# On large deviation properties of Brownian motion with dry friction

Yaming Chen<EMAIL_ADDRESS>Wolfram Just<EMAIL_ADDRESS>School of
Mathematical Sciences, Queen Mary University of London, London E1 4NS, United
Kingdom

(September 17, 2014)

###### Abstract

We investigate piecewise-linear stochastic models as with regards to the
probability distribution of functionals of the stochastic processes, a
question which occurs frequently in large deviation theory. The functionals
that we are looking into in detail are related to the time a stochastic
process spends at a phase space point or in a phase space region, as well as
to the motion with inertia. For a Langevin equation with discontinuous drift,
we extend the so-called backward Fokker-Planck technique for nonnegative
support functionals to arbitrary support functionals, to derive explicit
expressions for the moments of the functional. Explicit solutions for the
moments and for the distribution of the so-called local time, the occupation
time and the displacement are derived for the Brownian motion with dry
friction, including quantitative measures to characterize deviation from
Gaussian behavior in the asymptotic long time limit.

###### pacs:

02.50.–r, 05.40.–a, 46.55.+d, 46.65.+g

##  I Introduction

The study of functionals of stochastic processes attracts considerable
interest from various points of view, most notably in the context of large
deviation theory where the probability distribution of integrals or sums of
random variables result in a potential characterizing large deviations from
the mean and extreme events (see Touchette (2009) for a recent review). Some
simple functionals can characterize the time a process spends at a phase space
point or in a phase space region, or more simply such integrals may just
represent physical quantities like the position of an object Majumdar (2005).
Here we consider these functionals in the context of piecewise-smooth
stochastic systems. We use the motion of a Brownian particle subjected to dry
friction as an illustrative example where explicit expressions for the
distribution of functionals can be derived in a formally exact manner, and
where we can quantify the deviation from Gaussian behavior which prevails in
the asymptotic limit.

Functionals of a process $v(t)$ have been investigated intensively in the past
and have found numerous applications in physics Majumdar (2005). For instance,
two of the popular functionals are the local time
$\int_{0}^{t}\delta(v(\tau))d\tau$ and the occupation time
$\int_{0}^{t}\theta(v(\tau))d\tau$ (see, e.g., Sabhapandit et al. (2006)),
which describe how much time the process $v(t)$ has visited the origin and how
long it has taken positive values in the time window $[0,t]$, respectively.
Here $\delta(v)$ is the Dirac delta function and $\theta(v)$ is the step
function with $\theta(v)=1$ for $v>0$ and $\theta(v)=0$ for $v\leqslant 0$. In
addition, the area under the process, i.e., $\int_{0}^{t}v(\tau)d\tau$, which
is referred to as displacement if $v(t)$ denotes the velocity of an object, is
of particular importance due to its physical meaning.

To find statistical properties of functionals is usually nontrivial even for
some of the simplest functionals, such as the local time and the occupation
time. For the displacement, which is equivalent to solve a stochastic
differential equation with inertial term, one often restricts the study to the
overdamped case, in which limit the inertial term can be neglected Risken
(1989). To the best of our knowledge, the distributions of the displacement
are only available in closed analytic form for the cases of the pure diffusion
process and the Ornstein-Uhlenbeck process Kramers (1940). To compute the
distribution of a functional there are two significant achievements, i.e., the
celebrated Feynman-Kac formula developed by Kac Kac (1949) and the so-called
backward Fokker-Planck technique proposed by Majumdar and Comtet Majumdar and
Comtet (2002). The former enables one to obtain a corresponding Schrödinger
equation for the distribution of a functional and the later results in a
backward Fokker-Planck equation for the distribution. The advantage of the
later over the former is that the later depends only on the initial condition
of the process, no extra integral over $v$ is required to obtain the
distribution of the functional.

In this paper, we attempt to derive analytic results for some functionals of
the Brownian motion with dry (also called solid or Coulomb) friction,

$\dot{v}(t)=-\mu\sigma(v(t))-\gamma v(t)+\sqrt{D}\xi(t),$ (1)

which is used to describe the velocity of a solid object of unit mass
subjected to a dry friction in addition to a viscous drag and a random noise
(see, e.g., Kawarada and Hayakawa (2004); Hayakawa (2005); de Gennes (2005);
Touchette et al. (2010)). Here $\sigma(v)$, denoting the sign of $v$,
represents the dry friction force with coefficient $\mu>0$, $\gamma\geqslant
0$ denotes the viscous coefficient and $D>0$ is the strength of the Gaussian
white noise $\xi(t)$ characterized by

$\langle\xi(t)\rangle=0,\qquad\langle\xi(t)\xi(t^{\prime})\rangle=2\delta(t-t^{\prime}).$
(2)

The notation $\langle\cdots\rangle$ stands for the average over all possible
realizations of the noise. For this piecewise-linear stochastic model, the
transition probability distribution or propagator can be obtained by using
spectral decomposition methods Touchette et al. (2010) or Laplace transforms
Touchette et al. (2012). Especially, when $\gamma=0$, i.e., the model (1) with
pure dry friction (also call Brownian motion with two-valued drift), the
corresponding propagator is available in closed analytic form (see, e.g.,
Karatzas and Shreve (1984, 1991); Touchette et al. (2010)). The weak-noise
approximation of the model (1) has also been investigated in detail by using a
path integral approach Baule et al. (2010, 2011); Chen et al. (2013). In
addition, the first-passage time distributions of Eq. (1) can be derived by
using spectral decomposition methods or solving the backward Kolmogorov
equation (see Chen and Just (2014)). The dry friction effect in stochastic
settings is also observed in experiments, such as those investigated recently
by Chaudhury et al. Chaudhury and Mettu (2008); Goohpattader et al. (2009);
Goohpattader and Chaudhury (2010, 2012) and Gnoli et al. Gnoli et al. (2013a,
b). Since the motion of two solid objects over each other is ubiquitous in
nature Li et al. (2011); Baule and Sollich (2012, 2013); Weymouth et al.
(2013); Sano and Hayakawa (2014), the dry friction model (1) plays a
significant role at the interface between theory and experiment. Therefore, it
is interesting to have a close look at stochastic properties of functionals
whose underlying dynamics is governed by this model. In particular we will
focus on the local time, the occupation time and the displacement in this
paper.

It is worth noting that the local time and the occupation time of the pure dry
friction case [i.e., $\gamma=0$ in Eq. (1)] have been investigated in Majumdar
and Comtet (2002); Sabhapandit et al. (2006) even though the authors did not
attempt to study the dry friction effect in a stochastic setting. An integral
representation of the occupation time distribution of this simple model has
also been derived in Simpson and Kuske (2012). But the recursive relations for
the moments of the local time and the occupation time have not been given
explicitly. In addition, to the best of our knowledge analytic results of the
displacement for the pure dry friction case and functionals of the full model
(1) are not available in the literature.

The rest of this paper is arranged as follows. We first extend in Sec. II the
backward Fokker-Planck technique for a positive support functional to an
arbitrary support functional by replacing the double Laplace transform used in
Majumdar and Comtet (2002) with a Fourier-Laplace transform. Rather than
solving the backward equation directly, we then derive from this equation a
recursive ordinary differential equation (ODE) for the moments of the
functional. In Secs. III and IV, we show that the moments of the local time
and the occupation time are given explicitly by the solution of the
corresponding homogeneous ODE. Then we apply the results to solve the dry
friction model (1). In Sec. V, even though a simple formula is not available
for a general drift, we show that the moments of the displacement can be
obtained in the Laplace space for the dry friction model (1). Finally, results
are summarized in Sec. VI.

##  II Remarks on general piecewise-smooth stochastic systems

Let us consider the functional

$T=\int_{0}^{t}U(v(\tau))d\tau$ (3)

for an integrable kernel $U(v)$. The stochastic process $v(t)$ with initial
condition $v(0)=v_{0}$ is assumed to obey the following Langevin equation

$\dot{v}(t)=-\Phi^{\prime}(v(t))+\xi(t)$ (4)

with potential $\Phi(v)$, where $\xi(t)$ denotes the Gaussian white noise
defined by Eq. (2). In what follows, we will first discuss some general
properties of the distribution $p(T,t,v_{0})$ of the functional (3). In the
latter parts we are then going to derive relations for the moments
$M_{n}(t,v_{0})$ of $T$ which will be used in the applications.

###  II.1 Distribution of the functional

While it may be difficult to derive closed analytic expressions for the
distribution $p(T,t,v_{0})$ of the functional (3), some statements can be made
on the asymptotic properties in the limit of large $t$ if we assume that the
potential of the Langevin equation (4) is stable, i.e.,
$\Phi(v)\rightarrow\infty$ when $v\rightarrow\pm\infty$, and that the
correlation of the corresponding process decays sufficiently fast. Then a
simple heuristic argument suggests (see, e.g., Sabhapandit et al. (2006)) that
the distribution $p(T,t,v_{0})$ is simply Gaussian around its mean in the long
time limit. Thus the limiting distribution is characterized only by the first
and the second moments,

$p^{\text{asym}}(T,t,v_{0})=\frac{1}{\sqrt{2\pi\mathrm{Var}(T)}}\exp\left[-\frac{(T-M_{1}(t,v_{0}))^{2}}{2\mathrm{Var}(T)}\right],$
(5)

where $M_{1}(t,v_{0})$ and $\mathrm{Var}(T)$ stand for the first moment and
the variance of $T$, respectively. As stated in Sabhapandit et al. (2006), the
argument is as follows: At large time $t$, assuming the process $v(t)$ to be
mixing, the corresponding propagator $p(v,t|v_{0},0)$ tends to the stationary
distribution $\exp[-\Phi(v)]/Z$, where $Z$ is a normalization factor. Hence,
at time increments which exceed the correlation time of the system, the random
variables $U(v(\tau))-\langle U(v(\tau))\rangle$ are only weakly correlated.
Therefore, from the definition (3) we expect a central limit theorem to hold
in the limit of $t$ being much larger than the relaxation time and the
correlation time of the two aforementioned variables, resulting finally in a
Gaussian distribution for the functional $T$ around its mean. Hence, in order
to determine the limiting distribution of $T$ around its mean we only need to
obtain the first and the second moments, which can be attained by using the
results stated in the following subsections. Our arguments will be able to
predict higher-order cumulants as well so that we provide a quantitative tool
to benchmark the aforementioned reasoning.

For the subsequent investigations we will need the boundary conditions of the
distribution $p(T,t,v_{0})$ in the limit when $v_{0}$ tends to infinity. These
boundary conditions depend on the particular function $U(v)$ which enters the
functional $T$. They can be derived from the observation that a process
starting at infinity cannot cross the origin in finite time. For simplicity
let us discuss two particular choices, namely $U(v)=\delta(v)$ and
$U(v)=\theta(v)$, corresponding to the so-called local time

$T_{\text{loc}}=\int_{0}^{t}\delta(v(\tau))d\tau$ (6)

which measures the time that the process visits the origin, and the so-called
occupation time

$T_{\text{occ}}=\int_{0}^{t}\theta(v(\tau))d\tau$ (7)

which measures the time that the process takes positive values. Hence for the
distribution of the local time we have

$p_{\text{loc}}(T,t,v_{0}\rightarrow\pm\infty)=\delta(T),$ (8)

and for the occupation time the boundary condition reads

$\displaystyle p_{\text{occ}}(T,t,v_{0}\rightarrow\infty)=\delta(T-t),$
$\displaystyle p_{\text{occ}}(T,t,v_{0}\rightarrow-\infty)=\delta(T).$ (9)

The third quantity we are going to discuss in some detail will be the
displacement of the particle which can also be written as a functional with
the choice $U(v)=v$, i.e.,

$T_{\text{dis}}=\int_{0}^{t}v(\tau)d\tau.$ (10)

For the displacement, the argument of the boundary condition is slightly more
subtle. We expect that for $|v_{0}|\gg t$ the velocity of the particle does
not change considerably in the time window $[0,t]$ and that the displacement
should be at the scale of $v_{0}t$. Therefore, the corresponding boundary
condition can be formally approximated as
$p_{\text{dis}}(T,t,v_{0})\sim\delta(T-v_{0}t)$ in the asymptotic limit
$v_{0}\rightarrow\pm\infty$. Such a condition will be used later to obtain the
boundary condition for the moments of the displacement.

