsequent activation (the AV nodal recovery time) and is
usually taken as the bifurcation parameter. The variable R
k
represents a drift in the nodal
conduction time, and is sometimes taken to be constant. In this work, we consider R
k
as a
variable as in [Sun et al., 1995]. Note that the map f is piecewise smooth and is continuous
at the border A
b
:= 130ms.
4.1 Analysis of the border collision bifurcation
Numerical simulations indicate that the map (33) undergoes (some type of) supercritical
period doubling bifurcation as the bifurcation parameter S := H
k
is decreased through a
critical value (see Fig. 1). We show that this bifurcation is in fact a supercritical period
doubling BCB which occurs when the fixed point of the map hits the border A
b
= 130.
Let the fixed points of the map (33) be given by (A
?
?
(S),R
?
?
(S)) for A
k
<A
b
and
(A
?
+
(S),R
?
+
(S)) for A
k
>A
b
. Under normal conditions, the fixed point (A
?
?
(S),R
?
?
(S)) is
stable and it loses stability as S is decreased through a critical value S = S
b
where A
?
?
= A
b
.
Then, at criticality, the value of R
?
?
is denoted by R
b
.
10
Next, we calculate the limiting Jacobians on both sides of the border:
A=
?
?
?0.7e
?S
b
?
rec
?
R
b
?
fat
e
?(130+S
b
)
?
fat
e
?(130+S
b
)
?
fat
?
R
b
?
fat
e
?(130+S
b
)
?
fat
e
?(130+S
b
)
?
fat
?
?
(34)
and
B =
?
?
?3.0e
?S
b
?
rec
?
R
b
?
fat
e
?(130+S
b
)
?
fat
e
?(130+S
b
)
?
fat
?
R
b
?
fat
e
?(130+S
b
)
?
fat
e
?(130+S
b
)
?
fat
?
?
(35)
Also, the derivative of f with respect to S at (A
b
,R
b
,S
b
)is
parenleftbigg
b
1
b
2
parenrightbigg
=
?
?
?
110
?
rec
e
?S
b
?
rec
?
?
?
fat
e
?S
b
?
fat
?
R
b
?
fat
e
?(130+S
b
)
?
fat
?
?
?
fat
e
?S
b
?
fat
?
R
b
?
fat
e
?(130+S
b
)
?
fat
?
?
(36)
Next, the following parameter values are assumed (borrowed from [Sun et al., 1995]):
?
rec
= 70ms, ?
fat
= 30000ms, A
min
= 33ms, ? =0.3ms. For these parameter values,
S
b
=56.9078ms, R
b
=48.2108ms,
A=
parenleftbigg
?0.3121 0.9938
?0.0016 0.9938
parenrightbigg
,B=
parenleftbigg
?1.3322 0.9938
?0.0016 0.9938
parenrightbigg
and
parenleftbigg
b
1
b
2
parenrightbigg
=
parenleftbigg
?0.69861
?0.00161
parenrightbigg
.
The eigenvalues of A are ?
A1
= ?0.3109, ?
A2
=0.9926 (?
A
=0.6817, ?
A
= ?0.3086) and
those of B are ?
B1
= ?1.3315, ?
B2
=0.9931 (?
B
= ?0.3384 and ?
B
= ?1.3224). Note that
there is a discontinuous jump in the eigenvalues of the Jacobian matrix when the fixed point
hits the border at the critical parameter values S = S
b
. The occurrence of a period-doubling
border collision bifurcation at S
b
is now ascertained by applying Prop 2. The stability of the
period-2 orbit with one point in R
A
and the other in R
B
is determined by the eigenvalues
of AB. These eigenvalues are ?
AB
1
=0.4135 and ?
AB
2
=0.9867. This implies that a stable
period-2 orbit is born after the border collision. The supercritical period doubling BCB is
shown in the bifurcation diagram in Fig. 1. In the figure, the bifurcated solution departs in
a nonsmooth way from the nominal fixed point branch.
