Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban

Region

Jack Haddad and Ilya Ioslovich

Technion Sustainable Mobility and Robust Transportation (T-SMART) Laboratory,

Faculty of Civil and Environmental Engineering, Technion–Israel Institute of Technology, Haifa, Israel

Keywords:

Perimeter Trafﬁc Control, Macroscopic Fundamental Diagram, Optimal Control.

Abstract:

Trafﬁc ﬂow control has motivated many researchers since early decades of the 19th century. Recently, the

concept of a perimeter trafﬁc control for an urban region has been strengthened by a series of works, which

have shown that a perimeter controller, located at a region border, can manipulate the transfer ﬂows across

the border to maximize the total outﬂow of the region. The macroscopic fundamental diagram (MFD), that

relates average ﬂow with accumulation, is used to model the trafﬁc ﬂow dynamics in the region. Assuming

that the control inputs of the cross-border ﬂows are coupled, i.e. the border is always utilized over time for

transferring ﬂows by one of the two directions (from and towards the region), and that the urban region has

two trafﬁc ﬂow demands generated inside the region with internal and external destinations, and a generated

trafﬁc ﬂow outside the region with a destination to the region, the explicit formulation of the optimal feedback

control policy and a proof of optimality are provided. The proof is based on the modiﬁed Krotov-Bellman

sufﬁcient conditions of optimality, where the upper and lower bounds of state variables are calculated.

1 INTRODUCTION

In the last decade, network trafﬁc ﬂow modeling with

the Macroscopic Fundamental Diagram (MFD) repre-

sentation has intensively attracted the trafﬁc ﬂow and

control researchers. The MFD simpliﬁes the mod-

elling task of the trafﬁc ﬂow dynamics for large-scale

urban networks, as it provides aggregate relationships

between trafﬁc variables at an urban region.

The MFD provides a unimodal, low-scatter rela-

tionship between network vehicle density (veh/km)

or accumulation (veh) and network space-mean ﬂow

(outﬂow) (veh/hr) for different network regions, if

congestion is roughly homogeneous in the region.

The physical model of the MFD was initially pro-

posed by (Godfrey, 1969), but the theoretical ele-

ments for the existence of the MFD were provided

later by (Daganzo, 2007). The MFD was ﬁrst ob-

served with dynamic features in congested urban net-

work in Yokohama by (Geroliminis and Daganzo,

2008), and investigated using empirical or simulated

data by (Buisson and Ladier,2009; Ji et al., 2010; Ma-

zloumian et al., 2010; Daganzo et al., 2011; Gayah

and Daganzo, 2011; Zhang et al., 2013; Mahmassani

et al., 1987; Olszewski et al., 1995) and others.

Homogeneous networks with small variance of

link densities have a well-deﬁned MFD (as illustrated

in Fig. 1(a)), i.e. low scatter of ﬂows for the same

densities (or accumulations), (Geroliminis and Sun,

2011b; Mazloumian et al., 2010; Daganzo et al.,

2011; Knoop et al., 2013; Mahmassani et al., 2013).

Note that heterogeneous networks might not have

well-deﬁned MFDs, mainly in the decreasing part of

the MFD, as the scatter becomes higher as accumu-

lation increases and hysteresis phenomena has been

found to exist (Daganzo et al., 2011; Buisson and

Ladier, 2009; Saberi and Mahmassani, 2012; Geroli-

minis and Sun, 2011a). As a solution, these networks

might be partitioned into more homogeneous regions

with small variances of link densities, (Ji and Geroli-

minis, 2012). Note that the network topology, the sig-

nal timing plans of the signalized intersections inside

the region, and the infrastructure characteristics af-

fect the shape of the MFD, see e.g. (Geroliminis and

Boyacı, 2012).

The MFD concept has been utilized to introduce

control policies that aim at improving mobility and

decreasing delays in large urban networks, (Daganzo,

2007; Haddad and Geroliminis, 2012; Geroliminis

et al., 2013; Hajiahmadi et al., 2013; Haddad et al.,

2013; Aboudolas and Geroliminis, 2013; Keyvan-

Ekbatani et al., 2012; Knoop et al., 2012; Zhang et al.,

Haddad J. and Ioslovich I..

Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban Region.

DOI: 10.5220/0005009800140020

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 14-20

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

2013). E.g. perimeter control strategies, i.e. ma-

nipulating the transfer ﬂows at the perimeter border

of the urban region, have been introduced for single-

region cities in (Daganzo, 2007; Keyvan-Ekbatani

et al., 2012; Shraiber and Haddad, 2014), and for

multi-region cities in (Geroliminis et al., 2013; Had-

dad et al., 2013; Hajiahmadi et al., 2013; Aboudolas

and Geroliminis, 2013). In this paper, we deal with

perimeter control for a single urban region modelled

by an MFD.

