# Robust Queue Length Estimation for Ramp Metering
in a Connected Vehicle Environment

Yu Tang, Kaan Ozbay and Li Jin This work was in part supported by US NSF Award
CMMI-1949710, USDOT Award # 69A3551747124, NSFC Project 62103260 via the
C2SMART Center, NYU Tandon School of Engineering, SJTU UM Joint Institute, and
J. Wu & J. Sun Endowment Fund.Y. Tang is with C2SMART Center, Department of
Civil & Urban engineering, Tandon School of Engineering, New York University,
11201, USA<EMAIL_ADDRESS>Ozbay is with C2SMART Center, Department of Civil
& Urban engineering, Tandon School of Engineering, New York University, 11201,
USA<EMAIL_ADDRESS>Jin is with UM Joint Institute and Department of
Automation, Shanghai Jiao Tong University, Shanghai, 200240, China.
<EMAIL_ADDRESS>
###### Abstract

Connected vehicles (CVs) can provide numerous new data via vehicle-to-vehicle
or vehicle-to-infrastructure communication. These data can in turn be used to
facilitate real-time traffic state estimation. In this paper, we focus on ramp
queue length estimation in a connected vehicle environment, which improves
control design and implementation of ramp metering algorithms. One major
challenge of the estimation problem is that the CV data only represent partial
traffic observations and could introduce new uncertainties if real-time CV
penetration rates are unknown. To address this, we build our estimation
approach on both traditional freeway sensors and new CV data. We first
formulate a ramp queue model that considers i) variations in the penetration
rate and ii) noise in measurements. Then we develop a robust filter that
minimizes the impacts of these two kinds of uncertainties on queue estimation.
More importantly, we show that the designed filter has guaranteed long-term
estimation accuracy. It allows us to quantify in a theoretical way the
relationship between estimation error and fluctuation of CV penetration rates.
We also provide a series of simulation results to verify our approach.

##  I INTRODUCTION

Ramp metering is one of the major freeway management strategies that can be
used to alleviate both recurrent and non-recurrent traffic congestion. It
typically regulates on-ramp flows to improve mainline mobility, at the same
time keeping reasonable queue lengths at ramps [1, 2, 3]. Field evaluations
have demonstrated that well-designed ramp metering can significantly improve
traffic efficiency [4, 5, 6]. In current practice, ramp metering is
implemented mainly with the aid of induction loop detectors so that ramp
controllers can adjusted metered flows, based on real-time traffic volumes,
speed and occupancy [7].

It is also reported that the current control of ramp flows can be refined,
with smoothing metering rates and reducing traffic oscillation [8, 9, 10], if
on-ramp queue length is available to controllers. But the ramp queue length is
not easily acquired from widely-used induction loop detectors; it can only be
estimated from these sensors rather than be measured directly. The estimation
methods are usually built on deliberate assumptions of vehicle characteristics
and sensor locations [9, 11, 12, 13], and their applicability could be
restricted consequently.

The estimation problem could become more tractable, thanks to the emergence of
connected vehicles (CVs). In a connected vehicle environment, vehicle-to-
vehicle or vehicle-to-infrastructure communication yields substantial new
data, such as basic safety messages (BSMs), that contain detailed information
including vehicle position, speed, acceleration and so on [14]. Though these
data are only from partial observations of freeway traffic (mainly related to
CVs), it is still possible to convert them into traffic measurements [15, 16]
and enhance freeway state estimation [17, 18, 19, 20, 21, 22].

Figure 1: Problem setting: our target is to assess ramp queue length based on
induction loop detectors and Basic Safety Messages (BSMs) [16]. We assume that
induction loop detectors are deployed at both ends of the on-ramp, which is a
minimum requirement by ramp metering [23]; we also assume that road-side units
are deployed to collect BSMs. Note that BSMs contain information of vehicle
positions, and thus it is feasible to recognize those from CVs at the on-ramp
and to extract traffic data of the corresponding CVs [15, 16].

In this paper, we aim at developing a robust queue length estimation approach
that utilizes both traditional sensors (e.g. induction loop detectors) and
emerging CV data (e.g. BSMs); see Fig. 1. It should be pointed out that queue
length at a certain link may refer to i) distance from the first stopping
vehicle to the last one [24, 25] or ii) simply link-level vehicle number. The
latter suffices for ramp metering [12, 13, 11, 9], and following this
convention, we focus on estimating the vehicle number 111We use the terms
queue length/vehicle number interchangeably in this paper. between the stop
bar and the on-ramp entrance as shown in Fig. 1. To this end, we first
formulate a ramp queue model, a dynamic system that considers the fluctuating
penetration rate of CVs and noisy traffic measurements. Particularly, the
variation in the penetration rate induces model parameter uncertainty. Then we
propose a robust filter, based on the queue model, that produces the
estimation with performance guarantees, even under the parameter uncertainty
and the measurement noise.

