# Nonlinear Dual control based on Fast Moving Horizon estimation and Model
Predictive Control with an observability constraint

Emilien Flayac1, Girish Nair1, Iman Shames 2 1 Emilien Flayac
<EMAIL_ADDRESS>and Girish Nair<EMAIL_ADDRESS>are with the Department of Electrical & Electronic Engineering, University of
Melbourne, Melbourne, Australia 2Iman Shames<EMAIL_ADDRESS>is with
the School of Engineering, The Australian National University, Canberra,
Australia.This work received funding from the Australian Government, via grant
AUSMURIB000001 associated with ONR MURI grant N00014-19-1-2571

###### Abstract

This paper proposes an algorithm that combines Fast Moving Horizon Parameter
Estimation and Model Predictive Control subject to an observability constraint
designed to ensure a lower bound on the performance of the parameter
estimator. Output-feedback stability is proved through input-to-state
stability of the state/error system under a small noise and initial error
assumption. Numerical experiments have been carried out in the case of Active
Simultaneous Localisation and Mapping (SLAM).

##  I INTRODUCTION

Optimisation-based estimation techniques like Moving Horizon Estimation (MHE),
in which one tries to recover the trajectory of a system through solving an
optimisation problem, are arousing growing interests in both their theory
[17], [18] and their practical implementations [6]. System identification is
benefiting from these advances too, see [14] for example. The main advantage
of MHE is that it can deal with nonlinear systems and constraints while also
aiming for tractability. Indeed, only the measurements coming from a time
window of fixed size are used at each time step. A variation of MHE, called
Fast MHE, involves solving the associated optimisation problem partially. One
only performs a few iterations of a dedicated optimisation routine at each
time step and uses the current estimate as the initial guess for the next
problem. Much work has been done in this direction, see [1], [2], [13], [21]
and [22]. In all this work, the MHE scheme is enabled by the so-called _$N$
-step observability_ condition. It states that a small output error on a
rolling time window must lead to a small estimation error uniformly with
respect to time if one starts sufficiently close to the reference. These
conditions are also often assumed to hold uniformly with respect to the input
applied to the system. However, in the general nonlinear case, the input might
have an influence on the observability condition and thus the quality of the
estimation process. In adaptive control, where one tries to regulate a system
while also identifying its dynamical model, this phenomenon has been well
known since the seminal work of Feldbaum [8]. He stated that adaptive
controllers must be designed to guide the system in a standard way, as well as
to probe information and excite the system, which leads to good parameter
estimation. This is known as the _dual effect_ of the controller. See [16] for
a survey. It implies that the _separation principle_ , which claims that
independent design of the control and estimation schemes can be efficient,
does not hold in a general nonlinear case. In the context of Moving Horizon
Estimation, some work has been done to combine it with Model Predictive
Control (MPC). In the manner of MHE, MPC consists in solving a finite-horizon
optimal control problem on a rolling basis. In [5], a minimax MHE-MPC output
feedback scheme is presented although the observability assumption is assumed
to hold independently of the feedback scheme. In [7], the authors successfully
mix an Economic MPC scheme with a specific MHE technique that requires a high-
gain observer under feedback. However, the construction of such a feedback is
not specified in general and seems non-trivial. In [3], a MPC with a
Persistence of Excitation condition is combined with a recursive parameter
estimation algorithm. Still, the influence of the parameter on the dynamics is
supposed to be affine and a periodic persistent state trajectory is also
required. That is why, in this paper, we present a dual output-feedback scheme
for nonlinear discrete-time systems with noisy measurements, based on a fast
MHE algorithm for parameter estimation and on an MPC with a constraint on the
predicted _Observability Grammian_. The design of the controller and the
estimator are coupled in the sense that, at each time step, the MPC algorithm
aims to ensure good estimation performance at the following step through an
appropriate enforced observability condition. Contrary to [3], our method does
not require periodicity or any explicit property as the required excitation
coming from the input is generated by a general implicit constraint.

The remainder of the paper is structured as follows. Section II gathers
standard notations that will be used the entire paper. Section III describes
the setup of a nonlinear parametrised discrete-time system with noisy
measurements. Section IV presents a typical Moving Horizon Parameter
Estimation Problem and its fast implementation. In Section V, our dual MPC
scheme with an observability constraint is set. In Section VI, output-feedback
stability is established in terms of input-to-state stability of the
state/error system under a small noise and initial error assumption. Finally,
our estimation and control scheme is applied in Section VII to the problem of
Active Simultaneous Localisation and Mapping (SLAM).

##  II NOTATIONS

We respectively denote by $\mathbb{N}$ and $\mathbb{N}^{*}$ the set of non-
negative and positive integers. For some $n\in\mathbb{N}^{*}$ and
$m\in\mathbb{N}^{*}$ and $x\in\mathbb{R}^{n}$, $\|x\|$ denotes the Euclidian
norm of $x$. For $x\in\mathbb{R}^{n}$ and $R>0$, $\widebar{B}(x,R)$ denotes
the closed ball of radius $R$ centered at $x$. For $L\in\mathbb{N}^{*}$ and a
finite sequence of vectors $X\in{(\mathbb{R}^{n})}^{L}$,
$\|X\|=\max(\|X_{1}\|,\dots,\|X_{L}\|)$. For a bounded infinite sequence of
vectors $X\in(\mathbb{R}^{n})^{\mathbb{N}}$, $\|X\|_{\infty}$ denotes its
$\ell_{\infty}$-norm. For a linear operator $A$ from $\mathbb{R}^{n}$ to
$\mathbb{R}^{m}$, $\|A\|$ denotes the operator norm induced by the Euclidian
norm. For $i\in\mathbb{N}^{*}$, a function
$F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is called a $\mathcal{C}^{i}$
function if it is $i^{th}$-times continuously differentiable. Its
$i^{th}$-differential is denoted by $\nabla^{i}F$ with the simplification
$\nabla^{1}F=\nabla F$. In the sequel, $\succeq$ denotes the partial order on
positive semi-definite matrices and $I$ denotes the identity matrix. A
function $\alpha:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is called a
$\mathcal{K}$-function if it is continuous, increasing and such that
$\alpha(0)=0$. It is called a $\mathcal{K}_{\infty}$-function if it is also
unbounded. A function
$\beta:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is called
a $\mathcal{KL}$-function if for any $t\geq 0$, $\beta(\cdot,t)$ is a
$\mathcal{K}$-function and if for any $r\geq 0$, $\beta(r,\cdot)$ is
decreasing and converges to $0$ at infinity.

