# Minimal excludant over partitions into distinct parts

Prabh Simrat Kaur  Prabh Simrat Kaur, School of Mathematics
Thapar Institute of Engineering and Technology
Patiala 147004, India<EMAIL_ADDRESS>,  Subhash Chand Bhoria
Subhash Chand Bhoria, Pt. Chiranji Lal Sharma Government College, Urban
Estate, Sector-14, Karnal, Haryana 132001, India<EMAIL_ADDRESS>,
Pramod Eyyunni  Pramod Eyyunni, Discipline of Mathematics, Indian Institute of
Technology Indore, Simrol, Indore, Madhya Pradesh - 453552, India.
(Former) Indian Institute of Science Education and Research Berhampur,
Industrial Training Institute (ITI) Berhampur, Engineering School Road,
Berhampur, Odisha - 760010, India<EMAIL_ADDRESS>and  Bibekananda Maji
Bibekananda Maji
Discipline of Mathematics
Indian Institute of Technology Indore
Indore, Simrol, Madhya Pradesh 453552, India<EMAIL_ADDRESS>
###### Abstract.

The average size of the “smallest gap” of a partition was studied by Grabner
and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work
of Fraenkel and Peled, studied the concept of the “smallest gap” under the
name “minimal excludant” of a partition and rediscovered a result of Grabner
and Knopfmacher. In the present paper, we study the sum of the minimal
excludants over partitions into distinct parts, and interestingly we observe
that it has a nice connection with Ramanujan’s function $\sigma(q)$. As an
application, we derive a stronger version of a result of Uncu.

$2020$ Mathematics Subject Classification. Primary 11P81, 11P82, 05A17 ;
Secondary 05A19.
Keywords and phrases. Minimal excludant, MEX, Maximal excludant, Distinct
parts partition, Asymptotic formula.

##  1\. Introduction

Grabner and Knopfmacher [11] studied an interesting partition statistic under
the name ‘smallest gap’. They defined the smallest gap of an integer partition
as the least integer missing from the partition. Fraenkel and Peled introduced
the concept of a minimal excludant of a set $S$ of positive integers, namely,
the least positive integer missing from the set, denoted by
“$\textup{mex}(S)$”. Recently, in $2019$, Andrews and Newman explored the idea
of minimal excludant and in the process, rediscovered a result of Grabner and
Knopfmacher on “smallest gap”. They very naturally generalized this concept to
other arithmetic progressions in their two papers [4, 5]. Let us define

$\sigma\textup{mex}(n):=\sum_{\pi\in\mathcal{P}(n)}\textup{mex}(\pi),$ (1.1)

where $\mathcal{P}(n)$ denotes the collection of all integer partitions of
$n$. Interestingly, Andrews and Newman [4, Theorem 1.1] proved that

$\sigma\textup{mex}(n)=D_{2}(n),$ (1.2)

where $D_{2}(n)$ represents the number of two-colored partitions of $n$ into
distinct parts. The generating function of the above identity was, in fact,
obtained by Grabner and Knopfmacher [11, Theorem 3] in their work on the
smallest gap. They also obtained the following Hardy-Ramanujan-Rademacher type
exact formula for $\sigma\textup{mex}(n)$:

$\displaystyle\sigma\textup{mex}(n)=\frac{\pi}{2\sqrt{6\left(n+\frac{1}{12}\right)}}\sum_{k=1}^{\infty}\frac{A_{2k-1}(n)}{2k-1}I_{1}\left(\pi\frac{\sqrt{2\left(n+\frac{1}{12}\right)}}{\sqrt{3}(2k-1)}\right),$
(1.3)

where

$\displaystyle A_{k}(n)=\sum_{0\leq h<k\atop\gcd(h,k)=1}\exp\left(2\pi
i\left(s(h,k)-s(2h,k)-\frac{hn}{k}\right)\right),$

and $s(h,k)$ denotes the Dedekind sum, and $I_{1}$ denotes the modified Bessel
function of the first kind with index $1$. Moreover, $\sigma\textup{mex}(n)$
satisfies the following asymptotic formula:

$\displaystyle\sigma\textup{mex}(n)\sim\frac{1}{4\sqrt[4]{6n^{3}}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right)\quad\textrm{as}\,\,n\rightarrow\infty.$

A combinatorial proof of (1.2) has been obtained by Ballantine and Merca [6].
Recently, Chern [10] defined maximal excludant $``\textup{maex}(\pi)"$ as the
largest non-negative integer smaller than the largest part of $\pi$ that is
not a part of $\pi$. Analogous to (1.1), Chern defined

$\displaystyle\sigma\textup{maex}(n):=\sum_{\pi\in\mathcal{P}(n)}\textup{maex}(\pi).$

He [10, Theorem 1.1] obtained the following generating function identity for
$\sigma\textup{maex}(n)$:

$\displaystyle\sum_{n=1}^{\infty}\sigma\textup{maex}(n)q^{n}=\sum_{n=1}^{\infty}\frac{n}{(q;q)_{n-1}}\sum_{m=1}^{\infty}q^{m(n+1)}(-q;q)_{m-1}.$
(1.4)

Chern [10, Theorem 1.3] also established an asymptotic formula for
$\sigma\textup{maex}(n)$. More precisely, he showed that

$\displaystyle\sigma\textup{maex}(n)\sim\sigma
L(n),\quad\textrm{as}\,\,n\rightarrow\infty,$

where $\sigma L(n)$ denotes the sum of the largest parts of all partitions of
$n$. Kessler and Livingston showed that the following Hardy-Ramanujan type
asymptotic formula holds for $\sigma L(n)$:

$\displaystyle\sigma
L(n)\sim\frac{\log\frac{6n}{\pi^{2}}+2\gamma}{4\pi\sqrt{2n}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right),\quad\textrm{as}\,\,n\rightarrow\infty.$

Chern [10, Theorem 2.2] also derived a formula for $\sigma\textup{maex}(n)$,
that connects the divisor function and the coefficients of Ramanujan’s
$q$-series $\sigma(q)$, defined in (2.1) below.

In [4], Andrews and Newman studied another arithmetic function, namely,

$\displaystyle
a(n)=\sum_{\pi\in\mathcal{P}(n)\atop\textup{mex}(\pi)\,\textup{odd}}1.$

They111Note that there is a typo in [4, p. 250, Theorem 1.2], in which the
words odd and even are exchanged. observed that $\sigma\textup{mex}(n)\equiv
a(n)\pmod{2}$ and that $a(n)$ is almost always even and is odd exactly when
$n$ is of the form $j(3j\pm 1)$.

We study the questions raised by Andrews and Newman for the function
$\sigma\textup{mex}(n)$ in [4], but restricted to partitions into distinct
parts. We define the function $\sigma_{d}\textup{mex}(n)$ by

$\sigma_{d}\textup{mex}(n):=\sum_{\pi\in\mathcal{D}(n)}\textup{mex}(\pi),$
(1.5)

where $\mathcal{D}(n)$ denotes the collection of partitions of $n$ into
distinct parts. We also define

$\displaystyle
a_{d}(n):=\sum_{\pi\in\mathcal{D}(n)\atop\textup{mex}(\pi)\,\textup{odd}}1.$

In fact, the generating function for $a_{d}(n)$ was considered by Uncu [14] in
a different combinatorial context.

In the next section we state the main results.

##  2\. Main Results

Before stating the main results, let us begin with one of the most important
$q$-series of Ramanujan, which has been a constant source of study from the
point of view of both algebraic and analytic number theory. It is given by

$\displaystyle\sigma(q):=\sum_{n=0}^{\infty}\frac{q^{\frac{n(n+1)}{2}}}{(-q;q)_{n}}.$
(2.1)

Readers are encouraged to see [1], [2] and [3], for deeper results related to
$\sigma(q)$. Surprisingly, the generating function for
$\sigma_{d}\textup{mex}(n)$ is directly connected to $\sigma(q)$ as stated
below.

######  Theorem 2.1.

We have

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{mex}(n)q^{n}=(-q;q)_{\infty}\sigma(q).$

The next result gives us an asymptotic formula for
$\sigma_{d}\textup{mex}(n)$.

######  Theorem 2.2.

We have

$\displaystyle\sigma_{d}{\rm
mex}(n)\sim\frac{\exp\left(\pi\sqrt{\frac{n}{3}}\right)}{2(3n^{3})^{1/4}},\quad\textrm{as}\,\,n\rightarrow\infty.$
(2.2)

Now we will state a result due to Uncu [14, Theorem 3].

