CRONE control application for a toll plaza

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

CRONE control application for a toll plaza M. Uzunova ∗ D. Jolly ∗∗ E. Nikolov ∗∗∗ Univ Lille Nord de France, F-59000 Lille, CNRS, FRE 3304, F-59313 Valenciennes, UVHC, LAMIH, F-59313 Valenciennes, France (e-mail: [email protected]). ∗∗ Univ Lille Nord de France, F-59000 Lille, LGI2A, 62400 Bethune, France (e-mail: [email protected]). ∗∗∗ Automation Faculty, ANP, Technical University of Sofia, 1000 Sofia, Bulgaria (e-mail: [email protected]). ∗

Abstract: In this paper a non-integer robust control approach to determine the speed variation downstream to a toll plaza is proposed where the total flow exiting from the toll booths exceeds the capacity of the downstream highway. This can lead to congestion and reduced efficiency due to capacity drop. The proposed control approach is based on non-integer robust methodology. This control strategy assures the robust performances for the traffic velocity and maintains the density on the high ways under a constraint value with the main aim to assure fluid traffic flow. The traffic flow phenomenon is very complex and the main problem stay to avoid the traffic jams on the roads. Hence the purpose of this article is to apply non-integer control algorithm in the toll plaza for maintain the downstream traffic flow density under a constraint density and to assure fluid process. The interest stay to apply a control algorithm containing internal model with Smith structure for predict and reduce the time delay. More over we look to assure robust performances using the non-integer algorithm. Keywords: non-integer robust control, LWR macroscopic traffic flow model, tool plaza control, Smith predictor structure NOTATION

in the merge area close to a critical value that maximizes throughput. The performance of a toll gate can be sigLWR Lighthill, Whitham and Richards traffic model nificantly affected by a number of factors such as service capacity, vehicle arrival pattern, number of available gates, CRONE non-integer robust control and driver behavior. FD fundamental diagram This paper is organized as follows. In the second section PDE partial differential equation the macroscopic traffic flow model, for presenting the toll plaza physical process and the proposed Green function 1. INTRODUCTION method for solution are proposed. The 3 section is devoted to study and validate the nominal model in the time and Nowadays increasing of the transportation needs is ob- frequency domain. The section 4 describes the non-integer served that leads to different problems related to the order control conception. The study case, of the control congestions, saturation on the traffic network and delay in system regulations for toll plaza downstream traffic under the transportation. One of the solutions consists of control perturbation, is presented in section 5. Section 6 concludes strategies implementation for maintain the downstream the paper and presents the perspectives. density fluid and to avoid congestions, bottlenecks etc. The 2. MACROSCOPIC TRAFFIC FLOW MODEL main purpose of this article is to apply control algorithm GREEN FUNCTION METHOD FOR SOLUTION in order to maintain the downstream traffic flow density under a constraint value and to assure fluid process. This paper shows the interest to apply a control strategy which Modelling is very important phase for the synthesis of takes into account the influence of the perturbations, as control laws and observers. The precision of modelling a high upstream density coming from the toll plaza. In depends on the required objectives. For this reason, there this case the concept of the control system should have exists several kinds of models for the same process and the a robust performances for assure traffic flow under con- choice among those will depend on its use and of the obstraint value. Based on this reasoning a non-integer order jective control. For the control the selected model must be control strategy for traffic flow situated downstream in sufficiently simple to allow the realization of the real time a toll plaza is considered. This concept is proposed for control but enough precise to obtain the desired behavior. efficient regulation in cases where the total flow exceeds The modelling of the traffic flow is generally difficult and the capacity of the downstream highway and this leads for the high ways usually presented by the macroscopic to reduced efficiency of the automation system. Merging models where the traffic as is not observed as an individual traffic control aims at maintaining the number of vehicles behavior but as a movement of a group of vehicles. This

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10.3182/20120912-3-BG-2031.00012

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

time parameter, affecting on the traffic flow behaviour downstream on the toll plaza. Moreover this study consists to make observation and analysis of the stability and the performances of the nominal model, in the time and in the frequency domain and to conclude for the model behaviour under perturbations. All perturbation can provoke congestion traffic and bottlenecks. That leads to apply a control strategy on the toll plaza for regulate the downstream traffic affecting the control on the toll plaza level. Hence, the solution method efficiency is presented under comparisn analysiswith wildly used numerical methods results. For the start phenomenon (as toll plaza model) the fluid traffic is in equilibrium and for this nominal model only the left part of the fundamental diagram can obtained. The constraint is to have only points of functionalities situated in the fluid zone of the fundamental diagram Spiliopoulou (2010). These validation study shows the pertinence of the Green function approach for obtain the transfer function based on the PDE solution. The reconstruct fundamental diagram prove that the obtain transfer function model is enough precise to the real traffic flow behaviour downstream on the toll plaza. For the study case we have the follow initial conditions: road length is 1 [km], downstream of the toll plaza. The road parameters are L = 1 [km], ρmax = 120 [veh/km], ρcr = 60 [veh/km], constraint: density ρlim = 42 [veh/km]. For this model the parameters are: density ρ/ρmax = ρ0 = 0.4 [−], free speed Vf = 120 [km/h], calculated speed for the FD VfF D = 72 [km/h], gain kLW R = 0.013697, time constant TLW R = 0.013889 [s], delay τLW R = 0.013889 [s]

