Minimal error certificates for detection of faulty sensors using convex optimization

Transcription

1 Forty-Seenth Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 30 - October 2, 2009 Minimal error certificates for detection of faulty sensors using conex optimization Christian G. CLAUDEL, Matthieu NAHOUM and Alexandre M. BAYEN Abstract his article proposes a new method for sensor fault detection, applicable to systems modeled by conseration laws. he state of the system is modeled by a Hamilton-Jacobi equation, in which the Hamiltonian is uncertain. Using a Lax- Hopf formula, we show that any local measurement of the state of the system restricts the allowed set of possible alues of other local measurements. We derie these constraints explicitly for arbitrary Hamilton-Jacobi equations. We apply this framework to sensor fault detection, and pose the problem finding the minimal possible sensor error minimal error certificate as a set of conex programs. We illustrate the performance of the resulting algorithms for a highway traffic flow monitoring sensor network in the San-Francisco Bay Area. I. INRODUCION In control theory and information theory, the problems of system identification [14] and estimation [3] are closely linked. System identification uses data to infer the alue of parameters in the model of a system. Estimation takes a gien mathematical model for which the model parameters hae been obtained for instance using system identification and computes an estimate of the state of the system, from the model. In artificial intelligence, these two steps are respectiely referred to as learning and inference [12]. In control theory, traditional estimates are optimal in the least square sense, for example the Kalman filter recursiely proides a least square minimizer of the error functional between measurements and estimates. Extensions of the Kalman filter hae led to arious generalizations of these minimizers. When numerical alues for model parameters are collected as in system identification, they usually form a distribution. A possible approach to deal with distributions, is to maximize the log likelihood of the model, and to select the result as a constitutie model, subsequently used to do inference or estimation. An alternatie is to account for the uncertainty in the model by creating a family of models within a class, represented by the distribution. he present article follows a similar methodology for a class of nonlinear distributed parameter systems. More specifically, the article follows the framework and assumptions summarized below. C. Claudel is a Ph.D. Student, Department of Electrical Engineering and Computer Sciences, Uniersity of California, Berkeley claudel@eecs.berkeley.edu and a Research Intern, Mobile Internet Serices Systems, Nokia Research Center - Palo Alto. Corresponding Author. M. Nahoum is a M.Sc. Student, Systems Engineering, Department of Ciil and Enironmental Engineering, Uniersity of California, Berkeley CA, A. Bayen is an assistant Professor, Systems Engineering, Department of Ciil and Enironmental Engineering, Uniersity of California, Berkeley, CA, Learning set: We assume the aailability of a learning set. his set encompasses prior physical realizations of the system, known to be physically acceptable. Family of model parameters: he model describing the physical system of interest is described by a distributed parameter in the present case, a function. here exists a family of distributed model parameters which characterize the model for example a parametric family of functions, or the subset of a functional space. Certificate: We assume the aailability of a certificate, i.e. a procedure to label a model parameter as acceptable or not, based on the learning set. Measurements: Measurements of the state of the system are gien, which we seek to ealuate. An example of this framework is gien in section II for the specific case of highway traffic sensors, for illustratie purposes. Since a single model parameter does not fully capture the underlying physics of the system, it is in general impossible to exactly reconstruct the state of the system using the model and partial measurements of the state. Equialently, a partial measurement of the state will not completely constraint the alue of another partial measurement. In other words, the alue of a gien measurement only partially restricts the possible alues which other measurements could hae to be compatible. his is due to the fact that the system of interest is not described by a single model, but by a family of models. Since we want to detect faults in sensors measuring the state of the system, we want to check if the measurements are physically compatible with the family of models and the learning set certificate or not. he natural question to ask is thus: Gien some exact measurements, gien the family of models and the learning set certificate, can we find a set of necessary conditions that these measurements hae to satisfy to be compatible? As a corollary, can we compute the distance minimal error between noisy measurements and the set of measurements which satisfy the family of models and the learning set certificate? he contributions of the article are as follows. It soles the problem outlined aboe for a specific class of nonlinear partial differential equations PDEs constitutie models: the Hamilton-Jacobi HJ PDE. It makes the assumptions that the Hamiltonians of the HJ PDEs are not known but belong to a family. It deries necessary conditions that the state of the system must obey, assuming that the model parameters describing its eolution belong to the learning set. hese necessary conditions yield consistency conditions for the /09/$ IEEE 1177 Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