Different control approaches have been proposed

to solve perimeter control problems for single-region

cities. (Daganzo, 2007) has presented a bang-bang

control as an optimal control policy for an urban re-

gion. A Proportional-Integrator (PI) perimeter con-

troller has been designed for an urban region in

(Keyvan-Ekbatani et al., 2012). The formulated non-

linear system is linearized around a priori known set-

point chosen carefully within a value range in the un-

congested regime of the MFD having positive slope

and close to the critical state (total time spent) of the

MFD function. The work in (Keyvan-Ekbatani et al.,

2012) aims at regulating the dynamic system around

the desired chosen set-point, at which the system’s to-

tal time spent is minimized, in other words, the state

reference is the same as the set-point. Moreover, the

work in (Keyvan-Ekbatani et al., 2012) do not allow

direct consideration of the control constraints, but im-

pose them after the design process, e.g. adjusting or

ﬁne-tuning the controller gains.

In (Shraiber and Haddad, 2014), a robust perime-

ter controller has been designed for an urban region

with the MFD representation. The designed controller

is a ﬁxed PI-controller with proportional K

and inte-

grator K

gains, which stabilizes the linearized system

against MFD and parameter uncertainties. The ro-

bust controller is also designed to handle control con-

straints within the design level in a systematic way.

The results showed that the controller has performed

well for the whole state set, and not necessary for a

value range nearby a set-point.

In this paper, the optimal feedback control for a

perimeter trafﬁc ﬂow at an urban region is derived,

and a proof of optimality is provided with the help

of the modiﬁed Krotov-Bellman sufﬁcient conditions

of optimality. The region is assumed to be a ho-

mogeneous region having a well-deﬁned MFD with

two trafﬁc ﬂow demands generated inside the region

with internal and external destinations, and a gener-

ated trafﬁc ﬂow outside the region with a destination

to the region.

2 OPTIMAL PERIMETER

CONTROL: PROBLEM

DEFINITION

This paper deals with a perimeter control problem for

a homogeneous urban region having a well-deﬁned

MFD, schematically shown in Fig. 1. The ﬂow dy-

namic equations for a homogeneous urban region

have been already formulated in (Shraiber and Had-

dad, 2014), and they are brieﬂy presented as follows.

There are two state variables denoted by n

(t) and

(t) (veh), which respectively represent the num-

ber of vehicles traveling in the region with destina-

tion inside and outside the region at time t. The to-

tal accumulated number of the vehicles in the region

is n

(t) = n

(t) + n

(t). The MFD links the accu-

mulation, n

(t), and trip completion ﬂow, deﬁned as

the output ﬂow of the region. The MFD provides a

low-scatter relationship, if congestion is roughly ho-

mogeneous in the region. The MFD is denoted by



(t)



(veh/s), and it is assumed to be Lipschitz,

continuous, non-negative, and unimodal. This as-

sumption is based on many simulation and empiri-

cal results, e.g. in (Geroliminis and Daganzo, 2008).

The MFD is deﬁned as the trip completion ﬂow for

the region at n

(t): (i) the sum of a transfer ﬂow,

i.e. trips from the region with external destination

(outside the region), plus (ii) an internal ﬂow, i.e.

trips from the region with internal destination (in-

side the region). The transfer ﬂow is calculated cor-

responding to the ratio between accumulations, i.e.

(t)/n

(t)·G



(t)



, while the internal ﬂow is cal-

culated by n

(t)/n

(t) · G



(t)



The trafﬁc ﬂow demands generated in the region

with internal and external destinations are respec-

tively denoted by q

(t) and q

(t) (veh/s), while

(t) (veh/s) denotes a generated trafﬁc ﬂow outside

the region with destination to the region, as schemat-

ically shown in Fig. 1(b). Following (Shraiber and

Haddad, 2014), a perimeter control is introduced on

the border of the urban region, where its inputs are

coupled u(t) (− ) and 1 − u(t) and control the ratios

of ﬂows, 0 ≤ u(t) ≤ 1, that cross the border from in-

side to outside and from outside to inside the region at

time t, respectively, see Fig. 1(b). It is also assumed

that the perimeter control will not change the shape of

the MFDs. Note also that the internal ﬂow cannot be

controlled or restricted.

