Modelling and H∞ Control of Urban Transportation Network

13th IFAC Symposium on Control in Transportation Systems The International Federation of Automatic Control September 12-14, 2012. Sofia, Bulgaria

Modelling and H∞ Control of Urban Transportation Network Zhou Huide ∗ Rachid Bouyekhf ∗ Adbellah EL Moudni ∗ ∗ Laboratoire Syst`emes et Transports(SeT), UTBM, Belfort Technopˆ ole, 90010 Belfort Cedex, France (e-mail: [email protected], [email protected], [email protected]).

Abstract: In this paper, the controllability and H∞ control method of urban transportation network are considered. Firstly, the signal control of a network is modeled as a discrete-time state space model. It is found that the system is not completely controllable and hence, the system is decomposed into controllable subsystem and free one. To avoid congestion and reduce the impact of disturbance, H∞ control method is adapted and a traffic-responsive strategy under state and control constraints is constructed by means of Linear Matrix Inequality (LMI). Finally, a system including two intersections is studied in order to illustrate the result. Keywords: Urban transportation network, modelling, controllability, H∞ control, LMI, two intersections 1. INTRODUCTION

with an optimization algorithm provided for online application. Its major drawback is the bad performance in case of saturated traffic conditions. There is another kind of traffic-responsive strategies, like OPAC (Gartner, 1983), CRONOS (Boillot et al., 1992), RHODES (Mirchandani and Wang, 2005). These strategies optimize next few switch-over points in a future time horizon by using some complex algorithms. For example, OPAC applies Dynamic Programming (DP) for minimizing total intersection delay at the local intersection and synchronization in the network level. These strategies are conceptually applicable to a whole network, but they suffer from huge computational demand due to complex algorithms. Their real-time implementation is still limited to single intersections.

Urban transportation is one of the major aspects in modern cities. However, the rapid increase of traffic demand makes the congestion problem serious. Consequently, deployment of effective signal control strategy for the transportation network becomes a big challenge in the field. Traffic signal control can be made easily for a single isolated intersection and it has been carried out by various methods (e.g. Gazis and Potts, 1963; D’Ans and Gazis, 1976; Chang and Lin, 2000; Motawej et al., 2010). However, it is very difficult in the case of network wide signal control and it is probably the most serious problem facing traffic engineering. This situation has motivated many researchers to circumvent difficulties involved in traffic network signal control. In particular, Little et al. (1981) published a program called MAXBAND, which is a bandwidth optimization program for calculating signal timing plans. However this strategy is only applicable in under-saturated traffic conditions. Another notable work is TRANSYT-7F, which is a software package utilized for optimizing split, cycle, and offset timing parameters (Hale, 2005). The recent releases perform multi-period optimization by using the Genetic Algorithm (GA), which requires excessive running time and large memory on the computer. Computational efficiency limits the application of TANSYT-7F.

In this context, this paper tries to solve the signal control problem in case of network scale. By analyzing the controllability of the network model, we find that the system is not completely controllable. Hence we decompose the model to isolate the controllable part and construct traffic-responsive strategy for it. By considering the traffic demand as the disturbance, the design of a H∞ controller under state and control constraints is presented. Finally, an example is worked out to illustrate the results.

2. TRANSPORTATION NETWORK MODELLING

MAXBAND and TRANSYT-7F are both fixed-time strategies, which are based on historical rather than real-time data. However traffic demands are varying from time to time. Real-time data should be taken into account to deal with the traffic demand changes, which generates the need of traffic-responsive strategies. There have been some efforts made in this field. One of the most known works is SCOOT (split, cycle, and offset optimization technique), which was firstly presented in Bretherton et al. (1982). It is essentially the TRANSYT-7F traffic model 978-3-902823-13-7/12/$20.00 © 2012 IFAC

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The urban transportation system is the network of intersections. Generally the intersections are controlled by the traffic signals that decide which flows can pass intersections while others have to stop and generate queues. To make sure that every flow has the chance to pass, traffic signals change periodically. A signal cycle c is one repetition of the series of signal combinations in an intersection, and one of these signal combinations is called a phase (Diakaki et al., 2002). Figure 1 shows an intersection with two phases.