2 sensor measurements, which can be used directly to detect any inconsistency in a set of exact sensor measurements. he proposed method leads to a practical application. It poses the problem of finding the minimal possible error of the sensors in the L 1, L 2 and L sense as a set of conex programs, and demonstrates the applicability of the method to sensor network fault detection. he applicability of this framework is demonstrated with the example of a network of traffic sensors, by computing a certificate assessing if the minimal possible error of the sensors for a gien set of measurements is within the sensors certified range or not. he rest of the article is organized as follows. Section II presents the properties of the state considered in this article, as well as the HJ PDE which models it. Section III deries compatibility conditions between local state measurements, proided that these measurements are exact. We apply these results in section IV to the minimal error certificate problem, in which we find the minimal possible error associated with the measurements of a pair of sensors. II. SOLUIONS O SCALAR HAMILON-JACOBI EQUAIONS A. First order scalar hyperbolic conseration laws with concae Hamiltonians First order scalar hyperbolic conseration laws [11] are partial differential equations deried from the conseration principle of a physical quantity P for instance mass, electric charge or energy. If ρ, represents the density of P, and q, represents the flow of P, the conseration of P at location x X and for a time t [0, t max ] is expressed by: ρt, x + qt, x = 0 1 In the remainder of the article, we assume that the spatial domain X is defined by X := [ξ, χ], where ξ and χ represent the upstream and downstream boundaries of the domain. In numerous application fields such as thermodynamics, fluid mechanics or traffic flow theory, there exists a constitutie equation between the density ρ, and the flow q, of the form q, = ψρ,. his relation leads to the following form for equation 1: ρt, x + ψρt, x = 0 2 In conseration laws, the function ψ is known as flux function. In the context of Hamilton-Jacobi equations inestigated later, it is called Hamiltonian. In the remainder of the article, we assume that the Hamiltonian ψ is an upper semicontinuous concae function. Numerous fields make the assumption of a single alued ψ function to capture the dynamics of the system, whereas experimental data strongly suggests that a family of such ψ would be required to fully capture the state of the system. B. Hamilton-Jacobi equations with concae Hamiltonians Instead of soling equation 2 directly, we consider an equialent formulation obtained using the following standard ariable change: Definition 2.1: [Integral formulation of the conseration law]. Gien a density function ρ, and a flow function q,, we define the integral form M, as: M0, 0 := 0 Mt 2, x 2 Mt 1, x 1 := x 2 x 1 ρt 1, xdx + t 2 t 1 qt, x 2 dt, t 1, x 1, t 2, x 2 [0, t max ] X 2 3 In the remainder of this article, M, is denoted as state of our problem. he state eolution equation is the following Hamilton-Jacobi HJ PDE, obtained by integration of equation 2: Mt, x Mt, x ψ = 0 4 In the illustratie applications presented later, the state represents the cumulatie number of ehicles function [16], [9], [10], [5], which is a possible way of describing the flow of ehicles on a highway section. C. State estimation For clarity, we define two different functions which represent the true state of the system, and its estimate, obtained using a model as well as partial state information estimated state. Definition 2.2: [rue state and estimated state] he true state M, represents the state of the system, which could be obtained if measured by errorless sensors coering the entire space-time domain [0, t max ] X. he estimated state of the system represents the alue of the state inferred from partial knowledge of the true state denoted as alue condition on a subset D of [0, t max ] X, and the state eolution equation 4: M, soles equation 4 on [0, tmax ] X D 5 Mt, x = Mt, x, t, x D Note that D can be any subset of the space time domain [0, t max ] X. he set D is not necessarily included in the boundary [6] of [0, t max ] X, nor of empty interior [8]. he traditional Cauchy problem is defined by D := 0} X} [0, t max ] ξ}} [0, t max ] χ}}. he modeled state of the system depends upon the choice of the model parameters, i.e. the choice of the Hamiltonian ψ in the present case. Because of modeling errors, the true state is in general not identical to the estimated state, as can be seen in Figure 1, in the context of traffic. As can be seen in the figure, the solution M, of 4 with the proper alue of M, as initial and boundary conditions does not lead to M, eerywhere in the domain. his is the motiation of the present article Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