The vehicle-conservation equations in the urban

regions are given as follows (same equations (1) and

(2) in (Shraiber and Haddad, 2014)):

OptimalFeedbackControlforaPerimeterTrafﬁcFlowatanUrbanRegion

(t)) (veh/s)

Trip completion ﬂow

1,max

∗

Accumulation, n

(t) (veh)

(a)

1 − u(t)

(t)

u(t)

(b)

Figure 1: (a) A schematic MFD which is Lipschitz, con-

tinuous, non-negative, and unimodal function, (b) An urban

region with three trafﬁc demand q

(t), q

(t), and

a perimeter controller with inputs u(t) and 1− u(t).

(t)

= q

(t) + (1− u(t)) · q

(t)

−

(t)

· G



(t)



(1)

(t)

= q

(t) −

(t)

· G



(t)



· u(t). (2)

Let us now rewrite (1) to have a state equation cor-

responding to variable n

(t) instead of n

(t). The

reason for that is only technical as this simpliﬁes the

mathematical proofs given later in Section 3. By sum-

ming (1) and (2) and substituting n

(t) = n

(t) −

(t), one gets

(t)

= q

(t) + q

(t) −

(t) − n

(t)

· G



(t)



−



(t) +

(t)

· G



(t)





· u(t).

(3)

The optimal control problem aims at manipulating the

control input u(t) to optimize an objective J, subject

to (2) and (3). There are a variety of criteria that can

be chosen, e.g. the throughput of the transportation

network and the total network delay. In this paper,

the throughput of the transportation network in the re-

gion is chosen, which is deﬁned as the total number

of vehicles that complete their trips and reach their

destination during the time interval [t

], i.e.



(t)



dt, (4)

where t

and t

(s) are the starting and ﬁnal times of

the control process. Note that the deﬁned problem

here is an optimal control problem, while in (Shraiber

and Haddad, 2014) the problem is deﬁned as a regu-

lator control problem.

3 OPTIMAL CONTROL DESIGN

The MFD function G



(t)



is assumed to be uni-

modal with a single maximum value at n

∗

(veh), see

also Fig. 1(a), i.e.

∗

= argmax

(t)



(t)



. (5)

Let us denote u

(t) (−) as the steady-state control

input, which corresponds to a steady-state condition

/dt = 0, at n

(t) = n

∗

, i.e. the steady state control

input u

(t) is calculated from (3) with dn

/dt = 0 and

(t) = n

∗

, as follows:

(t) =



(t) + q

(t) −

∗

− n

(t)

∗

· G

∗

)





(t) +

(t)

∗

· G

∗

)



(6)

Note that u

(t) is a time dependent corresponding to

the trafﬁc demands q

(t), q

(t), and q

(t).

Theorem 3.1. The optimal feedback control u

∗

)

for the problem P1: max

u(t)

subject to (2) and (3)

is as follows:

If n

(t) 6= n

∗

, then

∗

) =

(

0 ∀n

(t) < n

∗

1 ∀n

(t) > n

∗

(7)

otherwise (n

(t) = n

∗

) =











(t) 0 ≤ u

(t) ≤ 1,

0 u

(t) < 0,

1 u

(t) > 1.

(8)

Proof. The proof is based on the unimodality assump-

tion of the function G

(t)). The unimodal func-

tion G

(t)) has the point-wise maximum for each

point t at n

(t) =

(t) if n

(t) ≤ n

∗

and n

(t) is an

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

upper bound of n

(t). Similarly, G

(t)) has the

point-wise minimum for each point t at n

(t) = n

(t)

if n

(t) ≥ n

∗

and n

(t) is a lower bound of n

(t). The

upper bound

(t) and the lower bound n

(t) have to

be found with respect to the dynamical state equa-

tions (2) and (3) with initial conditions n

) = n

1,0

) = n

12,0

(see Lemmas 3.3 and 3.4). To com-

plete the proof we shall use the sufﬁcient global op-

timality conditions in the form of modiﬁed Krotov-

Bellman conditions (see sections 3.1 and 3.2) and

prove Lemmas 3.2, 3.3, and 3.4 in Section 3.3.

According to (8), the optimal feedback control for

(t) = n

∗

is u

∗

) = u

(t), where u

(t) has to sat-

isfy the control constraint 0 ≤ u

(t) ≤ 1. However,

if u

(t) < 0 or u

(t) > 1, then the optimal feedback

control is respectively u

∗

) = 0 or u

∗

) = 1. Note

that in the latter two cases the state cannot be kept

at n

∗

, since if u

(t) < 0 the variable n

(t) will de-

crease from n

(t) = n

∗

even with all feasible u(t), but

the minimum decrease is achieved by u(t) = 0, and if

(t) > 1 then n

(t) will increase for all feasible u(t),

but the minimum increase is achieved by u(t) = 1.

