# On the Regret of Recursive Methods for
Discrete-Time Adaptive Control with Matched Uncertainty

Aren Karapetyan, Efe C. Balta, Anastasios Tsiamis, Andrea Iannelli, and John
Lygeros This work has been supported by the Swiss National Science Foundation
under NCCR Automation (grant agreement $51\text{NF}40\\_180545$), the European
Research Council under the ERC Advanced grant agreement $787845$ (OCAL) and by
the German Research Foundation (DFG) under Germany’s Excellence Strategy - EXC
2075 – 390740016. A. Karapetyan, A. Tsiamis and J. Lygeros are with the
Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH
Zürich), 8092 Zürich, Switzerland (E-mails: akarapetyan, atsiamis,
jlygeros@ethz.ch).E. C. Balta is with the Control and Automation Group,
inspire AG, 8005 Zürich, Switzerland, and with the Automatic Control
Laboratory<EMAIL_ADDRESS>Iannelli is with the Institute for Systems
Theory and Automatic Control, University of Stuttgart, Stuttgart 70569,
Germany (E-mail: andrea.iannelli@ist.uni-stuttgart.de).

###### Abstract

Continuous-time adaptive controllers for systems with a matched uncertainty
often comprise an online parameter estimator and a corresponding parameterized
controller to cancel the uncertainty. However, such methods are often
unimplementable, as they depend on an unobserved estimation error. We consider
the equivalent discrete-time setting with a causal information structure. We
propose a novel, online proximal point method-based adaptive controller, that
under a weak persistence of excitation (PE) condition is asymptotically stable
and achieves finite regret, scaling only with the time required to fulfill the
PE condition. We show the same also for the widely-used recursive least
squares with exponential forgetting controller under a stronger PE condition.

##  I Introduction

Adaptive control studies controllers that can adapt to or learn unmodelled
changes in the dynamics [1]. Adaptive controllers often guarantee perfect
asymptotic tracking or regulation, and parameter convergence [2, 3].
Concurrently with the developments in adaptive control, the theory of online
optimization was established to minimize an a priori unknown, sequentially
revealed cost [4]. The connections between the two have recently been
highlighted [5, 6], showing that adaptive controllers are closely linked to an
online cost optimization problem and are often proportional to an online
estimation cost gradient. In this work, we consider adaptive controllers that
aim to minimize a cost function while controlling the system [7]. Given such a
controller, we note that asymptotic stability alone does not automatically
provide performance guarantees on the cost [8, 9]. Instead, we provide finite-
time guarantees by characterizing the _regret_ [4] of our adaptive controller,
a concept borrowed from online learning to quantify the additional cost
incurred by the controller over a horizon due to partial model knowledge.

We consider systems with matched uncertainty [2], where the unknown dynamics
is parameterized by an unknown parameter vector and a known, state-dependant
feature matrix. In this setting, the uncertainty can be exactly canceled by
the control input given knowledge of the true parameter. We aim to find causal
adaptive controllers that estimate this parameter recursively while
stabilizing the system and minimizing regret. Most continuous-time adaptive
controllers in this setting utilize Lyapunov theory and provide only
asymptotic stability guarantees, see for example [1, 3, 2]. Using the same
tools, recently, a finite $\mathcal{O}(1)$ regret bound for linear systems is
derived in [10], and for nonlinear time-varying systems in [11]. In [11], the
authors also extend the results to a discrete-time setting, achieving a
$o(T)$, sublinear-in-time, bound with a causal formulation, and a
$\mathcal{O}(1)$ with a non-causal one. In contrast to our proposed method,
the Lyapunov-based adaptive controllers in [10, 11] are not realizable in
discrete time, as the parameter update depends on the current, unobserved
error, making it non-causal. The online gradient descent-based algorithm in
[11], although causal, achieves ${o}(T)$ regret even in the deterministic
case. The difficulty is inherent in the discrete-time formulation of the
problem, which introduces a one-step delay in the error observation. In
discrete time, the convenient Lyapunov function cancellation [2] no longer
holds and discrete-time Lyapunov functions are known to be harder to construct
[12].

The recursive learning algorithm for the adaptive controller can be set up to
relate to an online or “running” cost. This setting is studied in the time-
varying optimization literature providing asymptotic tracking and convergence
guarantees for online costs, e.g., [13, 14]. However, in this line of work,
the problem setup considers no underlying dynamics for the state, and the
regret generally is not studied.

We design causal, uncertainty-matching, recursive methods for discrete-time,
nonlinear, time-varying systems. Our contributions are threefold. First, we
introduce a recursive proximal learning (RPL) parameter estimation algorithm
based on the proximal point method [15]. Under a weak notion of persistence of
excitation (PE), RPL produces estimates of the unknown parameters that are
contractive with respect to the true one. Second, we show that RPL-based
adaptive control achieves finite regret scaling with the time required to
achieve the weak PE condition. By contrast, in [11] finite regret is only
achieved through access to a Lyapunov function or by a non-causal controller.
Finally, we analyze the regret of the recursive least-squares with forgetting
factor (RLSFF) [16] in this setting. We show that while both RPL and RLSFF
achieve finite regret with similar bounds, this is established for RPL under a
weaker PE condition. We demonstrate the algorithms on a discrete-time model
reference adaptive control (MRAC) [17] numerical example.

Notation: The sets of positive real numbers, positive integers, and non-
negative integers are denoted by $\mathbb{R}_{+}$, $\mathbb{N}_{+}$ and
$\mathbb{N}$, respectively. For a given vector $x$, its Euclidean norm is
denoted by $\|x\|$. The spectral norm of a square matrix $W$ is denoted by
$\|W\|$, and the largest and smallest singular values and eigenvalues by
$\sigma_{min}(W)$, $\sigma_{max}(W)$, and $\lambda_{min}(W)$,
$\lambda_{max}(W)$ respectively. For a positive scalar $r\in\mathbb{R}_{+}$,
we define the closed ball around the origin as
$\mathcal{B}_{r}:=\\{x\in\mathbb{R}^{n}\;|\;\|x\|\leq r\\}$.

##  II Problem Statement

We consider nonlinear dynamical systems with matched uncertainty of the form

$x_{k+1}=f_{k}(x_{k})+B_{k}(x_{k})\left(u_{k}-\alpha(x_{k})\right),$ (1)

where $f:\mathbb{N}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ models the
known nominal dynamics and satisfies $f_{k}({0})={0}$ for all
$k\in\mathbb{N}$,
$B_{k}:\mathbb{N}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$ is a
known input matrix with a bounded norm, and
$\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is an _unknown_ mapping to
the input space, capturing the uncertain dynamics of the model. We assume that
$\alpha$ can be exactly parameterized by possibly nonlinear, but known and
bounded basis matrices, $\phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{p\times
m}$, that map the state to a feature in $\mathbb{R}^{p\times m}$. The
uncertainty is then given by $\alpha(x_{k})=\phi^{\top}(x_{k})\theta^{\star}$,
where $\theta^{\star}\in\mathbb{R}^{p}$ is an unknown weighting vector111We
drop the explicit state dependence on state for $B_{k}:=B_{k}(x_{k})$ and
$\phi_{k}:=\phi(x_{k})$, wherever required for readability.. We also assume
$\|B_{k}(x_{k})\phi^{\top}(x_{k})\|\leq b$ for all $k\in\mathbb{N}$,
$x\in\mathbb{R}^{n}$ and some $b\in\mathbb{R}_{+}$. The tracking problem under
partial model knowledge can also be cast into this framework in the context of
MRAC, as shown in Section V.

We characterize the performance of a given controller for system (1) over a
time horizon $T$ by comparing it to a benchmark control signal
$u^{\star}=\begin{bmatrix}{u^{\star}_{0}}^{\top}&\ldots&{u^{\star}_{T-1}}^{\top}\end{bmatrix}^{\top}$
that cancels the uncertainty perfectly and is defined below. We use regret to
quantify the controller performance

$\mathcal{R}_{T}({u})=\sum_{k=0}^{T-1}c(x_{k})-c(x^{\star}_{k}),$ (2)

where
$u=\begin{bmatrix}u_{0}^{\top}&\ldots&u_{T-1}^{\top}\end{bmatrix}^{\top}$
denotes the control signal of a given controller, $x_{k}$ and $x^{\star}_{k}$
are state sequences due to the control signals $u$ and $u^{\star}$,
respectively, and $c:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a stage cost. We
restrict our analysis to locally Lipschitz continuous costs and assume for any
$R\in\mathbb{R}_{+}$, there exists a $L_{c}\in\mathbb{R}_{+}$, such that for
all $x,y\in\mathcal{B}_{R}$,

$\|c(x)-c(y)\|\leq L_{c}\|x-y\|.$ (3)

###### Assumption 1

(System Dynamics) The nominal dynamics $x_{k+1}=f_{k}(x_{k})$ are uniformly,
globally _E- $\delta$-ISS_ [18]. In other words, there exist
$c_{0},c_{w},r_{w}\in\mathbb{R}_{+}$ and $\rho\in(0,1)$, such that for any
$x_{0},y_{0}\in\mathbb{R}^{n}$, $w_{k}\in\mathcal{B}_{r_{w}}$, and all
$k\in\mathbb{N}$ the perturbed dynamics $y_{k+1}=f_{k}(y_{k})+w_{k}$ satisfy

$\|x_{k}-y_{k}\|\leq
c_{0}\rho^{k}\|x_{0}-y_{0}\|+c_{w}\sum_{i=0}^{k-1}\rho^{k-i-1}\|w_{i}\|.$

The _E- $\delta$-ISS_ property characterizes only the known and autonomous
part of the dynamics. It is known that if $f_{k}$ is smooth in the state and
is contractive, then it is also _E- $\delta$-ISS_ [19, 11]. If $f_{k}$ is not
smooth, then under milder conditions, we show in [20] that exponential
stability implies _E- $\delta$-ISS_.

###  II-A Online Estimation

We consider adaptive control laws that produce control inputs of the form
$u_{k}=\phi^{\top}(x_{k})\theta_{k}$ where $\theta_{k}\in\mathbb{R}^{p}$ is an
online estimate of $\theta^{\star}$ at time $k$. The closed-loop dynamics for
a controller of this form is given by

$x_{k+1}=f_{k}(x_{k})+\underbrace{B_{k}(x_{k})\phi(x_{k})^{\top}\left(\theta_{k}-\theta^{\star}\right)}_{w_{k}(x_{k})},$
(4)

where $w_{k}(x_{k})$ can be thought of as a state-dependant artificial
disturbance introduced due to an inexact uncertainty matching. One can also
consider a benchmark counterfactual control input
$u_{k}^{\star}:=\phi(x_{k}^{\star})^{\top}\theta^{\star}$ and the
corresponding signal $u^{\star}$ that perfectly cancels out the uncertainty in
(1) leading to a benchmark state evolution

$x_{k+1}^{\star}=f_{k}(x_{k}^{\star}).$ (5)

Note that $u^{\star}$ is a counterfactual policy that is not realizable, as
$\theta^{\star}$ is unknown. To simplify the analysis, we take
$x_{0}=x_{0}^{\star}$.

A standard setup in adaptive control design is to produce estimates
$\theta_{k}$ that minimize the magnitude of $w_{k}(x_{k})$, effectively
steering the state of (4) to that of (5) thanks to the _E- $\delta$-ISS_
property. This is often done by setting up a least squares estimation problem,
minimizing the following online estimation cost for all $k\in\mathbb{N}_{+}$

$\displaystyle\theta_{k}$
$\displaystyle=\operatorname*{arg\,min}_{\theta}h_{k-1}(\theta),$ (6)
$\displaystyle h_{k-1}(\theta):\\!$
$\displaystyle=\\!\frac{1}{2}\sum_{i=0}^{k-1}\|B_{i}\phi_{i}^{\top}\left(\theta-\theta^{\star}\right)\|^{2}=\frac{1}{2}\|\Phi_{k}{\theta}-Y_{k}\|^{2},$
$\displaystyle Y_{k}$ $\displaystyle=\begin{bmatrix}y_{0}\\\ y_{1}\\\
\vdots\\\
y_{k-1}\end{bmatrix},\qquad\Phi_{k}=\begin{bmatrix}B_{0}\phi_{0}^{\top}\\\
B_{1}\phi_{1}^{\top}\\\ \vdots\\\ B_{k-1}\phi^{\top}_{k-1}\end{bmatrix},$
$\displaystyle y_{k}$
$\displaystyle=x_{k+1}-f_{k}(x_{k})-B_{k}\phi_{k}^{\top}\theta_{k}=B_{k}\phi_{k}^{\top}\theta^{\star}.$

Note that at time $k$, the latest available online estimation cost is
$h_{k-1}$, as $y_{k}$ is not yet available. Solving (6) directly requires
computing an expensive matrix inverse at each time step and maintaining an
increasing memory. In this work, we study two recursive methods that alleviate
this issue, both optimizing an online cost related to (6). In particular, we
introduce and analyze a recursive proximal learning method in Section III, and
also study the well-established recursive least squares with forgetting factor
[16] in Section IV.

