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Perimeter Control for Two-region Urban Traffic System Based on Model Free Adaptive Predictive Control with Constraints
Proceedings, 1st IFAC Workshop on Control of Transportation Proceedings, 1st IFAC IFAC Workshop Workshop on on Control Control of of Transportation Transportation Proceedings, 1st Systems Proceedings, 1st Workshop of Transportation Systems Proceedings, 1st IFAC IFAC Workshop on Control Control of online Transportation Available at www.sciencedirect.com Systems Haifa, Israel, June 30 - July 1, 2019on Systems Proceedings, 1st IFAC Workshop Haifa, Israel, Israel, June June 30 -- July July 1, 2019 2019on Control of Transportation Systems Haifa, 30 1, Haifa, Israel, Israel, June June 30 30 -- July July 1, 1, 2019 2019 Systems Haifa, Haifa, Israel, June 30 - July 1, 2019
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Perimeter Control for Two-region Urban Traffic System Based on Model Free Perimeter Control for Two-region Urban Traffic Based on Model Free Adaptive Predictive Control withSystem Constraints Perimeter Control for Two-region Urban Traffic System Based on Model Free Adaptive Predictive Control withSystem Constraints Perimeter Control for Two-region Urban Traffic Based on Model Free Adaptive Predictive Control with Constraints , Ting Lei*. Control Zhongsheng HouConstraints ** * Adaptive Predictive with Ting Lei*. Zhongsheng Hou**,,* Ting Lei*. Zhongsheng Hou**,,* Ting Lei*. Zhongsheng Hou**,* * Advanced Control SystemsTing Laboratory, Beijing Jiaotong University, Beijing 100044, China Lei*. Zhongsheng Hou ** *
* Advanced Control Systems Laboratory, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]) * Advanced Control Systems Laboratory, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]) ** The School of Automation, Qingdao Qingdao 266071, * Advanced Control Systems Laboratory, BeijingUniversity, Jiaotong University, Beijing China 100044, China (e-mail: [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) (e-mail: [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) (e-mail: [email protected]; [email protected]) Abstract: Recent studies have shown that macroscopic urban traffic control, especially perimeter control, Abstract: Recent studies macroscopic urban traffic perimeter plays an important role inhave the shown field ofthat urban traffic control. In thiscontrol, paper, aespecially novel data driven control, control Abstract: Recent studies have shown that macroscopic urban traffic control, especially perimeter control, plays an important role in the field of urban traffic control. In this paper, a novel data driven control method named model free adaptive predictive control withtraffic constraints (C-MFAPC) is utilized for Abstract: Recent studies have shown that macroscopic urban control, especially perimeter control, plays an named important role free in the field ofpredictive urban traffic control. Inconstraints this paper, (C-MFAPC) a novel data is driven control method adaptive control for perimeter controlmodel for two-region urban traffictraffic system. Inwith this strategy, thea advantages of utilized model free plays an important role in the field of urban control. In this paper, novel data driven control method named free adaptive predictive controlInwith constraints is utilized for perimeter controlmodel for two-region urban traffic control system. this are strategy, the(C-MFAPC) advantages of model free adaptive named control (MFAC) andadaptive model predictive (MPC) combined. That is, theis controller can method free predictive control(MPC) constraints utilized for perimeter controlmodel for two-region urban traffic control system. Inwith this are strategy, the(C-MFAPC) advantages of model free adaptive control (MFAC) and model predictive combined. That is, the controller can be designed by (MFAC) merely using theurban I/O data of control the system, while the system output sequence canfree be perimeter control for two-region traffic system. In this strategy, the advantages of model adaptive control and model predictive (MPC) are combined. That is, the controller can be designed by merely usingmodel. the I/O data of the system, while the system output sequence can be predicted without the system Moreover, the constraints of perimeter control input and the urban adaptive control andmodel. model predictive (MPC) are combined. That is, the controller can be designed by (MFAC) merely using the I/O data of control the system, while the system output sequence can be predicted without the system Moreover, theframework, constraints of perimeter control input and the(MFD) urban traffic system’s output areusing bothmodel. considered. Inofthis macroscopic fundamental diagram be designed by merely the I/O data the system, while the system output sequence can be predicted without the system Moreover, constraints of perimeter control input and the urban theframework, macroscopic fundamental diagram (MFD) traffic system’s output are considered. In this is utilized to determine theboth desired vehicle accumulations in each region and control generateinput the output data of predicted without the system model. Moreover, the constraints of perimeter and the urban traffic system’s output are both considered. In this framework, macroscopic fundamental diagram (MFD) is to determine the desired vehicle of accumulations in each region andviagenerate the output data of theutilized urban traffic system.are The effectiveness the proposed method is tested simulation, and the(MFD) result traffic system’s output both considered. In this framework, macroscopic fundamental diagram is utilized to determine the desired vehicle of accumulations in each region andviagenerate the output of the urban system. The effectiveness the proposed method is tested simulation, and thedata result shows thattraffic it works better than MPCvehicle strategy, which is commonly used forand perimeter control. is utilized to determine the desired accumulations in each region generate the output data of the urban traffic system. The effectiveness of the proposed method is tested via simulation, and the result shows that it works better than MPC strategy, which is commonly used for perimeter control. the urban traffic system. The effectiveness of the proposed method is tested via simulation, and the result shows itModel works better than MPC strategy, whichControl) iswith commonly used perimeter control. © 2019,that IFAC (International Federation of Automatic Hosting by for Elsevier Ltd. All rights reserved. Keywords: free adaptive predictive control constraints (C-MFAPC), Perimeter control, shows that itModel works better than MPC strategy, control which iswith commonly used for perimeter control. Keywords: free adaptive predictive constraints (C-MFAPC), Perimeter control, Macroscopic fundamental diagram (MFD) Keywords: Model free adaptive predictive control with constraints (C-MFAPC), Perimeter control, Keywords: free adaptive MacroscopicModel fundamental diagrampredictive (MFD) control with constraints (C-MFAPC), Perimeter control, Keywords: Model free adaptive predictive control with constraints (C-MFAPC), Perimeter control, Macroscopic fundamental diagram (MFD) Macroscopic fundamental diagram (MFD) Macroscopic fundamental diagram (MFD) researches on perimeter control, such as LQR (Aboudolas 1. INTRODUCTION researches on perimeter as LQR (Aboudolas and Geroliminis, 2013), control, optimal such control (Haddad, 2017), 1. INTRODUCTION 1. INTRODUCTION researches on perimeter control, such as LQR (Aboudolas and Geroliminis, 2013), optimal control (Haddad, 2017), 1. INTRODUCTION MPC (Geroliminis et al.,control, 2013; Sirmatel and Geroliminis, solve the problem researches on perimeter such as LQR (Aboudolas Urban traffic control1.isINTRODUCTION an effective way to and Geroliminis, 2013), optimal control and (Haddad, 2017), MPC (Geroliminis ettheal.,existing 2013; perimeter Sirmatel Geroliminis, solve the problem 1.isINTRODUCTION Urban traffic control an effective way to 2018), etc. Most of control methods of urban traffic congestion. The traffic information of each 2018), and Geroliminis, 2013), optimal control and (Haddad, 2017), MPC (Geroliminis ettheal.,existing 2013; perimeter Sirmatel Geroliminis, Urban traffic control is an effective way to solve the problem etc. Most of control methods are based onMost the MFD-based trafficperimeter model. However, MFD is of urban traffic congestion. The traffic information of each link and intersection is required to be known in the micro MPC (Geroliminis et al., 2013; Sirmatel and Geroliminis, 2018), etc. of the existing control methods Urban traffic control is an effective way to solve the problem of urban traffic congestion. The traffic information of each are based on the MFD-based traffic networks, model. However, MFD is well-defined in which methods is highlink and intersection is required to known theof2011), micro 2018), etc.onMost of heterogeneous the existing control level traffic control (Aboudolas al.,be2009; Lin in et al., are based the MFD-based trafficperimeter model. However, MFD is of urban traffic congestion. Theettraffic information each not not well-defined in heterogeneous networks, which is highlink and intersection is required to be known in the micro and hasMFD-based hysteresis phenomena (Buisson andis Ladier, level traffic (Aboudolas al., 2009; Lintoin ethandle al., 2011), which is intersection toocontrol complicated, and etitto is be difficult the scattered are based on the traffic networks, model. However, MFD is not well-defined in heterogeneous which highlink and is required known the micro level traffic (Aboudolas etit al., 2009; Lintoethandle al., 2011), scattered and has hysteresis phenomena (Buisson andisLadier, The inaccurate traffic model will which lead tohighthe which is of toocontrol and isdistributions. difficult the 2009). problem thecomplicated, imbalanced vehicle Therefore, not well-defined in heterogeneous networks, scattered and has hysteresis phenomena (Buisson and Ladier, level traffic control (Aboudolas et al., 2009; Lin et al., 2011), which is of toothecomplicated, and it isdistributions. difficult to handle the 2009). inaccurate model will lead to the problem imbalanced vehicle Therefore, the controltraffic effects of the(Buisson above model-based urban traffic control at the and macro has attracted great scatteredThe and of has hysteresis phenomena and Ladier, which is of toothe complicated, it islevel difficult to handle the deterioration 2009). The inaccurate traffic model will lead to the problem imbalanced vehicle distributions. Therefore, deterioration of the control effects of the above model-based urban traffic control at the macro level has attracted great control methods. In (Ji and Geroliminis, 2012; Lopeztoet the al., 2009). The of inaccurate traffic model will lead interest ofofresearchers andthe engineers. In recent years, a macro problem the imbalanced vehicle distributions. Therefore, deterioration the control effects of the above model-based urban traffic control at macro level has attracted great control methods. In control (Ji and effects Geroliminis, 2012;networks Lopez et are al., interest of researchers and engineers. In recentdiagram years, a(MFD) macro 2017), the heterogeneous urban traffic level tool called macroscopic fundamental deterioration of the of the above model-based urban traffic control at macro level has attracted great 2017), control methods. In (Ji and Geroliminis, 2012;networks Lopez et are al., interest of researchers andthe engineers. In recent years, a(MFD) macro the into heterogeneous urban traffic partitioned some homogeneous regions to make the level tool called macroscopic fundamental diagram has been to and solve thefundamental problem of diagram setting of 2017), control methods. In (Ji homogeneous and Geroliminis, 2012;networks Lopez et are al., interest ofproposed researchers engineers. In recent years, point a(MFD) macro the into heterogeneous urban traffic level tool called macroscopic partitioned some regions to make the MFDs well-defined, but the homogeneous regions are has been proposed to solve the problem of setting point of urban traffic control, and some MFD-based perimeter control 2017), the heterogeneous urban traffic networks are partitioned into some homogeneous regions to make the level tool proposed called macroscopic (MFD) has been to solve thefundamental problem of diagram setting point of difficult MFDs well-defined, butperfectly, the homogeneous regions are to be partitioned andregions the modeling process urban traffic some MFD-based control partitioned into some but homogeneous toregions make the methods arecontrol, presented to the make the urban traffic load MFDs well-defined, the homogeneous are has been proposed toand solve problem ofperimeter setting point of difficult to be partitioned perfectly, and the modeling process urban traffic control, and some MFD-based perimeter control MFD well-defined, is also arduous.butperfectly, methods arecontrol, presented to make the urban trafficcontrol load of balanced. MFDs the homogeneous regions are difficult to be partitioned and the modeling process urban traffic and some MFD-based perimeter methods MFD is arduous. perfectly, and the modeling process balanced. are presented to make the urban traffic load of difficult to also be partitioned of MFD is also arduous. methods are presented to make the urban traffic load In view of the above shortcomings of the model-based balanced. MFD depicts a unimodal and low-scatter relationship of MFD is also arduous. balanced. In view of the shortcomings of the to model-based MFD depicts a unimodal and low-scatter relationship control above approaches, it is necessary develop a between the number of vehicles and trip completion flow in a perimeter In view of the above shortcomings of the to model-based MFD depicts a unimodal and low-scatter relationship perimeter control approaches, it is necessary develop between the number of vehicles and trip completion flow in a new control strategy without the model information. For thisa region or an urban traffic network. The definition of MFD In view of the above shortcomings of the model-based MFD depicts a unimodal and low-scatter relationship perimeter control approaches, it model is necessary to develop a between the number of vehicles and trip completion flow in a new control strategy without the information. For region or an urban traffic network. The definition of MFD purpose, model freeapproaches, adaptive control emerged as this thea was in (Godfrey, 1969; Herman and Prigogine, perimeter control is (MFAC) necessary to develop between of vehicles and trip flow in a new control strategy without control theit model information. For this regionprovided orthe annumber urban traffic network. Thecompletion definition of MFD purpose, model free adaptive (MFAC) emerged as the require. MFAC is originally proposed inemerged (Hou,For 1994), was provided in2007), (Godfrey, Herman Prigogine, new control strategy without the model information. 1979; Daganzo, and network. the1969; existence it and was verified in times purpose, model free adaptive control (MFAC) as this the region or an urban traffic The of definition of MFD was provided in (Godfrey, 1969; Herman and Prigogine, times require. MFAC is originally proposed in (Hou, 1994), and the systematic framework has been shaped in (Hou and 1979; Daganzo, 2007), and the existence of it was verified in purpose, model free adaptive control (MFAC) emerged as the (Geroliminis and Daganzo, 2008). MFD is an inherent times require. MFAC is originally proposed in (Hou, 1994), was provided (Godfrey, Herman Prigogine, 1979; Daganzo,in2007), and the1969; existence of it and was verified in Jin, and 2011; the systematic has beenhas shaped in (Hou and HouMFAC and framework Jin,is 2013). MFAC been successfully (Geroliminis and Daganzo, 2008). MFD an inherent characteristic of urban road network, and it isit is not affected by timestherequire. originally proposed in (Hou, 1994), and systematic framework has been shaped in (Hou and 1979; Daganzo, 2007), and the existence of was verified in (Geroliminis andurban Daganzo, 2008). and MFD inherent Jin, anddiscrete-time Jin, 2013). has MFAC been in successfully in Hou many nonlinear systems, suchand as characteristic of road network, is is notan affected by applied traffic demand, which makes it2008). helpful MFD foritmacroscopic urban and 2011; the systematic beenhas shaped (Hou Jin, 2011; Hou and framework Jin, 2013). MFAC has been successfully (Geroliminis and Daganzo, inherent characteristic of urban road network, and itmacroscopic is is notan affected by computer applied in many discrete-time nonlinear systems, such as traffic demand, which makes it helpful for urban communication networks (Chu et al., 2013), Jin, 2011; anddiscrete-time Jin, 2013). MFAC has systems, been successfully traffic perimeter control, since the detailed O-D are by no applied characteristic ofwhich urban road network, and is nottables affected in Hou many nonlinear such as traffic demand, makes it the helpful foritmacroscopic urban computer communication networks (Chu et al., 2013), traffic perimeter control, since detailed O-D tables are no nonlinear systems (Pang et al.,systems, 2016), and it as is applied in communication many discrete-time nonlinear such longer needed. which makes it helpful for macroscopic urban networked traffic demand, computer networks (Chu et al., 2013), traffic nonlinear systems (Pang(Lei et(Chu al., and2013), it is longer perimeter needed. control, since the detailed O-D tables are no networked also applicable for perimeter control and2016), Hou, 2018). computer communication networks et al., traffic perimeter control, since the detailed O-D tables are no networked nonlinear systemscontrol (Pang(Lei et al., and it is longer also applicable for perimeter and2016), Hou, 2018). In the needed. last decades, there have been already a number of networked nonlinear systemscontrol (Pang(Lei et al., 2016), and it is longer applicable for perimeter and Hou, 2018). In the needed. last decades, there have been already a number of also In this paper, a data-driven perimeter control method called also applicable for perimeter perimeter control (Lei and Hou, 2018). In the last decades, there have been already a number of model In this paper, a data-driven control method called adaptive predictive controlcontrol with constraints (Csupportedthere by thehave National Natural Science of In this free paper, a data-driven perimeter method called InThis thework lastwas decades, been already a Foundation number of model free adaptive predictive control with constraints (CThis work was supported by the National Natural Science Foundation of MFAPC) is presented. It combines the merits of MFAC and This under work was supported by the Natural Science Foundation of China Grants 61433002 and National 61833001, and in part by the Beijing In this free paper, a data-driven perimeter control method called model adaptive predictive control with constraints (CMFAPC) is presented. It combines the merits of MFAC and This work was supported by the National Natural Science Foundation of China under Grants 61433002 and 61833001, and in in part by by the Beijing Beijing This under work was supported by the National Natural Science Foundation of Natural Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and part the model free adaptive predictive control with constraints (CMFAPC) is presented. It combines the merits of MFAC and China under Grants 61433002 and 61833001, and in in part by by the Beijing Beijing This under work was supported by the National Natural Science Foundation of Natural Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and part the Natural Science Foundation under Grant W17E000020. MFAPC) is presented. It combines the merits of MFAC and Naturalunder Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and in part by the Beijing Natural Science Foundation under Grant W17E000020.
Copyright 2019 IFAC 25 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Natural Science Foundation under Grant W17E000020. Copyright 2019 IFAC 25 Copyright ©under 2019 responsibility IFAC 25 Control. Peer review© of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 25 Copyright © 25 10.1016/j.ifacol.2019.08.143 Copyright © 2019 IFAC 25
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MPC. On one hand, the control input sequence on the control horizon can be obtained by only using the I/O data of the urban traffic system. On the other hand, the system output sequence on the prediction horizon could be predicted without the traffic model information. MFAPC for singleinput and single-output (SISO) systems is proposed in (Hou and Jin, 2013). In this work, it is extended to the multipleinput and multiple-output (MIMO) form, and the I/O constraints are also taken into account.
M ii (k ) (nii (k ) / ni (k )) Gi (ni (k )),
(5)
Gi (ni (k )) ai ni (k ) bi ni (k ) ci ni (k ).
(6)
3
2
where the variables are defined as follows: i, j 1, 2
ni (k ) nij (k )
The main contribution of this paper is that a new data driven C-MFAPC strategy is introduced to deal with the problem of perimeter control for two-region urban traffic system, which combines the excellences of MFAC and MPC, and the system constraints are also considered. Under this scheme, the complex and difficult urban traffic modeling process can be avoided. Instead, only the I/O data are needed during the process of controller design. Meanwhile, the high uncertainties can be handled.
nii (k ) nicr nijam qij (k )
qii (k )
Gi (ni (k ))
The rest of this paper is organized as follows. The dynamics for the two-region urban traffic system is presented in Section 2. In Section 3, the C-MFAPC perimeter control strategy is presented. The numerical simulation results of the C-MFPAC method compared with no control (NC) and MPC strategies are described in Section 4. In Section 5, the conclusions are summarized.
ai , bi , ci
M ij (k )
M ii (k ) uij (k ) T
Symbols representing regions, The total number of vehicles travelling in region i at time step k, i.e., the vehicle accumulations in region i at time step k, Number of vehicles travelling in region i with destination to region j at time step k, Number of vehicles travelling in region i with destination in itself at time step k, The critical accumulation of region i, The jammed accumulation of region i, Exogenous traffic demands generated in region i with destination to region j at time step k, Endogenous traffic demands generated in region i with destination in itself at time step k, Trip completion flow for region i at time step k, i.e., the MFD of region i, The parameters of the MFDs, Transfer flow from region i to region j at time step k, Internal flow in region i at time step k, Perimeter control ratio from region i to region j at time step k, The sampling and control cycle length.
