A Macroscopic Traffic Data Assimilation Framework Based on Forier-Galerkin Method and Minima Estimation Tigran T. Tchrakian and Sergiy Zhk Abstract In this paper, we propose a new framework for macroscopic traffic state estimation based on the Forier- Galerkin projection method and minima state estimation approach. We assign a Forier-Galerkin redced model to a partial differential eqation describing a macroscopic model of traffic flow. Taking into accont a priori estimates for the projection error, we apply the minima method to constrct the state estimate for the redced model that gives s, in trn, the estimate of the Forier-Galerkin coefficients associated with a soltion if the original macroscopic model. We illstrate or approach with a nmerical eample. I. INTRODUCTION Macroscopic models of traffic flow can be sed together with traffic data in order to estimate the state of traffic on a link or a set of links. Macroscopic models are mathematically described by Partial Differential Eqations (PDEs). The basic idea is to rn the model, which otpts a soltion in time and space, and to obtain an estimate of the state by combining this soltion with the available data over the spatio-temporal domain. Past approaches have involved linearizing [] the model in order to apply Kalman filtering [2], [3], or sing the nonlinear model as it stands, and employing filtering techniqes sch as Etended Kalman Filtering [4] or Particle Filtering [5]. While each category of methods has its respective advantages, what they have in common is that they are applied to a redced model which is obtained from the original macroscopic model by local methods. Local methods se limited information from the spatial domain in order to compte the soltion at a given point in that domain. Specifically, all of the approaches listed above se finite differences as a soltion method for the nderlying PDE, where the soltion in a spatial cell is inflenced by the soltion in its neighboring cells. The reslting system is a set of difference eqations, each describing the evoltion of the state variable (typically density) at a given point in space. In contrast to local soltion methods, global methods se information from the entire spatial domain and deliver continos (in space) approimations for the soltion to the macroscopic model. This allows the soltion at any point in the domain to be compted. While local methods can captre shocks, and provide an adeqate soltion to the model, they have certain drawbacks when it comes to dataassimilation. Specifically, the discretization in space shold take into accont locations of the sensors in order to make it possible to relate the state of the redced model and The athors are with IBM Research Ireland, Damastown, Dblin 5, Ireland {tigran,sergiy.zhk}@ie.ibm.com the measrements. This becomes a problem if sensors are moving or their locations are given imprecisely. In this paper, we propose a global approach to traffic state estimation by sing a global method, namely the Forier- Galerkin method, to transform the macroscopic traffic model to a system of ordinary differential eqations, and then perform the data assimilation on the reslting system sing minima estimation. Forier-Galerkin method belongs to the class of so-called projection methods. It sggests to project a soltion of the macroscopic model onto a sbspace generated by {e in } n and then describe the evoltion of the projection coefficients only. We refer the reader to [6] for frther details on spectral projection methods. In order to captre shock formation and correctly track shocks we apply a modification of the vanishing viscosity method that allows s to efficiently deal with Gibbs phenomenon (see [7]). Forier-Galerkin method provides estimates of the projection error in L 2 -norm. We stress that there is no statistical information available on projection errors and therefore statistical ncertainty description of Kalman filter or particle filters does not apply. This motivated s to apply the minima approach. It allows to constrct a garanteed estimate of the state of a differential eqation with ncertain bt bonded initial condition and model errors (in or case these errors are associated with a projection error which is bonded in L 2 ). Minima approach is based pon the following idea: to minimize the worst-case estimation error. In other words, one comptes the estimation error for the worst-case realization of ncertain model error and initial condition and then defines the minima estimate as sch that has a minimal worst-case error. This shows that the minima estimate is robst to all possible realizations of ncertain parameters lying within the given bonding set. We refer the reader to [8], [9], [] and [] for the basic information on the minima framework. Discssion on robst projection methods by means of the minima state estimation approach can be fond in [2], [3], [4]. Or iterative state estimation approach is based on the following idea: we consider a given estimate of the state and se the bilinear strctre of the redced model corresponding to the conservation law in order to generate a new estimate. Each iteration of the process is based on the previos estimate and is obtained applying the minima approach to the linear state eqation with a parameter. One stage of the proposed procedre can be seen as a fied-point iteration: given the estimate on the previos iteration we constrct a linear system and estimate its state in order to prodce a minima state estimate which will be sed in the
