Randomly perturbed switching dynamics of a DC/DC converter

0.23 MB·

by  Chetan D. Pahlajani

RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC
CONVERTER
CHETAN D. PAHLAJANI
6
1 Abstract. In this paper, we study the effect of small Brownian noise on a switching dynamical
0 systemwhichmodelsafirst-orderdc/dcbuckconverter. Thestatevectorofthissystemcomprises
2 acontinuouscomponentwhosedynamicsswitch,basedontheon/off configuration ofthecircuit,
between two ordinary differential equations (ode), and a discrete component which keeps track of
n
theon/offconfigurations. Assumingthattheparametersandinitialconditionsoftheunperturbed
a
J system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this
system when the forcing input in the on state is subject to small white noise fluctuations of size
5
ε, 0<ε≪1. For the ensuing stochastic system whose dynamics switch at random times between

] a small noise stochastic differential equation (sde) and an ode, we prove a functional law of large
R numbers which states that in the limit of vanishing noise, the stochastic system converges to the
P underlyingdeterministic one on time horizons of order O(1/εν),0≤ν <2/3.
.
h
t
a
m
1. Introduction
[

1 Ordinary differential equations (ode) and dynamical systems play a fundamental role in mod-
v elling and analysis of various phenomena arising in science and engineering. In many applications,
3
however, the smooth evolution of the ode dynamics is punctuated by discrete instantaneous events
4
which give rise to switching or non-smooth behaviour. Examples include instantaneous switching
8
0 between different governing ode in a power electronic circuit [BKYY, BV01, dBGGV], discontinu-
0 ous change in velocity for an oscillator impacting a boundary[SH83, Nor91], etc. In such instances,
.
1 the dynamical system involves functions which are not smooth, but only piecewise-smooth in their
0
arguments. Such piecewise-smooth dynamical systems [dBBCK] display a wealth of phenomena
6
not seen in their smooth counterparts, and have hence been the subject of much current research.
1
: Dynamical systems arising in practice are almost always subject to random disturbances, owing
v
perhaps to fluctuating external forces, or uncertainties in the system, or unmodelled dynamics,
i
X
etc. A more accurate picture can therefore be obtained by modelling such systems (at least in the
r continuous-time case) using stochastic differential equations (sde); intuitively, this corresponds to
a
adding a “noise” term to the ode. For cases where the perturbing noise is small, it is natural to
ask whether the stochastic (perturbed)system converges to the deterministic (unperturbed)one in
the limit of vanishing noise, and if yes, how the asymptotic behaviour of the fluctuations may be
quantified. Such questions have played a significant role in the development of limit theorems for
stochastic processes; see, for instance [DZ98, EK86, FW12, PS08].
Although smooth dynamical systems perturbed by noise have been analysed in great depth over
the past few decades, the effect of random noise on non-smooth or switching dynamical systems
remains, with some exceptions (see, for instance, [CL07, CL11, HBB1, HBB2, SK1, SK2, SK3]),
Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar
382355, India
E-mail address: [email protected].
Date: January 6, 2016.
1
2 RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER
relatively unexplored. One of the challenges in such an undertaking is that even in the absence of
noise, the dynamics of switching systems can prove rather difficult to analyse. Part of the reason is
the frequently encountered intractability of such systems to analytic computation [BC, dBGGV],
even in cases when the component subsystems are linear.
Ourprimaryinterest is thestudyof stochastic processeswhich arisedueto smallrandompertur-
bations of (non-smooth) switching dynamical systems. These problems are of immediate relevance
in the analysis of dc/dc converters in power electronics—naturally susceptible to noise—in which
time-varying circuit topology leads to mathematical models characterised by switching between
different governing ode. In the purely deterministic setting, the dynamics of these systems have
been extensively studied, with much of the work focussing on buck converters [BKYY, dBGGV,
HDJ92, FO96]; these are circuits used to transform an input dc voltage to a lower output dc
voltage. Perhaps the simplest of these is the first-order buck converter; this is a system which
switches between two linear first-order ode. While this circuit is pleasantly amenable to some
explicit computation, it nevertheless displays rich dynamics in certain parameter regimes. Periodic
orbits, bifurcations and chaos for this converter have been studied in [BKYY, HDJ92].
In the present paper, we study small random perturbations of a switching dynamical system
which models a first-order buck converter. The state vector of this system comprises a continuous
component (the inductor current) governed by one of two different ode, and a discrete component
which takes values 1 or 0 depending on whether the circuit is in the on versus off configurations,
respectively. Assuming that the parameters and initial conditions of the unperturbed system have
been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when
the forcing (input dc voltage) is subject to small white noise fluctuations of size ε, with 0 <ε 1.