3 Fig. 1. Illustration of the difference between true state and modeled state. We consider here the traffic flow on a highway section near Emeryille, CA. he state of traffic is represented by a scalar function of time and space known as cumulatie number of ehicles function [16], [9], [10]. he upper left figure represents the true state of the system obtained in the present case using ideo cameras monitoring the entire domain of interest [5], [7]. he upper right figure represents the modeled state, reconstructed from the knowledge of the initial, upstream and downstream conditions illustrated in Figure 3. he difference in absolute alue between the true state and the modeled state is shown in the lower figure as a colormap. he dark areas represent areas where the true state is close to the modeled state, while the true state is far from the modeled state in pale areas. he fact that the state M, does not sole 4 exactly can also be inferred from Figure 2. Indeed, as can be seen from the figure, there is no single alued relation between Mt,x and Mt,x, due to the set alued nature of the past realizations. lower semicontinuous function ψ certificate such that O Hypψ certificate. Any concae and lower semicontinuous function ψ such that ψ ψ certificate pointwise is an acceptable model parameter. hese definitions are illustrated in Figure 2. In the present application of interest, the measurements and Mt,χ come from flow sensors, which measure Mt,ξ for all t [0, t max ]. Fact 2.4: [Physical properties of the state] he true state M, is assumed to satisfy the following properties: 1 M, is Lipschitz-continuous, and thus differentiable almost eerywhere. 2 he space and time deriaties of Mt, x are bounded and continuous almost eerywhere. he two aboe properties hold wheneer the flow Mt,x and density Mt,x associated with the conseration equation 1 are continuous almost eerywhere and bounded. D. Value conditions Soling equation 4 requires the knowledge of alue conditions for instance initial, boundary and/or internal conditions [5], [6] to characterize the existence and uniqueness of the solution. Definition 2.5: [Value condition] Let the true state M, be gien. A alue condition c, is a function of [0, t max ] X satisfying the following condition: ct, x = Mt, x, t, x Domc 6 where Domc is a subset of [0, t max ] X. A alue condition c, represents partial knowledge of the state M, on some domain Domc. Figure 3 illustrates the domains of definition of some alue conditions. Fig. 2. Illustration of an experimental flow-density diagram ψ. his plot represents the set of possible alues of flow and density obtained from an experimental data set [4]. he horizontal axis represents the density, whereas the ertical axis represents the flow. Each point in this plot has coordinates Mt,x, Mt,x for some t, x [0, t max] X. Note that Mt, x is a solution in the weak sense of 4 if and only if almost all points of coordinates Mt,x, Mt,x for all t, x [0, t max] X belong to the graph of some concae function ψ, which obiously cannot be the case in the present situation. Example 2.3: [Learning set, model parameters, certificate and measurements in the context of traffic flow sensing] In the context of traffic flow sensing, the learning set is the set of points } P :=, t, x [0, t max ] X. he conex hull of P is denoted O. he certificate is a concae and Mt,x, Mt,x Fig. 3. Illustration of the domains of the possible alue conditions used to construct the solution to the HJ PDE 4. he initial condition M 0 is defined on 0} X. he upstream and downstream boundary conditions are defined on [0, t max] ξ} and [0, t max] χ} respectiely. Internal conditions are defined on an empty interior domain included in ]0, + [ ]ξ, χ[. Other types of alue conditions exist, such as the hybrid conditions [8]. In this article, we focus on upstream and downstream boundary conditions, though the proposed algorithm can be extended to the case of internal conditions. Definition 2.6: [Boundary condition functions] he upstream and downstream boundary conditions, denoted by γ, and β, are alue conditions 6 defined on [0, t max ] ξ} and [0, t max ] χ} respectiely Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