This is explained as follows. Let us ﬁrst respectively

denote a(t) and b(t) as follows:

a(t) = q

(t) + q

(t) −

∗

− n

(t)

∗

· G

∗

(9)

b(t) = q

(t) +

(t)

∗

· G

∗

), (10)

then, (3) is rewritten as

(t)

= a(t) − b(t) · u(t) (11)

where b(t) ≥ 0, and a(t)−b(t)· u

(t) = 0. Therefore,

if u

(t) < 0 ≤ u(t) then dn

/dt = a(t) − b(t) · u(t) <

a(t) − b(t) · u

(t) = 0, and if u

(t) > 1 ≥ u(t) then

/dt = a(t) − b(t) · u(t) > a(t) − b(t) · u

(t) = 0.

3.1 Modiﬁed Krotov-Bellman Sufﬁcient

Conditions of Optimality

The Krotov-Bellman sufﬁcient conditions of optimal-

ity are summarized as follows. The reader can refer

to (Krotov, 1996) for further information. Given a dy-

namic system

= f(t, x,u), (12)

with state variables x(t), control inputs u(t), initial

conditions x(t

) = x

, and the following objective

function

minJ =

(t,x,u)dt, (13)

one can construct a function R(t, x,u) as follows

R(t, x,u) =

∂V

∂x

f(t, x,u) − f

(t,x,u) +

∂V

∂t

, (14)

where V(t,x) is assumed to be a continuous and dif-

ferentiable function. Taking into account that the full

time derivative of V(t,x) with respect to (12) is

∂V

∂x

f(t, x,u) +

∂V

∂t

, (15)

and substituting (15) and (14) in (13), one gets

J = V(t

,x(t

)) −V(t

) −

R(t, x,u)dt. (16)

The sufﬁcient Krotov-Bellman conditions of optimal-

ity are as follows: if there exists a pair (x

∗

) such

that x

∗

is the solution of the dynamic system

∗

= f(t, x

∗

) (17)

over the time interval [t

], and the following prop-

erties

∗

= argsup

R(t, x,u),

R(t, x,u

∗

) = µ(t), (18)

Θ = V(t

,x(t

)) = Constant,

hold, then this pair (x

∗

) is a global optimum solu-

tion. Note that µ(t) is any measurable bounded func-

tion of t. According to these sufﬁcient conditions of

optimality, the problem is reduced to a solution of

the nonlinear Krotov-Bellman PDE for the function

V(t,x).

In this paper, the modiﬁed Krotov-Bellman con-

ditions are proposed, where the maximization of

R(t, x,u) over u in (18) is replaced by the maximiza-

tion of R(t, x,u) over x, as follows:

∗

= argsup

R(t, x,u),

R(t, x

∗

,u) = µ(t) , (19)

Θ = V(t

,x(t

)) = Constant.

Note that in both variants (18) and (19), the resulting

function R(t,x,u) after maximization, i.e. R(t, x,u

∗

)

and R(t,x

∗

,u), respectively, will be a function of time

t only.

3.2 Application to the Maximum

throughput Objective

Applying the modiﬁed Krotov-Bellman conditions to

the problem (2)–(4), one gets

OptimalFeedbackControlforaPerimeterTrafﬁcFlowatanUrbanRegion

sup

(t),n

(t)



∂V

∂n



(t) + q

(t)

−

(t) − n

(t)

(t)) −



(t) +

(t)

· G

(t))



· u(t)



∂V

∂n

(t)



(t) −

(t)

· G

(t)) · u(t)



+ G

(t)) +

∂V

∂t



= µ(t) ,

(20)

Θ = V(t

),n

)) = Constant. (21)

Note that (20) and (21) are without taking into ac-

count the upper and lower bound constraints on state

variable n

(t), i.e.

(t) = n

(t) − n

(t) ≤ 0, (22)

(t) = n

(t) − n

(t) ≤ 0. (23)

Let

and λ

be the Lagrange multipliers for

(22) and (23), respectively. Now, let us choose

V(t,n

(t),n

(t)) = C, where C is a constant. Then,

imposing state constraints (22) and (23) on (20) and

(21) with V(t, n

(t),n

(t)) = C, one gets

sup

(t)

(t)) −

· g

(t) − λ

· g

(t)] = µ(t),

(24)

Θ = C. (25)