Actually uij (k ) M ij (k ) is the real transfer flow from region i to region j at time step k. However, there is a boundary capacity between the regions. The boundary capacity Cij ((n j (k )) is defined as below (Sirmatel and Geroliminis, 2018):
(a)
(b)
Fig. 1. The two-region urban traffic system and the MFDs: (a) the sketch of the road network; (b) the MFDs of the regions. 2. TRAFFIC DYNAMICS FOR TWO-REGION URBAN TRAFFIC NETWORK SYSTEM
(1)
nii (k 1) nii (k ) T (qii (k ) u ji (k ) M ji (k ) M ii (k )),
(2)
ni (k ) nii (k ) nij (k ),
(3)
M ij (k ) (nij (k ) / ni (k )) Gi (ni (k )),
(4)
where Cijmax represents the maximum value of the boundary capacity, and 0 1 . In consideration of the boundary capacity, one has uij (k ) Cij ((n j (k )) / M ij (k ) . Besides, uij ( k ) is also limited by the green time ratio of boundary intersections. Define , max , and min are the initial, maximum and minimum green
In this section, the urban traffic dynamics based on vehicle conservation is presented. As shown in Fig. 1, there is an urban traffic network partitioned into two regions with different MFDs. Region 1 locates at the city centre, while region 2 is on the periphery. The dynamic equations of the urban traffic system are the same with the ones in (Geroliminis et al., 2013) after being discretized via the firstorder Euler approach, which are shown as below. It is noteworthy that the urban traffic model is only utilized to generate the output data of the urban road network system, instead of perimeter controller design.
nij (k 1) nij (k ) T (qij (k ) uij (k ) M ij (k )),
Cijmax ,if 0 n j (k ) n jjam max Cij (n j (k )) Cij (7) Cijmax n j (k ), if n jjam n j (k ) n jjam , jam 1 (1 ) n j
time
ratios,
respectively.
Thus,
uijg,max max /
and
u min / are the theoretical approximation of the maximum and minimum perimeter control ratios, respectively. Therefore, one has umin uij (k ) uij ,max (k ) , g ij ,min
where umin uijg,min , and uij ,max (k ) min{uijg,max , Cij ((n j (k )) / M ij (k )} . 3. PERIMETER CONTROL FOR THE URBAN TRAFFIC SYSTEM As shown in the urban traffic dynamics (1)–(6), the tworegion urban road network system is a MIMO nonlinear system, which has two inputs and two outputs. In this section, 26
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Theorem 1 (Hou and Jin, 2013). Consider the nonlinear system (8) satisfying assumptions 1 and 2, there must exist a time-varying matrix pseudo Jacobian matrix (PJM) denoted by (k ) 22 , so that the system (8) can be transformed into the following equivalent CFDL data model:
C-MFAPC method based on the compact form dynamic linearization (CFDL) data model (Hou and Jin, 2013) is utilized to address the perimeter control problem of the tworegion urban traffic system. The scheme consists of two parts: dynamic linearization and perimeter controller design, which are introduced in the following respectively.
y(k 1) y(k ) (k )u(k ).
Since the urban traffic system is a MIMO nonlinear system, an equivalent description of the unknown nonlinear system can be provided by dynamic linearization technique (DLT). For the convenience of reading, the general form of CFDL data model for MIMO nonlinear systems is presented below.
, u(k nu )),
The objective function for estimating PJM at the current time step k is proposed below:
(8)
J ( (k )) || y(k ) y(k 1) (k )u(k 1) ||2 || (k ) ˆ(k 1) ||2 , (11) where 0 is a weighting factor to penalize excessive change of the estimation of the PJM. The following modified projection algorithm can be applied to estimate (k ) by minimizing (11):
n y nu 2
nonlinear function vector. The proposed two-region urban traffic system has two perimeter control inputs and two outputs, i.e., m=2. The inputs of the system are the perimeter control ratios between the regions, while the outputs are the number of vehicles in the regions. Thus, the input and output vectors are defined as u(k ) [u12 (k ), u21 (k )]T and y(k ) [n1 (k ), n2 (k )]T , respectively. Combining (1)–(6), one has
u ji (k ) M ji (k ) M ii (k )), i, j 1, 2, i j.
is
After the CFDL data model for the urban traffic network system is established, the C-MFAPC perimeter controller is designed as follows.
where u(k ) m and y(k ) m are the control input and the output vectors of the system at time step k, respectively. ny and nu are two unknown integers, and m f (...) [ f1 (...), f m (...)]T m is an unknown
ni (k 1) ni (k ) T ( qij (k ) qii (k ) uij ( k ) M ij ( k )
2 2
3.2 C-MFAPC Perimeter Controller Designing
Consider the following MIMO nonlinear system with m inputs and m outputs:
, y(k n y ), u(k ),
(10)
(k ) 12 (k ) where u(k ) u(k ) u(k 1) , and (k ) 11 21 (k ) 22 (k ) bounded for all the time step k.
3.1 Dynamic Linearization for the Urban Traffic System
y(k 1) f ( y(k ),
27
( y(k ) ˆ(k 1)u(k 1))uT (k 1) , (12) ˆ(k ) ˆ(k 1) || u(k 1) ||2 ˆii (k ) ˆii (1), if | ˆii (k ) | or | ˆii (k ) | M (13) or sign(ˆii ( k )) sign(ˆii (1)), i 1, 2. ˆij (k ) ˆij (1), if | ˆij (k ) | (14) or sign(ˆij (k )) sign(ˆij (1)), i, j 1, 2, i j.
(9)
where ˆ(k ) is the estimation value of (k ) , and ˆij (1) is the initial value of ˆij (k ) , i, j 1, 2 . (0, 2] is a weighting factor, is a small positive constant, M and are positive constants.
Comparing (8) and (9), it can be seen that ny and nu are all equal to 0 in the urban traffic system. In order to utilize the MFAPC scheme for the two-region urban traffic system, the following assumptions and theorem are proposed below:
Denote Ny and Nu are the prediction and control horizons, respectively, so that it can be obtained from (10) that
Assumption 1. The partial derivatives of f (...) with respective to every element of u(k ) are continuous.
Y (k 1) E (k ) y(k ) A(k )U (k ),
Assumption 2. The system (8) is generalized Lipschitz, i.e., for all k1 k2 , k1 , k2 0 , and u(k1 ) u(k2 ) , one has
(15)
where the variables are defined as below:
Y ( k 1) [ y( k 1) T , , y( k N y ) T ]T 2 N y U ( k ) [ u( k )T , , u( k N u 1)T ]T 2 Nu constant. 2Ny T Remark 1. In reality, the above assumptions imposed on the E ( k ) [ I 2 2 , I 2 2 , , I 2 2 ] urban traffic system are reasonable and acceptable. 0 0 0 ( k ) Assumption 1 is easy to be verified from the dynamics of the 0 0 ( k ) ( k 1) two-region urban traffic system (9), and it is a typical assumption for controller design for nonlinear systems. A(k ) ( k ) ( k 1) Assumption 2 is a physical constraint by the inherent ( k N u 1) characteristic of urban traffic system, i.e., finite change of perimeter control ratios cannot lead to infinite change of the ( k ) ( k 1) ( k N u 1) number of vehicles in the regions. Meanwhile, it is a physical constraint of the real system from the energy point of view. y(k1 1) y(k2 1) h u(k1 ) u(k2 ) , where h is a positive
27
2 N y 2 N u
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At each time step k, the following auto-regressive (AR) model is utilized to predict the PJM of the next Nu time steps:
ˆ(k j ) 1 (k )ˆ(k j 1) 2 (k )ˆ(k j 2)
or sign(ˆpp (k j )) sign(ˆpp (1)), p 1,2.
ˆpq (k j ) ˆpq (1), if | ˆpq (k j ) | or sign(ˆpq (k j )) sign(ˆpq (1)), p, q 1, 2, p q.
2 Nu
,
N jam [n
]
T T jam
,..., n
, Umin [u ,..., u ] T min
and Umax (k ) [umax (k ) ,..., umax (k Nu 1) ]
(17)
(18)
2 Nu
.
where g = [1,1,0,...,0]T . Thus, the perimeter control inputs u(k ) is obtained as follow:
u(k ) u(k 1) g T U (k ) .
(25)
The C-MFAPC perimeter control scheme is constructed of (11)–(25). The reset mechanism (13)–(14) and (17)–(18) are designed to make the ability for tracking time-varying parameters of the PJM estimation algorithm stronger.
, ˆ (k j n p )] , the PJM T
prediction model (16) can be transformed into the following form:
(k j ) (k )ˆ (k j 1),
T T
T T min
The optimization problem (21) with constraints of (24) is a convex quadratic programming (QP) problem, which can be solved in many ways, such as sequential quadratic programming (SQP), interior point method, etc. After U (k ) is calculated from (21)–(24), one has u(k ) = g T U (k ) ,
where i (k ) 22 , i=1,2,…,np, is the coefficient matrix, and np is a proper model order. np is usually chosen as 2-7 (Han, 1984). In this paper, np=2. Denote (k ) [1 (k ), , n p (k )] and ˆ (k j 1) [ˆ (k j 1),
,
2Ny
T jam
(16)
ˆpp (k j ) ˆpp (1), if | ˆpp (k j ) | or | ˆpp (k j ) | M
T
2 Nu
T
n p (k )ˆ(k j n p ), j 1, 2,..., N u ,
T
where U (k 1) [u(k 1)T , u(k )T , , u(k Nu 2)T ]T
(19)
where (k ) can be calculated below:
(k ) (k 1)
(ˆ(k ) (k 1)ˆ (k 1)) ˆ T (k 1), (20) || ˆ (k 1) ||2
where 0 is a weighting factor to penalize excessive change of the estimation of (k ) .