≪
Ourmainresultisafunctional law of large numbers(flln)whichstatesthat,asε 0,thesolution
ց
of the stochastically perturbed system converges to that of the underlying deterministic system,
over time horizons T of order O(1/εν) for any 0 ν < 2/3.
ε
≤
Part of the novelty of this work, in the context of the literature on switching diffusions (see, e.g.,
[BBG99, LM07, YZ1, YZ2]), is that the switching in our problem is not driven by a discrete-state
stochastic process (whose transitions may occur at a rate depending on the continuous component
of the state); rather, the switching occurs whenever the continuous component of the state hits a
threshold (on off), or upon the arrival of a time-periodic signal (off on). Our switching
→ →
is thus entirely determined by the continuous component, together with a periodic clock signal.
We also note that since the input dc voltage in the buck converter influences the inductor current
only in the on state [BKYY], the perturbed system has alternating stochastic and deterministic
evolutions: the dynamics switch at random times between an sde driven by a small Brownian
motion of size ε in the on state, and an ode in the off state. The import of our results is
that even in the presence of small stochastic perturbations, one may expect the buck converter to
function close to its desired operation for “reasonably long” times.
The rest of the paper is organised as follows. In Section 2, we describe the switching systems
(deterministic and stochastic) in some detail, and we pose our problem of interest. Next, in Section
3, we state our main result (Theorem 3.1) and outline the steps to the proof through a sequence of
auxiliary lemmas and propositions. A few of these auxiliary results are proved in Section 3, with
the remainder (the slightly lengthier ones) being deferred to Section 4.
2. Problem Description
In this section, we formulate our problem of interest. We start with a description, including the
governing ode’s and the switching mechanism, of a dynamical system modelling a first-order buck
converter in Section 2.1. Random perturbations of this system, which lead to a switching sde/ode
RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER 3
model, are discussed in Section 2.2. In Section 2.3, we obtain explicit formulas for solutions to
both the sde and the ode’s between switching times, and we piece these together at switching
times to obtain expressions describing the overall evolution of both the perturbed (stochastic) and
unperturbed (deterministic) switching systems. Finally, after showing in Section 2.4 how problem
parameters can be tuned and initial conditions chosen to ensure that the unperturbed system has
a stable periodic orbit, we pose our questions of interest.
Before proceeding further, we note that we have a hybrid system. Indeed, the full state of the
system is specified by a vector z = (x,y) taking values in Z , R 0,1 ; here, x R is the
× { } ∈
continuous component of the state—corresponding to the inductor current in the buck converter—
while the discrete component y takes values 1 or 0 depending on whether the switch is on or
off.
2.1. Deterministicswitchingsystem. Asnotedabove,thestateofoursystemattimet [0, )
∈ ∞
will be specified by a vector z(t) , (x(t),y(t)) taking values in Z , R 0,1 . We will assume
×{ }
that the dynamics of x(t) when y(t) = 1 (on configuration) are governed by the ode
dx
(1) = αonx+β,
dt −
while the dynamics of x(t) when y(t)= 0 (off configuration) are described by
dx
(2) = αoffx.
dt −
Here, β, αon, αoff are fixed positive parameters with β representing the (rescaled) input voltage of
an external power source, while αon and αoff denote the (rescaled) resistances in the on and off
configurations, respectively.1
The switching between the on and off configurations is effected as follows. A reference level
xref (0,β/αon) is fixed. Suppose the system starts in the on configuration, i.e., y(0) = 1, with
∈
x(0) (0,xref). The current x(t) increases according to (1), with y(t) staying at 1, until x(t) hits
∈
the level xref. At this point, an on off transition occurs: y(t) jumps to 0 and x(t) now evolves
→
according to (2). This continues until the next arrival of a periodic clock signal with period 1
(which arrives at times n N) triggers an off on transition: y(t) jumps back to 1, x(t) again
∈ →
evolves according to (1), and the cycle continues. Note that if a clock pulse arrives in the on
configuration, it is ignored. Of course, if one starts in the off configuration, x(t) evolves according
to (2) until the next clock pulse, at which point the system goes on, and the subsequent dynamics
are as described above. An important assumption in our analysis is that x(t) is continuous across
switching times.
