1. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 491
Dimensionless Approach to Multi-Parametric
Stability Analysis of Nonlinear Time-Periodic
Systems: Theory and Its Applications to
Switching Converters
Hao Zhang, Member, IEEE, Yuan Zhang, Student Member, IEEE, and Xikui Ma
Abstract—This paper proposes a dimensionless approach to
analyze the multi-parametric stability behavior of switching con-
verters, which can be characterized by a nonlinear time-periodic
(NTP) system. The main objective is to analyze how multiple
circuit parameters affect the stability patterns of the derived
NTP system and to simplify the parametric complexity of such
NTP system. In contrast to previous work, the proposed method
focuses on the parametric resultant relationships of the NTP
system in the sense of topological equivalence, and investigates
its stability in terms of the homeomorphic NTP system. Firstly,
an equivalent stability theory of NTP systems is proposed. Then,
based on the equivalent theory, a normalized map is introduced
and various interesting properties are derived so as to formulate
the dimensionless approach. Moreover, the approximate solution
of the NTP system in dimensionless parameter space is calculated
by using the Galerkin method, and its stability pattern is identified
with the help of eigenvalue analysis approach. Finally, a case
study of one-cycle controlled Zeta PFC converter is discussed
in detail to exemplify the application of the proposed method.
These analytical results agree well with those ones obtained from
experimental measurements.
Index Terms—dimensionless approach, equivalent stability,
multi-parametric stability analysis, nonlinear time-periodic
system, parametric resultant relationship, switching converter.
I. INTRODUCTION
AS A CLASS of typical piecewise smooth systems,
switching converters can exhibit a great variety of non-
linear behaviors such as period-doubling bifurcation, Hopf
bifurcation and chaos [1]–[17]. Due to the improperly de-
signed values of some practically relevant circuit parameters,
distortion behaviors can frequently occur, which will lead to
the rapid rise of device stress and the drastic degeneration of
system performance. Hence, the stability issue associated with
behavior boundaries in terms of some major circuit parameters
Manuscript received November 08, 2011; revised February 28, 2012; ac-
cepted March 18, 2012. Date of publication October 09, 2012; date of current
version January 24, 2013. This work was supported in part by the National Nat-
ural Science Foundation of China (Grant No. 51177118, 50607015), in part
by the Fundamental Research Funds for the Central Universities (Grant No.
XJJ20100058), and in part by the Creative Foundation of the State Key Lab of
Electrical Insulation & Power Equipment, China (Grant No. EIPE11301). This
paper was recommended by Associate Editor X. Li.
The authors are with the State Key Laboratory of Electrical Insulation and
Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University,
Xi’an 710049, China (e-mail: haozhang@xjtu.edu.cn).
Digital Object Identifier 10.1109/TCSI.2012.2215798
is very crucial to the dynamics knowledge of switching con-
verters, and has become an important goal and subject of much
on-going research [4]–[8], [11]–[13], [16], [17]. However, with
the ever-increasing energy conversion requirement of power
electronic systems, switching converters are developing toward
the trend of higher dimension and more complex structure,
such as two-stage or even multistage cascade (or parallel) con-
verter configuration [7], [8], [16]–[21]. Despite their gaining
increased acceptance in many applications of telecommunica-
tions, aeronautics, and new IC technologies [22]–[24], these
switching converter architectures have inevitably rise to the
great challenge of stability analysis.
Up to now, several typical approaches such as bifurcation
analysis, large- or small-signal analysis and Lyapunov func-
tion method have been proposed to identify stability patterns
of switching converters [11]–[17], [25]–[28]. Although these
previous results have greatly improved the understanding of
the system dynamical behaviors, they merely focused on single
parametric stability boundaries, which didn’t consider resultant
relationships among system parameters at all. For example, re-
sistor , capacitor and line cycle were usually regarded as
three independent parameters no matter how separately or to-
gether they were discussed in previous approaches. In fact, there
physically exist such resultant relationships among the three pa-
rameters as . Thus, if the resultant relationship isn’t un-
covered, these approaches will become too complex to enhance
the in-depth understanding of the systems dynamics, which re-
sults in the fact that some useful information essential to design-
oriented optimization is of great difficulty to obtain. Evidently,
these parametric resultant relationships are quite indispensable
to stability analysis. As far as switching converters of great com-
plexity and high dimension are concerned, the system stability
analysis will become more complicate, which is certainly asso-
ciated with these resultant relationships among multiple circuit
parameters. In this sense, the actual concept of multi-parametric
stability analysis should be based on the parametric resultant re-
lationships. Now we may ask the following question: Does there
exist an appropriate approach to characterize the parametric re-
sultant relationships? In the previous literatures on nonlinear
analysis of dc/dc converters [7], [14], a dimensionless technique
has slightly involved the related question. However, these di-
mensionless treatments failed to give a general non-dimension-
alized rule, and never established the parametric resultant rela-
tionships in the sense of the system stability. This reason is that
there exists a major drawback in such technique, which is the
1549-8328/$31.00 © 2012 IEEE

2. 492 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
lack of a rigorous mathematical theory to formulate the above
resultant relationships. Of course, this technique is not appli-
cable to multi-parametric stability analysis at all.
In this paper, a general dimensionless approach (DA) is sys-
tematically proposed to establish a theoretical basis for para-
metric resultant relationships, and illustrate its applications to
multi-parametric stability analysis of switching converters. The
proposed method begins with a nonlinear time-periodic (NTP)
system that can be regarded as a general model of practical
switching converters. Unlike the previous studies, it presents
a rigorous mathematical theory for switching converters in the
sense of topological equivalence, and investigates the stability
of a derived NTP system by means of its corresponding dimen-
sionless homeomorphic NTP system. For readability and effec-
tiveness of exposition, we illustrate the method and its salient
features using a typical Zeta PFC converter as an example. The
analysis is of help to us in doping out the resultant relationships
in such NTP system on earth, and more significantly, some an-
alytical results obtained here can potentially facilitate design of
switching converters for stable operation.
The paper is organized as follows. Section II gives detailed
definition of a nonlinear time-periodic system for practical
switching converters. Section III establishes the framework on
the equivalent stability theory of homeomorphic NTP systems.
