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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1
Optimal Perimeter Control for Two Urban Regions
With Macroscopic Fundamental Diagrams: A
Model Predictive Approach
Nikolas Geroliminis, Jack Haddad, and Mohsen Ramezani
Abstract—Recent analysis of empirical data from cities showed
that a macroscopic fundamental diagram (MFD) of urban traf-
fic provides for homogenous network regions a unimodal low-
scatter relationship between network vehicle density and network
space-mean flow. In this paper, the optimal perimeter control for
two-region urban cities is formulated with the use of MFDs. The
controllers operate on the border between the two regions and
manipulate the percentages of flows that transfer between the two
regions such that the number of trips that reach their destinations
is maximized. The optimal perimeter control problem is solved
by model predictive control, where the prediction model and the
plant (reality) are formulated by MFDs. Examples are presented
for different levels of congestion in the regions of the city and the
robustness of the controller is tested for different sizes of error
in the MFDs and different levels of noise in the traffic demand.
Moreover, two methods for smoothing the control sequences are
presented. Comparison results show that the performances of the
model predictive control are significantly better than a “greedy”
feedback control. The results in this paper can be extended to de-
velop efficient hierarchical control strategies for heterogeneously
congested cities.
Index Terms—Macroscopic fundamental diagrams (MFDs),
model predictive control (MPC), perimeter control.
I. INTRODUCTION
E
FFICIENT monitoring and traffic management of large-
scale urban networks still remain a challenge for both traf-
fic researchers and practitioners. A large urban network mainly
consists of the following two elements: 1) urban links and,
2) signalized intersections. Modeling the traffic flow dynamics
of each element in a large urban network with a large number
of links and intersections is a complex task. We have to model
the evolution of queues at each signalized intersection and
account for the queue dynamic interactions between adjacent
intersections, i.e., capturing the dynamics of propagation and
Manuscript received February 13, 2012; revised June 21, 2012; accepted
August 9, 2012. This work was supported in part by the Swiss National Science
Foundation under Grant 200021-13016. The Associate Editor for this paper was
B. De Schutter. (Corresponding author: Nikolas Geroliminis.)
The authors are with the Urban Transport Systems Laboratory, School of Ar-
chitecture, Civil and Environmental Engineering, École Polytechnique Fédérale
de Lausanne, 1015 Lausanne, Switzerland (e-mail: nikolas.geroliminis@
epfl.ch; jack.haddad@epfl.ch; mohsen.ramezani@epfl.ch).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TITS.2012.2216877
spillback of queues because of high demand. Even if this
task is completed, a centralized control approach would be
a very challenging task, not only because of the computa-
tional complexity but also because users might change their
travel patterns (e.g., time of departure, route choice, and mode
choice). Hence, instead of this micromodeling approach, the
macroscopic fundamental diagram (MFD) aims at simplifying
the micromodeling task of the urban network, where the col-
lective traffic flow dynamics of subnetworks capture the main
characteristics of traffic congestion, such as the evolution of
space-mean flows and densities in different regions of the city.
MFD can be utilized to introduce elegant control strategies to
improve mobility and decrease delays in large urban networks,
which local strategies cannot achieve.
The MFD of urban traffic provides for different network
regions a unimodal low-scatter relationship between network
vehicle density (veh/km) and network space-mean flow or
outflow (veh/h) if congestion is roughly homogeneous in the
region. Alternatively, the MFD links accumulation, which is
defined as the number of vehicles in the region, and trip
completion flow, which is defined as the output flow of the
region. Network flow or trip completion flow increases with
density or accumulation up to a critical point, while additional
vehicles in the network cause strong reductions in the flow.
The first theoretical proposition of such a physical model was
developed in [1], while similar approaches were also initiated
in [2] and [3]. The physical model of MFD was observed with
dynamic features in congested urban networks in Yokohama,
Japan, in [4]. This paper showed the following two important
properties of MFD that can be utilized for management and
control purposes: 1) some urban regions approximately exhibit
an MFD, and 2) the shape of the MFD is not very sensitive
to different demand patterns. Property 1 is important for mon-
itoring purposes, because flow can easily be observed with
different types of sensors, whereas outflow is more difficult.