10.3182/20120912-3-BG-2031.00014

CTS 2012 September 12-14, 2012. Sofia, Bulgaria

ith (resp, jth) queue. The flow di denotes the output from the ith queue to the outside.

Fig. 1. Two-phase intersection with four legs

Fig. 2. The flows related with ith queue

A cycle is divided into several phases and each phase comprises a loss time and an effective green time. According to Diakaki et al. (2002), loss time is the period which traffic flows can’t use; effective green time is the actual available time for movements. For instance, in the twophase intersection, the cycle is split to two phases, hence there are two effective green times (ge1 , ge2 ) and two loss times (lo1 , lo2 ).

The traffic flows are described by their flow rates. There are two important flow rates. Input flow rate qi is the rate of ri . It changes from cycle to cycle and is uncontrollable by signals. So in kth cycle, ri can be stated as ri (k) = qi (k)c (3) Saturation flow rate si is the rate of leaving flow from the ith queue without interruption during effective green time (Motawej et al., 2011). The saturation flow rates depend on the transportation conditions, which can be considered fixed generally. So the number of vehicles leaving ith queue in one cycle is si gi . Hence, for kth cycle we have: di (k) = si gi (k)λi,i (4) σi,j (k) = sj gj (k)λi,j (5) σj,i (k) = si gi (k)λj,i (6) where i, j ∈ {1, · · · , n}, the exit rate λi,i is the proportion of di to the leaving flow from ith queue, the exchange rate λj,i (resp, λi,j ) is the proportion of σj,i (resp, σi,j ) to Pnthe leaving flow from ith queue (resp, jth queue), and j=1 λj,i = 1. These rates can be considered fixed and known (Diakaki et al., 2002).

The loss times and the cycle are considered constant generally (Diakaki et al., 2002). So the sum of all effective green times in one intersection is constant. For example in the two-phase intersection above, the sum of two effective green times is ge1 + ge2 = c − (lo1 + lo2 ). As a result, there is only one independent variable. If ge1 is chosen as the control variable, and let gi denote the effective green time corresponding to the ith queue, there exists         0 1 ge1 g1 0   g2  ge1   1  g  = g  = −1 ge1 + c − (l + l ) o1 o2 e2 3 c − (lo1 + lo2 ) −1 ge2 g4

Now, let li be the sum of all exchange flows related to the ith queue. According to Figure 2, in kth cycle we have n n X X li (k) = σi,j (k) − σj,i (k)

All intersections have similar formulas. By combining them, we have g = Gu + ξ (1) where g ∈ Rn is the vector of effective green times of all queues, u ∈ Rm is the vector of all chosen independent control variables, G is the coefficient matrix, and ξ is a constant vector.

=

In order to fit physically reasonable signalization, the controller must respect the boundary conditions on gi , i.e., 0 ≤ gi ≤ c However, certain boundary points do not fit physically reasonable signalization settings, because each phase should have a positive effective green time. It is important to select two boundary values gi,min and gi,max so that gi,min ≤ gi ≤ gi,max . Therefore the limitation of control variables is given by the the following constraint set U = {u ∈ Rm / umin ≤ u ≤ umax } (2) T where umin = [u1,min , · · · , um,min ] and umax = [u1,max , · · · , um,max ]T are respectively the maximal and minimal values of the control variable u.

=

j=1,j6=i n X

j=1,j6=i n X

j=1,j6=i

sj gj (k)λi,j −

si gi (k)λj,i

j=1,j6=i

sj gj (k)λi,j − si gi (k)

n X

λj,i

j=1,j6=i

j=1,j6=i

With

n X

Pn

λj,i = 1, it follows n X sj gj (k)λi,j − si gi (k)(1 − λi,i ) li (k) = j=1

(7)

j=1,j6=i

Define a vector Li = [Li,1 , · · · , Li,n ]T in the following manner  −si (1 − λi,i ) : for i = j Li,j = sj λi,j : for i 6= j Then equation (7) can be rewritten as n X Li,j gj (k) + Li,i gi (k) li (k) =

In transportation network, vehicles are distributed in deferent queues, and the vehicle movements can be considered as traffic flows. Figure 2 illustrates all flows related with ith queue, i ∈ {1, · · · , n}. xi is the length of the ith queue. The flow ri denotes the input from the outside to the queue. The flow σi,j (resp, σj,i ), j 6= i, j ∈ {1, · · · , n}, represents the vehicles moving from the jth (resp, ith) queue to the