Note that Lagrange multipliers are taken with sign

minus because of maximization of J

. According to

Karush-Kuhn-Tucker (KKT) conditions all Lagrange

multipliers are non-negative and may have positive

values only when the corresponding constraint is

binding, i.e. non-redundant. From maximization of

(24), one gets

∂G

∂n

−

+ λ

= 0. (26)

This implies that n

(t) =

(t) when

∂G

∂n

> 0, i.e.

when n

(t) < n

∗

(because G



(t)



is assumed to

be unimodal with a single maximum value at n

∗

), and

(t) = n

(t) when

∂G

∂n

< 0, i.e. when n

(t) > n

∗

Note that Lagrange multipliers for upper and lower

constraints cannot be non-zero simultaneously, and

because of unimodality the only point when

∂G

∂n

= 0

is the point n

(t) = n

∗

3.3 Upper and Lower Bounds of State

Variable n

(t)

Recall that to complete the proof of Theorem 3.1, the

upper bound n

(t) and the lower bound n

(t) have to

be found with respect to the dynamical state equa-

tions (2) and (3) with initial conditions n

) = n

1,0

) = n

12,0

. In this section, it is explained how to

achieve the upper bound

(t) and the lower bound

(t), according to Lemmas 3.3 and 3.4, respectively.

But ﬁrst Lemma 3.2 is presented, which is taken from

(Krotov et al., 1971) and utilized to prove Lemmas 3.3

and 3.4.

Lemma 3.2. Consider an ODE system dx/dt =

f(x,u,t), where f(x,u,t) is Lipschitz and continuous

vector-function, x(t) = [x

(t),x

(t),.. . ,x

(t)]

are

the state variables, and u(t) are measurable bounded

control inputs. The upper bound of the solution x(t)

with initial conditions x(t

) = x

is denoted as

x(t).

Each component i of this bound can be calculated ac-

cording to the following equation

= sup

u,x

,...,x

i−1

i+1

,...,x

i−1

i+1

,...,x

,u,t) , (27)

with the initial condition

) = x

i,0

, where f

(·) is

the i-component of the vector-function f(·). The lower

bound x

(t) can be calculated in the same way by just

replacing sup by inf in (27).

Proof. The proof is a straightforward, however, it is

not presented in this paper. The reader can refer to

(Krotov et al., 1971) for an explicit proof.

Lemma 3.3. The upper bound for state variable n

(t)

is achieved with control input u(t) = 0.

Proof. Let us start with state variable n

(t). From

(2), it is clear that the second term in the right-hand

side is a non-positive and in particular it is equal to

zero for u(t) = 0. Therefore, the supremum of the

right-hand side will be achieved for u(t) = 0 and it

depends only on t. According to Lemma 3.2, an upper

bound

(t) is achieved with u(t) = 0. Now, let us

consider state variable n

(t). From (3) one can see

that the supremum over u(t) is achieved for u(t) =

0, then the right-hand side is a function of t and the

variables n

(t), n

(t). It has been shown that with

u(t) = 0 from (2) it follows that n

(t) =

(t). After

substitution n

(t) =

(t) in the right-hand side of

(3), it follows according to Lemma 3.2 that an upper

bound n

(t) = n

(t) is obtained for u(t) = 0.

Lemma 3.4. The lower bound for state variable n

(t)

is achieved with control input u(t) = 1.

Proof. The inﬁmum over u(t) of the right-hand side

of (3) is achieved for u(t) = 1. Substituting u(t) = 1

into (3), one gets

= q

(t) + q

(t) − G

(t)). (28)

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

It follows from Lemma 3.2 that the solution of (28) is

a lower bound n

(t).

Lemmas 3.3 and 3.4 complete the proof of The-

orem 3.1. One can add that if at some point in time

t = t

, the optimal trajectory will go left or right from

the value n

) = n

∗

, then we can split the problem

into two pieces (namely from t = t

to t = t

and

from t = t

to t = t

) and build new upper or lower

bounds respectively from the initial point n

) and

) = n

∗

4 CONCLUSIONS

The analytical solution for the optimalperimeter feed-

back control with the maximum throughput criterion

in an urban region has been derived and described.

The modiﬁed Krotov-Bellman sufﬁcient conditions of

optimality have been utilized for the proof of optimal-

ity. The resulting optimal control policy is oriented to

keep the state variable, i.e. the total number of the

moving vehicles in the region, as close as possible to

the critical accumulation, n

∗

, where the MFD value

is maximized. Though this optimal solution is in-

tuitively expected here it is rigorously proven. The

numerical simulations and comparison with existing

practices will be done in consequent papers.

ACKNOWLEDGEMENTS

This research was supported by Carl E. Schustak en-

ergy research and development fund.

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