Property 2 is important for control purposes, because efficient
active traffic management schemes can be developed without
a detailed knowledge of origin–destination (OD) tables. Other
investigations of MFD using empirical or simulated data can be
found in [5]–[8] and other papers, whereas routing strategies
that are based on MFD can be found in [9].
Recent studies [8], [10], [11] have shown that networks with
heterogeneous distribution of density exhibit network flows that
are smaller than those that approximately meet homogeneity
conditions (low spatial variance of link density), particularly
1524-9050/$31.00 © 2012 IEEE
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2 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
for high network densities. Moreover, networks with a small
variance of link densities have a well-defined MFD, i.e., low
scatter of flows for the same densities. One possible solution for
heterogeneous networks is that they might be partitioned into a
number of more homogeneous regions with small variances of
link densities, as each region will have a well-defined MFD. See
[12] for more information on network partitioning. In [7] and
[13], strong hysteresis phenomena in freeways that might not
disappear after partitioning is shown. Nevertheless, the work
in [12] showed that urban networks can be partitioned in a
way that decreases the degree of heterogeneity within clusters.
Partitions should not have a very small size, because the law
of large numbers will not apply, and high scatter might exist
in MFDs. In addition, a large number of partitions will not
allow the development of simple control strategies as shown
in this paper, because control might change the route choices,
and detailed ODs might be needed.
Management and control of multiregion MFDs systems can
improve urban mobility, prevent overcrowding, and relieve
congestion in cities. The optimal control policy was derived
for a single MFD system in [3]. The main logic behind this
policy is that it aims at decreasing inflows in regions with high
densities of destinations and points in the decreased part of an
MFD and manage the accumulation to maintain the flow in the
city at its maximum. However, in case of multiregion cities with
multiple centers of congestion and/or attraction, control policies
are more complicated and not well understood. For stability
analysis of controlling two urban regions, see [14].
Because of the scatter in the MFD, mainly in the congested
regime, errors are expected between the MFD model and the
plant (reality). Therefore, an optimal open-loop control for
the multiregion MFDs system would be a suboptimal solution
compared with the optimal closed-loop control. The closed-
loop control takes into account errors between the model
and the plant by utilizing feedback-monitored information.
Furthermore, the closed-loop control can tackle disturbances
for which the model was not designed, e.g., noise in the
traffic demand. The optimal closed-loop control is obtained by
implementing the model predictive control (MPC) framework.
A historical survey for industrial applications of MPC can be
found in [15], while theoretical issues of MPC can be found
in [16]–[20].
MPC is a receding horizon scheme, where, at each time
step, an optimal open-loop of the problem with finite hori-
zon is optimized; then, only the first controller is applied to
the plant, and the procedure is again carried out. A receding
horizon framework has been used for optimization in different
traffic control problems, for example, ramp metering of freeway
networks in [21] and [22], variable speed limits and route
guidance for freeway networks in [23] and [24], signal control
for large-scale urban networks in [25]–[27], and mixed urban
and freeway networks in [28]. The open-loop optimization in
the traffic MPC models, for example, in [23] and [24], uses
a direct simultaneous method to transcript it into a finite-
dimensional nonlinear programming through the discretization
of both control and state variables, whereas in [22] and [26], a
feasible direction algorithm is utilized to solve the open-loop
optimization problem. Overviews of different control applica-
Fig. 1. Two-region MFDs system. Two regions R
1
and R
2
with four traffic
demands q
11
(t), q
12
(t), q
21
(t),andq
22
(t) and two perimeter controllers
u
12
(t) and u
21
(t).
tions in transportation problems can be found, e.g., in [29]
and [30].