2.2. Random perturbations. We now suppose that the forcing term β in (1) is subjected to
small white noise perturbations of size ε, 0 < ε 1; for the buck converter, this corresponds to
≪
small random fluctuations in the input voltage. In this setting, the state of the system at time
t [0, ) is given by a stochastic process Zε , (Xε,Yε) taking values in Z. The dynamics of Xε
∈ ∞ t t t t
in the on configuration (Yε = 1) are now governed by the sde
t
(3) dXtε = (−αonXtε +β)dt+εdWt,
where W is a standard one-dimensional Brownian motion, while evolution of Xε in the off state
t t
(Yε = 0) is governed by the ode (2), as before. The switching mechanism is similar to that in the
t
unperturbed case, but with the stochastic processes Xε, Yε playing the roles of x(t), y(t). Note,
t t
1More precisely, β =Vin/L, αon =R/L and αoff =(R+rd)/L, where Vin is the input voltage, R denotes the load
resistance, and r is thediode resistance [BKYY].
d
4 RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER
in particular, that the times for on off transitions are given by passage times of Xε (governed
→ t
by (3)) to the level xref. As before, Xtε is assumed to be continuous across switching times.
2.3. Explicit formulas. The foregoing discussion makes clear how z(t) = (x(t),y(t)) and Zε =
t
(Xε,Yε) are to be obtained, once an initial condition z = (x ,y ) Z has been specified: the
t t 0 0 0 ∈
evolutions of x(t) and Xε are given, respectively, by concatenating solutions to (1) and (2), and
t
solutions to (3) and (2), at the respective switching times, maintaining continuity. The function
y(t) and the sample paths of Yε—which are piecewise constant and take values in 0,1 —will be
t { }
assumed to be right-continuous. Below, we obtain expressions for z(t) and Zε starting from initial
t
condition z0 = (x0,1) where x0 (0,xref). We note that starting with y0 = 1 entails no real loss
∈
of generality; indeed, as will become apparent, the expressions below can be easily modified to
accommodate the case when y = 0.
0
In the sequel, we will use 1 to denote the indicator function of the set A, and for real numbers
A
a,b, we let a b and a b denote the minimum and maximum of a and b, respectively.
∧ ∨
2.3.1. Solution of deterministic switching system. As indicated above, we fix an initial condition
z0 = (x0,1) with x0 ∈(0,xref). Let s0 , 0, and set x0,off(t) ≡ x0. Next, define x1,on(t), 1[s0,∞)(t)·
β/αon +(x0 β/αon)e−αont for t 0. Let t1 , inf t > 0 : x1,on(t) = xref be the first time that
− ≥ { }
x(cid:8)1,on(t) reaches level xref and(cid:9)define x1,off(t) , 1[t1,∞)(t)·xrefe−αoff(t−t1). Let s1 , inf{t > t1 : t ∈ Z}
be the time of arrival of the next clock pulse. The solution of the deterministic switching system
on the interval [s ,s ) is now given by x1(t), x1,on(t) 1 (t)+x1,off(t) 1 (t).
0 1 · [s0,t1) · [t1,s1)
In general, given the solution over [s ,s ), the solution xn(t) over [s ,s ) is obtained as
0 n−1 n−1 n
follows. We let
β β
xn,on(t) , 1 (t) + xn−1,off(s ) e−αon(t−sn−1) for t 0,
[sn−1,∞) · αon n−1 − αon ≥
(cid:26) (cid:18) (cid:19) (cid:27)
tn , inf t > sn−1 : xn,on(t) = xref ,
{ }
(4) xn,off(t) , 1[tn,∞)(t)·xrefe−αoff(t−tn) for t ≥ 0,
s , inf t > t : t Z ,
n n
{ ∈ }
xn(t) , xn,on(t) 1 (t)+xn,off(t) 1 (t).
· [sn−1,tn) · [tn,sn)
The evolution of the deterministic switching system over [0, ) is now given by
∞
(5) z(t) = (x(t),y(t)) where x(t) , xn(t), y(t), 1 (t).
[sn−1,tn)
n≥1 n≥1
X X
We have thus decomposed the evolution into a sequence of on/off cycles with the switching
times t and s correspondingto the n-th on off and off on transitions, respectively; next,
n n
→ →
we have solved the ode (1) and (2) between switching times, and then linked the pieces together
while maintaining continuity of x(t) at switching times.
2.3.2. Solution of stochastic switching system. We now provide a similar detailed construction of
the stochastic process Zε = (Xε,Yε) starting from the same initial condition z = (x ,1). Let
t t t 0 0
W = W ,F : 0 t < be a standard one-dimensional Brownian motion on the probability
t t
space {(Ω,F,P). W≤e intr∞od}uce, for each n N, the processes Xn,ε,on, Xn,ε,off, Xn,ε and random
switching times τε, σε, which are defined recu∈rsively as follows. Sett σε , 0tand defitne X0,ε,off x .
n n 0 t ≡ 0
Now, let Xt1,ε,on , 1[σ0ε,∞)(t)· β/αon +(x0−β/αon)e−αont+ε 0te−αon(t−u)dWu for t ≥ 0, and let
τ1ε , inf{t > 0 : Xt1,ε,on = xrnef} be the first passage time of XRt1,ε,on to level xroef. We next define
RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER 5
Xt1,ε,off , 1[τ1ε,∞)(t)·xref e−αoff(t−τ1ε) and let σ1ε , inf{t > τ1ε : t ∈ Z} be the time of arrival of the
next clock pulse. We now set X1,ε , X1,ε,on 1 (t)+X1,ε,off 1 (t).