###  II-B Regret

Speed or velocity-gradient controllers are introduced in [7] to solve the
continuous-time equivalent of the online estimation problem (6) in the
framework of (integral) goal-oriented control. It is shown that these methods
achieve finite estimation error in the limit, under the assumption that the
controller at time $t$ can depend on the estimation error
${\theta}_{t}-\theta^{\star}$. Recent works [10, 11] noted that if an
exponential Lyapunov function exists for the nominal system, a speed-gradient
descent-based uncertainty matching controller achieves finite quadratic cost
on the state. In particular, the suboptimal closed-loop state trajectory
$x_{t}$ satisfies

$J_{\mathrm{cont}}(x_{0},u):=\lim_{T\rightarrow\infty}\int_{0}^{T}\|x_{t}\|^{2}dt=\mathcal{O}(1).$

Moreover, the existence of the exponential Lyapunov function also implies that
$J_{\mathrm{cont}}(x_{0},u^{\star})$ for the optimal state evolution is
finite. Defining continuous-time regret as
$\mathcal{R}_{\mathrm{cont}}(u):=J_{\mathrm{cont}}(x_{0},u)-J_{\mathrm{cont}}(x_{0},u^{\star})$,
an $\mathcal{O}(1)$ bound and the extra cost accumulated by the adaptive
control law as compared to the benchmark follows directly. In [11] a discrete-
time version of the speed-gradient algorithm is introduced with a finite cost
bound, but with the same assumption that at time $k$ the error
$\theta_{k}-\theta^{\star}$ is available, making the controller non-causal.
Without this assumption, the authors achieve $o(T)$ sublinear regret using an
online learning toolkit. In this work, we consider causal controllers that
minimize (6) at time $k$ with access only to $y_{k-1}$. To characterize the
performance of such an adaptive, uncertainty-matching controller, we analyze
the regret (2).

##  III Recursive Proximal Learning

The recursive proximal learning algorithm corresponds to the
$\varepsilon$-scaled proximal operator of (6) evaluated at the _previous_
estimate

$\begin{split}\theta_{k}&=\mathrm{prox}_{\varepsilon
h_{k-1}}\left(\theta_{k-1}\right)\\\
&=\operatorname*{arg\,min}_{\theta}\left(h_{k-1}(\theta)+\frac{\varepsilon}{2}\|\theta-\theta_{k-1}\|^{2}\right),\end{split}$
(7)

for some $\theta_{0}\in\mathbb{R}^{p}$ and $\varepsilon\in\mathbb{R}_{+}$.
From the online learning perspective [4], $\theta_{k}$ can be thought of as
the minimizer of the latest available cost $g_{k-1}$ at time $k$

$g_{k-1}(\theta)=h_{k-1}(\theta)+\frac{\varepsilon}{2}\|\theta-\theta_{k-1}\|^{2}.$
(8)

While a closed-form solution to (7) exists, it requires memory increasing with
time. The following equivalent set of updates to (7) remedies both issues. In
particular, for all $k\in\mathbb{N}$, the following is equivalent to (7)

$\begin{split}P_{k+1}^{-1}&=P_{k}^{-1}+\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top},\\\
H_{k+1}&=H_{k}+\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top}\\\
s_{k+1}&=s_{k}+\phi_{k}B_{k}^{\top}y_{k}\\\
\theta_{k+1}&=\theta_{k}-P_{k+1}\left(H_{k+1}\theta_{k}-s_{k+1}\right),\end{split}$
(9)

with the initialization $H_{0}=\boldsymbol{0}_{p\times p}$, $s_{0}={0}$ and
$P_{0}^{-1}=\varepsilon I_{p}$.

###  III-A Contraction in Parameter Space

We define $T_{s}$-weak PE as follows.

###### Definition 1

A matrix signal $\phi_{k}:\mathbb{N}\rightarrow\mathbb{R}^{p\times n}$ is
called $T_{s}$-weak persistently exciting if for a $T_{s}\in\mathbb{N}$, there
exists a $\delta\in\mathbb{R}_{+}$ such that

$\displaystyle 0<\delta I_{p}\leq\sum_{i=0}^{T_{s}}\phi_{i}\phi_{i}^{\top}.$

We show below that under a $T_{s}$-weak PE assumption, the RPL update (7)
enjoys a contraction property on the parameter; moreover, under an additional
assumption, the lifted input vector $\Phi_{k}\theta_{k}$ is also contractive.

###### Assumption 2

There exists a $\beta\in\mathbb{R}_{+}$ such that
$\lim_{T\rightarrow\infty}\sum_{i=0}^{T}\phi_{i}B_{i}^{\top}B_{i}\phi_{i}^{\top}\leq\beta
I_{p}$

###### Lemma 1

Assume $B_{k}\phi_{k}^{\top}$ is $T_{s}$-weak persistently exciting, then the
RPL estimate (7) satisfies

$\begin{split}\|{\theta}_{k}-\theta^{\star}\|&\leq\eta\|{\theta}_{k-1}-\theta^{\star}\|,\qquad\forall
k\geq T_{s}\\\
\|{\theta}_{k}-\theta^{\star}\|&\leq\|{\theta}_{k-1}-\theta^{\star}\|,\qquad\forall
0<k<T_{s}\end{split}$ (10)

where $\eta=\frac{\varepsilon}{\delta+\varepsilon}\in(0,1)$. Moreover, if
Assumption 2 holds, and
$\varepsilon<\frac{\delta\sqrt{\delta}}{\sqrt{\beta}-\sqrt{\delta}}$ then
there exists a $\gamma\in(0,1)$ such that

$\displaystyle\|\Phi_{k+1}\left({\theta}_{k}-\theta^{\star}\right)\|$
$\displaystyle\leq\gamma\|\Phi_{k}\left({\theta}_{k-1}-\theta^{\star}\right)\|,\qquad\forall
k\geq T_{s},$
$\displaystyle\|\Phi_{k+1}\left({\theta}_{k}-\theta^{\star}\right)\|$
$\displaystyle\leq\sqrt{\beta}\|\theta_{0}-\theta^{\star}\|,\qquad\forall
0\leq k<T_{s}.$

###### Proof:

The proximal update (7) can equivalently be represented by a single online
Newton step update [21] on (8). At time $k$, it is equivalently represented by

$\theta_{k}=\theta_{k-1}-\left(\nabla^{2}g_{k-1}(\theta)\Bigr{|}_{\theta=\theta_{k-1}}\right)^{-1}\nabla
g_{k-1}(\theta)\Bigr{|}_{\theta=\theta_{k-1}}.$

Subtracting the true $\theta^{\star}$ from both sides, denoting
$\tilde{\theta}_{k}:=\theta_{k}-\theta^{\star}$, noting that
$Y_{k}=\Phi_{k}\theta^{\star}$ and recalling the definition of $h_{k-1}$ from
(6)

$\displaystyle\tilde{\theta}_{k}$
$\displaystyle=\tilde{\theta}_{k-1}-\left(\Phi_{k}^{\top}\Phi_{k}+\varepsilon
I_{p}\right)^{-1}\Phi_{k}^{\top}\left(\Phi_{k}\theta_{k-1}-Y_{k}\right)$
$\displaystyle=\tilde{\theta}_{k-1}-\left(\Phi_{k}^{\top}\Phi_{k}+\varepsilon
I_{p}\right)^{-1}\Phi_{k}^{\top}\Phi_{k}\tilde{\theta}_{k-1}$
$\displaystyle=\left(I_{p}-\left(\Phi_{k}^{\top}\Phi_{k}+\varepsilon
I_{p}\right)^{-1}\Phi_{k}^{\top}\Phi_{k}\right)\tilde{\theta}_{k-1}$
$\displaystyle:=\left(I_{p}-M_{k}\right)\tilde{\theta}_{k-1},$

where $M_{k}=\left(\Phi_{k}^{\top}\Phi_{k}+\varepsilon
I_{p}\right)^{-1}\Phi_{k}^{\top}\Phi_{k}$. Using the orthonormality property
of singular value decomposition of symmetric matrices and the weak persistence
of excitation of $B_{k}\phi_{k}$, for all $k\geq T_{s}$

$\displaystyle\sigma_{min}\left(M_{k}\right)$
$\displaystyle\geq\frac{\delta}{\varepsilon+\delta}\in(0,1),$
$\displaystyle\eta_{k}:=\|I_{p}-M_{k}\|$
$\displaystyle\leq\frac{\varepsilon}{\delta+\varepsilon}\in(0,1),$

obtaining the contraction result for parameter since
$\sigma_{max}(M_{k})\in(0,1)$. For all $0<k<T_{s}$, if the $T_{s}$-weak PE
condition is not fulfilled, hence $\sigma_{min}(M_{k})=0$ and
$\|\tilde{\theta}_{k}\|\leq\|\tilde{\theta}_{k-1}\|$.

To prove the second part, note that for a full column rank matrix $A$,
$\|A\tilde{\theta}_{k+1}\|=\|\tilde{\theta}_{k+1}\|_{A^{\top}A}$, and
${{\lambda_{min}(A^{\top}A)}}={\sigma_{min}(A^{\top}A)}$ , and likewise for
$\lambda_{max}$. Then, for all $k\geq T_{s}$

$\displaystyle\sigma_{min}\left(\left(\Phi_{k+1}^{\top}\Phi_{k+1}\right)^{-1}\right)\|\tilde{\theta}_{k}\|^{2}_{\Phi_{k+1}^{\top}\Phi_{k+1}}\leq\|\tilde{\theta}_{k}\|^{2}$
$\displaystyle\leq\eta^{2}\|\tilde{\theta}_{k-1}\|^{2}\leq\eta^{2}\sigma_{max}\left(\left(\Phi_{k}^{\top}\Phi_{k}\right)^{-1}\right)\|\tilde{\theta}_{k-1}\|_{\Phi_{k}^{\top}\Phi_{k}}^{2}.$

It follows that

$\displaystyle\|\Phi_{k+1}\tilde{\theta}_{k}\|\leq\eta\sqrt{\frac{\sigma_{max}\left(\Phi_{k+1}^{\top}\Phi_{k+1}\right)}{\sigma_{min}\left(\Phi_{k}^{\top}\Phi_{k}\right)}}\|\Phi_{k}\tilde{\theta}_{k-1}\|,$
(11)

where we used the fact that
$\sigma_{max}\left((A^{\top}A)^{-1}\right)=1/\sigma_{min}(A^{\top}A)$ and
$\sigma_{min}\left((A^{\top}A)^{-1}\right)=1/\sigma_{max}(A^{\top}A)$. The
proof is completed by noting that for all $k\geq T_{s}$

$\gamma:=\eta\sqrt{\frac{\sigma_{max}\left(\Phi_{k+1}^{\top}\Phi_{k+1}\right)}{\sigma_{min}\left(\Phi_{k}^{\top}\Phi_{k}\right)}}\leq\frac{\varepsilon\sqrt{\beta}}{\varepsilon\sqrt{\delta}+\delta\sqrt{\delta}}<1,$

given $\varepsilon<\frac{\delta\sqrt{\delta}}{\sqrt{\beta}-\sqrt{\delta}}$.
Finally, for all $k\in\mathbb{N}$,
$\|\Phi_{k+1}\tilde{\theta}_{k}\|\leq\|\Phi_{k+1}\|\|\tilde{\theta}_{k}\|\leq\sqrt{\beta}\|\tilde{\theta}_{0}\|$.
∎

###  III-B Closed-loop Analysis

Consider the suboptimal state evolution (4) under the RPL controller
$u_{k}^{\mathrm{RPL}}=\phi_{k}^{\top}\theta_{k}$, with $\theta_{k}$ given in
(7). We can represent it equivalently as

$x_{k+1}=f_{k}(x_{k})+\underbrace{S_{k}\left(\Phi_{k+1}{\theta}_{k}-\Phi_{k+1}\theta^{\star}\right)}_{w_{k}(x_{k})},$
(12)

where
$S_{k}:=\begin{bmatrix}\boldsymbol{0}&\ldots&\boldsymbol{0}&I_{n},\end{bmatrix}\in\mathbb{R}^{n\times
n(k+1)}$ is a selector matrix that selects the last component of
$\Phi_{k+1}\left({\theta}_{k}-\theta^{\star}\right)$, corresponding to
$B_{k}\phi_{k}^{\top}(\theta_{k}-\theta^{\star})$. Similarly, the optimal
state evolution under $u^{\star}$ is given by

$x_{k+1}^{\star}=f_{k}(x^{\star}_{k})+\underbrace{S_{k}\left(\Phi^{\star}_{k+1}\theta^{\star}-\Phi^{\star}_{k+1}\theta^{\star}\right)}_{\boldsymbol{0}},$
(13)

where $\Phi^{\star}_{k+1}:=\Phi(x^{\star}_{[0,\ldots,k]})$. In the following
theorem, we characterize the closed-loop system (12) in terms of asymptotic
stability and regret. In particular, we show that the regret of the RPL
controller scales with $\mathcal{O}\left(\|\tilde{\theta}_{0}\|T_{s}\right)$.