In this paper, the optimal perimeter control problem for two-
region urban cities is formulated, where the dynamic equations
are modeled according to their MFDs. Moreover, the optimal
control solution is obtained by applying the MPC framework.
The open-loop optimal control problem is solved using a direct
sequential method that discretizes only the control variables
with piecewise constant controls, whereas the state variables are
continuous and integrated using the state-of-the-art methods for
ordinary differential equation (ODE) solvers.
This paper is organized as follows. The control problem for
a two-region MFDs system is presented in Section II. Then,
in Section III, MPC is formulated, the parameters are tuned,
and the control laws of a greedy controller (GC) are presented.
Comparison results of case study examples are presented in
Section IV, showing the performance differences between MPC
and the GC. Finally, two different methods are introduced in
Section V to smooth the control sequences.
II. T
WO-REGION MACROSCOPIC FUNDAMENTAL
DIAGRAMS SYSTEM
In this paper, a heterogeneous traffic network that can be
partitioned into two homogeneous regions is considered. A
traffic network for a two-region system is schematically shown
in Fig. 1. Two regions R
i
, i = 1, 2, where each region has a
well-defined MFD are given as follows: 1) the periphery of the
center R
1
, and 2) the city center R
2
. Note that the geographical
relative position of these regions does not affect the dynamics
of the problem; for example, it can be two regions next to each
other. An endogenous traffic demand is defined as a flow in
which its origin and destination are the same region, whereas
the origin and destination of an exogenous traffic demand
are not the same. For the two-region system, there are two
endogenous traffic demands in R
1
, denoted by q
11
(t) (veh/s),
and in R
2
, denoted by q
22
(t) (veh/s), and two exogenous traffic
demands generated in R
1
and R
2
with destination to R
2
and R
1
,
denoted by q
12
(t) and q
21
(t) (veh/s), respectively. Correspond-
ing to the endogenous and exogenous traffic demands, four
accumulation states are used to model the dynamic equations,
n
ij
(t)(veh),i,j= 1, 2, where n
ij
(t) is the total number of
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GEROLIMINIS et al.: OPTIMAL PERIMETER CONTROL FOR TWO URBAN REGIONS WITH MFDs 3
vehicles in R
i
with destination to R
j
at time t. Let us denote
n
i
(t) (veh) as the accumulation or the total number of vehicles
in R
i
at time t, i.e., n
i
(t)=
j
n
ij
(t).
MFD is defined by G
i
(n
i
(t)) (veh/s), which is the trip
completion flow for region i at n
i
(t). The trip completion flow
for region i is the sum of transfer flows, i.e., trips from i with
destination j, i = j, plus the internal flow, i.e., trips from i
with destination i. The transfer flow from i with destination
to j is calculated corresponding to the ratio between accumu-
lations, i.e., M
ij
(t)=n
ij
(t)/n
i
(t) · G
i
(n
i
(t)), i = j, whereas
the internal flow from i with destination to i is calculated by
M
ii
(t)=n
ii
(t)/n
i
(t) · G
i
(n
i
(t)). These relationships assume
that the trip lengths for all trips within a region (internal or
external) are similar, i.e., the distance traveled per vehicle inside
a region is independent of the origin and destination of the
trip. For a description of different cases, refer to [31], which
will not alter the methodology. Simulation and empirical results
[4] show that the shape of MFD can be approximated by a
nonsymmetric unimodal curve skewed to the right, i.e., the
critical density, which maximizes the network flow, is smaller
than half the jammed density. Thus, we utilize a third-order
function of n
i
(t), e.g., G
i
(n
i
(t)) = a
i
· n
i
(t)
3
+ b
i
· n
i
(t)
2
+
c
i
· n
i
(t), where a
i
, b
i
, and c
i
are the estimated parameters.
In our formulated problem, the perimeter controllers, de-
noted by u
12
(t) and u
21
(t)(−), are introduced on the border
between the t wo regions, as shown in Fig. 1, where the purpose
is to control the transfer flows between the two regions such that
the total number of vehicles that complete their trips and reach
their destinations in the two-region MFDs system is maximized.