##  III PROBLEM SETUP

Consider the following nonlinear discrete-time partially observed dynamical
system, defined for any $t\geq 0$:

$\displaystyle x_{t+1}=f(x_{t},u_{t},\bar{p}),$ $\displaystyle
y_{t}^{0}=\begin{bmatrix}x_{t}^{0}\\\ y_{t}\end{bmatrix},$ (1)

where $x_{0}\in\mathbb{R}^{n_{x}}$ is given, $x_{t}^{0}=x_{t}+v^{0}_{t}$ and
$y_{t}=h(x_{t},\bar{p})+v_{t}$. Moreover, $x_{t}$ is a state variable valued
in $\mathbb{R}^{n_{x}}$, $\bar{p}$ is an unknown parameter valued in a given
set $\mathcal{P}\subset\mathbb{R}^{n_{p}}$ and $u_{t}$ is a control variable
valued in a given set $\mathcal{U}\subset\mathbb{R}^{n_{u}}$. Note that
$x_{t}^{0}$ is a measurement of the state, $y_{t}$ is an observation valued in
$\mathbb{R}^{n_{y}}$ involving also the parameter $\bar{p}$ and $v_{t}$ and
$v_{t}^{0}$ are unmodelled bounded disturbances on the observations
respectively valued in $\mathbb{R}^{n_{y}}$ and $\mathbb{R}^{n_{x}}$. The
dynamics of the system
$f:\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{u}}\times\mathbb{R}^{n_{p}}\rightarrow\mathbb{R}^{n_{x}}$
is such that $f(0,0,\bar{p})=0$ and
$h:\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{p}}\rightarrow\mathbb{R}^{n_{y}}$ is
the observation function.

###### Assumption III.1

Functions $f$ and $h$ are $\mathcal{C}^{3}$ functions, $\mathcal{U}$ is a
compact set and $\mathcal{P}$ is a closed convex set.

###### Assumption III.2

There exists $\nu>0$, such that for any $t\geq 0$, $\|v_{t}\|\leq\nu$ and ,
$\|v_{t}^{0}\|\leq\nu$.

In the rest of the paper, we assume that Assumption III.2 holds and $\nu$ is
given.

##  IV MOVING HORIZON PARAMETER ESTIMATION

###  IV-A Setup

In this section, we present a perturbed and unperturbed Moving Horizon
Parameter Estimation (MHPE) and show in Proposition IV.3 the existence and
uniqueness of the solutions of these problems as well as a bound between their
optimiser that holds under an observability condition. Fix
$L\in\mathbb{N}^{*}$. In the following, for any $t\geq 0$, any disturbance
sequence $v_{t,L}=(v_{t-L+1},\dots,v_{t})\in(\mathbb{R}^{n_{y}})^{L}$, any
control sequence
$u_{t,L}=(u_{t-L+1},\dots,u_{t-1})\in(\mathbb{R}^{n_{u}})^{L-1}$, any starting
state $x_{t-T+1}\in\mathbb{R}^{n_{x}}$, and any parameter
$p\in\mathbb{R}^{n_{p}}$, we define the cumulative output error, $C$ as
follows:

$\displaystyle
C(p,x_{t-L+1},u_{t,L},v_{t,L})=\sum_{k=t-L+1}^{t}\|y_{k}-h(x_{k},p)\|^{2}+\theta(p),$

where $\theta$ is a penalty function and the state sequence
$x_{t,L}=(x_{t-T+1},\dots,x_{t})$ follows the dynamics (1) with input
$u_{t,L}$ and initial condition $x_{t-L+1}$. We also define the noise-free
output error, $\widebar{C}$ as follows:

$\displaystyle\widebar{C}(p,x_{t-L+1},u_{t,L})$
$\displaystyle=C(p,x_{t-L+1},u_{t,L},0),$
$\displaystyle=\sum_{k=t-L+1}^{t}\|\bar{y}_{k}-h(x_{k},p)\|^{2}+\theta(p),$

where $\bar{y}_{k}=h(x_{k},\bar{p})$.

Thus, the MHPE problem can be defined as follows:

$\begin{array}[]{rrclcc}C^{*}(x_{t-L+1},u_{t,L},v_{t,L})=\displaystyle\underset{p\in\mathbb{R}^{n_{p}}}{\text{min}}&\lx@intercol
C(p,x_{t-L+1},u_{t,L},v_{t,L})\hfil\lx@intercol\\\
\textrm{s.t.}&x_{k+1}=f(x_{k},u_{k},p)\end{array}$ (2)

as well as the noise-free MHPE:

$\begin{array}[]{rrclcc}\widebar{C}^{*}(x_{t-L+1},u_{t,L})=\displaystyle\underset{p\in\mathbb{R}^{n_{p}}}{\text{min}}&\lx@intercol\widebar{C}(p,x_{t-L+1},u_{t,L})\hfil\lx@intercol\\\
\textrm{s.t.}&x_{k+1}=f(x_{k},u_{k},p)\end{array}$ (3)

###### Assumption IV.1

Function $\theta$ is a non-negative smooth convex penalty function such that
$\theta(p)=0$ for any $p\in\mathcal{P}$.

###### Lemma IV.1

Under Assumption III.1, for any $v_{t,L}\in(\mathbb{R}^{n_{y}})^{L}$,
$u_{t,L}\in(\mathbb{R}^{n_{u}})^{L}$, and $x_{t-T+1}\in\mathbb{R}^{n_{x}}$,

$\displaystyle\widebar{C}^{*}(x_{t-L+1},u_{t,L})$
$\displaystyle=\widebar{C}(\bar{p},x_{t-L+1},u_{t,L})=0,$
$\displaystyle\nabla_{p}\widebar{C}(\bar{p},x_{t-L+1},u_{t,L})$
$\displaystyle=0,$

and $C$ and $\widebar{C}$ are $\mathcal{C}^{3}$ functions. In particular,
$\bar{p}$ is a global minimiser of Problem (3).

###### Proof:

The first item follows from the definition and the non-negativity of $\bar{C}$
and the second one from the regularity assumptions. ∎

###### Proposition IV.2

Under Assumptions III.1 and IV.1, for any
$v_{t,L}\in(\mathbb{R}^{n_{y}})^{L}$, $u_{t,L}\in(\mathbb{R}^{n_{u}})^{L}$,
$x_{t-T+1}\in\mathbb{R}^{n_{x}}$, and $p\in\mathbb{R}^{n_{p}}$

$\displaystyle\nabla_{p}^{2}C(p,x_{t-L+1},u_{t,L},v_{t,L})$
$\displaystyle=\mathcal{O}(p,x_{t-L+1},u_{t,L})+\nabla^{2}\theta(p)$
$\displaystyle+\mathcal{R}(p,x_{t-L+1},u_{t,L},v_{t,L}),$
$\displaystyle\nabla_{p}^{2}\bar{C}(\bar{p},x_{t-L+1},u_{t,L})$
$\displaystyle=\mathcal{O}(\bar{p},x_{t-L+1},u_{t,L})+\nabla^{2}\theta(\bar{p}),$

where $\mathcal{O}=NN^{T}$ is called the _Observability Grammian_ with $N$
depending only on first order terms with respect to p and where
$\mathcal{R}(p,x_{t-L+1},u_{t,L},v_{t,L})$ is a second order term with respect
to p.

###### Proof:

This can be derived from classical online Moving Horizon Estimation results.
For example see [6] ∎

We now show the existence and uniqueness of solution of Problem (2). We first
need to introduce an assumption on the Observability Grammian at $\bar{p}$.