###### Theorem 1

Assume that (1) satisfies Assumption 1, and $B_{k}\phi_{k}$ is $T_{s}$-weak
persistently exciting, then the RPL adaptive controller

  1. (i)

renders the closed-loop system asymptotically stable,

  2. (ii)

achieves finite regret

$\mathcal{R}_{T}(\mathrm{RPL})\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\left(\frac{T_{s}}{1-\rho}\\!+\\!\frac{\rho^{T}\\!+\\!(1-\eta)\rho+\eta}{(1-\rho)^{2}(1-\eta)}\right),$

for the $\eta$ established in Lemma 1. If in addition, $B_{k}\phi_{k}$ also
satisfy Assumption 2 and
$\varepsilon<\frac{\delta\sqrt{\delta}}{\sqrt{\beta}-\sqrt{\delta}}$, then

$\mathcal{R}_{T}(\mathrm{RPL})\leq
c_{w}c_{p}L_{c}\|\tilde{\theta}_{0}\|\left(\frac{T_{s}}{1-\rho}\\!+\\!\frac{\rho^{T}\\!+\\!(1-\gamma)\rho+\gamma}{(1-\rho)^{2}(1-\gamma)}\right),$

where $c_{p}:=\|\Phi_{T_{s}}\|\leq\sqrt{\beta}$ and $\gamma$ is defined in
Lemma 1.

###### Proof:

To show (i), recall the definition of _E- $\delta$-ISS_, and the fact that
$f_{k}({0})={0}$ for all $k\in\mathbb{N}$, then consider the suboptimal state
evolution of the nonlinear system under the RPL adaptive controller (12) for
all $k\in\mathbb{N}_{+}$, recalling that
$\tilde{\theta}_{k}=\theta_{k}-\theta^{\star}$

$\begin{split}\|x_{k}\|&\leq
c_{0}\rho^{k}\|x_{0}\|+c_{w}\sum_{i=0}^{k-1}\rho^{k-i-1}\|B_{i}\phi_{i}\tilde{\theta}_{i}\|\\\
&<c_{0}\|x_{0}\|+\frac{c_{w}b\|\tilde{\theta}_{0}\|}{1-\rho},\end{split}$ (14)

where we use the submultiplicativity property of the norms, the bound on
$B_{k}\phi_{k}$ and non-expansivity of $\|\tilde{\theta}_{k}\|$ from Lemma 1.
The asymptotic stability of the closed-loop then follows from
$\lim_{k\rightarrow\infty}\|x_{k}\|=0$, since by Lemma 1
$\lim_{k\rightarrow\infty}\|\tilde{\theta}_{k}\|=0$, given the input-to-state
stability [22] of the closed-loop system (12).

Next, we show (ii), by first noting that since $x_{k}$ and $x_{k}^{\star}$ are
bounded from (14), it follows from (3)

$\displaystyle\mathcal{R}_{T}(\mathrm{RLS})\leq\underbrace{L_{c}\sum_{k=T_{s}+1}^{T-1}\|x_{k}-x_{k}^{\star}\|}_{\text{I}}+\underbrace{L_{c}\sum_{k=0}^{T_{s}}\|x_{k}-x_{k}^{\star}\|}_{\text{
II}},$

where II represents the cost in the initial $T_{s}$-long non-PE phase, and I
the cost improvement afterwards.

Using the _E- $\delta$-ISS_ condition on the nominal system and noting that
$x_{k}$ and $x_{k}^{*}$, given by (12) and $\eqref{eq:optimal_rpl}$,
respectively, are equal in the first time step, for all $k>T_{s}$

$\begin{split}&\|x_{k}-x_{k}^{\star}\|\leq
c_{w}b\sum_{i=0}^{k-1}\rho^{k-i-1}\|\tilde{\theta}_{i}\|\\\ &\leq
c_{w}b\sum_{i=1}^{T_{s}}\rho^{k-i}\|\tilde{\theta}_{i-1}\|+c_{w}b\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\|\tilde{\theta}_{i}\|\\\
&\leq
c_{w}b\rho^{k-T_{s}}\|\tilde{\theta}_{0}\|\sum_{i=1}^{T_{s}}\rho^{T_{s}-i}+c_{w}b\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\|\tilde{\theta}_{i}\|\\\
&\leq\rho
c_{w}c_{T_{s}}b\|\tilde{\theta}_{0}\|\rho^{k-T_{s}}+c_{w}b\|\tilde{\theta}_{0}\|\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\eta^{i-T_{s}+1},\end{split}$
(15)

where the second inequality follows from the boundedness of $B_{k}\phi_{k}$,
the third from the non-expansivity of $\|\tilde{\theta}_{k}\|$ from Lemma 1
and rearrangement of the constants, and the last one from the contraction of
$\|\tilde{\theta}_{k}\|$ and we define
$c_{T_{s}}=\frac{1-\rho^{T_{s}}}{1-\rho}$. For $0<k\leq T_{s}$, we may only
use the non-expansive result from Lemma 1 and hence

$\|x_{k}-x_{k}^{\star}\|\leq
c_{w}b\sum_{i=1}^{k}\rho^{k-i}\|\tilde{\theta}_{i-1}\|\leq
c_{w}b\|\tilde{\theta}_{0}\|\sum_{i=1}^{k}\rho^{k-i}.$ (16)

Using the Cauchy Product inequality defined for two finite series
$\\{a_{i}\\}_{i=1}^{T}$ and $\\{b_{i}\\}_{i=1}^{T}$

$\textstyle{\sum_{i=0}^{T}\left|\sum_{j=0}^{i}a_{j}b_{i-j}\right|\leq\left(\sum_{i=0}^{T}|a_{i}|\right)\left(\sum_{j=0}^{T}|b_{j}|\right)},$

and summing (16) and (15) over $T_{s}-1$ and $T-T_{s}-2$, respecitvely, we get

II $\displaystyle\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\sum_{k=0}^{T_{s}-1}\sum_{i=0}^{k}\rho^{k-i}\leq
c_{w}c_{T_{s}}T_{s}bL_{c}\|\tilde{\theta}_{0}\|$ I $\displaystyle\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\left(\rho
c_{T_{s}}\\!\sum_{k=1}^{T-T_{s}-1}\rho^{k}\\!+\\!\sum_{k=0}^{T-T_{s}-2}\sum_{i=0}^{k}\rho^{k-i}\eta^{i+1}\right)$
$\displaystyle\leq\rho
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\frac{\rho^{T}+\rho}{(1-\rho)^{2}}$
$\displaystyle\quad+c_{w}b\eta
L_{c}\|\tilde{\theta}_{0}\|\frac{\left(1-\rho^{T-T_{s}-1}\right)\left(1-\eta^{T-T_{s}-1}\right)}{\left(1-\rho\right)\left(1-\eta\right)}.$

The first regret bound is achieved by combining the terms.

Now if Assumption 2 is satisfied and
$\varepsilon<\frac{\delta\sqrt{\delta}}{\sqrt{\beta}-\sqrt{\delta}}$, one can
make use of the lifted input contraction in Lemma 1. In particular, for all
$k>T_{s}$, referring to (12) and (13) again

$\begin{split}&\|x_{k}-x_{k}^{\star}\|\leq
c_{w}\sum_{i=0}^{k-1}\rho^{k-i-1}\|w_{i}(x_{i})\|\\\ &\leq
c_{w}\|S_{k}\|\sum_{i=0}^{k-1}\rho^{k-i-1}\|\Phi_{i+1}\tilde{\theta}_{i}\|\\\
&\leq
c_{w}\sum_{i=1}^{T_{s}}\rho^{k-i}\|\Phi_{i}\tilde{\theta}_{i-1}\|+c_{w}\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\|\Phi_{i+1}\tilde{\theta}_{i}\|\\\
&\begin{split}\leq
c_{w}\sum_{i=1}^{T_{s}}\rho^{k-i}&\|\Phi_{i}\tilde{\theta}_{i-1}\|\\\
&+c_{w}\|\Phi_{T_{s}}\tilde{\theta}_{T_{s}-1}\|\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\gamma^{i-T_{s}+1}\end{split}\\\
&\begin{split}=c_{w}\rho^{k-T_{s}}\sum_{i=1}^{T_{s}}&\rho^{T_{s}-i}\|\Phi_{i}\tilde{\theta}_{i-1}\|\\\
&+c_{w}\|\Phi_{T_{s}}\tilde{\theta}_{T_{s}-1}\|\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\gamma^{i-T_{s}+1}\end{split}\\\
&\leq\rho
c_{w}c_{p}c_{T_{s}}\|\tilde{\theta}_{0}\|\rho^{k-T_{s}}+c_{w}c_{p}\|\tilde{\theta}_{0}\|\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\gamma^{i-T_{s}+1},\end{split}$
(17)

where the second inequality follows from the submultiplicative property of the
norms, the third from the fact that $\|S_{k}\|=1$, and the last two from Lemma
1 and by denoting $c_{T_{s}}:=\frac{1-\rho^{T_{s}}}{1-\rho}$. Similarly, for
$0<k\leq T_{s}$

$\|x_{k}-x_{k}^{\star}\|\leq
c_{w}\sum_{i=1}^{k}\rho^{k-i}\|\Phi_{i}\tilde{\theta}_{i-1}\|\leq
c_{w}c_{p}\|\tilde{\theta}_{0}\|\sum_{i=1}^{k}\rho^{k-i}$ (18)

We then obtain new bounds for I and II

II $\displaystyle\leq
c_{w}c_{p}\|\tilde{\theta}_{0}\|L_{c}\sum_{k=0}^{T_{s}-1}\sum_{i=0}^{k}\rho^{k-i}\leq
T_{s}\|\tilde{\theta}_{0}\|c_{w}c_{T_{s}}c_{p}L_{c},$

where the first inequality follows from (18) and the second one from the
Cauchy Product inequality. Using (17)

$\displaystyle\text{I}\\!\leq\\!c_{w}c_{p}\|\tilde{\theta}_{0}\|L_{c}\left(\rho
c_{T_{s}}\sum_{k=1}^{T-T_{s}-1}\rho^{k}\\!+\\!\sum_{k=0}^{T-T_{s}-2}\\!\sum_{i=0}^{k}\rho^{k-i}\gamma^{i+1}\right)$
$\displaystyle\leq\rho
c_{w}c_{p}c_{T_{s}}\|\tilde{\theta}_{0}\|L_{c}\frac{\rho-\rho^{T-T_{s}}}{1-\rho}$
$\displaystyle\qquad+c_{w}c_{p}\|\tilde{\theta}_{0}\|\gamma
L_{c}\frac{\left(1-\rho^{T-T_{s}-1}\right)\left(1-\gamma^{T-T_{s}-1}\right)}{\left(1-\rho\right)\left(1-\gamma\right)}.$

Noting that the first term in the above summation is upper bounded by $\rho
c_{w}c_{p}\|\tilde{\theta}_{0}\|L_{c}\left(\frac{\rho^{T}+\rho}{(1-\rho)^{2}}\right)$
and combining I and II results in the second stated bound.