######  Theorem 2.3.

Let $U(n)$ be the sequence of numbers defined by

$\displaystyle\sum_{n=0}^{\infty}U(n)q^{n}=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binom{n+1}{2}}}{(-q;q)_{n}}.$

Then $U(n)\geq 0$ for all $n\geq 0$.

Interestingly, we observe that the generating function for $U(n)$ and
$a_{d}(n)$ are indeed the same.

######  Theorem 2.4.

We have

$\sum_{n=0}^{\infty}U(n)q^{n}=\sum_{n=0}^{\infty}a_{d}(n)q^{n}=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binom{n+1}{2}}}{(-q;q)_{n}}.$
(2.3)

An immediate consequence of this result is

######  Corollary 2.5.

For any $n\geq 0$, we have

$\displaystyle U(n)>0\quad\textrm{except for}\,\,n=1.$

Andrews and Newman also defined $\textup{moex}(\pi)$ to be the smallest odd
integer missing from $\pi$. This naturally led them to define the function
$\sigma\textup{moex}(n)$, given by

$\sigma\textup{moex}(n):=\sum_{\pi\in\mathcal{P}(n)}\textup{moex}(\pi).$

We analogously define a quantity for partitions into distinct parts and study
its generating function. Define

$\sigma_{d}\textup{moex}(n):=\sum_{\pi\in\mathcal{D}(n)}\textup{moex}(\pi).$

######  Theorem 2.6.

The generating function for $\sigma_{d}\textup{moex}(n)$ is

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{moex}(n)q^{n}$
$\displaystyle=(-q;q)_{\infty}\left(1+2\sum_{n=1}^{\infty}\frac{q^{n^{2}}}{(-q;q^{2})_{n}}\right)$
$\displaystyle=(-q;q)_{\infty}\left(1+2\sum_{n=1}^{\infty}(-1)^{n-1}q^{n}(q^{2};q^{2})_{n-1}\right).$

In other words,

$\sum_{n=0}^{\infty}\sigma_{d}\textup{moex}(n)q^{n}=(-q;q)_{\infty}\left(1+\sigma^{*}(-q)\right),$

where

$\sigma^{*}(q):=2\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n^{2}}}{(q;q^{2})_{n}}.$

In literature, the $q$-series $\sigma(q)$ and $\sigma^{*}(q)$ are found to
appear simultaneously at many places. Andrews, Dyson and Hickerson [3] proved
that the coefficients of these two $q$-series are very small and related to
the arithmetic of the quadratic real field $\mathbb{Q}(\sqrt{6})$.

Next, similar to the definition (1.5), we define $\sigma_{d}\textrm{maex}(n)$
as

$\displaystyle\sigma_{d}\textrm{maex}(n):=\sum_{\pi\in\mathcal{D}(n)}\textrm{maex}(n).$

The next result provides us a generating function identity for
$\sigma_{d}\textrm{maex}(n)$.

######  Theorem 2.7.

We have

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{maex}(n)q^{n}=\sum_{k=1}^{\infty}k(-q;q)_{k-1}\sum_{m=1}^{\infty}q^{\frac{m(m+1)}{2}+km}.$

One can easily observe that Theorem 2.7 is an analogue of Chern’s identity
[10, (2)]. Before going on to the proofs of our main results, we establish an
auxiliary result in the next section.

##  3\. Preliminaries

For the proof of the asymptotic formula for $\sigma_{d}\textup{mex}(n)$,
namely, Theorem 2.2, we derive a monotonic property of
$\sigma_{d}\textup{mex}(n)$, which is itself of independent interest.

######  Proposition 3.1.

For $n\geq 7$, we have that

$\sigma_{d}\textup{mex}(n+1)>\sigma_{d}\textup{mex}(n).$

  * Proof.

We consider two cases depending on whether $n$ is a triangular number or not.

Case 1: Suppose $n\geq 7$ is not a triangular number, that is, $n\neq
1+2+\cdots+k$ for any positive integer $k$. We define a map $f$ from
$\mathcal{D}(n)$ to $\mathcal{D}(n+1)$ as follows: Consider a distinct parts
partition $\pi$ of $n$. It will be of the form

$\pi=(a_{1},a_{2},\dots,a_{k})\quad\text{with}\ a_{1}>a_{2}>\cdots>a_{k}\geq
1,\ a_{1}+a_{2}+\cdots+a_{k}=n.$ (3.1)

Then let $f(\pi):=\pi^{{}^{\prime}}=(a_{1}+1,a_{2},\dots,a_{k})$. Clearly,
this is a distinct parts partition of $n+1$. Also see that
$(a_{1}+1)-a_{2}\geq 2$, i.e., a partition of the form $f(\pi)$ has a gap of
at least two between its largest part and the next largest part. Moreover, $f$
preserves minimal excludants, that is
$\textup{mex}(\pi)=\textup{mex}(\pi^{{}^{\prime}})$. This is because as $n$ is
not a triangular number, $\pi=(a_{1},\dots,a_{k})$ cannot be the partition
$(k,k-1,\dots,1)$ and consequently, there exists an $i,\ 1\leq i\leq k$ such
that $a_{i}-a_{i+1}>1$ with the interpretation that $a_{k+1}=0$. Choosing
$i_{0}$ to be the largest integer $i$ for which $a_{i}-a_{i+1}>1$ implies that
$\textup{mex}(\pi)=\textup{mex}(\pi^{{}^{\prime}})=a_{i_{0}+1}+1$.

For example, if $n=9$, then the partition $\pi_{1}=(4,3,2)$ satisfies
$k=i_{0}=3$ with $\textup{mex}(\pi_{1})=\textup{mex}(f(\pi_{1}))=1$ and the
partition $\pi_{2}=(8,1)$ satisfies $k=2,i_{0}=1$ with
$\textup{mex}(\pi_{2})=\textup{mex}(f(\pi_{2}))=2$. Thus, we have matched up
the minimal excludants of partitions in $\mathcal{D}(n)$ with those of a
certain collection of partitions in $\mathcal{D}(n+1)$ and hence
$\sigma_{d}\textup{mex}(n)\leq\sigma_{d}\textup{mex}(n+1)$. We next exhibit a
partition in $\mathcal{D}(n+1)$ that does not lie in the image of $f$, thus
proving that the inequality
$\sigma_{d}\textup{mex}(n+1)>\sigma_{d}\textup{mex}(n)$ holds. The idea is to
get hold of a partition $\pi$ in which both $\ell(\pi)$ and $\ell(\pi)-1$
occur as parts so that it cannot be in the image of $f$.

If $n$ is odd: Consider the partition
$\lambda_{1}=\left(\frac{n+1}{2},\frac{n-1}{2},1\right)$ of $n+1$. Note that
$\frac{n+1}{2}-\frac{n-1}{2}=1$ and that $\lambda_{1}$ has distinct parts as
$(n-1)/2>1$.

If $n$ is even: Look at the partition
$\lambda_{2}=\left(\frac{n}{2}+1,\frac{n}{2}\right)$ of $n+1$, which is again
in $\mathcal{D}(n+1)\setminus f(\mathcal{D}(n))$.

Case 2: Assume that $n\geq 7$ is a triangular number, so that
$n=k+(k-1)+\cdots+1$ for a unique positive integer $k$. As in Case $1$, for
any $\pi$ in $\mathcal{D}(n)$ other than the partition $\mu=(k,k-1,\dots,1)$,
the partition $f(\pi)$ lying in $\mathcal{D}(n+1)$ will have the same minimal
excludant as $\pi$. But for $\mu$, whose minimal excludant is $k+1$, we see
that $\textup{mex}(f(\mu))=k$. This means that
$\sum_{\pi\in\mathcal{D}(n)}\textup{mex}(\pi)=1+\sum_{\pi\in
f(\mathcal{D}(n))}\textup{mex}(\pi)$. In other words, to prove the required
inequality we have to show that the contribution of the partitions from
$\mathcal{D}(n+1)\setminus f(\mathcal{D}(n))$ to the sum of minimal excludants
is at least $2$. We once again proceed according to the parity of $n$.

If $n$ is odd: Consider the partition
$\nu_{1}=\left(\frac{n+1}{2},\frac{n-1}{2},1\right)$ of $n+1$. Observe that
$(n-1)/2>2$ and so $\nu_{1}\in\mathcal{D}(n+1)\setminus f(\mathcal{D}(n))$
with its minimal excludant being $2$.

