Urban transportation network analysis from a thermodynamic perspective

Accepted Manuscript

Urban Transportation Network Analysis from a Thermodynamic Perspective Mohamad Sleiman, Rachid Bouyekhf, Adbellah El Moudni PII: DOI: Reference:

S0016-0032(18)30628-8 https://doi.org/10.1016/j.jfranklin.2018.09.033 FI 3651

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

9 May 2017 10 September 2018 25 September 2018

Please cite this article as: Mohamad Sleiman, Rachid Bouyekhf, Adbellah El Moudni, Urban Transportation Network Analysis from a Thermodynamic Perspective, Journal of the Franklin Institute (2018), doi: https://doi.org/10.1016/j.jfranklin.2018.09.033

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Urban Transportation Network Analysis from a Thermodynamic Perspective Mohamad Sleimana,∗, Rachid Bouyekhfa , Adbellah El Moudnia a

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Nanomedecine-Imagery-Therapeutics Lab, UTBM, Univ. Bourgogne Franche-Comt´e, 90010 Belfort Cedex, France

Abstract

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The paper explores the application of thermodynamic formalism to model and control transportation networks. Specifically, by considering the vehicles as the abstract energy supplied to the system, we show in certain circumstances that certain thermodynamic concepts such as temperature, thermal capacity and thermal equilibrium can have the corresponding notions in transportation context. In addition, despite the lack of a natural principle in transportation context that corresponds to the second law of thermodynamic as we will show, the most important thermodynamic notion, which is the entropy, can be also defined in order to measure the disorder of transportation systems. It is then shown that the state when all lanes have the same occupancy corresponds to the thermal equilibrium arising in isolated thermodynamic system. This equilibrium occupancy leads to a minimum entropy corresponding to a minimal disorder. Besides, by taking the transportation entropy as the storage function, a robust dissipativity based control strategy is presented to reduce the disorder and render the system better organized. Finally, an example is worked out to illustrate the results.

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1. Introduction

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Keywords: Transportation systems, thermodynamic systems, entropy, majorization theory, dissipativity theory, Linear Matrix Inequality (LMI)

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When we want to study transportation networks, we instantly meet computational difficulties that grow if the scale of the systems increases. This situation impede us from establishing clearly the impact of various factors on the behavior of the system. Transportation systems can be analyzed in various ways, depending on the complexity of the problem, but the main objective is to reduce complex computation and to develop techniques that can be easily implemented on a computer, to make possible an automatic analysis of modeled systems. This situation has motivated many researchers to develop methods and techniques in order to attenuate difficulties involved in transportation systems. In particular, the traffic flow theory has been the most widely used [1]. It ∗

Corresponding author. Email addresses: [email protected] (Mohamad Sleiman), [email protected] (Rachid Bouyekhf), [email protected] (Adbellah El Moudni) Preprint submitted to Journal of The Franklin Institute

October 18, 2018

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considers the vehicle movements as liquid flows, which matches the general impression of the traffic in modern cities. Hence, the traffic flow theory can effectively describe the macroscopic traffic phenomenons and has led to several well-known traffic signal control strategies, like TRANSYT [2], SCOOT [3], TUC [4, 5], etc. The traffic flow theory has various approaches for describing the behavior of such flows. Among them, the queueing theory has been applied to study the behavior of traffic flows near certain sections (e.g. traffic bottlenecks) where the demand exceeds the available capacity [6]. The human factors have also been considered in traffic flow theory. Specifically, the car following model examines the manner in which individual vehicles follow one another [7, 8]. This approach connects the microscopic behavior of individual vehicles and the macroscopic features of traffic flows. In addition, the kinematic wave theory has been also very popular to study the flow-density relationship from microscopic point of view [9, 10]. Its first-order approximation has led to the Cell Transmission Model (CTM) [11], which shows high accuracy in the simulation. Hence, CTM has been used in traffic estimation [12] and traffic signal control [13]. Beside the traffic flow theory, the applications of Petri nets (PNs) in modelling and control of transportation systems have been also conducted for over a decade [14, 15]. This graphical tool is useful for analyzing performances and assisting intelligent traffic control. Moreover, due to the rapid increase of traffic demands, many interesting studies related to the traffic signal control in the oversaturated situation can be found in [16], [17] and [18]. Furthermore, a number of numerical solution algorithms have been used, like genetic algorithms [19] and ant colony optimization [20]. However, these optimization approaches have been faced some difficulties, especially the real-time feasibility. In addition, the concept of NFD (network fundamental diagram) has been recently an issue of many investigations [21, 22]. The former used the NFD concept in order to propose a control strategy that maximizes the network outflow. The latter exploited the urban NFD concept and the gating measures in order to find a feedback control, which is able to improve mobility in the network. Another notable work in the field is the paper [23] where the authors developed a linear model predictive controller by using the link transmission model. On the other hand, the use of statistical thermodynamic to treat the nonphysical systems has attracted an attention since a long time in [24], [25] and [26]. In the first two papers, the authors attempted to maximize the Shannon entropy by using a probabilistic approach in order to attain the equilibrium in the communication network [24] or in the transportation network [25]. In the third paper [26], the physical entropy has been used in order to find a physical relevant solution in the case of non convergence problem. Besides the thermodynamic theory has been formulated for describing the state of the transportation network during the congestion period [27, 28] or for calculating some parameters like traffic flows and vehicle speeds [29]. However, in all these papers, the traffic light control has not been considered despite the fact that this control is the most important tool to manage the traffic flows in a transportation network. More recently, some works have established a similarity between transportation and thermodynamic systems [30, 31]. The authors in these works have just introduced the thermodynamic entropy in the transportation field and they have developed a traffic light feedback control to manage the traffic. But, the lacks in these papers can be summarized in three main points. Firstly, the authors have used the entropy in the transportation system without any mathematical proof demonstrated the legitimacy of its use. Secondly, the output flows to the outside are considered 2

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controllable by the effective green times. But, in reality, they represent together with the input flows the uncertainties in the system. Finally, the formulation of the supply function is incomplete. Indeed, the output flows to the outside have not been taken into consideration in these works. However, since the supply function is the mathematical expression of the relationship between the system and the outside, then the input and the output flows should be considered in the exchange of vehicles between the system and the outside. In this work, we propose a fairly and more realistic approach between the thermodynamic system and the transportation one. Indeed, by regarding the vehicles as the abstract energy supplied to the network, we show and prove that certain thermodynamic concepts can be applied to transportation systems. Among them, the most important notion is the entropy which can be thought as a measure of the system disorder. This paper is organized as follows: Section 2 shows the connections between transportation and thermodynamic systems. In particular, the basic concepts such as energy, temperature, thermal capacity are introduced in the transportation context. Section 3 gives a mathematical proof of transportation entropy in order to measure the system disorder. Section 4 shows that the notion of the thermal equilibrium can be also introduced to transportation systems. By considering the transportation entropy as the storage function, a dissipativity phenomenon is presented in Section 5 to reduce the disorder and hence improve the system organization. Finally, an example is worked out to demonstrate the performance of our proposed control in section 6. 2. Basic Concepts and Conservation of Vehicles

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2.1. Store-and-forward modelling For the sake of clarity and in order to render the paper self-contained, the store and forward model that is given here, has been already presented in our earlier work [32]. In a transportation system, the connected traffic streets provide a network for vehicles that can use it to move. In addition, the intersections in a network are composed from the connected streets and crossing areas between them [5]. For example, Figures 1(a) and 1(b) illustrate two common types of intersections that connect 4 and 3 streets respectively.