Previous studies have adopted Kalman filter for queue length estimation, which
mainly deals with measurement noise. Given only induction loop detectors,
Kalman filters were built on the assumptions of uniform vehicle length and
headway, internal sensors in the middle of on-ramps, and so on [12, 13, 11].
Though numerical examples have demonstrated that those filters could still
work given heterogeneous vehicles, e.g. with various length, the internal
sensors are indispensable. Such sensors, however, may be unavailable in
practice [23], impeding application of those estimation approaches. Kalman
filters were also developed in a connected vehicle environment [25, 22]. These
methods ignored the uncertainty of CV penetration rates, e.g. assuming the
rate estimated from other approaches [22], and thus failed to quantify in a
theoretical way how uncertain CV penetrate rates influence estimation
accuracy. To our best knowledge, limited work has discussed this problem.

Our major contribution is to introduce a robust filter with a performance
guarantee (Theorem 1). Here the robustness is two-fold: the filter minimizes
the impacts of both measurement noise and uncertain CV penetration rates on
queue length estimation. Moreover, Theorem 1 allows us to quantify the
relationship between estimation accuracy and the uncertain penetration rates.
We also demonstrate these via a series of simulation results. It should be
pointed out that our approach is related to robust $H_{\infty}$ filters.
Though the general theories of $H_{\infty}$ filters have been well developed
[26], it is unclear how to apply them to queue length estimation. As we will
see, the typical design of $H_{\infty}$ filters does not work in our problem;
we need to consider one of its extensions [27] and develop the filter based on
our ramp queue model.

The rest of the paper is organized as follows. Section II introduces the ramp
queue model. Section III presents our robust filter design. Section IV
provides a case study. Finally, Section V summarizes our work and discusses
future research.

##  II Modeling and formulation

We present the ramp queue model and formally state the estimation problem in
this section.

Consider queuing vehicles at the on-ramp as shown in Fig. 1. We denote by
$x_{\mathrm{all}}(t)$ (resp. $x_{\mathrm{cv}}(t)$ ) the number of total
vehicles (resp. CVs) between the stop bar and the detector A in Fig. 1.
Besides, we let $f_{\mathrm{all}}^{\mathrm{in}}(t)$ (resp.
$f_{\mathrm{all}}^{\mathrm{out}}(t)$) denote the total flow passing the
detector A (resp. detector B) in Fig. 1. Among them, we denote by
$f_{\mathrm{cv}}^{\mathrm{in}}(t)$ (resp. $f_{\mathrm{cv}}^{\mathrm{out}}(t)$)
the inflow (resp. outflow) of CVs. Clearly, it follows
$x_{\mathrm{all}}(t)\geq x_{\mathrm{cv}}(t)$,
$f_{\mathrm{all}}^{\mathrm{in}}(t)\geq f_{\mathrm{cv}}^{\mathrm{in}}(t)$ and
$f_{\mathrm{all}}^{\mathrm{out}}(t)\geq f_{\mathrm{cv}}^{\mathrm{out}}(t)$ for
$t=0,1,\cdots$. By the conservation law, we have

$\begin{bmatrix}x_{\mathrm{all}}(t+1)\\\
x_{\mathrm{cv}}(t+1)\end{bmatrix}=\begin{bmatrix}x_{\mathrm{all}}(t)\\\
x_{\mathrm{cv}}(t)\end{bmatrix}+\delta_{t}\begin{bmatrix}f^{\mathrm{in}}_{\mathrm{all}}(t)\\\
f^{\mathrm{in}}_{\mathrm{cv}}(t)\end{bmatrix}-\delta_{t}\begin{bmatrix}f^{\mathrm{out}}_{\mathrm{all}}(t)\\\
f^{\mathrm{out}}_{\mathrm{cv}}(t)\end{bmatrix},$ (1)

where $\delta_{t}$ is the cycle length of ramp metering, e.g. 30 seconds. That
is, the queue length is updated after each control cycle. It helps with
determining metering rates in the next cycle.

Though we do not explicitly model the relation between the vehicle numbers and
the flows (concrete models are available in [28]), we consider the inflows and
outflows subject to the following constraints

$\displaystyle 0\leq$ $\displaystyle
x_{\mathrm{all}}(t)+\delta_{t}(f_{\mathrm{all}}^{\mathrm{in}}(t)-f_{\mathrm{all}}^{\mathrm{out}}(t))\leq
Q,t=0,1,\cdots,$ (2a) $\displaystyle 0\leq$ $\displaystyle
x_{\mathrm{cv}}(t)+\delta_{t}(f_{\mathrm{cv}}^{\mathrm{in}}(t)-f_{\mathrm{cv}}^{\mathrm{out}}(t))\leq
Q,t=0,1,\cdots,$ (2b)

due to limited ramp space, where $Q$ is the maximum queue length. It indicates
$x_{\mathrm{all}}(t),x_{\mathrm{cv}}(t)\in[0,Q]$. For the sake of convenience,
we also require the CV flows $f_{\mathrm{cv}}^{\mathrm{in}}(t)$ and
$f_{\mathrm{cv}}^{\mathrm{out}}(t)$ for $t=0,1,\cdots$ such that

$\sum_{t=1}^{T}x_{\mathrm{cv}}(t)\to\infty,\text{ as }T\to\infty.$ (3)