Because the perimeter controllers exist only on the border
between the two regions, the internal flows cannot be controlled
or restricted, whereas the transfer flows are controlled by the
controllers such that only a ratio transfers at time t.The
perimeter controllers u
12
(t) and u
21
(t), where 0 ≤ u
12
(t) ≤ 1
and 0 ≤ u
21
(t) ≤ 1, are, respectively, the ratio of the transfer
flow that transfers from R
1
to R
2
and from R
2
to R
1
at time
t. I t is also assumed t hat these controllers will not change the
shape of the MFDs. Implementations of the controllers in real
networks are discussed i n the Conclusion of this paper.
The criterion is to maximize the output of the traffic network,
i.e., the number of vehicles that complete their trips and reach
their destinations. Therefore, the two-region MFDs control
problem with four state variables is formulated as follows
(similar to [32]):
J =max
u
12
(t),u
21
(t)
t
f
t
0
[M
11
(t)+M
22
(t)] dt (1)
subject to
dn
11
(t)
dt
= q
11
(t)+u
21
(t) · M
21
(t) − M
11
(t) (2)
dn
12
(t)
dt
= q
12
(t) − u
12
(t) · M
12
(t) (3)
dn
21
(t)
dt
= q
21
(t) − u
21
(t) · M
21
(t) (4)
dn
22
(t)
dt
= q
22
(t)+u
12
(t) · M
12
(t) − M
22
(t) (5)
0 ≤ n
11
(t)+n
12
(t) (6)
0 ≤ n
21
(t)+n
22
(t) (7)
n
11
(t)+ n
12
(t) ≤ n
1,jam
(8)
n
21
(t)+ n
22
(t) ≤ n
2,jam
(9)
u
min
≤ u
12
(t) ≤ u
max
(10)
u
min
≤ u
21
(t) ≤ u
max
(11)
n
11
(t
0
)=n
11,0
; n
12
(t
0
)=n
12,0
(12)
n
21
(t
0
)=n
21,0
; n
22
(t
0
)=n
22,0
where t
f
(s) is the final time, n
ij,0
,i,j= 1, 2, are the initial
accumulations at t
0
, n
1,jam
and n
2,jam
(veh) are the accumula-
tions at the jammed density in R
1
and R
2
, respectively, and
u
min
and u
max
are the lower and upper bounds for u
12
(t)
and u
21
(t), respectively. Recall that M
ij
(t)=n
ij
(t)/n
i
(t) ·
G
i
(n
i
(t)), i, j = 1, 2. Equations (2)–(5) are the conservation
of mass equations for n
ij
(t), whereas (6)–(9) are the l ower and
upper bound constraints on accumulations in R
1
and R
2
.
III. M
ODEL PREDICTIVE CONTROL FOR THE TWO -REGION
MAC ROS C OP IC FUNDAMENTAL DIAGRAMS PROBLEM
The two-region MFDs problem (1)–(12) aims at finding the
perimeter control inputs, i.e., ratios of transfer flows of R
1
and R
2
, that maximize the number of vehicles that complete
their trips (reach their destinations). This problem is an optimal
control problem with a nonlinear objective function (1) and
dynamic equations (2)–(5), inequality state constraints (6)–(9),
control constraints (10) and (11), and initial states (12). More-
over, errors are expected in the modeling because of the scatter
in the MFDs, mainly in the congested regime and of the de-
mand profile. Therefore, the optimal control problem is solved
by applying the MPC approach, which can handle the state
and control constraints and the errors in the MFDs modeling.
Furthermore, MPC is a real-time implementable solution that
can be utilized for real-time urban traffic applications.
MPC is a form of rolling horizon control in which the current
control inputs are obtained by solving a finite-horizon open-
loop optimal control problem at each time step, with a current
state feedback from the plant being the initial state of the
model, see Fig. 2. The open-loop optimization problem yields
a sequence of optimal control inputs after several iterations
of solving nonlinear programming, and the first control action
in this sequence is applied to the plant, then the procedure is
carried out again.