4.2 Feedback control of the period doubling border collision bi-
furcation
In past studies of control of the cardiac conduction model considered here, the control is
usually applied as a perturbation to the bifurcation parameter (the nodal recovery time)
S [Christini & Collins, 1996; Chen et al., 1998]. The state A
k
has been used in the feedback
loop by other researchers who developed control laws for this model (e.g., [Hall et al., 1997;
11
40 45 50 55 57 60 65
40
60
80
100
120
130
150
170
S
R                               A
Figure 1: Joint bifurcation diagram for A
k
and for R
k
for (33) with S := H
k
as bifurcation
parameter and ?
rec
= 70ms, ?
fat
= 30000ms, A
min
= 33ms and ? =0.3ms.
12
0 500 1000 1500 2000 2500 3000
0
20
40
60
80
100
120
130
140
160
Beat number k
R
k
                                     A
k
Figure 2: Iterations of map (33) showing the alternation in A
k
as a result of a supercritical
period doubling BCB (?
rec
= 70ms, ?
fat
= 30000ms, A
min
= 33ms, ? =0.3ms and S =
45ms).
13
Brandt et al., 1997; Chen et al., 1998]). We use the same measured signal in our feedback
design. Below, the control methods of Sec. 3 are used to quench the period doubling border
collision bifurcation, replacing the period doubled orbit by a stable fixed point. First,
simultaneous static feedback control is considered followed by simultaneous washout filter-
aided feedback control.
4.2.1 Simultaneous static feedback control
Next, a feedback control u
k
= ?
1
(A
k
? A
b
)+?
2
(R
k
? R
b
) is applied as a perturbation to S
on both sides of the border.
The limiting Jacobians of the controlled system to the left and right of the border are
given by
?
A=
parenleftbigg
?0.3121 0.9938
?0.0016 0.9938
parenrightbigg
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
A
+
parenleftbigg
?0.6986
?0.00161
parenrightbigg
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
b
parenleftbig
?
1
?
2
parenrightbig
and
?
B=
parenleftbigg
?1.3322 0.9938
?0.0016 0.9938
parenrightbigg
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
B
+
parenleftbigg
?0.6986
?0.00161
parenrightbigg
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
b
parenleftbig
?
1
?
2
parenrightbig
respectively. Using the results of Sec. 3.1, stabilizing control gains (?
1
,?
2
) are obtained by
solving (9)-(16). Figure 3 (a) shows all stabilizing gains (?
1
,?
2
) that satisfy (9)-(16), and
Fig. 3 (b) shows the bifurcation diagram of the controlled system with (?
1
,?
2
)=(?1, 0).
Figure 4 (a) shows the e?ectiveness of the control in quenching the period-2 orbit and
simultaneously stabilizing the unstable fixed point. The robustness of the control law with
respect to noise is demonstrated in Fig. 4 (b).
4.2.2 Washout filter-aided feedback
In the previous section, control of period doubling border collision bifurcation using static
feedback was considered. Static state feedback changes the operating conditions (fixed
points) of the open-loop system. This results in wasted control e?ort and may also result
in degrading system performance. Washout filter-aided linear feedback, on the other hand,
does not change the value of the fixed points of the open-loop system since the control
vanishes by nature at steady state. Adding a washout filter in the feedback loop provides
automatic tracking of the fixed point to be stabilized even in the presence of model uncer-
tainty or small parameter drift. This is valuable in applications where the parameters may
vary, which is particularly useful for the cardiac arrhythmia model considered in this paper.
Consider the cardiac model with simultaneous washout filter-aided feedback
parenleftbigg
A
k+1
R
k+1
parenrightbigg
=
?
?
?
?
?
?
