Supervisory Model Predictive Control for freeway traffic systems

Supervisory Model Predictive Control for Freeway Traffic Systems Antonella Ferrara, Simona Sacone, Silvia Siri

Abstract— The application of Model Predictive Control schemes to real world complex plants, such as freeway traffic systems, is sometimes limited because of the necessity of tuning the control action depending on the system operating condition, and the significant computational burden inherent to the methodology. The MPC-based scheme presented in this paper aims at overcoming the mentioned limitations by using a supervisory control approach. The supervisor, at each time step, chooses among three possible actions: i) the controller needs to be changed and the new control action is computed, ii) no change is made to the controller, but it is necessary to recompute the control action, iii) neither is the controller varied, nor it is necessary to recompute the control action and the already determined control law is kept. In other terms, the control sequence is recomputed according to an event-triggered mechanism in which the control action is properly tuned to the system conditions. In the paper, the proposed control scheme is developed with reference to a class of non-linear systems admitting a model of Mixed Logical Dynamical type, which is suitable to describe freeway traffic systems, and the inputto-state practical stability of the controlled system is proved. Finally, the application of the proposed control scheme to a freeway traffic system is discussed and studied via simulation.

I. I NTRODUCTION Model Predictive Control (MPC) is nowadays a reliable control technique widely used for process control and applications of different nature [1], [2], [3], among which the control of freeway traffic systems, as evidenced by the numerous papers which have appeared in the literature (see, for instance, [4], [5], [6], [7]). In spite of its effectiveness, MPC has a major drawback since the computational load to solve the optimization problem grows with the system complexity. This significantly limits the on-line usage of MPC algorithms especially in the context of freeway traffic control, where the number of states and control variables can be very high by virtue of the large scale geographical distribution of the system to control. A number of solutions have been proposed to overcome this limitation, leading to the formulation of Fast MPC algorithms [8], with the controller replaced by a lookup table computed off-line, and Plug and Play MPC schemes [9], in which it is possible to use simpler controllers by applying a decentralized synthesis procedure. This work has been supported by the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n. 257462 HYCON2 Network of excellence. A. Ferrara is with the Dept. of Electrical, Computer and Biomedical Engineering, University of Pavia, Italy

[email protected] S. Sacone and S. Siri are with the Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Italy

[email protected], [email protected]

In this paper, we propose a different way to reduce the computational burden of the MPC algorithms oriented to deal with traffic control systems. The idea is to rely on an eventtriggered algorithm inspired by [10], and to include it into a supervisory control scheme, giving rise to a supervisory event-triggered MPC scheme. Event-triggered control strategies ([11], [12]), in which sampling is dictated by the occurrence of events, rather than time-triggered, have become popular in recent years because of the difficulty, in many applications, to apply equidistant sampling for feedback control, due to limited communication resources or inadequate computation power. In our proposal, we design a supervisor so that the asynchronous sampling provided by the event-triggered formulation is exploited to reduce the number of times in which the optimization problem underlying the MPC strategy is solved, i.e. the control law is updated only when it is no longer adequate. On the other hand, a supervisory control scheme (see, for instance, [13], [14], [15], [16]) can be capable of choosing among a family of candidate controllers the one which is more likely to provide the best control performance at the current time step. In our paper, also this feature is effectively exploited. More specifically, relying on the output of a suitable monitoring signal generator, the supervisor not only triggers the recomputations of the control law only when strictly necessary, but also selects, at any sampling time, the appropriate controller to be used. Since our aim is to design a control scheme suitable for on-line usage in freeway traffic systems, we assume that the process dynamics could be affected by interacting physical laws, logical rules and operating constraints. A quite natural way to represent and predict the process behaviour, in such a case, is to adopt a Mixed Logical Dynamical (MLD) model (see [17]). In our proposal the MLD model is used both as a predictor in solving the MPC problem, and as a process emulator to accomplish the generation of an appropriate monitoring signal. The set of controllers, for the sake of uniformity, includes controllers that are all of MPC type, but the scheme is designed to host controllers even of different nature. The stability properties of the controlled system are discussed in the paper while the performance of the proposed scheme when applied to a freeway is analyzed in simulation. This paper is structured as follows. In Section II the proposed supervisory control scheme, with all its modules, is outlined. The stability properties of the controlled system are investigated in Section III. The application of the overall control scheme to freeway traffic systems is described in Section IV, while some conclusive remarks are outlined in Section V.