To compactly express Xn,ε,ontfor genteral·n,[σl0eε,tτ1εI) = I :t 0 ·t <[τ1ε,σ1ε) be the process defined by
t { t ≤ ∞}
I , teαonudW for t 0. Note that I is a continuous, square-integrable Gaussian martingale.
Wte now0 define, fuor each≥n N,
R ∈
Xnt,ε,on , 1[σnε−1,∞)(t)· αβon + Xnσnε−−11,ε,off− αβon e−αon(t−σnε−1)+εe−αont It−It∧σnε−1 ,
(cid:26) (cid:18) (cid:19) (cid:27)
τnε , inf{t > σnε−1 :Xnt,ε,on = xref}, (cid:16) (cid:17)
(6) Xnt,ε,off , 1[τnε,∞)(t)·xref e−αoff(t−τnε),
σε , inf t > τε :t Z ,
n { n ∈ }
Xn,ε , Xn,ε,on 1 (t)+Xn,ε,off 1 (t).
t t · [σnε−1,τnε) t · [τnε,σnε)
Our stochastic process of interest is now given by
(7) Zε , (Xε,Yε), where Xε , Xn,ε, Yε , 1 (t).
t t t t t t [σnε−1,τnε)
n≥1 n≥1
X X
Once again, the evolution comprises a sequence of on/off cycles, with the quantities above
admitting a natural interpretation which parallels the unperturbed (deterministic) case.
2.4. Stable periodic orbit. We now describe the assumptions on the problem parameters that
ensure the existence of a stable periodic solution to (4), (5). The argument proceeds by analysing
the stroboscopic map [BKYY] which takes the system state at one clock instant to the state at
the next. Map-based techniques are used extensively in analysing the switching dynamics of power
electronic circuits; see also [dBGGV, HDJ92].
Assumption 2.1. Fix xref > 0, 0 < αon < log2. Select β > 0 such that
eαon
(8) 2xrefαon < β < xrefαon.
eαon 1
(cid:18) − (cid:19)
Let αoff > 0 such that
β/αon xref
(9) αon < αoff < − αon.
xref
(cid:18) (cid:19)
Wenowdefineamapf :[0,xref] [0,xref]whichmapsx0 [0,xref]tothesolutionx(t)attime1,
→ ∈
subject to the initial condition being (x ,1), i.e., f : x x(1;x ,1). We are interested in the case
0 0 0
7→
when f is only piecewise smooth. Put another way, if we let xborder , β/αon +(xref β/αon)eαon
be the particular value of x0 for which the corresponding x1,on(t) satisfies x1,on(1) = x−ref, we would
like xborder (0,xref). It is easily seen that the upper bound on β in (8) ensures that such is indeed
∈
the case. The map f :x x(1;x ,1) is seen to be given by
0 0
7→
β + x β e−αon if 0 x xborder,
(10) f(x) , αon − αon ≤ ≤
xrefe−(cid:16)αoff β/α(cid:17)on−x αoff/αon if xborder < x xref.
 β/αon−xref ≤
(cid:16) (cid:17)
Proposition 2.2. Suppose Assumption 2.1 holds. Then, the mapping f : [0,xref] [0,xref] has a
→
unique fixed point x∗ which lies in the interval (xborder,xref). Further, f′(x∗) < 1, implying that
| |
x∗ is a stable fixed point of the discrete-time dynamical system x f(x ).
n n
7→
6 RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER
Proof. Leth(x) , f(x) x. Notethath(0) = f(0) >0, h(xref) = f(xref) xref = xref(e−αoff 1)< 0.
− − −
Sincehiscontinuous,thereexistsx∗ (0,xref)suchthath(x∗) = 0,i.e.,f(x∗) = x∗. Sincef(x)> x
∈
forallx [0,xborder],wemusthavex∗ (xborder,xref). Further,sincef(x)decreaseson(xborder,xref)
∈ ∈
as x increases over this same interval, f can have at most one fixed point. It is easily checked that
for x (xborder,xref), we have
∈
f′(x) = αofff(x) αoffxref < 1,
| | β αonx ≤ β αonxref
− −
where the last inequality follows from the upper bound on αoff in (9). This proves stability of
x∗. (cid:3)
We can now pose our principal questions of interest. Suppose z(), Zε are obtained from (5),
· ·
(7), respectively, with initial conditions z(0) = Zε = (x∗,1), where x∗ is as in Proposition 2.2.