∎

The theorem states that given the assumptions on the nominal system, both (12)
and (13) state trajectories converge to the origin. It is important to note
that no further restrictions are put on the stage costs $c$, other than local
Lipschitz continuity. Hence, the finite regret result of the theorem does not
imply the cost $\sum_{k=0}^{T-1}c(x_{k})$ is minimized, but the relative cost
performance is bounded. Moreover, it shows that this performance, in terms of
regret scales with the initial exploratory time period $T_{s}$, and cannot be
avoided, as during these transients the parameter updates are only
nonexpansive by Lemma 1. If the additional assumptions on $\varepsilon$ and
$B_{k}\phi_{k}$ are satisfied, then the constant term is updated from a
uniform bound $b$ to $c_{p}:=\|\Phi_{T_{s}}\|$, which relates to the
regression matrix only in the initial period.

In the following lemma, we show that Lipschitz continuity of the basis
matrices and $\phi({0})=\boldsymbol{0}_{p\times m}$ is a sufficient condition
to fulfill Assumption 2.

###### Lemma 2

Let $B_{k}\phi_{k}$ be $T_{s}$-weak persistently exciting,
$\phi({0})=\boldsymbol{0}_{p\times m}$, and the basis matrices $\phi$ be
$L-$Lipschitz continuous, such that there exists a $L\in\mathbb{R}_{+}$
satisfying

$\|\phi(x)-\phi(y)\|\leq L\|x-y\|.\quad\forall x,y\in\mathbb{R}^{n},$

then Assumption 2 holds.

###### Proof:

Given $T_{s}$-weak PE holds, we make use of (10), summing the first bound in
(14) over a horizon $T$ and taking the limit we get

$\displaystyle\operatorname*{\overline{lim}}_{T\rightarrow\infty}\sum_{k=0}^{T}\|x_{k}\|\leq\frac{c_{0}\|x_{0}\|}{1-\rho}+c_{w}b\sum_{k=1}^{T_{s}}\sum_{i=1}^{k-1}\rho^{k-i}\|\tilde{\theta}_{i-1}\|$
$\displaystyle<\operatorname*{\overline{lim}}_{T\rightarrow\infty}c_{w}b\|\tilde{\theta}_{0}\|\left(\rho
c_{T_{s}}\sum_{k=1}^{T-T_{s}-1}\rho^{k}+\sum_{k=0}^{T-T_{s}-2}\sum_{i=0}^{k}\rho^{k-i}\eta^{i+1}\right)$
$\displaystyle<\frac{\rho^{2}c_{w}c_{T_{s}}b\|\tilde{\theta}_{0}\|}{1-\rho}+\frac{c_{w}b\|\tilde{\theta}_{0}\|\eta}{(1-\rho)(1-\eta)},$

where the first inequality follows from the _E- $\delta$-ISS_ definition and
geometric series, the second from Lemma 1 and the last from the Cauchy Product
inequality. The result then follows from the boundedness of $B_{k}$ and
$\phi_{k}$ and $\|\phi(x)\|\leq L\|x\|$. ∎

##  IV RLS with Exponential Forgetting

The RLSFF algorithm minimizes the online cost

$\begin{split}g_{k-1}^{f}(\theta)=\frac{1}{2}\sum_{i=0}^{k-1}\lambda^{2(k-1-i)}&\|B_{i}\phi_{i}^{\top}\left(\theta-\theta^{\star}\right)\|\\\
&+\frac{\lambda^{2(k-1)}\varepsilon}{2}\|\theta_{0}-\theta^{\star}\|^{2},\end{split}$
(19)

for a fixed exponential forgetting factor $\lambda^{2}\in(0,1)$, where the
first term can be thought of as the discounted version of (6) and the second,
as a discounted regularizer on the _initial_ parameter error. The update can
be recursively implemented as follows [23] for all $k\in\mathbb{N}$

$\begin{split}P_{k+1}^{-1}&=\lambda^{2}P_{k}^{-1}+\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top},\\\
\theta_{k+1}&=\theta_{k}-P_{k+1}\phi_{k}B_{k}^{\top}(B_{k}\phi_{k}^{\top}\theta_{k}-y_{k}),\end{split}$
(20)

where $\theta_{0}\in\mathbb{R}^{p}$ is an initial guess and
$P_{0}^{-1}=\varepsilon I_{p}$. The convergence of the above update requires a
persistence of excitation condition in the sense of the following [16].

###### Definition 2

A matrix signal $\phi_{k}:\mathbb{N}\rightarrow\mathbb{R}^{p\times n}$ is
called persistently exciting if it is bounded and there exists a $\delta>0$
and $T_{s}\in\mathbb{N}$ such that for all $k_{0}\in\mathbb{N}$

$0<\delta I_{p}\leq\sum_{i=k_{0}}^{k_{0}+T_{s}}\phi_{i}\phi_{i}^{\top}.$

The PE condition in Definition 2, commonly found in the adaptive control
literature [1, 16] is a stronger version of the $T_{s}$-weak PE condition in
Definition 1. The latter requires the positive definiteness of the regressor
matrix only in the initial phase of the estimation, rather than in all time
intervals of length $T_{s}$.

###### Lemma 3

Assume $B_{k}\phi_{k}$ is persistently exciting then the RLSFF update (20)
satisfies

$\displaystyle\|\theta_{k}-\theta^{\star}\|$ $\displaystyle\leq
c_{r}\lambda^{k-T_{s}}\|\theta_{0}-\theta^{\star}\|,$ $\displaystyle\forall
k\geq T_{s}$ $\displaystyle\|\theta_{k}-\theta^{\star}\|$
$\displaystyle\leq\|\theta_{0}-\theta^{\star}\|,$ $\displaystyle\forall
0<k<T_{s}$

where
$c_{r}^{2}={\varepsilon\left(\lambda^{2T_{s}}-\lambda^{-2)}\right)}/\left(\delta\left(1-\lambda^{-2}\right)\right)$.

The non-expansive result follows by noting from (20)

$\tilde{\theta}_{k}=\left(I_{p}-P_{k+1}\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top}\right)\tilde{\theta}_{k-1},\quad\forall
k>0,$

and that
$\sigma_{min}\left(P_{k+1}\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top}\right)\geq
0$ and
$\sigma_{max}\left(P_{k+1}\phi_{k}B_{k}^{\top}B_{k}\phi_{k}^{\top}\right)<1$.
The proof for the exponential stability for $k\geq T_{s}$ can be found in [23]
and is hence omitted.

Consider the RLSFF controller $u_{k}^{\mathrm{RLS}}=\phi_{k}^{\top}\theta_{k}$
with $\theta_{k}$ given by the recursive law (20).

###### Theorem 2

Assume that (1) satisfies Assumption 1, and $B_{k}\phi_{k}$ is persistently
exciting, then the RLSFF adaptive controller

  1. (i)

renders the closed-loop system asymptotically stable

  2. (ii)

achieves finite regret

$\mathcal{R}_{T}(\mathrm{RLS})\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\left(\frac{T_{s}}{1-\rho}\\!+\\!\frac{c_{r}\left(\rho^{T}+1\right)}{(1-\rho)^{2}(1-\lambda)}\right).$

###### Proof:

Consider the benchmark state $x_{k}^{\star}$ evolution in (5) under perfect
uncertainty matching and the RLSFF closed-loop state given by (4) with
$\theta_{k}$ update in (20). Using the _E- $\delta$-ISS_ condition on the
nominal system and noting that $x_{k}$ and $x_{k}^{*}$ are equal in the first
time step, for all $k>T_{s}$

$\displaystyle\|x_{k}-x_{k}^{\star}\|\leq
c_{w}b\sum_{i=0}^{k-1}\rho^{k-i-1}\|\tilde{\theta}_{i}\|$
$\displaystyle\leq\rho
c_{w}c_{T_{s}}b\|\tilde{\theta}_{0}\|\rho^{k-T_{s}}+c_{w}c_{r}b\|\tilde{\theta}_{0}\|\sum_{i=T_{s}}^{k-1}\rho^{k-i-1}\lambda^{i-T_{s}},$

where the second inequality follows from the bound of $B_{k}\phi_{k}$ and
Lemma 3, and we define $c_{T_{s}}:=\frac{1-\rho^{T_{s}}}{1-\rho}$. For
$0<k\leq T_{s}$

$\|x_{k}-x_{k}^{\star}\|\leq
c_{w}b\sum_{i=1}^{k}\rho^{k-i}\|\tilde{\theta}_{i-1}\|\leq
c_{w}b\|\tilde{\theta}_{0}\|\sum_{i=1}^{k}\rho^{k-i}$

Similar to the proof of Theorem 1, the states, $x_{k}$ and $x_{k}^{\star}$ can
both be shown to be bounded and, hence

$\displaystyle\mathcal{R}_{T}(\mathrm{RLS})\leq\underbrace{L_{c}\sum_{k=T_{s}+1}^{T-1}\|x_{k}-x_{k}^{\star}\|}_{\text{I}}+\underbrace{L_{c}\sum_{k=0}^{T_{s}}\|x_{k}-x_{k}^{\star}\|}_{\text{
II}},$

where, as before, II represents the cost in the initial $T_{s}$ long PE phase,
and I the cost improvement afterward. Using the Cauchy Product inequality and
the inequalities above

II $\displaystyle\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\sum_{k=0}^{T_{s}-1}\sum_{i=0}^{k}\rho^{k-i}\leq
c_{w}c_{T_{s}}T_{s}bL_{c}\|\tilde{\theta}_{0}\|$ I $\displaystyle\leq
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\left(\rho
c_{T_{s}}\\!\sum_{k=1}^{T-T_{s}-1}\rho^{k}\\!+\\!\sum_{k=0}^{T-T_{s}-2}\sum_{i=0}^{k}\rho^{k-i}\lambda^{i}\right)$
$\displaystyle\leq\rho
c_{w}bL_{c}\|\tilde{\theta}_{0}\|\frac{\rho^{T}+\rho}{(1-\rho)^{2}}$
$\displaystyle\quad+c_{w}c_{r}bL_{c}\|\tilde{\theta}_{0}\|\frac{\left(1-\rho^{T-T_{s}-1}\right)\left(1-\lambda^{T-T_{s}-1}\right)}{\left(1-\rho\right)\left(1-\lambda\right)}.$

Combining the terms complestes the proof. The asymptotic stability proof
parallels that of Theorem 1 and is omitted. ∎

###  IV-A Analysis of the Bounds

Both the RPL and the RLSFF adaptive controllers achieve a finite regret
containing three separate terms of potential interest. A constant term, an
exponentially decaying one with rate $\rho$, and a linear term in the PE time
$T_{s}$. The latter is of most interest, which arises from the non-expansive
properties of both estimators and signifies that no improvement in the worst-
case can be expected until the time $T_{s}$ when the PE condition is met.
Comparing the regret bounds directly, when $\gamma<\lambda$ and $c_{r}>1$,
i.e. $\varepsilon>\delta$ for the RLSFF algorithm (20), then the upper bound
of the RLSFF regret is strictly larger than that of the modified RPL
controller.