Fig. 2. Traffic demands.
The cost function for obtaining U (k ) is as below: Ny
4. SIMULATION RESULTS
Nu 1
In this section, the performance of the proposed C-MFAPC strategy for the two-region urban traffic network system is tested via numerical simulation. As mentioned in Section 2, an urban road network is partitioned into two regions, and the MFD of region 2 is the same as the one in the downtown of Yokohama (Geroliminis et al., 2013), while the MFD of region 1 is 1.2 times as the one in region 2, i.e., a1=4.959×10−7, b1=9.938×10−7, c1=5.04×10−3, a2=4.133×10−7, b2=8.282×10−7, and c2=4.2×10−3, respectively. The jammed and critical accumulations of the two regions are the same, which are njam=10000(veh) and ncr=3400(veh), respectively. The boundary capacity parameters are 0.64 , C12max 3.2 (veh / s) and C21max 3.84 (veh / s) , respectively.
J ( U (k )) y(k i ) y (k i ) u(k j ) , (21) 2
*
i 1
2
j 0
is the desired vehicle accumulations of where y (k i ) the regions, 0 is a weighting factor to restrain the change of perimeter control inputs, and is a positive factor which is introduced to make the order of magnitude of the two items in (21) are in the same level. *
2
It is noteworthy that both the inputs and the outputs have constraints. On one hand, the vehicle accumulations in each region cannot be less than 0 or greater than nijam , i.e.
0 y(k i ) n jam , i 1,..., N y . where 0 = [0,0] , and n jam [n T
jam 1
(22)
n1 (0) 6400 , n11 (0) 3200 , n12 (0) 3200 , n2 (0) 6000 , n21 (0) 3000 , and n22 (0) 3000 . Both of the regions are suffering morning peak under the initially congested condition. The desired vehicle accumulations in each region is chosen as 0.97ncr in order to make the the traffic throughput maximized and ensure the robustness of the urban road network system. The initial, maximum and minimum green time ratios are 0.5 , max 0.9 and min 0.1 , respectively. Thus, uijg,max 1.8 The
jam T 2
, n ] . On the other hand, the perimeter control inputs are under the following constraints: umin u(k j ) umax (k j ), j 0,..., Nu 1.
where
umax (k j ) [u12,max (k j ), u21,max (k j )]
T
(23) ,
and
umin [umin , umin ] . Combining (15), (22) and (23), one has T
(k ) U (k ) N jam E (k ) y(k ) U (k ) E (k ) y(k ) - (k ) , U (k ) U max (k ) U (k 1) I U (k 1) U min -I U (k )
initial
number
of
vehicles
are:
and uijg,min 0.2 according to the definition of perimeter
(24)
control ratio. Other controller parameters are: 1000 , 3 , 1 , respectively. 28
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29
Fig. 3. Result of NC: (a) evolution of accumulation in each region; (b) perimeter control ratio of NC.
Fig. 4. Result of MPC: (a) evolution of accumulation in each region; (b) perimeter control ratio of MPC.
Fig. 5. Result of C-MFAPC: (a) evolution of accumulation in each region; (b) perimeter control ratio of C-MFAPC. The measurement noises in the demands, accumulations, and MFDs obey normal and uniform distribution, respectively, which are shown as below:
qij (k ) qij (k ) (1
(0, q2ij )),
(26)
ni (k ) ni (k ) (1
(0, n2i )),
(27)
Gi (ni (k )) Gi (ni (k )) (1
( Gi , Gi )),
network system under the same condition. NC is a benchmark for comparing the perimeter control effects, in which the perimeter control ratios are equal to 1 all the time (Aboudolas and Geroliminis, 2013). MPC is commonly used for perimeter control. The nonlinear urban traffic model (1)– (6) in Section 2 is utilized to design the MPC perimeter controller. The objective function and the constraints of MPC strategy are the same as the ones in (Geroliminis et al., 2013). The predictive horizon and control horizon of MPC and CMFAPC are both chosen as Np=5 and Nc=2.
(28)
where the error parameters are q2ij 0.1 , n2i 0.1 , and
G 0.3 , respectively.
The simulation results are shown in Fig. 3–5. Fig. 3 shows the result of NC. It can be seen from the figures that region 2 fell into jammed state after half time and the congestion maintained till the end, while the road space in region 1 was wasted due to there were few vehicles travelling in it.
i
The simulation duration lasts 4 hours including the morning peak hour, and the control cycle length is T=240s. Hence, the 4 hours of simulation time is divided into 60 time steps. The time-varying traffic demands are depicted in Fig. 2. The simulation is based on MATLAB 2015b.
Fig. 4 describes the result of MPC strategy. It can be seen from Fig. 4 (a) that the regions were no longer jammed, but both of the regions were still congested at the end of the simulation, since the accumulations in the regions were
Meanwhile, in order to compare with C-MFAPC strategy, no control (NC) and MPC are also tested in the same urban road 29
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greater than ncr.
volume control for interactive network traffic replay. Computer Networks, 57 (7), 1611–1629. Daganzo, C.F. (2007). Urban gridlock: Macroscopic modeling and mitigation approaches. Transportation Research Part B: Methodological, 41 (1), 49–62. Geroliminis, N. and Daganzo, C. F. (2008). Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings. Transportation Research Part B: Methodological, 42 (9), 759–770. Geroliminis, N., Haddad, J., and Ramezani, M. (2013). Optimal perimeter control for two urban regions with macroscopic fundamental diagrams: a model predictive approach. IEEE Transactions on Intelligent Transportation Systems, 14 (1), 348–359. Godfrey, J.W. (1969). The mechanism of a road network. Traffic Engineering and Control, 11 (7), 323–327. Haddad, J. (2017). Optimal coupled and decoupled perimeter control in one-region cities. Control Engineering Practice, 61, 134–148. Han, Z. (1984). On the identification of time-varying parameters in dynamic systems. Acta Automatica Sinica, 10 (4), 330–337. Herman, R. and Prigogine, I. (1979). A two-fluid approach to town traffic. Science, 204 (4389), 148–151. Hou, Z. (1994). The parameter identification, adaptive control and model free learning adaptive control for nonlinear systems. Ph.D. thesis, North-eastern University, Shenyang (In Chinese). Hou, Z. and Jin, S. (2011). Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems. IEEE Transactions on Neural Networks, 22 (12), 2173–2188. Hou, Z. and Jin, S. (2013). Model free adaptive control: theory and applications. CRC press, Florida. Ji, Y. and Geroliminis, N. (2012). On the spatial partitioning of urban transportation networks. Transportation Research Part B: Methodological, 46 (10), 1639–1656. Lei, T. and Hou, Z. (2018). Model free adaptive perimeter control for two-region urban traffic system with input and output constraints. In IEEE 7th Data Driven Control and Learning Systems Conference (DDCLS), 374–379, Enshi. Lin, S., Schutter, B.D., Xi, Y., and Hellendoorn, H. (2011). Fast model predictive control for urban road networks via MILP. IEEE Transactions on Intelligent Transportation Systems, 12 (3), 846–856. Lopez, C., Krishnakumari, P., Leclercq, L., Chiabaut, N., and Lint, H.V. (2017). Spatiotemporal partitioning of transportation network using travel time data. Transportation Research Record, 2623, 98–107. Pang, Z.H., Liu, G.P., Zhou, D., and Sun, D. (2016). Databased predictive control for networked nonlinear systems with network-induced delay and packet dropout. IEEE Transactions on Industrial Electronics, 63 (2), 1249– 1257. Sirmatel, I.I., & Geroliminis, N. (2018). Economic model predictive control of large-scale urban road networks via perimeter control and regional route guidance. IEEE Transactions on Intelligent Transportation Systems, 19 (4), 1112–1121.