0
For any fixed T N, do the dynamics of Zε converge to those of z() in a suitable sense as
• ∈ · ·
ε 0?
Ifցyes, can the results be strengthened to the case when T = T N grows to infinity, but
ε
• ∈
“not too fast”, as ε 0?
ց
In the next section, we will show that both these questions can be answered in the affirmative,
provided T = O(1/εν) with 0 ν < 2/3.
ε
≤
3. Main Result
Recall that the state space for the evolution of z(t) and Zε is Z = R 0,1 , which inherits
t ×{ }
the metric r(z ,z ) , x x 2+ y y 2 1/2 for z = (x ,y ), z = (x ,y ) Z, from R2.
1 2 1 2 1 2 1 1 1 2 2 2
If I is a closed subinterv|al o−f [0,| ),|we−let|D(I;Z) be the space of functions z :∈I Z which
(cid:8) ∞ (cid:9) →
are right-continuous with left limits. This space can be equipped with the Skorokhod metric d
I
[Bil, EK86], which renders it complete and separable. If z D(I;Z) and J is a closed subinterval
of I, then the restriction of z to J is an element of D(J;Z∈) which, for simplicity of notation, will
also be denoted by z. For our switching systems of interest, we note that the function z(t) in (5),
and the sample paths of the process Zε in (7), belong to D([0, );Z). Our goal here is to study
the convergence, as ε 0, of Zε to z()t in the space D([0,T ];Z∞) for time horizons T = O(1/εν)
ց · · ε ε
where 0 ν < 2/3.
We sta≤rt by defining the Skorokhod metric d on the space D(I;Z), where I = [0,T] for some
I
T > 0.2 Let Λ˜T be the set of all strictly increasing continuous mappings from [0,T] onto itself,3
and let ΛT be the set of functions λ Λ˜T for which
∈
λ(t) λ(s)
γT(λ) , sup log − < .
0≤s<t≤T(cid:12) t−s (cid:12) ∞
(cid:12) (cid:12)
For z , z D([0,T];Z), we now define (cid:12) (cid:12)
1 2
∈ (cid:12) (cid:12)
(11) d[0,T](z1,z2) , inf γT(λ) sup r(z1(t),z2(λ(t))) .
λ∈ΛT( ∨0≤t≤T )
Note that if du[0,T](z1,z2) , sup0≤t≤Tr(z1(t),z2(t)) is the uniform metric on D([0,T];Z), then
d[0,T](z1,z2) ≤ du[0,T](z1,z2). Indeed, the latter corresponds to the specific choice λ(t) ≡ t.
We now state our main result.
2See[Bil, EK86] for thecase I =[0,∞).
3Thus, we haveλ(0)=0 and λ(T)=T for all λ∈Λ˜T.
RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER 7
Theorem 3.1 (Main Theorem). Fix T N, 0 ν < 2/3. For ε (0,1), let T N such that
ε
T T/εν. Suppose z(), Zε are given by∈(5), (7)≤, respectively, with∈initial condition∈s z(0) = Zε =
ε ≤ · · 0
(x∗,1), where x∗ is as in Proposition 2.2. Then, for any p ∈ [1,∞), we have that d[0,Tε](Zε,z) → 0
in Lp, i.e.,
(12) εliցm0E d[0,Tε](Zε,z) p = 0.
(cid:2)(cid:0) (cid:1) (cid:3)
Remark 3.2. Of course, Theorem 3.1 implies that d[0,Tε](Zε,z) converges to 0 in probability, i.e.,
for any ϑ > 0, we have limεց0P d[0,Tε](Zε,z) ≥ ϑ = 0.
(cid:8) (cid:9)
To explain the intuition behind Theorem 3.1, we note that when ε 1, the likely behaviour of
≪
Xε is to closely track x(t). Therefore, one expects that with high probability, we have τε t ,
σεt= s = n for each 1 n T (at least if T is not too large). On this “good” event, a rnan≈domn
n n ≤ ≤ ε ε
time-deformation λ, for which γTε(λ) is small, can be used to align the jumps of Ytε and y(t) so
that Yλε(t) ≡ y(t). Continuity now ensures that Xλε(t) is close to x(t), and we get that d[0,Tε](Zε,z)
can be bounded above by a term which goes to zero as ε 0. It now remains to show that the
ց
probability of the complement of this event, i.e., the event where one or more of the τε differ from
n
t by a significant amount, is small.