##  V Numerical Simulation

In this section, we show that the MRAC problem can be cast into the considered
setting and provide a numerical example to showcase the performance of
discussed adaptive controllers. Consider the uncertainty-matched system

$x_{k+1}=Ax_{k}+B(u_{k}-\phi(x_{k})^{\top}\theta^{\star}),$ (21)

where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$,
$\phi(x_{k})$ is defined as before, and $\theta^{\star}\in\mathbb{R}^{p}$ is
unknown. The MRAC goal is to track the reference dynamics

$\bar{x}_{k+1}=A_{r}\bar{x}_{k}+B_{r}r_{k},$

where $\bar{x}_{k}\in\mathbb{R}^{n}$, is the ideal reference vector to be
tracked, $r_{k}\in\mathbb{R}^{m}$, is the bounded input command,
$B_{r}\in\mathbb{R}^{n\times m}$ and $A_{r}\in\mathbb{R}^{n\times n}$ is Schur
stable. As in [17], we assume $(A,B)$ to be controllable and $B$ to be full
column rank, and consider the input
$u_{k}=-K_{1}x_{k}+K_{2}r_{k}+\phi_{k}^{\top}\theta_{k}$ with matrices
$K_{1},K_{2}\in\mathbb{R}^{m\times n}$ chosen such that $A-BK_{1}=A_{r}$ and
$BK_{2}=B_{r}$ hold and $\theta_{k}$ being the parameter estimate. Denoting
$e_{k}=x_{k}-\bar{x}_{k}$, the error dynamics are given by

$e_{k+1}=A_{r}e_{k}+B\phi(x_{k})^{\top}\tilde{\theta}_{k},$ (22)

in the form of (1). Since $A_{r}$ is stable, the _E- $\delta$-ISS_ condition
is satisfied. For the cost we consider $c(x):=\|x\|^{2}$ [11], which is
locally Lipschitz since the closed-loop states are bounded.

We consider the following numerical example [17]

$A=\begin{bmatrix}1.0314&0.2526\\\
0.2526&1.0314\end{bmatrix},\;B=\begin{bmatrix}0.0314\\\ 0.2526\end{bmatrix},$

with the reference model dynamics

$A_{r}=\begin{bmatrix}-0.9929&0.2253\\\
-0.0569&0.8117\end{bmatrix},\;B_{r}=\begin{bmatrix}0.0314\\\
0.2526\end{bmatrix}.$

The true parameter is
$\theta^{\star}=\begin{bmatrix}-0.75&-0.50\end{bmatrix}^{\top}$ and
$\phi(x_{k})=x_{k}$. Figure 1 shows the tracking of the first component of
state $x_{k}$ for the above example under the RPL, RLSFF and the command
governor-based MRAC controller proposed in [17] without regret guarantees.

Figure 1: Comparison of adaptive controllers for the MRAC example in terms of
asymptotic tracking (on the top) and regret (on the bottom).

For all algorithms we initialize the estimate by
$\theta_{0}\\!=\\!\begin{bmatrix}5.00&-1.00\end{bmatrix}^{\top}$, take
$\varepsilon=1$ and start the simulation at
$x_{0}=\begin{bmatrix}0.2&0.2\end{bmatrix}^{\top}$. For RLSFF we used a
forgetting factor of $\lambda^{2}=0.99$, as lower values led to conditioning
problems. For the controller of [17] we used the same parameters as in Example
1, [17] and a tuned saturation value of $1.5$ for the command governor update
for best results. The superior performance of RPL is evident from Fig. 1,
despite a less stringent PE condition. The top figure shows the tracking
performance of the controllers while the bottom the regret for the quadratic
costs, which in this case is
$\mathcal{R}_{T}(u):=\sum_{k=0}^{T-1}\|e_{k}\|^{2}$, since by definition
$\bar{x}_{0}={x}_{0}$ and therefore $e^{\star}_{k}=0$ for all
$k\in\mathbb{N}$.

##  VI Conclusion

Two recursive learning algorithms were considered for the nonlinear adaptive
control problem with matched uncertainty in the context of regret
minimization. We showed that both the recursive least squares with forgetting
factor and the novel recursive proximal learning algorithm are asymptotically
stable and achieve finite regret scaling with the time required to achieve
persistence of excitation. Possible extensions include the consideration of
inexact basis matrices, bounded, non-stochastic noise, and the development of
goal-oriented controllers that instead of the estimation cost aim to minimize
the objective cost directly.

## References

  * [1] K. J. Åström and B. Wittenmark, _Adaptive control_. Addison Wesley Publishing Company, 1989.
  * [2] A. Astolfi, D. Karagiannis, and R. Ortega, _Nonlinear and adaptive control with applications_. Springer, 2008, vol. 187.
  * [3] K. S. Narendra and A. M. Annaswamy, _Stable adaptive systems_. Courier Corporation, 2012.
  * [4] N. Cesa-Bianchi and G. Lugosi, _Prediction, learning, and games_. Cambridge university press, 2006.
  * [5] M. Raginsky, A. Rakhlin, and S. Yüksel, “Online convex programming and regularization in adaptive control,” in _49th IEEE Conference on Decision and Control (CDC)_ , 2010, pp. 1957–1962.
  * [6] A. M. Annaswamy and A. L. Fradkov, “A historical perspective of adaptive control and learning,” _Annual Reviews in Control_ , vol. 52, pp. 18–41, 2021.
  * [7] A. Fradkov, I. Miroshnik, and V. Nikiforov, _Nonlinear and Adaptive Control of Complex Systems_. Springer Science & Business Media, 1999, vol. 491.
  * [8] M. Nonhoff and M. A. Müller, “On the relation between dynamic regret and closed-loop stability,” _Systems & Control Letters_, vol. 177, p. 105532, 2023.
  * [9] A. Karapetyan, A. Tsiamis, E. C. Balta, A. Iannelli, and J. Lygeros, “Implications of regret on stability of linear dynamical systems,” _IFAC-PapersOnLine_ , vol. 56, no. 2, pp. 2583–2588, 2023.
  * [10] J. E. Gaudio, T. E. Gibson, A. M. Annaswamy, M. A. Bolender, and E. Lavretsky, “Connections between adaptive control and optimization in machine learning,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_. IEEE, 2019, pp. 4563–4568.
  * [11] N. M. Boffi, S. Tu, and J.-J. E. Slotine, “Regret bounds for adaptive nonlinear control,” in _Learning for Dynamics and Control_. PMLR, 2021, pp. 471–483.
  * [12] S. Akhtar, R. Venugopal, and D. S. Bernstein, “Logarithmic lyapunov functions for direct adaptive stabilization with normalized adaptive laws,” _International Journal of Control_ , vol. 77, no. 7, pp. 630–638, 2004.
  * [13] A. Simonetto, A. Mokhtari, A. Koppel, G. Leus, and A. Ribeiro, “A class of prediction-correction methods for time-varying convex optimization,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 17, pp. 4576–4591, 2016.
  * [14] E. Dall’Anese, A. Simonetto, and A. Bernstein, “On the convergence of the inexact running Krasnosel’skii–mann method,” _IEEE Control Systems Letters_ , vol. 3, no. 3, pp. 613–618, 2019.
  * [15] N. Parikh, S. Boyd _et al._ , “Proximal algorithms,” _Foundations and trends® in Optimization_ , vol. 1, no. 3, pp. 127–239, 2014.
  * [16] R. M. Johnstone, C. R. Johnson Jr, R. R. Bitmead, and B. D. Anderson, “Exponential convergence of recursive least squares with exponential forgetting factor,” _Systems & Control Letters_, vol. 2, no. 2, pp. 77–82, 1982.
  * [17] K. M. Dogan, T. Yucelen, W. M. Haddad, and J. A. Muse, “Improving transient performance of discrete-time model reference adaptive control architectures,” _International Journal of Adaptive Control and Signal Processing_ , vol. 34, no. 7, pp. 901–918, 2020.
  * [18] D. Angeli, “A Lyapunov approach to incremental stability properties,” _IEEE Transactions on Automatic Control_ , vol. 47, no. 3, pp. 410–421, 2002.
  * [19] D. N. Tran, B. S. Rüffer, and C. M. Kellett, “Convergence properties for discrete-time nonlinear systems,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 8, pp. 3415–3422, 2018.
  * [20] A. Karapetyan, E. C. Balta, A. Iannelli, and J. Lygeros, “Closed-loop finite-time analysis of suboptimal online control,” _arXiv preprint arXiv:2312.05607_ , 2023.
  * [21] E. Hazan, A. Kalai, S. Kale, and A. Agarwal, “Logarithmic regret algorithms for online convex optimization,” in _International Conference on Computational Learning Theory_. Springer, 2006, pp. 499–513.
  * [22] Z.-P. Jiang and Y. Wang, “Input-to-state stability for discrete-time nonlinear systems,” _Automatica_ , vol. 37, no. 6, pp. 857–869, 2001.
  * [23] S. Brüggemann and R. R. Bitmead, “Exponential convergence of recursive least squares with forgetting factor for multiple-output systems,” _Automatica_ , vol. 124, p. 109389, 2021.


Discovering the QCD critical point will be a landmark in the QCD phase
diagram. Progress in both theory and experimental measurements is needed to
ascertain the existence and location of the QCD critical point. We discuss in
this section the outlook from the theoretical side on the study of critical
fluctuations along with the future opportunities with upcoming new
experimental facilities in the hunt for the QCD critical point.

Figure 34:  Ratio of fourth to second order cumulant ($C_{4}/C_{2}$) of net-
proton/proton distributions($\kappa\sigma^{2}$) in top 0-5% Au+Au collisions
from $\sqrt{s_{NN}}$ = 3.0 to 200 GeV from the STAR experiment along with
measurement on proton $C_{4}/C_{2}$ from the HADES experiment [105] for 0-10%
central Au+Au collisions at $\sqrt{s_{NN}}$ = 2.4 GeV with $|y|<$ 0.4. The
bars and blue bands on the data points represent the statistical and
systematic uncertainties, respectively. The projection of uncertainties on the
cumulant ratio $\kappa\sigma^{2}$ from the ongoing phase II of the BES program
at RHIC are shown in green band around unity. The double-headed arrow shows
the collision energy range ($\sqrt{s_{NN}}$ = 3 - 7.7 GeV) covered by the
STAR’s fixed target program (STAR FXT).

###  4.1 Theory

A concrete answer to the existence and location of the QCD critical point will
require a complete understanding of the dynamics of the system created in the
heavy-ion collisions. A quantitative theoretical framework that concerns the
dynamical evolution of the system formed in heavy-ion collisions till it
reaches its final state will be crucial to the search of QCD critical point.
Studies of fluctuations and their coupling with critical modes in stochastic-
hydrodynamics as well as hydro-kinetics have been ongoing. Stochastic fluid
dynamics generally involves event-by-event simulation of the viscous
relativistic fluid dynamical evolution with stochastic conservation law, where
energy-momentum tensor and conserved current 4-vector include Ideal + Viscous
+ Noise parts. On the other hand, the hydro-kinetics method is used to
calculate the time dependence of the two-point correlation functions of the
fluid dynamical fields using a set of linearized equations of motion.
Hydrodynamical fluctuations and their time evolution considering the effect of
critical slowing down are being studied [138, 139, 140, 141, 142, 143, 144,
145].

Another important aspect of stochastic fluid dynamics and hydro-kinetics study
of fluctuations is the particlization. Experimental observables measure
correlations of produced particles. The conversion from fluid degrees of
freedom to the particle degrees of freedom includes the effects like finite
particle statistics, limited acceptance and conservation of energy and global
charges at the freeze-out hypersurface.

Further, the thermodynamics of the system formed in these collisions can be
studied via comparison of the fluctuations measured in the experiments to the
various thermal model and lattice-QCD calculations [26, 146, 147]. Since the
net-particle distribution could be fully characterized by its all-order
moments, consistent thermal description of mean yields as well higher moments
becomes important to test the thermal nature of the whole system [148].
Deviation from thermal equilibrium might indicate interesting physics, such as
the presence of a QCD critical point.

###  4.2 Experiment

The observed non-monotonic collision energy dependence of net-proton
$C_{4}/C_{2}$ has been the most interesting result so far in the experimental
search of the QCD critical point. Recently STAR collaboration reported the
proton $C_{4}/C_{2}$ in Au+Au collisions at $\sqrt{s_{NN}}$ = 3 GeV from
STAR’s FXT program [149]. At such low energies, the number of event-by-event
anti-protons are negligibly. As seen from the Fig. 34, a suppression in
$C_{4}/C_{2}$ was observed compared to higher collision energies. This is
consistent with expectations from baryon number conservation and suggests that
matter is dominantly hadronic at such low collision energies. The non-
monotonic collision energy dependence of the $C_{4}/C_{2}$ measurements from
the BES-I program together with the observed suppression at $\sqrt{s_{NN}}$ =
3 GeV (FXT) indicates that the possible critical point could only exist at
collision energies higher than 3 GeV. Nonetheless, the measurements at
collision energies $\sqrt{s_{NN}}$ = 7.7 – 27 GeV suffer from large
statistical uncertainties which are the key energies that drive the non-
monotonic collision energy dependence trend. The measurement on proton
$C_{4}/C_{2}$ by HADES [105], for 0-10% central Au+Au collisions at
$\sqrt{s_{NN}}$ = 2.4 also has large uncertainties and require more event
statistics to serve as effective experimental baseline in the large $\mu_{B}$
region. Here, in this section, we discuss future prospects of fluctuation
measurements in experiments that are currently active and upcoming in the near
future.