If (3) does not hold, there exists $T_{0}>0$ such that $x_{\mathrm{cv}}(t)=0$
for all $t\geq T_{0}$. It means that there are none of queuing CVs at the on-
ramp after time $T_{0}$, and our method will not apply any more in that case.
Note that (3) implies

$\sum_{t=1}^{T}x_{\mathrm{all}}(t)\to\infty,\text{ as }T\to\infty.$ (4)

Obviously, there exists the following relationship between
$x_{\mathrm{all}}(t)$ and $x_{\mathrm{cv}}(t)$:

$x_{\mathrm{cv}}(t)=(\alpha+\theta(t))x_{\mathrm{all}}(t),$ (5)

where $\alpha+\theta(t)$ denotes the penetration rate of CVs at time $t$. We
use $\alpha$ to represent the market penetration rate and $\theta(t)$ to
denote the variation at the on-ramp. Besides, we make the assumption below.

######  Assumption 1.

The market penetration rate $\alpha\in(0,1]$ is known, but the fluctuation
$\theta(t)$ is unknown and bounded by

$||\theta(t)||_{\ell_{\infty}}\leq\Theta\in[0,1],$ (6)

where $||\theta(t)||_{\ell_{\infty}}:=\sup_{t\geq 0}|\theta(t)|$.

Note that the upper bound $\Theta$ can be roughly calibrated when we have
access to measurements of CV flows and total flows; see (7) below.

We consider the traffic data given by

$\begin{bmatrix}\tilde{f}_{\mathrm{all}}^{\mathrm{in}}(t)\\\
\tilde{f}_{\mathrm{all}}^{\mathrm{out}}(t)\\\
\tilde{f}_{\mathrm{cv}}^{\mathrm{in}}(t)\\\
\tilde{f}_{\mathrm{cv}}^{\mathrm{out}}(t)\\\
\tilde{x}_{\mathrm{cv}}(t)\end{bmatrix}=\begin{bmatrix}f_{\mathrm{all}}^{\mathrm{in}}(t)\\\
f_{\mathrm{all}}^{\mathrm{out}}(t)\\\ f_{\mathrm{cv}}^{\mathrm{in}}(t)\\\
f_{\mathrm{cv}}^{\mathrm{out}}(t)\\\ x_{\mathrm{cv}}(t)\end{bmatrix}+w(t),$
(7)

where $w(t)\in\mathbb{R}^{5}$ denotes unknown measurement noise,
$\tilde{f}_{\mathrm{all}}^{\mathrm{in}}(t)$ and
$\tilde{f}_{\mathrm{all}}^{\mathrm{out}}(t)$ are measured by induction loop
detectors, $\tilde{f}_{\mathrm{cv}}^{\mathrm{in}}(t)$,
$\tilde{f}_{\mathrm{cv}}^{\mathrm{out}}(t)$, $\tilde{x}_{\mathrm{cv}}(t)$ are
extracted from BSMs [15]. We only know the measured values due to the noise
$w(t)$, and we assume $w(t)$ to satisfy:

######  Assumption 2.

The noise $w(t)$ satisfies $||w||_{\ell_{2}}<\infty$, where

$||w||_{\ell_{2}}:=\Big{(}\sum_{t=0}^{\infty}||w(t)||_{2}^{2}\Big{)}^{1/2}=\Big{(}\sum_{t=0}^{\infty}w^{\mathrm{T}}(t)w(t)\Big{)}^{1/2}.$
(8)

Note that (8) implies that the noise has finite energy. It is a common
technique for modeling temporary noise [26].

Letting

$x(t):=\begin{bmatrix}x_{\mathrm{all}}(t)\\\
x_{\mathrm{cv}}(t)\end{bmatrix}\text{ and
}\tilde{f}(t):=\begin{bmatrix}\tilde{f}_{\mathrm{all}}^{\mathrm{in}}(t)\\\
\tilde{f}_{\mathrm{all}}^{\mathrm{out}}(t)\\\
\tilde{f}_{\mathrm{cv}}^{\mathrm{in}}(t)\\\
\tilde{f}_{\mathrm{cv}}^{\mathrm{out}}(t)\end{bmatrix},$

we reformulate (1), (5) and (7) into a dynamic system as follows:

$\displaystyle x(t+1)=$ $\displaystyle(A+\Delta
A(t))x(t)+B\tilde{f}(t)+Dw(t),$ (9a) $\displaystyle y(t)=$ $\displaystyle
Cx(t)+Ew(t)$ (9b)

with an unknown initial condition $x(0)\in[0,Q]^{2}$, where

$A=\begin{bmatrix}1&0\\\ \alpha&0\end{bmatrix},\Delta
A(t)=\begin{bmatrix}0&0\\\ \theta(t)&0\end{bmatrix},$
$B=-D=\begin{bmatrix}\delta_{t}&-\delta_{t}&0&0&0\\\
0&0&\delta_{t}&-\delta_{t}&0\end{bmatrix},$
$C=\begin{bmatrix}0&1\end{bmatrix},E=\begin{bmatrix}0&0&0&0&1\end{bmatrix}.$

Here $x(t)$ denotes the model state, $y(t)$ denotes the measured output, and
$\Delta A(t)$ represents the parameter uncertainty. Since the observed flows
$\tilde{f}(t)$ do not equal to the real flows
$f_{\mathrm{all}}^{\mathrm{in}}(t)$, $f_{\mathrm{all}}^{\mathrm{out}}(t)$,
$f_{\mathrm{cv}}^{\mathrm{in}}(t)$ and $f_{\mathrm{cv}}^{\mathrm{out}}(t)$, it
is necessary to include the measurement noise in the state equation (9a).