If $n$ is even: In this case, we look at two partitions given by
$\nu_{2}=(\frac{n}{2}+1,\frac{n}{2})$ and
$\nu_{3}=(\frac{n}{2},\frac{n}{2}-1,2)$. Clearly,
$\nu_{2}\in\mathcal{D}(n+1)\setminus f(\mathcal{D}(n))$ with minimal excludant
$1$. Again, note that $\nu_{3}$ has distinct parts since $\frac{n}{2}-1>2$ and
hence $\textup{mex}(\nu_{3})=1$.

With this, we complete the proof of the proposition. ∎

In the upcoming section, we provide the proofs of our results.

##  4\. Proofs of the main results

###  4.1. Proof of Theorem 2.1

####  4.1.1. First proof of Theorem 2.1

Let $p_{d}^{mex}(m,n)$ be the number of partitions of $n$ into distinct parts
whose minimal excludant is $m$. Then, we have

$\displaystyle\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}p_{d}^{mex}(m,n)z^{m}q^{n}$
$\displaystyle=\sum_{m=1}^{\infty}z^{m}q^{1}\cdot q^{2}\cdots
q^{m-1}\prod_{k=m+1}^{\infty}(1+q^{k})$
$\displaystyle=\sum_{m=1}^{\infty}z^{m}q^{\binom{m}{2}}\prod_{k=m+1}^{\infty}(1+q^{k})$
$\displaystyle=(-q;q)_{\infty}\sum_{m=1}^{\infty}\frac{z^{m}q^{\binom{m}{2}}}{(-q;q)_{m}}.$
(4.1)

Differentiating both sides of (4.1) with respect to $z$ and putting $z=1$, we
get

$\sum_{n=0}^{\infty}\left(\sum_{m=1}^{\infty}mp_{d}^{mex}(m,n)\right)q^{n}=(-q;q)_{\infty}\sum_{m=1}^{\infty}\frac{mq^{\binom{m}{2}}}{(-q;q)_{m}}.$

But
$\displaystyle\sum_{m=1}^{\infty}mp_{d}^{mex}(m,n)=\sigma_{d}\textup{mex}(n),$
the sum of minimal excludants in all the partitions of $n$ into distinct
parts. Thus,

$\sum_{n=0}^{\infty}\sigma_{d}\textup{mex}(n)q^{n}=(-q;q)_{\infty}\sum_{m=1}^{\infty}\frac{mq^{\binom{m}{2}}}{(-q;q)_{m}}.$
(4.2)

We now show that the sum on the right hand side of (4.2) is nothing but
Ramanujan’s series $\sigma(q)$. Start with

$\displaystyle\sum_{n=0}^{\infty}\frac{q^{n(n+1)/2}}{(-q;q)_{n}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(n+1)-n}{(-q;q)_{n}}q^{n(n+1)/2}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(n+1)q^{n(n+1)/2}}{(-q;q)_{n}}-\sum_{n=0}^{\infty}\frac{nq^{n(n+1)/2}}{(-q;q)_{n}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{nq^{n(n-1)/2}}{(-q;q)_{n-1}}-\sum_{n=1}^{\infty}\frac{nq^{n(n+1)/2}}{(-q;q)_{n}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{nq^{n(n-1)/2}}{(-q;q)_{n-1}}\left(1-\frac{q^{n}}{1+q^{n}}\right)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{nq^{n(n-1)/2}}{(-q;q)_{n}}.$

Therefore, from (4.2), we deduce that

$\sum_{n=0}^{\infty}\sigma_{d}\textup{mex}(n)q^{n}=(-q;q)_{\infty}\sigma(q).$
(4.3)

∎
We now give an alternate proof of (4.3), on the lines of Andrews and Newman’s
second proof of the generating function for $\sigma\textup{mex}(n)$. (see [4,
p. $251$])

####  4.1.2. Second proof of Theorem 2.1

Let $\mathcal{D}_{i}(n)$ denote the number of partitions of $n$ into distinct
parts for which $\textup{mex}(\pi)>i$. Then we claim that

$\mathcal{D}_{i}(n)=p_{d}\left(n-\frac{i(i+1)}{2},\ i\right),$ (4.4)

where $p_{d}(m,\ i)$ denotes the number of partitions of $m$ into distinct
parts with smallest part greater than $i$. To see this, start with a distinct
parts partition $\pi$ of $n$ with $\textup{mex}(\pi)>i$. By the definition of
minimal excludant, the integers $1$ through $i$ must all occur as parts in
$\pi$. Moreover, since $\pi$ is a distinct parts partition, each of the
numbers $1$ to $i$ appears exactly once in $\pi$. Subtract the quantity
$1+2+\cdots+i$ from $\pi$. This gives a distinct parts partition
$\pi^{\prime}$ of $n-(1+2+\cdots+i)$ (since we began with a distinct parts
partition $\pi$, removing some parts from it doesn’t affect its distinct
nature). Now, since $\pi$ has only one copy of each of $1$ to $i$,
$\pi^{\prime}$ will not have any parts less than or equal to $i$. Therefore
$\pi^{\prime}$ is a distinct parts partition of $n-\frac{i(i+1)}{2}$ with
smallest part greater than $i$.

Conversely, starting with a distinct parts partition $\lambda$ of
$n-\frac{i(i+1)}{2}$ with $s(\pi)>i$, we add the quantity $1+2+\cdots+i$ to
$\lambda$ to get a distinct parts partition $\lambda^{\prime}$ (since
$\lambda$ had no parts less than or equal to $i$) with the integers $1$ to $i$
all occurring as parts. This means that $\textup{mex}(\lambda^{\prime})>i$.
Hence, this bijection proves the claim in (4.4).

From the definition of $\mathcal{D}_{i}(n)$, $\sigma_{d}\textup{mex}(n)$ can
be expressed as

$\sigma_{d}\textup{mex}(n)=\sum_{i=0}^{\infty}\mathcal{D}_{i}(n),$ (4.5)

since each distinct parts partition $\pi$ with $\textup{mex}(\pi)=i$ is
counted $i$ times on the right hand side of (4.5), once in each of
$\mathcal{D}_{0}(n),\mathcal{D}_{1}(n),\dots,\mathcal{D}_{i-1}(n)$. On the
left hand side of (4.5), we add together the minimal excludants over all the
distinct parts partitions, thus each distinct parts partition $\pi$
contributes a weight $\textup{mex}(\pi)$ to it. Thus, on both sides of
equation (4.5), each distinct parts partition contributes the same number and
hence the identity holds.

Now, the generating function of distinct parts partitions with $s(\pi)>i$ is
simply

$\sum_{n=0}^{\infty}p_{d}(n,\ i)q^{n}=(-q^{i+1};q)_{\infty}.$

Therefore, $\mathcal{D}_{i}(n)$, which is the number of distinct parts
partitions of $n-\frac{i(i+1)}{2}$ with smallest part greater than $i$, will
be the coefficient of $q^{n-\frac{i(i+1)}{2}}$ in $(-q^{i+1};q)_{\infty}$.
Equivalently, this is the coefficient of $q^{n}$ in
$q^{\frac{i(i+1)}{2}}(-q^{i+1};q)_{\infty}$.

Thus,

$\sum_{n=0}^{\infty}\mathcal{D}_{i}(n)q^{n}=q^{\frac{i(i+1)}{2}}(-q^{i+1};q)_{\infty}=(-q;q)_{\infty}\frac{q^{\frac{i(i+1)}{2}}}{(-q;q)_{i}}.$

We are ready to obtain the generating function of $\sigma_{d}\textup{mex}(n)$.
Starting with (4.5), we get

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{mex}(n)q^{n}=\sum_{n=0}^{\infty}q^{n}\sum_{i=0}^{\infty}\mathcal{D}_{i}(n)$
$\displaystyle=\sum_{i=0}^{\infty}\sum_{n=0}^{\infty}\mathcal{D}_{i}(n)q^{n}$
$\displaystyle=\sum_{i=0}^{\infty}(-q;q)_{\infty}\frac{q^{\frac{i(i+1)}{2}}}{(-q;q)_{i}}$
$\displaystyle=(-q;q)_{\infty}\sum_{i=0}^{\infty}\frac{q^{\frac{i(i+1)}{2}}}{(-q;q)_{i}}=(-q;q)_{\infty}\sigma(q).$

∎

###  4.2. Proof of Theorem 2.2

We first state an important asymptotic result for coefficients of a power
series, due to Ingham [12].