4 E. Properties of the solutions to the HJ PDE problem In order to define the properties of the solution, we first need to define a conex transform of the Hamiltonian ψ as follows. Definition 2.7: [Conex transform] For a concae function ψ defined as preiously, the conex transform φ is gien by: φ u := sup p Domψ [p u + ψp] 7 In the remainder of this article, we use the results of [2], [7], [5], [6] to characterize the solution to equation 4 by a Lax-Hopf formula. he Lax-Hopf formula is deried in [2], [5], using results from iability and control theory. Proposition 2.8: [Lax-Hopf formula] he solution M c, associated with a lower semicontinuous alue condition c, can be computed using the following Lax-Hopf formula: M c t, x = inf ct, x + u + φ u u, Domφ R + 8 he Lax-Hopf formula proides an explicit solution of the problem defined in [2]. Equation 8 also implies a ery important inf-morphism property [2], [5], [6], [7], which is a key property used to build the algorithms used in this article. his property was initially deried using capture basins [2]. Proposition 2.9: [Inf-morphism property] Let us assume that the alue condition c is the minimum of a finite number of functions c i, namely: t, x [0, t max ] X, ct, x := min i I c it, x 9 With the aboe assumption, the solution M c defined by 8 can be written as: t, x [0, t max ] X, M c t, x = min M c i t, x 10 i I Proof he definition 9 of c, and the Lax-Hopf formula 8 imply: M ct, x = inf min c it, x + u + φ u u, Domφ R + i I 11 Equation 10 is obtained by reersing the order of the infimums in equation 11. he inf-morphism property is a practical tool to integrate new alue conditions for computing the modeled state M,. In addition, we can use it to break a complex problem inoling multiple alue conditions into a set of more tractable subproblems [5], [6], [8]. In the next section, we use 8 to derie a set of compatibility conditions that hae to be satisfied by the alue conditions obtained through measurements. hese conditions will enable us to construct conex optimization programs for sensor error analysis in section IV III. CONSRAINS OF HE MODEL ON HE VALUE CONDIIONS A. Upper estimate of the Hamiltonian We first need to define a particular class of Hamiltonians satisfying the following property. Proposition 3.1: [Learning set certificate]. For any gien state M satisfying the set of properties outlined in section 2.4, there exists a concae and upper semicontinuous function ψ certificate such that: Mt, x Mt, x t, x [0, t max] X, ψ certificate 12 Proof he proof of this proposition is immediate. Indeed, we can always choose for ψ certificate any constant function greater than the upper bound of Mt,x. Note that the choice of a function ψ certificate compatible with 12 is not unique. An example of choice of ψ certificate satisfying 12 is illustrated in Figure 2. B. Compatibility conditions We now show that, for any finite set of alue conditions, there exist compatibility conditions between solutions and alue conditions. Proposition 3.2: [Compatibility conditions] Let us define a set of alue condition functions c i, as in 6, an upper semicontinuous and concae Hamiltonian ψ certificate satisfying 12, and its associated conex transform φ as in 7. We also define a set of solutions M ci, associated with c i, as in 8. Gien these assumptions, the following set of conditions must be satisfied: M cj t, x c i t, x, t, x Domc i, 13 t, x Domc i, i [1, n], j [1, n] Proof Let us consider i [1, n], j [1, n] and t, x Domc i. We first express M ci t, x in terms of c i, using the Lax-Hopf formula 8: M ci t, x = inf c i t, x + u + φ u 14 u, Domφ R + In the aboe formula, φ is the conex transform of the Hamiltonian ψ certificate, which satisfies equation 7. Since the alue condition c i, satisfies 6 and t, x Domc i, we hae that Mt, x = c i t, x. Hence, we can write the inequality M cj t, x c i t, x as: inf Mt, x + u + φ u Mt, x,u [0,t max] Domφ 15 Since the deriaties of M, are continuous almost eerywhere and bounded, we can write: Mt, x + u + φ u Mt, x = Mt τ,x+τu 0 + u Mt τ,x+τu + φ u dτ 16 Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

5 Since ψ certificate is concae and upper semicontinuous, it is equal to its Legendre-Fenchel biconjugate. Hence, we hae [5], [8] that ψ certificate ρ = inf ρu + u Domφ φ u, and thus that ψ certificate ρ ρu + φ u for all u Domφ. his result enables us to derie the following inequality from equation 16: Mt, x + u + φ u Mt, x Mt τ,x+τu + ψ certificate 0 Mt τ,x+τu dτ 17 Since ψ certificate satisfies 12, we hae that Mt τ,x+τu + ψ certificate Mt τ,x+τu 0 for all τ, u [0, ] Domφ. Since > 0, the right hand side of equation 17 is nonnegatie, which implies the following inequality: Our objectie is to compute the minimal error certificate on the obsered data, which can be thought of as follows. he ector of obsered data s meas can be iewed as a point in a multidimensional space S. he set C of consistent data, that is the set of data points which obey 13 can also be iewed as a subset of S. he exact data s, which is unknown to us, is an element of C by Proposition 3.2. he distance d between s meas and the set C is defined by: d := inf s C s smeas Since s C, the error e := s s meas satisfies e d. Note that the error is unknown since the exact data s is unknown, but the smallest possible alue of e is d. his property is illustrated in Figure 5., u R + Domφ, Mt, x + u + φ u Mt, x 0 18 Equation 15 is obtained from equation 18 by taking the infimum oer, u R + Domφ, which completes the proof. We now present a practical application of the aboe compatibility conditions: the computation of the minimal error certificate for a pair of sensors. IV. APPLICAION: LINEAR/QUADRAIC PROGRAMMING FORMULAION OF HE MINIMAL ERROR CERIFICAE A. Problem definition PROBLEM Motiated by applications of sensor networks for highway traffic monitoring systems, we consider a stretch of highway located