II. P ROBLEM F ORMULATION uex

We consider a non-linear process P whose dynamics can be expressed as x(k + 1) = f (x(k), u(k)) y(k) = g (x(k)) +

ν ˆ

(1)

σ

T

−

Cσ

u

ν

P

+

ν ˜

M

µ

S

n

where k ∈ Z , x(k) ∈ X ⊂ R is the system state accessible for measurement, u(k) ∈ U ⊂ Rm is the control signal and y(k) ∈ Rp is the measured output. It is assumed that X and U are closed and bounded sets. System (1) can be affected by additive exogenous inputs, thus becoming x(k + 1) = f (x(k), u(k)) + uex (k) y(k) = g (x(k))

(2)

where uex (k) ∈ Rn and kuex (k)k ≤ υ, ∀k. We assume that system (2), starting from any initial condition and subject to whichever u(·) and uex (·), does not present finite time escape behaviours. When the evolution of the plant is influenced by logic rules, on/off inputs, piecewise linear functions, it can be useful to associate with (1) a Mixed Logical Dynamical model, namely Pˆ , described by the following set of relationships x(k + 1) = Ax(k) + B1 u(k) + B2 δ(k) + B3 z(k) y(k) = Cx(k) + D1 u(k) + D2 δ(k) + D3 z(k)

(3a) (3b)

E2 δ(k) + E3 z(k) ≤ E1 u(k) + E4 x(k) + E5

(3c)

rd

rc

where δ(k) ∈ {0, 1} and z(k) ∈ R represent respectively auxiliary logical and real variables. In MLD systems the auxiliary variables are used to represent logical conditions or nonlinear relationships by means of mixed linear equalities and inequalities (by exploiting the propositional logic). Note that, in the general MLD form, the state and input vectors can be composed of both real-valued and binary quantities. Anyway, in this work, the considered state and input vectors are only composed of real-valued quantities. Moreover, completely well-posed MLD systems, according to the definition provided in [17], are considered in this paper. This means that, given x(k) and u(k), the values of δ(k) and z(k) are uniquely defined through inequalities (3c). The MLD model (3) is extended to include the additive and bounded exogenous input uex (·), as follows. x(k + 1) = Ax(k) + B1 u(k) + B2 δ(k) + B3 z(k) + B6 uex (k) y(k) = Cx(k) + D1 u(k) + D2 δ(k) + D3 z(k) E2 δ(k) + E3 z(k) + E6 uex (k) ≤ E1 u(k) + E4 x(k) + E5

(4a) (4b) (4c)

The MLD system (4) will be used as an emulator/predictor of (2) in the proposed control scheme. To control the described class of systems, we propose a scheme which is inspired by the vast literature on supervisory control. The proposed control scheme is depicted in Fig. 1 and its main elements are: •

Pˆ

the set of controllers C;

uex

Fig. 1.