###  II.2 Moments of the functional

We start to derive one of the main results by assuming the potential $\Phi(v)$
to be continuous and to be smooth everywhere apart from $v=0$ where we allow
the potential to be not differentiable, so that the deterministic part of the
Langevin equation (4) has a discontinuity at $v=0$. More general cases of
piecewise-smooth systems could be considered with ease but the discussion of
this special case will be sufficient to uncover the general structure. In
addition, our particular choice also contains the dry friction model (1). For
a nonnegative support functional it has been shown by Majumdar and Comtet
Majumdar and Comtet (2002) that the double Laplace transform of the
distribution $p(T,t,v_{0})$ satisfies a backward equation (see Eq. (4) in
Majumdar and Comtet (2002)). In this paper we do not confine the discussion to
nonnegative functionals, i.e., $T$ can be negative. Hence, instead of using
the double Laplace transform we have to consider the Fourier-Laplace transform
of the distribution $p(T,t,v_{0})$, i.e.,

$r(k,s,v_{0})=\int_{-\infty}^{\infty}e^{-ikT}dT\int_{0}^{\infty}e^{-st}p(T,t,v_{0})dt.$
(11)

Similarly, using the method stated in Majumdar and Comtet (2002) (see also
Sabhapandit et al. (2006)) it is straightforward to show that $r(k,s,v_{0})$
satisfies the following backward equation

$\displaystyle\partial^{2}_{v_{0}}r(k,s,v_{0})-\Phi^{\prime}(v_{0})\partial_{v_{0}}r(k,s,v_{0})-[ikU(v_{0})+s]r(k,s,v_{0})+1=0.$
(12)

Let us assume that the $n$-th moment $M_{n}(t,v_{0})$ of $T$ exists so that
the moments of $T$ in the Laplace space are given by

$\displaystyle\widetilde{M}_{n}(s,v_{0})=\int_{0}^{\infty}e^{-st}M_{n}(t,v_{0})dt=\left.i^{n}\partial^{n}_{k}r(k,s,v_{0})\right|_{k=0}$
$\displaystyle\mbox{for }n\geqslant 1.$ (13)

Thus by acting the operator $i^{n}\partial^{n}_{k}$ on Eq. (12) and setting
$k=0$, we obtain a hierarchy of equations for $\widetilde{M}_{n}(s,v_{0})$,

$\displaystyle\partial^{2}_{v_{0}}\widetilde{M}_{n}(s,v_{0})-\Phi^{\prime}(v_{0})\partial_{v_{0}}\widetilde{M}_{n}(s,v_{0})-s\widetilde{M}_{n}(s,v_{0})=-nU(v_{0})\widetilde{M}_{n-1}(s,v_{0}),\qquad\widetilde{M}_{0}(s,v_{0})=r(0,s,v_{0})=1/s,$
(14)

which is easier to deal with than solving Eq. (12).

###  II.3 Boundary conditions

In order to solve Eq. (14), it is essential to know the boundary conditions
for the moments in the limit $v_{0}\rightarrow\pm\infty$. Using Eq. (8) and
the uniform convergence of the Laplace transform for $\mbox{Re}(s)>0$ it
follows that the moments of the local time obey

$\widetilde{M}^{\text{loc}}_{n}(s,v_{0}\rightarrow\pm\infty)=0\qquad\mbox{for
}\mbox{Re}(s)>0\mbox{ and }n\geqslant 1,$ (15)

whereas a similar argument and Eq. (9) yield for the moments of the occupation
time that

$\widetilde{M}^{\text{occ}}_{n}(s,v_{0}\rightarrow\infty)=\frac{n!}{s^{n+1}},\qquad\widetilde{M}^{\text{occ}}_{n}(s,v_{0}\rightarrow-\infty)=0$
(16)

for $\mbox{Re}(s)>0$ and $n\geqslant 1$. For the displacement, the reasoning
presented at the end of subsection II.1 suggests that the $n$-th moment of the
displacement as a function of $v_{0}$ should be bounded by a polynomial of
order $n$ in $v_{0}$. We will see later that this condition is sufficient to
determine the solution of the dry friction model (1).

###  II.4 Matching conditions

Since we are here concerned with a piecewise-smooth potential $\Phi(v)$, say
with a kink at $v=0$, we have to solve Eq. (14) for $v_{0}>0$ and $v_{0}<0$,
respectively. Even for a piecewise-smooth potential $\Phi(v)$ the solution of
Eq. (14) has to be at least continuous, so that we require continuity of the
moments at $v_{0}=0$, resulting in the first matching condition

$\widetilde{M}_{n}(s,0-)=\widetilde{M}_{n}(s,0+).$ (17)

As for the first derivative we may obtain a matching condition by integrating
Eq. (14) across the discontinuity at $v_{0}=0$. Because of the continuity of
the moments the right hand side will contribute only if the kernel $U(v)$ has
a $\delta$ singularity at $v=0$. Thus for the local time (6) the matching
condition reads

$\partial_{v_{0}}\widetilde{M}^{\text{loc}}_{n}(s,0+)-\partial_{v_{0}}\widetilde{M}^{\text{loc}}_{n}(s,0-)=-n\widetilde{M}^{\text{loc}}_{n-1}(s,0),$
(18)

whereas for any other kernel, e.g., for the occupation time and for the
displacement the matching condition just simplifies to

$\partial_{v_{0}}\widetilde{M}^{\text{occ}/\text{dis}}_{n}(s,0+)=\partial_{v_{0}}\widetilde{M}^{\text{occ}/\text{dis}}_{n}(s,0-).$
(19)

Here we have used the shorthand notation
$\partial_{v_{0}}\widetilde{M}_{n}(s,0\pm)$ to denote
$\partial_{v_{0}}\widetilde{M}_{n}(s,v_{0})|_{v_{0}\rightarrow 0\pm}$.

###  II.5 Structure of the solution

Let us briefly discuss how we are going to approach the analytic solution of
the hierarchy (14) for a general potential $\Phi(v)$ with nonanalyticity at
$v=0$. Suppose that the appropriate fundamental piecewise-smooth solution
$\varphi(s,v_{0})$ of the corresponding homogeneous ODE of Eq. (14) is known,
which vanishes for $\mbox{Re}(s)>0$ when $v_{0}\rightarrow\pm\infty$. The
solution obeys

$\left[\partial_{v_{0}}^{2}-\Phi^{\prime}(v_{0})\partial_{v_{0}}-s\right]\varphi(s,v_{0})=0$
(20)

for $v_{0}\neq 0$ and we assume $\varphi(s,v_{0})$ to be continuous at
$v_{0}=0$. To start with, we consider the hierarchy (14) separately on the two
domains $v_{0}>0$ and $v_{0}<0$. Given $\widetilde{M}_{0}(s,v_{0})=1/s$ [see
Eq. (14)] we denote by $\widetilde{M}^{p}_{1}(s,v_{0})$ a particular solution
of the inhomogeneous differential equation (for $n=1$) which fulfils the
boundary condition at infinity. By adding a multiple of the homogeneous
solution $\varphi(s,v_{0})$ we then obtain two branches with two constants of
integration which are adjusted according to the matching conditions (see
subsection II.4). Hence, at each level of the hierarchy we determine a
particular solution $\widetilde{M}^{p}_{n}(s,v_{0})$ and construct the general
solution, which already satisfies the boundary condition at infinity (see
subsection II.3), through

$\widetilde{M}_{n}(s,v_{0})=\widetilde{M}_{n}^{p}(s,v_{0})+\left\\{\begin{array}[]{ll}C^{+}_{n}\varphi(s,v_{0}),&v_{0}>0,\\\
C^{-}_{n}\varphi(s,v_{0}),&v_{0}<0.\end{array}\right.$ (21)

Finally we compute the coefficients $C^{\pm}_{n}$ with the help of the
matching conditions at $v_{0}=0$ (see subsection II.4). The relevance of the
trivial Eq. (21) comes from the observation that we just have to deal with the
matching conditions caused by the nonanalyticities of the potential.
Furthermore, in most of the cases considered here, we will be able to express
the particular solutions $\widetilde{M}^{p}_{n}(s,v_{0})$ in terms of the
homogeneous solution $\varphi(s,v_{0})$ as well, so that the entire structure
just requires solving Eq. (20).

We may not be able to do the inverse Laplace transform for the moments
analytically. However, in the long time limit the behavior of the moments is
dominated by the singular terms of the Laplace transform and those are often
not too difficult to evaluate. We can also resort to numerical Laplace
inversion, such as the so-called Talbot method Abate and Valkó (2004), which
usually gives accurate results even at a very short time.

In the next three sections, we will apply the formula (21) to the local time,
the occupation time, and the displacement. The analysis of the local time can
be largely done for general potential while we are going to derive explicit
formulas for the other two observables for the dry friction model (1) without
and with viscous friction, respectively.

##  III Local time

To begin with we first consider the local time problem, i.e., the setup with
kernel $U(v)=\delta(v)$. In this case it is possible to derive closed form
expressions for a general potential $\Phi(v)$ with discontinuity at $v=0$. We
will use this simple case to illustrate our approach.