between the upstream and downstream boundaries called ξ and χ respectiely, as illustrated in Figure 4. he state of the system is described here by the cumulatie number of ehicles function, which is formally defined in [9], [10], [15], [16]. wo fixed sensors located at ξ and χ measure the flow, i.e. the time deriatie of the state, as illustrated in Figure 4 Fig. 4. Illustration of the sensor layout. he traffic flow data is obtained from two fixed sensors arrows located at the upstream and downstream boundaries of the domain, and measuring the time deriatie of the state for all times t [0, t max]. Fig. 5. Illustration of the minimal error problem. his figure illustrates the problem of interest in the space of data points S. he set of physically possible elements of S i.e. the set of points which satisfy 3.2 is denoted as C arrow. he real alue of the data is a point s C, and is usually unknown. he minimal error of the sensors generating a data point s is measured by the distance d between s and C. If this distance is greater than the maximal admissible error for a pair of working sensors dashed line, then the point d is necessarily created by a faulty pair of sensors. Note howeer that the conerse does not imply that the sensors are not faulty. he maximal leel of error e m of a working sensor is usually established by the manufacturer as part of the sensor design. A distance d greater than e m implies an error e greater than e m, which indicates the failure of at least one of the sensors wrong data, bad hypothesis on the sensor placement,.... Howeer, note that a distance d lower than this threshold does not indicate that the sensors are working correctly. Indeed, the error e can be greater than e m een if d is lower than e m. he following section details the measurements and error norms. B. Sensor measurements In the remainder of this article, we assume that the upstream and downstream sensors generate the following measurements: Definition 4.1: [Sensor measurements] Let N be the set of integers [0, n max ] such that n max + 1 = t max, where 1181 Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

6 t max is a finite time horizon. he discrete time measurements of the upstream and downstream sensors are defined by: q meas in n, n N upstream sensor flow measurements qout meas 19 n, n N downstream sensor flow measurements he exact alues of the parameters are denoted by: qin n, n N true upstream flow alue q outn, n N true downstream flow alue 20 hey are not known in practice. he difference between sensor measurements and exact alues is determined by the sensor error, which is usually expressed using some norm, for instance in the L 1, L 2, or L sense. hese norms are defined for q in the case q out follows similarly as: e L1 := e L2 := e L n n := max i N q in qin meas qin meas qin qin meas q meas in q in qin meas q meas in 2 21 Using the slack ariables e and f, we can also express equation 21 as: n e L1 = min e q e in q meas in q s.t. in meas, i N and e qin qmeas in qin meas, i N 22 n e L2 = min e 23 s.t. e qin qmeas qin meas 2, i N e L = min f q f in q meas in qin s.t. meas, i N and f qin qmeas in qin meas, i N C. Decision ariables 24 he ector of decision ariables used later in the conex programs is defined as follows. Definition 4.2: [Decision ariables] Let Δ represent the alue of M0, ξ M0, χ. Let q in and q out be defined as in 20. Let e in, e out, f in and f out be slack ariables defined as in 22, 23 and 24, and associated respectiely with the upstream and downstream sensors. he ector of decision ariables of interest is defined as: X := Δ, q in 0,..., q in n, q out 0,..., q out n, e in 0,..., e in n, e out 0,..., e out n, f in, f out 25 We now express the physical constraints 13 for the specific problem of interest two fixed sensors in terms of the decision ariables. For this, we need to express the alue condition functions γ, and β, of Definition 2.6, and the solutions M γ, and M β, to 4 as a function of the decision ariables q in, q out. his is the object of the two following sections. D. Piecewise affine boundary conditions In this section, we define the piecewise affine boundary conditions obtained from sensor measurements. he sensor data is supposed to be sampled temporally, so that it takes piecewise affine alues on each sampling interal. hus, q in and q out are piecewise constant, and the boundary conditions functions integrated oer time are piecewise affine. his property is ery important, since the solution to 4 associated with piecewise affine boundary conditions can be computed semi-explicitly [6]. Definition 4.3: [Piecewise affine boundary conditions] Let the ector of decision ariables X be defined as in 25. he upstream and downstream boundary condition functions γ, and β, are defined as: γt, x = q in + q in nt n βt, x = Δ + q out + q outnt n if x = ξ and n N s. t. t [n, n + 1 ] if x = χ and n N s. t. t [n, n + 1 ] 26 he functions γ, and β, are piecewise affine. For all n N, the affine blocks γ n, and β n, are defined by: q in + q in nt n if x = ξ and γ nt, x:= t [n, n + 1 ] + otherwise Δ + q out + q outnt n if x = χ and β nt, x:= t [n, n + 1 ] + otherwise 27 Using equation 27, we can express 26 as: γ, = inf γ n, n N β, = inf β 28 n, n N We now compute the solutions M γ, and M β, respectiely associated with the alue conditions γ, and β, analytically using the method deeloped in [6]. E. Solutions associated with affine boundary conditions For this specific application, we assume that the Hamiltonian ψ certificate is gien by the following formula: ρ if ρ kc ψ certificate ρ := wk m ρ if k c ρ Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