The proposed supervisory event-triggered control scheme

the process model, or the actual process in a practical implementation, P ; ˆ; • the emulator/predictor P • the monitoring signal generator M ; • the supervisor S. In the control scheme ν(k) is an extended output vector which includes both the state vector x(k) and the output  T vector y(k), i.e. ν(k) = x(k)T ; y(k)T . Analogously νˆ(k) is the extended output vector of the emulator/predictor, i.e.  T νˆ(k) = x ˆ(k)T ; yˆ(k)T . We assume that the exogenous bounded input signal uex (k), which acts as input to the process, is also available to the emulator/predictor. Note that, in spite of some affinities, the idea behind our proposal differs from the one underlying classical supervisory control. In that case, the aim is to select at any time instant the candidate controller to be used in the control scheme, and the choice is made relying on the monitoring signals which account for a measure of the discrepancies between the unknown plant and the various model structures that the estimator can incorporate. So, once the most appropriate model has been identified, the associated controller is used in the scheme. In other terms, the set of controllers contains a candidate controller for any element of the set P of indexes of possible plant models, P being a finite index set. In our case, we do not need an estimator, but an emulator of the plant model (i.e., the MLD system), capable of representing the different plant behaviours through the different configurations of the auxiliary logical and continuous variables. The emulator is used to generate the error signal ν˜(k) = ν(k) − νˆ(k) which is used by the supervisor to decide, at any sampling time, whether to compute the control action or to use the control samples already present in a buffer. The supervisor S is based on a set of switching conditions (to decide whether to change the controller) and a set of triggering conditions (to decide whether to recompute the control action), which will be illustrated in the following. •

A. Set of Controllers We consider a finite set of MPC controllers C = {C1 , C2 , . . . , CKc } as well as a controller selection index σ(k), taking value in the set Σ = {1, . . . , Kc }. In this paper, we assume that all the controllers of the set C are of MPC type; however, the proposed framework has a general

validity, in the sense that controllers of different nature can be present in C. Then, each controller is computed by solving a Finite-Horizon Optimal Control Problem (FHOCP) and the different controllers in set C differ for the FHOCP formulation. Moreover, in the FHOCP statement the following concepts of equilibrium state and equilibrium pair are used. Definition 1: A vector xe ∈ X is an equilibrium state for the MLD system (3) and for the input vector ue ∈ U if, starting at time step k0 ∈ Z with x(k0 ) = xe and with u(k) = ue , ∀k ≥ k0 , it is x(k) = xe , ∀k ≥ k0 , ∀k0 ∈ Z. The pair (xe , ue ) is denoted as an equilibrium pair.  In the following, vectors δ e and z e denote the admissible values of the auxiliary variables with reference to the equilibrium pair (xe , ue ). The problem to be solved at the generic k-th time step to determine the controller Cσ , with σ ∈ Σ, can be stated as Problem 1: Given a system in MLD form (3) and its initial conditions x(k), find the optimal control sequence u(h), h = k, . . . , k + Kpσ − 1, minimizing the cost function Jσ (k) =

k+Kpσ −1 

X

kx(h) − xe k2Q1σ + ku(h) − ue k2Q2σ

h=k

+kδ(h) − δ e k2Q3σ + kz(h) − z e k2Q4σ



(5)

subject to (3a), (3c) and x(k + Kpσ ) = xe

(6)

where kxk2Q := xT Qx, and Q1σ , Q2σ , Q3σ , Q4σ are symmetric positive definite matrices.  The control law derived by solving Problem 1 within a MPC framework is named mixed integer predictive control (MIPC) law. In [17] it has already been proved that the MIPC law stabilizes the system defined by (3). Let us define the control law obtained at time step k as a solution to Problem 1, with reference to σ ∈ Σ, as u◦σ (k+h|k), h = 0, . . . , Kpσ −1. It is worth noting that the controllers differ in the choice of the weights used in the cost function and in the length of the MPC optimization horizon. Thus, any member of the set of controllers is uniquely specified by the parameter set Cσ = {Kpσ , Q1σ , Q2σ , Q3σ , Q4σ }. B. Monitoring Signal Generator The monitoring signal generator receives as input the error signal ν˜(k) and returns the monitoring signal µ(k) as a function of such an error. Any well defined measure of the error can be a reasonable monitoring signal. For instance, one could have µ(k) = k˜ ν (k)k or µ(k) =  2 T 2 ν˜1 (k) ν˜22 (k) . . . ν˜n+p (k) . Note that a monitoring signal generated by filtering k˜ ν (k)k with a first-order discrete-time filter is also a good choice. C. Supervisor In analogy with the classical supervisory control, the supervisor is in charge of choosing the controller Cσ to be used at time step k to generate u(k), and then produces at