###  III.1 General drift

As the matching condition (18) takes care of the singularities at $v=0$ we
only need to consider the hierarchy (14) for nonvanishing arguments, i.e., we
can confine our study to the homogeneous equation (20). Hence in Eq. (21) the
particular solution vanishes and the moments are expressed in terms of the
fundamental solution $\varphi(s,v_{0})$. The coefficients follow immediately
from the matching conditions at $v_{0}=0$ [see Eqs. (17) and (18)] and we
arrive at the recursive relation

$\widetilde{M}^{\text{loc}}_{n}(s,v_{0})=\frac{n\varphi(s,v_{0})\widetilde{M}^{\text{loc}}_{n-1}(s,0)}{\partial_{v_{0}}\varphi(s,0-)-\partial_{v_{0}}\varphi(s,0+)}\qquad\mbox{for
}n\geqslant 1.$ (22)

Therefore, the higher moments are determined by the first moment via

$\widetilde{M}^{\text{loc}}_{n}(s,v_{0})=n!\big{[}s\widetilde{M}^{\text{loc}}_{1}(s,0)\big{]}^{n-1}\widetilde{M}^{\text{loc}}_{1}(s,v_{0}),$
(23)

where

$\widetilde{M}^{\text{loc}}_{1}(s,v_{0})=\frac{\varphi(s,v_{0})/s}{\partial_{v_{0}}\varphi(s,0-)-\partial_{v_{0}}\varphi(s,0+)}.$
(24)

This simple relation enables one to obtain the distribution of the local time
explicitly in the Fourier-Laplace space, which reads [see Eq. (11)]

$\displaystyle r(k,s,v_{0})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{(-ik)^{n}}{n!}\widetilde{M}^{\text{loc}}_{n}(s,v_{0})$
(25) $\displaystyle=$
$\displaystyle\frac{\widetilde{M}^{\text{loc}}_{1}(s,v_{0})}{s\widetilde{M}^{\text{loc}}_{1}(s,0)}\frac{1}{1+iks\widetilde{M}^{\text{loc}}_{1}(s,0)}.$

Then inverting the Fourier transform with respect to $k$ yields the Laplace
transform of the distribution for the local time

$\widetilde{p}_{\text{loc}}(T,s,v_{0})=\frac{\widetilde{M}^{\text{loc}}_{1}(s,v_{0})}{s^{2}[\widetilde{M}^{\text{loc}}_{1}(s,0)]^{2}}\exp\left(-\frac{T}{s\widetilde{M}^{\text{loc}}_{1}(s,0)}\right)$
(26)

for $T\geqslant 0$, which indicates that the distribution is fully determined
by the Laplace transform of its first moment. The result (26) extends that
obtained in Sabhapandit et al. (2006) to be valid for arbitrary initial
condition $v_{0}$ [see Eq. (30) therein]. In general, we may not be able to do
the inverse Laplace transform for Eq. (26) analytically. However, we can
resort to numerical inversion (e.g., the Talbot method Abate and Valkó (2004))
to produce the distribution $p_{\text{loc}}(T,t,v_{0})$ numerically,
especially for short time $t$.

###  III.2 Pure dry friction

In order to make further progress and to illustrate the effectiveness of the
approach let us consider the particular model (1) with only dry friction
($\gamma=0$). In this case, we let $\mu=D=1$. Other values can be covered by
using the appropriate rescaling

$x=\mu v/D,\qquad\tau=\mu^{2}t/D.$ (27)

Hence the pure dry friction case can be written in the form (4) with a stable
potential

$\Phi(v)=|v|.$ (28)

The appropriate fundamental solution of the homogeneous ODE (20) corresponding
to Eq. (14) is

$\varphi(s,v_{0})=e^{(1-\sqrt{1+4s})|v_{0}|/2},$ (29)

which vanishes for $\mbox{Re}(s)>0$ when $v_{0}\rightarrow\pm\infty$. Then,
from Eqs. (26) and (24) we obtain the local time distribution in the Laplace
domain,

$\widetilde{p}_{\text{loc}}(T,s,v_{0})=\frac{\sqrt{1+4s}-1}{s}e^{-(\sqrt{1+4s}-1)(T+|v_{0}|/2)}$
(30)

for $T\geqslant 0$. In this case we can even perform the inverse Laplace
transform explicitly. From the table of Laplace transforms in Abramowitz and
Stegun (1968) we have the identity

$\mathcal{L}^{-1}\left(\frac{e^{-k\sqrt{\alpha}}}{b+\sqrt{\alpha}}\right)=\frac{e^{-k^{2}/(4t)}}{\sqrt{\pi
t}}-be^{bk+b^{2}t}\mathrm{erfc}\left(b\sqrt{t}+\frac{k}{2\sqrt{t}}\right),$
(31)

where $\mathrm{erfc}(z)$ is the complementary error function. Hence, using the
shifting property of Laplace transform, we obtain from Eq. (30) the inverse
Laplace transform, i.e, the local time distribution

$\displaystyle p_{\text{loc}}(T,t,v_{0})$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi t}}e^{-(2T+|v_{0}|-t)^{2}/(4t)}$ (32)
$\displaystyle-e^{2T+|v_{0}|}\mathrm{erfc}\left(\frac{\sqrt{t}}{2}+\frac{2T+|v_{0}|}{2\sqrt{t}}\right)$

for $T\geqslant 0$, which generalizes the result in Sabhapandit et al. (2006)
to be valid for any $v_{0}$ [see Eq. (A4) therein].

In most of cases the inverse Laplace transform cannot be performed
analytically. Thus let us discuss how the previous analysis based on moments
[see, e.g., Eq. (5)] matches with the exact expression (32). We will also use
the opportunity to illustrate the method to derive asymptotic results, which
will be used later in cases where no exact distribution function is available.
Using Eqs. (23), (24) and (29) the Laplace transform of the moments of the
local time are explicitly given by

$\widetilde{M}^{\text{loc}}_{n}(s,v_{0})=\frac{n!}{4^{n}s^{n+1}}(\sqrt{1+4s}+1)^{n}e^{(1-\sqrt{1+4s})|v_{0}|/2}.$
(33)

The expressions have a pole at $s=0$ and a branch cut on the real axis for
$s<-1/4$. As for the inverse transform we can use the standard asymptotic
result (see, e.g., Murray (1974)) that the time dependence of
$M_{n}^{\text{loc}}(t,v_{0})$ is determined by the pole at $s=0$ up to
contributions of $O(e^{-(1/4-o)t})$. Here the notation $o$ stands for an
arbitrary small positive correction to $1/4$. This correction is due to a
power-law correction to the leading exponential behavior. The same notation
will be used in the rest of this paper. Hence the moments in the asymptotic
long time limit read

$\displaystyle M^{\text{loc}}_{n}(t,v_{0})$ (34) $\displaystyle=$
$\displaystyle
4^{-n}\left.\frac{d^{n}}{ds^{n}}\left[(\sqrt{1+4s}+1)^{n}e^{(1-\sqrt{1+4s})|v_{0}|/2+st}\right]\right|_{s=0}$
$\displaystyle+O(e^{-(1/4-o)t}).$

In particular, for the first two moments we have the simple expressions

$\displaystyle M^{\text{loc}}_{1}(t,v_{0})$ $\displaystyle=$
$\displaystyle\frac{t}{2}+\frac{1-|v_{0}|}{2}+O(e^{-(1/4-o)t}),$ (35)
$\displaystyle M^{\text{loc}}_{2}(t,v_{0})$ $\displaystyle=$
$\displaystyle\frac{t^{2}}{4}+\frac{(2-|v_{0}|)t}{2}+\frac{v_{0}^{2}-2|v_{0}|-2}{4}$
(36) $\displaystyle+O(e^{-(1/4-o)t}),$

which result in the variance

$\text{Var}(T_{\text{loc}})=\frac{t}{2}-\frac{3}{4}+O(e^{-(1/4-o)t}).$ (37)

Finally the limiting distribution (5) reads

$\displaystyle p_{\text{loc}}^{\text{asym}}(T,t,v_{0})$ (38) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi(t-3/2)}}\exp\left[-\frac{(2T-t+|v_{0}|-1)^{2}}{4t-6}\right]$
$\displaystyle+O(e^{-(1/4-o)t}).$

As seen in Fig. 1, this limiting distribution in leading order matches well
with the exact analytic expression (32) at large time $t$.

Figure 1: (Color online) Local time distribution of the pure dry friction case
[see Eqs. (4) and (28)] for $v_{0}=0$ and two different times: (a) $t=5$ and
(b) $t=20$. Lines correspond to the leading order limiting distribution (38),
and points to the analytic expression (32). Deviations from Gaussian behavior
are noticeable for small time only.

From the perspective of large deviation results it is of interest to look at
the asymptotic properties of the higher order cumulamnts as well. Of course
they can be worked out rather straightforwardly from the expression (34). For
instance, for the third and the fourth cumulants we obtain

$\displaystyle K^{\text{loc}}_{3}=7/4+O(e^{-(1/4-o)t}),$ $\displaystyle
K^{\text{loc}}_{4}=-45/8+O(e^{-(1/4-o)t}),$ (39)

a result which shows that both are negligible compared to the first two
moments [see Eqs. (35) and (36)] in the asymptotic limit of large $t$. Hence
this simple indicator confirms that the large deviation function of the local
time, defined by
$p_{\text{loc}}(t\vartheta,t,v_{0})\sim\exp(-t\phi(\vartheta))$ is quadratic
(see also Fig. 1), a feature which of course can be confirmed as well if one
employs the exact analytic expression of the distribution function (32) and
evaluates $t^{-1}\ln p_{\text{loc}}(t\vartheta,t,v_{0})$ in the limit
$t\rightarrow\infty$ (see also Sabhapandit et al. (2006)).

###  III.3 Dry and viscous friction

Let us now add viscous friction to the setup discussed in the previous
subsection. Intuitively one probably would not expect any major change as a
linear viscous force is unlikely to have any impact on the predominantly
Gaussian behavior found previously. For the full force case (1), we let
$\gamma=D=1$ without loss of generality. Other values are covered by the
rescaling

$x=(\gamma/D)^{1/2}v,\qquad\tau=\gamma t.$ (40)

Hence the full model can be written in the form (4) with a stable potential

$\Phi(v)=(|v|+\mu)^{2}/2.$ (41)

The appropriate fundamental solution of the homogeneous ODE (20) is given by

$\displaystyle\varphi(s,v_{0})=e^{(|v_{0}|+\mu)^{2}/4}D_{-s}(|v_{0}|+\mu),$
(42)

which vanishes for $\mbox{Re}(s)>0$ in the limit $v_{0}\rightarrow\pm\infty$.
Here we use the symbol $D_{\nu}(z)$ to denote the parabolic cylinder functions
Buchholz (1969).

As already stated in the previous subsection we now employ the general
expression (21) without any particular inhomogeneous solution and use the
matching conditions (17) and (18) to compute the Laplace transform of the
moments. In fact the general expressions Eqs. (23) and (24) immediately yield

$\displaystyle\widetilde{M}^{\text{loc}}_{n}(s,v_{0})$ $\displaystyle=$
$\displaystyle\frac{n!}{2^{n}s^{n+1}}\left(\frac{D_{-s}(\mu)}{D_{-s-1}(\mu)}\right)^{n-1}$
(43)
$\displaystyle\times\frac{e^{(|v_{0}|+\mu)^{2}/4}D_{-s}(|v_{0}|+\mu)}{e^{\mu^{2}/4}D_{-s-1}(\mu)}.$

For real argument $z$ the parabolic cylinder function fulfils $D_{\nu}(z)\neq
0$ for any $\nu$ with nonvanishing imaginary part and $D_{\nu}(z)>0$ for any
negative real value of $\nu$. Hence, we conclude that all the singularities of
Eq. (43) lie on the nonpositive real axis as $\mu>0$. In addition all these
singularities are poles. The largest negative pole which determines the
asymptotic properties of the moments is given by

$s_{0}=\max\\{s:D_{-s-1}(\mu)=0\\}<-1.$ (44)

The dependence of this value on the dry friction coefficient $\mu$ is
displayed in Fig. 2 and turns out to be a monotonically decreasing function.
It is now again straightforward to write down the leading order asymptotic
expansion of the inverse Laplace transform, i.e., of the time dependent
moment, resulting in

$\displaystyle
M^{\text{loc}}_{n}(t,v_{0})=\left.\frac{e^{(|v_{0}|+\mu)^{2}/4}}{2^{n}e^{\mu^{2}/4}}\frac{d^{n}}{ds^{n}}\left[\left(\frac{D_{-s}(\mu)}{D_{-s-1}(\mu)}\right)^{n-1}\frac{D_{-s}(|v_{0}|+\mu)}{D_{-s-1}(\mu)}e^{st}\right]\right|_{s=0}+O(e^{(s_{0}+o)t}),$
(45)

where $s_{0}+o<0$.  In particular the leading terms of the the first two
moments in the long time limit are given explicitly as follows,