In Section IV, based on the proposed equivalent stability theory,
a systematic representation of the DA for multi-parametric sta-
bility analysis is discussed in terms of dimensionless parameter
sets. In Section V, stability patterns of the Zeta PFC converter
are analyzed as an example to exemplify the proposed method.
In Section VI, experimental results and power unbalance anal-
ysis are presented for verification purposes. In Section VII,
some interesting behavior boundaries of the converter are
illustrated. Finally, some remarkable conclusions are arrived
at in Section VIII. The paper also contains three Appendixes,
which address the proofs of the equivalent stability theorem
(A), and provide details of coefficient matrices (B) as well as
constraint equations (C) for the Zeta PFC converter.
II. DEFINITION OF NTP SYSTEMS FOR SWITCHING
CONVERTERS
If the piece-wise smooth feature of switching converters is
replaced by their overall smooth trait after a standard averaging
approach [2], then their low-frequency dynamical behaviors can
be described by the following differential equation:
(1)
where is a nonlinear continuous smooth function defined
in with an initial condition
and represent the time interval and the variable domain, re-
spectively.
Moreover, if the right-hand side function of (1) satisfies
for , where is the least period, then
we define such system as a NTP system.
It is worth noting that these converter systems are modeled by
the low-frequency model as shown in (1), which is, of course,
not applicable to the fast-scale dynamics at the time scale of
switching frequency. In this paper, we focus on low-frequency
dynamics of the switching converters with periodic force inputs
or outputs.
Note that the solution in (1) is usually a non-zero one.
Thus, for the convenience of later discussion, we introduce the
following transformation:
(2)
where is the perturbed solution of . Substituting (2)
into (1), becomes the equilibrium point for the following
system:
(3)
Since
(4)
(5)
Subtracting each side of (5) from the corresponding side of
(4), one obtains
(6)
combining (3) with (6), we get the transformed system as
follows:
(7)
where for . Hence, (7) is also
a NTP system, which holds the same period of (1). For brevity,
denotes the perturbed solution of (7) with an initial
condition .
the stability properties of such as (uniform) stability and
(uniform) asymptotical stability are fully equivalent to those of
the equilibrium point , which can be accurately expressed
by the Lyapunov definitions [29]. Moreover, the following the-
orem on the relation between the local stability of a NTP system
and its corresponding linearized system is relevant to our sub-
sequent study. For conciseness, we refer the readers to [30] for
a detailed proof.
Theorem 2.1: Assume that is sufficiently smooth (i.e., at
least ) for the NTP system (1), and
• let be its transformed zero solution system
defined in (7);
• let be the Jacobian matrix of
with respect to , evaluated at the solution .
Suppose that there exists a common period between
and , and then for the linear equation with the following
periodic coefficients:
(8)
if is an asymptotically stable equilibrium point of (8), the
solution of (1) is asymptotically stable.
Remark 2.1: The least period of and are denoted
by and , respectively. As indicated in [31], don’t need
to be the same as . However, the two least periods should
be commensurable, namely, satisfy , where
and are the least positive integers if available. Thereby, we

3. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 493
define as the common period for and .
In particular, if is equal to , one obtains .
III. THEOREMS ON EQUIVALENT STABILITY OF NTP SYSTEMS
A. Time Reparameterization
In general, the stability of the NTP system (1) depends on its
structural parameters. For the sake of clarity, the NTP system
(1) is rewritten as the following parameter-dependent system:
(9)
where is a time-invariant circuit parameter vector,
is a parameter set defined in and is of period
defined in . The unperturbed solution of (9) is periodic,
satisfying for .
As is mentioned above, there exists a common period for
the NTP system , which is generally related to the external
forcing period of the system. However, such periodic feature
fails to be explicitly expressed in (9), and accordingly the con-
nection between the forcing period and the natural periods1 of
the system cannot be easily obtained. Note that the stability is
a systematic property with time approaching infinity. Thus, for
the NTP system , the information relevant to transient and
time- dependent behaviors can be omitted, while only the in-
formation on the stability of cycles needs to be preserved. it is
necessary to introduce a time reparameterization process for the
NTP system [32]. Suppose that is a smooth positive
number with respect to and there exists an invertible map
(10)
where and . Then, is transformed into
the following time-parameterized system:
(11)
where denotes the derivative of with respect to . Obvi-
ously, the periods of and satisfies
and .
B. Theorems on Equivalent Stability
In this subsection, we consider the stability properties of NTP
system in the sense of topological equivalence. Note that
homeomorphic spaces possess the same topological property
and there is certainly no intrinsic difference among homeomor-
phic spaces [32], [33]. However, in terms of generality and con-
venience, there can be a significant difference. By using a proper
homeomorphism, a physical problem of high parametric com-
plexity may be homeomorphic to an easier and more general
one with reduced parametric complexity so that its topological
properties are more easily determined.
There are, however, two basic questions to be appropriately
answered. One is whether the stability of a NTP system is
equivalent to that of its homeomorphic NTP system; the other
is whether there exists a proper homeomorphism retaining the
equivalent stability. Based on the two questions and the above
1Roughly speaking, a natural period is the intrinsic time scale of a given
system related to its own physical parameters, describing the time process for a
studied phenomenon.
time reparametrization, we propose the following two theo-
rems on the equivalent stability of NTP systems as sufficient
conditions. Detailed proofs are given in Appendix A.
Theorem 3.1: Consider the NTP system . Suppose that
is stable in the neighborhood of its cycle when . If
there exist two invertible maps and satisfying that
• possesses a smooth positive function such that
where ;
• is an invertible linear map as such
that where is the corresponding matrix.
and are the two domains defined in .
Let be the product map of and with the form of
(12)
Then, the following three statements hold.
1) The mapped system is a NTP system
with a parameter vector , and is the vector function from
to .
2) For the NTP systems and , their cycles are stable iff
they are uniformly stable.
3) The (uniformly) stability of in the neighborhood of
is equivalent to the (uniformly) stability of original system
in the neighborhood of .