This scheme of feedback control, i.e., the feedback loop
of states from the plant to the model as initial states for the
optimization, can handle errors between the prediction model
and the plant.
A. Two-Region MFDs Prediction Model
and Optimization Problem
The MPC controller obtains the optimal control sequence
for the current horizon by solving an optimization problem
formulated with the prediction model; see the bottom of Fig. 2.
The prediction model used in the MPC scheme is formulated
with (2)–(5). The dynamic equations predict the evolution of
accumulations for the two regions with MFDs, given the initial
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4 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 2. MPC scheme for the two-region MFDs system.
accumulations and future values of perimeter control inputs and
demand.
In this paper, we follow the direct methods for solving the
optimization problem (other solution methods include dynamic
programming and indirect methods). The direct methods are
most commonly used because of their applicability and robust-
ness, where their basic principle is “first discretize and then
optimize”. These methods can handle inequality constraints and
use state-of-the-art methods for nonlinear problem solvers.
The open-loop optimal control problem is solved using the
direct sequential method, also referred to as single shooting or
control vector parameterization in the literature, e.g., [33] and
[34]. The direct sequential method transcripts the open-loop
optimal control problem into a finite-dimensional nonlinear
problem through the discretization of the control variables
only with piecewise constant controls, whereas the ODEs are
embedded in the nonlinear problem, i.e., numerical integration
is used between the time steps. A schematic description of the
direct sequential method is shown in Fig. 3. Note the continuous
dynamics of the state variables n
ij
(t), i, j = 1, 2. Let N
p
(−)
be the finite-dimensional horizon, which starts from the current
control step k
c
. At each discrete time step k, k
c
≤ k ≤ k
c
+
N
p
− 1, there are two perimeter control inputs, u
12
(k) and
u
21
(k), which are assumed to be constant during the time
period t
k−1
≤ t ≤ t
k
. For online computational complexity, the
number of control inputs that should be optimized are reduced
to a horizon that is smaller than N
p
, called the control horizon
N
c
, where N
c
≤ N
p
. The rest of the control variables, u
12
(k)
and u
21
(k) for k
c
+ N
c
≤ k ≤ k
c
+ N
p
− 1, are assumed to be
equal to the control inputs at the end of the control horizon.
Following the direct sequential method, the control vector is
discretized, and the two-region MFDs optimal control problem
(1)–(12) is approximated by a finite-dimensional nonlinear
programming problem in the piecewise constant control inputs.
First, the equations of the prediction model (2)–(5) are rewritten
Fig. 3. Direct sequential method for solving the open-loop optimization
problem.
in a compact form with discrete control variables at time step
k
c
with finite-dimensional N
p
as follows:
dn(t)
dt
= f (n(t), u(k), q(t))
t
k−1
≤ t ≤ t
k
k = k
c
,k
c
+ 1,...,k
c
+ N
p
− 1 (13)
where n (t)=[n
11
(t),n
12
(t),n
21
(t),n
22
(t)]
T
, q(t)=[q
11
(t),
q
12
(t),q
21
(t),q
22
(t)]
T
, and u(k)=[u
12
(k),u
21
(k)]
T
. Then,
the Lagrange form (1) is transferred into the Mayer form by
introducing an additional state variable z(t) and an additional
differential equation dz(t)/dt. Moreover, the path constraints
(6)–(9) must hold for all t (continuous variable), where t
0
≤
t ≤ t
f
; hence, the number of constraints would be infinite.