(a) Four streets

(b) Three streets

Figure 1: Examples of intersections

Moreover, a street can be separated into many traffic lanes depending on the direction of vehicles. For example, the 4 streets in the intersection illustrated in Figure 1(a) are all split into 3

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two lanes corresponding to two opposite directions. On the other hand, in Figure 1(b), all streets consist of three lanes (Note that in this intersection, the lanes are split based on both the current and the potential directions of vehicles). A street may have to more than one traffic signals, but a lane can only have one. Hence, the traffic lanes are more appropriate to be considered as the traffic units areas rather than the streets. For each lane, we choose the number of vehicles within it as its state variable. By applying the traffic flow theory [1] for each lane, the dynamics of the number of vehicles depend on the traffic flows in it. Now, consider the following network represented in Figure 2. This network includes n traffic lanes, n > 0.

Figure 2: The traffic flows related with the lane i

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By using the store-and-forward model where the lanes are considered as reservoirs which respect the vehicle conservation law, the change of xi is given by (For more details see [33] and [32])   xi (k + 1) = xi (k) + T li (k) + ri (k) − di (k) (1)

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where xi is the number of vehicles in the lane i, T denotes the control interval, the flow ri (in veh/h) denotes the input flow, the flow di (in veh/h) is the output flow and li is the sum of all exchange flows related with lane i, which means li (k) =

n X

j=1, j,i


σi, j (k) − σ j,i (k)

(2)

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where σi, j (in veh/h) (resp, σ j,i ), j , i, i, j ∈ {1, · · · , n}, represents the vehicles travel from the lane j (resp, i) to the lane i (resp, j). It is important to note that, only the terms corresponding to actual links between lanes of the network are made explicit in equation (2). Otherwise, all the σi, j for non-existing links do not appear in the equations. The model (1) makes sense only if the state variables xi (k) remain non-negative for all k ∈ N. The flows ri , di and σi, j are also defined to be non-negative on the non-negative orthant. In vector form, the transportation system can be modeled by the following equation   x(k + 1) = x(k) + T l(k) + r(k) − d(k) , ∀k ∈ N (3)

where x = [x1 , · · · , xn ]> , l = [l1 , · · · , ln ]> , r = [r1 , · · · , rn ]> and d = [d1 , · · · , dn ]> . 4

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It is important to note that without any specific assumption, (3) is a general model, which fits any kind of transportation system in any circumstance. Based on this general model, we will explore the nature of transportation system from thermodynamic point of view in the sequel.

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2.2. Introduction to Thermodynamic System The fundamental concept for analyzing thermodynamic systems is the concept of energy. Let us assume that a matter with a unique temperature is called a subsystem, the thermodynamic system consists of a set of connected subsystems. Each of them can store a certain quantity of energy and can exchange energy with other subsystems. Let the energy stored by the subsystems be their state variables. The dynamics of these states are determined by the energy flows between the subsystems and their surroundings. Now, according to the discrete-time thermodynamic model presented by [34], consider a thermodynamic system including n subsystems as shown in Figure 3. The i-th subsystem is denoted by Ψi , i ∈ {1, · · · , n}, and its stored energy is denoted by Ei . Let Ei∗ > 0 be the thermal capacity of Ψi , then the absolute temperature of Ψi is given by (4)

Figure 3: Thermodynamic system

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T ia = Ei /Ei∗

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In Figure 3, the energy flow supplied to the i-th subsystem from the outside is denoted by rie . The flow σei, j (resp, σej,i ), i , j, i, j ∈ {1, · · · , n}, represents the transmission of energy from the j-th (resp, i-th) subsystem to the i-th (resp, j-th) subsystem. The flow die , i ∈ {1, · · · , n}, represents the energy loss from the i-th subsystem to the outside. By combining the input, exchange and output energy flows, the dynamic of the energy Ei stored by Ψi can be given by Ei (k + 1) = Ei (k) +

n X

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σei, j (k) − σej,i (k) + rie (k) − die (k)

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which infers the state-space model of the thermodynamic system in vector form E(k + 1) = E(k) + le (k) + re (k) − de (k) 5

(6)

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where E(k) = [E1 (k), · · · , En (k)]T is the vector of the energy stored by all subsystems at the beginning of k-th interval, re = [r1e , · · · , rne ]T , de = [d1e , · · · , dne ]T , and le = [l1e , · · · , lne ]T represent all exchange flows such that n X e (σei, j − σej,i ), i ∈ {1, · · · , n} (7) li = j=1, j,i

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2.3. Correspondences of Basic Concepts Now, by comparing the transportation model (3) and the thermodynamic model (6), it is clear that these two systems have many common aspects. In particular, the vehicles perform very similarly as the energy does in the thermodynamic system. Hence, the vehicles can be considered as the abstract transportation “energy”. Based on this fundamental analogy, the thermodynamic concepts can be introduced into the transportation context. Each traffic lane can contain certain amount of vehicles. The vehicles move between the connected lanes or between a lane and the outside. The lanes are like the subsystems in the thermodynamic context. They store vehicles and exchange them with their surroundings. Hence, the traffic lane is the corresponding notion of the thermodynamic subsystem and the vehicle number xi corresponds to the energy Ei , i ∈ {1, · · · , n}, stored in thermodynamic subsystems . The temperature is also an important thermodynamic concept. In order to find its correspondence in transportation context, the corresponding notion of the thermal capacity should be firstly determined. Since the lengths of traffic lanes are fixed, there exists a maximal capacity for each lane to contain vehicles. Let xi∗ , i ∈ {1, · · · , n}, be the capacity of the lane i. It is not difficult to observe that the appropriate corresponding notion of the thermal capacity Ei∗ is the lane capacity xi∗ . Furthermore, we can define the occupancy of a lane as the proportion of the vehicles number to the capacity, which is given by fi =

xi , xi∗

i ∈ {1, · · · , n}

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Compared with the equation (4), the occupancies fi obviously correspond to the absolute temperatures T ia , i ∈ {1, · · · , n}. In summary, Table 1 lists the correspondences of the basic concepts between thermodynamic and transportation systems. This analogy will provide us the opportunity to introduce thermodynamic principles into transportation context as well. Table 1: Correspondence of basic concepts

Thermodynamic Concepts energy subsystem energy stored in a subsystem thermal capacity temperature

Transportation Concepts vehicles traffic lane number of vehicles in a lane lane capacity occupancy

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2.4. Conservation of Vehicle (First Principle) The first law of thermodynamics, called the conservation of energy [35], indicates that the energy can not be created or destroyed, it can only change forms or be transferred. Specifically, the change of the energy stored in any thermodynamic system equals exactly the amount of the net energy exchange with its surroundings. This principle also holds for the transportation system, because the change of vehicles within a system equals exactly the amount of the net traffic flows between the system and the outside. To show this principle more clearly, let  , [1, · · · , 1]> be the vector whose all components are equal to 1, and define n X > xi (9) U, x= i=1

as the total number of the vehicles within the transportation network. Let

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Q ,  > (r − d)

(10)

be the net vehicles exchange with it surrounding. Then, we have ∀k ∈ N ∆U(k) = Q(k)

(11)

where ∆U(k) = U(k + 1) − U(k). Indeed, from (3), we have

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  U(k + 1) =  > x(k + 1) =  > x(k) +  > l(k) +  > r(k) − d(k) = U(k) +  > l(k) + Q(k)

It follows

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∆U(k) − Q(k) =  > l(k)

Now, observe that  > l(k) =

n X

(12) (13)

li (k) and since

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j=1

n  X  li (k) = σi, j (k) − σ j,i (k)

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j=1, j,i

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in view of (2), hence we get >

 l(k) = =

n  X n X

i=1 j=1, j,i n n X X i=1 j=1, j,i

But

n n X X

σi, j (k) −

σi, j (k) −

σ j,i (k) =

i=1 j=1, j,i

n X

j=1, j,i n n X X

σ j,i (k)

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i=1 j=1, j,i

n n X X

j=1 i=1,i, j

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 σ j,i (k)

σ j,i (k)

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It follows >

 l(k) =

n n X X

i=1 j=1, j,i

σi, j (k) −

n n X X

σ j,i (k) = 0

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j=1 i=1,i, j

U(k2 ) = U(k1 ) +

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which is the desired result. This result indicates that the total vehicles in the transportation system depends only on the traffic flows between the system and the outside, the exchange flows between different lanes can not change the total number of the vehicles within the network. Furthermore, (11) implies (18)