Note that Theorem 3.1 deals with the Lyapunov stability of
. However, for practical applications, it is always desirable
that the operating point of switching converters would even-
tually converge to the original steady state once the perturba-
tion factors vanish, i.e., the switching converters are asymptoti-
cally stable. the following asymptotic equivalent stability theory
should be proposed.
Theorem 3.2: Consider the NTP system . Suppose that
is asymptotically stable in the neighborhood of its cycle
when . If the product map is of the same definition
shown in Theorem 3.1, then besides the first statement 1) pro-
posed in Theorem 3.1, the following two statements hold.
1) For the NTP systems and , their cycles are asymptot-
ically stable iff they are uniformly asymptotically stable.
2) The (uniformly) asymptotic stability of in the neighbor-
hood of is equivalent to the (uniformly) asymptotic
stability of in the neighborhood of .
Here Theorem 3.1 and Theorem 3.2 are referred as the equiv-
alent stability theorem for the NTP systems. It should be high-
lighted that Theorems 3.1 and 3.2 give the sufficient conditions
for the equivalent stability properties of NTP systems. Namely,
the product map can be selected as a combined invertible map
with and . According to the proofs, a significant feature of
is the boundedness of its submap . However, it is not true
for every homeomorphism in general2. the first question pro-
posed earlier in this subsection will not be fully established. A
necessary and sufficient condition may possibly exist for certain
bounded operators, which is not the focus of the interest herein.
Note that the stability behavior of depends on the param-
eters assigned in . By using the proper map , a param-
eter-dependent map can be induced between the original pa-
rameter set and the mapped parameter set
i.e., (where and can be equal or not),
2For instance, the tangent homeomorphism belongs to an unbounded operator
defined in a bounded region due to its unbounded image with
.

4. 494 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
which maps the cycles of at onto the cycles of at
, preserving the direction of time. Moreover, if
the dimension between and satisfies , then the
dimension of original parameter space of will be reduced,
which may simplify the parametric complexity of the original
NTP system. As will be shown shortly, the main function of the
proposed theorems lie in this.
C. Parametric Dimensionality Reduction
In the following subsection, some discussions on how to
choose the invertible matrix in the map are provided in
order to guarantee a simplified parameter space with reduced
dimension. For clarity, the parameter vectors and are written
as follows:
(13)
(14)
where and .
Since the coefficients in (9) contains the elements and
, a feasible way to reduce the number of the elements in
is to represent by using certain combinations of and
such as their sums or products, i.e.,
or . Moreover, according to the Buckingham
theorem [35], if the dynamical behavior of the NTP system
depends upon dimensional variables3, then by taking
a suitable transformation to remove their units in order to yield
some dimensionless variables, the system can be reduced to only
dimensionless variables, where the reduction number
depends upon the problem complexity
that equals to the number of its fundamental dimensions. Note
that and are the dimensional circuit parameters of the
switching system, which are embodied as coefficients in (9).
Thus, a compact form of can be selected as an invertible diag-
onal matrix, which transforms the dimensional state vector
into the dimensionless state vector
as follows:
(15)
where and the invertible matrix is written as
(16)
Consequently, the key point of selecting is to choose a
proper which should ensure that the circuit state variable
can be transformed into the dimensionless state variable ,
and is usually a combination of and as aforemen-
tioned.
IV. THE DA FOR MULTI-PARAMETRIC STABILITY ANALYSIS
A. Representation of the DA
As discussed in Section III, by removing units from the NTP
system with physical quantities to yield its dimensionless vari-
ables, one can get an equivalent NTP system of reduced param-
eter dimension. an apparent purpose of the DA is to reduce the
3The variables here are extensional notions, which include the system’s state
variables and the circuit parameters. For the NTP system (9) studied in this
paper, they consist of the state vector and the parameter vector .
number and complexity of variables by shaping them into di-
mensionless forms. However, a more significant role lies in re-
vealing some fundamental properties of the complex system and
giving new insights into the resultant relationships among var-
ious parameters, which affect its dynamical behaviors.
A famous example to illustrate the above role is the dimen-
sionless Reynolds number in fluid dynamics [35], which
is used to characterize the fluid in a pipe being either laminar
flow or turbulent flow. Note that Reynolds number involves a
combination of various factors into a dimensionless variable
, and subsequently analyze the given physical phenom-
enon in an equivalent manner. Inspired by this dimensionless
treatment, it becomes possible to investigate the parameter-de-
pendent NTP system in its dimensionless space. This formu-
lation, together with the equivalent stability theory, brings about
the complete establishment of DA as follows.
Firstly, the time reparameterization function in the product
map should represent the characteristic period, wherein the
fundamental dynamics of the system are captured. Since we are
mainly concern with the low-frequency dynamics of switching
converters, can be selected as the reciprocal of the common
period in the system under study, i.e., .
Secondly, we choose the invertible matrix as shown in (16).
Then, by using the map , we get the dimensionless variable
in the product space . Based on Theorems 3.1 and
3.2, the original dimensional NTP system is transformed into
the following dimensionless NTP system:
(17)
where is the dimensionless parameter vector. The common
period of and equals to 1 here.
Finally, we define the product map and
the induced parameter-dependent map as the nor-
malized map and the parameter normalized map, respectively.
As highlighted earlier, the dimension of original variable space
containing state variables and circuit parameters can be reduced
now, and its reduction number equals to the number of
the fundamental dimensions of the NTP system.
B. Some Properties of
So far, we have derived the dimensionless NTP system. Note
that is an invertible square matrix and is a smooth pos-
itive number. Referring to Theorems 3.1 and 3.2, the (asymp-
totic) stability of the dimensionless system near is fully
equivalent to the stability properties of the dimensional system
near . one can perform the multi-parametric stability
analysis in the dimensionless NTP system, which not only re-
tains equivalent stability properties but helps to establish the
parametric resultant relationships on the system stability. Sev-
eral straightforward properties of the map , regarding to the
parametric resultant relationships, can be naturally held as fol-
lows and are stated without proof.
1) For such that is (asymptotically) stable, then
the dimensionless parameter vector can
yield to be (asymptotically) stable.
2) Assume that there exists a dimensionless parameter vector
such that is (asymptotically) stable. Then, it
follows that for such that would make
be (asymptotically) stable.

5. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 495
3) Considers the stability boundaries of and
of . If there exists a critical parameter vector
, then . Moreover, if there exists
, then it follows that such that
for .
Note that can establish the relationship between the dimen-
sionless stability boundary and its dimensional boundary .
Thereby, the parametric resultant relationships among all groups
of circuit parameters affecting the system stability can be visual-
ized by plotting a few behavior boundaries in dimensionless pa-
rameter space. However, boundary margins are often expressed
as 3-D surfaces or 2-D curves. For effectiveness of exposition,
we suppose the parameter vectors and can be divided into
several subvectors, such as and , where
there is no coupling relation between and . Thus, we get the
parameter normalized submap ,
where and are defined in (18)
(18)
the original parameter sets (or ) and (or ) can be
divided into the following independent subsets (or ) and
(or ), which satisfy
(19)
(20)
Hence, the above properties of the map are also suitable for
its submap . As such, the multi-parametric stability analysis
based on the parametric resultant relationships can be carried
out by means of several families of dimensionless parameters.
C. Stability of Dimensionless Periodic Solutions
In this subsection, we employ the Galerkin method to get the
approximate solution of (9), and investigate its stability patterns
via an eigenvalue analysis approach.
1) Dimensionless Periodic Solutions: Note that the state
variables of a switching converter generally consist of the
voltage across capacitors and the current through induc-
tors in its power and control stages. Thus, the state vector
of (9) can be decomposed into where
and are the power
state vector and the control state vector, respectively. Likewise,
the corresponding dimensionless state vector of (17) can
be represented as , where
and . For convenience, we construct the
following diagonal matrix as . Multiplying
by both side of (17) simultaneously, we obtain
(21)
Then, we add the first rows of (21) on each side and the
following equation is readily acquired:
(22)
where denotes the th element of and is the
corresponding th row of the . At first glance, there
are at least state variables in (22), which seems difficult to
get their periodic solutions. However, there lie often some extra
constraints among these variables based on
some fundamental circuit theories and feedback control features
of the system. Suppose that there are constraint equations
for the system (9) with state variables of the following form:
...
(23)
Putting (23) into (22), the state variables in (22) can be then
reduced to an unique and independent variable, which is denoted
by . the multivariable NTP system is simplified into a single-
variable nonlinear differential equation which only contains
as follows:
(24)
where is a differential operator. Since is a continuous peri-
odic vector of period 1, (24) can be solved in the time interval
[0, 1]. It is well-known that the solution of a given NTP system
can be formulated via the Galerkin method [36]. The key idea of
Galerkin method is based on the fact that any function in a
function space can be uniquely expressed as a linear combi-
nation of the linear independent basis function . Thus, an
approximate solution of is written as
(25)
where is the coefficient for the th linear independent basis
function. Then, the residual error satisfying the differential
equation (24) with terms of the sum (25) is defined as
(26)
According to [36], the residual error should be orthogonal to
each basis function as follows:
(27)

6. 496 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
Due to the orthogonality of the trigonometric function set, we
can simply choose as
(28)
By substituting (26) into (27), one obtains
(29)
where .
To utilize the approach, all we need to do is to solve these
algebraic equations for the coefficients in (29).
According to [36], the Galerkin approximation would
converge to the exact solution of if is sufficiently large.
For of order , it is feasible to neglect high order basis
functions beyond the th if a required accuracy is satisfied.
Here we simply denote the Galerkin approximation as
if no confusion arises. Then, the other solutions of can
be easily calculated by putting into (23). As such, the numer-
ical solution is obtained.
2) Stability of Periodic Solutions: So far, we have got the pe-
riodic solution of the NTP system (17). According to Theorem
2.1, the stability of can be investigated by its corresponding
linearization system. Suppose in (17) satisfies ,
then the asymptotic stability of can be equivalently char-
acterized by its linear-time periodic equation as follows:
(30)
where denotes the Jacobian matrix at . In general,
an eigenvalue analysis approach [37] can be one of the most
powerful tools to analyze the stability properties of (30). The
approach plays an important role in investigating the relation
between the stability patterns and the eigenvalues of the mon-
odromy matrix for (30). Since detailed derivations of a mon-
odromy matrix for a given linear periodic equation has been
proposed in our earlier work [12], we omit the explicit form of
the matrix here and only give the results of the calculated eigen-
values as follows:
(31)
where the eigenvalues obtained in (31) can provide enough in-
formation to analyze the stability of the cycle and to iden-
tify its bifurcation patterns when the parameter varied. As in-
dicated in [38], if , for all , the solu-
tion is asymptotically stable. However, if , for some
(at least one), then the associated solution is unstable. Besides,
the unstable behaviors of the system depend on the manner in
which the eigenvalues leave the unit cycle. Specifically, if a pair
of complex conjugate eigenvalues across the unit circle, then a
Neimark-Sacker bifurcation pattern occurs.
V. APPLICATION OF THE DA TO ZETA PFC CONVERTER
A. Zeta PFC Converter and Its Mathematical Model
1) One-Cycle Controlled Zeta PFC Converter: The one-
cycle controlled (OCC) Zeta PFC converter as shown in Fig. 1
contains the power stage, the output voltage controller and the
Fig. 1. One-cycle controlled Zeta PFC converter. The parameter values are
V–70 V, mH–1.8 mH, mH, F,
F, V, k
k –7 k , k , nF, kHz.
OCC PWM modulator. Since detailed circuit configuration and
operation have been illustrated in [39], [40], we only give the
following remarks on its operation. Firstly, the system is re-
stricted to CCM operation owing to the fact that almost all of
practical medium and high power supplies usually work in this
operation. Secondly, we focus on the case that once reaches
during one switching cycle, the integrator will be reset to the
normal operation. Here, we denote as ,
where and represent the rms value and the line voltage
frequency, respectively.
Note that the Zeta PFC converter (without the voltage con-
troller) can be modeled as
(32)
where equals to
and equals to . Note that the coefficient
matrix can be easily obtained from Kirchoff’s
laws, and the voltage controller can be formulated by
(33)
where and
. Here
and denote the dc gain and time constant of the
feedback network, respectively. By combining (32) and (33),
the closed-loop model of the OCC Zeta PFC converter is for-
mulated as follows:
(34)
where . The coefficient matrices
and stand for
and respec-
tively, and their detailed expressions are given in Appendix B.

7. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 497
Furthermore, since the OCC block is a built-in PWM modu-
lator, the duty cycle can be dictated as follows: [39], [40]
(35)
we get the following general form:
(36)
Substituting (36) into (34), we obtain the mathematical model
of the system as follows:
(37)
Note that the Zeta PFC converter is a parameter-dependent
NTP system as shown in (9), where its common period equals
to the periods of and , i.e., . For
brevity, we omit the average overbar for the corresponding
state variables in the following discussions.
2) Dimensionless Mathematical Model: Here we apply the
DA derived in Section IV to get the dimensionless mathematical
model of (37). First, we choose as , i.e., and
select as . Then, by
taking upon (36), we obtain
(38)
Likewise, by applying to (37), we get the following dimen-
sionless NTP system (after rearranging):
(39)
where and
. The explicit form of (39) is shown in (40),
shown at the bottom of the page, where the dimensionless
parameters are specifically expressed as
(41)
where and in (13) and (14) are
denoted as follows, respectively:
(42)
(43)
the parametric resultant relationships affecting the stability
of the Zeta PFC converter are revealed by the equations shown
in (41) after the DA, and the dimension reduction equals to the
number of the fundamental dimensions of the NTP system. In
fact, the terms of in (41) can be
regarded as the natural periods of the system, which will be
discussed later.
Although is a dimensional parameter, it is often fixed as a
constant in practical applications. Hence, we omit in and
use the simplified one in the rest of the discussions. Thus, the
parameter subvectors and defined in Section IV can be
written as
(44)
where the parameter subsets and
satisfy (19) and (20).
Moreover, the constraint equations among and
can be easily obtained as follows, and their detailed derivation
are given in Appendix III:
(45)
By combining (22), (38) with (45), we get the explicit form
of (24) as shown in
(46)
(40)

8. 498 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
Fig. 2. Comparison of the dimensionless solution waveforms by using
numerical simulation performed in MATLAB/SIMULINK (solid line) and the
Galerkin Approach of the DA model (dashed line). (a) . (b) .
Due to the limit of length, we cannot provide the detailed forms
of its partial derivation and only give the result of for
the converter as shown in (47).
(47)
B. Eigenvalue Analysis Using the DA
1) Periodic Solutions by the Galerkin Approach: Here, the
dimensionless parameters are listed as follows:
and
, which the corresponding dimensional param-
eters are V, k and mH. Here we
choose because the accuracy of the approximate solu-
tion from the Galerkin approach can be ensured adequately in
most cases. Thereby, the dimensionless solutions and are
calculated as follows
(48)
(49)
From (48) and (49), it follows that these analytical results
agree very well with those ones obtained from the numerical
simulation, as shown in Figs. 2(a) and (b).
2) Eigenvalue Loci: Now, we use the derived monodromy
matrix to investigate the possible bifurcation pattern of such
NTP system. Specifically, we investigate the movement of the
eigenvalues as the dimensionless parameters and
are varied, which correspond to the variation of and
. Table I shows the changing trend of the five eigenvalues as
and are varied, respectively. We clearly observe that
the loci of a pair of complex conjugate eigenvalues and
begin to cross the unit circle as increases to 44.3733 (i.e.,
arrives at 5.348 k when V, mH),
moves away from 0.041391 (i.e., removes from 60 V as
Fig. 3. Movement of the eigenvalues with the increase of , where Fig. 3(a)
shows the trend of the loci of eigenvalues as is increased from 34.9842 to
44.4329 and Fig. 3(b) depicts the detailed loci corresponding to Table I(A).
TABLE I
EIGENVALUES FOR DIFFERENT VALUES OF DIMENSIONLESS PARAMETERS
AND IN THE ZETA PFC CONVERTER WITH OCC (A) WITH
AND , (B) WITH
AND , (C) WITH AND
k mH) and reaches
(i.e., reaches 1.77855 mH when V, k ),
while other eigenvalues and still stay inside the unit
cycle. Thus, this implies Neimark-Sacker bifurcation occurs. To
make the movement of the eigenvalues more intuitive, we plot
their loci as is increased (see Fig. 3). Here, we omit the other
two figures for and to save space since they are similar
to Fig. 3.
VI. EXPERIMENTAL RESULTS AND POWER UNBALANCE
ANALYSIS
A. Experimental Observation
To further validate these results, experimental tests of the
OCC Zeta PFC converter were carried out. Experimental wave-
forms are presented in Fig. 4, which show that oscillatory insta-
bility occurs as the parameter values of and are smaller
or the value of is larger than one in normal operation. These
results are in good agreement with the theoretical predictions

9. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 499
Fig. 4. Experimental waveforms of the one-cycle controlled Zeta PFC con-
verter ( : 1 A/div; : 50 V/div; time: 5 ms/div). (a) Stable operation for
k V, mH. (b) Oscillatory instability for
k V, mH, which manifests as medium-fre-
quency oscillation of . (c) Stable operation for V, k
mH. (d) Oscillatory instability for mH, V,
k , which manifests as medium-frequency oscillation of .
presented in Table I. Fig. 4(a) shows the measured waveforms of
the output voltage and the rectified input current for k
when V, mH, where the system is stable for
W. Fig. 4(b) shows these measured waveforms as
decreases to 5.3 k and the system becomes unstable, where
increases to 115 W. As the input voltage is increased from
60 V to 70 V when k mH, the measured
waveforms (see Fig. 4(c)) become normal again compared to
those shown in Fig. 4(b), and increases to 125 W. However,
the oscillatory instability appears again as increases to 1.8
mH when k V (see Fig. 4(d)), and
remains near at 125 W.