any time step k a controller selection index σ(k). In the case when σ(k) = σ(k − 1), the supervisor has also to decide whether to recompute the control law (solving the MPC optimization problem associated with Cσ ) or to use the already computed control law. In particular, in this latter case, the control variable u(k) is the sample following, in the buffer which includes all the elements of the control sequence determined the last time the optimization problem was solved, the one used at time step k − 1. This second task is accomplished by the supervisor by setting the controller activation signal T (k) to 1 in case the recomputation is necessary, to 0 otherwise. To determine the value of the controller selection index σ(k), the supervisor relies on switching conditions basically depending on the system state; instead, to determine the controller activation signal T (k), the supervisor considers a set of triggering conditions based on the signal µ(k). Such triggering conditions can be expressed as a single condition, i.e., for instance, ( 1 µ(k) ≥ α T (k) = (7) 0 otherwise where α is a design constant parameter to be specified. The switching conditions, on the other hand, are used to choose the controller to be applied to the plant depending on the actual system state and on suitable expressions necessary to guarantee some stability properties of the overall control scheme. Let us assume that the set X ⊂ Rn can be partitioned into card(Σ) subsets, denoted as Xσ , each of which is associated with a specific value of σ ∈ Σ. The switching condition allowing σ(k) = σ ¯ in our case is x(k) ∈ Xσ¯ ∧ Jσ¯ (k)−Jσ(k−1) (k−1) ≤ −α3 (kx(k)k)+α4 (υ) (8) where α3 (·) is a K∞ -function and α4 (·) is a K-function. III. S TABILITY A NALYSIS Without loss of generality, from now on let us suppose that the origin is an equilibrium pair for system (3) and that the corresponding admissible values of the auxiliary variables δ e and z e are null. The aim of the present section is to discuss the stability properties of the proposed control scheme, by verifying in particular the property of input-to-state practical stability to which the following definitions are related. Definition 2 (from [18]): A system x(k + 1) = ϕ(x(k), w(k))

(9)

where x(k) is the system state and w(k) is an input disturbance such that kw(k)k ≤ ω, ∀k, is input-to-state practically stable (ISpS) if there exist a KL function β(·, ·), a K-function γ(·) and a non-negative constant c such that kx(k)k ≤ β(kx0 k, k) + γ(ω) + c,

∀k

(10) 

Definition 3 (from [19]): A continuous function V (·) : ℜn −→ ℜ+ is a ISpS-Lyapunov function for system (9)

if there exist K∞ -functions α1 (·), α2 (·) and α3 (·), a Kfunction α4 (·) and non-negative constants c1 and c2 such that α1 (kx(k)k) ≤ V (x(k)) ≤ α2 (kx(k)k) + c1 ,

∀k

V (ϕ(x(k),w(k))) − V (x(k)) ≤ − α3 (kx(k)k) + α4 (kw(k)k) + c2 ,

∀k

(11) (12) 

According to the previous definitions, in the following results we will denote with w(k) the input signal including both the control signal and the exogenous input. Now ignore the supervisor and only consider the application of MPC based on a predictor of MLD type to system (2). With reference to this case, the following result can be proved. Theorem 1: Consider a generic time step k and a generic σ ∈ Σ; if the state x(k) is known and is such that a feasible solution of Problem 1 formulated on (3) exists, system (2) subject to the application of the MIPC law is ISpS.  Sketch of the Proof: Let us consider the optimal cost function (5) as a candidate ISpS-Lyapunov function and let us denote Jσ (k) as V (x(k)). First of all, it is possible to verify that V (x(k)) ≥ λmin (Q4σ )kx(k)k2 where λmin (Q4σ ) is the minimum eigenvalue of matrix Q4σ . Then, by some algebraic computations, it is possible to show that V (x(k)) ≤ ψ1 kx(k)k2 + ψ2 kx(k)k + ψ3 , where ψ1 , ψ2 and ψ3 are suitable constants. Hence, condition (11) is verified. In order to verify condition (12), suppose to optimally solve the FHOCP at time step k for a generic σ and to determine the control strategy u◦σ (k + h|k), h = 0, . . . , Kpσ − 1. Moreover, define a control sequence u ˜σ (k + h) = u◦σ (k + h|k), h = 1, . . . , Kpσ − 1, and u ˜σ (k + Kpσ ) = 0. Let J n (x◦ (k + 1|k)) denote the cost function obtained starting from the state x◦ (k + 1|k) by applying the control sequence u ˜σ (k + h), h = 1, . . . , Kpσ , to the nominal system (3) and J d (x(k + 1)) denote the cost function starting from x(k + 1) by applying the same control sequence to system (2). Then, it can be written J d (x(k + 1)) − V (x(k)) = J d (x(k + 1)) − J n (x◦ (k + 1|k)) + J n (x◦ (k + 1|k)) − V (x(k))