net iteration as a starting point. A similar algorithm was proposed for image motion estimation in [5]. Althogh we do not present a rigoros convergence proof, the nmerical reslts look promising. The remainder of the paper is organized as follows: in Section II, we describe the flow model sed in the work, and in Section III, we eplain briefly how finite-difference methods have previosly been sed to obtain a system for state estimation, and motivate the se of global soltion methods. In Section IV, we describe the Forier-Galerkin method and se it to derive a redced model for the traffic PDE. In Section V, we otline the data-assimilation procedre for the redced model. Nmerical reslts are presented in Section VI where and we eplain advantages of or method over the approaches based on local methods. We describe ftre work in Section VII. II. FLOW MODEL We employ the Lighthill-Whitham-Richards [6], [7] (LWR) model. This is the standard eqilibrim traffic flow model consisting of a scalar conservation law, with initial data (, t) t + f((, t)) = () () = (, ) (2) where : R R + R is the traffic density, R and t R + are the independent variables, space and time respectively, and f : R R is the fl fnction. A typical fl fnction is that of Greenshields [8], given by ( f() = V m ) (3) m where the constants V m and m are the maimm speed and the maimm density respectively. III. FINITE-DIFFERENCE-BASED STATE ESTIMATION A standard way of solving this model is to se finite differences. Using that approach, the density soltion obtained is a grid fnction. That is, at each comptational time-step, t, the soltion given by the finite difference method is j i = (i, j t), i =...,,... ; j =,, 2,... (4) Defining the vector U j = j i, i =...,,..., an n- level difference scheme is one for which the soltion U j is constrcted from U j, U j 2,..., U j n+. Difference schemes which have been sed to solve () (2) are typically two-level methods, where the soltion at a given time-step is constrcted only from the soltion at the previos time-step. Also, the schemes applied to this problem sally reqire limited information from the spatial domain. For a first-order Godnov scheme applied to a scalar conservation law, the soltion j i depends on either j i+ or j i, depending on the signal speed. The se of limited information is why sch schemes are termed local methods. For a two-level method, the scheme can be written as U j+ = HU j, (5) where H is the difference operator. For traffic flow, the signal speed, f (), will change in sign depending on the density,, so the difference operator will need to change the direction of spatial differencing as appropriate. As a reslt, H will depend on the grid-fnction U j, meaning (5) is not a linear system. However, attempts have been made to linearize sch operators. Mñoz et al. [] took sch an approach, and sed a piecewise linear fl fnction to obtain a switched-linear system. The nmber of linear systems they switch between depends on some assmptions made abot the nmber of congested and free-flow regions which can eist on a single link. This linearized model was sed for estimation and control [2], [3] with the assmption that at most one of each region (congested/free-flow) can eist on a link. This reslted in a system which switched between five different linear systems, for which they sed a Mitre Kalman Filter for the estimation/control. The measrements in that work are of the state (with some additive noise), so the transformation from observation space to state space is trivial; in the case where all components of the state vector, U j can be measred, the observation operator is the identity matri. In general, thogh, sensors are sparse, and the observation space is of lower dimension than the state space. The observations mst be taken at grid-points, meaning that the sch an approach works best for fied sensors, and the spatial discretization shold be chosen sch that the grid-points coincide with the sensor locations. Another well-known work on traffic data assimilation is that by Herrera and Bayen [9], who sed a Newtonian relaation method, which is a heristic method that relaes the dynamical model towards the observations. That scheme is sitable for mobile sensors throgh the se of an additional term in the flow model, which acts as either a sorce or a sink, in order to rela the conservation law towards the measrements from GPS-eqipped vehicles. This method also falls into the local category becase if its se of finite differences to discretize the model. Therefore, the incorporation of the measrements into the model is affected by the choice of spatial discretization. In the following section, we describe spectral (global) soltion methods, and will later show how sing the reslting system for estimation allows s to take measrements from any point in space, nlike in finite-difference-based methods. IV. GALERKIN METHOD FOR LWR MODEL We consider the initial vale problem defined by () and (2), where [, 2π] and t. We also restrict or attention to periodic bondary conditions, (, t) = (2π, t). In the Forier-Galerkin method, soltions are soght in the space, span{e in } n [6]: N (, t) = Defining the residal, n= R N (, t) = N t a n (t)e in. (6) + f( N) (7)