Given the queuing system (9a)-(9b), our goal is to design a robust filter that
estimates the state $x(t)$ given $y(t)$ and $\tilde{f}(t)$ but without knowing
the initial condition $x(0)$, the noise $w(t)$ and the parameter uncertainty
$\Delta A(t)$.

##  III Filter Design

We introduce the proposed filter design in this section. Consider a filter
given by

$\displaystyle\hat{x}(t+1)=$ $\displaystyle
A\hat{x}(t)+B\tilde{f}(t)+L(y(t)-\hat{y}(t)),$ (10a)
$\displaystyle\hat{y}(t)=$ $\displaystyle C\hat{x}(t),$ (10b)

with an initial condition $\hat{x}(0)\in[0,Q]^{2}$, where $L$ is the gain to
be designed. Note that we can manually specify the initial condition
$\hat{x}(0)$.

Define the estimation error

$e(t)=x(t)-\hat{x}(t).$ (11)

Plugging (9a)-(9b) and (10a)-(10b) into (11) gives

$e(t+1)=(A-LC)e(t)+\Delta A(t)x(t)+(D-LE)w(t).$ (12)

We aim at designing the filter gain $L$ so that the estimation error $e(t)$
satisfies the following criterion [27]. That is, there exists
$\mu_{1},\mu_{2}>0$ and some positive definite function
$\gamma:[0,Q]^{2}\to\mathbb{R}_{\geq 0}$ such that

$\displaystyle\sum_{t=0}^{T}||e(t)||_{2}^{2}\leq$
$\displaystyle\mu_{1}\sum_{t=0}^{T}||x(t)||_{2}^{2}+\mu_{2}\sum_{t=0}^{T}||w(t)||_{2}^{2}$
$\displaystyle+\gamma(x(0),\hat{x}(0)),~{}T=0,1,\cdots,$ (13)

given any initial condition $x(0)$ and $\hat{x}(0)$, the input $u(t)$, the
noise $w(t)$ and the parameter uncertainty $\Delta A(t)$.

The criterion (13) is an extension of the typical objective of designing
$H_{\infty}$ filters [26]. We consider it because it is not easy to implement
the conventional design of $H_{\infty}$ filters in our problem. The major
difficult is that we need to additionally assume $\tilde{f}(t)=0$ when
$x(t)=0$, which unreasonably indicates that the inflows equal zero when there
are no queuing vehicles at the on-ramp.

To understand the criterion (13), we divide the both sides by
$\sum_{t=0}^{T}||x(t)||_{2}^{2}$ and obtain

$\frac{\sum_{t=0}^{T}||e(t)||_{2}^{2}}{\sum_{t=0}^{T}||x(t)||_{2}^{2}}\leq\mu_{1}+\mu_{2}\frac{\sum_{t=0}^{T}||w(t)||_{2}^{2}}{{\sum_{t=0}^{T}||x(t)||_{2}^{2}}}+\frac{\gamma(x(0),\hat{x}(0))}{{\sum_{t=0}^{T}||x(t)||_{2}^{2}}}.$
(14)

Noting i) $||w||_{\ell^{2}}^{2}<\infty$ by Assumption 2, ii)
$\gamma(x(0),\hat{x}(0))<\infty$ owing to $x(0),\hat{x}(0)\in[0,Q]^{2}$ and
iii) $||x||_{\ell^{2}}^{2}=\infty$ by (4), we let $T\to\infty$ and reduce (14)
to

$\frac{||e||_{\ell^{2}}^{2}}{||x||_{\ell^{2}}^{2}}\leq\mu_{1}.$ (15)

Thus, if (13) holds, the long-term relative error is bounded by $\mu_{1}$, and
an upper bound of the long-term error rate is given by $\sqrt{\mu_{1}}$ (or a
lower bound of the long-term estimation accuracy is given by
$1-\sqrt{\mu_{1}}$) [27]. It indicates that we can minimize $\mu_{1}$ by
designing the filter gain $L$. Now, we turn to $\mu_{2}$. Recalling (14), we
can conclude that $\mu_{2}$ is related to convergence speed: smaller $\mu_{2}$
indicates faster convergence.

For the filter design, we have the following result:

######  Theorem 1.

Consider the queuing system (9a)-(9b) with the filter (10a)-(10b). If there
exists a symmetric positive definite matrix $P\succ 0$, a vector
$R\in\mathbb{R}^{2}$ and positive scalars $\mu_{1},\mu_{2},\mu_{3}>0$ such
that

$\begin{bmatrix}-\mu_{3}I&\Theta P&0&0&0\\\ *&-P&PA-RC&0&PD-RE\\\
*&*&-P+I&0&0\\\ *&*&*&(\mu_{3}-\mu_{1})I&0\\\
*&*&*&*&-\mu_{2}I\end{bmatrix}\prec 0,$ (16)

where $I$ denotes the identity matrix, then the filter gain is given by

$L=P^{-1}R,$ (17)

and the filter (10a)-(10b) satisfies (13).