(13)

By analysing separately the two terms J d (x(k + 1)) − J n (x◦ (k + 1|k)) and J n (x◦ (k + 1|k)) − V (x(k)) and by exploiting the fact that a generic function kv1 k2Qv −kv2 k2Qv is Lipschitz in a closed and bounded domain, it is possible to obtain V (x(k + 1)) − V (x(k)) ≤ J d (x(k + 1)) − V (x(k)) (14) ≤ −α3 (kx(k)k) + α4 (kw(k)k) + c2 as in condition (12). As a second step, it is necessary to prove that the ISpS property is maintained even when the control is implemented in the event-triggered fashion. Let us gather in set Υ = {ki ∈ Z+ : T (ki ) = 1}, ki+1 > ki , the time steps in which the FHOCP is solved. This means that in a generic time step k ∈ / Υ, ki < k < ki+1 , ki , ki+1 ∈ Υ, the MIPC law determined at time step ki is adopted. Of course, it must be

ki+1 − ki ≤ Kpσ(ki ) − 1. The control law resulting from the application of the event-triggered policy is named hereafter mixed integer event-triggered predictive control (MIETPC) law. It is possible to prove the following result. Theorem 2: Consider a generic time step k and a generic value σ ∈ Σ; if the state x(k) is known and is such that a feasible solution of Problem 1 formulated on (3) exists, system (2) subject to the application of the MIETPC law is ISpS.  Sketch of the Proof: First of all it is proved that, considering a time step ki ∈ Υ, it holds V (x(ki +j))−V (x(ki )) ≤ −α3 (kx(ki )k)+α4 (ω)+c2 (15) ∀j = 1, . . . , ki+1 − ki − 1. The case j = 1 is directly proved by Theorem 1, while the cases j > 1 are proved following the same line of reasoning of the proof of Theorem 1. Then, condition (10) must be verified, taking inspiration from the proof of Theorem 1 in [21]. By virtue of (15), the results reported in the proof of Theorem 1 and by exploiting Lemma 3.5 in [22], condition (10) is firstly verified for all time steps ki ∈ Υ. Then, with some algebra, condition (10) is verified also for k ∈ / Υ. Finally, one has to guarantee that the supervisory eventtriggered MPC still ensures the ISpS of the controlled MLD system, as stated in the following theorem. Theorem 3: System (2) subject to the application of the supervisory event-triggered MPC law is ISpS.  Sketch of the Proof: On the basis of Theorem 1 and 2, it is possible to claim that whichever controller in the set C is applied at a certain time instant, the ISpS of the controlled system is guaranteed. One still has to prove that the switching among the various controlled systems results in a ISpS system. The proof of this fact can be developed by observing that the switching conditions (8) allow to guarantee that (12) is fulfilled also in the time steps in which a switching between two different controllers takes place. IV. C ASE S TUDY: H IGHWAY T RAFFIC C ONTROL Now we come back to the application problem from which we received inspiration to formulate the proposed control scheme, i.e. freeway traffic control. We consider the firstorder macroscopic traffic model known as Cell Transmission Model (CTM), which is based on the subdivision of the freeway into cells and on the discretization of the time horizon [23]. Let N be the number of cells and let K be the number of time steps ; let T denote the sample time and Li denote the length of cell i. Let us define the following quantities referred to a generic time step k: • ρi (k) traffic density of cell i [veh/km]; + − • Φi (k) total flow entering, Φi (k) total flow exiting cell i [veh/h]; • φi (k) mainstream flow entering cell i from cell i − 1 [veh/h]; • li (k) queue length in the on-ramp of cell i [veh]; • ri (k) flow entering cell i from the on-ramp [veh/h]; • di (k) on-ramp demand referred to cell i [veh/h]; • si (k) flow exiting cell i through the off-ramp [veh/h];