$\displaystyle
M^{\text{loc}}_{1}(t,v_{0})=\frac{D_{0}(\mu)t}{2D_{-1}(\mu)}+\frac{D_{0}(\mu)}{2D_{-1}(\mu)}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu+|v_{0}|)}{D_{0}(\mu+|v_{0}|)}\right)+O(e^{(s_{0}+o)t}),$
(46) $\displaystyle
M^{\text{loc}}_{2}(t,v_{0})=\frac{D_{0}^{2}(\mu)t^{2}}{4D_{-1}^{2}(\mu)}+\frac{D_{0}^{2}(\mu)}{2D_{-1}^{2}(\mu)}\left(2\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}-\frac{D_{0}^{(1,0)}(\mu+|v_{0}|)}{D_{0}(\mu+|v_{0}|)}\right)t+O(1).$
(47)

Here the symbol $D^{(1,0)}_{\Lambda}(z)$ denotes the derivative of the
parabolic cylinder function with respect to its index, i.e.,
$\left.\partial_{\nu}D_{\nu}(z)\right|_{\nu=\Lambda}$. Thus we obtain for the
leading order term of the variance the expression

$\mathrm{Var}(T_{\text{loc}})=\frac{D_{0}^{2}(\mu)}{2D_{-1}^{2}(\mu)}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)t+O(1).$
(48)

Here we have stated the results for the second moment and the variance up to
$O(1)$ as the constant term is too cumbersome to be written down explicitly.
Of course, including the constant term the expression is again correct up to
$O(e^{(s_{0}+o)t})$. As shown in Fig. 3, the corresponding asymptotic
distribution (5) matches well with the numerical evaluation of Eq. (26) for
times which are sufficiently large.

Figure 2: (Color online) Maximal solution $s=s_{0}$ (44) of the equation
$D_{-s-1}(\mu)=0$ as a function of $\mu$.

Figure 3: (Color online) Local time distribution of the dry and viscous
friction case [see Eqs. (4) and (41)] for $v_{0}=0$, $\mu=1$ and two different
times: (a) $t=5$ and (b) $t=30$. Lines corresponds to the leading-order
limiting distribution (5), and points to the numerical evaluation of Eq. (26).

To quantify the accuracy of the asymptotic Gaussian distribution we can now
check as well for the size of the higher order cumulants. From Eq. (43) we can
easily confirm that to leading order the third and the fourth cumulants
satisfy

$K_{3}^{\text{loc}}=c_{3}^{\text{loc}}(\mu)t+O(1),\qquad
K_{4}^{\text{loc}}=c_{4}^{\text{loc}}(\mu)t+O(1),$ (49)

where both coefficients $c_{3}^{\text{loc}}(\mu)$ and
$c_{4}^{\text{loc}}(\mu)$ are independent of the initial value $v_{0}$.
Intuitively, we expect that we recover the result of the previous subsection
[see Eq. (39)] if the dry friction term dominates the viscous friction force.
That means that the coefficients $c_{3}^{\text{loc}}(\mu)$ and
$c_{4}^{\text{loc}}(\mu)$ in Eq. (49) which govern the linear time dependence
of the cumulants decay to zero for large $\mu$ [see also Eq. (39)]. Indeed,
numerical evaluation of the coefficients confirms what one expect intuitively
(see Fig. 4). It is however remarkable and somehow counterintuitive that the
addition of a viscous force results in large cumulants as compared to the pure
dry friction case [see Eq. 39)]. The linear increase of the cumulants with
time indicates that the entirely Gaussian behavior of large fluctuations of
the pure dry friction model, i.e., a strictly quadratic large deviation
function, is modified by non-Gaussian contributions due to viscous damping. In
fact, Fig. 4 shows a resonance like phenomenon, where the deviations from
Gaussian behavior become maximal at some intermediate value of $\mu$. The
fourth order cumulant becomes maximal in absolute value if the relative
strength of dry and viscous friction attains a certain optimal value.

Figure 4: (Color online) Coefficients $c^{\text{loc}}_{3}(\mu)$ and
$c^{\text{loc}}_{4}(\mu)$ in the third and the fourth cumulants of the local
time of the dry and viscous friction case [see Eq. (49)].

##  IV Occupation time

As we have seen in the previous section, largely closed analytic expressions
can be derived for the distribution function of the local time. To check the
robustness of the conclusions against the change of the observable we now
consider the occupation time (7). We will see that we can again provide a
largely analytic account if we still follow the strategy outlined in the
previous section.

###  IV.1 General drift

Let us first discuss to which extent we can derive properties of the
occupation time distribution and its moments without specifying the drift. As
pointed out in Eq. (21) the construction of particular solutions of the
inhomogeneous moment hierarchy (18) is the key, if we assume a fundamental
solution of the homogeneous equation (20) to be given.

We first observe that the kernel $U(v)=\theta(v)$ vanishes for $v<0$, so that
for a particular solution of the moment hierarchy (18), we may assume
$\widetilde{M}_{n}^{p}(s,v_{0})=0$ for $v_{0}<0$. Furthermore, the first
equation of the hierarchy ($n=1$) can be solved easily to yield

$\widetilde{M}_{1}^{p}(s,v_{0})=\left\\{\begin{array}[]{lll}1/s^{2}&&\mbox{for
}v_{0}>0,\\\ 0&&\mbox{for }v_{0}<0.\end{array}\right.$ (50)

Now to compute the higher order contributions $\widetilde{M}_{n}^{p}(s,v_{0})$
recursively we observe that the $m$-th order derivative of the homogeneous
equation (20) with respect to $s$ just resembles the equations of the moment
hierarchy, i.e.,

$\left[\partial_{v_{0}}^{2}-\Phi^{\prime}(v_{0})\partial_{v_{0}}-s\right]\partial_{s}^{m}\varphi(s,v_{0})=m\partial_{s}^{m-1}\varphi(s,v_{0}).$
(51)

Hence $\widetilde{M}_{n}^{p}(s,v_{0})$ can be expressed as a linear
combination of partial derivatives of the fundamental solution. Using such an
inductive argument and Eq. (21) we finally arrive at the expression for the
Laplace transform of the $n$-th moment of the occupation time,

$\displaystyle\widetilde{M}^{\text{occ}}_{n}(s,v_{0})=\left\\{\begin{array}[]{lll}n!/s^{n+1}+\sum_{m=0}^{n-1}(-1)^{m}{n\choose
m}C^{+}_{n-m}\partial^{m}_{s}\varphi(s,v_{0})&&\mbox{for }v_{0}>0,\\\
C^{-}_{n}\varphi(s,v_{0})&&\mbox{for }v_{0}<0.\end{array}\right.$ (54)

The coefficients $C^{+}_{n}$ and $C^{-}_{n}$ are determined by the matching
condition at $v_{0}=0$ [see Eqs. (17) and (19)]. Thus, we obtain the explicit
formulas

$\displaystyle
C^{\pm}_{1}=\frac{\partial_{v_{0}}\varphi(s,0\mp)}{s^{2}\varphi(s,0)[\partial_{v_{0}}\varphi(s,0+)-\partial_{v_{0}}\varphi(s,0-)]},$
(55)

and for $n>1$

$\displaystyle C^{\pm}_{n}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}\sum_{m=1}^{n-1}(-1)^{m}{n\choose
m}C^{+}_{n-m}[\partial_{v_{0}}\varphi(s,0\mp)\partial^{m}_{s}\varphi(s,0)$
(56)
$\displaystyle-\varphi(s,0)\partial^{m}_{s}\partial_{v_{0}}\varphi(s,0+)]+\frac{n!\partial_{v_{0}}\varphi(s,0\mp)}{s^{n+1}}\Bigg{\\}}$
$\displaystyle\Big{/}\big{\\{}\varphi(s,0)[\partial_{v_{0}}\varphi(s,0+)-\partial_{v_{0}}\varphi(s,0-)]\big{\\}}.$

While such expressions are certainly more involved than those for the local
time problem we can still express the entire moment problem in terms of the
fundamental solution of the homogeneous equation (20).

###  IV.2 Pure dry friction

For the model with only dry friction, i.e., with the the potential (28), the
fundamental solution of the corresponding homogeneous system is given by Eq.
(29). Hence, it is straightforward to obtain the moments of the occupation
time in the Laplace space by using the formula (54). The first two moments are
given explicitly by

$\displaystyle\widetilde{M}^{\text{occ}}_{1}(s,v_{0})=\frac{1}{2s^{2}}\left\\{\begin{array}[]{lll}2-e^{-(\sqrt{1+4s}-1)v_{0}/2}&&\mbox{for
}v_{0}>0,\\\ e^{(\sqrt{1+4s}-1)v_{0}/2}&&\mbox{for
}v_{0}<0,\end{array}\right.$ (59)
$\widetilde{M}^{\text{occ}}_{2}(s,v_{0})=\frac{1}{4s^{3}}\left\\{\begin{split}&8-\frac{5\sqrt{4a+1}+4v_{0}s+1}{\sqrt{4s+1}}e^{-\left(\sqrt{4s+1}-1\right)v_{0}/2}&&\mbox{for
}v_{0}>0,\\\
&\frac{3\sqrt{4s+1}-1}{\sqrt{4s+1}}e^{\left(\sqrt{4s+1}-1\right)v_{0}/2}&&\mbox{for
}v_{0}<0.\end{split}\right.$ (60)

The analytic structure of these expressions is solely determined by the
analytic structure of the fundamental solution and thus coincides with what we
found for the local time, namely a pole at $s=0$ and a branch cut for
$s<-1/4$. Thus the asymptotic expansion of the inverse Laplace transform
results in the long time behavior of the moments,

$\displaystyle
M^{\text{occ}}_{1}(t,v_{0})=\frac{t}{2}+\frac{v_{0}}{2}+O(e^{-(1/4-o)t}),$
(61) $\displaystyle
M^{\text{occ}}_{2}(t,v_{0})=\frac{t^{2}}{4}+\frac{1+v_{0}}{2}t+\frac{v_{0}^{2}-6}{4}+O(e^{-(1/4-o)t}).$
(62)

The variance of the occupation time is easily computed as

$\text{Var}(T_{\text{occ}})=\frac{t}{2}-\frac{3}{2}+O(e^{-(1/4-o)t}),$ (63)

and the limiting distribution (5) reads

$\displaystyle p_{\text{occ}}^{\text{asym}}(T,t,v_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi(t-3)}}\exp\left[-\frac{(2T-t-
v_{0})^{2}}{4t-12}\right]$ (64) $\displaystyle+O(e^{-(1/4-o)t}).$

At large time, this limiting distribution matches well with the Monte Carlo
simulation of the corresponding Langevin equation by using the Euler-Maruyama
scheme Kloeden and Platen (1992) (see Fig. 5).  As shown in Chen et al.
(2013), the application of this scheme is stable for piecewise-smooth
stochastic differential equations and only requires that we choose the
integration time-step small enough.