####  4.2.1 STAR BES-II and FXT

In order to make precise measurements in the collision energy range $7.7$ GeV
$\leq\sqrt{s_{NN}}\leq 19.6$ GeV, phase II of the beam energy scan (BES-II)
has been ongoing using the STAR detector at RHIC [150]. As compared to the
presented results from the BES-I program, additional collision energy at
$\sqrt{s_{NN}}$ = 9.2 GeV from the BES-II program will be available for
fluctuation measurements with large statistics. Table 4 contains the event
statistics for each collision energy being collected in the ongoing BES-II
program and the corresponding $\mu_{B}$ at chemical freeze-out.

Table 4: Event statistics to be collected in Au+Au collisions for $7.7$ GeV $\leq\sqrt{s_{NN}}\leq 19.6$ GeV from the ongoing BES-II program at RHIC and the corresponding $\mu_{B}$ at chemical freeze-out.  $\sqrt{s_{NN}}$ (GeV) | Events (millions) |  $\mu_{B}$ (MeV)
---|---|---
7.7 | 100 | 398
9.2 | 160 | 355
11.5 | 230 | 287
14.5 | 300 | 264
19.6 | 400 | 188

The statistical uncertainties of the measurements in BES-II will be
significantly reduced as compared to current measurements. The magnitude of
projected statistical uncertainties from BES-II measurements are shown in Fig.
34. In addition to precision measurements of cumulants, the goal is to also
improve the quality of the potential critical signal. Several detector
upgrades have been made in the STAR detector at RHIC. Upgrades such as that of
the inner chambers of the TPC, the event plane detector, the end-cap time of
flight detector, bring new scopes to the fluctuation measurements, like
enlarged rapidity coverage (from $|\eta|<1$ to $|\eta|<1.5$), improved
particle identification and centrality definition at forward rapidity. The QCD
critical point induced power law behaviour of the fluctuation measurements can
be tested using the extended rapidity coverage [151]. STAR fixed target (FXT)
program will carry Au+Au collisions from $\sqrt{s_{NN}}$ = 3.0 -7.7 GeV (shown
in the Fig. 34) allowing to scan the QCD phase diagram up to $\mu_{B}$ = 720
MeV. The collision energies, event statistics and the corresponding $\mu_{B}$
values for the STAR FXT program is listed in table 5.

Table 5: Event statistics to be collected in Au+Au collisions from the ongoing STAR FXT program at RHIC and the corresponding $\mu_{B}$ at chemical freeze-out.  FXT Energy (GeV) | Events (millions) |  $\mu_{B}$ (MeV)
---|---|---
7.7 | 160 | 398
6.2 | 120 | 487
5.2 | 100 | 541
4.5 | 100 | 589
3.9 | 120 | 633
3.5 | 120 | 666
3.2 | 200 | 699
3.0 | 260 | 720

At beam energies above $\sqrt{s_{NN}}\geq$ 19.6 GeV, the reaction rates of
STAR are limited to a thousand Hertz (Hz) by the TPC read-out, and drop down
to a few Hz for energies below $\sqrt{s_{NN}}$ = 8 GeV in collider mode due to
low luminosity. Future new experiments, which are designed for operation at
high rates, large acceptance, and the state-of-the-art particle
identification, at the energy region where baryon density is high, will be
needed in the search for the critical point. The new facilities for studying
high baryon density matter includes (a) Nuclotron-based Ion Collider fAcility
(NICA) at the Joint Institute for Nuclear Research (JINR), Dubna, Russia
[152], (b) Compressed Baryonic Matter (CBM) at Facility for Anti-proton and
Ion Research (FAIR), Darmstadt, Germany [153], (c) CSR External-target
Experiment (CEE) at High Intensity heavy-ion Accelerator Facility (HIAF),
Huizhou, China [154], and (d) Heavy-Ion program at Japan Proton Accelerator
Research Complex (JPARC-HI), Tokai, Japan [155].

Figure 35: Interaction rates from various heavy-ion collision facilities [165,
55]: BES-II at RHIC (7.7 $<\sqrt{s_{NN}}<$ 19.6 GeV, solid red circles), STAR
FXT (solid blue squares), NICA (red triangles), CBM at FAIR (solid green
squares), HADES (open circles), HIAF (solid diamonds), and JPARC-HI (solid
stars) [155, 166].

####  4.2.2 NICA

Two dedicated experiments are designed at NICA for the study of QCD phase
diagram: a fixed target experiment, Baryonic Matter at Nuclotron (BM$@$N)
[156], which will cover the lower energy region, and the Multi-Purpose
Detector (MPD) [157] which will run in collider mode. The MPD experiment
includes critical point search using fluctuation measurements among its
physics priorities [158]. The MPD is a $4\pi$ spectrometer which can detect
charged hadrons, electrons and photons in heavy-ion collisions in the energy
range of the NICA collider. The detector setup includes the Central Detector
(CD) covering $\pm 2$ units in pseudo-rapidity ($\eta$), Time-Projection
Chamber (TPC), barrel Time-Of-Flight system (TOF), Zero-Degree Calorimeter
(ZDC) and Fast Forward Detector (FFD) [159]. The TPC is the main tracking
detector of the MPD and will provide tracking and PID with resolution better
than 8% in the range $|\eta|<1.2$, and momentum resolution for charge
particles better than 3% in $0.1<p_{T}<1$ GeV/c. The cylindrical part of TOF
also has full azimuthal and $|\eta|<1.2$ coverage and will allow PID in the
$p_{T}$ range 0.1 – 2 GeV/c for charged hadrons. Combining the measurements
from TPC and TOF will provide an efficient $\pi/K$ PID separation up to 1.5
GeV/c and $\pi/p$ separation up to 3 GeV/c [160].

Initial run of NICA will be done with Bi+Bi collisions at $\sqrt{s_{NN}}$ =
9.2 AGeV [161]. Au+Au collisions are planned with beam energies spanning from
$\sqrt{s_{NN}}$ = 12 - 27 GeV with luminosity $L\geq 10^{30}cm^{-2}s^{-1}$ and
$\sqrt{s_{NN}}$ = 4 - 11 GeV with average luminosity $L\geq
10^{27}cm^{-2}s^{-1}$. The interaction rate of 6 kHz for minimum bias is
planned for the higher collision energies but decreases to about 10 Hz because
of low luminosity at $\sqrt{s_{NN}}$= 4 GeV. A technique to sustain the beam
luminosity in low energies for prolonged period of time, called the electron
cooling, which is also being used in BES-II at STAR, will be employed.

####  4.2.3 Compressed Baryonic Matter (CBM) experiment

The CBM experiment is a fixed-target detector capable of identifying hadrons,
leptons ($e$, $\mu$) and photons. The interaction rate will be higher for the
CBM experiment at FAIR. The experiment will operate over the energy range,
$\sqrt{s_{NN}}$= 2.7 - 4.9 GeV. The goal of the CBM experiment at SIS100 is to
discover fundamental properties of QCD matter at high baryon density using
rare probes. The experiment plans to perform high-precision study of higher-
order fluctuations at various beam energies in baryon density region of 500
MeV $<\mu_{B}<$ 800 MeV. In order to achieve high precision, the measurements
will be performed at reaction rates up to 10 MHz. The CBM detector acceptance
of polar emission angles between 2.5 and 25 degrees allows it to cover mid-
rapidity and the forward rapidity for symmetric collision systems over the
FAIR energy range. The lab pseudo-rapidity range coverage of the CBM detector
system is $1.5<\eta<3.8$. The detector setup includes Micro Vertex Detector
(MVD), Silicon Tracking System (STS), Ring Imaging Cherenkov Detector (RICH),
Transition Radiation Detector (TRD), and Time-of-Flight Detector (TOF) [162,
163]. The collision vertex of the event will be provided by the MVD. The STS
inside the dipole field records the momentum information of the charged
particle tracks. Particle identification will be done by the TOF. Measurement
of centrality of collision will be possible with Project Spectator Detector
(PSD). Feasibility study of performing fluctuation measurements at CBM are
ongoing and show promising prospects [164].

The new experiments will collect enormous event statistics due to high
interaction rates. Large event statistics will be beneficial for higher-order
fluctuation measurements. Handling such rates would demand fast and state-of-
the-art detectors, which are currently being envisaged. The interaction rates
from various upcoming experiments are summarised in the Fig. 35 [165, 55]. In
addition, understanding the systematics of fluctuation measurements is
crucial. For example, determining the centrality from different detectors (PSD
at CBM and EPD at STAR) than those used for PID, enlarging the kinematic
acceptance for PID (iTPC at STAR) will help us to study the measurements
better. Due to low multiplicity of produced particles in low energy
collisions, centrality definition using charge particles or energy deposition
measurements suffer from poor resolution. This could result in a artificial
correlation in fluctuation measurements as discussed in subsection 3.3.5. So,
proper investigation of this effect should be studied in the fluctuation
measurements. As nearly half of the net-baryon is carried out by
(anti-)neutrons, fluctuations of neutron number, in principle, should also
reflect criticality. Adding neutrons to proton fluctuations would serve as a
better proxy of net-baryon number. Other important factors will be the
detector efficiencies and acceptance for various hadrons at low energies,
especially in the fixed target configuration. Uniform acceptance of protons at
mid-rapidity coverage would be crucial for studies on the critical point via
proton number fluctuations.

####  4.2.4 Acceptance: collider vs fixed target

Figure 36: y-$p_{T}$ acceptance figure of protons measured in Au+Au collisions
in the STAR experiment. The upper panel is from the collision in the fixed-
target mode at $\sqrt{s_{NN}}$ = 3.0 GeV. The bottom panel represents the
collision in collider mode at $\sqrt{s_{NN}}$ = 54.4 GeV.

In nuclear collision experiments, for a given detector setup, the mode of
collisions determines the phase space acceptance for the charged particles.
For example, as shown in Fig. 36, the acceptance of protons is asymmetric in
fixed target mode (upper panel) as compared to the collider mode (lower
panel), where it is symmetric.

In the collider mode, the mid-rapidity is always at zero, and this allows STAR
TPC to perform mid-rapidity measurements at all collision energies. However,
in the fixed target mode, the location of mid-rapidity depends on collision
energy. The acceptance of the STAR detector allows coverage of mid-rapidity
distributions of protons in fixed target mode at $\sqrt{s_{NN}}$ = 3 GeV,
while at higher fixed target energies, additional detectors like iTPC (inner -
TPC) and eTOF (extended - TOF) are needed for the phase-space coverage around
the mid-rapidity.

##  5 Summary

In the absence of any first-principle theoretical calculations on the location
of the QCD critical point, one has to rely on predictions from QCD-based
theories to search for critical signals in heavy-ion collisions. One such
calculation is done with the linear sigma model, which predicts a non-
monotonic dependence of kurtosis of number fluctuations when approaching the
critical point. Experimental measurement on the net-proton cumulant ratio
$C_{4}/C_{2}$ in 0-5% Au+Au collisions, shows an evidence of a non-monotonic
collision energy dependence of the measurements in the range $\sqrt{s_{NN}}$ =
7.7 - 200 GeV.

The normalised fourth-order cumulant ratio of proton shows collision energy
dependence similar to net-proton measurements, which is attributed to the
presence of four-proton correlations, the anti-proton, on the other hand, have
a weak dependence on collision energy. None of the model calculations are able
to reproduce the observed energy dependence of the net-proton cumulant
$C_{4}/C_{2}$. Comparison with a wide pool of baseline calculations suggests
that the 0-5% net-proton $C_{4}/C_{2}$ measurements deviate below and above
the baseline calculations at a level of 2-3 $\sigma$ at various collision
energies. The exhibited non-monotonicity of net-proton cumulants with respect
to Skellam baseline, UrQMD baseline, and HRG canonical ensemble calculations
are found to be of 3.1-3.3 $\sigma$ significance. The recent measurement of
proton $C_{4}/C_{2}$ in Au+Au collisions at $\sqrt{s_{NN}}$ = 3 GeV show
suppression for 0-5% centrality and is consistent with the expectation from
the baryon number conservation effect suggesting the dominance of hadronic
matter in such low energies. The suppression of proton cumulant ratio
$C_{4}/C_{2}$ at 3 GeV, together with the reported non-monotonic collision
energy dependence of net-proton $C_{4}/C_{2}$ at higher energies, indicates
that the possible critical point could exist at collision energies higher than
3 GeV.