The proof of Theorem 1 is available in Appendix -A. Note that $X\succ 0$
(resp. $X\prec 0$) denotes that $X$ is a positive (resp. negative) definite
matrix, and $X\succeq 0$ (resp. $X\preceq 0$) implies that $X$ is a semi-
positive (resp. semi-negative) definite matrix. We also use “$*$” to simplify
notations of symmetric matrices, e.g.

$\begin{bmatrix}X_{11}&X_{12}\\\
*&X_{22}\end{bmatrix}:=\begin{bmatrix}X_{11}&X_{12}\\\
X^{\mathrm{T}}_{12}&X_{22}\end{bmatrix}.$

Recalling the elaboration of (13), we can first specify $\beta$ and then solve

$(\mathrm{P}_{1})~{}\min_{\mu_{1},\mu_{2},\mu_{3}>0,P,R\succ
0}~{}\mu_{1}+\beta\mu_{2}~{}s.t.~{}\eqref{eq_thm}.$ (18)

Since $\mu_{1}$ is related to long-term estimation error, we can quantify, by
solving $\mathrm{P}_{1}$, the relationship between the long-term estimation
accuracy and the uncertainty bound $\Theta$. Note that $\mathrm{P}_{1}$
belongs to convex programming, and hence it can be easily solved by Mosek
[29].

##  IV Case Study

We present a real-world case study, based on a model developed using the
microscopic traffic simulation software, SUMO, to illustrate our robust filter
design. We first present a sensitivity analysis that reveals impact of the
penetration rates of CVs and measurement noise. Then we focus on comparison of
various estimation methods.

(a) Selected on-ramp.

(b) SUMO setting.

Figure 2: Simulation model developed in SUMO.

We consider an on-ramp of Interstate 210 Eastbound (I-210 E) in Los Angeles,
California; see Fig. 2(a). The ramp meter, detectors A and B at the on-ramp
are set based on the real locations (see Fig. 2(b)), and we have the maximum
queue length $Q_{\max}=32$. We also deploy detector M at the mainline to
implement the following ramp metering algorithm ALINEA [4]:

$\displaystyle r_{i}(t)=$
$\displaystyle\mathrm{mid}\\{\underline{R},r_{i}(t-1)+K_{i}(\tilde{o}_{\mathrm{M}}-o_{\mathrm{M}}(t)),\bar{R}\\}$
(19a) $\displaystyle r_{o}(t)=$
$\displaystyle\begin{cases}\underline{R}&\text{if
}o_{\mathrm{A}}(t)<\tilde{o}_{\mathrm{A}}\\\ \bar{R}&\text{if
}o_{\mathrm{A}}(t)\geq\tilde{o}_{\mathrm{A}}\end{cases}$ (19b) $\displaystyle
r(t)=$ $\displaystyle\max\\{r_{i}(t),r_{o}(t)\\}$ (19c)

where $r_{i}(t)$ denotes the metered flow at time $t$ given by integral
control with gain $K_{i}$, $r_{o}(t)$ denotes the metered flow at time $t$
given by the queue-override strategy, $r(t)$ is the selected metered flow as
control input. Briefly speaking, (19a) determines the metered flow in order to
keep the mainline occupancy $o_{\mathrm{M}}(t)$, measured by detector M, close
to the critical occupancy $\tilde{o}_{\mathrm{M}}$, and it also clips the
metered flow with the minimum (resp. maximum) metered flow $\underline{R}$
(resp. $\bar{R}$) to avoid the well-known wind-up phenomenon [10], where the
$\mathrm{mid}$ operator takes the middle value of all the members. (19b)-(19c)
indicate that when the occupancy $o_{\mathrm{A}}(t)$ at the ramp entrance,
measured by detector A, is higher than the threshold $\tilde{o}_{\mathrm{A}}$,
it is necessary to choose the maximum metered flow to discharge queuing
vehicles at the on-ramp as quickly as possible. Since we adopt the one-car-per
green policy [1], we finally convert the metered flow $r(t)$ into red time to
realize ramp metering in SUMO. It should be pointed out the algorithm above
does not depend on ramp queue length. The reason lies in that we focus on
evaluating ramp queue length estimation here, and this algorithm provides the
same benchmark when we compare different estimation methods.

The traffic demands considered are illustrated in Fig 3. For vehicle
composition, we assume that $95\%$ are passenger vehicles, $3\%$ are light-
duty trucks and the remaining are heavy-duty trucks. We consider the period
13:00-20:00 since the peak hours of I-210 E are usually from 14:00-19:00. We
use the first half an hour as the warm-up period and start the estimation at
13:30.

(a) Upstream mainline demand.

(b) On-ramp demand.

Figure 3: Traffic demands.

###  IV-A Sensitivity analysis

(a) Impacts on $\mu_{1}$.