######  Proposition 4.1.

Let $A(q)=\sum_{n=0}^{\infty}a(n)q^{n}$ be a power series with radius of
convergence $1$. Assume that $\\{a(n)\\}$ is a weakly increasing sequence of
non-negative real numbers. If there are constants $\alpha,\beta\in\mathbb{R}$,
and $C>0$ such that

$\displaystyle
A(e^{-t})\sim\alpha\,t^{\beta}\exp\left(\frac{C}{t}\right),\,\,\text{
as}\,\,t\rightarrow 0^{+},$

then we have

$\displaystyle
a(n)\sim\frac{\alpha}{2\sqrt{\pi}}\frac{C^{\frac{2\beta+1}{4}}}{n^{\frac{2\beta+3}{4}}}\exp\left(2\sqrt{Cn}\right),\,\,\text{as}\,\,n\rightarrow\infty.$
(4.6)

  * Proof of Theorem 2.2.

Recall that the generating function for $\sigma_{d}{\rm mex}(n)$ is given by
Theorem 2.1, namely,

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{mex}(n)q^{n}=(-q;q)_{\infty}\sigma(q):=B(q).$

In a famous work on quantum modular forms, Zagier [15, p. 7] pointed out that,
for $t\rightarrow 0^{+}$,

$\displaystyle\sigma\left(e^{-t}\right)=2-2t+5t^{2}-\frac{55}{3}t^{3}+\frac{1073}{12}t^{4}-\frac{32671}{60}t^{5}+\cdots.$
(4.7)

Now, using the transformation formula for the Dedekind’s eta-function, one can
show that, for $t\rightarrow 0^{+}$,

$\displaystyle\frac{1}{\left(e^{-t};e^{-t}\right)_{\infty}}\sim\sqrt{\frac{t}{2\pi}}\exp\left(\frac{\pi^{2}}{6t}\right).$
(4.8)

Euler’s identity suggests that

$\displaystyle(-q;q)_{\infty}=\frac{1}{(q;q^{2})_{\infty}}=\frac{\left(q^{2};q^{2}\right)_{\infty}}{(q;q)_{\infty}}.$
(4.9)

Therefore, in view of (4.8) and (4.9), we can derive, for $t\rightarrow
0^{+}$,

$\displaystyle\left(-e^{-t};e^{-t}\right)_{\infty}\sim\frac{1}{\sqrt{2}}\exp\left(\frac{\pi^{2}}{12t}\right).$
(4.10)

Finally, combining (4.7) and (4.10), we arrive at

$\displaystyle
B\left(e^{-t}\right)\sim\sqrt{2}\exp\left(\frac{\pi^{2}}{12t}\right),\quad{\rm
as}\,\,t\rightarrow 0^{+}.$

Again, by Proposition 3.1 we know that the sequence $\\{\sigma_{d}{\rm
mex}(n)\\}$, for $n\geq 7$, is an increasing sequence of positive integers.
Now we are ready to invoke Proposition 4.1, with $\alpha=\sqrt{2},\beta=0$ and
$C=\displaystyle\frac{\pi^{2}}{12}$. Substituting these constants in (4.6), we
obtain

$\displaystyle\sigma_{d}{\rm
mex}(n)\sim\frac{1}{2(3n^{3})^{1/4}}\exp\left(\pi\sqrt{\frac{n}{3}}\right),\quad\text{as}\,\,n\rightarrow\infty.$

This finishes the proof. ∎

In the next subsection, we provide proofs of other results.

###  4.3. Proofs of other results

  * Proof of Theorem 2.4.

Recall that $a_{d}(n)$ counts the number of distinct parts partitions of $n$
with an odd minimal excludant. So the least integer missing from such a
partition can only be of the form $2n+1$ for some $n\geq 0$. And all the
integers from 1 through $2n$ should occur exactly once and for the integers
greater than $2n+1$, they may occur at most once. Putting this together, we
may write

$\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}p_{d}^{omex}(m,n)z^{m}q^{n}=\sum_{k=0}^{\infty}z^{2k+1}q^{1}\cdot
q^{2}\cdots q^{2k}\prod_{\ell=2k+2}^{\infty}(1+q^{\ell}),$ (4.11)

where $p_{d}^{omex}(m,n)$ denotes the number of distinct parts partitions of
$n$ with an odd minimal excludant $m$. Putting $z=1$ in (4.11), we get

$\sum_{n=0}^{\infty}a_{d}(n)q^{n}=\sum_{k=0}^{\infty}q^{\binom{2k+1}{2}}\prod_{\ell=2k+2}^{\infty}(1+q^{\ell})=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{q^{\binom{2n+1}{2}}}{(-q;q)_{2n+1}}.$
(4.12)

Consider the rightmost sum in (4.12). Rewriting it, we get

$\displaystyle\sum_{n=0}^{\infty}\frac{q^{\binom{2n+1}{2}}}{(-q;q)_{2n+1}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{1+\cdots+2n}}{(1+q)\cdots(1+q^{2n+1})}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{1+\cdots+2n}}{(1+q)\cdots(1+q^{2n})}\left\\{1-\frac{q^{2n+1}}{1+q^{2n+1}}\right\\}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{1+\cdots+2n}}{(1+q)\cdots(1+q^{2n})}-\sum_{n=0}^{\infty}\frac{q^{1+\cdots+(2n+1)}}{(1+q)\cdots(1+q^{2n+1})}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{\binom{2n+1}{2}}}{(-q;q)_{2n}}-\sum_{n=0}^{\infty}\frac{q^{\binom{2n+2}{2}}}{(-q;q)_{2n+1}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binom{n+1}{2}}}{(-q;q)_{n}}.$

Putting this in (4.12), we see that

$\sum_{n=0}^{\infty}a_{d}(n)q^{n}=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binom{n+1}{2}}}{(-q;q)_{n}}.$

This completes the proof. ∎

  * Proof of Corollary 2.5.

We note that $a_{d}(n)$ is non-negative for all $n\geq 0$, since it counts
certain kind of partitions. Moreover, for $n>1$, we always have a partition of
$n$ into distinct parts with an odd minimal excludant, namely, the partition
$n$, where the minimal excludant is $1$. So $a_{d}(n)>0$ for all $n>1$, and
hence by (2.3), we conclude that $U(n)>0$ for all $n>1$. ∎

Uncu [14, Theorem 3.2], remarks that the infinite series in (2.3) is a false
theta function studied by Rogers. Further, in the same paper, he gives a
combinatorial explanation of the fact that the coefficients on the right hand
side of (2.3) are non-negative. But, by Corollary 2.5, via our interpretation
in terms of minimal excludant, we have shown that all but one of the
coefficients, namely $a_{d}(1)$, are infact positive.

  * Proof of Theorem 2.6.

Let $p_{d}^{moex}(m,n)$ denote the number of distinct parts partitions $\pi$
of $n$ with $\textup{moex}(\pi)=m$. Consider the following double sum

$\displaystyle\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}p_{d}^{moex}(m,n)z^{m}q^{n}$
$\displaystyle=(-q^{2};q^{2})_{\infty}\sum_{k=0}^{\infty}z^{2k+1}q^{1+3+\cdots+(2k-1)}(-q^{2k+3};q^{2})_{\infty}$
$\displaystyle=(-q^{2};q^{2})_{\infty}\sum_{k=0}^{\infty}z^{2k+1}q^{k^{2}}(-q^{2k+3};q^{2})_{\infty}$
$\displaystyle=(-q;q^{2})_{\infty}(-q^{2};q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{z^{2k+1}q^{k^{2}}(-q^{2k+3};q^{2})_{\infty}}{(-q;q^{2})_{\infty}}$
$\displaystyle=(-q;q)_{\infty}\sum_{k=0}^{\infty}\frac{z^{2k+1}q^{k^{2}}}{(-q;q^{2})_{k+1}}.$

Differentiate with respect to $z$ and then put $z=1$ to get

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{moex}(n)q^{n}$
$\displaystyle=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(2n+1)q^{n^{2}}}{(-q;q^{2})_{n+1}}$
$\displaystyle=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(2n+1)q^{n^{2}}}{(-q;q^{2})_{n}}\left\\{1-\frac{q^{2n+1}}{1+q^{2n+1}}\right\\}$
$\displaystyle=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(2n+1)q^{n^{2}}}{(-q;q^{2})_{n}}-(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(2n+1)q^{(n+1)^{2}}}{(-q;q^{2})_{n+1}}$
$\displaystyle=(-q;q)_{\infty}\sum_{n=0}^{\infty}\frac{(2n+1)q^{n^{2}}}{(-q;q^{2})_{n}}-(-q;q)_{\infty}\sum_{n=1}^{\infty}\frac{(2n-1)q^{n^{2}}}{(-q;q^{2})_{n}}$
$\displaystyle=(-q;q)_{\infty}+2(-q;q)_{\infty}\sum_{n=1}^{\infty}\frac{q^{n^{2}}}{(-q;q^{2})_{n}}.$

Alternatively, using Chern’s result [10, Proposition 2.1] with $x=1,y=-1$
gives us

$\sum_{n=0}^{\infty}\sigma_{d}\textup{moex}(n)q^{n}=(-q;q)_{\infty}+2(-q;q)_{\infty}\sum_{n=1}^{\infty}(-1)^{n-1}q^{n}(q^{2};q^{2})_{n-1}.$

∎

  * Proof of Theorem 2.7.

