Modeling, control and robustness analysis for a road sector

2010 Management and Control of Production Logistics University of Coimbra, Portugal September 8-10, 2010

Modeling, control and robustness analysis for a road sector Catalin Dimon(1), Geneviève Dauphin-Tanguy(2), Dumitru Popescu(1) (1)Department of Automatic Control and System Engineering, University Politehnica of Bucharest 313 Splaiul Independentei, 060042, Bucharest, Romania (2)LAGIS, Ecole Centrale de Lille, Cité Scientifique - BP 48 - 59651 Villeneuve d’Ascq Cedex E-mail: [email protected]

Abstract: When modeling a real process the effects of nonlinearities must be taken into account, especially when designing a control algorithm. Such is the case of a road traffic sector, which will be considered from a macroscopic point of view. An input-output model is proposed, based on experimental data, and then a control algorithm is used to obtain imposed performances for the closed loop system. The robustness of control is taken into account, by assessing the modulus margin. Based on this indicator the robustness of the control law is analyzed in order to find what nonlinearities it can cope with. Following, we try to prove that the system can be controlled using the same algorithm, when different types of perturbations affect the system. Keywords: modeling, control, robustness, sampling time, traffic.

1. INTRODUCTION The problem of modeling and controlling the traffic flow has been tackled by many researchers, but has not yet reached a satisfactory level and thus further analysis and improvements are researched. In the literature, we can find a considerable number of papers covering different traffic related aspects, such as modeling, simulation and control solutions [1, 2]. When modeling road traffic, there are two main approaches that are frequently found, depending on the level of detail used to describe the road sector: macroscopic and microscopic [3, 4]. Recent studies concerning the difficulties encountered when modeling traffic flow have also tried to make a hybrid model by combining the macroscopic and microscopic approaches, in order to adapt the traffic flow models to small scale problems [5]. However, in order to control a road traffic sector, the macroscopic point of view seems the most adequate choice. It doesn’t require a large set of variables compared to the other type of models, and thus it leads to a shorter decision time. Furthermore, it permits a general representation of a traffic network, offering access to a global view of the entire system. The macroscopic models are usually derived from hydrodynamic models, resulting in describing the traffic as a flow of vehicles. The main variables used for describing its behavior are density, flow and speed of vehicles. The law used to describe the traffic flow is that of vehicles conservation, to which are added equilibrium and flow dynamics relations between the main variables.

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The traffic control can be differentiated in two categories [6]: • controlling traffic flow in urban areas, concerning the rules and control algorithms used for optimizing the time cycles for traffic lights in order to improve the fluidity of traffic flow ; • controlling traffic flow for road networks and highway areas, concerning control techniques used for avoiding congestion problems and improving highway efficiency. We will try to combine the two, aiming at finding an algorithm for controlling traffic cycles in order to avoid congestion problems. 2. MODELING A ROAD SECTOR 2.1 Continuous model Since a first order model is used, there will also be made reference to the initial macroscopic model, developed by Lighthill and Whitham (1955) and Richards (1956), also known as the LWR model [7]. To describe the behavior of a road sector, it is considered the similarity with the flow of a fluid through a pipe, as it is represented in Fig.1 [8, 9]. The parameters used to describe the model are the following: - L , sector length; - C , sector capacity (depending on L); - p , inside pressure; - pext , exterior pressure;

ɺ e , input flow ; - m ɺ s , output flow. - m

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pext mɺ e

mɺ s

Fig.1. One-way road sector, considered from a hydrodynamic perspective Starting from the equations that describe the flow of a fluid through the section of a pipe [10], a non-linear model is obtained, given by the following set of equations:

where p0 is the nominal pressure inside the road sector, which will be obtained from the nominal functioning regime of the non-linear model. By using the Matlab/Simulink simulation environment, the open loop responses of the linearized model was compared with the one from the non-linear model, resulting that the linearized model, mlin = m0 + δ m , approximates the behavior of the non-linear model. The linear transfer function relating

F ( s) =

mɺ = mɺ e − mɺ s 1 m C mɺ s = k p − pext p=

(1)

In order to simplify the model, it is made the assumption that pext = 0 , meaning that there aren’t any restrictions upon the exit. The case of a variable exterior pressure will be analyzed later in the paper, considering it as an external perturbation. We are interested in a reference vehicle flow value which will assure a good flow of vehicles on a city road. By taking into account the speed limit for such a road, a reference speed value of v0 = 50 km / h will be assumed. From the equation of the LWR model, describing the average speed of all vehicles on a road sector, we have the relation:

δm

τP τ Ps +1

to

δ mɺ e

is: (4)

with τ P ≅ 72 , determined by considering the time constant as a RC element, where C is the capacity of the road sector and R is the system’s resistive component, obtained from (1), using the nominal pressure and flow values, as follows:

R=

2 p0 mɺ 0

(5)

By assuming for the closed loop system an aperiodic response, a PI controller was chosen [10], given by the expression:

 1  (6) H ( s ) = K R 1 +   Ti s  The parameters K R = 0.1 and Ti = τ P = 72 were fixed so

(2)

that the closed loop model will assure a response time of about 10 seconds.