(b) Impacts on $\mu_{2}$.

Figure 4: Sensitivity analysis of market penetration rate of CVs, $\alpha$ and
uncertainty bound $\Theta$.

The following presents sensitivity analysis of CV penetration rates and
measurement noise. We first solve the problem $\mathrm{P}_{1}$ given different
combinations of $\alpha$ and $\Theta$. We choose the weight $\beta=0.1$ since
we put priority to the error rate. The results are presented in Fig. 4. We
show $\sqrt{\mu_{1}}$ in Fig. (4(a)) since it denotes an upper bound of long-
term error rate. It is reasonable to see that when the fluctuation is mild
with small $\Theta$, the error upper bound is relatively low. We also present
$\log{\mu_{2}}$ in Fig. (4(b)). We consider $\log{\mu_{2}}$ since it could be
large when $\alpha$ is small but $\Theta$ is large. Recalling that $\mu_{2}$
is related to convergence speed, we conclude that the convergence could be
vary slow, given small $\alpha$ and large $\Theta$, when the measurement noise
occurs.

Then, we analyze the filter performance in the simulation. Note that we only
specify the CV market penetration rate in advance and do not manipulate the
variation bound $\Theta$, which requires us to estimate $\Theta$. It turns out
that $\theta(t)$ are relatively scattered given any
$\alpha\in\\{0.1,0.3,\cdots,0.9\\}$ and it is not easy to design the filter
gain $L$ if we consider the bound $\Theta$ satisfying (6). Thus we consider
$\Theta$ such that $|\theta(t)|\leq\Theta$ for enough $t$. We find that
$\Theta=0.08$ suffices for $\alpha\in\\{0.1,0.3,\cdots,0.9\\}$, which is used
to derive the filter gain. To test our robust filter, we also assume that the
measurement noise happens from 15:00-17:00. Concretely, we assume that
$f^{\mathrm{out}}_{\mathrm{cv}}(t)$, $f^{\mathrm{in}}_{\mathrm{cv}}(t)$,
$f^{\mathrm{out}}_{\mathrm{all}}(t)$, $f^{\mathrm{out}}_{\mathrm{all}}(t)$ (in
unit of veh/h) are subject to bounded noise uniformly distributed on
$[-60~{}\mathrm{veh/h},60~{}\mathrm{veh/h}]$ and $x_{\mathrm{cv}}(t)$ (in unit
of veh) is subject to bounded noise uniformly distributed on
$[-2~{}\mathrm{veh},2~{}\mathrm{veh}]$. The result is summarized in Table I.
We consider the upper bound $\sqrt{\mu_{1}}$, $\sqrt{\hat{\mu}_{1}}$ and the
root mean square Error (RMSE), where $\hat{\mu}_{1}$ is obtained from the
simulation:

$\hat{\mu}_{1}=\sum_{t=0}^{T}||e(t)||_{2}^{2}/||x(t)||_{2}^{2}.$ (20)

Clearly, $\sqrt{\mu_{1}}$ denotes a theoretical upper bound and
$\sqrt{\hat{\mu}_{1}}$ denotes an empirical evaluation. It is reasonable to
find that given higher CV penetration rate, our robust filter yields lower
error.

TABLE I: Sensitivity analysis of CV penetration rates. Market penetration rate | 0.1 | 0.3 | 0.5 | 0.7 | 0.9
---|---|---|---|---|---
$\sqrt{\mu_{1}}$ | 1 | 0.4491 | 0.3164 | 0.2712 | 0.2537
$\sqrt{\hat{\mu}_{1}}$ | 0.6540 | 0.3708 | 0.2701 | 0.1845 | 0.1288
RMSE | 9.629 | 5.459 | 3.976 | 2.716 | 1.896

We also test our filter given different uniformly bounded noise. We consider
the flows $f^{\mathrm{out}}_{\mathrm{cv}}(t)$,
$f^{\mathrm{in}}_{\mathrm{cv}}(t)$, $f^{\mathrm{out}}_{\mathrm{all}}(t)$,
$f^{\mathrm{out}}_{\mathrm{all}}(t)$ are subjected to bounded noise $[-U,U]$
from 15:00-17:00. The result is given in Table II. Noting the on-ramp flow
shown in Fig. 3(b), we conclude that our filter is robust even when the noise
is relatively large, e.g. $U=180$ veh/h.

TABLE II: Sensitivity analysis of measurement noise. Noise bound $U$ (veh/h) | 60 | 90 | 120 | 150 | 180
---|---|---|---|---|---
$\sqrt{\hat{\mu}_{1}}$ ($\alpha=0.1$) | 0.6540 | 0.6542 | 0.6546 | 0.6550 | 0.6554
$\sqrt{\hat{\mu}_{1}}$ ($\alpha=0.5$) | 0.2701 | 0.2708 | 0.2717 | 0.2728 | 0.2742
$\sqrt{\hat{\mu}_{1}}$ ($\alpha=0.9$) | 0.1288 | 0.1307 | 0.1330 | 0.1357 | 0.1389

###  IV-B Estimation comparison

We consider another two estimation methods as our baselines. The first one is
given by