######  Definition IV.1 (L-step observability after feedback)

For $M>0$, $\bar{\delta}>0$ and $t\geq L-1$, system (1) is said to be _L-step
observable after feedback_ on $\widebar{B}(0,M)$ at time $t$ with level
$\bar{\delta}$, iff $x_{t-L+1}\in\widebar{B}(0,M)$ implies that there exists a
control sequence $u_{t,L}\in{\mathcal{U}}^{L}$ that satisfies:

$\displaystyle\mathcal{O}(\bar{p},x_{t-L+1},u_{t,L})\succeq\bar{\delta}I.$ (4)

######  Proposition IV.3 (Local existence and uniqueness)

For $M>0$ and $\bar{\delta}>0$, assume that system (1) is $L$-step observable
after feedback on $\widebar{B}(0,M)$ at time $t$ with level $\bar{\delta}$.
Under Assumptions III.1, III.2 IV.1 there exist ${K}(M,\nu)\geq 0$ that is non
decreasing with respect to $\nu$ such that if:

$\displaystyle K(M,\nu)\nu<\bar{\delta}$ (5)

then for any $x_{t-L+1}\in\widebar{B}(0,M)$ and $u_{t,L}\in{\mathcal{U}}^{L}$
satisfying Equation (4) and for any $\|v_{t,L}\|\leq\nu$,
$C(\cdot,x_{t-L+1},u_{t,L},v_{t,L})$ admits a unique minimiser on
$\widebar{B}(\bar{p},\nu)$ denoted by $p^{*}_{t,\nu}$. Moreover,
$p^{*}_{t,\nu}$ is a strict local minimiser of Problem (2).

###### Proof:

The result follows from (4) and the local Lipschitz property of $\mathcal{O}$,
$\mathcal{R}$ and $\theta$ around $(x_{t-L+1},\bar{p},v_{t,L})$

∎

The existence and uniqueness result is standard, see [9]. The major point of
Proposition IV.3 is that the existence and uniqueness result that holds under
(5) is time invariant. Indeed, Proposition IV.3 means that if the noise on the
measurements is sufficient small, Problem (2) has a locally unique solution
that is close to $\bar{p}$ uniformly with respect to $x_{t-L+1}$, $u_{t,L}$
and $v_{t,L}$. In the following, $p^{*}_{t,\nu}$ will denoted by $p^{*}_{t}$
when it exists.

###  IV-B Fast estimation algorithm

Classical MHE algorithms aim to compute $p^{*}_{t}$ precisely which may
require to perform numerous iterations of some optimisation routine,
$\psi_{t}:\mathbb{R}^{n_{p}}\rightarrow\mathbb{R}^{n_{p}}$. As a consequence,
cheap techniques where one computes an estimate of $p^{*}_{t}$ denoted by
$\hat{p}_{t}$ by performing a few iterations of the optimisation routine have
been introduced. See [6] for a review on fast MHE. Therefore, we make the
following assumption:

######  Assumption IV.2 (Optimisation algorithm)

There exist $\bar{r}>0$, $\gamma>0$ such that for any $t\geq L-1$, if
$p^{*}_{t}$ exists then for any $p\in\widebar{B}(\bar{p},r)$:

$\displaystyle\|\psi_{t}(p)-p^{*}_{t}\|\leq\gamma\|p-p_{t}^{*}\|$ (6)

Under Assumption IV.2, for any $t\geq L-1$ and some
$\hat{p}_{L-1}\in\widebar{B}(\bar{p},\bar{r})$, we define the online estimate
of $\bar{p}$ as follows:

$\displaystyle\hat{p}_{t+1}=\psi_{t}(\hat{p}_{t}).$ (7)

Note that $\hat{p}_{t}$ can be defined without requiring $p^{*}_{t}$ to exist.
The goal of the following is to design a controller that satisfies Equation
(4) along the trajectories of (1) while also stabilising it at $0$.

##  V OBSERVABILITY SEEKING MODEL PREDICTIVE CONTROL

The controller presented in this section is composed of two parts: a
controller that is input-to-state stabilising with respect to perturbation on
the control and a non-destabilising observability-seeking controller. We first
recall the notion of Regional ISS stability for nonlinear discrete-time
systems and then define our ISS stabilising controller.

###  V-A Regional ISS-stability for nonlinear discrete time systems

The definitions from this section are taken from [15]. For
$(n,m)\in(\mathbb{N}^{*})^{2}$, consider a system of the following form for
$t\geq 0$:

$\displaystyle\chi_{t+1}=g_{t}(\chi_{t},w_{t}),$ (8)

where $\chi_{t}\in\mathbb{R}^{n}$ is the state and
$w_{t}\in\mathcal{W}\subset\mathbb{R}^{m}$ is a disturbance such that
$0\in\mathcal{W}$ and
$g_{t}:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ satisfies
$g(0,0)=0$.

######  Definition V.1 (Robustly positively invariant (RPI) set)

Let $\Omega\subset\mathbb{R}^{n}$. The set $\Omega$ is called a _(uniformly)
Robustly Positively Invariant_ set for system (8) iff for any $t\geq 0$,
$\chi\in\Omega$ and $w\in\mathcal{W}$, $g_{t}(\chi,w)\in\Omega$.

######  Definition V.2 (Regional Input-to-State Stability)

Consider a RPI set $\Omega$ for System (8) such that
$0\in\textrm{int}(\Omega)$. System (8) is said to be _Input-to-State Stable
(ISS) in $\Omega$_ if there exist a $\mathcal{K}\mathcal{L}$-function
$\beta:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ and a
$\mathcal{K}$-function $\gamma:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such
that for any bounded sequence $w\in\mathcal{W}^{\mathbb{N}}$ and
$\chi_{0}\in\Omega$ and any $t\geq 0$:

$\displaystyle\|\chi_{t}\|\leq\beta(\|\chi_{0}\|,t)+\gamma(\|w\|_{\infty})$
(9)

If (9) holds for $\Omega=\mathbb{R}^{n}$, then System (8) is said to be
_globally ISS_.

######  Definition V.3 (Regional ISS-Lyapunov function)

Consider a RPI set $\Omega$ for System (8) such that
$0\in\textrm{int}(\Omega)$. A function
$V:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ is called a _ISS-Lyapunov function
in $\Omega$ _ if there exist two $\mathcal{K}_{\infty}$ functions
$\underline{\alpha}$ and $\bar{\alpha}$ such that for any $\chi\in\Omega$:

$\displaystyle\underline{\alpha}(\|\chi\|)\leq
V(\chi)\leq\bar{\alpha}(\|\chi\|),$

and there exist a $\mathcal{K}_{\infty}$-function $\alpha$ and a
$\mathcal{K}$-function $\sigma$ such that for any $t\geq 0$,
$\chi_{0}\in\Omega$, one has along the trajectories of system (8):

$\displaystyle
V(\chi_{t+1})-V(\chi_{t})\leq-\alpha(\|\chi_{t}\|)+\sigma(\|w_{t}\|).$

If these items hold for $\Omega=\mathbb{R}^{n}$, $V$ is called a _global ISS-
Lyapunov function_.

###### Proposition V.1

For any RPI set $\Omega$, if system (8) admits a ISS-Lyapunov function in
$\Omega$ then it is ISS in $\Omega$.

###### Proof:

See [15] ∎

In the sequel, we assume that there exists a feedback controller that makes
system (1) globally ISS with respect to disturbances in the input with the
nominal parameter $\bar{p}$.

###### Assumption V.1

There exist a globally Lipschitz nominal state-feedback
$\kappa:\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{p}}\rightarrow\mathcal{U}$ with
Lipschitz constant $L_{\kappa}$ such that $\kappa(0,\bar{p})=0$ and a function
$V:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{+}$ such that the system defined
for any $t\geq 0$, $x_{0}\in\mathbb{R}^{n_{x}}$ and
$d_{t}\in\mathbb{R}^{n_{u}}$ by:

$\displaystyle x_{t+1}=f(x_{t},\kappa(x_{t},\bar{p})+d_{t},\bar{p}),$ (10)

admits $V$ as a global ISS-Lyapunov function.