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for any k1 , k2 ∈ N, k2 > k1 . Hence, according to [36], the conservation of vehicles also implies that the transportation system (3) is lossless with respect to the storage function (9) and the supply function (10). 3. Transportation Entropy

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3.1. Second Law of Thermodynamics In the thermodynamic system, the directions and the quantities of the energy movements must follow some specific rules. For example, without heating equipment, a hot beverage will definitely become cooler in a cold room. In other words, the energy can not move from cold air to the hot beverage but in the opposite direction. This phenomena is consistent with the second law of thermodynamics (Clausius statement, [37]):

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Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

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This principle is usually connected with a special concept of entropy. The second law of thermodynamics can be also described in terms of how the entropy of thermodynamic system changes. In particular, Clausius proposed that the increase of the system entropy due to the input energy Qe from the environment is Qe /T sa , where T sa is the shifted absolute temperature at the spot where the energy transmission happens [37]. However, for any thermodynamic system, the actual increase of entropy is bigger than or equals to the one supplied by the environment. In other words, any thermodynamic system is creating entropy itself. This phenomenon can be presented by the following Clausius inequality Z dQe (19) ∆ψ ≥ T sa or its discrete-time version [34] ψ(k + 1) ≥ ψ(k) +

n X

Qei (k) , ∀k sa T i (k + 1) i=1 8

∈N

(20)

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where ψ is the entropy of the thermodynamic system, Qei (k) is the net energy that the i-th subsystem absorbs from the outside during the interval k at the shifted absolute temperature of the i-th subsystem T isa (k + 1) at the end of interval k, i ∈ {1, · · · , n}. Furthermore, since the entropy is opposite to the capacity of the thermodynamic system to do useful work, it is also regarded as the measure of the system disorder [38]. More precisely, bigger entropy indicates that the system is worse organized. Hence, the Clausius inequality also implies that the thermodynamic system always tends to become worse organized. For the discrete-time thermodynamic model (6), based on the Clausius inequality (20), [34] presented the formula of its entropy as follows ψ(E) , E ∗> ln(a + Pe E) −  > E ∗ ln a

(21)

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where ψ(E) is the entropy, E ∗ = [E1∗ , · · · , En∗ ]> , a is a positive scalar which represents the difference between the shifted absolute temperature and the absolute temperature defined in (4) such that T isa = a + T ia , i ∈ {1, · · · , n}, and Pe is a diagonal matrix with diagonal elements 1/Ei∗ . Here, we use the notation ln(x) = [ln(x1 ), · · · , ln(xn )]> . Note that this formula is identical with the Boltzmann entropy expression for statistical thermodynamics. Now, since the entropy (21) is not easy to apply in control issues, [34] also introduced a dual notion to entropy, called ectropy, to measure the system order. The dual inequality to (20) is given by n X φ(k + 1) − φ(k) ≤ Qei (k)T ia (k + 1), ∀k ∈ N (22)

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i=0

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where φ is the ectropy of the thermodynamic system, Qei (k)T ia (k+1) is the input ectropy of i-th subsystem supplied by the environment with the net input energy Qei (k), and T ia (k + 1) is the absolute temperature of i-th subsystem at the end of interval k. This inequality is also called anti-Clausius inequality, which, like Clausius inequality, is consistent with the second law of thermodynamics. Anti-Clausius inequality indicates that in any circumstance, any thermodynamic system is destroying its ectropy so that the increase of the ectropy is always less than or equal to the one supplied from its surroundings. Opposite with the entropy, the ectropy measures the capacity of the system to do useful work, and bigger ectropy corresponds to better organization of the system. For the discrete-time thermodynamic model (6), based on the anti-Clausius inequality (22), [34] presented the formula of the ectropy as follows φ(E) ,

1 > E Pe E 2

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Contrary to the energy, the entropy and the ectropy are both non-conservative notions. Specifically, in the isolated system, the sum of energy is always constant, but the entropy tends to increase while the ectropy tends to decrease. Besides, it is important to note that the ectropy (23) has the form of Lyapunov function, which is more useful in control issues than the entropy. 3.2. Transportation Entropy The correspondence between transportation systems and thermodynamic systems encourages us to introduce the entropy concept into transportation context in order to evaluate the system 9

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performances. However, the problem is that there is no natural principle in transportation context that corresponds to the second law of thermodynamics. As a matter of fact, the directions of vehicles are determined by the drivers, there is no such rule that the vehicles only move from more crowded lanes to less crowded ones. Hence, we can not introduce transportation entropy based on certain principles as it is defined in the thermodynamic context. Nevertheless, based on the correspondences of certain concepts, such as energy and temperature, we can still simply introduce the formulas in the thermodynamic system to present the transportation entropy as the measure of the system disorder. To do this, the significations of orders in these two systems must be firstly compared. Indeed, in thermodynamic context, higher order means higher capacity to do useful work, which is related with bigger temperature differences between subsystems. For example, in a thermodynamic system including two subsystems, if more energy concentrates on one single subsystem to generate bigger temperature difference, the quantity of the potential energy movements is bigger, which means that the system can generate more useful work. In this case, the system is regarded as better organized and has higher order. But on the converse, if the differences between the lane occupancies are bigger, the order of the transportation system is lower. Indeed, for a general transportation network, if more vehicles concentrate in only a few lanes, it has bigger probability to generate congestion in these lanes and the other lanes are wasted. In this case, the system is regarded as worse organized and has lower order. Moreover, the energy input to the thermodynamic system brings more capacity to do useful work, which increases the order. However, in transportation context, the vehicle input brings more potential opportunity to generate congestions, which decrease the order. For an isolated thermodynamic system without energy input, the disorder (entropy) can only increase until it reaches equilibrium. On the converse, if, at a certain moment, the transportation system has no vehicle input, the system will keep dissipating the present vehicles and consequently become better organized. In conclusion, the order signification of these two systems is opposite, which means that the disorder in the transportation system corresponds to the order in the thermodynamic system. In other words, the transportation entropy (resp, ectropy) should correspond to the thermodynamic ectropy (resp, entropy). Therefore, according to the thermodynamic ectropy and entropy formulas (23), (21), the transportation entropy and ectropy can be defined respectively as 1 > x Px 2 φ(x) , x∗> ln(a + Px) −  > x∗ ln a

ψ(x) ,

(24) (25)

where ψ(x) is transportation entropy, φ(x) is transportation ectropy, a is a positive scalar, P is the diagonal matrix with the diagonal elements 1/xi∗ , and x∗ = [x1∗ , · · · , xn∗ ]> . The formula of the transportation entropy (24) comes only from the thermodynamic system, but we have not shown its signification in the transportation context. Indeed, Figure 4 shows the well known fundamental diagram of traffic flow based on kinetic wave theory, where ρm is the jam density, ν is the speed and νm is the free-flow speed [39]. It is observed that the traffic flow and the speed can be considered as functions of the traffic density. More precisely, bigger traffic density 10

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Traffic Density

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Traffic Flow

generates the smaller speed and hence prolongs the delay. Indeed, despite the habits of drivers, any vehicle need spend more time to pass more crowded lanes.