=

j=1,j6=i n X

Li,j gj (k) =

(8)

LTi g(k)

j=1

Let xi (k) be the ith queue length at the beginning of cycle k. According to the equations (3), (4) and (8), the change 73

CTS 2012 September 12-14, 2012. Sofia, Bulgaria

of the queue lengths in kth cycle can be represented by the following equation xi (k + 1) = xi (k) + li (k) + ri (k) − σi,i (k) = xi (k) + LTi g(k) + qi (k)c − si λi,i gi (k) T

So the transportation system described by equation (11) is not completely controllable, and the controllability dimension is λ. This analysis indicates that the traffic signals can not completely control the transportation network. This matches the experience that if the vehicles which enter the system are too many, only traffic signal control can not prevent congestions, and if there are very few vehicles, even the normal fixed-time signal control can effectively evacuate the traffic. So, the system (denoted as Σ) should be decomposed to two subsystems:

(9)

T

Now, define x = [x1 , · · · , xn ] and q = [q1 , · · · , qn ] , then by combining the above equations, the whole network can be equivalently restated as a state space expression x(k + 1) = x(k) + Lg(k) + q(k)c − Dg(k) = x(k) + q(k)c + (L − D)g(k) where L is a matrix whose ith rows is LTi and D is a diagonal matrix with diagonal elements si λi,i . Since g = Gu + ξ in view of (1), it follows x(k + 1) = x(k) + q(k)c + (L − D)(Gu(k) + ξ) = x(k) + q(k)c + (L − D)Gu(k) + (L − D)ξ = x(k) + q(k)c + Bu(k) + h (10) where B = (L − D)G and h = (L − D)ξ. Now, according to Diakaki et al. (2002), there exists a nominal situation where the input of each queue equals its output. Mathematically, this means q N c + BuN + h = 0, where q N is the nominal input flow rates and uN represents the nominal control variables. This yields h = −q N c − BuN , and hence the equation (10) can be restated as

(1) Controllable subsystem, denoted as Σc , is completely controllable, whose dimension is λ; (2) Uncontrollable subsystem, denoted as Σu , can’t be controlled by v, whose dimension is (n − λ). According to the analysis above, the state variables of uncontrollable subsystem Σu should have no relationship with v. Indeed, if a vector θ ∈ Rn can make B T θ = 0, we have θ T x(k + 1) = θ T x(k) + θ T ω(k) Obviously θ T x is such an uncontrollable variable. Since rank(B) = λ, there must exist exactly (n − λ) linear independent vector solutions of the equation B T θ = 0 (Lax, 2007), these vectors correspond to (n − λ) state variables of Σu . On the other hand, there must be λ linear independent vectors, which are not solution of B T θ = 0 and therefore correspond to the state variables of controllable subsystem Σc .

x(k + 1) = x(k) + q(k)c + Bu(k) − q N c − BuN = x(k) + (q(k) − q N )c + B(u(k) − uN ) Finally, define v(k) = u(k) − uN as the new control variable and ω = (q(k) − q N )c, the equation (10) yields the state space representation of our system x(k + 1) = x(k) + Bv(k) + ω(k) (11) And the control constraint becomes U = {v ∈ Rm / − v 2 ≤ v ≤ v 1 , v 1 , v 2 > 0} (12) where v 1 = umax − uN , v 2 = uN − umin . Since q(k) is unknown vector, ω(k) is the difference between the real input and the nominal one, which can be considered as disturbance vector. It is assumed that the disturbance vector belongs to Ω = {w ∈ Rn / kω(k)k ≤ d} (13)

For the following decomposition, a set of linear independent vectors, H = {θ 1 , · · · , θ n }, is chosen such that (1) these vectors are orthogonal, which means  0 : i 6= j θ Ti θ j = i, j ∈ {1, · · · , n} 1:i = j (2) B T θ = 6 0, for i ∈ {1, · · · , λ}; T (3) B θ = 0, for i ∈ {λ + 1, · · · , n}. These characters appear that H can be regrouped to two sets, and two matrices are defined as

Note that, it makes no sense to speak of negative queue. Hence, the state x(k) ≥ 0 for all k ∈ N. On the other hand, our model doesn’t consider the congestion situation, hence the queue lengths are always beneath their capacities. Let x∗i , i ∈ {1, · · · , n} be the capacity of ith queue length. The system (11) has physical meaning only if x belongs to the region of admissible states X = {x ∈ Rn / 0 ≤ x ≤ x∗ } (14) ∗ ∗ where x = [x1 , · · · , x∗n ]T .