process is based on a fluid mechanic principle and similar to the movement of fluid in a tube. In the following this paper proposes the solution of this model in order to obtain the transfer function of the model, to study the stability and the performances of this system and to apply a noninteger control strategy. The toll plaza traffic flow model using the LWR model is presented by (1) Del Castillo (1995), Lighthill (1955) and Lighthill (2002), Abouaissa et al. (2006). The LWR model is a part of the macroscopic traffic flow models based on the fluid mechanics. The physical model is presenting by a non-linear hyperbolic partial differential equation P.D.E. (2) demonstrates the distribution of the vehicles in the highs ways. The model is obtain using the conservation law, that means that number of vehicles measured in one moment and one point on the road, will be measured in another point on the road after an elapsed time depending of the velocity parameter. This model takes part of the class of the distributed systems and of the class of the uncertainty process because of the unpredictable variation of the model’s parameters. The LWR model was completed by the relation called ”fundamental diagram”, obtained experimentally, as approximation of all the possible equilibrium states of the traffic. Hence, we will use the parabolic diagram of Greenschield for the solution procedure. ∂Q (x, t) ∂ρ (x, t) + =0 ∂x ∂x Q (x, t) = ρV(ρ (x, t))   ρ (x, t)   V (ρ (x, t)) = Vf 1 − ρmax    ρ (x, t) ∂ρ (x, t) ∂ρ (x, t)   + Vf 1 −  ∂t  ρmax  ∂x  ∂Vf 1 − ρ(x,t)  ρmax  +ρ (x, t) =0 ∂x     

(1)

Two different variation of the upstream density as an input of the simulation model are considered. That leads to eject downstream of the toll plaza the full number of vehicles for the system without applied control. The first signal is a constant value equal of the constraint value of the density. The second signal represents high density variations (rush hour or traffic under perturbations). This saturation can provoke a bottleneck and congestion traffic, because the measured density exceeds the critical density value. Fig.1 shows the output density measured on the 1st kilometer and the reconstructed left part of the fundamental diagram when saturation on the road is observed Kotsialos (2004), Papageorgiou (2003), Papageorgiou (2008). The distribution of the vehicles and the behavior of the flow for the both cases (under the constraint and under perturbation) are presented. The reconstructed left part of the fundamental diagram is only for the perturbed system. The high upstream density provokes congestion traffic, downstream of the toll plaza. In the following based on frequency synthesis in Nyquist plot shown on Fig.2 present the system performances. This study gives us information about the influence of the perturbation on the system margins. For the control system reduced stability margins (gain margin) for the perturbed model are obtained.

(2)

As result, using the Green function method (1) and under equilibrium constraint, a delay transfer function (3) is obtqined, depending of the time, the space and the velocity Oustaloup (1991), Uzunova (2010), Uzunova (2008), Uzunova (2008). This function shown that there are no loses in the model. After equivalent transformations the classical form of the transfer function (4) is reached: G (x, x0 , p) =

u (x − x0 ) −p e V (x, p)

(x−x0 ) V (x,p)




e−pτ

 

G (x, x0 , p) = 1 1
kLW R (T 2 2 p + 1) τLW R p + τLW R p + 1 LW R

(3)

(4)

The parameter of the obtained transfert function are : static gain kLW R , time constant TLW R and time delay τLW R

Using this plant simulation study the model behavior can observed (downstream on the road) under perturbations (high upstream density). That leads to reduced stability margins and to assure robust properties we need to apply robust control strategy on the toll plaza, in order to limit the coming upstream flow under the constrain values. This perturbed plant model is presented by : (i) variation of the plant parameters under internal (parametrical)

3. NOMINAL MODEL STUDY The first study in this paper consists to verify the nominal model behaviour. As mentioned in the previous section the obtained result is delay transfer function. In the present study case it consists to observe the influence on the 61

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

Fig. 1. Output density distribution and reconstruction of the fundamental diagram

Fig. 2. Nyquist plot of stability for the traffic flow model under perturbations

perturbations and/or (ii) variation of the plant structure under the external perturbations. That proves the need of control strategy which will assure the stability of the system opposing to the perturbations and will assure invariance of the plant parameters and robustness. For the plant transfer function, presented by delay function, the propose control system contain Smith predictor using the non-integer robust order algorithm for synthesis. 4. NON-INTEGER ORDER CONTROL CONCEPTION This section propose the structural scheme and the control algorithm synthesis according to the purposes in the present paper. The considered control system contain noninteger order strategy with Smith predictor. The structural scheme of the control system is presented in Fig.3.