Figure 4: Fundamental diagram of traffic flow

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With this in mind, the fundamental question now is that, given two states x(k1 ) and x(k2 ) of the network with k1 < k2 , what does mathematically it mean to say that one is more disordered than the other? The first observation is that if x(k2 ) ≤ x(k1 ) (that is, xi (k2 ) ≤ xi (k1 ), i = 1, · · · , n), then the number of vehicles of each lane decreases between instants k1 and k2 . This implies in view of fundamental diagram, that the density decreases and hence the speed of vehicles increases in each lane that leads to the greatest order in the system. Since x(k2 ) ≤ x(k1 ) and P is diagonal matrix with positive elements, it follows ψ(x(k2 )) = 12 x(k2 )> Px(k2 ) ≤ ψ(x(k1 )) = 21 x(k1 )> Px(k1 ) and hence, in this case the entropy can be thought of as a measure of the tendency of the system to lose disorder. In the general case, the question can be answered by mean of the majorization theory which was introduced by Hardy, Littlewood, and Polya [40]. It is a formalization of the concept of diversity in the components of vectors. Over the past decades, majorization theory has found applications in disciplines ranging from statistics, probability theory and economics, mathematical genetics, quantum mechanics to linear algebra and geometry. For more additional details, the readers can refer to the excellent book on the subject of [41]. This concept can be explained as follows. For any vector x ∈ Rn , denote the decreasing xi by

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x[1] ≥ x[2] ≥ · · · ≥ x[n]

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This means that x[1] is the greatest element in x, x[2] is the second-greatest element and so on.

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Definition 1. ([41]) A vector x ∈ Rn is said to be weakly-submajorized by a vector y ∈ Rn or y is said to weakly-submajorizes x, in symbols x ≺ y, if r r X X x[i] ≤ y[i] , ∀r = 1, · · · , n (27) 1

1

In other words

x[1] ≤ y[1] x[1] + x[2] ≤ y[1] + y[2] .. . n n X X x[i] ≤ y[i] 1

1

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If

Pn 1

x[i] =

Pn

1 y[i] ,

x is said to be majorized by y.

Λi j ≥ 0,

n X j=1

Λi j ≤ 1,

n X i=1

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It is clear that if x ≤ y (that is, xi (k2 ) ≤ xi (k1 ), i = 1, · · · , n) then x is weakly-submajorized by y. Hence, definition 1 covers the case mentioned above. Besides, it is clear that if x is a positive P vector, then x ≺ ( ni=1 xi , 0, · · · , 0). The concept of majorization has the substantial advantage that it provides an easy and obvious computational procedure for comparing two vectors, but it provides little light on the meaning of the ordering. A more practical approach is via doubly substochastic matrices. A matrix Λ ∈ Rn×n is said to be doubly substochastic if Λi j ≤ 1

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P P If nj=1 Λi j = ni=1 Λi j = 1, Λ is said to be doubly stochastic. Using the above notions, a characterization of the concept of the weak submajorization is given in the following lemma (See [41], A.4 page 14) Lemma 1. Let x ∈ Rn+ and y ∈ Rn+ be positive vectors. Then the following statements are equivalent.

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1. x is weakly submajorized by y 2. There exists a doubly substochastic matrix Λ such that x = Λy P P 3. The inequality ni=1 Θ(xi ) ≤ ni=1 Θ(yi ) holds for any increasing order-preserving function Θ.

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For majorization theory, the order-preserving functions are called Schur-convex. It follows immediately form this lemma that the meaning of the ordering is now clear. Indeed, from item 2 each component xi is a linear combination of y1 , · · · , yn . This implies that x is an average of y using weights which add up to less than unity. This means that all components xi are more nearly equal than the ones yi . In other words, x represents a more uniform distribution than y (see [41]). Item 3 is useful to identify the class of functions consistent with a given ordering. It means that if x is weakly submajorized by y then the inequality associated with y will be greater than the inequality associated with x by a class of real-valued evaluation functions Θ. This is the measurement of the degree of inequality of a distribution. P x2 In the following, we will show that weak submajorization and the entropy ψ(x) = ni=1 xi∗ are i connected, in that both offer approaches to the problem of quantifying what it means for a network state to be more disordered than another. Indeed, if we assume for k2 ≥ k1 that the state of the P network x(k2 ) is weakly submajorized by the state x(k1 ) and since the relation ri=1 x[i] measures the concentration of the r greatest components of x, r = 1, · · · , n, inequalities (27) imply that x(k2 ) represents lower degree of concentration of vehicles than x(k1 ). In other words, the distribution of vehicles in the state x(k2 ) is more spread out in the network than that of x(k1 ). Therefore, the system is more organized when the state x(k2 ) is reached, which leads to low possibility to emerge x2 congestions at the instant k2 . Now, let ψi (xi ) = xi∗ , clearly, ψi (xi ) is order-preserving convex i

12

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function and increasing in positive orthant. From item 3 of lemma 1 we get n X i=1

ψi (xi (k2 )) ≤

n X

ψi (xi (k1 ))

(30)

i=1

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P Since ψ(x(k)) = ni=1 ψi (xi (k)), it follows ψ(x(k2 )) ≤ ψ(x(k1 )). This means that smaller entropy corresponds to more uniform distribution of vehicles. Conversely, if ψ(x(k2 )) ≤ ψ(x(k1 )), x(k2 ) is weakly submajorized by x(k1 ) in view of 3) of lemma 1. Consequently, x(k2 ) represents a more even distribution of vehicles in the network than x(k1 ). Hence, the order emanating from the distribution x(k2 ) is greater than the one from x(k1 ). Consequently, we can get rise the conclusion that the transportation entropy can be thought of as a measure of the system disorder. 4. Occupancy Equilibrium

PT

ED

M

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For an isolated thermodynamic system without any input (re (k) ≡ 0) and output (de (k) ≡ 0), energy flows if a pair of connected subsystems have different temperatures. The energy transmission will emerge between them so that their temperatures tend to approximate. After enough amount of time, the isolated system will reach certain state when the temperature is spatially and temporally uniform. In this state, the system will lose all its capacity to produce any useful work. Such particular state is called thermal equilibrium [35]. For an isolated thermodynamic system in thermal equilibrium, its ectropy would be a minimum and entropy would be a maximum giving rise to a state of absolute disorder. Since the lane occupancies correspond to the temperatures, this concept of thermal equilibrium can be easily introduced into the transportation context to denote the state when the occupancies of all traffic lanes are the same. Such state is called the occupancy equilibrium in this paper. It can be denoted by the following expression x¯n x¯1 ¯ ∗ = · · · = ∗ = f = constant x1 xn

(31)

AC

CE

The occupancy equilibrium indicates that no particular concentration of vehicles exists in the system and hence the possibility to emerge congestion is extremely low (except the situation where f¯ = 1, which is immensely rare in an urban transportation network). If a transportation network can be isolated in certain circumstances, its entropy (resp, ectropy) is minimal (resp, maximal) when the system reaches the occupancy equilibrium. In other words, for isolated network, this state corresponds to the best organization of the system. This observation is stated in the following proposition.

Proposition 1. Suppose the transportation network is isolated (i.e., r = 0 and d = 0). Let N be the total number of vehicles in the system. Let ψ(x) and φ(x) be the entropy and ectropy of the system given by (24) and (25) respectively, and define D , {x ∈ Rn+ / > x = N}, where  > = (1, 1, · · · 1). Then, N arg min(ψ(x)) = arg max(φ(x)) = x¯ = > ∗ x∗ (32) x∈D x∈D  x 13

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Proof. The proof is identical to the proof of Proposition 3.17 of [34] for thermodynamic systems. Hence it is omitted here.