In the rest of the paper, we assume that Assumption V.1 holds for system (1).
We fix $\kappa$ and $V$ as in Assumption V.1 and the associated
$\underline{\alpha}$, $\bar{\alpha}$, $\alpha$ and $\sigma$ from Definition
V.3.

###  V-B Model Predictive Control with maximum level of observability

The purpose of this section is to design an MPC controller that corrects the
feedback controller $\kappa$ and ensures that Assumption IV.1 is satisfied one
step forward in the future while also keeping the system stable. We assume
that the control horizon is one even if it means to consider augmented
controls and observations. Thus, the predicted Observability Grammian
$\mathcal{O}$ will be considered on $[t-L+2,t+1]$. Notice that $\mathcal{O}$
as defined in Proposition IV.2, can also be seen as a function of some state
sequence $x_{t,L}$, some input sequence $u_{t,L}$ and some parameter $p$ even
if $x_{t,L}$ does not satisfy (1). Specifically, in this section, we consider
$\mathcal{O}$ at the measured states $x^{0}_{t,L}$. With a slight abuse of
notation, $\mathcal{O}$ can be decomposed by definition as follows, for any
$p\in\mathbb{R}^{n_{p}}$:

$\displaystyle\mathcal{O}(p,x^{0}_{t+1,L},u_{t+1,L})$
$\displaystyle=\Gamma(p,x^{0}_{t,L-1},u_{t,L-1})$ (11)
$\displaystyle+S(p,x^{0}_{t+1,L},u_{t+1,L}),$

where $\Gamma(p,x^{0}_{t,L-1})\succeq 0$ depends only on the parameter and the
present and past measured state trajectory and
$S(p,x^{0}_{t+1,L},u_{t+1,L})\succeq 0$ depends on the whole trajectory of
measured states and the parameter. By omitting the dependency on the past
trajectory and injecting (1), we define
$S_{f}(p,x_{t}^{0},u_{t})=S(p,f(x_{t}^{0},u_{t},p),u_{t})$. The MPC problem
can then be formulated as follows for any $t\geq L-1$ and some $0<\mu<1$,
${\delta}^{\prime}>0$:

—s—[2] δ,u_t^obs -δ+c(u_t) &κ(x_t^0,^p_t)+u_t^obs&∈U &
∥u_t^obs∥≤σ^-1(μ2α(12∥x_t^0 ∥)) & ^Γ_t+
S_f(^p_t,x_t^0,κ(x_t^0,^p_t)&+u_t^obs)⪰δI &δ≥δ’ where
$\widehat{\Gamma}_{t}=\Gamma(\hat{p}_{t},x^{0}_{t,L-1},u_{t,L-1})$ is given
and $u_{t,L-1}$ denotes the sequence of previous controls. In the following,
we fix a initial control sequence $\hat{u}_{L-1,L}$. Note that Constraint
(V-B) ensures control admissibility, Constraint (V-B) ensures that
$u_{t}^{\text{obs}}$ does not destabilise system (14), Constraint (V-B)
ensures that $\delta$ is lower bound on the smallest eigenvalue on the
Observability Grammian, Constraint (V-B) ensures that $\delta$ is bounded from
below by $\bar{\delta}$ uniformly in $t$. The cost (V-B) is a combination on
the lower bound on the Observability Grammian and a cost on
$u_{t}^{\text{obs}}$ denoted by $c$. In the sequel, we make a feasibility
assumption on Problem (V-B).

###### Assumption V.2

For any $M>0$, there exists ${\delta}^{\prime}(M)>0$ and $0<\mu(M)<1$ such
that for any $t\geq L-1$ if $\|x_{t-L+2}\|\leq M$ then Problem (V-B) is
feasible.

In the rest of the paper, we suppose that Assumption V.2 holds and we denote
by $\hat{u}_{t}^{\text{obs}}(x_{t,L}^{0},\hat{p}_{t},M)$ a feasible point of
Problem (V-B) for any $M>0$. Finally, for $M>0$, the total control applied to
system, denoted by $\hat{u}_{t}$ for any $t\geq L-1$, can be written as
follows:

$\displaystyle\hat{u}_{t}(x_{t,L}^{0},\hat{p}_{t},M)=\kappa({x}_{t}^{0},\hat{p}_{t})+\hat{u}_{t}^{\text{obs}}(x_{t,L}^{0},\hat{p}_{t},M).$
(12)

###### Remark V.1

Note that when it is defined the sequence of input $\hat{u}_{t,L}$ satisfies
by construction:

$\displaystyle\mathcal{O}^{0}=\Gamma(\hat{p}_{t},x^{0}_{t,L-1},\hat{u}_{t,L-1})+S_{f}(\hat{p}_{t},x^{0}_{t},\hat{u}_{t})\succeq\delta^{\prime}I$
(13)

Problem (V-B) is a nonlinear Semi-Definite Program because of Constraint (V-B)
and is generally very hard to solve. However, thanks to Constraints (V-B) and
(V-B), it will be sufficient for $\hat{u}_{t}^{\text{obs}}$ to just be
feasible in order to maintain observability. Assumption V.2 can be seen as a
one step reachability assumption of the set defined by Constraints (V-B) and
(V-B) using small inputs from $\mathcal{U}$.

##  VI OUTPUT-FEEDBACK STABILITY

For $t\geq L-1$, let $e_{t}=\hat{p}_{t}-\bar{p}$ be the parameter estimation
error. Roughly speaking, we show in this section that under the control law
(12), both $x_{t}$ and $e_{t}$ satisfy some ISS property for small initial
estimation error and small measurement noise. We first introduce the augmented
system state/error. For any $t\geq L-1$, provided that for any $L-1\leq k\leq
t$, $\|x_{t-L+1}\|\leq M$ for some $M>0$ and that $\hat{u}_{k}$, $p^{*}_{k+1}$
exist we consider an augmented state $\chi_{t}=(x_{t},e_{t})$ and an augmented
dynamics
$g_{t}:\mathbb{R}^{n_{x}+n_{p}}\times\mathbb{R}^{n_{x}+n_{p}}\rightarrow\mathbb{R}^{n_{x}+n_{p}}$
such that for any $\chi_{L-1}=(x_{L-1},e_{L-1})\in\mathbb{R}^{n_{x}+n_{p}}$,
and $t\geq L-1$, the state/error system can be written as follows:

$\displaystyle\chi_{t+1}=g_{t}(\chi_{t},w_{t}),$ (14)

where
$g_{t}(\chi_{t},w_{t})=\begin{bmatrix}f(x_{t},\hat{u}_{t}(M,x_{t}^{0}),\bar{p})\\\
\psi_{t}(\bar{p}+e_{t})\end{bmatrix}$ and
$w_{t}=(p^{*}_{t+1}-\bar{p},v_{t}^{0})$. Note that the expression of
$g_{t}(\chi_{t},w_{t})$ does not depend explicitly of $w_{t}$. However,
$w_{t}$ is a disturbance term that will be used to represent the effect of the
measurement noise in the right-hand side of (14) in the proof. We now define
the candidate ISS Lyapunov function $W$ as follows, for any
$\chi=(x,e)\in\mathbb{R}^{n_{x}+n_{p}}$ and some $\lambda>0$:

$\displaystyle W(\chi)=V(x)+\lambda\sigma(\|e\|)$ (15)

We first introduce a technical assumption on $\sigma$.