However, several methods are efficient in dealing with path
constraints in the sequential method, e.g., transcription as in-
tegral constraints. The optimization problem is now formulated
as follows:
min
u(k
c
),u(k
c
+1),...,u(k
c
+N
p
−1)
−z(t
k
c
+N
p
−1
) (14)
subject to
dn(t)
dt
= f(n(t), u(k), q(t)) (15)
dz(t)
dt
= M
11
(t)+M
22
(t) (16)
u
min
≤ u(k) ≤ u
max
(17)
where t
k−1
≤t≤t
k
k =k
c
,k
c
+1,...,k
c
+N
p
−1
u(k)=u(k
c
+N
c
−1) k =k
c
+N
c
,...,k
c
+N
p
−1 (18)
k
c
+N
p
−1
k=k
c
t
k
t
k−1
max {0; −n
11
(t) − n
12
(t)}
2
dt ≤ (19)
k
c
+N
p
−1
k=k
c
t
k
t
k−1
max {0; −n
21
(t) − n
22
(t)}
2
dt ≤ (20)
k
c
+N
p
−1
k=k
c
t
k
t
k−1
max{0; n
11
(t)+n
12
(t)−n
1,jam
}
2
dt≤ (21)
k
c
+N
p
−1
k=k
c
t
k
t
k−1
max{0; n
21
(t)+n
22
(t)−n
2,jam
}
2
dt≤ (22)
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GEROLIMINIS et al.: OPTIMAL PERIMETER CONTROL FOR TWO URBAN REGIONS WITH MFDs 5
where u
min
=[u
min
,u
min
]
T
, and u
max
=[u
max
,u
max
]
T
.
Note that the path constraints (6)–(9) are reformulated as inte-
gral constraints (19)–(22), respectively, with relaxation, where
>0 is a small nonnegative constant.
B. Two-Region MFDs Plant
The dynamic equations of the two-region MFDs plant differ
from the prediction model (2)–(5), because they include errors
in the MFDs for both regions and noise in the traffic demand,
see the top of Fig. 2 for an illustration. Hence, the evolutions of
the accumulations over time are not the same for the prediction
model and the plant, which are considered to have different
magnitudes and profile (biased and unbiased). Thus, the plant
and the model accumulation, MFD, and demand can signifi-
cantly differ.
1) Errors in MFDs: Let us denote the MFDs with errors
for R
1
and R
2
,by
˜
G
1
and
˜
G
2
, respectively. The errors in the
MFDs result in errors in accumulations of the plant, which
are denoted by ˜n
ij
(t), i, j = 1, 2, to distinguish them from
the accumulations of the prediction model n
ij
(t). Note that
˜n
i
(t)=
j
˜n
ij
(t). The variance of the MFD increases with the
accumulations in the region, as described in [10]; see the top of
Fig. 2. Reasons for this variance are asymmetric OD and route
choices, which increase the heterogeneity in the distribution of
congestion within a region. It is assumed that the variance of
the MFD is uniformly distributed, where the error at step k is
calculated at the time instant t
k−1
as
ε (˜n
1
(t
k−1
)) ∼ U (−α
1
· ˜n
1
(t
k−1
),α
1
· ˜n
1
(t
k−1
)) (23)
ε (˜n
2
(t
k−1
)) ∼ U (−α
2
· ˜n
2
(t
k−1
),α
2
· ˜n
2
(t
k−1
)) (24)
where α
1
and α
2
(1/s) are given parameters. It is assumed that
the errors ε(˜n
1
(t
k−1
)) and ε(˜n
2
(t
k−1
)) (veh/s) are constant
during the time step t
k−1
≤ t ≤ t
k
; therefore, the MFDs of the
plant
˜
G
1
and
˜
G
2
for t
k−1
≤ t ≤ t
k
are
˜
G
1
(˜n
1
(t)) = G
1
(˜n
1
(t)) + ε(˜n
1
(t
k−1
)) (25)
˜
G
2
(˜n
2
(t)) = G
2
(˜n
2
(t)) + ε(˜n
2
(t
k−1
)) . (26)
2) Unbiased and Biased Noise in the Demand: The follow-
ing two different types of noise in the demand are considered:
1) unbiased noise with Gaussian distribution and 2) biased
noise with a sudden jump in the demand profile for a time
period. In both cases, let us denote the traffic demand q (t)
with noise as
˜
q(t)=[˜q
11
(t), ˜q
12
(t), ˜q
21
(t), ˜q
22
(t)]
T
. Unbiased
demand noise represents random and recurrent variations of the
demand from day to day because of travel patterns, whereas
biased demand noises might represent cases of nonrecurrent
events (e.g., special events or accidents).