and reqiring it to be orthogonal to span{e in } n, we obtain the coefficients a n (t). However, soltions to () can develop shock-discontinities, even for smooth initial data. This will give rise to strong oscillations which will spread to the entire spatial domain, casing the Forier-Galerkin method to fail. Tadmor [7] showed that the addition of spectral viscosity to a scalar conservation law can help overcome this problem, and allow, moreover, the correct entropy soltion to be obtained. The conservation law with viscosity is t + f() = ε, (8) where ε is a constant. Sbstitting (6) into (8), and sing the fl fnction (3), where we set V m = m =, leads to the residal, R N (, t) = 2 + ε n= ȧ n (t)e in + +k k n= +k n= n 2 a n (t)e in. n= ina n (t)e in ika n k (t)a k (t)e in Forcing the orthogonal projection of this residal onto span{e in } n to vanish, we obtain the following system of ODEs, da n dt = k= n k n =.... (9) 2ika n k (t)a k (t) ina n (t) εn 2 a n (t), () The last term on the right hand side is de to the viscosity. Tadmor [7] showed that for scalar conservation laws, the addition of nmerical viscosity on the higher modes leads to convergence to the niqe entropy soltion. Ths, the viscosity term is only activated for n/2 m, where m is some threshold wave nmber. This will be discssed in frther detail in Section VI. The initial coefficients can be obtained from the data (2) by compting the integral, 2π a n () = ()e in d, n =.... 2π () In order to proceed with the data assimilation, we now epress the system as a differential eqation, and define an observation operator. We write () as dx dt = A(X)X, (2) where X : R C N+ is a vector containing the N + epansion coefficients, and A : C N+ C N+ is constrcted sing (). The observation eqation is given by Y (t) = HX(t) + η(t), (3) where Y : R R M is a vector of density measrements, and H : C N+ R M is the observation operator, which is a mapping from the comple state space of epansion coefficients to the real observation space of measred densities. This operator is constrcted sing (6), and the M density measrements can be taken anywhere on the interval, [, 2π]. In the following section, we will se the system (2) (3) for data assimilation. V. DATA ASSIMILATION In this section we describe the eperimental data assimilation algorithm sed in this paper. It is based on the minima state estimation framework. We briefly recall it here. Assme that the state of the process of interest (a vector of projection coefficients in or case) at time instant t is described by the vector X(t) and the evoltion of X(t) over time is governed by a system of differential eqations: dx dt = A(p)X + em (t), X() = X + e b (), (4) where e m represents model error and e b describes the error in the initial condition, p is a parameter, possibly depending on the state vector X. In or case the term e m addresses the projection error introdced by the Galerkin method. In other words, e m models an impact of higher order Forier modes onto those described by the Galerkin model (2). Since (, t) is at least sqare-integrable in space for every t we can assert that: ) the vector of projection coefficients X(t) is bonded in L 2 ; 2) the error in the initial condition e b is bonded in R n. One can prove (we refer the reader to [6, p.25] for frther details on estimating the projection error) sing properties of the conservation law (8) that the projection error e m is bonded in L 2 together with initial condition. Now, assming that the vector of density observations Y (t) is related to the vector of projection coefficients X(t) throgh the relation (3) and the observation error η is bonded we propose the following ncertainty description: Q e b 2 R n + T Qe m 2 R + n R η 2 Rmdt, (5) where Q, Q, R are symmetric positive definite matrices representing worst-case bonds on the energy of ncertain parameters. Now we are ready to state eqations for the minima estimate X and minima gain K for the linear eqation (4) with a parameter: dk dt = A(p)K + KA (p) + Q KH R HK, K() = Q d X dt = A(p) X + K H R (Y (t) H X(t)), X() = X. (6)