$\hat{x}_{\mathrm{all}}(t)=\frac{2\tilde{x}_{\mathrm{cv}}(t)}{\tilde{f}_{\mathrm{cv}}^{\mathrm{in}}(t)/\tilde{f}_{\mathrm{all}}^{\mathrm{in}}(t)+\tilde{f}_{\mathrm{cv}}^{\mathrm{out}}(t)/\tilde{f}_{\mathrm{all}}^{\mathrm{out}}(t)}.$
(21)

It evaluates the CV penetration rate at the on-ramp based on the inflows and
outflows and converts the observed $\tilde{x}_{\mathrm{cv}}(t)$ to
$\hat{x}_{\mathrm{all}}(t)$. The second one was proposed based on the Kalman
filter [11], given by

$\displaystyle\hat{x}_{\mathrm{all}}(t)=$
$\displaystyle\hat{x}_{\mathrm{all}}(t-1)+\delta_{t}(\tilde{f}_{\mathrm{all}}^{\mathrm{in}}(t)-\tilde{f}_{\mathrm{all}}^{\mathrm{out}}(t))$
$\displaystyle+K_{f}(Q_{\mathrm{bb}}\frac{\bar{L}_{\mathrm{veh}}}{\bar{L}_{\mathrm{veh}}+L_{\mathrm{d}}}o_{\mathrm{A}}(t-1)-\hat{x}_{\mathrm{all}}(t-1)),$
(22)

where $K_{f}$ is the filter gain, $\bar{L}_{\mathrm{veh}}$ is average physical
length of vehicles and $L_{\mathrm{d}}$ denotes sensor length. In the
simulation, we set $K_{f}=0.1$ [11],. It should be pointed out that (22)
typically relies on internal detectors, e.g. induction loop sensors between
detectors A and B in Fig. 2(b). However, we assume that such kind of sensors
is not available and use detector A instead to provide the occupancy
measurement $o_{\mathrm{A}}(t-1)$.

We propose five scenarios for the purpose of comparison. The first and the
second scenarios adopt the estimator (21), but are given CV penetration rates
$\alpha=0.25$ and $\alpha=0.5$, respectively. In scenario 3, the Kalman filter
(22) is considered. We apply our robust filter in the last two scenarios, also
given different CV penetration rates, $\alpha=0.25$ and $\alpha=0.5$,
respectively. For each scenario, we also add measurement noise between
15:00-17:00 in the simulation.

(a) Scenario 1: estimator (21) when $\alpha=0.25$.

(b) Scenario 2: estimator (21) when $\alpha=0.50$.

(c) Scenario 3: Kalman filter (22) with gain $K_{f}=0.1$[11].

(d) Scenario 4: proposed robust filter when $\alpha=0.25$.

(e) Scenario 5: proposed robust filter when $\alpha=0.50$.

Figure 5: Comparison of different estimation methods.

The results are shown in Fig. 5. Clearly, the estimator (21) is sort of open-
loop and thus produces noisier estimation than our robust filter does, given
the same CV market penetration rate (see Fig. 5(a) vs. Fig. 5(d), and Fig.
5(b) vs. Fig. 5(e)). Besides, it is not surprising to see that higher CV
penetration rates benefit the estimation (see Fig. 5(a) vs. Fig. 5(b), and
Fig. 5(d) vs. Fig. 5(e)). When there are none of internal sensors, the Kalman
filter (22) could yield really poor performance. As indicated in Table III, it
achieves relatively large $\sqrt{\hat{\mu}_{1}}$ and RMSE.

TABLE III: Performance summary. | $\sqrt{\hat{\mu}_{1}}$ | RMSE
---|---|---
Scenario 1 | 0.5767 | 8.490
Scenario 2 | 0.3526 | 5.191
Scenario 3 | 0.8205 | 12.079
Scenario 4 | 0.4248 | 6.253
Scenario 5 | 0.2701 | 3.976

##  V Concluding remarks

In this paper, we investigated ramp queue estimation that combines information
from traditional infrastructure sensors and emerging CV data. We first
presented a ramp queue model that takes measurement noise and uncertain CV
penetration rates into account. Then we used the theory of extended
$H_{\infty}$ filters to propose a robust filter for our ramp queue model and
to show the performance guarantee of our filter (Theorem 1). It turns out that
Theorem 1 can be used to reveal how the uncertainty of penetration rates
influences long-term estimation accuracy. Possible future directions include
integrating our filters into localized/coordinated ramp control and analyzing
their control performance.