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Now, observe that if the state x¯ is reached, then xx¯∗i =  >Nx∗ = f¯, for all i = 1, · · · , n, which i corresponds to the occupancy equilibrium. This occupancy equilibrium leads to the minimal disorder and maximal order for the transportation network. Hence, for an isolated system the study of control problems should concern this state as an important objective (see proposition 3). 5. Dissipativity based control of the entropy

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In transportation context, the input vehicles bring disorder to the system. This supplied disorder depends not only on the net vehicles exchange between the network and the outside T (r(k) − d(k)), but also on the distribution of them. The vehicles which enter the more crowded lanes bring more disorder than the ones with the same quantity which enter the less crowded lanes. Since the occupancies represent how the lanes are crowed, it is natural to measure the supplied entropy with respect to the input flows T (r(k) − d(k)) and occupancy factors. Let f = [ f1 , · · · , fn ]> be the vector of all occupancies. Note that, because f (k) corresponds to the beginning of the interval k while f (k + 1) corresponds to the end of it, the supplied entropy in the interval k should be with respect to f (k + 1) rather than f (k). So, we define
S (k) = T f > (k + 1) r(k) − d(k)

(33)

M

as the supplied entropy in the interval k. Furthermore, since f (k) = Px(k) in view of (8), the supplied entropy is restated as
S (k) = T x> (k + 1)P r(k) − d(k)

ED

(34)

PT

Now, as in the anti-Clausius inequality (22), we propose its corresponding version in transportation context as follows ∆ψ(x(k)) = ψ(x(k + 1)) − ψ(x(k)) ≤ S (k),

∀k ∈ N

(35)

AC

CE

If this inequality is satisfied, the variation of the storage of the system disorder is always smaller than or equal to the supplied one from the outside. In other words, the transportation system has the tendency to decrease its disorder and become better organized. Furthermore, according to [42] and [43], (35) also implies the dissipativity of the system with the entropy (24) as the storage function and with (34) as the supply function. In this case, (35) is called the dissipation inequality. Figure 5 illustrates this dissipativity of system disorder. The input flows r bring disorder to the transportation system to make it worse organized. At the same time, the appropriate traffic signal control dissipates the traffic so that the system stores only a part of this input disorder ψ. Unfortunately, contrary to thermodynamic systems where the dissipation of temperature is naturally verified, the dissipativity phenomenon does not exist naturally in transportation context. To show this, observe from equations (3) and (35) that 14

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Figure 5: Dissipativity of transportation entropy

  1 1 > x (k + 1)Px(k + 1) − x> (k)Px(k) = x> (k + 1)P x(k + 1) − x(k) − T l(k) 2 2 >   1 − x(k + 1) − x(k) P x(k + 1) − x(k) + T x> (k + 1)Pl(k) (36) 2

M

∆ψ(x(k)) =

PT

ED

Since P is positive definite and x(k + 1) − x(k) − T l(k) = T (r(k) − d(k)), it follows from (34) that  >  1 ∆ψ(x(k)) = S (k) + T x> (k + 1)Pl(k) − x(k + 1) − x(k) P x(k + 1) − x(k) 2 ≤ S (k) + T x> (k + 1)Pl(k) (37) Let

AC

CE

g(x(k)) = x> (k + 1)Pl(k) (38) Pn It follows ∆ψ(x(k)) ≤ S (k)+g(x(k)). Since li (k) = j=1, j,i (σi, j (k)−σ j,i (k)) and P = diag(x1∗ , · · · , xn∗ ), we get n n n X xi (k + 1)li (k) X X xi (k + 1)(σi, j (k) − σ j,i (k)) g(x(k)) = = xi∗ xi∗ i=1 j=1, j,i i=1

=

n−1 X n  X xi (k + 1) i=1 j=i+1

xi∗

−

 x j (k + 1)  σi, j (k) − σ j,i (k) ∗ xj

(39)


x (k+1)
Now, if we suppose that xi (k+1) − j x∗ σi, j (k) − σ j,i (k) ≤ 0 for all i , j and for all k ∈ N, xi∗ j then g(x(k)) ≤ 0 so that ∆ψ(x(k)) ≤ S (k), ∀k ∈ N, hence the transportation system is dissipative. 15

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Unfortunately this assumption does not exist in reality because it implies that the more occupied lanes give more vehicles to the less occupied ones which is not realistic in transportation systems. To show this, recall that the occupancy of lane i is defined by fi = xx∗i and the flow σi, j (resp, σ j,i ), i j , i, represents the vehicles travel from the lane j (resp, i) to the lane i (resp, j). Now, observe that if g(x(k)) ≤ 0 then there are two cases to consider: xi (k+1) xi∗

− j x∗ ≥ 0 and σi, j (k) − σ j,i (k) ≤ 0. In this case we have xi (k+1) ≥ j x∗ and xi∗ j j σi, j (k) ≤ σ j,i (k). It follows from the first inequality, that the lane i is more occupied than the lane j and gives more vehicles to it in view of the second inequality.

x (k+1)

x (k+1)

Case 2:

xi (k+1) xi∗

x (k+1)

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Case 1:

x (k+1)

− j x∗ ≤ 0 and σi, j (k) − σ j,i (k) ≥ 0. In this case we have xi (k+1) ≤ j x∗ and xi∗ j j σi, j (k) ≥ σ j,i (k). Similarly to case one, the lane j is more occupied than the lane i and gives more vehicles to it.

ED

M

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Besides, even the situation where σi, j (k) = σ j,i (k) for all k ∈ N (lanes i and j have the same exchange between them) is not realistic because the directions of vehicles are determined by the drivers. However, it is known that the traffic signals have the ability to control the traffic flows. Therefore, it is possible that under certain control schemes, the traffic signals can satisfy the dissipation inequality (35) without satisfying the assumption that g(x(k)) ≤ 0. This is the main concern in the following. Now, in order to introduce the notion of signal control to the system, we use the basic idea of the store-and-forward model where the total outflow rate of any lane is considered as its average value which leads to [33] n X ui si , ∀i ∈ {1, · · · , n} (40) σ j,i = c j=1, j,i

σi, j = λi, j

CE

PT

where ui is the effective green time interval of lane i which comprises the amber signal length beside the green signal and si is the saturation flow rate. Thus, by using the turning rates λi, j ∈ [0, 1] from lane j towards lane i, the inflow rates are given by (see[33]) ujsj c

(41)

AC

P where nj=1 λi, j = 1. Note that this approach models the system into a linear form and hence significantly simplifies the problem. It is inevitable that such simplification also leads to a few consequences as follows (see [5]) 1. The step T cannot be shorter than the cycle time c, hence the oscillations of vehicle numbers due to the green/red commutations cannot be described by the model 2. The equation (40) implies that the outflows of all lanes are considered saturated, the undersaturated and over-saturated (spill-back) situations are not concerned by the model. 3. The equation (41) imply that the turning rates are predetermined, hence the model cannot consider the uncertainties of these parameters. 16

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However, despite these consequences, the model is qualified for a number of highly efficient optimization and control methods for large-scale networks as demonstrated by several studies such as TUC [4, 5], MPC [44], [45], [46]. Now, by replacing (40), (41) in equation (2), li (k) can be rewritten as (Fore more details see [32])

j=1

n X

σ j,i (k) =

j=1, j,i

n ujsj ui si X λ j,i = λi, j − c c j=1, j,i j=1, j,i

n n X X ujsj ui si λi, j − λ j,i c c j=1, j,i j=1, j,i

λ j,i = 1, it follows

n X ujsj ui si li (k) = λi, j − (1 − λi,i ) c c j=1, j,i

Now, if we put Bi, j = Then li (k) can be rewritten as li (k) =

n X

(

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Pn

j=1, j,i n X

σi, j (k) −

− sci (1 − λi,i ) : sj λ : c i, j

M

Or,

n X

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li (k) =

Bi, j u j (k) + Bi,i ui (k) =

j=1, j,i

for i = j for i , j

n X

Bi, j u j (k) = B>i u(k)

(42)

(43)

(44)

(45)

j=1

ED

This in turn transforms equation (3) into


x(k + 1) = x(k) + Bu(k) + T r(k) − d(k)

(46)

AC

CE

PT

where u = [u1 , · · · , un ]> is the control signal of the network and B is a n × n matrix whose i-th rows is given by T B>i . More details of this store-and-forward modelling are seen in [4] and [47] and [32]. Next, it is important to note that the flows ri and di are not always measurable and then they are not controllable. Indeed, they represent the uncertainties in the model and they are usually considered as constant nominal values, see [45] ,[48] and [32] . However, in this paper they are assumed unknown, time-varying, and bounded, i.e, R = {r ∈ Rn / kr(k)k2 ≤ rm , rm > 0} D = {d ∈ Rn / kd(k)k2 ≤ dm , dm > 0}