n
Our thoughts are organised as follows. First, we introduce an additional scale δ 0 to quantify
ց
proximity ofτε tot ; wewilllater take δ = ες forsuitableς > 0. Now, forε,δ (0,1), setGε,δ , Ω,
n n ∈ 0
and for n 1, define
≥
Gε,δ , ω Gε,δ : τε(ω) t δ
n { ∈ n−1 | n − n|≤ }
Bε,δ , ω Gε,δ : τε(ω) t > δ = Gε,δ Gε,δ.
n { ∈ n−1 | n − n| } n−1\ n
ε,δ ε,δ ε,δ ε,δ
Note that the G ’s are decreasing, i.e., G G ... and that the B ’s are pairwise disjoint.
n 0 ⊃ 1 ⊃ n
Consequently, Tε Gε,δ = Gε,δ and P Tε Bε,δ = Tε P Bε,δ .
∩n=1 n Tε ∪n=1 n n=1 n
We now outline theprincipalsteps i(cid:16)n provingT(cid:17)heorem 3.1.(cid:16)First(cid:17), in Proposition 3.3, we derive a
P
path-wiseestimateford[0,Tε](Zε,z)anditspositivepowers. Thisresultassuresusthatourquantity
ε,δ
of interest is indeed small on the event G and of order 1 on its complement. Then,in Proposition
T
ε
3.6, we obtain an upperboundon P Bε,δ in terms of the tail of the standard normal distribution.
n
These two propositions enable us to(cid:16)comp(cid:17)lete the proof of Theorem 3.1. Both Propositions 3.3 and
3.6 are proved through a series of Lemmas; the proofs of the latter are deferred to Section 4.
Westartbyintroducingsomenotation. Letton , tn sn−1 andtoff , sn tn denotethefractions
− −
of time in each interval [n,n+1] for which the deterministic system is in the on and off states
respectively, and let tmin , ton toff.
∧
Proposition 3.3. For every p > 0, there exists a constant C > 0 such that for all ε,δ (0,1),
p
Tε N satisfying 0 < δ tmin/(4Tε), we have ∈
∈ ≤
(13) (d[0,Tε](Zε,z))p ≤ Cp((Tεδ)p +δp+1Ω\GεT,εδ +εpTpε0≤stu≤pTε|Wt|p).
8 RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER
To prove Proposition 3.3, we will employ the (random) time-deformation λε : Ω ΛT defined
→ ε
by
τε(ω) s
(14) λε(ω) , 1 (t) s + n − n−1 (t s )
t [sn−1,tn) · n−1 t s − n−1
n≥1 (cid:26) (cid:18) n− n−1 (cid:19) (cid:27)
X
s τε(ω)
+ 1 (t) τε(ω)+ n− n (t t ) if ω Gε,δ,
[tn,sn) · n s t − n ∈ Tε
n≥1 (cid:26) (cid:18) n− n (cid:19) (cid:27)
X
and
(15) λε(ω) , t if ω Ω Gε,δ.
t ∈ \ Tε
Note that in actuality, λε = λε,δ. However, we have suppressed the δ-dependence to reduce clutter
and also because we will eventually take δ = δ(ε). The first step is to show that γT (λε) is small;
ε
this is accomplished in Lemma 3.4 below. Next, in Lemma 3.5, we estimate sup0≤t≤Tε|Xλεεt −x(t)|p
and sup0≤t≤Tε|Yλεεt −y(t)|p for p > 0.
Lemma 3.4. Let Tε N. If 0 < δ tmin/(4Tε), then for each ω Ω, we have
∈ ≤ ∈
4T δ
(16) γTε(λε(ω)) ≤ tmεin .
Lemma 3.5. For every p > 0, there exists a constant c > 0 such that for all ε,δ (0,1), we have
p
∈
sup Xε x(t)p c δp1 +1 +εpTp sup W p ,
(17) 0≤t≤Tε| λεt − | ≤ p( GεT,εδ Ω\GεT,εδ ε0≤t≤Tε| t| )
sup Yε y(t)p 1 .
0≤t≤Tε| λεt − | ≤ Ω\GεT,εδ
Lemmas 3.4 and 3.5 are proved in Section 4. We now have
Proof of Proposition 3.3. Let λε ΛT be as in (14), (15). It is easily seen from (11) that
∈ ε
(d[0,Tε](Zε,z))p ≤ 3p (γTε(λε))p+sup0≤t≤Tε|Xλεεt −x(t)|p+sup0≤t≤Tε|Yλεεt −y(t)|p . The claim
(13) now follows fromnLemmas 3.4, 3.5. o (cid:3)
We now estimate P Bε,δ . Let T denote the right tail of the normal distribution; i.e.,
n
(cid:16) (cid:17)
1 ∞
T (x) , e−t2/2dt for x 0.
