Highway traffic estimation of improved precision using ...

Highway traffic estimation of improved precision using the Derivative-free nonlinear Kalman Filter Gerasimos Rigatos∗ , Pierluigi Siano† , Nikolaos Zervos∗∗ and Alexey Melkikh‡ ∗

Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece, email: [email protected] † Department of Industrial Engineering, University of Salerno, 84084, Fisciano, Italy, email: [email protected] ∗∗ Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece, email: [email protected] ‡ Institute of Physics and Technology, Ural Federal University, 620002, Yekaterinburg, Russia, email: [email protected]

Abstract. The paper proves that the PDE dynamic model of the highway traffic is a differentially flat one and by applying spatial discretization its shows that the model’s transformation into an equivalent linear canonical state-space form is possible. For the latter representation of the traffic’s dynamics, state estimation is performed with the use of the Derivative-free nonlinear Kalman Filter. The proposed filter consists of the Kalman Filter recursion applied on the transformed state-space model of the highway traffic. Moreover, it makes use of an inverse transformation, based again on differential flatness theory which enables to obtain estimates of the state variables of the initial nonlinear PDE model. By avoiding approximate linearizations and the truncation of nonlinear terms from the PDE model of the traffic’s dynamics the proposed filtering methods outperforms, in terms of accuracy, other nonlinear estimators such as the Extended Kalman Filter. The article’s theoretical findings are confirmed through simulation experiments. Keywords: highway traffic estimation, nonlinear PDE dynamics, differential flatness theory, nonlinear estimation, distributed parameter systems, Extended Kalman Filter, Derivative-free nonlinear Kalman Filter. PACS: 07.05.Dz Control systems, 07.07.Tw Servo and control equipment; robots, 07.50.Qx Signal processing electronics, 02.30.Yy Control theory

INTRODUCTION In this paper the Payne-Whitham PDE model is considered [1]-[6]. By applying spatial discretization this PDE model is turned into an equivalent description which consists of a set of coupled nonlinear ordinary differential equations. Thus, a state-space description is obtained. Next, it is proven that this state-space model of the Payne-Whitham PDE is a differentially flat one. This means that all its state variables and its control inputs can be expressed as differential functions of a specific algebraic variable which is the system’s flat output [7]-[10]. The differential flatness properties of the traffic model indicate also that it is possible to transform it into a linear canonical form, for which state estimation can be performed with the use of the standard Kalman Filter recursion. Unlike EKF, the Kalman Filter estimation based on the transformed linearized model of the PDE does not require the computation of Jacobian matrices and partial derivatives and thus is known as Derivative-free nonlinear Kalman Filter [11]-[12]. Actually, the transformation of the Payne-Whitham PDE into the aforementioned linear canonical state-space form is an exact linearization that is free of numerical approximation errors. Consequently, estimation of the traffic’s dynamics with the use of the Derivative-free nonlinear Kalman Filter is numerically more accurate and computationally more robust than the estimation which is performed by the Extended Kalman Filter.

TRAFFIC MODELING WITH THE USE OF PDES Scalar models of the traffic flow Macroscopic data modeling considers that the motion of vehicles in highways is a continuous-time phenomenon and has been inspired from hydrodynamics. For estimating traffic dynamics, both one-dimensional and higher dimensional PDE models have been proposed. Classical 1D PDE models of traffic consider the traffic state at a point x and at time instant t to be represented by the density function ρ (x,t) [3]-[4]. The evolution of the density function ρ (x,t)

is frequently given by the LWR (Lighthill-Whitham-Richards) PDE, which expresses the conservation of vehicles on road links: ∂ρ ∂t

ρ) + ∂ Q( ∂x = 0

(1)

where Q() is a flux function, which denotes the flux of vehicles associated with density ρ , at a stationary state. The initial conditions for the PDE are ρ0 . The flux depends of the density and on the mean value of the vehicles’ speed according to the relation Q(ρ ) = q = ρ v = ρ V (ρ )

(2)

where v is the vehicles’ flow speed and V (ρ ) is a nonlinear function to be defined next. Several parametric flux functions have been proposed. Such a function is the so-called Greenshields flux function, which expresses an linear relationship between density and speed: ρ ) Q(ρ ) = vmax ρ (1 − ρmax

(3)

ρ

and thus V (ρ ) = vmax (1 − ρmax ), while vmax denotes the free flow speed and ρmax the jam density [3]-[4]. There are also

variations of the flux function based on an exponential relation between density and flow Q(ρ ) = ρ vmax exp(− α1 ( ρρc )n ).