ɺ (t ) is the flow rate, ρ (t ) is the density and v(t ) where m is the average speed, which gives the desired flow value. A road sector of length L = 500 m was considered, which for

The closed loop responses for the linear and non-linear model are shown in Fig.2. The output error between linear and nonlinear models, considered for the variation in the reference signal (from t = 500 s ) is shown in Fig.3.

mɺ (t ) = ρ (t ) v (t )

an average vehicle length of 5m would accommodate

C = 100 vehicles. This can be translated in a maximal density of ρ 0 = 0.2 veh / m . Based on experimental data, the considered reference will be m0 = 20 veh and nominal

ɺ 0 ≅ 0.5 veh / s . flow value will be m For the following tests it is assumed a variation of approximately 10% from the nominal value stated above: δ m = 2 veh . By linearizing the non-linear part in (1) around a static functioning point, conserving only the first order factor, the following linearized equation is obtained for the sector in Fig.1:

δ mɺ s =

mɺ k δ p, with k = 0 2 p0 p0

(3)

Fig. 2. Closed loop response for the linearized model (blue) and the nonlinear model (magenta), with the reference signal (yellow)

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Fig. 3. Closed loop output error between linearized model response and nonlinear model response

Fig. 4. Closed loop output error between linearized discrete model response and nonlinear model response, using the discrete controller

2.2 Discrete model

3. ROBUSTNESS ANALYSIS

The goal is to find a discrete equivalent for the linear model, as well as for its controller. The discrete model can be computed as follows:

 1 − e −Ts  F (z ) = Z  F (s )   s  −1

(7)

where T is the chosen sample period.

3.1 Modulus margin In order to determine the robustness of the control algorithm, we use as an indicator the modulus margin ( ∆M ), which represents the minimal distance between the Nyquist plot of the open loop linearized model and the “critical point” (-1,0j) [11]. It can be approximated using the absolute value of the output-perturbation sensibility function, as follows:

Ts

By using the transformation z = e , it becomes:

∆M = min 1 + H OL (e − jωT ) ω∈ℝ

−   −1 τ 1 − e  z  F ( z −1 ) =  T

−

= min ω∈ℝ

(8)

T

τ

1− e z

−1

1 1 = − jωT ) max S yp (e− jωT ) S yp (e ω∈ℝ

Given a system:

For the sample period T = 10sec , which keeps for the discrete model similar performances to those of the continuous model, it was obtained:

F ( z −1 ) =

9.337 z 1 − 0.8703 z −1

(9)

Similar, for the controller:

H ( z −1 ) ≅

B( z −1 ) A( z −1 )

and its associated controller:

−1

F ( z −1 ) ≅

(11)

H ( z −1 ) =

D ( z −1 ) C ( z −1 )

the output-perturbation sensibility function can be written as:

0.1 − 0.08611 z 1 − z −1

−1

A( z −1 )C ( z −1 ) S yp ( z ) = A( z −1 )C ( z −1 ) + B ( z −1 ) D ( z −1 )

(10)

In Matlab/Simulink, the responses from the linearized discrete model and from the non-linear model were compared, using in both cases the discrete control algorithm. The output error is shown in Fig.4 and it can be noticed that is similar to that from the continuous case, in Fig.3.

−1

(12)

By using the computed values from our discrete model and controller, the following expression was obtained:

309

S yp ( z −1 ) ≅

1 − 1.87 z −1 + 0.8703 z −2 1 − 1.857 z −1 + 0.8592 z −2

(13)

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In the frequency domain the sensibility function can be written as:

S yp (e− jωT ) ≅

1 − 1.87 e − jωT + 0.8703 e − j 2ωT (14) 1 − 1.857 e − jωT + 0.8592 e − j 2ωT

which can be developed further by using Euler’s formula. From the definition of the modulus margin, it was needed to compute the maximum value of the output-perturbation sensibility function, and we obtained:

max S yp (e − jωT ) ≅ 1.1065

(15)

ω∈ℝ

By using the discrete controllers corresponding to different sampling periods, the system response for the non-linear model was simulated. The results are illustrated in Fig.6. As can be noticed, the model’s output varies with the sampling period. By taking into account that a linearized model, that behaves slightly different from the real system, was used, the maximal admissible value for the sampling time should be T = 10sec , the biggest value for which a desired output is obtained. The applied command for the previous sampling periods is shown in Fig.7. For T = 1s , T = 10s and T = 20s the output speed stabilizes at the desired value of 50 km / h ,

which leads to the value for the modulus margin:

∆M ≅ 0.9935 .

(16)

while for T = 30s and T = 40s it oscillates continuously. Given the variations, T = 10s is used for further analysis.