Suppose $\pi_{d}$ is a distinct parts partition of $n$ with maximal excludant
$k$. Note that $k\geq 1$, since partitions with maximal excludant $0$ do not
contribute to the sum
$\sum_{\pi\in\mathcal{D}(n)}\textup{maex}(\pi)=\sigma_{d}\textup{maex}(n)$. We
can divide $\pi_{d}$ into two components, ${\pi_{d}}^{\prime}$ and
${\pi_{d}}^{\prime\prime}$:
The first component ${\pi_{d}}^{\prime}$ is a distinct parts partition with
parts $\leq k-1$; and the second one ${\pi_{d}}^{\prime\prime}$ is a gapfree
distinct parts partition with $s(\pi)=k+1$, i.e., each integer between
$s(\pi)$ and $\ell(\pi)$ also occurs as a part. Observe that the second
component ${\pi_{d}}^{\prime\prime}$ upon conjugation gives a gapfree
partition in which the smallest part $s(\pi)=1$ and the largest part
$\ell(\pi)$ appears exactly $k+1$ times and all other parts appear exactly
once. We consider a two variable generating function $D(z,q)$ for
$p_{d,k}(n)$, the number of distinct parts partitions of $n$ with maximal
excludant $k$. In $D(z,q)$, the exponent of $z$ indicates the maximal
excludant of a partition $\pi_{d}$ into distinct parts, and the exponent of
$q$, as always, keeps track of the number being partitioned by $\pi_{d}$.

$\displaystyle
D(z,q):=\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}p_{d,k}(n)z^{k}q^{n}=\sum_{k=1}^{\infty}(-q;q)_{k-1}z^{k}\sum_{m=1}^{\infty}q^{1+2+...+(m-1)+(k+1)m}.$

Now differentiating $D(z,q)$ with respect to $z$ and substituting $z=1$, we
get the generating function for $\sigma_{d}\textup{maex}(n)$,

$\displaystyle\sum_{n=0}^{\infty}\sigma_{d}\textup{maex}(n)q^{n}=\sum_{k=1}^{\infty}k(-q;q)_{k-1}\sum_{m=1}^{\infty}q^{\frac{m(m+1)}{2}+km}.$

∎

##  5\. Concluding Remarks

Inspired by the work of Andrews and Newman, in the current paper, we studied
the minimal excludant over partitions into distinct parts. We have proved that
the generating function for $\sigma_{d}{\rm mex}(n)$ is the product of the
generating function for distinct parts partition function and Ramanujan’s
well-known $q$-series $\sigma(q)$. We also established a Hardy-Ramanujan type
asymptotic formula for $\sigma_{d}{\rm mex}(n)$. It would be interesting to
find a Hardy-Ramanujan-Rademacher type exact formula for $\sigma_{d}{\rm
mex}(n)$, analogous to the result (1.3) of Grabner and Knopfmacher for
$\sigma{\rm mex}(n)$.

We also examined $a_{d}(n)$, which counts the number of distinct parts
partitions with an odd minimal excludant. Quite surprisingly, we have observed
that the generating function for $a_{d}(n)$ has been studied by Uncu in a
different context, which immediately improved Uncu’s result [14, Theorem 3].
Subsequently, we studied $\sigma_{d}{\rm moex}(n)$ and its generating function
has been expressed as the product of the generating function for the distinct
parts partition function and $1+\sigma^{*}(-q)$. It is interesting that the
function $\sigma^{*}(q)$ has mostly been seen to appear in the vicinity of
$\sigma(q)$. Recently, using the theory of modular forms, Barman and Singh [7,
8], and Chakraborty and Ray [9] found interesting congruence properties and
density results for Mex-related partition functions. Readers are encouraged to
see the paper of da Silva and Sellers [13] for parity results and congruence
properties related to Mex-related partition functions of Andrews and Newman.

Acknowledgements. We sincerely thank the anonymous referee for providing many
useful suggestions, particularly, for outlining ideas for the proof of Theorem
2.2. The first author wants to thank Prof. Meenakshi Rana for her continuous
support. The second author wants to thank the Department of Mathematics, Pt.
Chiranji Lal Sharma Government College, Karnal for a conducive research
environment. The third author is a SERB National Post Doctoral Fellow (NPDF)
supported by the fellowship PDF/2021/001090 and would like to thank SERB for
the same. In addition, he wishes to thank IIT Indore for its conducive
research environs and also his former employer IISER Berhampur for providing
flexibility in working conditions. The last author wishes to thank SERB for
the Start-Up Research Grant SRG/2020/000144.

## References

  * [1] G. E. Andrews, _Ramanujan’s “Lost” Notebook V: Euler’s partition identity_ , Adv. Math. 61 (1986), 156–164.
  * [2] G.E. Andrews, _Questions and conjectures in partition theory_ , Amer. Math. Monthly 93 (1986), 708–711.
  * [3] G.E. Andrews, F.J. Dyson and D. Hickerson, _Partitions and indefinite quadratic forms_ , Invent. Math. 91 (1988), 391–407.
  * [4] G. E. Andrews, D. Newman, _Partitions and the minimal excludant_ , Ann. Comb. 23 (2019), no. 2, 249–254.
  * [5] G. E. Andrews, D. Newman, _The minimal excludant in integer partitions_ , J. Integer Seq. 23 (2020), Article 20.2.3.
  * [6] C. Ballantine, M. Merca, _Combinatorial proof of the minimal excludant theorem_ , Int. J. Number Theory, 17, No. 8, 1765–1779 (2021).
  * [7] R. Barman, A. Singh, _On Mex-related partition functions of Andrews and Newman_ , Res. Number Theory, 7, 53 (2021).
  * [8] R. Barman, A. Singh, _Mex-related partition functions of Andrews and Newman,_ J. Integer Seq., 24 (2021), Article 21.6.3.
  * [9] K. Chakraborty, C. Ray, _Distribution of generalized Mex-related integer partitions_ , Hardy-Ramanujan Journal, Hardy-Ramanujan Society (Special Commemorative volume in honour of Srinivasa Ramanujan), 43 (2021), 122–128.
  * [10] S. Chern, _Partitions and the maximal excludant_ , Electron. J. Combin. 28 (2021), no. 3, Paper No. 3.13.
  * [11] P. J. Grabner, A. Knopfmacher, _Analysis of some new partition statistics_ , Ramanujan J. 12(3) (2006), 439–454.
  * [12] A. E. Ingham, _A Tauberian theorem for partitions_ , Ann. of Math. (2) 42 (1941), 1075–1090.
  * [13] R. da Silva, J. A. Sellers, _Parity considerations for mex-related partition functions of Andrews and Newman_ , J. Integer Seq. 23, no. 5 (2020), Article 20.5.7
  * [14] A. K. Uncu, _On a weighted spin of the Lebesgue identity_ , Mathematical Aspects of Computer and Information Sciences 2019, 273–279.
  * [15] D. Zagier, Quantum modular forms, in: _Quanta of maths_ , 659–675, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010.