(47)

where, k k2 is the usual Euclidean norm on Rn . Note that, in this model, the state x and control u should respect their physical limitations. Firstly, the effective green time is bounded between its maximal umax and minimal umin values i.e. U = {u ∈ Rm /

umin ≤ u ≤ umax }

17

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In addition, since congestion situations are not taken into consideration by the modelling, the model is no longer valid if the number of vehicles exceeds the capacity of each lane xi∗ . Therefore, the model (46) is valide only if x belongs to the following region X = {x ∈ Rn / 0 ≤ x ≤ x∗ }

(49)

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where x∗ = [x1∗ , · · · , xn∗ ]T . Now, the robust dissipativity based control problem is stated as follows: find a feedback control law u(x) such that for all r ∈ R and for all d ∈ D, the closed loop system is dissipative. In other words, according to (38) find u(x) such that 
> g(x(k)) = x> (k + 1)Pl(k) = x(k) + Bu(k) + T r(k) − d(k) Pl(k) 
 = u> (k)BT PBu(k) + u> (k)B> P (x(k) + T r(k) − d(k) ≤ 0, ∀r ∈ R, d ∈ D (50)

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In the following, we will provide the way to construct a feedback control law that achieves the
dissipativity. To make the idea clear, put M = B> PB, y = 12 B> Px and w = T2 B> P r(k) − d(k) . It follows g(u) = u> Mu + 2u> y + 2u> w (51) Now, by Cauchy-Schwarz inequality we have u> w ≤ |u> w| ≤ kuk2 kwk2 . Furthermore, we have in view of (47)

ED

M



T T kwk2 = k B> P r(k) − d(k) k2 ≤ kB> Pk2 k r(k) − d(k) k2 2 2
T > ≤ kB Pk2 kr(k)k2 + kd(k)k2 2
T > ≤ kB Pk2 rm + dm , wm 2

(52)

CE

PT

This and u> w ≤ |u> w| ≤ kuk2 kwk2 yield u> w ≤ kuk2 wm . Since, kuk2 ≤ kuk1 , where kuk1 is the 1-norm and kuk1 =  > u because u > 0, where  > = (1, · · · , 1), it follows, u> w ≤  > uwm and hence (51) implies g(u) ≤ u> Mu + 2u> (y + wm ) , gm (u) (53)

AC

Consequently, if gm (u) ≤ 0 then g(u) ≤ 0 and the system would be dissipative. The problem P now is to find u such that gm (u) ≤ 0, ∀r ∈ R, d ∈ D. Indeed, since ni=1 li (k) =  > l =  > Bu = 0 for all u then  > B = 0, which implies that the rows of B are linearly dependent. Consequently B is singular so M is symmetric positive semidefinite and all its eigenvalues are nonnegative. Hence, there exists an orthogonal matrix K (K > = K −1 ) such that M = KCK > , where C is a diagonal matrix of the form C = diag(ξ1 ; · · · , ξr , 0, · · · , 0) (54)

where ξi > 0 are the positive eigenvalues of M and r is the rank of M. The orthonormal columns of K are the eigenvectors of all eigenvalues of M. Let D = diag(ξ1 ; · · · , ξr ), D is invertible and M can be rewritten as ! ! ! 1 1 Ir 0 D2 0 D2 0 K> (55) M=K 0 0 0 In−r 0 In−r 18

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1

D2 0

J=K

!

(56)

J>

(57)

0 In−r

then J −1 is well defined because K is orthogonal and Ir 0 0 0

M=J

!

In this case, gm (u) can be rewritten as >

gm (u) = u J >

−1

!

Ir 0 0 0

J > u + 2cu> (y + wm )

>

Let z = J u and θ = J (y + wm ). Because −z gm (z) = z

0 0 0 In−r

!

! Ir 0 z + 2z> θ = z> z + 2z> θ − z> 0 0

0 0 0 In−r

≤ z> z + 2z> θ = (z + θ)> (z + θ) − θ> θ , f (θ) Let R ∈ Rn×n and put z = Rθ, it follows

(58)

z ≤ 0 il follows

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>

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If we set

!

z

(60)

M

h i f (θ) = (z + θ)> (z + θ) − θ> θ = θ> (R + I)> (R + I) − I θ

(59)

ED

Hence, f (θ) ≤ 0 for all θ ∈ Rn if and only if (R + I)> (R + I) − I is negative semidefinite, which is equivalent to the following LMI (see appendix) ! I (R + I)> 0 (61) (R + I) I

CE

PT

Summarizing, from z = Rθ, z = J > u, θ = J −1 (y + wm ) and y = 12 B> Px we have our feedback control law that achieves the dissipativity 1 −1 u = (J > )−1 RJ −1 B> Px + (J > ) RJ −1 wm 2

(62)

This leads to the following theorem.

AC

Theorem 1. Let R be a solution of the following LMI ! I (R + I)> 0 (R + I) I

and let H = 21 (J > )−1 RJ −1 B> P, Γ = (J > )−1 RJ −1 wm . Then the feedback control law u = Hx + Γ

(63)

renders the system robustly dissipative with respect to entropy (24) as the storage function and the supply function (34) for all r ∈ R and for all d ∈ D. 19

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Next, observe that if the matrix R is chosen to be symmetric (i.e., R> = R), then f (θ) ≤ 0 if and only if −2 ≤ λi ≤ 0, for each eigenvalue λi of R. To show this, it suffices to observe that because R is symmetric it is always diagonalizable. Let R = FLF > be the eigendecomposition of R, where F > = F −1 and L is the diagonal matrix whose diagonal elements are the eigenvalues of R. We have (R + I)> (R + I) − I = R2 + 2R = F(L2 + 2L)F >

(64)

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It follows f (θ) ≤ 0 if and only if (L2 + 2L) is negative semidefinite. Since L2 + 2L = diag(λ21 + 2λ1 , · · · , λ2n + 2λn ), this infers (L2 + 2L) is negative semidefinite if and only if λ2i + 2λi ≤ 0, ∀i, which implies that −2 ≤ λi ≤ 0, ∀i = 1, · · · , n. Summarizing, we have the following result. Theorem 2. Let R be a symmetric matrix such that −2 ≤ λi ≤ 0, ∀i = 1, · · · , n, where λi is an eigenvalue of R and let H = 21 (J > )−1 RJ −1 B> P, Γ = (J > )−1 RJ −1 wm . Then the feedback control law u = Hx + Γ

(65)

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renders the system robustly dissipative with respect to entropy (24) as the storage function and the supply function (34) for all r ∈ R and for all d ∈ D.

M

Remark 1. From the theorem just stated, it follows immediately that, for any arbitrarily selected R that satisfies −2 ≤ λi ≤ 0, ∀i = 1, · · · , n, the feedback control defined by (65) exhausts the class of control laws which render the system dissipative. Hence, theorem 2 gives explicit but non-unique solutions of the robust dissipativity-based control problem.

ED

Now, once the gain matrix H and the vector Γ are given, the feedback control u = Hx+Γ should not violate the constraint (48). This means that every trajectory x(k; xo ) inside the admissible set X does not leave the following region for any instant k ∈ N despite disturbances: U x = {v ∈ Rm /

− π2 ≤ Hx ≤ π1 }

(66)

PT

where π1 = umax − Γ and π2 = Γ − umin . In other words, if the feedback control ensures X ⊆ U x then the constraints on control are verified. To achieve this, we make use of a result which will be recalled now for readers convenience.

AC

CE

Lemma 2 ([49]). Let F ∈ Rq×n and G ∈ R p×n be two matrices. The system of inequalities F x ≤ δ is satisfied by any vector of the non-empty convex polyhedron defined by the system Cx ≤ η if and only if there exists a matrix Q ∈ R p×q with non-negative coefficients satisfying the following conditions QC = F Qη ≤ δ

(67) (68)

Now, observe that constraints on the state (49) and the control vector (66) can be rewritten as the following general polyhedral forms respectively: ! ! n In x∗ o n X= x∈R / x≤ (69) −In 0 20

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n

! ! π1 o H x≤ π2 −H

n

Let

K11 K12 K21 K22

K=

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It follows from (71) and (72) that :

!