B. Power Unbalance Analysis
As discussed in [20], the Zeta PFC converter belongs to the
Type I-IIA configuration, in which the power flows between and
within the power stage are cataloged into the high-frequency
power flow and the low-frequency power flow. Figs. 5(a) and
(b) show the equivalent high-frequency power flow diagrams
for the Zeta PFC converter under CCM operation. Firstly, when
the equivalent synchronized switches for converters 1 and 2 turn
on, the inductor of converter 1 absorbs energy from the input,
while transfers energy through of converter 2 into the
load with low-frequency buffering capacitor . Then, after the
switches turn off, starts to release the energy absorbed in
the previous on-time interval to , which will retransfer that
energy to converter 2 as shown in Fig. 5(b). Also, releases
energy to and .
the system operation can be regarded as the process of con-
stantly absorbing and releasing energy in the form of charging
and discharging for inductors and capacitors within high-fre-
quency cycles, which results in an overall low-frequency power
flow. It is clear that if the values of and are rel-
atively small, the dynamical response of energy delivery will
be rather fast. One can see shortly that the transferring speed of
energy around the high-frequency, i.e., the power flow response
Fig. 5. Equivalent high-frequency power flow diagrams for the Zeta power
stage under CCM operation. The switches of the converter 1 and 2 are synchro-
nized by the same switching sequence. (a) Power flows as the switches turn on.
(b) Power flows as the switches turn off.
(PFR) of the power stage does affect the system stability. Gen-
erally, the low-frequency power flow lies in two different man-
ners. Firstly, if the power demand of the load is met by rapid PFR
through a well-designed power stage which accomplishes the
rapid PFR requirement, then the high-frequency power flow or-
derly occurs. Thus, the system has stable low-frequency power
flow, and its instantaneous power balance relation among the
input power , the output power and
the buffered power in satisfy
(50)
However, if the dynamical responses of the inductors and ca-
pacitors around the high-frequency fail to meet the rapid PFR
requirement, the buffering capacitor will lose the role of ab-
sorbing (filling) the instantaneous power surplus (deficit). Then,
the power balance relation (50) will be untenable, which leads
to instantaneous power unbalance. The residual power
in and equals
(51)
which causes the oscillatory surge of the system with an un-
stable low-frequency power flow.
The above discussions lead to an interesting idea of the power
unbalance analysis, i.e., the instantaneous power unbalance re-
sults in an oscillatory manner of the system and such oscilla-
tory dynamics tend to attenuate the power unbalance scenario.
Specifically, when decreases from 7 k to 5.3 k , the output
power demand is increased from 80 W to 115 W. If (50) can be
met by the rapid PFR, the system is stable. Unfortunately, since
the selected parameters and fail to meet such require-
ment, the power unbalance mode shown in (51) occurs with a
large magnitude of oscillations depicted in Fig. 4(b). However,
as increases from 60 V to 70 V, the PFR is accelerated and
the power unbalance mode ceases as shown in Fig. 4(c). More-
over, if the PFR slows down by the increase of , the power
unbalance case also occurs due to an unmatched power require-
ment, which leads to the oscillation shown in Fig. 4(d).
From the above power unbalance analysis, the aforemen-
tioned parametric resultant relationships can be regarded as a
compromised effect on the power balance of the system, which
is subjected to major circuit parameters in the power stage, the
voltage controller and the input/reference voltages.
C. Dimensionless Distributed Discharge Time Constants
Note that and represent the discharge (or charge)
time constants of the well-known and fundamental

10. 500 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
circuits. Now, if the time constants and are rela-
tively small, the discharge (or charge) process will be faster than
that of large time constants. That is to say, the PFR of the in-
ductor and the capacitor will be rapider during the charge
or discharge process. Inspired by such concept, we define the
dimensionless variables and shown in (41) as di-
mensionless distributed discharge time constants , i.e.,
and as follows:
(52)
As aforementioned, the system stability is associated with the
circuit parameters in the power stage, the voltage controller and
the input/reference voltages, which result in a resultant effect on
the power balance of the system. In fact, the can be used
to describe such effect in the power stage. Moreover, the di-
mensionless parameters and for the voltage controller
and the dimensionless parameter for the input/reference
voltages can also be applied to formulate this effect. We draw
some interim conclusions on the multi-parametric stability of
the system according to the above discussion.
• The increase of corresponding to the increase of
will reduce the discharge (or charge) process in the power
stage, which results in a slower PFR. The power unbalance
case will arise because of the unmatched power require-
ment, which leads to oscillatory instability of the system.
• The decease of corresponding to the increase of will
cause an increasing demand of the output power. If the
increasing power demand fails to be met by the power stage
with a given , then the power unbalance case occurs,
which results in the oscillatory instability.
• The increase of corresponding to the decrease of
will enlarge the discrepancy of power supply and demand.
When the relatively “increasing” power demand, com-
pared with the decreasing power supply, fails to be met
by a given power stage, the oscillatory instability
emerges due to the power unbalance mode.
VII. FURTHER APPLICATION
In this section, we will apply the method developed in this
paper to visualize the qualitative behaviors of the system both
in the dimensionless and dimensional parameter spaces, and
present the boundary surface and curves of stable region and
oscillatory unstable region in terms of practically relevant pa-
rameters. Specifically, theoretical dimensionless boundaries are
shown in Figs. 6(a) and 7(a), which is obtained by using the
analytical method derived in Section V. Using the parameter
normalized map, we get the circuit parameter boundary sur-
face as shown in Figs. 6(b), 7(b) and 8. Also, we take a few
cross sections from the boundary surface as indicative boundary
curves shown in Figs. 6(c)–(d) and 7(c)–(d). Experimental data
are plotted along with the analytical results for verification pur-
pose. To emphasize the parametric resultant relationships on the
stability types of the system, the following general discussions
are made.
Firstly, the oscillatory instability is prone to occur for rela-
tively large values of or small values of or . In general,
a relatively small can increase the PFR and enlarge
Fig. 6. Stability boundaries for the Zeta PFC converter with OCC plotted in the
parameter space and . (a) Boundary surface under
different and for and
. (b) Boundary surface plotted in under different
for F, k and V. (C) Cross
section curves for F under different . (d) Cross section curves for
mH under different .