The result of C-MFAPC is depicted in Fig. 5. As shown in Fig. 5 (a), the congestion phenomenon in both regions was disappeared. Instead, the number of vehicles in the two regions were both near the desired value, which made the traffic throughput maximized. In addition, the total time spent (TTS) of the urban traffic network is introduced to evaluate the perimeter control performance of the above strategies, which is calculated as K
2
TTS T ni (k ) . The results are compared in Table 1, k 1 i 1
where the improvement is relative to NC, expressed in percentages. As shown in Table 1, the proposed C-MFAPC method worked better than NC and MPC, since the TTS under C-MFAPC strategy is less than the ones under the other two approaches. Table 1. TTS for the entire urban traffic network Strategy
TTS (s)
NC
1.2040×108
MPC
1.0649×108
11.55
7
41.28
C-MFAPC
7.0695×10
Improvement (%)
5. CONCLUSION A new data driven control method called C-MFAPC is introduced to address the problem of perimeter control for two-region urban road network system. A remarkable advantage of the proposed strategy is that only the I/O data of the system is utilized to design the perimeter controller, and the urban traffic model is no longer needed. Meanwhile, the I/O constraints are considered to make it more practical. The simulation results show that this strategy is superior to MPC under model errors. In the future work, the desired output of each region can be determined via urban traffic big data, instead of the timeconsuming and laborious establishment of MFDs. Meanwhile, the the proposed perimeter control method can be further applied in multi-region urban road network systems. REFERENCES Aboudolas, K. and Geroliminis, N. (2013). Perimeter and boundary flow control in multi-reservoir heterogeneous networks. Transportation Research Part B: Methodological, 55 (9), 265–281. Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2009). Store-and-forward based methods for the signal control problem in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 17 (2), 163–174. Buisson, C. and Ladier, C. (2009). Exploring the impact of homogeneity of traffic measurements on the existence of macroscopic fundamental diagrams. Transportation Research Record, 137 (2124), 127–136. Chu, W., Guan, X., Cai, Z., and Gao, L. (2013). Real-time 30
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IFAC PapersOnLine 52-6 (2019) 25–30
Perimeter Control for Two-region Urban Traffic System Based on Model Free Perimeter Control for Two-region Urban Traffic Based on Model Free Adaptive Predictive Control withSystem Constraints Perimeter Control for Two-region Urban Traffic System Based on Model Free Adaptive Predictive Control withSystem Constraints Perimeter Control for Two-region Urban Traffic Based on Model Free Adaptive Predictive Control with Constraints , Ting Lei*. Control Zhongsheng HouConstraints ** * Adaptive Predictive with Ting Lei*. Zhongsheng Hou**,,* Ting Lei*. Zhongsheng Hou**,,* Ting Lei*. Zhongsheng Hou**,* * Advanced Control SystemsTing Laboratory, Beijing Jiaotong University, Beijing 100044, China Lei*. Zhongsheng Hou ** *
* Advanced Control Systems Laboratory, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]) * Advanced Control Systems Laboratory, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]) ** The School of Automation, Qingdao Qingdao 266071, * Advanced Control Systems Laboratory, BeijingUniversity, Jiaotong University, Beijing China 100044, China (e-mail: [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) (e-mail: [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) ** The School of Automation, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]) (e-mail: [email protected]; [email protected]) Abstract: Recent studies have shown that macroscopic urban traffic control, especially perimeter control, Abstract: Recent studies macroscopic urban traffic perimeter plays an important role inhave the shown field ofthat urban traffic control. In thiscontrol, paper, aespecially novel data driven control, control Abstract: Recent studies have shown that macroscopic urban traffic control, especially perimeter control, plays an important role in the field of urban traffic control. In this paper, a novel data driven control method named model free adaptive predictive control withtraffic constraints (C-MFAPC) is utilized for Abstract: Recent studies have shown that macroscopic urban control, especially perimeter control, plays an named important role free in the field ofpredictive urban traffic control. Inconstraints this paper, (C-MFAPC) a novel data is driven control method adaptive control for perimeter controlmodel for two-region urban traffictraffic system. Inwith this strategy, thea advantages of utilized model free plays an important role in the field of urban control. In this paper, novel data driven control method named free adaptive predictive controlInwith constraints is utilized for perimeter controlmodel for two-region urban traffic control system. this are strategy, the(C-MFAPC) advantages of model free adaptive named control (MFAC) andadaptive model predictive (MPC) combined. That is, theis controller can method free predictive control(MPC) constraints utilized for perimeter controlmodel for two-region urban traffic control system. Inwith this are strategy, the(C-MFAPC) advantages of model free adaptive control (MFAC) and model predictive combined. That is, the controller can be designed by (MFAC) merely using theurban I/O data of control the system, while the system output sequence canfree be perimeter control for two-region traffic system. In this strategy, the advantages of model adaptive control and model predictive (MPC) are combined. That is, the controller can be designed by merely usingmodel. the I/O data of the system, while the system output sequence can be predicted without the system Moreover, the constraints of perimeter control input and the urban adaptive control andmodel. model predictive (MPC) are combined. That is, the controller can be designed by (MFAC) merely using the I/O data of control the system, while the system output sequence can be predicted without the system Moreover, theframework, constraints of perimeter control input and the(MFD) urban traffic system’s output areusing bothmodel. considered. Inofthis macroscopic fundamental diagram be designed by merely the I/O data the system, while the system output sequence can be predicted without the system Moreover, constraints of perimeter control input and the urban theframework, macroscopic fundamental diagram (MFD) traffic system’s output are considered. In this is utilized to determine theboth desired vehicle accumulations in each region and control generateinput the output data of predicted without the system model. Moreover, the constraints of perimeter and the urban traffic system’s output are both considered. In this framework, macroscopic fundamental diagram (MFD) is to determine the desired vehicle of accumulations in each region andviagenerate the output data of theutilized urban traffic system.are The effectiveness the proposed method is tested simulation, and the(MFD) result traffic system’s output both considered. In this framework, macroscopic fundamental diagram is utilized to determine the desired vehicle of accumulations in each region andviagenerate the output of the urban system. The effectiveness the proposed method is tested simulation, and thedata result shows thattraffic it works better than MPCvehicle strategy, which is commonly used forand perimeter control. is utilized to determine the desired accumulations in each region generate the output data of the urban traffic system. The effectiveness of the proposed method is tested via simulation, and the result shows that it works better than MPC strategy, which is commonly used for perimeter control. the urban traffic system. The effectiveness of the proposed method is tested via simulation, and the result shows itModel works better than MPC strategy, whichControl) iswith commonly used perimeter control. © 2019,that IFAC (International Federation of Automatic Hosting by for Elsevier Ltd. All rights reserved. Keywords: free adaptive predictive control constraints (C-MFAPC), Perimeter control, shows that itModel works better than MPC strategy, control which iswith commonly used for perimeter control. Keywords: free adaptive predictive constraints (C-MFAPC), Perimeter control, Macroscopic fundamental diagram (MFD) Keywords: Model free adaptive predictive control with constraints (C-MFAPC), Perimeter control, Keywords: free adaptive MacroscopicModel fundamental diagrampredictive (MFD) control with constraints (C-MFAPC), Perimeter control, Keywords: Model free adaptive predictive control with constraints (C-MFAPC), Perimeter control, Macroscopic fundamental diagram (MFD) Macroscopic fundamental diagram (MFD) Macroscopic fundamental diagram (MFD) researches on perimeter control, such as LQR (Aboudolas 1. INTRODUCTION researches on perimeter as LQR (Aboudolas and Geroliminis, 2013), control, optimal such control (Haddad, 2017), 1. INTRODUCTION 1. INTRODUCTION researches on perimeter control, such as LQR (Aboudolas and Geroliminis, 2013), optimal control (Haddad, 2017), 1. INTRODUCTION MPC (Geroliminis et al.,control, 2013; Sirmatel and Geroliminis, solve the problem researches on perimeter such as LQR (Aboudolas Urban traffic control1.isINTRODUCTION an effective way to and Geroliminis, 2013), optimal control and (Haddad, 2017), MPC (Geroliminis ettheal.,existing 2013; perimeter Sirmatel Geroliminis, solve the problem 1.isINTRODUCTION Urban traffic control an effective way to 2018), etc. Most of control methods of urban traffic congestion. The traffic information of each 2018), and Geroliminis, 2013), optimal control and (Haddad, 2017), MPC (Geroliminis ettheal.,existing 2013; perimeter Sirmatel Geroliminis, Urban traffic control is an effective way to solve the problem etc. Most of control methods are based onMost the MFD-based trafficperimeter model. However, MFD is of urban traffic congestion. The traffic information of each link and intersection is required to be known in the micro MPC (Geroliminis et al., 2013; Sirmatel and Geroliminis, 2018), etc. of the existing control methods Urban traffic control is an effective way to solve the problem of urban traffic congestion. The traffic information of each are based on the MFD-based traffic networks, model. However, MFD is well-defined in which methods is highlink and intersection is required to known theof2011), micro 2018), etc.onMost of heterogeneous the existing control level traffic control (Aboudolas al.,be2009; Lin in et al., are based the MFD-based trafficperimeter model. However, MFD is of urban traffic congestion. Theettraffic information each not not well-defined in heterogeneous networks, which is highlink and intersection is required to be known in the micro and hasMFD-based hysteresis phenomena (Buisson andis Ladier, level traffic (Aboudolas al., 2009; Lintoin ethandle al., 2011), which is intersection toocontrol complicated, and etitto is be difficult the scattered are based on the traffic networks, model. However, MFD is not well-defined in heterogeneous which highlink and is required known the micro level traffic (Aboudolas etit al., 2009; Lintoethandle al., 2011), scattered and has hysteresis phenomena (Buisson andisLadier, The inaccurate traffic model will which lead tohighthe which is of toocontrol and isdistributions. difficult the 2009). problem thecomplicated, imbalanced vehicle Therefore, not well-defined in heterogeneous networks, scattered and has hysteresis phenomena (Buisson and Ladier, level traffic control (Aboudolas et al., 2009; Lin et al., 2011), which is of toothecomplicated, and it isdistributions. difficult to handle the 2009). inaccurate model will lead to the problem imbalanced vehicle Therefore, the controltraffic effects of the(Buisson above model-based urban traffic control at the and macro has attracted great scatteredThe and of has hysteresis phenomena and Ladier, which is of toothe complicated, it islevel difficult to handle the deterioration 2009). The inaccurate traffic model will lead to the problem imbalanced vehicle distributions. Therefore, deterioration of the control effects of the above model-based urban traffic control at the macro