Figure 5: (Color online) Occupation time distribution of the pure dry friction
case [see Eqs. (4) and (28)] for $v_{0}=0$ and two different times: (a) $t=10$
and (b) $t=20$. Lines correspond to the leading-order asymptotic distribution
(64), and points to the Monte Carlo simulation of the corresponding Langevin
equation by using the Euler-Maruyama scheme with time-step $\Delta t=0.0001$
and an ensemble of $10^{6}$ realizations.  With the chosen step-size and
ensemble the standard numerical errors of the Euler-Maruyama scheme (see Ref.
Gardiner (2009)) are within the resolution of the figure, apart from the
extreme tails of the distribution where the statistics becomes poor.

To estimate whether there are corrections to the leading asymptotic Gaussian
behavior of large deviations [see Eq. (64)] we just work out, as before, the
third and the fourth cumulants

$K^{\text{occ}}_{3}=O(e^{-(1/4-o)t}),\quad
K^{\text{occ}}_{4}=-45/4+O(e^{-(1/4-o)t}).$ (65)

As these quantities show no linear growth with time both become negligible for
large deviation properties and we expect again a purely quadratic large
deviation function (see also Simpson and Kuske (2012)). Thus as for the local
time [see Eq. (39)] Gaussian behavior prevails.

###  IV.3 Dry and viscous friction

The inclusion of a viscous force results in the Langevin equation (4) with the
potential (41). As for the local time the knowledge of the fundamental
solution (42) is sufficient to evaluate the moments of the occupation time by
Eqs. (54), (55), and (56). Even though the involvement of the parabolic
cylinder function makes the computations slightly cumbersome it is possible to
write down explicit results for the low order moments. For the Laplace
transform of the first two moments we obtain

$\displaystyle\widetilde{M}^{\text{occ}}_{1}(s,v_{0})=\frac{1}{2s^{2}D_{-s}(\mu)}\left\\{\begin{array}[]{lll}2D_{-s}(\mu)-e^{v_{0}^{2}/4+\mu
v_{0}/2}D_{-s}(\mu+v_{0})&&\mbox{for }v_{0}>0,\\\ e^{v_{0}^{2}/4-\mu
v_{0}/2}D_{-s}(\mu-v_{0})&&\mbox{for }v_{0}<0,\end{array}\right.$ (68)
$\displaystyle\widetilde{M}^{\text{occ}}_{2}(s,v_{0})$ $\displaystyle=$
$\displaystyle\frac{e^{v_{0}^{2}/4+\mu
v_{0}/2}D_{-s}(\mu+v_{0})}{2s^{3}D_{-s-1}(\mu)D^{2}_{-s}(\mu)}\Big{\\{}D_{-s}(\mu)\left[sD^{(1,0)}_{-s-1}(\mu)-2D_{-s-1}(\mu)\right]+D_{-s-1}(\mu)\left[sD^{(1,0)}_{-s}(\mu)-D_{-s}(\mu)\right]\Big{\\}}$
(69) $\displaystyle+\frac{2}{s^{3}}-\frac{e^{v_{0}^{2}/4+\mu
v_{0}/2}}{s^{2}D_{-s}(\mu)}D_{-s}^{(1,0)}(\mu+v_{0})\quad\mbox{for }v_{0}>0,$
$\displaystyle\widetilde{M}^{\text{occ}}_{2}(s,v_{0})=\frac{e^{v_{0}^{2}/4-\mu
v_{0}/2}D_{-s}(\mu-
v_{0})}{2s^{3}D_{-s-1}(\mu)D_{-s}^{2}(\mu)}\Big{\\{}D_{-s}(\mu)\Big{[}sD^{(1,0)}_{-s-1}(\mu)+D_{-s-1}(\mu)\Big{]}-sD_{-s-1}(\mu)D^{(1,0)}_{-s}(\mu)\Big{\\}}\quad\mbox{for
}v_{0}<0.$ (70)

For the inversion of the Laplace transform in the asymptotic limit of large
time we again have to analyze the singularities of these expressions. Poles
appear at $s=0$ and at values where the parabolic cylinder functions vanish.
It is a rather obvious consequence of the properties of the parabolic cylinder
functions (see subsection III.3) that all these singularities are poles on the
nonpositive real axis and that the largest negative pole appears at $s_{0}+1$,
where $s_{0}$ has been introduced in Eq. (45) (see also Fig. 2). Hence the
standard asymptotic expansion of the inverse Laplace transform results in

$\displaystyle
M^{\text{occ}}_{1}(t,v_{0})=\frac{t}{2}+\frac{\sigma(v_{0})}{2}\left(\frac{D_{0}^{(1,0)}(\mu+|v_{0}|)}{D_{0}(\mu+|v_{0}|)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)+O(e^{(s_{0}+1+o)t}),$
(71) $\displaystyle
M^{\text{occ}}_{2}(t,v_{0})=\frac{t^{2}}{4}+\frac{\sigma(v_{0})}{2}\left(\frac{D_{0}^{(1,0)}(\mu+|v_{0}|)}{D_{0}(\mu+|v_{0}|)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)t+\frac{1}{2}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)t+O(1),$
(72)

where $s_{0}+1+o<0$.  Thus the variance of the occupation time reads

$\mathrm{Var}(T_{\text{occ}})=\frac{1}{2}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)t+O(1).$
(73)

Here we have used, as before, the symbol $D_{\nu}^{(1,0)}$ to denote the
derivative of the parabolic cylinder function with respect to its index. The
first moment (71) and the variance (73) determine the asymptotic behavior of
the limiting distribution (5) which, for large values of $t$, matches well
with the Monte Carlo simulation of the corresponding Langevin equation, as
shown in Fig. 6.

Figure 6: (Color online) Occupation time distribution of the dry and viscous
friction case [see Eqs. (4) and (41)] for $v_{0}=0$, $\mu=1$ and two different
times: (a) $t=5$ and (b) $t=20$. Lines correspond to the leading-order
asymptotic distribution (5), and points to the Monte Carlo simulation of the
corresponding Langevin equation by using the Euler-Maruyama scheme with time-
step $\Delta t=0.0001$ and an ensemble of $10^{6}$ realizations.

In order to quantify deviations from the Gaussian limit we can again check the
values of the third and the fourth cumulants, which are evaluated as

$K^{\text{occ}}_{3}=O(1),\qquad
K^{\text{occ}}_{4}=c_{4}^{\text{occ}}(\mu)t+O(1),$ (74)

where $c_{4}^{\text{occ}}(\mu)$ is independent of the initial value $v_{0}$.
While it is quite straightforward to express such a coefficient in terms of
parabolic cylinder functions we just focus here on a graphical discussion of
the result. As mentioned in the previous section we expect the cumulants to
approach the dry friction case [see Eq. (65)] in the limit when $\mu$ tends to
infinity. Indeed the coefficient $c^{\text{occ}}_{4}(\mu)$ in Eq. (74) decays
to zero, as shown in Fig. 7. Furthermore, the occurrence of a contribution
which is linear in time means that deviations from a purely quadratic large
deviation function appear due to the inclusion of the viscous force, as it was
the case for the local time. Hence such a feature does not seem to be related
to the particular functional that is considered.

Figure 7: (Color online) Coefficient $c^{\text{occ}}_{4}(\mu)$ in the fourth
cumulant of the occupation time of dry and viscous friction case [see Eq.
(74)].

##  V Displacement

Finally we investigate the distribution of the displacement of the object,
which as well can be expressed in terms of the functional (3) with the choice
$U(v)=v$. In this case, particular solutions $\widetilde{M}_{n}^{p}(s,v_{0})$
of the corresponding inhomogeneous ODE (14) are not easy to write down in
general. So a general formula like Eq. (23) or (54) is not available.
Fortunately, for the dry friction model (1) we will see in the following
subsections that particular solutions can be constructed rather
straightforwardly for the pure dry friction case and the full model,
respectively.

###  V.1 Pure dry friction

Let us consider Eq. (14) with the potential (28) and $U(v)=v$ for $n=1$. On
either of the domain $v_{0}>0$ or $v_{0}<0$, it is obvious that a particular
polynomial solution is given by

$\widetilde{M}_{1}^{p}(s,v_{0})=\frac{sv_{0}-\sigma(v_{0})}{s^{3}}.$ (75)

Hence, using Eq. (29) for the homogeneous solution we obtain from Eq. (21) the
expression for the first moment. The two constants are then determined by the
two matching conditions (17) and (19) so that we arrive at

$\displaystyle\widetilde{M}^{\text{dis}}_{1}(s,v_{0})$ $\displaystyle=$
$\displaystyle\frac{sv_{0}-\sigma(v_{0})}{s^{3}}+\frac{\sigma(v_{0})}{s^{3}}\varphi(s,v_{0})$
(76) $\displaystyle=$
$\displaystyle\frac{sv_{0}-\sigma(v_{0})}{s^{3}}+\frac{\sigma(v_{0})}{s^{3}}e^{(1-\sqrt{1+4s})|v_{0}|/2}.$

For the moment of the next order we have to determine a particular solution of
the inhomogeneous equation (14) for $n=2$, i.e., where the right hand side is
essentially given by Eq. (76). While that is certainly a straightforward task
we can uncover the underlying algebraic structure by reducing all the
manipulations to the properties of the fundamental solution (29). Since

$\displaystyle|v_{0}|\varphi(s,v_{0})$ $\displaystyle=$
$\displaystyle|v_{0}|e^{(1-\sqrt{1+4s})|v_{0}|/2}$ (77) $\displaystyle=$
$\displaystyle-\sqrt{4s+1}\partial_{s}e^{(1-\sqrt{1+4s})|v_{0}|/2}$
$\displaystyle=$ $\displaystyle-\sqrt{4s+1}\partial_{s}\varphi(s,v_{0}),$

we can use the property (51) for $m=1$ to construct one particular solution of
Eq. (14) for $n=2$ as

$\displaystyle\widetilde{M}_{2}^{p}(s,v_{0})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1+4s}v_{0}^{2}+2|v_{0}|}{s^{3}(1+4s)}e^{(1-\sqrt{1+4s})|v_{0}|/2}$
(78) $\displaystyle+\frac{(2s|v_{0}|-3)^{2}+8s+3}{2s^{5}}.$

The first term in this expression is caused by the second term in the
inhomogeneity (76), while the second term is a polynomial part due to Eq.
(75). For this contribution we have used the property (77). Finally the second
moment is determined by Eq. (21) and the matching conditions (17) and (19),
resulting in

$\displaystyle\widetilde{M}^{\text{dis}}_{2}(s,v_{0})$ $\displaystyle=$
$\displaystyle-\frac{(11s+3)\left(\sqrt{4s+1}+1\right)}{s^{5}(4s+1)}e^{(1-\sqrt{1+4s})|v_{0}|/2}$
(79) $\displaystyle+\widetilde{M}_{2}^{p}(s,v_{0}).$

Higher order moments can now be constructed easily by using these algebraic
steps repeatedly.