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(70) Ux = x ∈ R /   H and C = −IInn , then according to Lemma 2, the condition X ⊆ U x is Hence, if we take F = −H equivalent to the existence of the non-negative matrix K ∈ R2n×2n such that + ! ! H In (71) = K −H −In ! ! π1 x∗ ≤ (72) K π2 0

K11 − K12 K22 − K21 K11 x∗ K21 x∗ which infers that

=H =H ≤ π1 ≤ π2

M

H = K11 − K12 = K22 − K21

(73)

(74) (75) (76) (77) (78)

ED

In order to simplify the linear conditions (78) and to reduce the number of variables, define the matrices H + and H − by H + = (Hi+j ), where Hi+j = sup(Hi j , 0) −

It is clear that

PT

H =

(Hi−j ),

where

H + ≥ 0;

CE

Now, from (78) and (81) we have

Hi−j

= − inf(Hi j , 0) = sup(−Hi j , 0)

H − ≥ 0;

H = H+ − H−

H = H + − H − = K11 − K12 = K22 − K21

(79) (80) (81) (82)

AC

Since the decomposition H = H + − H − is minimal in view of sup and inf operators, it follows H + ≤ K11 H − ≤ K12

and and

H + ≤ K22 H − ≤ K21

These together with inequalities (76)-(77) imply ! ∗! ! ∗! ! x x π1 H+ O K11 O ≤ ≤ O K21 x∗ O H − x∗ π2 Now, we are prepared to prove 21

(83) (84)

(85)

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Proposition 2. The necessary and sufficient condition to verify the inclusion X ⊆ U x is the existence of two non-negative matrices N1 and N2 such that (86) (87) (88)

N1 − N2 = H N1 x∗ ≤ π1 N2 x∗ ≤ π2

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Proof. Necessity: It can be proved by simply letting N1 = H + , N2 = H − and ! H+ H− K= H− H+

(89)

in (71) and (72) and use Lemma 2. Sufficiency: Assume that all conditions of the theorem are satisfied and let x ∈ X, i.e.,

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0 ≤ x ≤ x∗

(90)

0 ≤ N1 x ≤ N1 x∗ −N2 x∗ ≤ −N2 x ≤ 0

(91) (92)

M

Premultiplying this inequality by N1 and −N2 gives

(93)

Addition of these inequalities leads to

−N2 x∗ ≤ (N1 − N2 )x ≤ N1 x∗

ED

According to conditions (87) and (88), it follows (94)

−π2 ≤ Hx ≤ π1

PT

which implies x ∈ U x . The proof is completed.

CE

Now, in order to cast the constraints (87) and (88) in LMIs, observe that they are equivalent to π1i − N1i x∗ ≥ 0, π2i − N2i x∗ ≥ 0,

∀i = 1 · · · , n

(95)

AC

where N1i and N2i are the i-th row of the matrices N1 and N2 respectively. Since N1 ≥ 0, N2 ≥ 0 and x∗ ≥ 0, then by Cauchy-Schwarz inequality it follows π1i − N1i x∗ ≥ π1i − kN1i k2 kx∗ k2 ≥ π1i − kN1i k22 kx∗ k2 ,

∀i = 1 · · · , n

(96)

Hence π1i − kN1i k22 kx∗ k2 ≥ 0 implies π1i − N1i x∗ ≥ 0. With this in mind, the inequality π1i − kN1i k22 kx∗ k2 ≥ 0 is equivalent to kxπ∗1ik2 − N1i N1i> ≥ 0, which is equivalent to the following LMI (see appendix)  π1i  N1i   kx∗ k2     0, ∀i = 1 · · · , n (97)   > N1i In 22

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Similarly for inequality (88), we have  π2i  N2i   kx∗ k2     0 =⇒ π2i − N2i x∗ ≥ 0,   N2i> In

∀i = 1 · · · , n

(98)

In summary, the final robust dissipativity-based control is presented in the following theorem.

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Theorem 3. Let R be a symmetric matrix such that −2 ≤ λi ≤ 0, ∀i = 1, · · · , n, where λi is an eigenvalue of R and let H = 21 (J > )−1 RJ −1 B> P, Γ = (J > )−1 RJ −1 wm . If there exist two non-negative matrices N1 and N2 solutions of the following LMI   π1i  N1i   kx∗ k2      O   >   N1i  In    0, subject to N1i − N2i = Hi ; i = 1, · · · , n  π2i (99) N2i    kx∗ k2     O    N2i> In

where Hi is the i-th row of the matrix H and O is the null matrix, then under the feedback control law u = Hx + Γ (100)

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1. The transportation system is robustly dissipative with respect to entropy (24) as the storage function and the supply function (34) for all r ∈ R and for all d ∈ D.

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2. X ⊆ U x .

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Remark 2. Since N1 , N2 and vector x∗ have positive elements, inequality (87) and (88) can never be satisfied if π1 and π2 are negatives. Hence, any attempt for the synthesis of the state feedback control law that renders the system dissipative, should first determine if such constraints requirement on π1 and π2 are fulfilled or not. This can be done by several choice of R such that −2 ≤ λi ≤ 0, ∀i = 1, · · · , n, where λi is an eigenvalue of R. If π1 ≤ 0 and/or π2 ≤ 0 for any R, we can adjust umin and umax , and if this does not help also, one possible solution is to apply the following saturation function   u : ui ≤ ui,min    i,min ui : ui,min ≤ ui ≤ ui,max sat(ui ) =  (101)    u : u ≥u i,max

i

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where ui is the value derived from the feedback (65) and sat(ui ) is the actual control applied to real system.

Next, if the transportation system is isolated, (i.e., r ≡ 0 and d ≡ 0 so that S (k) = 0 and Γ = 0), under the control law (65) the system (46) becomes x(k + 1) = x(k) + T l(k) = x(k) + BHx(k) 23

(102)

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> for any given initial state In this case the occupancy equilibrium x¯ = f¯x∗ = f¯P−1  where f¯ =  >x(0) x∗ x(0) < x∗ , is an equilibrium state. Indeed, we must only show that BH x¯ = 0. To do this, observe that f¯ f¯ H x¯ = (J > )−1 RJ −1 B> PP−1  = (J > )−1 RJ −1 B>  (103) 2 2 This and  > B = 0 yield H x¯ = 0 (104)

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Hence, x¯ is an equilibrium state for isolated transportation systems. To show the Lyapunov stability of the equilibrium state x¯, consider V(x) = 21 (x − x¯)> P(x − x¯) as a Lyapunov function candidate. From x¯ = f¯x∗ = f¯P−1 , we have

(105)

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1 1 1 V(x) = (x − f¯P−1 )> P(x − f¯P−1 ) = x> Px + f¯2  > P−1 PP−1  − f¯ > P−1 Px 2 2 2 1 ¯> = ψ(x) + f  x¯ − f¯ > x 2

According to the conservation of vehicles, the total number of vehicles  > x = N stays constant for isolated transportation systems. Thus, we have V(x) = ψ(x) − 21 f¯N and ∆V(x) = ∆ψ(x). It follows in view of S (k) = 0 ∆V(x(k)) = ∆ψ(x(k)) ≤ 0, ∀k ∈ N (106)

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which establishes the Lyapunov stability of the equilibrium state x¯. Furthermore, if the matrix R is chosen such that −2 < λi < 0, ∀i = 1, · · · , n, where λi is an eigenvalue of R, then f (θ) < 0 so that ∆ψ(x(k)) < 0. This implies that the equilibrium state x¯ is asymptotically stable. Summarizing, we have the following proposition.