R1 = [θ 1 , · · · , θ λ ] ∈ Rn×λ n×(n−λ)

R2 = [θ λ+1 , · · · , θ n ] ∈ R

λ = rank(B IB · · · I

(17) (18)

For a while, a proposition is stated for subsequent development. Proposition 1. The matrices R1 , R2 defined in the equations (17, 18) have some characters as follows (1) rank(RT1 B) = λ; (2) RT2 B = 0.

For further control issue, the controllability of the above model is considered. According to the difference equation (11), the controllable dimension of the system is n−1

(16)

Proof. Firstly, according to the equation (17), we have

B) = rank(B)

RT1 B = [θ 1 , · · · , θ λ ]T B = [B T θ 1 , · · · , B T θ λ ]T

The matrix B ∈ Rn×m , where n is the number of all queues and m is the number of control variables which correspond to the independent green times. Since the number of queues is obviously bigger than the number of green times, we have λ = rank(B) ≤ min(m, n) = m < n (15)

= [φ1 , · · · , φλ ]T ∈ Rλ×m where φi = B T θ i , i ∈ {1, · · · , λ}. Hence the rank of RT1 B depends on the linear dependency of φ1 , · · · , φλ . Considering the equation 74

CTS 2012 September 12-14, 2012. Sofia, Bulgaria

a 1 φ1 + · · · + a λ φλ = 0

(19)

T

⇒ B [a1 θ 1 + · · · + aλ θ λ ] = 0 where a1 , · · · , aλ are coefficients. According to the characters of H, except zero vector, any linear combination of θ 1 , · · · , θ λ is not the solution of B T θ = 0. So the equation (19) has only one solution a1 = · · · = aλ = 0. That means φ1 , · · · , φλ are linear independent, hence rank(RT1 B) = λ. Then since the vectors, θ λ+1 , · · · , θ n , are the solutions of B T θ = 0, we obviously have RT2 B = 0. The proof is completed.

Fig. 3. Standard H∞ problem where x is the states, the dimensions of x, ω, u, z and y are n, m1 , m2 , p1 and p2 respectively. The H∞ problem is to find a proper feedback controller to make the H∞ norm of close-loop transfer function less than a given positive number γ, which leads to kzk2 /kωk2 < γ.

With this proposition and the analysis above, we have that xc = RT1 x ∈ Rλ (resp, xu = RT2 x ∈ Rn−λ ) is the vector of state variables of the controllable subsystem Σc (resp, uncontrollable subsystem Σu ). By multiplying the matrices R1 and R2 respectively in the two sides of the equation (11), the difference equations of two subsystems are generated as follows Σc : xc (k + 1) = xc (k) + B c v(k) + ω c (k) (20) Σu : xu (k + 1) = xu (k) + ω u (k) (21)

Suppose the matrix D 22 = 0 and the feedback control law u = Ky, where K ∈ Rm2 ×p2 is the gain matrix, the close-loop form of the system can be stated as x(k + 1) = Acl x(k) + B cl ω(k) (27) z(k) = C cl x(k) + D cl ω(k) where Acl = A + B 2 KC 2 , B cl = B 1 + B 2 KD 21 , C cl = C 1 + D 12 KC 2 , and D cl = D 11 + D 12 KD 21 . In this case, according to Gahinet and Apkarian (1994) the solution of H∞ problem is that there exists a positive definite matrix P such that the following LMI is feasible   −P −1 ∗ ∗ ∗  AT −P ∗ ∗   cl ≺0 (28)  B cl 0 −γI ∗  0 C cl D cl −γI