As for the analytic structure, the approach described above clearly shows that
the moments inherit their analytic structure from the homogeneous solution.
Thus the first two moments have a pole at $s=0$ and a branch cut for $s<-1/4$.
The usual asymptotic expansion for the inverse Laplace transform results in

$\displaystyle
M^{\text{dis}}_{1}(t,v_{0})=v_{0}(1+|v_{0}|/2)+O(e^{-(1/4-o)t}),$ (80)
$\displaystyle M^{\text{dis}}_{2}(t,v_{0})$ $\displaystyle=$ $\displaystyle
10t+v_{0}^{4}/4+5|v_{0}|^{3}/3+5v_{0}^{2}-54$ (81)
$\displaystyle+O(e^{-(1/4-o)t}).$

Hence we obtain for the variance of the displacement

$\displaystyle\text{Var}(T_{\text{dis}})$ $\displaystyle=$ $\displaystyle
10t+2|v_{0}|^{3}/3+4v_{0}^{2}-54+O(e^{-(1/4-o)t})$ (82) $\displaystyle=$
$\displaystyle\sigma^{2}(t)+O(e^{-(1/4-o)t}),$

and the limiting distribution (5) reads

$\displaystyle p_{\text{dis}}^{\text{asym}}(T,t,v_{0})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{\sqrt{2\pi\sigma^{2}(t)}}\exp\left[-\frac{(T-v_{0}-v_{0}|v_{0}|/2)^{2}}{2\sigma^{2}(t)}\right]$
(83) $\displaystyle+O(e^{-(1/4-o)t}).$

Around the mean of the displacement, this asymptotic expression matches well
with the Monte Carlo simulation of the corresponding Langevin equation, as
shown in Fig. 8.

Figure 8: (Color online) Displacement distribution of the pure dry friction
case [see Eqs. (4) and (28)] for $v_{0}=0$ and two different times: (a)
$t=300$ and (b) $t=800$. Lines correspond to the leading-order asymptotic
distribution (83), and points to the Monte Carlo simulation of the
corresponding Langevin equation by using the Euler-Maruyama scheme with time-
step $\Delta t=0.001$ and an ensemble of $10^{6}$ realizations.

As already mentioned, the recursive procedure described at the beginning of
this subsection allows also for the computation of the higher order moments,
and thus for the evaluation of the higher order cumulants. However, as the
formulas are too cumbersome a software for symbolic calculation, such as
Mathematica, is useful to handle the expressions. We finally obtain for the
third and the fourth cumulants in the long time limit,

$K^{\text{dis}}_{3}=\left\\{\begin{array}[]{lll}O(1)&&\mbox{for }v_{0}\neq
0,\\\ 0&&\mbox{for }v_{0}=0,\end{array}\right.\qquad
K^{\text{dis}}_{4}=21120t+O(1).$ (84)

In this case the higher order cumulants show a linear time dependence already
in the pure dry friction model. Hence the corresponding large deviation
function does not show a plain quadratic form any more and the strictly
Gaussian behavior does not show up. Such a feature is as well clearly visible
in Fig. 8. It would be worth to explore whether this difference to the
previous cases is related to a functional with an unbounded kernel.

###  V.2 Dry and viscous friction

We now apply the procedure described in the previous subsection to analyze the
distribution of the displacement in a model with dry and viscous friction. To
begin with one particular polynomial solution of Eq. (14) with $U(v)=v$,
potential (41), and $n=1$ is given by

$\widetilde{M}_{1}^{p}(s,v_{0})=\frac{sv_{0}-\mu\sigma(v_{0})}{s^{2}(s+1)}.$
(85)

Hence using the expression for the general solution (21) and the matching
conditions at the discontinuity to determine the constants of integration we
arrive at [see Eq. (21)]

$\widetilde{M}^{\text{dis}}_{1}(s,v_{0})=\frac{sv_{0}-\mu\sigma(v_{0})}{s^{2}(s+1)}+\mu\sigma(v_{0})\frac{\varphi(s,v_{0})}{s^{2}(s+1)\varphi(s,0)}=\frac{sv_{0}-\mu\sigma(v_{0})}{s^{2}(s+1)}+\mu\sigma(v_{0})\frac{e^{(\mu+|v_{0}|)^{2}/4}D_{-s}(\mu+|v_{0}|)}{s^{2}(s+1)e^{\mu^{2}/4}D_{-s}(\mu)}.$
(86)

For the moment of the next order we need to integrate the inhomogeneous
equation (14) where the right hand side is essentially given by the first
order moment (86). As before we can make use of algebraic properties of the
fundamental solution $\varphi(s,v_{0})$ to construct
$\widetilde{M}^{p}_{2}(s,v_{0})$. Using the recurrence relation of the
parabolic cylinder function (see, e.g., Cuyt et al. (2008)), we have

$\displaystyle|v_{0}|\varphi(s,v_{0})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!|v_{0}|e^{(\mu+|v_{0}|)^{2}/4}D_{-s}(\mu+|v_{0}|)=e^{(\mu+|v_{0}|)^{2}/4}\big{[}D_{-s+1}(\mu+|v_{0}|)-sD_{-s-1}(\mu+|v_{0}|)-\mu
D_{-s}(\mu+|v_{0}|)\big{]}$ (87) $\displaystyle=$
$\displaystyle\\!\\!\varphi(s-1,v_{0})-s\varphi(s+1,v_{0})-\mu\varphi(s,v_{0}),$

so that the inhomogeneous part contains a polynomial in $v_{0}$ and the
fundamental solutions $\varphi(s,v_{0})$ and $\varphi(s\pm 1,v_{0})$. If we
now employ a suitable linear combination of Eq. (51) and the obvious identity
[see Eq. (20)]

$\left[\partial_{v_{0}}^{2}-\Phi^{\prime}(v_{0})\partial_{v_{0}}-s\right]\varphi(s+m,v_{0})=m\varphi(s+m,v_{0}),$
(88)

then a particular solution can be written in terms of the fundamental solution
and a polynomial, namely

$\displaystyle\widetilde{M}_{2}^{p}(s,v_{0})$ $\displaystyle=$
$\displaystyle\frac{2\mu
e^{(\mu+|v_{0}|)^{2}/4}}{s^{2}(s+1)e^{\mu^{2}/4}D_{-s}(\mu)}\big{[}D_{-s+1}(\mu+|v_{0}|)+sD_{-s-1}(\mu+|v_{0}|)+\mu\partial_{s}D_{-s}(\mu+|v_{0}|)\big{]}$
(89)
$\displaystyle+\frac{2v_{0}^{2}}{s(s+1)(s+2)}-\frac{2\mu(3s+2)|v_{0}|}{s^{2}(s+1)^{2}(s+2)}+\frac{4(s+\mu^{2})(s+1)+2\mu^{2}s}{s^{3}(s+1)^{2}(s+2)},$

where the first term is caused by the second term in Eq. (86) and the last
three terms are a polynomial part due to Eq. (85). As before, using the
matching conditions we then arrive at

$\displaystyle\widetilde{M}^{\text{dis}}_{2}(s,v_{0})$ $\displaystyle=$
$\displaystyle\widetilde{M}^{p}_{2}(s,v_{0})-\frac{2\mu
e^{(|v_{0}|+\mu)^{2}/4}D_{-s}(|v_{0}|+\mu)}{s^{3}(s+1)^{2}(s+2)e^{\mu^{2}/4}D_{-s}(\mu)D_{-s-1}(\mu)}\Big{\\{}\mu(s+1)(s+2)\left[D_{-s-1}(\mu)-sD^{(1,0)}_{-s-1}(\mu)\right]$
(90)
$\displaystyle+s(s^{2}+2s+2)D_{-s}(\mu)+s(s+1)^{2}(s+2)D_{-s-2}(\mu)\Big{\\}}.$

Higher order moments can be obtained by this algebraic method [see Eqs. (51),
(87) and (88)] in a straightforward way. By expansion, we can check that the
points $s=-1$ and $s=-2$ in Eqs. (86) and (90) are removable singularities. As
for the previous functionals the singularities of the first two moments lie on
the nonpositive real axis, and the subleading pole is given by $s_{0}+1$,
where $s_{0}$ is defined in Eq. (45). Hence, the standard asymptotic Laplace
inversion yields for the moments in the time domain

$\displaystyle
M^{\text{dis}}_{1}(t,v_{0})=v_{0}+\mu\sigma(v_{0})\left(\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}-\frac{D_{0}^{(1,0)}(\mu+|v_{0}|)}{D_{0}(\mu+|v_{0}|)}\right)+O(e^{(s_{0}+1+o)t}),$
(91) $\displaystyle
M^{\text{dis}}_{2}(t,v_{0})=2\left[1+\mu^{2}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)-2\mu\frac{D_{-2}(\mu)}{D_{-1}(\mu)}\right]t+O(1),$
(92)

where $s_{0}+1+o<0$.  Therefore the variance reads

$\mathrm{Var}(T_{\text{dis}})=2\left[1+\mu^{2}\left(\frac{D_{-1}^{(1,0)}(\mu)}{D_{-1}(\mu)}-\frac{D_{0}^{(1,0)}(\mu)}{D_{0}(\mu)}\right)-2\mu\frac{D_{-2}(\mu)}{D_{-1}(\mu)}\right]t+O(1).$
(93)

As shown in Fig. 9, the corresponding limiting distribution (5) in leading
order matches well with the Monte Carlo simulation of the corresponding
Langevin equation.

Figure 9: (Color online) Displacement distribution of the dry and viscous
friction case [see Eqs. (4) and (41)] for $v_{0}=0$, $\mu=1$ and two different
times: (a) $t=20$ and (b) $t=50$. Lines correspond to the leading-order
asymptotic distribution of Eq. (5), and points to the Monte Carlo simulation
of the corresponding Langevin equation by using the Euler-Maruyama scheme with
time-step $\Delta t=0.001$ and an ensemble of $10^{6}$ realizations.

As mentioned above, the higher moments can also be obtained analytically if
one uses the properties (51), (87) and (88). But we resort to computer algebra
packages to perform the corresponding tedious formal calculations. Finally, we
confirm that the third and the fourth cumulants satisfy

$K^{\text{dis}}_{3}=\left\\{\begin{array}[]{lll}O(1)&&\mbox{for }v_{0}\neq
0,\\\ 0&&\mbox{for }v_{0}=0,\end{array}\right.\qquad
K^{\text{dis}}_{4}=c_{4}^{\text{dis}}(\mu)t+O(1),$ (94)

which are consistent with those of the pure dry friction case [see Eq. (84)].
Here the coefficient $c_{4}^{\text{dis}}(\mu)$, which is shown in Fig. 10 as a
function of $\mu$, does not depend on the initial value $v_{0}$. From our
analytic results we recover that in the case $\mu=0$ the third and the fourth
cumulants vanish as the dry and viscous friction model reduces to the
Ornstein-Uhlenbeck process, for which the displacement distribution is
strictly Gaussian. In addition, $c_{4}^{\text{dis}}(\mu)$ almost vanishes in
the region $1\leqslant\mu\leqslant 3$ (see Fig. 10) so that deviations from
Gaussian behavior are hardly noticeable (see Fig. 9).

Figure 10: (Color online) Coefficient $c_{4}^{\text{dis}}(\mu)$ in the fourth
cumulant of the displacement of dry and viscous friction case [see Eq. (94)].

##  VI Conclusion

Large deviation properties of functionals, here integrals over a stochastic
process, have been analyzed for piecewise-smooth stochastic models. We have,
in particular, considered the Brownian motion with dry and viscous friction as
an illustrative example for our approach, where we have derived explicit
results for three particular choices of a functional, the local time which
measures the crossing of the discontinuity, the occupation time measuring the
direction of movement of an object, and the spatial displacement.