Fig. 7. Stability boundaries for the Zeta PFC converter with OCC plotted in
the parameter space and . (a) Boundary surface for
.
and . (b) Boundary surface for V and
mH under different . (C) Cross section curves for
ms under different . (d) Cross section curves for under different
.
the stable operation region of the system. However, the oppo-
site scenarios for the small and , which enlarge
the unstable margin, can be attributed to the insufficient energy
delivery and buffering in the power stage.
Secondly, the variation of the load resistance plays
a twofold role in the change trend of the
. Specifically, as decreases, the
and increase, while the

11. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 501
Fig. 8. Stability boundaries for the Zeta PFC converter with OCC plotted in
the dimensional parameter space for mH, k
under different . Here, the critical bifurcation points of for
and equals to 0.04386 and 0.04135, respectively.
and also decrease. In fact, the increase of can en-
large the stable operation boundary, and the increase of
together with the decrease of and will reduce the stable
boundary. the system stability will be a trade-off between the
above two opposite trends. As a result, the impact of load
resistance on the system stability is not significant as shown
in Figs. 6(c), (d) and 7. For this reason, extensive behavior
boundary as being a variable has not been analyzed herein.
Thirdly, the input/reference voltages have a significant ef-
fect on the system stability, and the oscillatory manner of the
system can be eliminated by increasing the input voltage or de-
creasing the reference voltage. a power balance mode can be ac-
complished by increasing the input voltage to improve the input
power supply, or by decreasing the reference voltage to reduce
the output power demand.
Finally, the oscillatory instability here tends to occur for rela-
tively small as shown in Fig. 8. Since the output power de-
mand arises as becomes smaller, the power unbalance case
is more prone to emerging under a poor power matching degree,
such as a lower input voltage. In fact, a relatively large and
corresponding to small and will aggravate the power
unbalance mode drastically. However, no matter what the values
of is, the oscillatory boundary region is not almost affected.
This reason is that its corresponding dimensionless parameter
has little effect on the system power delivery.
VIII. CONCLUSION
A general methodology for investigating parametric resultant
relationships of parameter-dependant NTP systems in the home-
omorphic space has been proposed systematically, which leads
to the DA for multi-parametric stability analysis of switching
converters. We start with the concept of NTP systems which
formulates the general mathematical model of switching con-
verters. Then, we prove that under a proper map, the stability
analysis of the original NTP system can be simplified to that
of the homeomorphic NTP system with a lower parameter di-
mension, but possesses equivalent stability properties. Subse-
quently, the DA is carried out based on the derived map, and a
specific example of the Zeta PFC converter is finally given to
validate the proposed method.
It is shown that the dimension of circuit parameter space can
be reduced by using the DA, and the parametric resultant re-
lationships are carried out by means of influencing the power
unbalance modes of the system, which can be analyzed by sev-
eral families of dimensionless parameters. In contrast to pre-
vious works, the parametric resultant relationships of the NTP
system have been fully considered in the sense of topological
equivalence. It has been highlighted that the proposed method is
capable of both simplifying the system parametric complexity
and revealing how these resultant relationships affect the sta-
bility patterns.
Another important aspect of our study here is to identify the
behavior boundaries by means of major circuit parameters, from
which plenty of useful information can be used to give some
design-oriented guidelines for stable operation. Compared with
previous results, multi-parametric stability behavior boundaries
in the form of 2-D curve or 3-D surf are displayed as one line
or surface rather than one point in a mapped parameter space.
That is to say, the proposed method can provide a family of cir-
cuit parameters rather than a set of ones so that optimal design
of circuit parameters becomes possible. Although the approach
is applied to the Zeta PFC converter as an example, the method-
ology of using homeomorphic space map and dimensionless pa-
rameter set is applicable to the multi-parametric stability anal-
ysis of other switching converter circuits, such as the two-stage
PFC converter studied in [13].
APPENDIX A
PROOFS
For the proofs, we need two auxiliary results. One is con-
cerned with the equivalent relation between stability and uni-
form stability for the zero solution of (7), and the other is re-
lated to the equivalent relation between its asymptotic stability
and uniformly asymptotic stability.
Lemma 1 ([34], Theorem 1.8.9): For the NTP system (7), the
equilibrium of (7) is stable iff it is uniformly stable.
Lemma 2 ([34], Theorem 1.8.11): For the NTP system (7), its
zero solution is asymptotically stable iff it is uniformly
asymptotically stable.
Remark L.1: Lemma 1 implies that if the NTP system (7) is
stable over and , then for
( is independent of ), as long as
, the perturbed solution of (7) with the initial state
will satisfy for . Thus, the (uniform)
stability of the zero solution can be fully characterized by
the case of ordering . Similar to Lemma 1, the (uniformly)
asymptotic stability of the zero solution can also be
thoroughly investigated by ordering for Lemma 2.
A. Proof of Theorem 3.1
For notational simplicity, the parameter vector in (9) is
omitted. Proof of each statement in Theorem 3.1 is given as fol-
lows.
1) As discussed in Section III, the periodic feature of the orig-
inal system still lies in the map , i.e.,
and . By applying the map to the
NTP system (11), we obtain the mapped system whose
vector field satisfies for .

12. 502 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013
Note that and
. This implies that the mapped system is also
a NTP system.
2) The zero solution systems for and for are
described by the following expressions, respectively:
(53)
(54)
where and
for . As mentioned in Section II, the (uniform)
stability of and would be fully equivalent to
and respectively. Based on Lemma 1, the cycles
of and are stable iff they are uniformly stable.
3) Note that the proof of the third statement 3) in Theorem
3.1 can be reduced to proving the (uniform) equivalent sta-
bility of and of the corresponding NTP
system and . For notational simplicity, de-
notes the (uniform) equivalent stability between and ,
and does the same relation between and .
Let be a transition system after the map with the
form of
(55)
Since is a NTP system, is also be a NTP system.
Hence, the proof of can be decomposed to prove
and .
: Note that is a NTP system. Considering
Lemma 1, the equilibrium of (53) is stable, if and
only if it is uniformly stable. Following the Remark L.1
one can get for such that
(56)
Let be zero, thus the statement (56) can be completely
equivalent for such that
(57)
Obviously, for , it follows that
such that
(58)
Note that is a NTP system. Using Lemma 1, one can
obtain for such that
(59)
Hence, the (uniform) stability of of is estab-
lished, and vice versa, i.e., .