?
parenleftbigg
A
min
+ R
k+1
+ (201 ? 0.7A
k
)e
?(S+u
k
)/?
rec
R
k
e
?(A
k
+(S+u
k
))/?
fat
+ ?e
?(S+u
k
)/?
fat
parenrightbigg
, for A
k
? 130
parenleftbigg
A
min
+ R
k+1
+ (500 ? 3.0A
k
)e
?(S+u
k
)/?
rec
R
k
e
?(A
k
+(S+u
k
))/?
fat
+ ?e
?(S+u
k
)/?
fat
parenrightbigg
, for A
k
> 130
(37)
14
?6 ?4 ?2 0 2 4 6
?2.5
?2
?1.5
?1
?0.5
0
?
2
?
1
45 50 55 60 65
110
115
120
125
130
135
140
S
A
(a) (b)
Figure 3: (a) Stabilizing simultaneous control gain pairs satisfying (9)-(16) are within
the shaded region in the figure; (b) Bifurcation diagram of the controlled system using
simultaneous state feedback with control gains (?
1
,?
2
)=(?1, 0).
0 500 1000 1500 2000 2500 3000
95
100
105
110
115
120
125
130
135
140
145
Beat number k
A
k
off on off off on on 
0 500 1000 1500 2000 2500 3000
95
100
105
110
115
120
125
130
135
140
145
Beat number k
A
k
off on off on off 
on 
(a) (b)
Figure 4: Time series resulting from map (33) with simultaneous linear state feedback
control applied at beat number n = 500. The control is switched o? and on every 500 beats
to show the e?ectiveness of the controller (S = 48ms and (?
1
,?
2
)=(?1, 0)). (a) without
noise; (b) with zero mean, ? =0.5ms white Gaussian noise added to S.
15
w
k+1
= A
k
+(1? d)w
k
(38)
z
k
= A
k
? dw
k
(39)
u
k
= ?
1
z
k
(40)
The limiting Jacobian matrices of the controlled system to the left and right of the border
are given by
?
A =
?
?
?0.31208? 0.69860?
1
0.99379 0.69860?
1
d
?0.001597? 0.001607?
1
0.99379 0.001607?
1
d
10? d
?
?
and
?
B =
?
?
?1.33223? 0.69860?
1
0.99379 0.69860?
1
d
?0.001597? 0.001607?
1
0.99379 0.001607?
1
d
10? d
?
?
respectively. Note that only one washout filter was used in the feedback loop. In general,
the number of washout filters needed can be any number between 1 and the dimension of
the system. In some cases, such as the cardiac model considered here, a single washout filter
su?ces.
Stabilizing washout filter-aided feedback parameters are obtained using Proposition 6.
Figure 5 shows the region of stabilizing control parameters ?
1
, d, which was obtained using
the LMI toolbox in Matlab.
Next, simultaneous static feedback and simultaneous washout filter-aided feedback are
compared. Figure 6 shows the bifurcation diagram of the closed-loop system for both
simultaneous static feedback and simultaneous washout filter-aided feedback. Note that
the (stabilized) fixed point of the closed-loop system using washout filter-aided feedback
coincides with the open-loop (unstable) fixed point. However, the (stabilized) fixed point
of the closed-loop system using static state feedback di?ers from the open-loop (unstable)
fixed point. This is also evident from Fig. 7 and Fig. 8 which show that the control e?ort
becomes zero in steady state when a washout filter is employed, whereas when static state
feedback is used, the control e?ort approaches a constant value di?erent from zero.
References
Abed, E.H , Wang, H.O. & Chen, R.C. [1994], ?Stabilization of period doubling bifurcations
and implications for control of chaos,? Physica D, 70, 154-164.
Banerjee, S. & Grebogi, C. [1999], ?Border collision bifurcations in two-dimensional piece-
wise smooth maps,? Physical Review E, 59(4), 4052-4061.
Bernardo, M. di, Feigin, M.I., Hogan, S.J. & Homer, M.E. [1999], ?Local analysis of C-
bifurcations in n-dimensional piecewise smooth dynamical systems,? Chaos, Solitons
and Fractals, 10(11), 1881-1908.
16
0 0.3 0.6 0.9 1.2
?1.8
?1.5
?1.2
?0.9
?0.6
?0.3
d
?