###### Assumption VI.1

Function $\sigma$ is $\mathcal{K}_{\infty}$ and $s$-homogeneous for some $s>0$
meaning that for any $\lambda_{1}>0$ and $r\geq 0$,
$\sigma(\lambda_{1}r)=\lambda_{1}^{s}\sigma(r)$.

Note that under Assumption VI.1, for any $\chi\in\mathbb{R}^{n_{x}+n_{p}}$:

$\displaystyle\underline{\alpha}_{1}(\|\chi\|)\leq W(\chi)$
$\displaystyle\leq\bar{\alpha}_{1}(\|\chi\|),$ (16)

where for any $r\geq 0$,
$\underline{\alpha}_{1}(r)=\min(\underline{\alpha},\lambda\sigma)(\frac{1}{2}r)$
and $\bar{\alpha}_{1}(r)=\max(\bar{\alpha},\lambda\sigma)(r)$. Note that
$\underline{\alpha}_{1}$ and $\bar{\alpha}_{1}$ are
$\mathcal{K}_{\infty}$-functions as $\underline{\alpha}$, $\bar{\alpha}$ and
$\sigma$ are $\mathcal{K}_{\infty}$-functions.

###### Assumption VI.2

There exists $M_{0}$ and $\bar{\delta}_{0}$ such that $\|x_{L-1,L}\|\leq
M_{0}$ and System (1) is _L-step observable after feedback_ on
$\widebar{B}(0,M_{0})$ at time $L-1$ with level $\bar{\delta}_{0}$.

For any $R>0$, we set
$\Omega_{R}=\\{\chi\in\mathbb{R}^{n_{x}+n_{p}}|W(\chi)\leq R\\}$ We can now
state the main result of the section.

###### Theorem VI.1

Under Assumptions III.1, III.2, IV.1, IV.2, V.2 and VI.2, there exist
$M_{0}>0$, $\bar{\delta}_{0}>0$ $\widebar{R}>0$, $\bar{\delta}>0$, $r_{0}>0$,
$s>0$, ${r}_{max}>0$, $\lambda>0$, $\nu_{\max}>0$, $0<\gamma<\frac{1}{2}$, a
$\mathcal{K}_{\infty}$-function $\alpha_{1}$ and a $\mathcal{K}$-function
$\sigma_{1}$ such that if the following are satisfied:

$\displaystyle\nu$ $\displaystyle\leq\nu_{\max},$ $\displaystyle r_{0}$
$\displaystyle\leq r_{\max},$ $\displaystyle
K(M_{0},\nu)\nu<\min(\bar{\delta}_{0},\bar{\delta}),$ (17)

with $K(M_{0},\nu)$ from Proposition IV.3 and if $\|x_{L-1,L}\|\leq M_{0}$,
and
$\chi_{L-1}=(x_{L-1},e_{L-1})\in\Omega_{\widebar{R}}\cap(\mathbb{R}^{n_{x}}\times\widebar{B}(0,r_{0}))$
then for any $t\geq L-1$, $\hat{u}_{t}$ and $p_{t}^{*}$ are well defined and
the following hold:

$\displaystyle\|x_{t,L}\|\leq M_{0},$
$\displaystyle\|p_{t}^{*}-\bar{p}\|\leq\nu,$
$\displaystyle\chi_{t}\in\Omega_{\widebar{R}}$ $\displaystyle\|e_{t}\|\leq
r_{0},$ (18) $\displaystyle\|e_{t+1}\|$
$\displaystyle\leq\gamma\|e_{t}\|+(1+\gamma)\|p_{t+1}^{*}-\bar{p}\|$ (19)
$\displaystyle W(\chi_{t+1})$ $\displaystyle\leq
W(\chi_{t})-\alpha_{1}(\|\chi_{t}\|)+\sigma_{1}(\|w_{t}\|).$ (20)

###### Proof:

For the sake of conciseness, we only provide a sketch of proof. It is made by
strong induction. The initialisation is ensured by Assumption VI.2. Then, by
assuming that (18)-(20) hold for any $L\leq k\leq t$, one can prove that
$\hat{u}_{t+1}$ and $p_{t+1}^{*}$ are well defined by invoking Proposition
(IV.3), Assumption V.2 and Constraint (V-B). One further gets (19) from (6)
and (7) by the triangle inequality. Finally, from Assumption V.1 and
Constraint (V-B) and several manipulation of $\mathcal{K}$-functions, one gets
(20) which leads to (18) at time $t+1$ thanks to the strong induction
hypothesis and Remark 3.7 of [12].

∎

One can further notice from the non-decreasing property of $K$ that the set of
conditions (17) is feasible uniformly with respect to $t$ for $\nu$ and
$r_{0}$ sufficiently small.

###### Corollary VI.2

Under the assumptions and the settings of Theorem VI.1, there exist
$\widebar{R}>0$ and $r_{0}>0$ such that system (14) is ISS on
$\Omega_{\widebar{R}}\cap(\mathbb{R}^{n_{x}}\times\widebar{B}(0,r_{0}))$.
Moreover,

$\displaystyle\limsup_{t\rightarrow+\infty}{\|e_{t}\|}\leq\frac{1+\gamma}{1-\gamma}\nu.$
(21)

###### Proof:

From Equations (18) and (19), it is clear that
$\Omega_{\widebar{R}}\cap(\mathbb{R}^{n_{x}}\times\widebar{B}(0,r_{0}))$ is a
RPI set. The ISS property follows from (16), (20) and Proposition V.1.
Equation (21) is a direct consequence of Equation (19). ∎

Theorem VI.1 states that both the estimation error and the state of system (1)
are ultimately bounded by the magnitude of the measurement noise $\nu$.
Corollary VI.2 expresses this fact in terms of a Regional ISS property of
system (14). It also gives a more accurate ultimate bound on $e_{t}$

##  VII APPLICATION: BEARING-ONLY ACTIVE SLAM

In this section we consider a 2D robot represented by position variables
$x=(x_{1},x_{2})$ and following a first order dynamics defined for any
$t\in\mathbb{N}$ by:

$\displaystyle x_{t+1}=x_{t}+\Delta u_{t},$ (22)

where $u_{t}$ is a velocity input and $\Delta>0$. The unknown parameter in
this context is the 2D position of a landmark
$\bar{p}=(\bar{p}_{1},\bar{p}_{2})$. The state $x$ and the parameter $\bar{p}$
are supposed to be observed through noisy bearing angle measurements, denoted
by $y$ and which can be written as follows for any $t\geq 0$:

$\displaystyle
y_{t}=h(x_{t},\bar{p})+v_{t}=\frac{\bar{p}-x_{t}}{\|\bar{p}-x_{t}\|}+v_{t},$
(23)

with $\|v_{t}\|\leq\nu$. In this section, we focus on the sensor-centric view
of Simultaneous Localisation and Mapping (SLAM) in which the state of the
system $x_{t}$ is supposed to be already estimated up to some level of
precision. Therefore, one can assume, as in section III, that a noisy
measurement of $x_{t}$, denoted by $x^{0}_{t}$, is available so that
$x^{0}_{t}=x_{t}+v^{0}_{t}$ with $\|v^{0}_{t}\|\leq\nu$. The estimation
algorithm presented in Problem (2) and Equation (7) can be thought of as an
online version of a Graph SLAM, see [20]. With this in mind, the controller
(12) can be seen as an Active SLAM controller aiming to ensure the good
quality of Algorithm (7) through the resolution of Problem (V-B) and
especially Constraint (V-B), see [4] for more details. The controller $\kappa$
has be chosen as a smoothly saturated linear input and the Observability
Grammian $\mathcal{O}$ can be written as follows for any $t\geq L-1$:

$\mathcal{O}(\bar{p},x_{t+1,L},u_{t+1,L})=\sum_{k=t-L+1}^{t}H(x_{k},\bar{p})H^{T}(x_{k},\bar{p}),$

where:

$\displaystyle
H(x,\bar{p})=\frac{1}{\|x-\bar{p}\|^{2}}\begin{bmatrix}x_{2}-\bar{p}_{2}\\\
-(x_{1}-\bar{p}_{1})\end{bmatrix}.$

The simulation has been carried out in MATLAB using the PENLAB toolbox for
nonlinear Programming and Semi-Definite Programming to solve Problem (2) and
(V-B), see [10]. Note that $\psi_{t}$ in this context represents $10$
iterations of the optimisation routine. Figure 2 represents a horizontal
trajectory where only the controller $\kappa$ has been applied and a
trajectory resulting from (12). Figure 1 represents the corresponding
estimation errors in the landmark position. Both the trajectories have been
simulated with the following choice of parameters: $\bar{p}=(5,8)$,
$\Delta=0.1$, $L=10$, $\nu=0.03$, $\delta^{\prime}=1$, $\mu=0.5$, $c=0$,
$\hat{p}_{L-1}=(3,10)$, $x_{0}=(5,10)$. One can see that in this case the
horizontal trajectory does not allow the proper resolution of the MHPE problem
and leads to a diverging estimate. On the contrary, the observability-seeking
trajectory exhibits piece-wise circular behaviours which are known to ensure
observability [11], [19] and leads to good estimation performance.

Figure 1: Plot of the estimation error in the landmarks position for a
horizontal trajectory and a observability-seeking trajectory  Figure 2:
Comparison between a horizontal trajectory and an observability-seeking
trajectory obtained from the controller (12)

##  VIII CONCLUSION

In this paper, an output-feedback algorithms for adaptive control based on a
fast MHPE scheme and an MPC with a constraint on the Observability Grammian
has proposed. The closed-loop stability of the system has been proved in terms
of input-to-state stability of the augmented system composed of the original
state and the parameter estimation error. The method has been numerically
tested and validated on the nonlinear application of Active SLAM.

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# What if PSR J1910-5959A is an observable self-lensing binary?

Man Ho Chan<EMAIL_ADDRESS>Chak Man Lee  Department of Science and
Environmental Studies, The Education University of Hong Kong
Tai Po, New Territories, Hong Kong, China

###### Abstract

In a binary, when the orbital plane of the companion star is almost edge-on
along the line-of-sight direction, this would produce an observable self-
gravitational lensing effect, which would slightly increase the overall
optical intensity of the binary. However, the probability of getting one
observable self-lensing binary (SLB) is very low. There are only five observed
SLBs so far and all of them are eclipsing binaries. In this article, we
theoretically show that the neutron star-white dwarf (NS-WD) binary PSR
J1910-5959A could be an observable non-eclipsing SLB. It might be the first
binary showing both periodic optical amplification and Shapiro time delay of
radio signals, which is useful to verify our understanding about gravitational
lensing in relativistic binaries. Moreover, we show that the observed peak
amplification limit of the PSR J1910-5959A can help constrain the radius of
the WD, which is a crucial parameter to examine the mass-radius and
temperature-radius relationship for helium WD.

##  I Introduction

A massive compact object, such as a white dwarf, neutron star or black hole,
can act as a lens to bend light to produce gravitational lensing effect. In
fact, most of the lensing images observed previously originate from the light
of stars and galaxies located very far away from the gravitational lens.
Nevertheless, there exist some special compact binaries in which the orbital
planes are almost edge-on along the line of sight (i.e. the inclination angle
is $i\approx 90^{\circ}$). In these cases, the light of the companion stars
would be bent by the compact objects in the same binary to give the self-
gravitational lensing effect (Maeder, ; Rahvar, ). These special binaries are
called self-lensing binaries (SLBs) (Rahvar, ). The total optical intensity of
these SLBs would be increased by a certain percentage when the angular
distance between the companion star and the central compact object is small
(Rahvar, ; Sorabella, ).

The SLBs were discussed half a century ago for the studies of ‘collapsed
stars’ (Trimble, ). However, the self-gravitational lensing effect is a very
rare event because it requires the orbital inclination angle very close to
$90^{\circ}$. Using Kepler photometric data, Kruse and Agol (Kruse, ) reported
the first observed SLB, KOI-3278, which is a white dwarf (WD) eclipsing
binary. This binary consists of a G-type star and a WD on a 88-day orbit, in
which a 5-hour pulse of 0.1% optical amplification occurs every orbital
period. The periodic amplification pulses can be used to constrain the mass of
the WD ($\approx 0.63\,M_{\odot}$) in the binary. Kawahara et al. (Kawahara, )
and Masuda et al. (Masuda, ) then reported additional four SLBs (KIC 3835482,
KIC 12254688, KIC 12254688, KIC 845411) with amplification periods ranging
from 419 to 728 days. They modeled the radial velocities of these four SLBs,
and combined the binary mass function with the estimated primary masses to
derive the WD masses, all having $\sim 0.5-0.6\,M_{\odot}$ except KIC 845411
with an extremely small WD mass of 0.2 $M_{\odot}$ moving around a sun-like
star in an orbit more than ten times wider than theoretically expected
(Masuda2020, ). Intuitively, during the eclipsing moment, some of the light
from the companion star is blocked by the compact star so that the luminosity
is dropped, like the cases of transiting planets. However, the gravitational
lensing effect due to the compact star would bend the surrounding light to
increase the total luminosity along the line of sight, even outweighing and
compensating the luminosity drop during the eclipsing moment. Theoretically,
the luminosity amplification would be the largest during the eclipsing moment.
For the five SLBs observed, the percentage increase in optical luminosity
ranges from 0.05% to 0.15% during the eclipsing moment (Kawahara, ; Masuda, ;
Sorabella3, ).

So far all of the observed SLBs are WD binaries. Although a recent study has
reported a possible SLB involving a supermassive black hole, more data are
required to confirm this (Sorabella2, ). In fact, these rare SLBs are useful
for us to examine the understanding of binaries. For example, Sorabella et al.
(Sorabella, ) suggests to use the self-lensing effect in X-ray binaries to
measure the mass of the compact objects and the relevant parameters of the
X-ray binaries. Therefore, studying SLBs can help reveal the astrophysical
unknowns in binaries and verify our understanding of gravitational lensing.