To cope with general underlying stochastic models, we have extended the
backward Fokker-Planck technique developed in Majumdar and Comtet (2002) to
provide analytic solutions for a general support functional. For a generic
Langevin equation, the hierarchy of differential equations for the moments of
functionals have been derived and solutions have been provided for the local
time and the occupation time in terms of the solution of the corresponding
homogeneous equation.

Our study of a piecewise-linear model, i.e., the Brownian motion with pure dry
friction serves as a simple case study. For all three functionals the Laplace
transform of the moments can be calculated analytically. From such results
asymptotic properties in the time domain are easily extracted. In principle
one could even invert the transform analytically as in this case the
propagator of the corresponding Fokker-Planck equation is known in closed
analytic form (see, e.g., Touchette et al. (2010)). The results confirm that
the distribution of the functionals become strictly Gaussian in the asymptotic
limit, for the local and for the occupation time, as also clearly demonstrated
by comparing the analytic results with Monte Carlo simulations. Higher order
cumulants contribute for the displacement in the long time limit, so that a
strictly quadratic large deviation function gets modified by these terms.

The inclusion of an additional viscous force introduces, somehow
counterintuitively, non-Gaussian behavior for the large deviation properties
of the local time and the occupation time. In addition, our analytic results
confirm the numerical analysis provided in Menzel and Goldenfeld (2011) by
solving the corresponding Fokker-Planck equation numerically. For that purpose
we need to specialize our general expressions to the case $v_{0}=0$ and
translate our expressions, given in nondimensional units, to the original
scale (1) via (see Touchette et al. (2010))

$\mu\rightarrow\mu/(\gamma D)^{1/2},\quad M_{2}(t,v_{0})\rightarrow
DM_{2}(\gamma t,(\gamma/D)^{1/2}v_{0})/\gamma.$ (95)

In Fig. 11 it is clearly visible that our analytic expressions confirm the
numerical results presented in Menzel and Goldenfeld (2011) (see Figs. 5 and 9
therein) for the second moment and the distribution. As we have been able to
work out higher order cumulants as well we can even quantify deviations from
the predominantly observed Gaussian behavior.

Figure 11: (Color online) Analytic results of the dry and viscous friction
case in the original scaling system (1) for $\gamma=1$ and $v_{0}=0$. (a) The
second moment in leading order obtained from Eq. (92) via the rescaling (95);
(b) the limiting displacement distribution in leading order obtained from Eqs.
(5), (91) and (93) via the rescaling (95). Lines correspond to the results of
$\mu=6$ and $D=5.4$, and points to $\mu=1.1$ and $D=1$. These two sets of
arguments are properly chosen according to a fluctuation dissipation relation
presented in Menzel and Goldenfeld (2011), ensuring that the numerical results
coincide with each other.

The analytic method and the results obtained here allow for a rather detailed
study on how fluctuations and related large deviation properties are affected
by an underlying piecewise-smooth stochastic dynamics. Even a simple case,
e.g., the extension of the model considered here by an additional constant
bias (see, e.g., Touchette et al. (2010)) seems to be promising to analyze in
detail, as for instance with regards to stick-slip transitions on the one
hand, and even to benchmark theoretical investigations with real experiments
Chaudhury and Mettu (2008); Goohpattader et al. (2009); Goohpattader and
Chaudhury (2010, 2012) on the other.

###### Acknowledgements.

Y.C. was supported by the Chinese Scholarship Council. We thank Hugo Touchette
for useful discussions at the beginning of this research.

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Output tracking based on extended observer for nonlinear uncertain systems

Xinhua Wang, Zengqiang Chen, and Zhuzhi Yuan

Department of Automation, Nankai University, Tianjin, China
<EMAIL_ADDRESS>
_Abstract:_ A high-gain extended observer is designed for a class of nonlinear
uncertain systems. This observer has the ability of estimating system
uncertainty, and it can be used to estimate the derivatives of signal up to
order $n$. The controller based on this extended observer can make the
tracking error and its derivatives converge to zero rapidly even when
uncertainties and disturbances exist. The result of simulation indicates that
this method has satisfactory control performance for nonlinear uncertain
systems.

_Keywords:_ nonlinear uncertain system, high-gain extended observer,
estimating uncertainty, tracking error

1\. Introduction

Nonlinear control is a main research field in control theory and engineering
[1-5]. Output tracking problem of nonlinear uncertain systems is a hot topic
of current research, and many control methods have been proposed. In practice,
the derivatives of the tracked signal are generally unknown, which increases
the difficulty of controller design. Output tracking problems with the
assumption of known tracked signal and its derivatives were considered in
[3-5]. Many state observers are poor at observing nonlinear systems and tend
to converge slowly. In [3], a high-gain observer and a sliding mode control
are used for output feedback of nonlinear systems, and the derivatives of
tracked signal are assumed to be known. However, for this method, the obvious
tracking error exists for tracking control of uncertain systems. In [6,7], the
proposed extended state observer has precise estimation performance and strong
ability of disturbance rejection. However, the system stability was not
considered.

In this paper, for a class of nonlinear uncertain systems, a high-gain
extended observer is presented to estimate the system uncertainty and the
unknown states. The observer has rapid convergence rate and accurate
estimation, and the system stability of the extended observer is proved.
Furthermore, a controller is designed based on the extended observer to make
the convergence rate and accuracy of output tracking errors meet the control
requirements.

2\. Problem analysis

The following nonlinear uncertain system is considered:

$\left\\{\begin{array}[]{c}x^{\left(n\right)}=f\left(x,\dot{x},\cdots,x^{\left(n-1\right)},t\right)+b\cdot
u,\\\ \lx@intercol <EMAIL_ADDRESS>(1)

where, the function $f\left(x,\dot{x},\cdots,x^{\left(n-1\right)},t\right)$
includes the uncertainties and disturbances, and its first-order derivative
exists; $u$ is the control input; $b$ is a non-zero constant; the reference
signal is $y_{d}$, and its derivatives are unknown.

Define $x_{1}=x$, $x_{2}=\dot{x}$, $x_{3}=\dot{x}_{2}=x^{\left(2\right)}$, and
$x_{n}=\dot{x}_{n-1}=x^{\left(n-1\right)}$. Then,
$\dot{x}_{n}=x^{\left(n\right)}=f\left(x_{1},x_{2},\cdots,x_{n},t\right)+b\cdot
u$.

Also, define $x_{n+1}=f\left(x_{1},x_{2},\cdots,x_{n},t\right)$,
$\dot{x}_{n+1}=f^{\left(1\right)}\left(x_{1},x_{2},\cdots,x_{n},t\right)=g\left(x_{1},x_{2},\cdots,x_{n},t\right)$,
and
$\left|g\left(x_{1},x_{2},\cdots,x_{n},t\right)\right|+\left|y_{d}^{\left(n+1\right)}\right|\leq
M$, where, $0\leq M<+\infty$. Therefore, the system (1) can be expressed by

$\left\\{\begin{array}[]{l}\dot{x}_{1}=x_{2}\\\ \vdots\\\
\dot{x}_{n-1}=x_{n}\\\ \dot{x}_{n}=x_{n+1}+b\cdot u\\\
\dot{x}_{n+1}=g\left(x_{1},x_{2},\cdots,x_{n},t\right)\\\
y=x_{1}\end{array}\right.$ (2)

Then, the system error is

$\left\\{\begin{array}[]{l}\dot{e}_{1}=e_{2}\\\ \vdots\\\
\dot{e}_{n-1}=e_{n}\\\ \dot{e}_{n}=e_{n+1}+b\cdot u\\\
\dot{e}_{n+1}=g\left(x_{1},x_{2},\cdots,x_{n},t\right)-y_{d}^{\left(n+1\right)}\end{array}\right.$
(3)

where,

$\displaystyle e\left(t\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}e_{1}&\cdots&e_{n+1}\end{array}\right]^{T}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}x_{1}-y_{d}&\cdots&x_{n}-y_{d}^{\left(n-1\right)}&x_{n+1}-y_{d}^{\left(n\right)}\end{array}\right]^{T}$

In (2), subtracting $y_{d}^{\left(n\right)}$ from both sides, we can get
$\dot{e}_{n}=e_{n+1}+b\cdot u$ in (3). $e_{1}=x_{1}-y_{d}$ is known, and the
other variables are unknown.

3\. Design of extended observer

Choose the positive real numbers $\lambda_{1}$, $\lambda_{2}$, $\cdots$,
$\lambda_{n+1}$ that are not equal to each other, and make
$\underset{i=1}{\overset{n+1}{\Pi}}\left(s+\lambda_{i}\right)=0$ and
$s^{n+1}+h_{1}s^{n}+\cdots+h_{n}s+h_{n+1}=0$ equal. Define

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\min\left\\{\lambda_{1},\lambda_{2},\cdots,\lambda_{n+1}\right\\},$
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}-\frac{h_{1}}{\varepsilon}&1&\cdots&0\\\
\vdots&&\ddots&\\\ -\frac{h_{n}}{\varepsilon^{n}}&0&\cdots&1\\\
-\frac{h_{n+1}}{\varepsilon^{n+1}}&0&\cdots&0\end{array}\right],\text{
}B=\left[\begin{array}[]{c}0\\\ \vdots\\\ 0\\\ 1\end{array}\right]$ (14)
$\displaystyle\varepsilon$ $\displaystyle\in$ $\displaystyle\left(0,1\right)$

For the above matrix $A$, there exist the Vandermonde matrix $T$ such that

$A=T\cdot\text{diag}\left[-\frac{h_{1}}{\varepsilon},\cdots,-\frac{h_{n+1}}{\varepsilon^{n+1}}\right]\cdot
T^{-1}$ (15)

Theorem 1: For the system error (3), the extended observer is designed as
follows:

$\left\\{\begin{array}[]{l}\dot{\widehat{e}}_{1}=\widehat{e}_{2}-\frac{h_{1}}{\varepsilon}\left(\widehat{e}_{1}-e_{1}\right)\\\
\lx@intercol\hfil\vdots\hfil\lx@intercol\\\
\dot{\widehat{e}}_{n}=\widehat{e}_{n+1}-\frac{h_{n}}{\varepsilon^{n}}\left(\widehat{e}_{1}-e_{1}\right)+b\cdot
u\\\
\dot{\widehat{e}}_{n+1}=-\frac{h_{n+1}}{\varepsilon^{n+1}}\left(\widehat{e}_{1}-e_{1}\right)\end{array}\right.$
(16)

Therefore, we get the following conclusions:

1)

$\underset{\varepsilon\rightarrow
0^{+}}{\lim}\left\|\delta\left(t\right)\right\|=0$ (17)

2) When a $\varepsilon\in\left(0,1\right)$ is selected, we get

$\underset{t\rightarrow+\infty}{\lim}\left\|\delta\left(t\right)\right\|\leq\frac{\varepsilon
M}{\lambda}\left\|T\right\|\left\|T^{-1}\right\|$ (18)

where,

$\displaystyle\delta\left(t\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\delta_{1}&\cdots&\delta_{n+1}\end{array}\right]^{T}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\widehat{e}_{1}-e_{1}&\cdots&\widehat{e}_{n+1}-e_{n+1}\end{array}\right]^{T},$
$\displaystyle\widehat{e}\left(t\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\widehat{e}_{1}&\cdots&\widehat{e}_{n+1}\end{array}\right]^{T},$

$\varepsilon\in\left(0,1\right)$ is the perturbation parameter.