# Speed-Adaptive Model-Free Lateral Control for Automated Cars

Marcos Moreno-Gonzalez  Antonio Artuñedo  Jorge Villagra  Cédric Join  Michel
Fliess  Centre for Automation and Robotics (CSIC-UPM), ctra. de Campo Real, km
0,200, 28500 Arganda del Rey, Spain, (e-mail: {marcos.moreno,
antonio.artunedo, jorge.villagra}@csic.es). CRAN (CNRS, UMR 7039), Universitée
de Lorraine, BP 239, 54506 Vandoeuvre-lès-Nancy, France  LIX (CNRS, UMR 7161),
École Polytechnique, 91128 Palaiseau, France, (e-mail:
<EMAIL_ADDRESS>A.L.I.E.N., 7 rue Maurice Barrès, 54330
Vézelise, France, (e-mail: {cedric.join<EMAIL_ADDRESS>
###### Abstract

In order to increase the number of situations in which an intelligent vehicle
can operate without human intervention, lateral control is required to
accurately guide it in a reference trajectory regardless of the shape of the
road or the longitudinal speed. Some studies address this problem by tuning a
controller for low and high speeds and including an output adaptation law. In
this paper, a strategy framed in the Model-Free Control paradigm is presented
to laterally control the vehicle over a wide speed range. Tracking quality,
system stability and passenger comfort are thoroughly analyzed and compared to
similar control structures. The results obtained both in simulation and with a
real vehicle show that the developed strategy tracks a large number of
trajectories with high degree of accuracy, safety and comfort.

###### keywords:

Autonomous vehicles, Model-free control, Adaptive control applications

††thanks: This work is partially supported by the joint CNRS (France)-CSIC
(Spain) PICS Program through the project UbiMFC (CoopIntEer 203 073)

##  1 Introduction

Over the last years, autonomous driving capabilities have been developed, but
in order to increase the situations in which the vehicle operates without
driver intervention, accurately controlling the vehicle in any scenario is
essential.

In a decoupled control architecture, lateral control keeps the vehicle on the
path without losing stability or impairing passenger comfort, regardless of
the road or speed. This problem has been addressed with different approaches,
the first being to improve the available model of the vehicle in order to
better tune a model-based regulator. Another is the design of controllers for
different longitudinal speeds, and then provide a fitting law between their
different parameters or outputs. To avoid the problems induced by a complex
system identification, some approaches opted to rely on model-independent
strategies.

In this paper, a new lateral control strategy for autonomous vehicles is
presented within the Model-Free Control (MFC) paradigm (Fliess and Join,
2013). The proposed control law is based on the adaptation of one of the key
parameters of MFC as a function of driving speed, allowing thus the vehicle to
cope with a variety of situations without having to re-tune the controller. To
thoroughly evaluate the potential of this strategy, metrics of tracking
quality, stability of the feedback system and passenger comfort are defined. A
multi-objective optimization has been applied in simulation to determine which
control structure provides the best trade-off among the three criteria in a
wide spectrum of situations. The main contribution of the work relies on the
introduction of an easy-to-implement variation of MFC, which has proven to be
very effective for lateral control of automated vehicles.

The rest of the paper is structured as follows. Section II presents a brief
review of the lateral control strategies in the literature. A theoretical
introduction to Model-Free Control is presented in Section III. Section IV
motivates the proposed Speed-Adaptive Model-Free Control strategy. The results
from simulation and real-world tests are presented in Section V. Finally, the
last section draws some concluding remarks and the references.

##  2 State of the art

Lateral control of autonomous vehicles is studied from different approaches,
most of them based on a somewhat realistic model of the vehicle, being the
single-track model the most popular (Arifin et al., 2019), which is linear and
assumes a constant longitudinal velocity. Some real applications show that
this simplification may be inappropriate to control the vehicle in any
scenario. To overcome these limitations, different strategies have been
proposed in the last years. Zainal et al. (2017) applied the single-track
model to fit two PID regulators, one for low and one for high speeds; Liu et
al. (2018) used it to synthesize a robust LQR. Alternatively, Zanon et al.
(2014) rely on a more complex model to develop a Model Predictive Control
(MPC) strategy, and Laghmara et al. (2019) focus on jointly solving the path
planning and control problems; but this strategies are mainly tested on
specific situations.

Real vehicles have complex dynamics that vary with speed and steering angle,
with strong non-linearities, couplings between lateral and longitudinal
dynamics and variability of parameters that are already difficult to
characterize; consequently, it is extremely hard to find a realistic model for
a large spectrum of driving situations. As a result, the potential of control
strategies that do not rely on a vehicle dynamic model has catched attention.

Fuzzy control is a good example of these model-free techniques, as it absorbs
some of the variability of the system parameters and its formulation is
intuitive, but difficult to tune optimally over a wide working range. Two
fuzzy regulators were integrated and validated in traffic-based driving
environments in Godoy et al. (2015); other works (Jin et al., 2017) confirmed
the capabilities of fuzzy logic for lateral control. Another approach is pure
pursuit control (Park et al., 2014), which is based on a kinematic model of
the vehicle, but its performance degrades when high velocities or
accelerations are requested.

The MFC framework evoked in the introduction was successfully applied in
vehicle longitudinal control (Villagra et al., 2009) or in lateral control for
low-speed AGVs (Villagra and Herrero-Perez, 2012). Alternatively, in Menhour
et al. (2013) the flatness theory (Fliess et al., 1995), which allows finding
differentially flat outputs for non-linear systems, is applied to implement
the lateral control of a vehicle together with a model-free feedback
controller. This approach exhibited very good performance in simulation, but
its deployment in real vehicles requires measurements that cannot be obtained
with commercial sensors. Alternatively, (Wang et al., 2022) proposes an
adaptation mechanism for MFC and apply it on a scale car, but the resulting
adaptation dynamics is too slow for automated vehicles driving on real roads.

##  3 Model-free control principles

Fliess and Join (2013) state that the system dynamics can be approximated by
an ultra-local model

$y\textsuperscript{(n)}=F+\alpha\cdot u$ (1)

in which the linear relationship between the input $u$ and the nth derivative
of the output $y$ is fitted by a variable $F$ that absorbs model errors and
system disturbances, and where the ratio constant $\alpha$ is a design
parameter.

The control loop is closed by an intelligent PID controller, iPID controller
(usually iP or iPD):

$u=\frac{1}{\alpha}\cdot\left(-F+y_{r}^{(n)}+K_{p}\,e+K_{i}\int{e}+K_{d}\,\dot{e}\right)$
(2)

where $u$ is the control action, suffix $r$ means reference, $e$ is the
tracking error and $K_{p}$, $K_{i}$ and $K_{d}$ are the control parameters,
emulating those of a PID controller. The term $F$ must be estimated in real
time, for this purpose, it can be assumed to be the same between consecutive
instants and can be estimated from (1) as follows:

$\hat{F}(t_{k})=\hat{y}^{(n)}(t_{k})-\alpha\cdot u(t_{k-1})$ (3)

where $\hat{F}$ is the estimator of $F$, $t_{k}$ is the current instant and
$\hat{y}^{(n)}$ is the filtered nth derivative of $y$.

###### Remark 1

Note that the error dynamics derived from (1) and (2) can be expressed as
$f(e,K_{p},K_{i},K_{d})=\hat{F}-F$. If the estimation of $F$ is good enough
($\hat{F}\approx F$), then the system dynamics could be made asymptotically
stable through an appropriate choice of the control parameters.

##  4 Speed-adaptive lateral control

The parameter $\alpha$ defines in a certain way the aggressiveness of the
iP(D) controller, since the higher is $\alpha$, the smaller the increase in
the control action between sampling instants. Therefore, varying $\alpha$
might adapt the controller aggressiveness to different driving situations.

(a) Trajectory reference and tracking

(b) Reference and vehicle speed

(c) Steering wheel angle in (2)

Figure 1: Comparison between high and low alpha in low speed curves and higher
speed straight stretches

Fig. 1 shows the same MFC regulator with two different $\alpha$: the one with
a low value performs better at low speed curves but becomes highly oscillating
at a stretch where higher speed is allowed; the configuration with high
$\alpha$ is stable for the straight path (with little oscillation) but does
worse tracking at curves. This finding motivated the introduction of a model-
free controller whose $\alpha$ varies as a function of speed $v$. This
controller has a base $\alpha_{0}$ which is kept up to a given speed $v_{0}$,
after which it is increased proportionally to speed variation with a constant
$K_{\alpha}$:

$\alpha=\max\left\\{\alpha_{0},\,K_{\alpha}\cdot(v-v_{0})+\alpha_{0}\right\\}$
(4)

This $\alpha$ variation law allows to obtain a (i) more aggressive behaviour
in urban environments and (ii) smoother actions on the highway, where the
oscillations can impair comfort and lead to system instability due to the high
speed.