level has attracted great control methods. In (Ji and Geroliminis, 2012; Lopeztoet the al., 2009). The of inaccurate traffic model will lead interest ofofresearchers andthe engineers. In recent years, a macro problem the imbalanced vehicle distributions. Therefore, deterioration the control effects of the above model-based urban traffic control at macro level has attracted great control methods. In control (Ji and effects Geroliminis, 2012;networks Lopez et are al., interest of researchers and engineers. In recentdiagram years, a(MFD) macro 2017), the heterogeneous urban traffic level tool called macroscopic fundamental deterioration of the of the above model-based urban traffic control at macro level has attracted great 2017), control methods. In (Ji and Geroliminis, 2012;networks Lopez et are al., interest of researchers andthe engineers. In recent years, a(MFD) macro the into heterogeneous urban traffic partitioned some homogeneous regions to make the level tool called macroscopic fundamental diagram has been to and solve thefundamental problem of diagram setting of 2017), control methods. In (Ji homogeneous and Geroliminis, 2012;networks Lopez et are al., interest ofproposed researchers engineers. In recent years, point a(MFD) macro the into heterogeneous urban traffic level tool called macroscopic partitioned some regions to make the MFDs well-defined, but the homogeneous regions are has been proposed to solve the problem of setting point of urban traffic control, and some MFD-based perimeter control 2017), the heterogeneous urban traffic networks are partitioned into some homogeneous regions to make the level tool proposed called macroscopic (MFD) has been to solve thefundamental problem of diagram setting point of difficult MFDs well-defined, butperfectly, the homogeneous regions are to be partitioned andregions the modeling process urban traffic some MFD-based control partitioned into some but homogeneous toregions make the methods arecontrol, presented to the make the urban traffic load MFDs well-defined, the homogeneous are has been proposed toand solve problem ofperimeter setting point of difficult to be partitioned perfectly, and the modeling process urban traffic control, and some MFD-based perimeter control MFD well-defined, is also arduous.butperfectly, methods arecontrol, presented to make the urban trafficcontrol load of balanced. MFDs the homogeneous regions are difficult to be partitioned and the modeling process urban traffic and some MFD-based perimeter methods MFD is arduous. perfectly, and the modeling process balanced. are presented to make the urban traffic load of difficult to also be partitioned of MFD is also arduous. methods are presented to make the urban traffic load In view of the above shortcomings of the model-based balanced. MFD depicts a unimodal and low-scatter relationship of MFD is also arduous. balanced. In view of the shortcomings of the to model-based MFD depicts a unimodal and low-scatter relationship control above approaches, it is necessary develop a between the number of vehicles and trip completion flow in a perimeter In view of the above shortcomings of the to model-based MFD depicts a unimodal and low-scatter relationship perimeter control approaches, it is necessary develop between the number of vehicles and trip completion flow in a new control strategy without the model information. For thisa region or an urban traffic network. The definition of MFD In view of the above shortcomings of the model-based MFD depicts a unimodal and low-scatter relationship perimeter control approaches, it model is necessary to develop a between the number of vehicles and trip completion flow in a new control strategy without the information. For region or an urban traffic network. The definition of MFD purpose, model freeapproaches, adaptive control emerged as this thea was in (Godfrey, 1969; Herman and Prigogine, perimeter control is (MFAC) necessary to develop between of vehicles and trip flow in a new control strategy without control theit model information. For this regionprovided orthe annumber urban traffic network. Thecompletion definition of MFD purpose, model free adaptive (MFAC) emerged as the require. MFAC is originally proposed inemerged (Hou,For 1994), was provided in2007), (Godfrey, Herman Prigogine, new control strategy without the model information. 1979; Daganzo, and network. the1969; existence it and was verified in times purpose, model free adaptive control (MFAC) as this the region or an urban traffic The of definition of MFD was provided in (Godfrey, 1969; Herman and Prigogine, times require. MFAC is originally proposed in (Hou, 1994), and the systematic framework has been shaped in (Hou and 1979; Daganzo, 2007), and the existence of it was verified in purpose, model free adaptive control (MFAC) emerged as the (Geroliminis and Daganzo, 2008). MFD is an inherent times require. MFAC is originally proposed in (Hou, 1994), was provided (Godfrey, Herman Prigogine, 1979; Daganzo,in2007), and the1969; existence of it and was verified in Jin, and 2011; the systematic has beenhas shaped in (Hou and HouMFAC and framework Jin,is 2013). MFAC been successfully (Geroliminis and Daganzo, 2008). MFD an inherent characteristic of urban road network, and it isit is not affected by timestherequire. originally proposed in (Hou, 1994), and systematic framework has been shaped in (Hou and 1979; Daganzo, 2007), and the existence of was verified in (Geroliminis andurban Daganzo, 2008). and MFD inherent Jin, anddiscrete-time Jin, 2013). has MFAC been in successfully in Hou many nonlinear systems, suchand as characteristic of road network, is is notan affected by applied traffic demand, which makes it2008). helpful MFD foritmacroscopic urban and 2011; the systematic beenhas shaped (Hou Jin, 2011; Hou and framework Jin, 2013). MFAC has been successfully (Geroliminis and Daganzo, inherent characteristic of urban road network, and itmacroscopic is is notan affected by computer applied in many discrete-time nonlinear systems, such as traffic demand, which makes it helpful for urban communication networks (Chu et al., 2013), Jin, 2011; anddiscrete-time Jin, 2013). MFAC has systems, been successfully traffic perimeter control, since the detailed O-D are by no applied characteristic ofwhich urban road network, and is nottables affected in Hou many nonlinear such as traffic demand, makes it the helpful foritmacroscopic urban computer communication networks (Chu et al., 2013), traffic perimeter control, since detailed O-D tables are no nonlinear systems (Pang et al.,systems, 2016), and it as is applied in communication many discrete-time nonlinear such longer needed. which makes it helpful for macroscopic urban networked traffic demand, computer networks (Chu et al., 2013), traffic nonlinear systems (Pang(Lei et(Chu al., and2013), it is longer perimeter needed. control, since the detailed O-D tables are no networked also applicable for perimeter control and2016), Hou, 2018). computer communication networks et al., traffic perimeter control, since the detailed O-D tables are no networked nonlinear systemscontrol (Pang(Lei et al., and it is longer also applicable for perimeter and2016), Hou, 2018). In the needed. last decades, there have been already a number of networked nonlinear systemscontrol (Pang(Lei et al., 2016), and it is longer applicable for perimeter and Hou, 2018). In the needed. last decades, there have been already a number of also In this paper, a data-driven perimeter control method called also applicable for perimeter perimeter control (Lei and Hou, 2018). In the last decades, there have been already a number of model In this paper, a data-driven control method called adaptive predictive controlcontrol with constraints (Csupportedthere by thehave National Natural Science of In this free paper, a data-driven perimeter method called InThis thework lastwas decades, been already a Foundation number of model free adaptive predictive control with constraints (CThis work was supported by the National Natural Science Foundation of MFAPC) is presented. It combines the merits of MFAC and This under work was supported by the Natural Science Foundation of China Grants 61433002 and National 61833001, and in part by the Beijing In this free paper, a data-driven perimeter control method called model adaptive predictive control with constraints (CMFAPC) is presented. It combines the merits of MFAC and This work was supported by the National Natural Science Foundation of China under Grants 61433002 and 61833001, and in in part by by the Beijing Beijing This under work was supported by the National Natural Science Foundation of Natural Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and part the model free adaptive predictive control with constraints (CMFAPC) is presented. It combines the merits of MFAC and China under Grants 61433002 and 61833001, and in in part by by the Beijing Beijing This under work was supported by the National Natural Science Foundation of Natural Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and part the Natural Science Foundation under Grant W17E000020. MFAPC) is presented. It combines the merits of MFAC and Naturalunder Science Foundation under Grant W17E000020. China Grants 61433002 and 61833001, and in part by the Beijing Natural Science Foundation under Grant W17E000020.
Copyright 2019 IFAC 25 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Natural Science Foundation under Grant W17E000020. Copyright 2019 IFAC 25 Copyright ©under 2019 responsibility IFAC 25 Control. Peer review© of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 25 Copyright © 25 10.1016/j.ifacol.2019.08.143 Copyright © 2019 IFAC 25
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Ting Lei et al. / IFAC PapersOnLine 52-6 (2019) 25–30
MPC. On one hand, the control input sequence on the control horizon can be obtained by only using the I/O data of the urban traffic system. On the other hand, the system output sequence on the prediction horizon could be predicted without the traffic model information. MFAPC for singleinput and single-output (SISO) systems is proposed in (Hou and Jin, 2013). In this work, it is extended to the multipleinput and multiple-output (MIMO) form, and the I/O constraints are also taken into account.
M ii (k ) (nii (k ) / ni (k )) Gi (ni (k )),
(5)
Gi (ni (k )) ai ni (k ) bi ni (k ) ci ni (k ).
(6)
3
2
where the variables are defined as follows: i, j 1, 2
ni (k ) nij (k )
The main contribution of this paper is that a new data driven C-MFAPC strategy is introduced to deal with the problem of perimeter control for two-region urban traffic system, which combines the excellences of MFAC and MPC, and the system constraints are also considered. Under this scheme, the complex and difficult urban traffic modeling process can be avoided. Instead, only the I/O data are needed during the process of controller design. Meanwhile, the high uncertainties can be handled.
nii (k ) nicr nijam qij (k )
qii (k )
Gi (ni (k ))
The rest of this paper is organized as follows. The dynamics for the two-region urban traffic system is presented in Section 2. In Section 3, the C-MFAPC perimeter control strategy is presented. The numerical simulation results of the C-MFPAC method compared with no control (NC) and MPC strategies are described in Section 4. In Section 5, the conclusions are summarized.
ai , bi , ci
M ij (k )
M ii (k ) uij (k ) T
Symbols representing regions, The total number of vehicles travelling in region i at time step k, i.e., the vehicle accumulations in region i at time step k, Number of vehicles travelling in region i with destination to region j at time step k, Number of vehicles travelling in region i with destination in itself at time step k, The critical accumulation of region i, The jammed accumulation of region i, Exogenous traffic demands generated in region i with destination to region j at time step k, Endogenous traffic demands generated in region i with destination in itself at time step k, Trip completion flow for region i at time step k, i.e., the MFD of region i, The parameters of the MFDs, Transfer flow from region i to region j at time step k, Internal flow in region i at time step k, Perimeter control ratio from region i to region j at time step k, The sampling and control cycle length.