_Proof:_ The system error between (4) and (3) is

$\dot{\delta}\left(t\right)=A\delta\left(t\right)+B\left(-g\left(x_{1},x_{2},\cdots,x_{n},t\right)-y_{d}^{\left(n+1\right)}\right)$
(22)

Then, the solution to (9) can be expressed by

$\delta\left(t\right)=\exp\left(A\cdot
t\right)\delta\left(0\right)+\int_{0}^{t}\exp\left(A\left(t-\tau\right)\right)\left(-g\left(x_{1},x_{2},\cdots,x_{n},t\right)-y_{d}^{\left(n+1\right)}\right)d\tau
B$ (23)

Therefore, we get

$\displaystyle\delta\left(t\right)$ $\displaystyle=$
$\displaystyle\left\|\exp\left(A\cdot
t\right)\right\|\left\|\delta\left(0\right)\right\|+M\left\|\int_{0}^{t}\exp\left(A\left(t-\tau\right)\right)d\tau\right\|\left\|B\right\|$
(24) $\displaystyle\leq$
$\displaystyle\left\|T\cdot\text{diag}\left\\{\exp\left(-\frac{\lambda_{1}}{\varepsilon}t\right),\cdots,\exp\left(-\frac{\lambda_{n+1}}{\varepsilon}t\right)\right\\}\cdot
T^{-1}\right\|\left\|\delta\left(0\right)\right\|$
$\displaystyle+M\int_{0}^{t}\left\|\exp\left(A\left(t-\tau\right)\right)\right\|d\tau\left\|B\right\|$
$\displaystyle\leq$
$\displaystyle\left\|T\right\|\left\|T^{-1}\right\|\exp\left(-\frac{\lambda}{\varepsilon}t\right)\left\|\delta\left(0\right)\right\|$
$\displaystyle+\left\|T\right\|\left\|T^{-1}\right\|M\int_{0}^{t}\exp\left(-\frac{\lambda}{\varepsilon}\left(t-\tau\right)\right)d\tau\left\|B\right\|$
$\displaystyle\leq$
$\displaystyle\left\|T\right\|\left\|T^{-1}\right\|\left[\exp\left(-\frac{\lambda}{\varepsilon}t\right)\left\|\delta\left(0\right)\right\|+M\frac{\varepsilon}{\lambda}\left(1-\exp\left(\frac{\lambda}{\varepsilon}t\right)\right)\left\|B\right\|\right]$

Because $\left\|B\right\|=1$ and $\left\|\delta\left(0\right)\right\|$ is
bounded, $\underset{\varepsilon\rightarrow
0^{+}}{\lim}\left\|\delta\left(t\right)\right\|=0$. When
$\varepsilon\in\left(0,1\right)$ is selected, from (11), we can get
$\underset{t\rightarrow+\infty}{\lim}\left\|\delta\left(t\right)\right\|\leq\frac{\varepsilon
M}{\lambda}\left\|T\right\|\left\|T^{-1}\right\|$. This concludes the proof.
$\blacksquare$

The aim of observer design is to make $\widehat{e}_{1}\rightarrow e_{1}$,
$\cdots$, $\widehat{e}_{n+1}\rightarrow e_{n+1}$. This extended observer can
be used to estimate the system uncertainty and signal derivatives up to order
$n$. Although the convergence speed of common linear extended observers are
slower than that of nonlinear extended observers in the neighborhood of the
equilibrium, the use of high gains in the observer can speed up the
convergence. In addition to estimation of unknown error variables and system
uncertainty in (3), the controller $u$ is designed according to the observer
estimation to implement the system tracking.

4\. Controller design

Theorem 2: For the system error (3) and the extended observer (6), a sliding
variable is select as

$\sigma\left(t\right)=\widehat{e}_{n}+a_{n-1}\widehat{e}_{n-1}+\cdots+a_{1}\widehat{e}_{1}$
(25)

where, the polynomial $s^{n-1}+a_{n-1}s^{n-2}+\cdots+a_{1}=0$ is Hurwitz. The
controller is designed as

$\displaystyle u$ $\displaystyle=$
$\displaystyle-b^{-1}\left(U_{0}\text{sign}\left(\sigma\left(t\right)\right)-\left(\frac{h_{n}}{\varepsilon^{n}}+a_{n-1}\frac{h_{n-1}}{\varepsilon^{n-1}}+\cdots+a_{1}\frac{h_{1}}{\varepsilon^{1}}\right)\left(\widehat{e}_{1}\rightarrow
e_{1}\right)\right.$ (26)
$\displaystyle\left.+\widehat{e}_{n+1}+a_{n-1}\widehat{e}_{n}+\cdots+a_{1}\widehat{e}_{2}\right)$

Then, we can get

$\underset{t\rightarrow\infty}{\lim}\left\|e\left(t\right)\right\|\leq
k_{p}\sqrt{\varepsilon}$ (27)

where, $k_{p}$ and $U_{0}$ are the positive constants.

_Proof:_ Select a Lyapunov function candidate as
$V=\frac{1}{2}\sigma^{2}\left(t\right)$. Then, we get

$\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\sigma\left(t\right)\left(\dot{\widehat{e}}_{n}+a_{n-1}\dot{\widehat{e}}_{n-1}+\cdots+a_{1}\dot{\widehat{e}}_{1}\right)$
(28) $\displaystyle=$
$\displaystyle\sigma\left(t\right)\left(\widehat{e}_{n+1}-\frac{h_{n}}{\varepsilon^{n}}\left(\widehat{e}_{1}-e_{1}\right)+bu+a_{n-1}\left(\widehat{e}_{n}-\frac{h_{n-1}}{\varepsilon^{n-1}}\left(\widehat{e}_{1}-e_{1}\right)\right)+\cdots+a_{1}\left(\widehat{e}_{2}-\frac{h_{1}}{\varepsilon}\left(\widehat{e}_{1}-e_{1}\right)\right)\right)$
$\displaystyle=$ $\displaystyle-
U_{0}\left|\sigma\left(t\right)\right|=-\sqrt{2}U_{0}V^{\frac{1}{2}}$

Therefore, there exist a finite time $T_{0}$, for $t\geq T_{0}$, the variables
are in the sliding surface $\sigma\left(t\right)=0$ [8]. From (3) and
$\sigma\left(t\right)=0$, for $t\geq T_{0}$, we get

$\displaystyle\dot{e}_{n-1}$ $\displaystyle=$ $\displaystyle
e_{n}=\widehat{e}_{n}-\delta_{n}=-\left(a_{n-1}\widehat{e}_{n-1}+\cdots+a_{1}\widehat{e}_{1}\right)-\delta_{n}$
(29) $\displaystyle=$
$\displaystyle-\left\\{a_{1}\left(e_{1}+\delta_{1}\right)+\cdots+a_{n-1}\left(e_{n-1}+\delta_{n-1}\right)\right\\}-\delta_{n}$
$\displaystyle=$ $\displaystyle-a_{1}e_{1}-\cdots-
a_{n-1}e_{n-1}-a_{1}\delta_{1}-\cdots-a_{n-1}\delta_{n-1}-\delta_{n}$

From (3) and (16), we get

$\dot{\widetilde{e}}=\widetilde{A}\cdot\widetilde{e}+H\cdot\delta\left(t\right)$
(30)

where,

$\displaystyle\widetilde{e}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}e_{1}&\cdots&e_{n-1}\end{array}\right]^{T},\text{
}\delta\left(t\right)=\left[\begin{array}[]{ccc}\delta_{1}&\cdots&\delta_{n+1}\end{array}\right]^{T},$
(33) $\displaystyle\widetilde{A}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&\cdots&&1\\\
-a_{1}&-a_{2}&\cdots&-a_{n-1}\end{array}\right],$ (38) $\displaystyle H$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{ccccc}0&\cdots&0&0&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&0&0\\\
-a_{1}&\cdots&-a_{n-1}&-1&0\end{array}\right]$ (43)

Because both $\widetilde{A}$ and $A$ are Hurwitz, for the given positive-
definite matrices $Q_{1}$ and $Q_{2}$, there exist the positive-define
matrices $P_{1}$ and $P_{2}$, such that

$P_{1}\widetilde{A}+\widetilde{A}^{T}P_{1}=-Q_{1}\text{,
}P_{2}A+A^{T}P_{2}=-Q_{2}$ (44)

Define $\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right)=\widetilde{e}^{T}P_{1}\widetilde{e}+\delta^{T}\left(t\right)P_{2}\delta\left(t\right)$.
Taking derivative for $\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right)$ along the solutions of equations (9) and (17),
we get

$\dot{\Phi}\left(\widetilde{e},\text{
}\delta\left(t\right)\right)<-\eta_{1}\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right),\text{ }\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right)>r_{1}\varepsilon$ (45)

where, $\eta_{1}$ and $r_{1}$ are the positive constants. Select
$r_{2}>r_{1}$, and define

$\Omega=\left\\{\left.\widetilde{e},\text{
}\delta\left(t\right)\right|\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right)\leq r_{2}\varepsilon\right\\}$ (46)

Then, we can find that, there exists a finite time $t_{1}$, for $t>t_{1}$,
such that $\Phi\left(\widetilde{e},\text{
}\delta\left(t\right)\right)\in\Omega$. Therefore, from (16) and
$\underset{t\rightarrow+\infty}{\lim}\left\|\delta\left(t\right)\right\|\leq\frac{\varepsilon
M}{\lambda}\left\|T\right\|\left\|T^{-1}\right\|$, we can get
$\underset{t\rightarrow\infty}{\lim}\left\|e\left(t\right)\right\|\leq
k_{p}\sqrt{\varepsilon}$, where, $k_{p}$ is a positive constant. This
concludes the proof. $\blacksquare$

5\. Simulation example

The following system is considered:

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle x_{2}$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$
$\displaystyle\cos\frac{\pi}{2}x_{1}-x_{1}^{1/3}-4x_{2}^{1/3}+u$
$\displaystyle y$ $\displaystyle=$ $\displaystyle x_{1}$

The reference signal $y_{d}=2\sin t$. From (2), (3), (6) and $e_{1}=y-y_{d}$,
the designed extended observer is as follows:

$\begin{array}[]{l}\dot{\widehat{e}}_{1}=\widehat{e}_{2}-\frac{6}{0.1}\left(\widehat{e}_{1}-e_{1}\right)\\\
\dot{\widehat{e}}_{2}=\widehat{e}_{3}-\frac{11}{0.1^{2}}\left(\widehat{e}_{1}-e_{1}\right)+u\\\
\dot{\widehat{e}}_{3}=-\frac{6}{0.1^{3}}\left(\widehat{e}_{1}-e_{1}\right)\end{array}$

Select the sliding variable
$\sigma\left(t\right)=\widehat{e}_{2}+\widehat{e}_{1}$, and the controller is
designed as

$u=-\left(4\text{sign}\left(\sigma\left(t\right)\right)-\left(\frac{11}{0.1^{2}}+\frac{6}{0.1}\right)\left(\widehat{e}_{1}-e_{1}\right)+\widehat{e}_{3}+\widehat{e}_{2}\right)$

The plot of output tracking errors is shown in Figure 1.

Figure 1: System output tracking errors.

6\. Conclusion

This paper has presented a method of output tracking control based on extended
observer. From the theoretical analysis and simulation, the convergence rate
and precision of estimation and control are satisfactory. The future job is to
design observer and controller for nonlinear non-minimum-phase systems using
extended observer and the centre-manifold theory.

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