7 he function ψ certificate defined by 29 is both concae and upper semicontinuous indeed, continuous. It is commonly used in traffic flow modeling [9], [16]. Its parameters, w, k c and k m are chosen such that the condition 12 is satisfied on all aailable experimental data. his last condition enables us to use the constraints 13 for our present application. he conex transform φ of ψ certificate is gien [2], [5], [6] by: φ kc u + if u [, w] u = 30 + otherwise Since γ, = inf γ n, and β, = inf β n,, n N n N the inf-morphism property 10 implies: Mγ, = inf M γ n, n N M β, = inf M 31 β n, n N where M γn, χ and M βn, ξ are the solutions to 4 associated with γ n, and β n, defined by 28. hese solutions can be computed analytically using the framework deeloped in [6]. Proposition 4.4: [Explicit computation of the solutions] he solutions M γn, and M βn, associated with γ n, and β n, are gien by the following explicit formulas: i=k+1 + if t n + x ξ k [0, n 1] q in + q in nt n M γn t, x= if n + x ξ t n x ξ n q in + k ct n + 1 otherwise k [0, n 1], + if t n + χ x w χ x Δ + q out + q outnt w n + χ x kc + w M βn t, x= w if n + χ x χ x t n w w χ x Δ + q out + k ct n + 1 w otherwise 32 Proof his proposition is an instantiation of the result of [6] for the specific Hamiltonian 29. F. Expression of the compatibility conditions he conditions 13 can be expressed in the present case as: M γ t, ξ γt, ξ t [0, t max ] ii M β t, χ βt, χ t [0, t max ] iii M γ t, χ βt, χ t [0, t max ] M β t, ξ γt, ξ t [0, t max ] 33 In the present case, checking that the constraints 33 are satisfied amounts to checking that a set of linear inequalities is feasible. he two following propositions enable us to find the expression of 33 as linear inequalities Proposition 4.5: [Consistency check of a single boundary condition] he conditions 33 and ii are satisfied if and only if the following inequalities are satisfied: qin n k c n N 34 ii q out n k c n N Proof Let us choose t [n, n+1 ]. he function M γ t, ξ can be computed by combining 31 and 32: M γ t, [ ξ = k min min q in + t k + 1 k c, k [0,] ] q in + q in nt n 35 Since γt, ξ = q in +q in nt n, we hae that M γ t, ξ γt, ξ for all t [n, n + 1 ] if and only if: q in + t n q in n t k + 1 k c 0 36 By choosing t = n, equation 36 yields the necessary condition i=k+1 q in n k 1 k c 0 37 Choosing k = n 2 in 37 yields the necessary condition q in n 1 k c. Similarly, choosing k = n 3 yields q in n 2 k c. his can be extended triially by induction for all n N, and yields inequality of 34. Conersely, the condition of 34 is sufficient. Indeed, condition 36 can be rewritten as: i=k+1 q in k c + t n q in n k c 0 k [0, n 1] 38 Since > 0 and t n 0, the inequality 38 is triially satisfied when condition of 34 is satisfied, which completes the proof of. he proof of ii is identical to the proof of. Proposition 4.6: [Consistency check of multiple boundary conditions] We assume that the conditions 34 hold. he conditions 33 iii are satisfied if and only if the following inequalities are satisfied: Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

8 ii n 1 q in + q in n χ ξ n n Δ + q out n N n+ 1 q in Δ + q out +q out n + n + n N iii Δ + w n N Δ + n w 1 n n + q out + q out n w q in q out n+ w 1 q in χ ξ w n +q in n + n + n + w w w n N 39 Proof he conditions 33 can be expressed in the present case as: Mγ t, χ βt, χ t [0, t max ] ii M β t, ξ γt, ξ t [0, t max ] 40 We only proe that the constraint of 40 yields inequalities and ii of 39. Let us fix t [n +, n ]. We first express the solution M γ t, χ using 32 and the inf-morphism property 31: M γt, χ = [ min q in + q in nt n, k ] 41 min q in + k ct k + 1 k [0,] Since q in k k c for all k N, we hae that k q in +q in nt n q in +k c t k + 1 for all k [0, n 1]. Hence, we hae that M γ t, χ = q in + q in nt χ ξ n 42 hus, M γ t, χ is affine for all t [n +, n+1 + ], for all n N. By construction, βt, χ is also affine for all t [n, n + 1 ], for all n N. Hence, checking that condition of 40 is satisfied amounts to checking that Mγ n, χ βn, χ n N M γ n +, χ βn +, χ n N 43 Indeed, checking that the continuous and piecewise affine function M γ t, χ βt, χ is positie for all t [0, t max ] amounts to checking that it is positie on all its kinks. In addition, remarking that t [n, n + 1 ] if and only if n = t and t [n +, n ] if and only t if n =, we can express the functions M γ, χ and βt, χ explicitly as: M γ t, χ= t 1 t 1 βt, χ= Δ + q in + q in t t χ ξ t, t [0, t max ] q out + q out t t t t [0, t max ] 44 he conditions and ii of equation 39 are obtained from 43, using formula 44 for expressing the boundary conditions. One can similarly proe that inequality ii of 40 yields inequalities iii and of 39. G. Conex programming formulation of the minimal