where B c = RT1 B, ω c = RT1 ω ∈ Rλ and ω u = RT2 ω ∈ Rn−λ are disturbances of two subsystems. The control constraint (12) doesn’t change. And the domain (14) of x can generate the domains of xc and xu Xc = {xc ∈ Rλ / − xc2 ≤ xc ≤ xc1 , xc1 , xc2 > 0} (22) Xu = {xu ∈ Rn−λ / − xu2 ≤ xu ≤ xu1 ,

xu1 , xu2 > 0} (23) Now, since xu is not controlled by v, the set Xu must be positively invariant in order to ensure physical property of the system. That is xu (0) ∈ Xu implies xu (k) ∈ Xu for all k. Indeed, observe that k−1 X ω u (i) xu (k) = xu (0) +

where ∗ represents the transpose of the element across the diagonal, and the symbol ≺ means that the left matrix is negative definite. 3.2 Control of Transportation Network

To reduce kxc k and bound the impact of ω c , H∞ control method is applied to the controllable subsystem Σc . Figure 4 illustrates this problem with the difference equation (20) and the feedback control law v(k) = K c xc (k).

i=0

Hence if xu (0) ∈ Xu then xu (k) ∈ Xu is equivalent to −xu2 − xu (0) ≤

k−1 X

ω u (i) ≤ xu1 − xu (0)

(24)

i=0

On the other hand, the region of the disturbance (13) can generate the domain of ω c as follows Ωc = {ω c ∈ Rλ / kω c k2 ≤ dc } (25) Since the traffic signals have no influence to the uncontrollable subsystem (21), it is the controllable subsystem (20) which should be concerned by control issue.

Fig. 4. H∞ problem of Σc Comparing with the standard H∞ problem, in system Σc , the state is xc , the disturbance input is ω c , the control input is v, the controlled output and the measured one are both xc (in kth cycle z(k) = xc (k + 1), y(k) = xc (k)). So according to (26), the efficient matrices of the system becomes A = I B1 = I B2 = Bc C 1 = I D 11 = I D 12 = B c C 2 = I D 21 = 0 D 22 = 0 Furthermore, with the feedback control law v(k) = K c xc (k), the efficient matrices of the close-loop form (27) are given as follows Acl = I + B c K c B cl = I C cl = I + B c K c D cl = I

3. H∞ CONTROL 3.1 Review of H∞ Control According to Gahinet and Apkarian (1994), the standard H∞ problem deals with the system illustrated in Figure 3, which includes disturbance inputs ω, control inputs u, controlled outputs z and measured outputs y. Considering the linear discrete-time case, the system is stated by the state-space expression x(k + 1) = Ax(k) + B 1 ω(k) + B 2 u(k) z(k) = C 1 x(k) + D 11 ω(k) + D 12 u(k) (26) y(k) = C 2 x(k) + D 21 ω(k) + D 22 u(k) 75

CTS 2012 September 12-14, 2012. Sofia, Bulgaria

Hence the LMI (28) becomes   −P −1 ∗ ∗ ∗ I + K T B T −P ∗ ∗   ≺0 c c  I 0 −γI ∗  0 I + B c K c I −γI

4. EXAMPLE Consider a transportation system including two intersections shown in Figure 5. Every intersection has two phases and hence there are four effective green times g1,e1 , g1,e2 , g2,e1 and g2,e2 . Note that, g1,e1 (resp, g1,e2 ) is related to the phase that x1 and x2 (resp, x3 and x4 ) are approved to pass. g2,e1 and g2,e2 follow the same manner. Choosing g1,e1 and g2,e1 as the control variables, we have u = [g1,e1 , g2,e1 ]T .

(29)