In this article, we show theoretically that the neutron star-white dwarf (NS-
WD) binary PSR J1910-5959A could be another observable SLB. One special
feature is that this potential SLB might be the first non-eclipsing SLB
involving two compact objects, which is different from the observed five SLBs.
This can help verify our understanding about gravitational lensing in a
relativistic binary. On the other hand, the periodic amplification due to
self-gravitational lensing depends sensitively on the binary parameters.
Therefore, observing the periodic amplification or obtaining its limit can put
constraints on the binary parameters, which can provide useful information on
the evolution of NS-WD binaries and discriminate between alternative models
for helium white dwarf structure and cooling (Corongiu, ). Although the major
scope of this article is not suggesting any concrete observational plan for
verification, we will also briefly discuss some possible ways of observations
for future studies.

##  II Theory of SLBs

The theory of SLBs has been developed for several decades (Maeder, ; Rahvar,
). When the companion star is passing behind the compact object in the binary,
the gravitational lensing effect becomes significant. Lensing images of the
companion star would appear so that the total optical luminosity of the binary
is increased (luminosity of the binary + luminosity of the images) (see the
figures and discussion in Sorabella et al. (Sorabella, )). When the companion
star and the compact object are in complete alignment along the line of sight,
an Einstein ring would be formed around the compact object so that the optical
luminosity would be the largest. The radius of the Einstein ring formed is
given by (Rahvar, )

$R_{\rm
E}=\sqrt{\frac{4GM_{p}a}{c^{2}(1+\frac{a}{D})}}\approx\sqrt{\frac{4GM_{p}a}{c^{2}}},$
(1)

where $M_{p}$ is the mass of the compact object, $D$ is the distance to the
binary, and $a$ is the orbital semi-major axis of the companion star:

$a=\left[\frac{G(M_{p}+m_{\rm star})}{\omega^{2}}\right]^{1/3}.$ (2)

Here, $m_{\rm star}$ is the mass of the companion star and $\omega$ is the
angular speed which can be determined by the orbital period $T$ of the binary.

The amplification of luminosity depends on the impact parameter $P_{*}$ and
the angular separation between the companion star and the central compact
object $u$ (in units of the angular Einstein radius) (Rahvar, ):

$\displaystyle A[u,P_{*}]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left[\frac{u+P_{*}}{P_{*}^{2}}\sqrt{4+(u-P_{*})^{2}}E(k)\right.$
(3)
$\displaystyle\left.-\frac{u-P_{*}}{P_{*}^{2}}\frac{8+u^{2}-P_{*}^{2}}{\sqrt{4+(u-P_{*})^{2}}}K(k)\right.$
$\displaystyle\left.+\frac{4(u-P_{*})^{2}}{P_{*}^{2}(u+P_{*})}\frac{1+P_{*}^{2}}{\sqrt{4+(u-P_{*})^{2}}}\Pi(n;k)\right]$

where $E(k)$, $K(k)$ and $\Pi(n;k)$ are the complete elliptic integrals of the
first, second and third kind, respectively, in which $n$, $k$ are defined by

$\displaystyle n$ $\displaystyle=$
$\displaystyle\frac{4uP_{*}}{(u+P_{*})^{2}}$ (4) $\displaystyle k$
$\displaystyle=$ $\displaystyle\sqrt{\frac{4n}{4+(u-P_{*})^{2}}}.$ (5)

Here $u$ is a dimensionless parameter, which can be defined as

$u=\frac{a}{R_{\rm
E}}\sqrt{\sin^{2}\omega(t-t_{0})+\cos^{2}i\cos^{2}\omega(t-t_{0})},$ (6)

with $t_{0}$ being an arbitrary time at the moment when $u$ is the minimum and
$P_{*}$ is the impact parameter defined as

$P_{*}=\frac{R_{\rm star}}{R_{\rm E}},$ (7)

where $R_{\rm star}$ is the radius of the companion star.

For the eclipsing moment of the SLB (i.e. $u\approx 0$), the amplification
would be the maximum and Eq. (3) would reduce to (Rahvar, )

$A_{\rm max}=A[0,P_{*}]=\sqrt{1+\frac{4}{P_{*}^{2}}}.$ (8)

##  III Predicted Results

Consider a well-known NS-WD binary PSR J1910-5959A. This binary has been
studied for a long time and the orbital inclination angle is believed to be
very close to $90^{\circ}$ (Corongiu3, ; Corongiu2012, ). However, a recent
study shows that the orbital inclination angle is $i=88.90^{+0.15}_{-0.14}$
deg (Corongiu, ). By applying the Keplerian relation in Eq. (2) with the
relevant parameters in (Corongiu, ), the orbital semi-major axis of the binary
is $a\approx 3.14\times 10^{6}$ km. Since $a\cos i=60300^{+7700}_{-8200}$ km,
it is not an eclipsing binary as the largest radius of the WD constrained is
$R_{\rm star}\approx 43100$ km (i.e. $a\cos i>R_{\rm star}$). Nevertheless, we
find that the angular separation is probably small enough to have a
significant self-lensing amplification in optical luminosity. There exists a
minimum value of $u_{\rm min}$ such that the peak amplification might be
observable. Therefore, this might be the first observable non-eclipsing SLB.

The parameters of the binary such as orbital period $T$, $M_{p}$, $m_{\rm
star}$, and $i$ have been determined very precisely (Corongiu, ). The ranges
of the uncertainties for the ELL1 and ELL1H binary models considered in
(Corongiu, ) are very small (see Table 1). The orbital eccentricity is very
small so that the orbit is close to circular (Corongiu2012, ). However, there
is one uncertain parameter, $R_{\rm star}$, which is model-dependent and has
not been constrained well (Corongiu, ). Previous model has obtained $R_{\rm
star}=(0.058\pm 0.004)R_{\odot}$ (Corongiu2012, ) while a recent study got
$R_{\rm star}=(0.0359\pm 0.0004)R_{\odot}$ (Corongiu, ). In the followings, we
will use both values of $R_{\rm star}$ and involve the updated uncertainties
to predict the amplification.

In Fig. 1, we can see that the amplification sensitively depends on $R_{\rm
star}$. Since different values of $R_{\rm star}$ would also give different
values of $u_{\rm min}$, the peak amplification observed would be quite
different. In Fig. 2, by including the uncertainties of the parameters of the
benchmark model ELL1, we show the amplification profiles for two different
values of $R_{\rm star}$. The peak amplification can range from 1.0001 to
1.0015 (i.e. increased by 0.01% to 0.15%). The peak amplification of the
previous observable SLBs ranges from 0.05% to 0.15% (Kawahara, ; Masuda, ).
Moreover, one special feature is that the amplification function is close to a
Gaussian function for the non-eclipsing SLB while the previous observable SLBs
manifest a rectangular peak function due to the longer eclipsing time. That
would make the non-eclipsing SLB showing a periodic tiny sharp increase in
luminosity when $u=u_{\rm min}$. However, the duration of the sharp increase
in luminosity is about 150-200 s (see Fig. 2), which might not be easy to
identify. The period of the peak amplification is same as the orbital period
$T=0.837113489987(3)$ day (Corongiu, ).