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Proposition 3. Consider the isolated transportation system (i.e., r ≡ 0, d ≡ 0) (107)

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x(k + 1) = x(k) + T l(k) = x(k) + Bu

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Let R be a symmetric matrix such that −2 < λi < 0, ∀i = 1, · · · , n, where λi is an eigenvalue of R. Define H = 21 (J > )−1 RJ −1 B> P. Then under the feedback control law u = Hx, the equilibrium > state x¯ = f¯x∗ = f¯P−1  is asymptotically stable, where f¯ =  >x(0) for any given initial state x∗ ∗ x(0) < x .

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6. Example

To present the performances of our proposal strategy, consider now the following example of transportation network including 8 intersections, as shown in Figure 6. Each intersection has four phases and four linked routes. Hence, there are 32 lanes and 32 effective green times. The parameters given here are: The cycle c = 90s. The maximal and minimal green times are 50s and 15s respectively. The saturated flow rates in horizontal and vertical directions are 0.4veh/s. The turning rates are λi,i = 0.3 and λi, j = 0.7 respectively. The lanes capacities x∗ = [40, 35, 40, 37, 30, 40, 40, 30, 35, 42, 32, 42, 40, 40, 35, 40, 40, 35, 40, 37, 44, 30, 40, 30, 35, 40, 40, 40, 40, 30, 30, 42]T . The initial queue lengths are taken as x(0) = [30, 20, 22, 19, 15, 26, 24

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39, 20, 30, 35, 22, 19, 12, 26, 28, 39, 30, 30, 22, 19, 25, 15, 39, 20, 30, 35, 22, 19, 18, 10, 28, 39]T . Furthermore, in order to illustrate the behaviors of the control methods in presence of the
disturbances, the supplied function S (k) = T f > (k + 1) r(k) − d(k) is given randomly by certain function in Matlab for lanes which are located at network borders by respecting the lanes capacity x∗ .

Figure 6: Eight intersections network

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In order to show that our system is feedback dissipative, the difference between ∆ψ and the supplied function S is considered at first. To this end, two simulations for 100 cycles are realized. According to theorem 3, in the first simulation (figure 7), the matrix R is chosen such as all its diagonal elements are equal to −1. In this case we have π1 ≥ 0 and π2 ≥ 0, so that the strategy in Theorem 3 can be implemented.

Figure 7: Results of simulation 1

One can immediately observe that the difference ∆ψ − S is always negative. This shows that our feedback control (65) renders the system dissipative with respect to the supplied function S and hence, forces the system to be more organized. 25

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Table 2: Effective green time variation

2 17.6 24.5 22.4 24.1 24.5 29.9 29.2 39.1 29.3 39 29.4 25.3 39 29.1 29.1 32.2

3 17.8 24.5 22.4 29.1 24.5 30.2 29.2 29.2 29.4 37.8 25.1 26.4 38.8 29.3 29.5 31.8

4 19.5 24.5 22.5 29 24.3 29.2 27 28.2 29 38.7 29 25.1 39.1 29.1 29.1 29.9

5 17.8 24.6 22.7 29 24.6 28 28 39 29.4 38.9 29.4 25 38.7 29.3 29.5 38.1

6 17.6 24.4 22.4 29.2 24.2 30.1 29.3 38.8 29.5 38.1 29.3 25.2 38.9 29.4 29.5 39.1

7 17.6 24.5 22.8 29 24.5 30.1 29 38.9 29.4 39.2 29.2 25.4 39 29.4 29.3 39

8 17.4 24.6 23 29.1 24.7 30.3 29 39 29.5 39 29 25.5 39.1 29.5 29 39

9 17.8 24.8 23 29.1 25 30.3 29.2 38.9 29.2 38.9 29.2 25.5 38.7 29 29.2 38.9

10 18 24.8 22.8 29 24.8 30.4 29 38.8 29.1 38.8 29 25.2 39 29 29.4 39

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1 18.1 24.4 22.7 29.1 24.2 29.9 29.1 39 29 39 29.3 25.2 39 29.2 30 35.8

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cycle

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In addition, Table 2 shows few values of the control u for 16 lanes during the first ten cycles. It is clear that the control u respects their boundary conditions where the effective green times are always between its minimal 15s and its maximal 50s values. In the second simulation, we modified the matrix R in order to obtain π1 <0 and π2 <0. Hence, the saturated control presented in the Remark 2 is applied. Figure 8 gives the results of this situation.

Figure 8: Results of simulation 2

As in figure 7, the difference ∆ψ − S is always negative. Hence, the same remarks remain valid for the saturated control. 26

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Next, in order to show the stabilizability of the occupancy equilibrium for isolated transportation system, we consider the feedback control given in Proposition 3. Figure 9 displays the evolution of the state vector under the proposed control. We see that each lane of our system reaches its

Figure 9: Results of simulation 3

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occupancy equilibrium after some cycles. Now, in order to highlight the performance advantage of our control strategy, we proposed a comparison between the capacity of our proposed control and the capacity of the standard control used in the majority of transportation network, to force the number of vehicles to not exceed the lanes capacities. Indeed, the effective green time is generally considered fixed between 30 and 40 seconds. This duration is a generality and can undergo some limited variations. In this context, let us begin by applying our proposed control to the system (46). As a result, we have obtained the following figure 10.

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Figure 10: Evolution of state x under our proposed control

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In this figure, each sub-figure represents the variation of the number of vehicles in the lanes which have the same capacity. Since we have chosen 7 different values for the lane capacity x∗ in the parameters of the example, hence we have 7 sub-figures. In the following table, we show the different lanes with their correspond capacities. Number of lane 21 10,12,32 1,3,6,7,13,14,16,17,19,23,26,27,28,29 4,20 2,9,15,18,25 11 5,8,22,24,30,31

capacity (veh) 44 42 40 37 35 32 30

Table 3: capacities of lanes and their corresponding colors

Now, if we go back to the result of the simulations, we can see that each curve of state varies while remaining under the capacity of its correspond lane. Consequently, the number of vehicles 28

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in the lanes does not exceed the capacities of these latter. On the other hand, by applying the standard control 40s to the system (46), the following result has been obtained

Figure 11: Evolution of state x under the standard control

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One can remark that many curves exceed their correspond lanes capacities. Hence, the constraint on the state (49) is not respected and then we can say that this control is not able to prevent the congestion in the network. In conclusion, these simulations show the efficiency of our control strategy. Firstly, it renders the system dissipative and secondly the constraints on the state and control are respected. 7. Conclusion

This paper has presented an analysis of the transportation network based on thermodynamic point of view. By considering traffic lanes as thermodynamic sub-systems and the vehicles as the abstract energy supplied to them, certain similarities have been established between thermodynamic and transportation systems. 29

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Appendix A. The Linear Matrix Inequality

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Though the first law of thermodynamic is naturally verified in transportation context, the second thermodynamic principle does not work. Besides, the significations of order in the two systems are opposite, which means that the thermodynamic ectropy corresponds to the transportation entropy. We have shown that such transportation entropy is a measure of disorder of the system and hence may provide deep insight in the analysis of transportation control problems. In particular, a robust dissipativity phenomenon that reduces the system disorder and hence renders the system better organized has been presented. Although this phenomenon doesn’t exist naturally in transportation context, a feedback control strategy has been constructed to attain such objective by means of Linear Matrix Inequalities (LMI).

According to [50], a linear matrix inequality (LMI) is an expression of the form xi Fi  0

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where x ∈ Rm and the symmetric matrices Fi = FiT ∈ Rn×n , i = 0, · · · , m, are given. The symbol  means that F(x) is positive definite. The non-strict LMI can be also defined as F(x)  0

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The LMI represents a wide variety of convex constraints on x because linear inequalities, quadratic inequalities, matrix norm inequalities, etc can all be cast into the form of LMI. In particular, nonlinear but convex inequalities are converted to LMI usually by using the following Schur complement lemma (see [50]).

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Lemma 3. Let S be a symmetric matrix of the form A BT S = B C

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and C is invertible. Then S  0 if and only if C  0 and A − BT C −1 B  0. References

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