where P is a positive definite matrix, γ is a positive scalar. In this paper, we choose P = I for further development. We then have the following theorem. Theorem 2. If there exists a matrix K c solution of the LMI condition   −I ∗ ∗ ∗ T T I + K c B c −I ∗ ∗  (30)  ≺0 I 0 −γI ∗ 0 I + B c K c I −γI then the control v = K c xc renders kxc k2 /kω c k2 < γ. Now, once the gain matrix K c is given, the feedback control v = K c xc should not violate the constraint (12). This means that every trajectory xc (k; xc (0)) inside the region Xc does not leave the following region for any instant k ∈ N Uxc = {xc ∈ Rλ / − v 2 ≤ K c xc ≤ v 1 } (31) In other words, the feedback control must ensure the following inclusion Xc ⊆ Uxc (32) Now let K c1 = sup(K c , 0) and K c2 = −inf (K c , 0), we have K c = (K c1 − K c2 ), which yields K c xc = (K c1 − K c2 )xc ≤ K c1 xc1 + K c2 xc2 K c xc = (K c1 − K c2 )xc ≥ −K c1 xc2 − K c2 xc1 Hence, if there exist the non-negative matrices K c1 and K c2 such that K c1 xc1 + K c2 xc2 ≤ v 1 (33) K c2 xc1 + K c1 xc2 ≤ v 2 (34) the constraint (31) is satisfied. By summarizing, one way of determining K c1 and K c2 is by solving the following optimization min γ (35) γ,K c1 ,K c2 subject to K c1 ≥ 0, K c2 ≥ 0, K c = K c1 − K c2 ; LMI constraint (30); linear constraints (33) and (34).

Fig. 5. Two-intersection system Other parameters are given here: the cycle c = 90 s; the loss times are ignored for simplicity; the maximal and minimal green times are 70 s and 20 s respectively; all saturated flow rates are 0.6 veh/s; the exchange rates r5,1 = r2,6 = 0.9, and consequently the exit rates r1,1 = r6,6 = 0.1; the queue capacities x∗ = [80, 60, 70, 70, 60, 80, 70, 70]T veh; the nominal input flow rates q N = [0.3, 0.03, 0.3, 0.3, 0.03, 0.3, 0.3, 0.3 ]T veh/s, which generates the nominal green times uN = [45, 45]T s. Hence, the state space model of the system is x(k + 1) = x(k) + Bv(k) + ω(k) (36) where x(k) is the vector of queue lengths, v(k) = u(k) − uN is control vector, ω(k) = (q(k) − q N )c is disturbance vector, and the matrix   −0.6 0 −0.6 0.54   0.6 0     0.6 0   B=  0.54 −0.6 veh/s    0 −0.6   0 0.6 0 0.6 The control constraints for this system are U = {v ∈ R2 / − v 2 ≤ v ≤ v 1 } (37) where v 1 = [25, 25]T s, and v 2 = [25, 25]T s. The state vector belongs to the region X = {x ∈ R8 / 0 ≤ x ≤ x∗ } (38)

We are now in position to state the following result. Theorem 3. Suppose that inequalities (24) are satisfied. If there exists matrices K c1 and K c2 solution of the optimal problem (35), then the feedback control v = (K c1 − K c2 )xc guarantees the following (1) kxc k2 /kω c k2 < γ; (2) the constraint of control is respected.

The controllable dimension λ = rank(B) = 2. The matrices R1 ∈ R8×2 and R2 ∈ R8×6 are constructed with the manner in section 2. Then, we have the controllable state xc = RT1 x and the uncontrollable one xu = RT2 x.

Furthermore with kω c k2 ≤ dc , we have kxc k2 ≤ γdc . Hence if dc ≤ γ −1 inf (xc1 , xc2 ), this H∞ control respects the constraint xc ∈ Xc .

By using the CVX, a package for specifying and solving convex programs (Grant and Boyd, 2011) in MATLAB, we solve the control problem to get the minimal γ = 3.99, and the corresponding control law v = (K c1 − K c2 )xc with the gain matrices

It is important to note that this optimization with LMI and linear conditions is a convex programming problem, which can be effectively solved by using standard numerical tools (Grant and Boyd, 2011). 76

CTS 2012 September 12-14, 2012. Sofia, Bulgaria

K c1 =

  0.22 0 s/veh 0.06 0

K c2 =

control has been adapted and a state feedback control under state and control constraints has been solved by means of a LMI formulation.

  0 0.01 s/veh 0 0.21

Now, the performance of the proposed control method is illustrated in the simulation studies of the system. Suppose the initial queue lengths are x(0) = [35, 40, 45, 35, 40, 15, 58, 63]T veh. The system with nominal inputs (which means there is no disturbance) is firstly simulated. Table 1 shows the effective green time variation. It is observed that, effective green times differ from cycle to cycle. Meanwhile, as illustrated in Figure 6, the application of feedback control eliminates the controllable state xc in 20 cycles.