It should be mentioned that this is a very good value. Generally, in literature it is recommended to assure for a system a modulus margin:

∆M ≥ 0.5

(17) that will also imply satisfactory values for the gain and phase margins [11]. The modulus margin for the discrete model given by (16) would suggest that the model can be considered as a good approximation for its continuous counterpart. In the case that the obtained value of the modulus margin is unsatisfactory, it −1

could be improved by additional polynomials, H C ( z ) and

H D ( z −1 ) , added to the controller, such that the amplitude of the sensibility function is reduced [11]. In Fig.5 there is shown the dependency between the modulus margin and the sampling time. It can be noticed that as the sampling time value increases the value of the modulus margin decreases. The modulus margin becomes inferior to the desired value, given by (17), when the sampling time gets above 10 seconds. Thus we can say that the robustness of the system is assured for a sampling time value of maximum T = 10sec .

Fig. 6. Non-linear model response for different sampling periods: yellow–1s, magenta–10s, cyan–20s, red–30s, green–40s

Modulus margin for different sampling times 1 0.9 0.8

Modulus margin

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 1.2 log10(T)

1.4

1.6

1.8

2

Fig. 5. Modulus margin dependency on sampling time

Fig. 7. Generated command for different sampling times: yellow–1s, magenta–10s, cyan–20s, red–30s, green–40s 310

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3.2 Parameter variation

3.3 Exterior pressure

For the previous simulation a reference speed value of v0 = 50 km / h was considered. By changing this value the robustness of the controller can be analyzed. Were taken as references speed limits of 30 km / h , 60 km / h and

In the beginning it was assumed that the exterior pressure is zero. Following, it will be considered as a perturbation affecting the system. Supposing the exit of the sector is a parking area of limited capacity, the more vehicles are exiting the sector the bigger the exterior pressure gets. Nonetheless, in order to have a vehicle flow, the following condition has to be verified at all times:

80 km / h . The obtained results are similar to what we obtained in Fig.6 for the corresponding sampling time. As expected, the command (input flow) increases as the speed limit increases. The obtained results are shown in Fig. 8.

p > pext

(18)

The exterior pressure was considered varying between minimum and maximum values. The sector length is L = 500 m , implying an average capacity

C = 100 vehicles , and the reference for the desired number of vehicles is set to m0 = 20 veh . The control law from (10) is used. The obtained results are represented in Fig.9, Fig.10 and Fig.11. Firstly, it should be mentioned that the system assures the desired reference (number of vehicles) even in the presence of exterior pressure. The exterior pressure is considered varying between its minimum ( pext = 0 ) and maximum ( pext = 1 ) values. The maximum value can be interpreted as a red light at the end of road that prevents any vehicle from exiting. As can be seen in Fig.9, after the disappearance of exterior pressure (when the light goes green) the system is able to return at the reference value. Fig. 8. Generated command for different speed values: yellow–30km/h, magenta–50km/h, cyan–60km/h, red– 80km/h Until now, it was always assumed a road sector of length L = 500 m . By considering this parameter variable the robustness of the proposed control law can be also verified against parameter uncertainties affecting the model. We considered the values in Table 1; it can be noticed that the system’s output (number of vehicles) remains the same. The applied command also remains the same whatever the sector length and it is identical with the command obtained initially for the sampling period of T = 10s .

In Fig.10 it can be noticed that the output flow decreases as the exterior pressure increases. This is expected since the flow determines the number of vehicles exiting the sector. For example, if we consider a parking area at the end of the road sector, while the parking area gets filled, the number of vehicles that it can accommodate decreases.

Table 1. Output flow for different sector lengths Length

Capacity

Output

100

20

20

300

60

20

500

100

20

700

140

20

1000

200

20

Fig. 9. System output (number of vehicles) for variable exterior pressure: yellow – reference signal, magenta – system output, cyan – exterior pressure 311

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flow. Since the resulting model was a nonlinear one, in order to obtain a control algorithm its linear equivalent was determined. By using the linearized model to approximate the behavior of the road sector a control law was determined that assured certain imposed performances. Further, it is showed the importance and influence of choosing an adequate sampling period, while trying to conserve the system’s robustness. For future research we plan to further develop the control algorithms by increasing the model complexity such that we can obtain a better approximation of the real system. Also by studying the behavior of multiple interconnected sectors, there are new restrictions that have to be dealt with. REFERENCES Fig. 10. Output flow for variable exterior pressure: yellow – output flow, magenta – exterior pressure

Fig. 11. Generated command for variable exterior pressure: yellow – command, magenta – exterior pressure The generated command from Fig.11 has the same shape as in the previous simulations. When the exterior pressure increases, the command decreases as it determines the input flow for the road sector. Aiming to assure the desired reference value, it is normal to cut the input flow until an output flow determines the decrease of the number of vehicles present on the road. 4. CONCLUSIONS The behavior of traffic was analyzed from a macroscopic approach, considering the flow of vehicles from a one-way road sector. In order to describe the dynamics of the road sector, an input-output flow approach was chosen. The sector was modeled by studying the behavior of hydrodynamic systems and making a parallel with the behavior of traffic

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