• •

Di (k) demand, Si (k) supply of cell i [veh/h]; βi ∈ [0, 1) split ratio, Fi capacity [veh/h], ρ¯i jam density [veh/km], wi congestion wave speed [km/h], vi free flow speed [km/h], pi ∈ [0, 1] priority of on-ramp flow with respect to the mainstream flow of cell i.

The dynamic model is given by the state equations for the traffic density ρi (k) and the queue length li (k), i = 1, . . . , N , k = 1, . . . , K   T + − ρi (k + 1) = ρi (k) + Φ (k) − Φi (k) (16) Li i   li (k + 1) = li (k) + T di (k) − ri (k)) (17) where

Φ+ i (k) = φi (k) + ri (k)

(18)

Φ− i (k) = φi+1 (k) + si (k)

(19)

si (k) =

βi φi+1 (k) 1 − βi

The mainstream and on-ramp flows are obtained as Di−1 (k) + di (k) +

li (k) ≤ Si (k) T

then φi (k) = Di−1 (k),

ri (k) = di (k) +

i=1

h=k

(20)

Referring to cell i, it is useful to define the demand of cell i − 1 and the supply of cell i, as follows  Di−1 (k) = min (1 − βi−1 )vi−1 ρi−1 (k), Fi−1 (21)  Si (k) = min wi (¯ ρi − ρi (k)), Fi (22) If

three sets of binary auxiliary variables δi,j (h), j = 1, . . . , 3, i = 1, . . . , N , h = k, . . . , k + Kpσ − 1, and five sets of real auxiliary variables zi,j (h), j = 1, . . . , 5, i = 1, . . . , N , h = k, . . . , k + Kpσ − 1. Then, the finite-horizon optimal control problem to be solved at the generic time step k can be stated as follows. Problem 2: Given the initial conditions on the density and the queue length ρi (k) and li (k), i = 1, . . . , N , the demand of the cell before the first one D0 (h), h = k, . . . , k + Kpσ − 1, the supply of the cell after the last one SN +1 (h), h = k, . . . , k + Kpσ − 1, and the on-ramp demands di (h), i = 1, . . . , N , h = k, . . . , k + Kpσ − 1, find the optimal control variables ri (h), i = 1, . . . , N , h = k, . . . , k + Kpσ − 1, minimizing the cost function k+Kpσ −1 N  X X ρ l J(k) = γσ,i (ρi (h) − ρei )2 + γσ,i (li (h) − lie )2

li (k) T

+

r γσ,i (ri (h)

−

rie )2

+

3 X

δ e 2 γσ,i,j (δi,j (h) − δi,j )

j=1

+

5 X

z e 2 γσ,i,j (zi,j (h) − zi,j )

j=1

 (24)

subject to the CTM model in MLD form.  The proposed control scheme has been applied to a freeway stretch composed of 5 cells with two ramps, between cells 2 and 3, and between cells 3 and 4. The data used in the simulation campaign realized for testing the effectiveness of the proposed control scheme have been inspired from [25] and are referred to a real case study.