There are several issues that deserve further investigation. One of them is to find better evaluation of transportation system performance, and more precise analysis of the system constraints are also needed. Future work will deal with these issues. REFERENCES Boillot, F., Blosseville, J.M., Lesort, J.B., Motyka, V., Papageorgiou, M., and Sellam, S. (1992). Optimal signal control of urban traffic networks. In Road Traffic Monitoring, 1992 (IEE Conf. Pub. 355). IET. Bretherton, R.D., Hunt, P., Robertson, D., and Royle, M. (1982). The SCOOT on-line traffic signal optimisation technique. Traffic Engineering and Control, 23(4), 190– 192. Chang, T. and Lin, J. (2000). Optimal signal timing for an oversaturated intersection. Transportation Research Part B: Methodological, 34(6), 471–491. D’Ans, G.C. and Gazis, D.C. (1976). Optimal control of oversaturated Store-and-Forward transportation networks. Transportation Science, 10(1), 1 –19. Diakaki, C., Papageorgiou, M., and Aboudolas, K. (2002). A multivariable regulator approach to traffic-responsive network-wide signal control. Control Engineering Practice, 10(2), 183–195. Gahinet, P. and Apkarian, P. (1994). A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 4(4), 421–448. Gartner, N.H. (1983). OPAC: A demand-responsive strategy for traffic signal control. In Transportation Research Record 906. U.S. Dept. Transp., Washington, DC. Gazis, D. and Potts, R. (1963). The oversaturated intersection. In Proceedings of the Second International Symposium on the Theory of Road Traffic Flow, 221– 237. Grant, M. and Boyd, S. (2011). CVX: Matlab software for disciplined convex programming, version 1.21. Hale, D. (2005). Traffic Network Study Tool: TRANSYT7F, United States Version. McTrans Center in the University of Florida, Gainesville. Lax, P.D. (2007). Linear algebra and its applications. Wiley-Interscience. Little, J.D.C., Kelson, M.D., and Gartner, N.H. (1981). MAXBAND: a program for setting signals on arteries and triangular networks. In Transportation Research Record 795. U.S. Dept. Transp., Washington, DC. Mirchandani, P. and Wang, F. (2005). RHODES to intelligent transportation systems. IEEE Intelligent Systems, 20(1), 10– 15. Motawej, F., Bouyekhf, R., and El Moudni, A. (2010). Energy-based control for an over-saturated three-phase intersection. In Proceedings of the 12th IFAC Symposium on Large Scale Systems: Theory and Applications. France. Motawej, F., Bouyekhf, R., and El Moudni, A. (2011). A dissipativity-based approach to traffic signal control for an over-saturated intersection. Journal of the Franklin Institute, 348(4), 703–717.

Table 1. Effective green time variation cycle g1,e1 ( s) g2,e1 ( s) cycle g1,e1 ( s) g2,e1 ( s)

1 36 33 9 44.4 43.9

2 38.5 36 10 44.5 44.2

3 40.3 38.3 11 44.7 44.4

4 41.7 40 12 44.8 44.5

5 42.6 41.3 13 44.8 44.7

6 43.3 42.2 14 44.9 44.7

7 43.8 42.9 15 44.9 44.8

8 44.1 43.5 16 44.9 44.9

60 x 40

x

c,1 c,2

x

c

20 0 −20 −40

0

5

10 cycle

15

20

Fig. 6. xc trajectory in case of nominal input Then we add a persistent disturbance ω c = [1, 1]T . The system is simulated and xc is illustrated in Figure 7. It is observed that after increasing in the beginning, xc arrives a stable state, and the maximal kxc k2 /kω c k2 in this whole period is 3.58, less than γ. 4 3

x

c

2 1 xc,1

0

x

c,2

−1 20

25

30 cycle

35

40

Fig. 7. xc trajectory in case of persistent disturbance 5. CONCLUSIONS This paper presented a transportation network model and constructed a traffic-responsive strategy by applying H∞ control method. More precisely, the controllability of the model has been studied and it is found that the system is not completely controllable. So we isolated the controllable part and chose it as the objet of control issue. Since the traffic demand is considered as disturbance, the H∞ 77