The experimental results on net-charge and net-kaon cumulant ratio
$C_{4}/C_{2}$ in most central 0-5% collisions have large uncertainties
associated with the measurements and show a flat collision energy dependence
with most of the data points showing agreement with the Skellam baseline
within uncertainties. The current net-$\Lambda$ fluctuation results are only
up to the $3^{rd}$ order and do not shown any signs of critical fluctuations.
Addition of net-$\Lambda$’s to net-kaon fluctuation could serve as a better
proxy for net-strangeness fluctuations as most of the strangeness is carried
by kaons and $\Lambda$’s in the system.

Negative values were observed from measurements of the fifth and sixth-order
net-proton cumulant ratios, $C_{5}/C_{1}$ and $C_{6}/C_{2}$ for 0-40% Au+Au
collisions at most of the collision energies, although with large
uncertainties. $C_{6}/C_{2}$ becomes increasingly negative with decreasing
collision energy. At $\sqrt{s_{NN}}$ = 200 GeV, the $C_{6}/C_{2}$ measurements
progressively become negative from peripheral to central collisions. These
observations, albeit with large uncertainties, are qualitatively consistent
with the trend and sign predicted by LQCD calculations for the ratios of
fifth-to-first and sixth-to-second order baryon number susceptibilities for
QCD matter in the range $\mu_{B}<$ 110 MeV. In contrast, positive
$C_{5}/C_{1}$ and $C_{6}/C_{2}$ were reported in peripheral (70-80%)
collisions at all energies. Multiplicity dependence studies of the two ratios
in p+p collisions at $\sqrt{s_{NN}}$ = 200 GeV also yeild positive sign.
Current measurements on higher order factorial cumulants at $\sqrt{s_{NN}}$ =
7.7 GeV indicate that more event statistics is needed for the search of first-
order phase transition in the high $\mu_{B}$ ($\sim$ 400 MeV) region.

The system formed in the heavy-ion collision is of femto-scale level and
evolves dynamically with time. Hence, it is quite challenging both
theoretically and experimentally to probe into the physics of QCD critical
point. Theoretical calculations considering the effects of non-equilibrium and
critical slowing down near the CP will guide the experimental search for the
CP. As a first step, one tries to establish the possibility of the existence
of the QCD critical region, which could span about 100 MeV range in baryon
chemical potential. The current result on net-proton kurtosis is an important
step in that direction as these are the first fluctuation measurements
exhibiting non-monotonic variation with collision energy. To reduce the
statistical uncertainties on the current net-particle cumulant ratios
measurements at lower collision energies and improve the quality of the
observed signal, the BES-II program has been carried out at RHIC. The STAR FXT
program, along with several future experiments like, NICA and CBM, will
collect large event statistics and explore the baryon-rich energy region. The
overlapping $\mu_{B}$ region covered by these experiments will allow for
independent confirmation of possible critical signals. With the state-of-the-
art detector technology at the helm, the upcoming experiments carry huge
potentials to discover the elusive critical point and map the QCD phase
diagram with utmost precision.

Acknowledgements: The authors thank the colleagues from STAR and ALICE
collaborations for several discussions. B.M. was supported in part by the J C
Bose Fellowship from Department of Science of Technology and Department of
Atomic Energy, Government of India.