else  li (k) , φi (k) = mid Di−1 (k), Si (k) − di (k) − T  (1 − pi )Si (k)   li (k) ri (k) = mid di (k) + , Si (k) − Di−1 (k), pi Si (k) T (23) where the function mid returns the middle value. The finite-horizon optimal control problems to be solved within the proposed supervisory event-triggered MPC schemes adopt the CTM model for the predictions, i.e. the CTM model has to be included in the problem as a set of constraints. To this end, the CTM model is rewritten in MLD form, by introducing suitable sets of inequalities and auxiliary variables, in order to avoid the nonlinearities present in (21), (22) and (23). For space limitations, we do not report all the inequalities and the auxiliary variables present in the MLD formulation (for more details refer to [24]). It is only useful to specify that, in the FHOCP to be solved at a generic time step k over a prediction horizon Kpσ , the state variables are ρi (h) and li (h), i = 1, . . . , N , h = k + 1, . . . , k + Kpσ , the control variables are ri (h), i = 1, . . . , N , h = k, . . . , k + Kpσ − 1, and there are

200

150

100

50

0 5 4 3

Cells

2 1

Fig. 2.

0

5

10

15

20

25

30

35

40

45

50

55

60

Time steps

Traffic density in the no-control case.

The reported results refer to a case in which the traffic demand in the on-ramps is equal to 1000 [veh/h] in the first 20 time steps, then it increases up to 3000 [veh/h] in the next 20 time steps, and then it comes again to be equal to 1000 [veh/h]. As it can be seen in Fig. 2, this demand yields a severe congestion in cells 2 − 4. The supervisory control scheme acts by switching between different controllers, which become active depending on the values of the system state variables in the different cells. The parameters characterizing each controller with index σ ρ l r δ are the weights of the cost function, i.e. γσ,i , γσ,i , γσ,i , γσ,i,j ,

200

150

100

50

0 5 4 3

Cells

Fig. 3.

2 1

0

5

10

15

20

30

25

35

40

45

50

55

60

Time steps

Traffic density in the controlled case.

100

no−control case controlled case

90 80 70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

Time steps

Fig. 4.

Total length of the queues in no-control and controlled case.

z j = 1, . . . , 3, and γσ,i,j , j = 1, . . . , 5, i = 1, . . . , N and the prediction horizon Kpσ . As reported in Fig. 3 the proposed control scheme significantly reduces the traffic densities, completely avoiding the congestion situation. Of course, the control action obtains this result by reducing the traffic flows entering the freeways and this induces an increase in the overall queue lengths at on-ramps (see Fig. 4). Such an increase can anyway be considered acceptable if compared with the corresponding reduction in the traffic densities inside the freeway stretch.

V. C ONCLUSIONS This paper presents a supervisory event-triggered MPC scheme oriented to deal with freeway traffic systems. In the scheme a MLD model is used both as a predictor in solving the MPC problem, and as a process emulator to provide a suitable input to the supervisor. This latter has the role of selecting, at any sampling step, the appropriate controller to be used, as well as of triggering the update of the control law only when strictly necessary, in order to alleviate the computational burden with respect to classical MPC schemes. In the paper, the ISpS of the control scheme is formally proved, while its performance assessment when applied to a freeway traffic system is carried out in simulation. R EFERENCES [1] J.B. Rawlings, D.Q. Mayne, Model predictive control: theory and design, Nob Hill Pub., 2009.