√2π ≥
Zx
A simple integration by parts yields
3
(18) T (x) x2e−x2/2, for x 1.
≤ √2π ≥
Proposition 3.6. There exists δ (0,1) and K > 0 such that whenever 0 < δ < δ , ε (0,1),
0 0
∈ ∈
and n 1, we have
≥
δ
(19) P Bε,δ 3T K .
n ≤ ε
(cid:18) (cid:19)
(cid:16) (cid:17)
RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER 9
ε,δ ε,δ,− ε,δ,+
To prove this proposition, we write the event B as the disjoint union B B where
n n n
∪
Bε,δ,− , ω Gε,δ : τε(ω) < t δ and Bε,δ,+ , ω Gε,δ : τε(ω) > t +δ . The quantities
n { ∈ n−1 n n − } n { ∈ n−1 n n }
P(Bε,δ,−) and P(Bε,δ,+) are now estimated separately in Lemmas 3.7 and 3.8 below, whose proofs
n n
are given in Section 4.
Lemma 3.7. There exists K > 0 such that for any n 1, ε,δ (0,1), we have
−
≥ ∈
δ
(20) P Bε,δ,− 2T K .
n ≤ −ε
(cid:18) (cid:19)
(cid:16) (cid:17)
Lemma 3.8. There exists δ (0,1) and K > 0 such that whenever 0 < δ < δ , ε (0,1),
+ + +
∈ ∈
n 1, we have
≥
δ
(21) P Bε,δ,+ T K .
n ≤ +ε
(cid:18) (cid:19)
(cid:16) (cid:17)
We now provide
Proof of Proposition 3.6. From Lemmas 3.7 and 3.8, we take K , K K , δ , δ . Now, noting
− + 0 +
that T is strictly decreasing, we use (20) and (21) to get (19). ∧ (cid:3)
Finally, we have
Proof of Theorem 3.1. Fix p [1, ) and let δ , ες where ν < ς < 1. By the Burkholder-Davis-
∈ ∞
Gundyinequalities [KS91, Theorem3.3.28], thereexists auniversalpositive constantk suchthat
p/2
E sup0≤t≤Tε|Wt|p ≤ kp/2Tpε/2. Noting that E[1Ω\GεT,εδ] = Tn=ε1P(Bnε,δ), we see from Propositions
3.3(cid:2) and 3.6 that fo(cid:3)r ε (0,1) small enough,4 P
∈
3T 1
E[(d[0,Tε](Zε,z))p]≤ Cp Tpε(ς−ν)p +εςp+ εν T Kε1−ς +kp/2T3p/2ε(1−3ν/2)p .
(cid:26) (cid:18) (cid:19) (cid:27)
Since 0 ν < 2/3 and ν < ς < 1, straightforward calculations using (18) yield (12). (cid:3)
≤
4. Proofs of Lemmas
Proof of Lemma 3.4. For ω ∈ Ω\GεT,εδ, we have γTε(λε(ω)) = 0, implying (16). So, fix ω ∈ GεT,εδ.
Note that the function λε(ω) is piecewise-linear with “corners” at 0 = s < t < s < <
· 0 1 1 ···
tTε < sTε = Tε. Since ω ∈ GεT,εδ, we have τnε(ω) ∈ [tn −δ,tn +δ] for 1 ≤ n ≤ Tε. Recalling that
λε (ω) = τε(ω), it is now easy to see that
tn n
λε (ω) λε (ω) λε (ω) λε (ω) δ
(22) max tn − sn−1 1 sn − tn 1 .
1≤n≤Tε(cid:26)(cid:12) tn−sn−1 − (cid:12)∨(cid:12) sn−tn − (cid:12)(cid:27)≤ tmin
Now,let0 s < t T(cid:12)(cid:12)εandlet u0,u1,...,uk(cid:12)(cid:12) be(cid:12)(cid:12)asequentialenumera(cid:12)(cid:12)tionofallcornersstarting
≤ ≤ (cid:12) { (cid:12)} (cid:12) (cid:12)
just to the left of s and ending just to the right of t, i.e., u s < u < < u < t u . By
0 1 k−1 k
≤ ··· ≤
the triangle inequality, we have
λε(ω) λε (ω)
λε(ω) λε(ω) (t s) t u t − uk−1 1
| t − s − − | ≤ | − k−1|(cid:12) t uk−1 − (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
k−1 (cid:12) λε (ω) λε (ω) (cid:12) λε (ω) λε(ω)
+ u u (cid:12) ui − ui−1 1(cid:12) + u s u1 − s 1 .