The unbiased noise in the demand is assumed to have
Gaussian distribution as follows:
˜q
ij
(t)=max
q
ij
(t)+N (0,σ
2
ij
), 0
(27)
where i, j = 1, 2, and σ
2
ij
(veh
2
/s
2
) is the variance for the
traffic demand q
ij
(t).
Substituting the MFDs with errors (25)–(26) and the demand
with noises
˜
q(t) in the dynamic equations (2)–(5), we get the
two-region MFDs plant in a compact form (see also Fig. 2) as
d
˜
n(t)
dt
=
˜
f (
˜
n(t), u(k),
˜
q(t), ε(k)) (28)
where ε(k)=[ε(˜n
1
(t
k−1
)),ε(˜n
2
(t
k−1
))]
T
.
C. Greedy Controller (GC)
To investigate and estimate the performance of the MPC
controller, comparison results are done with a GC for different
levels and types of errors. GC is a state feedback control in
which its policy i s determined by the current accumulations
n
1
(t) and n
2
(t).Letn
1,cr
and n
2,cr
(veh) be the accumu-
lations that maximize G
1
and G
2
, respectively. GC is de-
signed according to the following policy: if both regions are
uncongested, i.e., n
1
(t) ≤ n
1,cr
and n
2
(t) ≤ n
2,cr
, then both
controllers should maximize the transfer flows, and therefore,
[u
12
(t),u
21
(t)] = [u
max
,u
max
]. If one region is congested and
the other region is uncongested, i.e., n
1
(t) ≤ n
1,cr
and n
2
(t) >
n
2,cr
,orn
1
(t) >n
1,cr
and n
2
(t) ≤ n
2,cr
, then the controllers
should minimize the transfer flow to the congested region and
maximize the transfer flow to the uncongested region. If both
regions are congested, i.e., n
1
(t) >n
1,cr
and n
2
(t) >n
2,cr
,
then the controllers should minimize the transfer flow to the
“more congested” region and maximize the transfer flow to
the “less congested” region; for example, if n
1
(t)/n
1,jam
>
n
2
(t)/n
2,jam
, then R
1
is more congested than R
2
, and there-
fore, [u
12
(t),u
21
(t)] = [u
max
,u
min
]. The GC law is summa-
rized in Table I.
D. Tuning the Prediction and Control Horizon Parameters
The performance of the MPC controller is affected by
the prediction horizon N
p
and the control horizon N
c
.The
prediction horizon N
p
should be large enough such that the
model can accurately predict the accumulations of the plant
corresponding to the control inputs. Increasing the prediction
horizon improves the performances of the MPC controller;
however, a large N
p
increases the optimization computing
time, which may add some barriers for online implementation,
i.e., the control actions cannot be implemented in the current
step if the computing time that corresponds to a large N
p
is
larger than the time duration of the control time step. Similar
considerations with regard to the tradeoff between computation
complexity and results should accurately be done for the control
horizon N
c
.
The perimeter controllers can be actuated by signalized inter-
sections that are placed in the border between the two regions of
the urban network, i.e., the perimeter control sequences can be
applied by choosing appropriate timing plans for the signalized
intersections. The effect of perimeter control to the rest of the
network and its MFDs will be discussed later. Let us assume
that the signalized intersections have a fixed common cycle
length, e.g., is equal to 60 (s). Then, the time duration of the
time step k
c
is set to be equal to the length of the cycle, i.e.,
t
k
− t
k−1
= 60. This duration is much larger than the time