error certificate problem Proposition 4.7: [Minimal error of a sensor] he minimal error ζ in respectiely ζ out of the fixed upstream respectiely downstream sensor in the L 1 sense is the solution to the following linear program LP: ζ in/out := min e in/out n e in q in q in meas q in meas, i N e in q in q in meas q in meas, i N s.t. e out q out q out meas q out meas, i N e out q out q out meas, i N q out meas 34 holds 39 holds 45 he minimal error η in respectiely η out of the fixed upstream respectiely downstream sensor in the L 2 sense is the solution to the following conex quadratically constrained quadratic program QCQP: η in/out := min e in/out n e in q in q in meas q in meas 2, i N s.t. e out q out q out meas q out meas 2, i N 34 holds 39 holds 46 he minimal error θ in respectiely θ out of the fixed upstream respectiely downstream sensor in the L sense is the solution to the following LP: 1184 Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

9 θ in/out := min f s.t. f q in/out q in/out meas q in/out meas, i N f q in/out q in/out meas q in/out meas, i N 34 holds 39 holds 47 Proof he proof follows from the definition of the error terms 22, 23 and 24, and from the inequality constraints 13 that is, from 34 and 39 with our assumptions, since the Hamiltonian 29 satisfies 12. Note that the inequalities 34 and 39 are linear in terms of the decision ariable 25. H. Fault detection in traffic sensor networks We apply the aboe results to sensors of the PeMS system [1], which is a network of loop detectors measuring traffic on California highways. he PeMS system is one of the data feeds currently integrated in the Mobile Millennium traffic monitoring system [17], [18], operated jointly by Nokia and UC Berkeley. One of the main challenges arising when using data from the PeMS system is the automated identification of the mislocated or faulty sensors. Preious approaches such as [13] hae successfully implemented sensor fault detection algorithms based on statistical correlation with adjacent sensors. In the present case, the use of a flow model enables us to proide a fault certificate, by proing that the leel of error in a pair of sensors exceeds its design alue. he main difficulty soled by this approach is the identification of realistic looking faulty data. he data in question, when looked at for a single sensor is not abnormal. Howeer, when checked with the data from other sensors, the approach deeloped earlier enables the identification of incompatibility, thus resulting in the detection of faults, which results from the proper use of the model and the method. We sole the fault detection problem by applying the LP 47 on all pairs of consecutie sensors present on the highway network. We first create a learning set based on the data of a subset of traffic sensors which are known to be alid. his learning set is shown in Figure 2. We then choose a function ψ certificate of the form 29 which satisfies the condition 12. his function ψ certificate will be our learning set certificate. We assume that the maximal allowable error of a PeMS sensor is 15%. here are multiple sources of uncertainty arising when dealing with loop detectors, such as paement depth, loop layout, which typically creates maximal errors of this magnitude. he maximal allowable error on a pair of PeMS sensors will thus be set to 30% in the considered scenario. As an application, we consider the results of 47 for fie consecutie sensors, labeled , , , and respectiely, as illustrated in Figure 6. For each one of the four adjacent pairs of sensors, we compute the minimal alue of the error θ in +θ out in equation 47, during a one month period at the frequency of one day. he distribution of these results is shown in Figure 6. Figure 6 shows that there is no indication of malfunction for the first and the last pairs of sensors. Note that the success to the minimal error test does not guarantee that a pair of sensors is working, since the actual error of the pair of sensors can be aboe the maximal allowable error. he second and third pairs exhibit L errors that are higher than 30%, which indicates a malfunction of the corresponding pairs. Further analysis has shown that the pair is passing the minimal error test, and thus that sensor is likely either failing or incorrectly mapped. Fig. 6. Faulty sensor detection. We consider here the traffic flow on highway I880-S near Oakland, CA. he linear program 47 is run each day on a one-month period. he sensors of interests are highlighted in the top figure, and their corresponding minimal error distribution oer the onemonth period is represented in the four bottom figures. Bottom: he top left and bottom right subfigures represents the minimal errors of the pair and hese minimal errors fall in the allowable range. In contrast, the minimal errors of the pair and are aboe the allowable range. his means that there must exist a fault in one of the sensors , , or he present method is currently implemented in the Mobile Millennium system, as a filtering method for the PeMS feed to the system. he results hae shown to greatly improe the performance of the system [18] Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.