Fig. 3. Structural scheme of the non-integer control system with Smith predictor The control algorithm uses the CRONE approximation Nikolov (2008), Uzunova (2008) of the non-integer order operator. The control method is based on a frequency approximation of the non-integer order algorithm (7) and (8), using a polynomial distribution of zeros and poles,where α and η are non-integer constantes, Φ is the cloose loop transfer function, Grat - rational part of the model function, RN E - non-integer order control algorithm.

The form of close-loop system transfer function and control algorithm are reached (5) and (6) : R (p) =

RN E (p) ∗
1 + RN E (p) Grat (p) 1 − e−pτ

(5)

∗

R (p) Grat (p) e−pτ Φ (p) = 1 + R (p) Grat (p)

(6) 62

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

α ˆ (p) = C R (p) = (p) , 0 < α < 1 ⇔ R

N Y 1+

i=−N

1+

p ωi p ωi′

ω′ logλ ωi+1 = i+1 = λη ⇔ α = logλη ωi ωi′

during a rush hour is presented when the upstream flow exceeds: (i) the critical density for the road geometry and/or (ii) the constrain density imposed by the condition to assure fluid traffic. The technological scheme of the control procedure is presented on Fig.5. For this first order LWR model the velocity is calculated as ration from the two state variables - density and flow.

(7)

(8)

The obtained structure of the P I α Dα robust control algorithm is presented as following:

The main aim is to assure robust performances of the system behavior trough the control system for space of time, which means between low and high frequency range, called ”general pattern”. That assures constant the parameters of the open-loop system, using the polynomial report as presenter on Fig.4. In this case the phase and the gain margins are invariant between the low and the high frequency range and assure the corresponding behavior for the regulated system.

Fig. 5. Technological scheme of traffic process In order to control the upstream density on the 1st km providing from the tool plaza (slow back loop) and the downstream density on level of the on-ramp (fast back loop) the control scheme for the system is realized with cascade structural Fig.6.

Fig. 4. Bode diagram for the robust properties for PID control algorithm

Fig. 6. Structural control scheme for the system: cascade model

5. CASE STUDY OF TOLL PLAZA TRAFFIC CONTROL FOR A ROAD SEGMENT WITH ON-RAMP

In the input of the simulation model the density variation is as presented in Fig.7: for the constant value limited by the constraint conditions in the system (considered as a normal period traffic circulation) and for the rush hour period (perturbed system under excitation).

This section proposes the study of the control system for regulation of toll plaza for maintain the downstream density under constraint value. For the control system robust strategy control at the toll plaza level is affected. The study consists to observe the traffic flow behavior for a six kilometer segment containing an on-ramp on the 5th km. The on-ramp flow is considered as a perturbation for the system. This additional flow will increase the main stream road density and that can provoke saturation on the road and overflow, exceeding the constraint value. The measurement of the density is in three points: on the first and on the sixth kilometer for the control system and on the on-ramp level.

These simulation results show the possibility to have congestion traffic during the rush hours. For the obtained control systems the results are shown in Fig.8 for the time response and Fig.9 the Nyquist frequency response. From the simulation results (Fig.7)-(Fig.9) we can conclude the following : (i) the output traffic flow satisfy the constraint in the system, (ii) the control algorithm assure the purpose to have fluid traffic flow downstream of the toll plaza, (iii) the non-integer order control algorithm assure the robust properties, (iv) invariant system margins are assured and the general pattern in Nyquist plot can be observed, (v) the control system increases: the performances, the phase and the gain margins.

The considered control system is under perturbation and our aim is to check the performances of the robust noninteger order control algorithm. The density variation 63

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

Fig. 8. Density variation for the control system: nonperturbed and perturbed control system

Fig. 7. Density variation for the simulation model: nonperturbed and perturbed simulation model

6. CONCLUSION The presented paper presents a study case of toll plaza problems. For this system a non-integer order control with Smith predictor under constrains to maintain fluid downstream traffic flow is applied. This work is follows the reasoning to obtain downstream density under a constraint value by measuring the upstream density and controlling the flow injected on tool plaza level. The obtain results show that the robust control system satisfy the robust stability and performances and stay in the margins defined by the model constraint. The time responses give information about the stability and the density variation on the road. The results are compared with the wildly used numerical results by the reconstructed fundamental diagram which shows the flow/density relation and prove the effectiveness of the used analytical method of solution for the partial differential equations.

Fig. 9. Nyquist frequency response of the non-integer control system under perturbation

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