Moreover, if we fix the value of $R_{\rm star}=0.0359R_{\odot}$, the two
models ELL1 and ELL1H give a smaller range of peak amplification (1.0001 to
1.0005, see Fig. 3). Therefore, the large range of the possible peak
amplification shown in Fig. 2 mainly originates from the uncertainty in
$R_{\rm star}$. This shows that the observed peak amplification limit can help
constrain the value of $R_{\rm star}$, which is a crucial parameter to examine
the mass-radius relationship and the temperature-radius relationship for
helium WD (Bassa, ; Corongiu, ). Note that we did not include the uncertain
limb-darkening effect in the above peak amplification calculations (Sorabella,
).

To observe this predicted very small amplification, we need to use a telescope
which has extremely good sensitivity. In fact, PSR J1910-5959A is a very dim
object. Previous optical observations using the Hubble Space Telescope (HST)
and the European Southern Observatory find that its V-band magnitude is
$V=22.13$ mag only (Bassa2, ; Ferraro, ; Bassa, ). Therefore, it is not easy
to observe this binary and even detect its luminosity change due to lensing
effect. Nevertheless, among most of the current telescopes, the James Webb
Space Telescope (JWST) may be the only one which can achieve the required
sensitivity to detect the luminosity change. Consider observing J1910-5959A by
the filter F150W of the JWST as an example. Taking an exposure time 12.5 h can
achieve the limit of the AB magnitude to 30.2 (Leung, ), which is equivalent
to the root-mean-square amplification variation $\Delta A\approx 0.00059$. For
this observational scenario, we can use 251.1 days to observe the binary 300
times. Only 2.5 mins are required for each observation to generate a single
data point. Observing 300 times can accumulate the required 12.5 h exposure
time. We can see in Fig. 2 that the assumed amplification peak $A=1.001$ (i.e.
0.1% peak amplification) can be marginally differentiable. However, such a
long observation time is a very costly observation. Also, the estimation above
did not account for the noise scaling and other effects such as limb-
darkening. The actual required observation time must be longer than the one
calculated above. Therefore, using optical observations might not be very
practical, unless we have a better quality telescope in the future or we have
secured a very long observation time using JWST. Nevertheless, such a
possibility for observing this self-lensing effect should not be undermined as
the advancement of observational techniques is extremely fast in current era.

On the other hand, another interesting property of this lensing binary is that
the WD can also affect the radio transmission of the NS due to Shapiro time
delay when the NS is behind the WD and the visual angle between them is
minimum (Corongiu2012, ). When $u$ is close to $u_{\rm min}$, the General
Relativistic effect would cause a tiny time dilation (i.e. the Shapiro time
delay) of the radio signals emitted from the NS such that we can constrain
some of the parameters using the radio data (Corongiu2012, ). Therefore, PSR
J1910-5959A would be the first SLB which can both demonstrate periodic optical
amplification and Shapiro time delay in radio signals. Moreover, since the
radius of the Einstein ring $R_{E}^{\prime}=\sqrt{4Gm_{\rm star}a/c^{2}}\sim
2000$ km is smaller than the size of the WD ($R_{\rm star}>26000$ km), no
radio pulse amplification would be resulted when the NS is behind the WD.

Figure 1: The amplification as a function of $u$ for two different values of
$R_{\rm star}$. Figure 2: The amplification as a function of time $t$. The
solid lines, dashed lines and dotted lines represent the average, maximum and
minimum amplification functions respectively for two possible values of
$R_{\rm star}$. Here, we have assumed the ELL1 binary model, including the
uncertainties of the binary parameters. The grey data points with error bars
represent the simulated amplification function using the JWST F150W filter
(assuming the peak $A=1.001$). Figure 3: The amplification as a function of
time $t$ for the ELL1 and ELL1H models. The solid line, dashed line and dotted
line represent the average, maximum and minimum amplification functions
respectively. Here, we have assumed $R_{\rm star}=0.0359R_{\odot}$ and
considered the uncertainties of the binary parameters.

##  IV Discussion

In this study, we specifically discuss the potential SLB PSR J1910-5959A and
see how probable we can observe the self-lensing effect (i.e. the amplified
luminosity). Previous five observed SLB can give amplified luminosity factor
ranging from 1.0005 to 1.0015 (Kawahara, ; Masuda, ; Sorabella3, ). We show
that the binary system PSR J1910-5959A can give a predicted amplified
luminosity factor ranging from 1.0001 to 1.0015, which might be able to
manifest an observable periodic tiny sharp increase in luminosity due to the
self-lensing effect. Although using a high-quality telescope (e.g. JWST) to
record the luminosity change is a possible way to identify the sixth SLB and
the first SLB involving NS, the required observation time might be too long so
that it would be a very costly observation. Nevertheless, our results here
indicate a possibility for observing this self-lensing effect, which might be
achievable due to the fast advancement of observational techniques.

Apart from the optical amplification, detecting the effect on radio pulse is
another way to examine this potential SLB. It is possible for us to detect the
Shapiro time delay of the NS when $u$ is close to $u_{\rm min}$. The Shapiro
time delay of the radio signals from the NS in PSR J1910-5959A has been
detected previously (Corongiu2012, ) and it is also one of the most important
self-lensing effects. Therefore, this makes PSR J1910-5959A very special as it
might give both optical amplification and Shapiro time delay in the same
binary.

Besides, observing the potential luminosity amplification and precise
measurement of the Shapiro time delay can help constrain the relevant
parameters of PSR J1910-5959A. Even if we cannot observe any peak
amplification, we can get a maximum limit of $A$ so that we can further
constrain the values of $R_{\rm star}$ and other useful parameters (e.g. $i$,
$m_{\rm star}$). These parameters can help constrain our understanding about
the helium WD relationships (e.g. mass-radius relationship) (Bassa, ), the
evolution of NS-WD binaries, and the models for helium WD structure and
cooling. Hence, observing the alleged self-lensing effect can provide another
independent way to investigate the mysteries in PSR J1910-5959A.

As illustrated above, we need a long observation time for JWST to observe the
potential peak amplification. Some future telescopes might be able to observe
the potential peak amplification or constrain the limit of $A$ for PSR
J1910-5959A. For example, the Characterising Exoplanets Satellite (CHEOPS),
launched in 2019, has similar missions with the existing Transiting Exoplanet
Survey Satellite (TESS) (Benz, ). It can concentrate on single objects, which
is more flexible than the TESS for observing PSR J1910-5959A (Southworth, ).
Future telescopes like Planetary Transits and Oscillations of stars (PLATO),
which will be launched in 2026, can provide more accurate photometry
measurement for binaries (Rauer, ). Therefore, we anticipate that future
observations of PSR J1910-5959A using high-quality optical telescopes can
provide new insights on gravitational lensing and NS-WD binaries.

Table 1: The binary parameters of PSR J1910-5959A for the ELL1 and ELL1H binary models (Corongiu, ). Parameter | ELL1 | ELL1H
---|---|---
$M_{p}$ ($M_{\odot}$) | $1.556_{-0.076}^{+0.067}$ | $1.541^{+0.080}_{-0.088}$
$m_{\rm star}$ ($M_{\odot}$) | $0.202\pm 0.006$ | $0.201\pm 0.007$
$i$ (deg) | $88.90_{-0.14}^{+0.15}$ | $88.9\pm 0.2$
$T$ (day) | 0.837113489987(3) | 0.837113489970(4)

##  V Acknowledgements

We thank the referee for useful comments and suggestions. The work described
in this paper was partially supported by a grant from the Research Grants
Council of the Hong Kong Special Administrative Region, China (Project No.
EdUHK 18300922).

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