## References

  * [1] K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74, 014001 (2011)
  * [2] P. Braun-Munzinger and J. Wambach, Rev. Mod. Phys. 81, 1031-1050 (2009)
  * [3] L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83-100 (2007)
  * [4] E. V. Shuryak, Phys. Rept. 61, 71-158 (1980)
  * [5] P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 172301 (2007)
  * [6] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005)
  * [7] L. P. Csernai, J. I. Kapusta and L. D. McLerran, Phys. Rev. Lett. 97, 152303 (2006)
  * [8] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature 443, 675-678 (2006)
  * [9] K. Fukushima and V. Skokov, Prog. Part. Nucl. Phys. 96 (2017), 154-199 doi:10.1016/j.ppnp.2017.05.002 [arXiv:1705.00718 [hep-ph]].
  * [10] A. Bazavov et al. [HotQCD], Phys. Lett. B 795, 15-21 (2019)
  * [11] H. T. Ding et al. [HotQCD], Phys. Rev. Lett. 123, no.6, 062002 (2019)
  * [12] L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt and S. Sharma, [arXiv:2111.12599 [hep-lat]].
  * [13] S. Ejiri, Phys. Rev. D 78, 074507 (2008)
  * [14] E. S. Bowman and J. I. Kapusta, Phys. Rev. C 79, 015202 (2009)
  * [15] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. Lett. 81, 4816-4819 (1998)
  * [16] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D 60, 114028 (1999)
  * [17] M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009)
  * [18] V. A. Goy, V. Bornyakov, D. Boyda, A. Molochkov, A. Nakamura, A. Nikolaev and V. Zakharov, PTEP 2017, no.3, 031D01 (2017)
  * [19] A. Bazavov et al. [HotQCD], Phys. Rev. D 96, no.7, 074510 (2017)
  * [20] J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C 73, 034905 (2006)
  * [21] A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167-199 (2006)
  * [22] P. Braun-Munzinger and J. Stachel, Nature 448, 302 (2007)
  * [23] M. Asakawa, S. Ejiri and M. Kitazawa, Phys. Rev. Lett. 103, 262301 (2009)
  * [24] B. Berdnikov and K. Rajagopal, Phys. Rev. D 61, 105017 (2000)
  * [25] M. A. Stephanov, Phys. Rev. Lett. 107, 052301 (2011)
  * [26] S. Gupta, X. Luo, B. Mohanty, H. G. Ritter and N. Xu, Science 332, 1525 (2011)
  * [27] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. Lett. 113, 092301 (2014)
  * [28] L. Adamczyk et al. [STAR], Phys. Lett. B 785, 551-560 (2018)
  * [29] M. M. Aggarwal et al. [STAR Collaboration], Phys. Rev. Lett. 105, 022302 (2010)
  * [30] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. Lett. 112, 032302 (2014)
  * [31] J. Adam et al. et al. [STAR Collaboration], Phys. Rev. Lett. 126, 092301 (2021)
  * [32] M. Abdallah et al. [STAR], Phys. Rev. C 104, no.2, 024902 (2021)
  * [33] Y. Hatta and M. A. Stephanov, Phys. Rev. Lett. 91, 102003 (2003) [erratum: Phys. Rev. Lett. 91, 129901 (2003)]
  * [34] M. Kitazawa and M. Asakawa, Phys. Rev. C 86, 024904 (2012) [erratum: Phys. Rev. C 86, 069902 (2012)]
  * [35] Z. Fodor and S. D. Katz, JHEP 04, 050 (2004)
  * [36] R. V. Gavai and S. Gupta, Phys. Rev. D 68, 034506 (2003)
  * [37] O. Philipsen, Nucl. Phys. A 830, 713C-716C (2009)
  * [38] Z. Fodor and S. D. Katz, JHEP 03, 014 (2002)
  * [39] S. Datta, R. V. Gavai and S. Gupta, Phys. Rev. D 95, no.5, 054512 (2017)
  * [40] C. S. Fischer, J. Luecker and C. A. Welzbacher, Phys. Rev. D 90, no.3, 034022 (2014)
  * [41] X. y. Xin, S. x. Qin and Y. x. Liu, Phys. Rev. D 90, no.7, 076006 (2014)
  * [42] M. Asakawa and K. Yazaki, Nucl. Phys. A 504, 668-684 (1989)
  * [43] O. Scavenius, A. Mocsy, I. N. Mishustin and D. H. Rischke, Phys. Rev. C 64, 045202 (2001)
  * [44] Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003)
  * [45] N. G. Antoniou and A. S. Kapoyannis, Phys. Lett. B 563, 165-172 (2003)
  * [46] W. j. Fu, J. M. Pawlowski and F. Rennecke, Phys. Rev. D 101, no.5, 054032 (2020)
  * [47] P. Isserstedt, M. Buballa, C. S. Fischer and P. J. Gunkel, Phys. Rev. D 100, no.7, 074011 (2019)
  * [48] A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov and J. J. M. Verbaarschot, Phys. Rev. D 58, 096007 (1998)
  * [49] A. Barducci, R. Casalbuoni, G. Pettini and R. Gatto, Phys. Rev. D 49, 426-436 (1994)
  * [50] F. Gao and J. M. Pawlowski, Phys. Rev. D 102, no.3, 034027 (2020)
  * [51] F. Gao and J. M. Pawlowski, Phys. Lett. B 820, 136584 (2021)
  * [52] R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti and R. Rougemont, Phys. Rev. D 96, no.9, 096026 (2017)
  * [53] R. Hagedorn, Nuovo Cim. Suppl. 3, 147-186 (1965) CERN-TH-520.
  * [54] R. J. Baxter, Exactly solved models in statistical mechanics
  * [55] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov and N. Xu, Phys. Rept. 853, 1-87 (2020)
  * [56] J. W. Chen, J. Deng, H. Kohyama and L. Labun, Phys. Rev. D 93, no.3, 034037 (2016)
  * [57] W. Fan et al. , [arXiv:1608.07903[hep-ph]].
  * [58] B. Friman, F. Karsch, K. Redlich and V. Skokov, Eur. Phys. J. C 71, 1694 (2011)
  * [59] B. Friman, Nucl. Phys. A 928, 198-208 (2014)
  * [60] S. Mukherjee, R. Venugopalan and Y. Yin, Phys. Rev. C 92, no.3, 034912 (2015)
  * [61] W. j. Fu, X. Luo, J. M. Pawlowski, F. Rennecke, R. Wen and S. Yin, [arXiv:2101.06035 [hep-ph]].
  * [62] A. Bzdak, V. Koch, Phys. Rev. C (R) 100, 014901 (2019)
  * [63] A. Bzdak, V. Koch, D. Oliinychenko and J. Steinheimer, Phys. Rev. C 98, no.5, 054901 (2018)
  * [64] A. Bzdak, V. Koch and N. Strodthoff, Phys. Rev. C 95, no.5, 054906 (2017)
  * [65] A. Bazavov, D. Bollweg, H. T. Ding, P. Enns, J. Goswami, P. Hegde, O. Kaczmarek, F. Karsch, R. Larsen and S. Mukherjee, et al. Phys. Rev. D 101, no.7, 074502 (2020)
  * [66] R. V. Gavai and S. Gupta, Phys. Lett. B 696, 459 (2011)
  * [67] N. M. Abdelwahab et al. [STAR], Phys. Rev. C 92, no.2, 021901 (2015)
  * [68] L. Adamczyk et al. [STAR], Phys. Rev. C 94, no.2, 024909 (2016)
  * [69] J. Adam et al. [STAR], Phys. Rev. C 99, no.4, 044918 (2019)
  * [70] A. Bzdak, V. Koch and V. Skokov, Phys. Rev. C 87, no.1, 014901 (2013)
  * [71] P. Braun-Munzinger, A. Rustamov and J. Stachel, Nucl. Phys. A 960, 114-130 (2017)
  * [72] X. Luo, B. Mohanty and N. Xu, Nucl. Phys. A 931, 808-813 (2014)
  * [73] N. K. Behera, R. K. Nayak and S. Dash, Phys. Rev. C 101, no.6, 064903 (2020)
  * [74] Y. Zhang, S. He, H. Liu, Z. Yang and X. Luo, Phys. Rev. C 101, no.3, 034909 (2020)
  * [75] F. Karsch and K. Redlich, Phys. Lett. B 695, 136-142 (2011)
  * [76] P. Garg, D. K. Mishra, P. K. Netrakanti, B. Mohanty, A. K. Mohanty, B. K. Singh and N. Xu, Phys. Lett. B 726, 691-696 (2013)
  * [77] M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst, S. Soff, L. Bravina, M. Belkacem, H. Weber and H. Stoecker, et al. J. Phys. G 25, 1859-1896 (1999)
  * [78] J. Fu, Phys. Lett. B 722, 144-150 (2013)
  * [79] A. Bhattacharyya, S. Das, S. K. Ghosh, R. Ray and S. Samanta, Phys. Rev. C 90, no.3, 034909 (2014)
  * [80] A. Bhattacharyya, R. Ray, S. Samanta and S. Sur, Phys. Rev. C 91, no.4, 041901 (2015)
  * [81] V. Vovchenko, A. Pasztor, Z. Fodor, S. D. Katz and H. Stoecker, Phys. Lett. B 775, 71-78 (2017)
  * [82] S. Samanta and B. Mohanty, [arXiv:1905.09311 [hep-ph]].
  * [83] P. Braun-Munzinger, B. Friman, K. Redlich, A. Rustamov and J. Stachel, Nucl. Phys. A 1008, 122141 (2021)
  * [84] A. Pandav, D. Mallick and B. Mohanty, Nucl. Phys. A 991, 121608 (2019)
  * [85] X. Luo, J. Phys. G 39, 025008 (2012)
  * [86] X. Luo, Phys. Rev. C 91, no.3, 034907 (2015) [erratum: Phys. Rev. C 94, no.5, 059901 (2016)]
  * [87] A. DasGupta, Asymptotic Theory of Statistics and Probability, Published by Springer-Verlag New York.
  * [88] B. Efron, The Annals of Statistics 7 p1-26(1979)
  * [89] B. Efron, Computers and the Theory of Statistics: Thinking the Unthinkable; SIAM Review, Published by Society for Industrial and Applied Mathematics (url: http://www.jstor.org/stable/2030104.)
  * [90] K. H. Ackermann et al. [STAR], Nucl. Instrum. Meth. A 499, 624-632 (2003)
  * [91] L. Adamczyk et al. [STAR], Phys. Rev. C 96, no.4, 044904 (2017)
  * [92] H. Bichsel, Nucl. Instrum. Meth. A 562, 154-197 (2006)
  * [93] M. Kitazawa and M. Asakawa, Phys. Rev. C 85, 021901 (2012)
  * [94] W. Czyz , LC. Maximon , Annals Phys. 52, 59 (1969)
  * [95] A. Bialas,M. Bleszynski,W. Czyz , Acta Physiol. Pol. B8: 389(1977)
  * [96] M. L. Miller, K. Reygers, S. J. Sanders and P. Steinberg, Ann. Rev. Nucl. Part. Sci. 57, 205-243 (2007)
  * [97] D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121 (2001)
  * [98] B. B. Back et al. [PHOBOS], Phys. Rev. C 70, 021902 (2004)
  * [99] B. Abelev et al. [ALICE], Phys. Rev. C 88, no.4, 044909 (2013)
  * [100] X. Luo, J. Xu, B. Mohanty and N. Xu, J. Phys. G 40, 105104 (2013)
  * [101] V. Skokov, B. Friman and K. Redlich, Phys. Rev. C 88, 034911 (2013)
  * [102] T. Sugiura, T. Nonaka and S. Esumi, Phys. Rev. C 100, no.4, 044904 (2019)
  * [103] X. N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501-3516 (1991)
  * [104] C. Adler et al. [STAR], Phys. Rev. Lett. 86, 4778 (2001) [erratum: Phys. Rev. Lett. 90, 119903 (2003)]
  * [105] J. Adamczewski-Musch et al. [HADES], Phys. Rev. C 102, no.2, 024914 (2020)
  * [106] M. S. Abdallah et al. [STAR], [arXiv:2112.00240 [nucl-ex]].
  * [107] T. Nonaka, M. Kitazawa and S. Esumi, Nucl. Instrum. Meth. A 984 (2020), 164632 doi:10.1016/j.nima.2020.164632 [arXiv:2006.15809 [physics.data-an]].
  * [108] Y. Zhang, Y. Huang, T. Nonaka and X. Luo, [arXiv:2108.10134 [physics.data-an]].
  * [109] M.S Abdallah et al. [STAR Collaboration], [arXiv:2105.14698 [nucl-ex]].
  * [110] A. Bzdak and V. Koch, Phys. Rev. C 86, 044904 (2012)
  * [111] T. Nonaka, M. Kitazawa and S. Esumi, Phys. Rev. C 95, no. 6, 064912 (2017)
  * [112] X. Luo and T. Nonaka, Phys. Rev. C 99, no.4, 044917 (2019)
  * [113] P. Garg, D. K. Mishra, P. K. Netrakanti, A. K. Mohanty and B. Mohanty, J. Phys. G 40, 055103 (2013)
  * [114] A. Bzdak, R. Holzmann and V. Koch, Phys. Rev. C 94, no.6, 064907 (2016)
  * [115] T. Nonaka, M. Kitazawa and S. Esumi, Nucl. Instrum. Meth. A 906, 10-17 (2018)
  * [116] B. I. Abelev et al. [STAR], Phys. Rev. C 81, 024911 (2010)
  * [117] A. Chatterjee, Y. Zhang, J. Zeng, N. R. Sahoo and X. Luo, Phys. Rev. C 101, no.3, 034902 (2020)
  * [118] Y. Zhang, S. He, H. Liu, Z. Yang and X. Luo, Phys. Rev. C 101, no.3, 034909 (2020)
  * [119] R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th Ed. Oliver and Boyd, Edinburgh and London 1963.
  * [120] A. Adare et al. [PHENIX], Phys. Rev. C 93, no.1, 011901 (2016)
  * [121] M. Sakaida, M. Asakawa, H. Fujii and M. Kitazawa, Phys. Rev. C 95, no.6, 064905 (2017)
  * [122] J. Brewer, S. Mukherjee, K. Rajagopal and Y. Yin, Phys. Rev. C 98, no.6, 061901 (2018)
  * [123] S. Acharya et al. [ALICE], Phys. Lett. B 807, 135564 (2020)
  * [124] S. Borsanyi, Z. Fodor, J. N. Guenther, S. K. Katz, K. K. Szabo, A. Pasztor, I. Portillo and C. Ratti, JHEP 10, 205 (2018)
  * [125] A. Pandav, [STAR Collaboration], CPOD 2021 conference, plenary talk (17th March 2021, Session 3)
  * [126] R. Nishitani, [STAR Collaboration], CPOD 2021 conference, plenary talk (17th March 2021, Session 3)
  * [127] J. Adam et al. [STAR], Phys. Rev. C 102, no.2, 024903 (2020)
  * [128] R. Bellwied, S. Borsanyi, Z. Fodor, J. N. Guenther, J. Noronha-Hostler, P. Parotto, A. Pasztor, C. Ratti and J. M. Stafford, Phys. Rev. D 101, no.3, 034506 (2020) doi:10.1103/PhysRevD.101.034506 [arXiv:1910.14592 [hep-lat]].
  * [129] Z. Fecková, J. Steinheimer, B. Tomášik and M. Bleicher, Phys. Rev. C 93, no.5, 054906 (2016)
  * [130] E. Shuryak and J. M. Torres-Rincon, Eur. Phys. J. A 56 (2020) no.9, 241 doi:10.1140/epja/s10050-020-00244-3 [arXiv:2005.14216 [nucl-th]].
  * [131] G. S. Bali, F. Bruckmann, G. Endrödi, S. D. Katz and A. Schäfer, JHEP 08, 177 (2014)
  * [132] E. M. Ilgenfritz, M. Muller-Preussker, B. Petersson and A. Schreiber, Phys. Rev. D 89, no.5, 054512 (2014)
  * [133] V. G. Bornyakov, P. V. Buividovich, N. Cundy, O. A. Kochetkov and A. Schäfer, Phys. Rev. D 90, no.3, 034501 (2014)
  * [134] A. Tomiya, H. T. Ding, X. D. Wang, Y. Zhang, S. Mukherjee and C. Schmidt, PoS LATTICE2018, 163 (2019)
  * [135] H. T. Ding, S. T. Li, Q. Shi and X. D. Wang, Eur. Phys. J. A 57, no.6, 202 (2021)
  * [136] A. Bazavov et al. [HotQCD], Phys. Rev. D 86, 034509 (2012)
  * [137] J. Adamczewski-Musch et al. [HADES], Phys. Rev. C 102, no.2, 024914 (2020)
  * [138] J. I. Kapusta, B. Muller and M. Stephanov, Phys. Rev. C 85, 054906 (2012)
  * [139] M. Bluhm, M. Nahrgang, T. Schäfer and S. A. Bass, EPJ Web Conf. 171, 16004 (2018)
  * [140] M. Singh, C. Shen, S. McDonald, S. Jeon and C. Gale, Nucl. Phys. A 982, 319-322 (2019)
  * [141] M. Nahrgang and M. Bluhm, Phys. Rev. D 102, no.9, 094017 (2020)
  * [142] M. Nahrgang, S. Leupold, C. Herold and M. Bleicher, Phys. Rev. C 84, 024912 (2011)
  * [143] C. Herold, M. Nahrgang, Y. Yan and C. Kobdaj, Phys. Rev. C 93, no.2, 021902 (2016)
  * [144] M. Sakaida, M. Asakawa, H. Fujii, M. Kitazawa, Phys. Rev. C (R) 95, 064905 (2017) 10.1103/PhysRevC.95.064905
  * [145] X. An, G. Basar, M. Stephanov and H. U. Yee, Phys. Rev. C 100, no.2, 024910 (2019)
  * [146] P. Alba, W. Alberico, R. Bellwied, M. Bluhm, V. Mantovani Sarti, M. Nahrgang and C. Ratti, Phys. Lett. B 738, 305-310 (2014)
  * [147] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, Phys. Rev. Lett. 111, 062005 (2013)
  * [148] S. Gupta, D. Mallick, D. K. Mishra, B. Mohanty and N. Xu, Nucl. Phys. A 1005, 121987 (2021)
  * [149] M. Abdallah et al. [STAR], [arXiv:2112.00240 [nucl-ex]].
  * [150] STAR BES-II White paper [STAR], Studying the Phase Diagram of QCD Matter at RHIC,
https://drupal.star.bnl.gov/STAR/files/BES$\\_$WPII$\\_$ver6.9$\\_$Cover.pdf

  * [151] B. Ling and M. A. Stephanov, Phys. Rev. C 93, no.3, 034915 (2016)
  * [152] N. S. Geraksiev [NICA/MPD], J. Phys. Conf. Ser. 1390, no.1, 012121 (2019)
  * [153] T. Ablyazimov et al. [CBM], Eur. Phys. J. A 53, no.3, 60 (2017)
  * [154] S. Ruan, J. Yang, J. Zhang, G. Shen, H. Ren, J. Liu, J. Shangguan, X. Zhang, J. Zhang and L. Mao, et al. Nucl. Instrum. Meth. A 892, 53-58 (2018)
  * [155] H. Sako et al., Letter of Intent for J-PARC Heavy-Ion Program,
http://j-parc.jp/researcher/Hadron/en/pac$\\_$1607/pdf/LoI$\\_$2016-16.pdf

  * [156] M. Kapishin et al., Eur. Phys. J. A, 52, 213 (2016)
  * [157] V. Golovatyuk et al., Eur. Phys. J. A, 52, 212 (2016)
  * [158] V. Kekelidze et al. [NICA], Nucl. Part. Phys. Proc. 273-275, 170-174 (2016)
  * [159] V. Golovatyuk, M. Kapishin, V. Kekelidze, V. Kolesnikov, O. Rogachevky and A. Sorin, J. Phys. Conf. Ser. 668, no.1, 012015 (2016)
  * [160] N. S. Geraksiev [MPD], J. Phys. Conf. Ser. 1023, no.1, 012030 (2018)
  * [161] A. Kisiel, [NICA], SQM 2021 conference, plenary talk (22nd May 2021)
  * [162] P. Senger [CBM], Nucl. Phys. A 967, 892-895 (2017)
  * [163] P. Senger, JPS Conf. Proc. 32, 010092 (2020)
  * [164] S. Samanta [CBM], Nucl. Phys. A 1005, 121896 (2021)
  * [165] T. Galatyuk, Nucl. Phys. A 982, 163-169 (2019)
  * [166] T. Hachiya, Int. J. Mod. Phys. E 29, no.11, 2040005 (2020)