##  5 Results

In this section, the control parameter space is explored in simulation
(section 5.3) using a high-fidelity vehicle model (section 5.4). This
evaluation is done with respect to three metrics (defined in 5.2) and is
applied to the proposed feedback control approach, a standard MFC and a PID in
a wide set of driving contexts (described in section 5.1). To confirm the
suitability of the estimated performance potential for each controller
(obtained in 5.5), experimental trials have been prepared and conducted in an
automated prototype, and are reported in section 5.6.

###  5.1 Benchmark description

Fig. 2 shows an aerial view of the three circuits used in the simulations,
which are designed to cover a wide range of realistic driving scenarios. They
have straight sections, wide and sharp curves and different dynamic
constraints, as can be seen in Table 1. The reference trajectories are
generated for each path applying the method for acceleration-limited speed
planning proposed in Artuñedo et al. (2022).

(a) Trajectory $T_{1}$

(b) Trajectory $T_{2}$

(c) Trajectory $T_{3}$

Figure 2: Benchmark trajectories Table 1: Dynamic constraints of each trajectory Trajectory | $T_{1}$ | $T_{2}$ | $T_{3}$
---|---|---|---
Maximum speed ($km/h$) | 35 | 100 | 70
Maximum long. acceleration ($m/s^{2}$) | 0.4 | 1.5 | 2.0
Maximum long. deceleration ($m/s^{2}$) | 0.7 | 2.0 | 2.0
Maximum lat. acceleration ($m/s^{2}$) | 1.0 | 4.0 | 2.0

###  5.2 Metrics description

The integral absolute lateral error (IAE) is used to evaluate the tracking
quality. But it is observed that the classical IAU (Integral of the Absolute
control action) is not suitable for analyzing the system dynamics, the
frequency spectrum of the feedback control action is used to define two
different performance indicators:

  1. 1.

$M_{\epsilon}$ quantifies the low frequency oscillations of the control
action, which can lead to vehicle instability, as well as impair the comfort
of the occupants.

  2. 2.

$M_{\zeta}$ quantifies the high frequency oscillations of the control action,
which do not destabilize the system per se, but cause discomfort to the
vehicle occupants.

The values of both metrics, $M_{\epsilon}$ and $M_{\zeta}$, are computed from
two separated frequency bands, experimentally identified: $\epsilon$ (1.1-4
Hz) and $\zeta$ (4-10 Hz), respectively. Note that 20 Hz is the control
frequency in this work. A high-pass filter is firstly applied to the feedback
control action with a cutoff frequency of 0.5 Hz and 4 Hz respectively. Then,
the spectrum is calculated by applying the Short-time Fourier transform with
5-second overlapping sections.

$M_{\epsilon}$ is finally calculated as the mean of the maximum power spectrum
at each section, considering a threshold and a scale factor to balance the
magnitude of both metrics:

$M_{\epsilon}=\frac{1}{n}\sum_{i=1}^{n}s_{\epsilon}\cdot\max{(10\cdot\log{P_{\epsilon,i}}+\lambda_{\epsilon})}$

where $n$ is the amount of 5-second sections, $P_{\epsilon,i}$ is the spectrum
power of band $\epsilon$ in section $i$, $s_{\epsilon}$ is a scale factor
($s_{\epsilon}=0.015$) and $\lambda_{\epsilon}$ is a threshold in dB
($\lambda_{\epsilon}=80dB$). Note that only straight and long sections where
the path curvature is below $0.01$ m-1 (drawn in orange in Fig. 2) are
considered due to the high sensitivity of $M_{\epsilon}$.

$M_{\zeta}$ is obtained with the maximum (instead of the mean) power among all
sections, using a scale factor of $s_{\zeta}=0.04$ and the same threshold
$\lambda_{\zeta}=80dB$. This particularity is motivated by the low equivalence
found in experimental tests between what is intended to be reflected by this
metric and the value obtained with the mean. However, when the maximum power
is considered, controllers that exhibit high frequency oscillations in any
section of the test trajectory are penalised with high values of $M_{\zeta}$.

To summarise, IAE is the chosen indicator of the reference tracking quality,
$M_{\epsilon}$ measures the (in)stability margin of the controller and
$M_{\zeta}$ reflects passenger discomfort.

###  5.3 Control parameter space exploration

Design trade-offs between tracking accuracy and control action safety and
softness depend on driving dynamics. It is therefore difficult to determine a
unique benchmark to assess the performance of the controller. Besides, it is
also hard to infer the control parameters of a given control structure for a
precise value of a metric. To overcome this, a multi-objective optimization
problem is defined:

$\underset{\mathbf{\dot{x}}=\mathbf{f}(\mathbf{x},\,\mathbf{p}),\,\mathbf{p}\in[\mathbf{p}_{min},\,\mathbf{p}_{max}]}{\min}\;\mathbf{J}(\mathbf{p})$

where the set of objective functions is $\mathbf{J(p)}=(\max($IAE${}_{i}),$
$\,\max(M_{\epsilon,i}),\,\max(M_{\zeta,i}))^{T}$, $i=1..3$ is the number of
circuit $T_{i}$ and $\mathbf{p}$ is the set of control parameters that will be
compared in Section 5.5. Note that the functions to minimize are constrained
by the system dynamics and the control parameters are bounded by design.

To solve this problem, the MATLAB’s ParetoSearch algorithm is used. A Pareto
front is obtained for each controller showing its potential to minimize each
objective. With the maximum acceptable values for the optimization objectives
and the Pareto front of each controller, a region is defined. The smaller the
region, the greater the potential of a controller; thus, several controllers
can be compared, even if they have very different characteristics. The
acceptable optimization objective zone has been defined experimentally: It is
observed that an IAE greater than 0.35 meters implies poor tracking in the
curves; similarly, an $M_{\epsilon}$ greater than 0.25 implies the possibility
of system instability on test trajectories; and a $M_{\zeta}$ greater than 0.7
leads to a loss of passenger comfort.

In order to make the results more general, every controller is simulated in
the three benchmark trajectories $T_{i}$, which cover a wide operation range.
The three quality metrics are obtained for each trajectory, and the maximum of
each metric is considered in the optimization. It is thus ensured that the
controller will not be less stable, comfortable or accurate in any situation.

###  5.4 Vehicle model used in simulation

The model used in simulations mimics with a high degree of fidelity the
experimental platform used, as it has 14 degrees of freedom (6 for the vehicle
body motion: longitudinal, lateral, vertical, roll, pitch, and yaw; and 8 for
the wheels: vertical motion and spin of each wheel).

The power-train modeling comprises: (i) the engine, whose torque map is
modeled from experimental measurements; (ii) the gearbox, which includes the
same drive ratios and gear shifting logic as the real vehicle; (iii) the
resistance torques coming from braking system and wind and gravitational
forces. The tire behaviour was reproduced with the Pacejka tire model (Pacejka
and Bakker, 1992).

An external actuation system has been added to the vehicle’s assisted
steering, which is modeled as in Lee et al. (2018). Its main parameters, such
as inertia or backlash, have been identified from extensive field tests.
Moreover, the small noise coming from the localization system has been
characterized and included in the model.

###  5.5 Simulation results

All controllers implemented include a feedforward term $\delta_{ff}$ (Godoy et
al., 2015): $\delta_{ff}=R_{S}\cdot\arctan{\left(L\cdot\kappa\right)}$, where
$L$ is the wheelbase, $\kappa$ is the path curvature and $R_{S}$ is the
steering ratio. Note that this feed-forward term relies on a simplified
kinematic vehicle model and the path curvature. The total steering control
action is $\delta_{t}=\delta_{ff}+\delta_{max}\cdot u_{fb}$, where
$u_{fb}\in[-1,1]$ is the normalized feedback control action and $\delta_{max}$
is the maximum steering angle.