We note that for any soltion to (4) the following estimate holds: X(t) X(t) 2 R K(t), where K(t) denotes n the maimal eigenvale of K. Now, we se the following algorithm to estimate the state of the non-linear eqation (2). Assme that X represents an estimate of the projection coefficients X (obtained, for instance, solving the conservation law () for given initial and bondary conditions). We set p = X in (4) and denote by X 2 the minima estimate obtained from (6) with p = X. In sch a way we get a mapping of a given estimate X into the corresponding minima estimate X 2. Now, thinking of this mapping as a contraction we approimate its fied point, that is we assign to the minima estimate p = X 2 a new minima estimate X 3 and repeat this procedre ntil the convergence is achieved. This gives s a seqence of estimates { X k }. Assming that this seqence is convergent we can define the estimate of the non-linear eqation (2) as a limiting point of { X k }. The following section shows the nmerical reslts on the convergence of the proposed iterative estimation procedre. VI. NUMERICAL IMPLEMENTATION AND RESULTS Is this section, we present some nmerical reslts for the Forier-Galerkin redced model and minima data assimilation, demonstrating the shock captring capabilities of the model. A. Soltion of Forier-Galerkin redced model We integrate the system () in time sing a 4-th order Rnge-Ktta scheme (RK4). Note that RK4 does not conserve the integral of the density over the domain and this method is, therefore, non-sitable for long-term simlations. We first get the initial coefficients by compting the integral (). This can be done by qadratres or by sing Fast Forier Transform (FFT). We start with the initial data, () = e 2( )2, (7) which is a Gassian, centred at =. The viscosity term in () is only activated for the higher modes [7], n/2 m, where m is the wave nmber beyond which the viscosity is added. This can be achieved by replacing the viscosity term with H[ n/2 m]εn 2 a n (t), where H[ ] is the heaviside step fnction. Tadmor [7] solved brgers eqation sing εm.25. Having obtained the N + initial coefficients, a n (), for the initial data, (7), we solve the system, (), sing RK4 with N = and m = 3. Fig. shows the soltion at different times of the initial vale problem with periodic bondary conditions. Also shown is the soltion to the same problem sing a first-order Godnov scheme. Fig. a shows the smooth initial profile, and Fig. b shows the profile steepening at t =., and a shock beginning to form. We see that oscillations start to form here. In Fig. c, we see that at t =, the soltion is a shock wave and a rarefaction wave, and that oscillations are still present. However, the soltion is still stable. Fig. d shows the soltion at t = 5.7. The oscillations have been damped ot, ecept near the shock. However, since the Forier-Galerkin soltion still coincides with the Godnov soltion, the former appears to captre shocks sccessflly. The inclsion of periodic bondary conditions is evident from the last figre. B. Data assimilation reslts To perform the data assimilation, we rn the model with initial data (7) ntil t = 5.7, and se the Godnov soltion of the initial vale problem, () and (7), to generate the measrements. Then we pertrb the measrements artificially with additive noise which has error covariance matri R = Id (so we allow qite large noise in observations), Id stands for identity matri. In other words, we se the Godnov soltion as grond trth, sampling vales of its soltion and assimilating them into the Forier-Galerkin model. We assme that we can take M measrements at points {, 2,..., M }, where < < 2 <... < M 2π, and we choose the Godnov spatial discretization in sch a way that it coincides with the locations of the M observations. At each time-step of RK4, we constrct the operators, A and H, and we se (6) for the estimation with Q = Id, Q = Id (this shows that we trst or Forier- Galerkin model). Fig 2 shows the reslts after few iterations of the algorithm presented in Section V. Also shown are the pertrbed and npertrbed observations. We observe a good match between the estimate and the measrements, which demonstrates the nmerical convergence of the iteration process for the chosen case. We note that the iteration process might fail if the initial gess was far off the measrements. This point is left for ftre research. An important property of the observation operator, H, is that it can be constrcted from (6) in sch a way as to map from the state space to an observation space in which the measrements are taken at any points on the interval [, 2π]. This is in contrast to finite difference-based methods, which reqire measrements from fied points in space, which ideally shold coincide with the grid-points of the spatial discretization. VII. CONCLUSIONS AND FUTURE WORK We distingish between finite-difference-based traffic state estimation, and estimation based on global methods, sch as the Forier-Galerkin projection method presented here. The approach looks promising. The continos-in-space reconstrction does not reqire the observation locations to coincide with grid-points, giving the method the ability to take measrements from any part of the spatial domain. This wold appear to give the method a considerable advantage over finite-difference-based methods, in which measrements mst be taken at grid-points. One of the main points for the ftre research is to address non-periodic bondary conditions that makes the redced model mch more attractive for practical applications. Another important direction is to stdy sfficient conditions for the convergence of the proposed iterative algorithm.