[2] R. Scattolini, Architectures for distributed and hierarchical Model Predictive Control - A review, Journal of Process Control, vol. 19, 2009, pp 723-731. [3] L. Gr¨une, J. Pannek, Nonlinear Model Predictive Control - Theory and Algorithms, Springer, 2011. [4] A. Kotsialos, M. Papageorgiou, Efficiency and equity properties of freeway network-wide ramp metering with AMOC, Transportation Research C, vol. 12, 2004, pp 401-420. [5] T. Bellemans, B. De Schutter, B. De Moor, Model predictive control for ramp metering of motorway traffic: A case study, Control Engineering Practice, vol. 14, 2006, pp 757-767. [6] J.R. Frejo and E.F. Camacho, Global versus local MPC algorithms in freeway traffic control with Ramp Metering and Variable Speed limits, IEEE Transactions on Intelligent Transportation Systems, vol. 13, 2012, pp 1556-1565. [7] A. Muralidharan, R. Horowitz, Optimal control of freeway networks based on the Link Node Cell transmission model, Proc. of American Control Conference, Montreal, Canada, 2012, pp 5769-5774. [8] Y. Wang, S. Boyd, Fast Model Predictive Control Using Online Optimization, IEEE Transactions on Control Systems Technology, vol. 18, 2010, pp 267-278. [9] S. Riverso, M. Farina, and G. Ferrari-Trecate, Plug-and-play decentralized model predictive control, Proc. of the 51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2012, pp 4193-4198. [10] A. Ferrara, A. Nai Oleari, S. Sacone, S. Siri, An event-triggered Model Predictive Control scheme for freeway systems, Proc. of the 51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2012, pp 6975-6982. [11] A. Eqtami, D.V. Dimarogonas and K.J. Kyriakopoulos, Novel EventTriggered Strategies for Model Predictive Controllers, Proc. of the 50th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2011, pp 3392-3397. [12] W.P.M.H. Heemels, K.H. Johansson, P. Tabuada, An Introduction to Event-triggered and Self-triggered Control, Proc. of the 51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2012, pp 3270-3285. [13] J.P. Hespanha, D. Liberzon and A.S. Morse, Supervision of integralinput-to-state stabilizing controllers, Automatica, vol. 38, 2002, pp 1327-1335. [14] J.P. Hespanha, D. Liberzon, and A.S. Morse, Hysteresis-based switching algorithms for supervisory control of uncertain systems, Automatica, vol. 39, 2003, pp 263-272. [15] A.S. Morse, Supervisory control of families of linear set-point controllers, Part 2: Robustness, IEEE Transactions on Automatic Control, vol. 42, 1997, pp 1500-1515. [16] L. Vu, D. Liberzon, Supervisory Control of Uncertain Linear TimeVarying Systems, IEEE Transactions on Automatic Control, vol. 56, 2011, pp 27-42. [17] A. Bemporad and M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica, vol. 35, 1999, pp 407-427. [18] E.D. Sontag, Y. Wang, New characterizations of Input-to-State Stability, IEEE Transactions on Automatic Control, vol. 41, 1996, pp 1283-1294. [19] D. Lim´on, T. Alamo, D.M. Raimondo, D. Mu˜noz de la Pe˜na, J.M. Bravo, A. Ferramosca and E.F. Camacho, Input-to-State Stability: A Unifying Framework for Robust Model Predictive Control, in L. Magni et al. (Eds.): Nonlinear Model Predictive Control, Springer-Verlag, 2009, pp 1-26. [20] A. Bemporad, G. Ferrari-Trecate and M. Morari, Observability and Controllability of Piecewise Affine and Hybrid Systems, IEEE Transactions on Automatic Control, vol. 45, 2000, pp 1864-1876. [21] D.E. Quevedo and D.Neˇsi´c, Input-to-state stability of packetized predictive control over unreliable networks affected by packet-dropouts, IEEE Transactions on Automatic Control, vol. 56, 2011, pp 370-375. [22] Z.P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica, vol. 37, 2001, pp 857-869. [23] C.F. Daganzo, The Cell Transmission Model, Part II: Network Traffic, Transportation Research Part B, vol. 29, 1995, pp 79-93. [24] A. Ferrara, S. Sacone, S. Siri, Event-based control of freeway systems, Proc. of the IEEE International Conference on Systems, Man, and Cybernetics, Manchester, UK, 2013, to appear. [25] C. Canudas-de-Wit, L.L. Ojeda, A.Y. Kibangou, Graph constrainedCTM observer design for the Grenoble south ring, Proc. of 13th IFAC Symposium on Control in Transportation Systems, Sofia, Bulgaria, 2012, pp 26-31.