: Note that the zero solution of of is
(uniform) stable. Then, for the essential condition part of
(59), one obtains
(60)
where is the matrix defined in Theorem 3.1. According
to the comparability of matrix norm, we get
(61)
Since , then we obtain
(62)
Therefore, substituting (61) and (62) into (61), we obtain
(63)
Denote as , then can be written as .
For conciseness, also denote as . Hence,
it follows that for such
that
(64)
Therefore, the (uniform) stability of the zero solution
of is established, and vice versa, i.e., . Fi-
nally, the statement 3) of Theorem 3.1 is valid.
B. Proof of Theorem 3.2
According to the proof of Theorem 3.1, we find that its first
statement addressing the periodic character of can be directly
applied to Theorem 3.2. Hence, we only need to give the proofs
for other two statements of Theorem 3.2.
1) Reviewing the proof of the Statement 2) of Theorem 3.1,
we find that the zero solution systems and are two
NTP systems. By applying Lemma 2, it follows that the
cycles of and are asymptotically stable, if and only
if they are uniformly asymptotically stable.
2) Note that the proof of the Statement 2) of Theorem 3.2 can
be similarly performed as that of the Statement 3) in The-
orem 3.1. For clearance, we redefine the (uniform) asymp-
totic stability equivalence between and as ,
and the same relation for and as . Let
be the same transition system defined in the previous Proof.
Hence, the proof of can also be decomposed to
prove and .
: Note that is a NTP system. Then consid-
ering Lemma 1, the zero solution of (53) is asymp-
totically stable, if and only if it is uniformly asymptoti-
cally stable. As mentioned in Remark L.1, one gets for
such that
(65)
Let be zero, thus (65) is completely equivalent for
such that
(66)
For , it follows that for
and such that
(67)

13. ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 503
Using Lemma 2, one obtains for and
such that
(68)
Hence, the (uniform) asymptotic stability of of is
established, and vice versa, i.e., .
: Note that of is (uniform) asymptotically
stable. Then, for the essential condition part of (68), one obtains
the same argument as (60). Also, the same form of (61) can
be get for asymptotical stability. Following the same notations
for and can be written as ,
which is denoted by . Hence, it follows that for
such that
(69)
the (uniform) asymptotic stability of the equilibrium of
is established, and vice versa, i.e., . This concludes
the proof of Statement 2) of Theorem 3.2.
APPENDIX B
The coefficient matrices , and for the
one-cycle controlled Zeta PFC converter are written as follows:
(70)
(71)
(72)
APPENDIX C
According to [39], [40], the voltage across the capacitor
and , i.e., and can be written as
(73)
Substituting (35) into (73), one obtains
(74)
Consider the Kirchhoff’s current law for the node in Fig. 1,
the relation between and can be obtained as
(75)
For the OCC controller with a voltage follower, can be
expressed as follows
(76)
applying the map to (73), (74), (75) and (76), we will
readily get the constraint equations among and as
shown in (45).
ACKNOWLEDGMENT
The authors would like to thank the Associate Editor and
the anonymous reviewers for their helpful suggestions and con-
structive comments which have led to significant improvement
in the presentation of this paper. They also would like to thank
Prof. C. K. Tse (Hong Kong Polytechnic University), Dr. Y. Ma
(UPMC, France) and Dr. P. Chen (EPFL, Switzerland), for their
valuable discussions and insightful feedback.
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Hao Zhang (M’06) was born in Shaanxi Province,
China, in 1973. He received the B.E. degree from
Xi’an University of Science & Technology, Xi’an,
China, in 1996, the M.Sc. degree from Xi’an Univer-
sity of Technology, Xi’an, China, in 2002, and the
Ph.D. degree in electrical engineering (with honors)
from Xi’an Jiaotong University, Xi’an, China, in
2005.
From December 2004 to June 2005, he was a Re-
search Assistant with Hong Kong Polytechnic Uni-
versity, Hong Kong. Since 2007, he has been an As-
sociate Professor, School of Electrical Engineering, Xi’an Jiaotong University.
During the academic year 2010–2011, he was a Visiting Professor of the Center
for Power Electronics Systems (CPES), Virginia Tech. His research interests
are in complex behaviors of distributed power systems, and power electronics
interfaces in micro-grid systems.
Yuan Zhang (S’11) was born in Chaohu, Anhui
Province, China, in 1989. He received the B.E.
degree (hons.) in electrical engineering in 2010 from
Xi’an Jiaotong University, Xi’an, China, where he
is currently pursuing the M.E. degree in the areas of
modeling, control and design of practical switching
power converters, grid-connected inverters of power
electronics interfaces in micro-grids.
Mr. Zhang was the recipient of many top schol-
arships and awards during his undergraduate and
graduate studies. His B.E. dissertation also won the
Distinguished Thesis Award from Xi’an Jiaotong University in 2010. He is a
member of IEEE Power & Energy Society and Chinese Physics Society.
Xikui Ma was born in Shaanxi, China, in 1958. He
received the B.E. and M.Sc. degrees, both in elec-
trical engineering, from Xi’an Jiaotong University,
China, in 1982 and 1985, respectively.
In 1985, he joined the Faculty of Electrical
Engineering, Xi’an Jiaotong University, where he
is currently the Chair Professor of the Electromag-
netic Fields and Microwave Techniques Research
Group. During the academic year 1994–1995, he
was a Visiting Scientist at the Power Devices and
Systems Research Group, Department of Electrical
Engineering and Computer, University of Toronto, Toronto, ON, Canada. His
research interests include electromagnetic field theory and its applications,
analytical and numerical methods in solving electromagnetic problems, mod-
eling of magnetic components, chaotic dynamics and its applications in power
electronics, and the applications of digital control in power electronics. He has
been actively involved in more than 25 research and development projects. He
is also the author or coauthor of more than 190 technical papers, and also the
author of five books in electromagnetic fields, including Electromagnetic Field
Theory and Its Applications (Xi’an: Xi’an Jiaotong University Press, 2000).
Prof. Ma received the Best Teacher Award from Xi’an Jiaotong University in
1999.