1
Figure 5: Stabilizing simultaneous washout filter-aided feedback control parameters are
within the shaded region.
17
45 50 55 60 65
110
120
130
140
150
S
A
Washout filter?aided 
feedback             
Static feedback 
Without control
Figure 6: Bifurcation diagram of closed-loop system, comparing static feedback control
(?
1
= ?1, ?
2
= 0) with washout filter-aided feedback control (?
1
= ?1, d =0.1). The (red)
dotted lines represent the open-loop bifurcation diagram.
18
0 500 1000 1500 2000 2500 3000
80
100
120
140
A
k
0 500 1000 1500 2000 2500 3000
?10
?5
0
5
10
Beat number k
u
k
(a) 
(b) 
Figure 7: Time series of controlled alternan model with static state feedback applied at beat
number 500 (?
1
= ?1, ?
2
=0andS = 48), (a) Conduction time A
k
, (b) Control input u
k
.
19
0 500 1000 1500 2000 2500 3000
80
100
120
140
A
k
0 500 1000 1500 2000 2500 3000
?5
0
5
10
Beat number k
u
k
(a) 
(b) 
Figure 8: Time series of controlled alternan model with washout filter-aided feedback applied
at beat number 500 (?
1
= ?1, d =0.1andS = 48), (a) Conduction time A
k
, (b) Control
input u
k
.
20
Brandt, M.E., Shih, H.T. & Chen, G. [1997], ?Linear time-delay feedback control of a
pathological rhythm in a cardiac conduction model,? Physical Review E, 56(2), R1334?
R1337.
Chay, T.R. [1995], ?Bifurcations in heart rhythms,? Internat. J. Bifurcation and Chaos,
5(6), 1439-1486.
Chen, D., Wang, H.O. & Chin, W. [1998], ?Suppressing cardiac alternans: Analysis and
control of a border-collision bifurcation in a cardiac conduction model,? Proc. IEEE
Internat. Symp. Circuits and Systems, 3, 635-638.
Christini, D.J. & Collins, J.J. [1996], ?Using chaos control and tracking to suppress a
pathological nonchaotic rhythms in cardiac model,? Physical Review E, 53(1), R49?
R51.
Christini, D.J., Stein, K.M., Markowitz, S.M., Mittal, S., Slotwiner, D.J., Scheiner, M.A.,
Iwai, S. & Lerman, B.B. [2001], ?Nonlinear-dynamical arrhythmia control in humans,?
P NATL ACAD SCI USA, 98(10), 5827-5832.
Hall, K., Christini, D.J., Tremblay, M., Collins, J.J., Glass, L. & Billette, J. [1997], ?Dy-
namic control of cardiac alternans,? Phys. Rev. Lett., 78(23), 4518-4521.
Hall, K. & Christini, D.J. [2001], ?Restricted feedback control of one-dimensional maps?,
Phys. Rev. E., 63(4), 1-9.
Hassouneh M.A. [2003], ?Feedback Control of Border Collision Bifurcations in Piecewise
Smooth Discrete-Time Systems,? Ph.D. Dissertation, University of Maryland, College
Park.
Hassouneh, M.A. & Abed, E.H. [2003], ?Lyapunov and LMI analysis and control of border
collision bifurcations,? in preparation.
Nusse, H.E. & Yorke, J.A. [1992], ?Border-collision bifurcations including period two to
period three for piecewise smooth systems,? Physica D, 57, 39-57.
Nusse, H.E., Ott, E. & Yorke, J.A. [1994], ?Border-collision bifurcations: An explanation
for observed bifurcation phenomena,? Phys. Rev. E, 49, 1073-1076.
Ott, E., Grebogi, C. & Yorke, J.A. [1990], ?Controlling chaos,? Phys. Rev. Lett., 64, 1196-
1199.
Sun, J., Amellal, F., Glass, L. & Billette, J. [1995], ?Alternans and period-doubling bifur-
cations in atrioventricular nodal conduction,? J. Theor. Biology, 173, 79-91.
21