.2 time =..2 time =. Forier Galerkin Godnov.8.8.6.6.4.4.2.2.2 2 3 4 5 6.2 2 3 4 5 6 (a) Initial profile (b) Shock forming at t =..2 time =. Forier Galerkin Godnov.2 time = 5.7 Forier Galerkin Godnov.8.8.6.6.4.4.2.2.2 2 3 4 5 6.2 2 3 4 5 6 (c) Shock-rarefaction soltion at t = Fig. : Nmerical soltion to LWR with initial data (7) (d) Soltion at t = 5.7 showing periodic B.C. Technically it wold be desirable to se an energy preserving time integrator for the redced model. VIII. ACKNOWLEDGMENTS This work was carried ot as part of the CARBOTRAF project, which is fnded by the Eropean Union Seventh Framework Programme FP7/27-23 nder grant agreement nmber 287867. REFERENCES [] L. Mñoz, X. Sn, R. Horowitz, and L. Alvarez, Traffic density estimation with the Cell Transmission Model, in American Control Conference, 23. Proceedings of the 23, vol. 5, 23, pp. 375 3755 vol.5. [2] X. Sn, L. Mñoz, and R. Horowitz, Highway traffic state estimation sing improved mitre Kalman filters for effective ramp metering control, in Decision and Control, 23. Proceedings. 42nd IEEE Conference on, vol. 6, 23, pp. 6333 6338 Vol.6. [3], Mitre Kalman filter based highway congestion mode and vehicle density estimator and its application, in American Control Conference, 24. Proceedings of the 24, 24. [4] Y. Wang and M. Papageorgio, Real-time freeway traffic state estimation based on etended Kalman filter: a general approach, Transportation Research Part B: Methodological, vol. 39, no. 2, pp. 4 67, 25. [5] L. Mihaylova, R. Boel, and A. Hegyi, Freeway traffic estimation within particle filtering framework, Atomatica, vol. 43, no. 2, pp. 29 3, 27. [Online]. Available: http://www.sciencedirect.com/science/article/pii/s5986376 [6] J. Hesthaven, D. Gottlieb, and S. Gottlieb, Spectral Methods for Time- Dependent Problems. Cambridge: Cambridge Univ. Press, 27. [7] E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM Jornal on Nmerical Analysis, vol. 26, no., pp. 3 44, 989. [Online]. Available: http://epbs.siam.org/doi/abs/.37/7263 [8] M. Milanese and R. Tempo, Optimal algorithms theory for robst estimation and prediction, IEEE Trans. Atomat. Control, vol. 3, no. 8, pp. 73 738, 985. [9] F. L. Chernosko, State Estimation for Dynamic Systems. Boca Raton, FL: CRC, 994. [] A. Nakonechny, A minima estimate for fnctionals of the soltions of operator eqations, Arch. Math. (Brno), vol. 4, no., pp. 55 59, 978. [] A. Krzhanski and I. Vályi, Ellipsoidal calcls for estimation and control, ser. Systems & Control: Fondations & Applications. Boston, MA: Birkhäser Boston Inc., 997. [2] S. Zhk, Minima state estimation for linear stationary differential-
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