###  -A Proof of Theorem 1

Consider the following Lyapunov function:

$V(e(t))=e^{\mathrm{T}}(t)Pe(t).$ (23)

where $P\succ 0$. We also define

$\xi(t):=\begin{bmatrix}e(t)&x(t)&w(t)\end{bmatrix}^{\mathrm{T}}$ (24)

and

$W:=\begin{bmatrix}A-LC&\Delta A(t)&D-LE\end{bmatrix}.$ (25)

It follows

$\displaystyle\Delta V(e(t)):=$ $\displaystyle V(e(t+1))-V(e(t))$ (26)
$\displaystyle=$
$\displaystyle\xi^{\mathrm{T}}(t)W^{\mathrm{T}}PW\xi(t)-e^{\mathrm{T}}Pe(t).$
(27)

Then we obtain

$\displaystyle\Delta
V(e(t))+e^{\mathrm{T}}(t)e(t)-\mu_{1}x^{\mathrm{T}}(t)x(t)-\mu_{2}w^{\mathrm{T}}(t)w(t)$
$\displaystyle=$ $\displaystyle\xi^{\mathrm{T}}(t)(W^{\mathrm{T}}PW+U)\xi(t),$
(28)

where

$U:=\begin{bmatrix}-P+I&0&0\\\ 0&-\mu_{1}I&0\\\ 0&0&-\mu_{2}I\end{bmatrix}.$
(29)

Summing both sides of (28) over $t=0,1,\cdots,T$ yields

$\displaystyle\sum_{t=0}^{T}||e(t)||_{2}^{2}-\mu_{1}\sum_{t=0}^{T}||x(t)||_{2}^{2}-\mu_{2}\sum_{t=0}^{T}||w(t)||_{2}^{2}$
$\displaystyle=$
$\displaystyle-V(e(T))+\sum_{t=0}^{T}\xi^{\mathrm{T}}(W^{\mathrm{T}}PW+U)\xi(t)+V(e(0)).$

Noting $V(e(T))>0$ and $\gamma(x(0),\hat{x}(0)):=V(e(0))$, we conclude that
the filter satisfies (13) if $W^{\mathrm{T}}PW+U\prec 0$.

Next, we apply the Schur complement lemma:

######  Lemma 1 (Schur complement lemma [30]).

Let $X$ be a symmetric matrix of real numbers given by

$X=\begin{bmatrix}A&B\\\ B^{\mathrm{T}}&C\end{bmatrix}.$

If $A$ is invertible, then $X\prec 0$ if and only if

$A\prec 0,~{}C-B^{\mathrm{T}}A^{-1}B\prec 0.$

Noting

$W^{\mathrm{T}}PW+U=U-W^{\mathrm{T}}P(-P)^{-1}PW,$

we conclude by Lemma 1 that $W^{\mathrm{T}}PW+U\prec 0$ requires

$\begin{bmatrix}-P&PW\\\ W^{\mathrm{T}}P&U\end{bmatrix}\prec 0.$ (30)

Recalling (25) and (29), we obtain

$\displaystyle\begin{bmatrix}-P&PW\\\
W^{\mathrm{T}}P&U\end{bmatrix}=\tilde{P}F(t)\tilde{I}+\tilde{I}^{\mathrm{T}}F^{\mathrm{T}}(t)\tilde{P}^{\mathrm{T}}$
$\displaystyle+\begin{bmatrix}-P&PA-RC&0&PD-RE\\\ *&-P+I&0&0\\\
*&*&-\mu_{1}I&0\\\ *&*&*&-\mu_{2}I\end{bmatrix}$ (31)

where

$\displaystyle R:=$ $\displaystyle PL$ (32a) $\displaystyle\tilde{P}:=$
$\displaystyle\begin{bmatrix}\Theta P&0&0&0\end{bmatrix}^{\mathrm{T}},$ (32b)
$\displaystyle\tilde{I}:=$ $\displaystyle\begin{bmatrix}0&0&I&0\end{bmatrix},$
(32c) $\displaystyle F(t):=$ $\displaystyle\Delta A(t)/\Theta.$ (32d)

######  Lemma 2 (Young’s inequality [30]).

Let $X\in\mathbb{R}^{m\times n}$ and $Y\in\mathbb{R}^{n\times m}$. Then for
any scalar $\kappa>0$ and any $F\in\mathbb{R}^{n\times n}$ such that
$F^{\mathrm{T}}F\preceq I$, there holds

$XFY+Y^{\mathrm{T}}F^{\mathrm{T}}X^{\mathrm{T}}\preceq\kappa^{-1}XX^{\mathrm{T}}+\kappa
Y^{\mathrm{T}}Y.$ (33)

Noting $F^{\mathrm{T}}(t)F(t)\preceq I$, we use Lemma 2 to obtain

$\tilde{P}F(t)\tilde{I}+\tilde{I}^{\mathrm{T}}F^{\mathrm{T}}(t)\tilde{P}^{\mathrm{T}}\preceq\mu_{3}^{-1}\tilde{P}\tilde{P}^{\mathrm{T}}+\mu_{3}\tilde{I}^{\mathrm{T}}\tilde{I}.$
(34)

Combining (31) and (34), we find that if

$\displaystyle\begin{bmatrix}-P&PA-RC&0&PD-RE\\\ *&-P+I&0&0\\\
*&*&(\mu_{3}-\mu_{1})I&0\\\ *&*&*&-\mu_{2}I\end{bmatrix}$
$\displaystyle+\mu_{3}^{-1}\tilde{P}\tilde{P}^{\mathrm{T}}\prec 0,$ (35)

then (30) holds, which indicates that the filter with the gain $L=P^{-1}R$
satisfies the criterion (13).

Finally, using Lemma 1 again, we obtain (35) if (16) holds. ∎

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