10 V. CONCLUSION his article presents a new method for fault detection in networks of sensors monitoring systems modeled by Hamilton-Jacobi partial differential equations. he state of the system of interest does not usually follow a partial differential equation characterized by a well known parameter, but by a distribution of parameters. Based on the solution methods of Hamilton-Jacobi equations, we show that any local knowledge of the state proides constraints on the possible alues of the state in the space-time domain. We subsequently use this fact to pose the problem of minimal error certificate of a pair of sensors as a set of conex programs. he minimal error certificate is the minimal possible error of the sensors gien a set of measurements. It is a ery effectie tool for detecting inconsistencies. When the error is aboe a preset threshold depending on the sensor type, the corresponding set of sensors is identified as faulty. his work has been successfully implemented in the Mobile Millennium [18] system, and is part of the consistency check methods used in this system. It checks the consistency of the data generated by more than 1400 sensors eery 30 seconds in real time with streaming data for all PeMS sensors in northern California. ACKNOWLEDGMENS he authors are grateful to Jean Pierre Aubin, who deeloped the mathematical framework used in this article, for his ision, his guidance and his scientific generosity. We thank Patrick Saint-Pierre for his help on setting the first ersion of the code used for this study and for fruitful conersation. he algorithms deeloped in this article are using technology produced by the company VIMADES. [11] L. C. EVANS. Partial Differential Equations. American Mathematical Society, Proidence, RI, [12]. HASIE, R. IBSHIRANI, and J. H. FRIEDMAN. he Elements of Statistical Learning. Springer, corrected edition, July [13] J. KWON, C. CHEN, and P. VARAIYA. Statistical methods for detecting spatial configuration errors in traffic sureillance sensors. ransportation Research Record: Journal of the ransportation Research Board, : , [14] L. LJUNG. System identification: theory for the user. Prentice-Hall, Inc. Upper Saddle Rier, NJ, USA, [15] K. MOSKOWIZ. Discussion of freeway leel of serice as influenced by olume and capacity characteristics by D.R. Drew and C. J. Keese. Highway Research Record, 99:43 44, [16] G. F. NEWELL. A simplified theory of kinematic waes in highway traffic, part I: general theory. ransporation Research B, 27B4: , [17] D. WORK, O.P. OSSAVAINEN, S. BLANDIN, A. BAYEN,. IWUCHUKWU, and K. RACON. An ensemble Kalman filtering approach to highway traffic estimation using GPS enabled mobile deices. In Proceedings of the 47th IEEE Conference on Decision and Control, pages , Cancun, Mexico, December [18] REFERENCES [1] [2] J.-P. AUBIN, A. M. BAYEN, and P. SAIN-PIERRE. Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints. In press: SIAM Journal on Control and Optimization, 475: , [3] Y. BAR-SHALOM, X.R. LI, and. KIRUBARAJAN. Estimation with applications to tracking and naigation. Wiley New York, [4]. CHOE, A. SKABARDONIS, and P. VARAIYA. Freeway performance measurement system: operational analysis tool. ransportation Research Record: Journal of the ransportation Research Board, :67 75, [5] C. G. CLAUDEL and A. M. BAYEN. Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: theory. Submitted to IEEE ransactions on Automatic Control, 2008, resubmitted in reised form, [6] C. G. CLAUDEL and A. M. BAYEN. Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: computational methods. Conditionally accepted to IEEE ransactions on Automatic Control, [7] C. G. CLAUDEL and A. M. BAYEN. Guaranteed bounds for traffic flow parameters estimation using mixed Lagrangian-Eulerian sensing. In Proceedings of the 46th Annual Allerton Conference on Communication, Control, and Computing, Allerton, IL, Sep [8] C. G. CLAUDEL and A. M. BAYEN. Solutions to switched Hamilton- Jacobi equations and conseration laws using hybrid components. In M. Egerstedt and B. Mishra, editors, Hybrid Systems: Computation and Control, number 4981 in Lecture Notes in Computer Science, pages Springer Verlag, Saint Louis, MO, April [9] C. F. DAGANZO. A ariational formulation of kinematic waes: basic theory and complex boundary conditions. ransporation Research B, 39B2: , [10] C. F. DAGANZO. On the ariational theory of traffic flow: wellposedness, duality and applications. Networks and heterogeneous media, 1: , Authorized licensed use limited to: Uni of Calif Berkeley. Downloaded on January 29, 2010 at 14:23 from IEEE Xplore. Restrictions apply.