i i−1 1
| − | u u − | − | u s −
i=2 (cid:12) i − i−1 (cid:12) (cid:12) 1− (cid:12)
X (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
4Onecan check that 0<ε<(cid:0)t4mTin(cid:1)1/(ς−ν)∧δ(cid:12)01/ς will suffice. (cid:12) (cid:12) (cid:12)
10 RANDOMLY PERTURBED SWITCHING DYNAMICS OF A DC/DC CONVERTER
Noting that t u , u u ,..., u s are less than t s , recalling the piecewise-linear
k−1 k−1 k−2 1
| − | | − | | − | | − |
nature of λε(ω), and using (22), we get
t
λε(ω) λε(ω) k λε (ω) λε (ω) 2T δ
t − s 1 ui − ui−1 1 ε .
(cid:12) t−s − (cid:12) ≤ i=1(cid:12) ui−ui−1 − (cid:12) ≤ tmin
(cid:12) (cid:12) X(cid:12) (cid:12)
Thus, (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
2T δ λε(ω) λε(ω) 2T δ
log 1 ε log t − s log 1+ ε .
− tmin ≤ t s ≤ tmin
(cid:18) (cid:19) (cid:18) − (cid:19) (cid:18) (cid:19)
Using the estimate log(1 x) 2x for x 1/2 [Bil, pp. 127], we get that for 0 < δ tmin/4Tε,
| ± |≤ | | | |≤ ≤
we have
4T δ λε(ω) λε(ω) 4T δ
ε log t − s ε .
− tmin ≤ t s ≤ tmin
(cid:18) − (cid:19)
Since s,t are arbitrary, (16) follows. (cid:3)
Proof of Lemma 3.5. We first bound sup0≤t≤Tε|Xλεεt −x(t)|p. Write Xλεεt −x(t) = 3i=1Lit,ε where
L1t,ε , 1[σnε−1,τnε)(λεt)· αβon + Xnσnε−−11,ε,off− αβon e−αon(λεt−σnε−1) P
n≥1 (cid:26) (cid:18) (cid:19) (cid:27)
X
β β
1 (t) + xn−1,off(s ) e−αon(t−sn−1) ,
− [sn−1,tn) · αon n−1 − αon
n≥1 (cid:26) (cid:18) (cid:19) (cid:27)
X
L2t,ε , 1[τnε,σnε)(λεt)·xref e−αoff(λεt−τnε)− 1[tn,sn)(t)·xrefe−αoff(t−tn),
n≥1 n≥1
X X
L3t,ε , 1[σnε−1,τnε)(λεt)·εe−αonλεt Iλεt −Iλεt∧σnε−1 .
nX≥1 (cid:16) (cid:17)
We start by noting that L1,ε(ω), L2,ε(ω) are bounded for all (t,ω) [0,T ] Ω. We will now
t t ∈ ε ×
show that for ω Gε,δ, L1,ε , L2,ε are in fact of order δ. We note that if ω Gε,δ, then
λε(ω) [σε (ω),∈τε(ωT))ε iff|tt |[s| t ,|t ) for all 1 n T . Thus, we have for each t∈ [0T,εT ],
t ∈ n−1 n ∈ n−1 n ≤ ≤ ε ∈ ε
1GεT,εδ ·L1t,ε = 1GεT,εδ ·n≥11[sn−1,tn)(t)·(cid:26)(cid:18)Xnσnε−−11,ε,off− αβon(cid:19)e−αon(λεt−σnε−1)
X
β
xn−1,off(s ) e−αon(t−sn−1)
n−1
− − αon
(cid:18) (cid:19) (cid:27)
= 1GεT,εδ · 1[sn−1,tn)(t)· Xnσnε−−11,ε,off−xn−1,off(sn−1) e−αon(λεt−σnε−1)
nX≥1 n(cid:16) (cid:17)
β
+ xn−1,off(sn−1) e−αon(λεt−σnε−1) e−αon(t−sn−1) .
− αon −
(cid:18) (cid:19) (cid:27)
(cid:16) (cid:17)
It is now easy to see that
β
(cid:12)1GεT,εδ ·L1t,ε(cid:12) ≤ 1GεT,εδ ·nX≥11[sn−1,tn)(t)·(cid:26)xrefαoffδ+(cid:18)αon −x∗(cid:19)αonδ(cid:27).
(cid:12) (cid:12)
Hence, there e(cid:12)xists K >(cid:12)0 such that for all t [0,T ],
1 ε
∈
(23) L1,ε(ω) 1 (ω)K δ+1 (ω)K .
| t | ≤ GεT,εδ 1 Ω\GεT,εδ 1