Three strategies have been evaluated: a discrete PID, an iPD (hereinafter MFC)
and a Speed-Adaptive iPD (SAMFC). The two latter control laws use a second
order ultra-local model (Fliess and Join, 2021), so that the discrete form of
(2) with no integral action yields:

$\displaystyle u_{iPD}(t_{k})$ $\displaystyle=$
$\displaystyle\frac{-\hat{F}(t_{k})+\ddot{y}_{1r}(t_{k})+K_{p}\,e(t_{k})+K_{d}\,\hat{\dot{e}}(t_{k})}{\alpha}$
$\displaystyle\hat{\dot{e}}(t_{k})$ $\displaystyle=$
$\displaystyle\dot{y}_{1r}(t_{k})-\hat{\dot{y}}_{1}(t_{k})$ (5)
$\displaystyle\hat{\dot{y}}_{1}(t_{k})$ $\displaystyle=$
$\displaystyle\frac{2y_{1}(t_{k})-2y_{1}(t_{k-1})-(T_{s}-2T_{c})\hat{\dot{y}}_{1}(t_{k-1})}{T_{s}+2T_{c}}$

where $y_{1}$ is the lateral deviation, $\hat{\dot{y}}_{1}$ is the estimated
derivative of the vehicle’s lateral deviation, $\hat{\dot{e}}$ is the filtered
estimation of the tracking error derivative, $T_{s}$ is the sample time and
$T_{c}$ is the derivative filter parameter.

Note that $\alpha$, $K_{p}$ and $K_{d}$ are the parameters that are varied in
the optimization of the MFC controller. For SAMFC, the constant $\alpha$ is
replaced by (4), so that $\alpha_{0}$, $K_{\alpha}$ and $v_{0}$ are included
in the design parameter set.

The PID controller is expressed as a Z-transform function

$U_{PID}(z)=\left(K_{p}+\frac{K_{i}\cdot T_{s}}{z-1}+\frac{K_{d}\cdot
N}{1+N\cdot T_{s}\frac{1}{z-1}}\right)E(z)$ (6)

being $K_{p}$, $K_{i}$, $K_{d}$ gains and filter parameter coefficient $N$ the
parameters varied in the multi-objective optimization.

Fig. 3 shows the resulting Pareto fronts, including a 3D view and its
projections on the IAE-$M_{\epsilon}$ and IAE-$M_{\zeta}$ planes. Remark that
while the Pareto front of the basic MFC regulator is very similar to the front
of the PID controller, the front of the SAMFC shows a significant reduction on
the three metrics at the same time. Fig. 3 also shows that the Pareto fronts
of the tested MFC and PID intersect and intersperse within the operation
range, making it difficult to compare both strategies.

The SAMFC regulator is able to cancel $M_{\epsilon}$ within the operation
range defined, as can be seen in Fig. 3b, in contrast to the MFC and PID
regulators, whose minimum $M_{\epsilon}$ is 0.05 approximately. While inside
the operation range MFC and PID have a minimum IAE of 0.13 and 0.17 meters
respectively, the SAMFC regulator reach 0.06 meters. On the other hand, the
three strategies offer several configurations where $M_{\zeta}$ is cancelled,
as Fig. 3c shows, meaning the passenger comfort is assured.

Table 2 gathers the volume of the region defined by the Pareto front of the
three strategies and the bounds of the operating region. As the points of the
fronts are not equally distributed, the volume is obtained by numerical
integration over the lineally interpolated surface between the points. The
smaller the volume of the region (VUP), the higher the optimization potential
of the controller structure, so, as can be seen in the table, SAMFC has a
significantly better performance than its competitors.

(a) 3D view

(b) IAE vs. Mϵ

(c) IAE vs. Mζ

Figure 3: 3D Pareto front for MFC, PID and SAMFC Table 2: Volume under the Pareto front (VUP) Control strategy | PID (6) | MFC (5) | SAMFC (5)+(4)
---|---|---|---
VUP | 0.0427 | 0.0404 | 0.0135

###  5.6 Experimental results

Figure 4: Experimental platform

The vehicle used in the experiments is a Citroën DS3 which includes hardware
modifications for the automated control of throttle, brake, gearbox and
steering systems (see Fig. 4). Its localization relies on a RTK-DGPS receiver
and on-board vehicle speed, accelerations and yaw rate sensors. An on-board
computer with an Intel Core i7-3610QE and 8Gb RAM is used for control. It is
also equipped with sensors to perceive the environment (Artuñedo et al.,
2019).

From the acceptable operating region defined in section 5.3, the parameter set
that yields the best tracking quality is selected for each controller. The
values are shown in Table 3. Note that this decision yields for this specific
system a MFC controller configuration with $K_{p}=0$.

Table 3: Parameters set for each controller Contr. | $K_{p}$ | $K_{d}$ | $N$ | ${\alpha}/{\alpha}_{0}$ | $K_{\alpha}$ | $v_{0}$
---|---|---|---|---|---|---
PID | 0.2216 | 0.0367 | 5.0 | - | - | -
MFC | 0 | 19.28 | - | 1409 | - | -
SAMFC | 0.5625 | 2.688 | - | 57.15 | 9.547 | 26.83
Table 4: Setups tested in the experimental platform Trajectory | $S_{1}$ | $S_{2}$
---|---|---
Max speed ($km/h$) | 35 | 56
Max long. acceleration ($m/s^{2}$) | 0.4 | 1.0
Max long. deceleration ($m/s^{2}$) | 0.7 | 2.0
Max lat. acceleration ($m/s^{2}$) | 1.0 | 2.0

The path of $T_{1}$ (see Fig. 2a) was used to generate two different testing
trajectories using the constraints in Table 4. Then, the controllers were
evaluated on both trajectories.

Table 5: Results obtained for each controller and trajectory Controller | $S_{1}$ | $S_{2}$
---|---|---
IAE | $M_{\epsilon}$ | $M_{\zeta}$ | IAE | $M_{\epsilon}$ | $M_{\zeta}$
PID (6) | 0.16 | 0.002 | 0.17 | 0.15 | 0.132 | 0.79
MFC (5) | 0.12 | 0.017 | 0.00 | 0.10 | 0.151 | 0.25
| SAMFC (5)$+$(4)
---
0.03 | 0.246 | 0.68 | 0.03 | 0.220 | 0.79
Figure 5: Tracking results on trajectory $S_{1}$

The paths followed by the vehicle to track $S_{1}$ are depicted in Fig. 5. As
shown, all tested controllers are able to complete the circuit with small
lateral errors. Nevertheless, some differences can be appreciated in the
sharpest curves, where the Maximum Lateral Error (MLE) is significantly
reduced by the SAMFC. In addition, Table 5 shows clear differences in the
tracking quality indicator (IAE, in meters): SAMFC obtains a reduction of more
than 70% with respect to MFC and PID. In order to analyze the performance in
more detail, Fig. 7 and Fig. 6 shows the control action and the lateral error
over time, respectively, where $\delta_{sw}$ the measured steering wheel
angle.

None of the controllers exhibit uncomfortable or near-unstable behavior, as it
is confirmed by $M_{\epsilon}$ and $M_{\zeta}$. They remain within the limits
of the operating region, with the exception of $M_{\zeta}$ for SAMFC, that is
slightly higher than expected due to minor unmodeled behaviours. However, the
performance of SAMFC in the longest straight section of $S_{1}$ presents less
low-frequency oscillations than MFC and PID, as can be observed between
instants t = 47 s and t = 60 s in Fig. 7. Note that the SAMFC is able to
achieve a very good tracking at low speeds (sharp turns) while limiting the
amplitude of low frequency oscillations at higher speeds, resulting in a more
stable behaviour.

Figure 6: Lateral error of each controller

(a) PID

(b) MFC

(c) SAMFC

Figure 7: Control actions of each controller

Finally, it is worth highlighting that the adaptation mechanism of $\alpha$ is
simple to implement and decoupled from the feedback gains, which makes the
method easier to tune than a parameter varying PID.

## Concluding remarks

The aim of this work was to obtain a lateral control strategy with a wide
operation range in order to increase the situations in which an autonomous
vehicle navigates without intervention. A minimum level of safety and comfort
were imposed. To that end, a systematic procedure to explore the controllers
performance has been applied on trajectories with different shapes and dynamic
constraints.

Speed-Adaptive MFC has demonstrated both in simulation and in real-world tests
that it can outperform other controllers with a similar structure, such as a
PID and a MFC. Furthermore, the changes introduced to the basic MFC do not
significantly increase the controller tuning difficulty, which, together with
the simple theoretical basis of MFC, results in an intuitive and efficient
control strategy.

Future work will assess the tracking capabilities and robustness of SAMFC with
respect to other model-based control strategies, such as Model Predictive
Control.

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