Actually uij (k ) M ij (k ) is the real transfer flow from region i to region j at time step k. However, there is a boundary capacity between the regions. The boundary capacity Cij ((n j (k )) is defined as below (Sirmatel and Geroliminis, 2018):
(a)
(b)
Fig. 1. The two-region urban traffic system and the MFDs: (a) the sketch of the road network; (b) the MFDs of the regions. 2. TRAFFIC DYNAMICS FOR TWO-REGION URBAN TRAFFIC NETWORK SYSTEM
(1)
nii (k 1) nii (k ) T (qii (k ) u ji (k ) M ji (k ) M ii (k )),
(2)
ni (k ) nii (k ) nij (k ),
(3)
M ij (k ) (nij (k ) / ni (k )) Gi (ni (k )),
(4)
where Cijmax represents the maximum value of the boundary capacity, and 0 1 . In consideration of the boundary capacity, one has uij (k ) Cij ((n j (k )) / M ij (k ) . Besides, uij ( k ) is also limited by the green time ratio of boundary intersections. Define , max , and min are the initial, maximum and minimum green
In this section, the urban traffic dynamics based on vehicle conservation is presented. As shown in Fig. 1, there is an urban traffic network partitioned into two regions with different MFDs. Region 1 locates at the city centre, while region 2 is on the periphery. The dynamic equations of the urban traffic system are the same with the ones in (Geroliminis et al., 2013) after being discretized via the firstorder Euler approach, which are shown as below. It is noteworthy that the urban traffic model is only utilized to generate the output data of the urban road network system, instead of perimeter controller design.
nij (k 1) nij (k ) T (qij (k ) uij (k ) M ij (k )),
Cijmax ,if 0 n j (k ) n jjam max Cij (n j (k )) Cij (7) Cijmax n j (k ), if n jjam n j (k ) n jjam , jam 1 (1 ) n j
time
ratios,
respectively.
Thus,
uijg,max max /
and
u min / are the theoretical approximation of the maximum and minimum perimeter control ratios, respectively. Therefore, one has umin uij (k ) uij ,max (k ) , g ij ,min
where umin uijg,min , and uij ,max (k ) min{uijg,max , Cij ((n j (k )) / M ij (k )} . 3. PERIMETER CONTROL FOR THE URBAN TRAFFIC SYSTEM As shown in the urban traffic dynamics (1)–(6), the tworegion urban road network system is a MIMO nonlinear system, which has two inputs and two outputs. In this section, 26
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Theorem 1 (Hou and Jin, 2013). Consider the nonlinear system (8) satisfying assumptions 1 and 2, there must exist a time-varying matrix pseudo Jacobian matrix (PJM) denoted by (k ) 22 , so that the system (8) can be transformed into the following equivalent CFDL data model:
C-MFAPC method based on the compact form dynamic linearization (CFDL) data model (Hou and Jin, 2013) is utilized to address the perimeter control problem of the tworegion urban traffic system. The scheme consists of two parts: dynamic linearization and perimeter controller design, which are introduced in the following respectively.
y(k 1) y(k ) (k )u(k ).
Since the urban traffic system is a MIMO nonlinear system, an equivalent description of the unknown nonlinear system can be provided by dynamic linearization technique (DLT). For the convenience of reading, the general form of CFDL data model for MIMO nonlinear systems is presented below.
, u(k nu )),
The objective function for estimating PJM at the current time step k is proposed below:
(8)
J ( (k )) || y(k ) y(k 1) (k )u(k 1) ||2 || (k ) ˆ(k 1) ||2 , (11) where 0 is a weighting factor to penalize excessive change of the estimation of the PJM. The following modified projection algorithm can be applied to estimate (k ) by minimizing (11):
n y nu 2
nonlinear function vector. The proposed two-region urban traffic system has two perimeter control inputs and two outputs, i.e., m=2. The inputs of the system are the perimeter control ratios between the regions, while the outputs are the number of vehicles in the regions. Thus, the input and output vectors are defined as u(k ) [u12 (k ), u21 (k )]T and y(k ) [n1 (k ), n2 (k )]T , respectively. Combining (1)–(6), one has
u ji (k ) M ji (k ) M ii (k )), i, j 1, 2, i j.
is
After the CFDL data model for the urban traffic network system is established, the C-MFAPC perimeter controller is designed as follows.
where u(k ) m and y(k ) m are the control input and the output vectors of the system at time step k, respectively. ny and nu are two unknown integers, and m f (...) [ f1 (...), f m (...)]T m is an unknown
ni (k 1) ni (k ) T ( qij (k ) qii (k ) uij ( k ) M ij ( k )
2 2
3.2 C-MFAPC Perimeter Controller Designing
Consider the following MIMO nonlinear system with m inputs and m outputs:
, y(k n y ), u(k ),
(10)
(k ) 12 (k ) where u(k ) u(k ) u(k 1) , and (k ) 11 21 (k ) 22 (k ) bounded for all the time step k.
3.1 Dynamic Linearization for the Urban Traffic System
y(k 1) f ( y(k ),
27
( y(k ) ˆ(k 1)u(k 1))uT (k 1) , (12) ˆ(k ) ˆ(k 1) || u(k 1) ||2 ˆii (k ) ˆii (1), if | ˆii (k ) | or | ˆii (k ) | M (13) or sign(ˆii ( k )) sign(ˆii (1)), i 1, 2. ˆij (k ) ˆij (1), if | ˆij (k ) | (14) or sign(ˆij (k )) sign(ˆij (1)), i, j 1, 2, i j.
(9)
where ˆ(k ) is the estimation value of (k ) , and ˆij (1) is the initial value of ˆij (k ) , i, j 1, 2 . (0, 2] is a weighting factor, is a small positive constant, M and are positive constants.
Comparing (8) and (9), it can be seen that ny and nu are all equal to 0 in the urban traffic system. In order to utilize the MFAPC scheme for the two-region urban traffic system, the following assumptions and theorem are proposed below:
Denote Ny and Nu are the prediction and control horizons, respectively, so that it can be obtained from (10) that
Assumption 1. The partial derivatives of f (...) with respective to every element of u(k ) are continuous.
Y (k 1) E (k ) y(k ) A(k )U (k ),
Assumption 2. The system (8) is generalized Lipschitz, i.e., for all k1 k2 , k1 , k2 0 , and u(k1 ) u(k2 ) , one has
(15)
where the variables are defined as below:
Y ( k 1) [ y( k 1) T , , y( k N y ) T ]T 2 N y U ( k ) [ u( k )T , , u( k N u 1)T ]T 2 Nu constant. 2Ny T Remark 1. In reality, the above assumptions imposed on the E ( k ) [ I 2 2 , I 2 2 , , I 2 2 ] urban traffic system are reasonable and acceptable. 0 0 0 ( k ) Assumption 1 is easy to be verified from the dynamics of the 0 0 ( k ) ( k 1) two-region urban traffic system (9), and it is a typical assumption for controller design for nonlinear systems. A(k ) ( k ) ( k 1) Assumption 2 is a physical constraint by the inherent ( k N u 1) characteristic of urban traffic system, i.e., finite change of perimeter control ratios cannot lead to infinite change of the ( k ) ( k 1) ( k N u 1) number of vehicles in the regions. Meanwhile, it is a physical constraint of the real system from the energy point of view. y(k1 1) y(k2 1) h u(k1 ) u(k2 ) , where h is a positive
27
2 N y 2 N u
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At each time step k, the following auto-regressive (AR) model is utilized to predict the PJM of the next Nu time steps:
ˆ(k j ) 1 (k )ˆ(k j 1) 2 (k )ˆ(k j 2)
or sign(ˆpp (k j )) sign(ˆpp (1)), p 1,2.
ˆpq (k j ) ˆpq (1), if | ˆpq (k j ) | or sign(ˆpq (k j )) sign(ˆpq (1)), p, q 1, 2, p q.
2 Nu
,
N jam [n
]
T T jam
,..., n
, Umin [u ,..., u ] T min
and Umax (k ) [umax (k ) ,..., umax (k Nu 1) ]
(17)
(18)
2 Nu
.
where g = [1,1,0,...,0]T . Thus, the perimeter control inputs u(k ) is obtained as follow:
u(k ) u(k 1) g T U (k ) .
(25)
The C-MFAPC perimeter control scheme is constructed of (11)–(25). The reset mechanism (13)–(14) and (17)–(18) are designed to make the ability for tracking time-varying parameters of the PJM estimation algorithm stronger.
, ˆ (k j n p )] , the PJM T
prediction model (16) can be transformed into the following form:
(k j ) (k )ˆ (k j 1),
T T
T T min
The optimization problem (21) with constraints of (24) is a convex quadratic programming (QP) problem, which can be solved in many ways, such as sequential quadratic programming (SQP), interior point method, etc. After U (k ) is calculated from (21)–(24), one has u(k ) = g T U (k ) ,
where i (k ) 22 , i=1,2,…,np, is the coefficient matrix, and np is a proper model order. np is usually chosen as 2-7 (Han, 1984). In this paper, np=2. Denote (k ) [1 (k ), , n p (k )] and ˆ (k j 1) [ˆ (k j 1),
,
2Ny
T jam
(16)
ˆpp (k j ) ˆpp (1), if | ˆpp (k j ) | or | ˆpp (k j ) | M
T
2 Nu
T
n p (k )ˆ(k j n p ), j 1, 2,..., N u ,
T
where U (k 1) [u(k 1)T , u(k )T , , u(k Nu 2)T ]T
(19)
where (k ) can be calculated below:
(k ) (k 1)
(ˆ(k ) (k 1)ˆ (k 1)) ˆ T (k 1), (20) || ˆ (k 1) ||2
where 0 is a weighting factor to penalize excessive change of the estimation of (k ) .
Fig. 2. Traffic demands.
The cost function for obtaining U (k ) is as below: Ny
4. SIMULATION RESULTS
Nu 1
In this section, the performance of the proposed C-MFAPC strategy for the two-region urban traffic network system is tested via numerical simulation. As mentioned in Section 2, an urban road network is partitioned into two regions, and the MFD of region 2 is the same as the one in the downtown of Yokohama (Geroliminis et al., 2013), while the MFD of region 1 is 1.2 times as the one in region 2, i.e., a1=4.959×10−7, b1=9.938×10−7, c1=5.04×10−3, a2=4.133×10−7, b2=8.282×10−7, and c2=4.2×10−3, respectively. The jammed and critical accumulations of the two regions are the same, which are njam=10000(veh) and ncr=3400(veh), respectively. The boundary capacity parameters are 0.64 , C12max 3.2 (veh / s) and C21max 3.84 (veh / s) , respectively.
J ( U (k )) y(k i ) y (k i ) u(k j ) , (21) 2
*
i 1
2
j 0
is the desired vehicle accumulations of where y (k i ) the regions, 0 is a weighting factor to restrain the change of perimeter control inputs, and is a positive factor which is introduced to make the order of magnitude of the two items in (21) are in the same level. *
2
It is noteworthy that both the inputs and the outputs have constraints. On one hand, the vehicle accumulations in each region cannot be less than 0 or greater than nijam , i.e.
0 y(k i ) n jam , i 1,..., N y . where 0 = [0,0] , and n jam [n T
jam 1
(22)
n1 (0) 6400 , n11 (0) 3200 , n12 (0) 3200 , n2 (0) 6000 , n21 (0) 3000 , and n22 (0) 3000 . Both of the regions are suffering morning peak under the initially congested condition. The desired vehicle accumulations in each region is chosen as 0.97ncr in order to make the the traffic throughput maximized and ensure the robustness of the urban road network system. The initial, maximum and minimum green time ratios are 0.5 , max 0.9 and min 0.1 , respectively. Thus, uijg,max 1.8 The
jam T 2
, n ] . On the other hand, the perimeter control inputs are under the following constraints: umin u(k j ) umax (k j ), j 0,..., Nu 1.
where
umax (k j ) [u12,max (k j ), u21,max (k j )]
T
(23) ,
and
umin [umin , umin ] . Combining (15), (22) and (23), one has T
(k ) U (k ) N jam E (k ) y(k ) U (k ) E (k ) y(k ) - (k ) , U (k ) U max (k ) U (k 1) I U (k 1) U min -I U (k )
initial
number
of
vehicles
are:
and uijg,min 0.2 according to the definition of perimeter
(24)
control ratio. Other controller parameters are: 1000 , 3 , 1 , respectively. 28
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29
Fig. 3. Result of NC: (a) evolution of accumulation in each region; (b) perimeter control ratio of NC.
Fig. 4. Result of MPC: (a) evolution of accumulation in each region; (b) perimeter control ratio of MPC.
Fig. 5. Result of C-MFAPC: (a) evolution of accumulation in each region; (b) perimeter control ratio of C-MFAPC. The measurement noises in the demands, accumulations, and MFDs obey normal and uniform distribution, respectively, which are shown as below:
qij (k ) qij (k ) (1
(0, q2ij )),
(26)
ni (k ) ni (k ) (1
(0, n2i )),
(27)
Gi (ni (k )) Gi (ni (k )) (1
( Gi , Gi )),
network system under the same condition. NC is a benchmark for comparing the perimeter control effects, in which the perimeter control ratios are equal to 1 all the time (Aboudolas and Geroliminis, 2013). MPC is commonly used for perimeter control. The nonlinear urban traffic model (1)– (6) in Section 2 is utilized to design the MPC perimeter controller. The objective function and the constraints of MPC strategy are the same as the ones in (Geroliminis et al., 2013). The predictive horizon and control horizon of MPC and CMFAPC are both chosen as Np=5 and Nc=2.
(28)
where the error parameters are q2ij 0.1 , n2i 0.1 , and
G 0.3 , respectively.
The simulation results are shown in Fig. 3–5. Fig. 3 shows the result of NC. It can be seen from the figures that region 2 fell into jammed state after half time and the congestion maintained till the end, while the road space in region 1 was wasted due to there were few vehicles travelling in it.
i
The simulation duration lasts 4 hours including the morning peak hour, and the control cycle length is T=240s. Hence, the 4 hours of simulation time is divided into 60 time steps. The time-varying traffic demands are depicted in Fig. 2. The simulation is based on MATLAB 2015b.
Fig. 4 describes the result of MPC strategy. It can be seen from Fig. 4 (a) that the regions were no longer jammed, but both of the regions were still congested at the end of the simulation, since the accumulations in the regions were
Meanwhile, in order to compare with C-MFAPC strategy, no control (NC) and MPC are also tested in the same urban road 29
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greater than ncr.
volume control for interactive network traffic replay. Computer Networks, 57 (7), 1611–1629. Daganzo, C.F. (2007). Urban gridlock: Macroscopic modeling and mitigation approaches. Transportation Research Part B: Methodological, 41 (1), 49–62. Geroliminis, N. and Daganzo, C. F. (2008). Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings. Transportation Research Part B: Methodological, 42 (9), 759–770. Geroliminis, N., Haddad, J., and Ramezani, M. (2013). Optimal perimeter control for two urban regions with macroscopic fundamental diagrams: a model predictive approach. IEEE Transactions on Intelligent Transportation Systems, 14 (1), 348–359. Godfrey, J.W. (1969). The mechanism of a road network. Traffic Engineering and Control, 11 (7), 323–327. Haddad, J. (2017). Optimal coupled and decoupled perimeter control in one-region cities. Control Engineering Practice, 61, 134–148. Han, Z. (1984). On the identification of time-varying parameters in dynamic systems. Acta Automatica Sinica, 10 (4), 330–337. Herman, R. and Prigogine, I. (1979). A two-fluid approach to town traffic. Science, 204 (4389), 148–151. Hou, Z. (1994). The parameter identification, adaptive control and model free learning adaptive control for nonlinear systems. Ph.D. thesis, North-eastern University, Shenyang (In Chinese). Hou, Z. and Jin, S. (2011). Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems. IEEE Transactions on Neural Networks, 22 (12), 2173–2188. Hou, Z. and Jin, S. (2013). Model free adaptive control: theory and applications. CRC press, Florida. Ji, Y. and Geroliminis, N. (2012). On the spatial partitioning of urban transportation networks. Transportation Research Part B: Methodological, 46 (10), 1639–1656. Lei, T. and Hou, Z. (2018). Model free adaptive perimeter control for two-region urban traffic system with input and output constraints. In IEEE 7th Data Driven Control and Learning Systems Conference (DDCLS), 374–379, Enshi. Lin, S., Schutter, B.D., Xi, Y., and Hellendoorn, H. (2011). Fast model predictive control for urban road networks via MILP. IEEE Transactions on Intelligent Transportation Systems, 12 (3), 846–856. Lopez, C., Krishnakumari, P., Leclercq, L., Chiabaut, N., and Lint, H.V. (2017). Spatiotemporal partitioning of transportation network using travel time data. Transportation Research Record, 2623, 98–107. Pang, Z.H., Liu, G.P., Zhou, D., and Sun, D. (2016). Databased predictive control for networked nonlinear systems with network-induced delay and packet dropout. IEEE Transactions on Industrial Electronics, 63 (2), 1249– 1257. Sirmatel, I.I., & Geroliminis, N. (2018). Economic model predictive control of large-scale urban road networks via perimeter control and regional route guidance. IEEE Transactions on Intelligent Transportation Systems, 19 (4), 1112–1121.
The result of C-MFAPC is depicted in Fig. 5. As shown in Fig. 5 (a), the congestion phenomenon in both regions was disappeared. Instead, the number of vehicles in the two regions were both near the desired value, which made the traffic throughput maximized. In addition, the total time spent (TTS) of the urban traffic network is introduced to evaluate the perimeter control performance of the above strategies, which is calculated as K
2
TTS T ni (k ) . The results are compared in Table 1, k 1 i 1
where the improvement is relative to NC, expressed in percentages. As shown in Table 1, the proposed C-MFAPC method worked better than NC and MPC, since the TTS under C-MFAPC strategy is less than the ones under the other two approaches. Table 1. TTS for the entire urban traffic network Strategy
TTS (s)
NC
1.2040×108
MPC
1.0649×108
11.55
7
41.28
C-MFAPC
7.0695×10
Improvement (%)
5. CONCLUSION A new data driven control method called C-MFAPC is introduced to address the problem of perimeter control for two-region urban road network system. A remarkable advantage of the proposed strategy is that only the I/O data of the system is utilized to design the perimeter controller, and the urban traffic model is no longer needed. Meanwhile, the I/O constraints are considered to make it more practical. The simulation results show that this strategy is superior to MPC under model errors. In the future work, the desired output of each region can be determined via urban traffic big data, instead of the timeconsuming and laborious establishment of MFDs. Meanwhile, the the proposed perimeter control method can be further applied in multi-region urban road network systems. REFERENCES Aboudolas, K. and Geroliminis, N. (2013). Perimeter and boundary flow control in multi-reservoir heterogeneous networks. Transportation Research Part B: Methodological, 55 (9), 265–281. Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2009). Store-and-forward based methods for the signal control problem in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 17 (2), 163–174. Buisson, C. and Ladier, C. (2009). Exploring the impact of homogeneity of traffic measurements on the existence of macroscopic fundamental diagrams. Transportation Research Record, 137 (2124), 127–136. Chu, W., Guan, X., Cai, Z., and Gao, L. (2013). Real-time 30