Non-scalar models of the traffic flow In case of non-scalar models of the traffic flow, additional degrees of freedom are included in the model of the traffic’s PDEs. Such a description is the Payne-Whitham model [3]-[4] ∂ρ ∂q ∂t + ∂ x = 0 2 c0 ∂ ρ V (ρ )−v ∂v ∂v τ ∂t + v ∂ x + ρ ∂ x =

(4)

Using a set of n equidistant points xi i = 1, · · · , n along the x-axis, discretization of the model of Eq. (4) is performed as follows: qi −qi−1 ∂ ρi ∂ t = − ∆x 2 c0 ρi −ρi−1 vi −vi−1 ∂ vi ∆x ∂ t = −vi ∆x − ρi q i = ρ i vi

+ V (ρiτ)−vi

(5)

The associated boundary conditions of the PDE are: ρ (x0 ,t) and v(x0 ,t).

ESTIMATION OF THE PAYNE-WHITHAM MODEL WITH THE DERIVATIVE-FREE NONLINEAR KALMAN FILTER Differential flatness of the Payne-Whitham PDE Next, the following state variables are defined (with reference to the grid point xi ): zi,1 = ρ (xi ) and zi,2 = v(xi ). Differential flatness theory has been applied for solving several problems of control and estimation in distributed parameter systems [13]-[15]. In the case of the Payme-Whitham traffic model, the flat output of the system is taken to be the vector Z f = [z1,1 , z2,1 , z3,1 , · · ·, zn−1,1 , zn,1 ]T . It can be proven that all state variables of the system can be written as functions of the flat output and its derivatives. By applying semi-discretization to the 1-st couple of equations on the Payne-Whitham model it holds [16] ∂ z1,2 ∂t

∂ z1,1 z1,1 z1,2 −ρ0 v0 ∆x ∂t = − V (z )−z z −v c2 z −ρ = −z1,2 1,2∆x 0 − zi,10 i,1∆x 0 + i,1τ i,2

(6)

By solving the first row of the above equation with respect to z1,2 one finally gets z1,2 =

(∆x˙z1,1 −ρ0 v0 ) −z1,1

(7)

and using that z1,1 is by definition a component of the flat output vector as well as that the boundary conditions ρ0 and v0 are constants, one gets that z1,2 is a function of the flat output and its derivatives, or equivalently z1,2 = f1 (Z f , Z˙ f ). For the 2-nd couple of equations of the Payne-Whitham model it holds ∂ z2,2 ∂t

∂ z2,1 z2,1 z2,2 −z1,1 z1,2 ∆x ∂t = − c20 z2,1 −z1,1 V (z )−z z −z + 2,1τ 2,2 = −z2,2 2,2∆x 1,2 − z2,1 ∆x

(8)

By solving the first row of the above equation with respect to z2,2 one finally gets z2,2 =

(∆x˙z2,1 −z1,1 z1,2 ) −z2,1

(9)

and using that z2,1 and z1,1 are by definition components of the flat output vector as well as that z1,2 is a differential function of the flat output, one gets that z2,2 is a function of the flat output and its derivatives, or equivalently z2,2 = f2 (Z f , Z˙ f ). For the 3-rd couple of equations on the Payne-Whitham model it holds ∂ z3,2 ∂t

∂ z3,1 z3,1 z3,2 −z2,1 z2,2 ∆x ∂t = − z −z c20 z3,1 −z2,1 V (z )−z = −z3,2 3,2∆x 2,2 − z3,1 + 3,1τ 3,2 ∆x

(10)

By solving the first row of the above equation with respect to z3,2 one finally gets z3,2 =

(∆x˙z3,1 −z2,1 z2,2 ) −z3,1

(11)

and using that z3,1 and z2,1 are by definition components of the flat output vector as well as that z2,2 is a differential function of the flat output, one gets that z3,2 is a function of the flat output and its derivatives, or equivalently z3,2 = f3 (Z f , Z˙ f ). Continuing in a similar manner one has that for the n − 1-th couple of equations on the PayneWhitham model it holds ∂ zn−1,2 ∂t

∂ zn−1,1 ∂t

=

zn−1,1 z3,2 −zn−2,1 zn−2,2 ∆x c20 zn−1,1 −zn−2,1 z −z + −zn−1,2 n−1,2∆x n−2,2 − zn−1,1 ∆x V (zn−1,1 )−zn−1,2 + τ

=−

(12)

By solving the first row of the above equation with respect to zn−1,2 one finally gets zn−1,2 =

(∆x˙zn−1,1 −zn−2,1 zn−2,2 ) −zn−1,1

(13)

and using that zn−1,1 and zn−2,1 are by definition components of the flat output vector as well as that zn−2,2 is a differential function of the flat output, one gets that zn−1,2 is a function of the flat output and its derivatives, or equivalently zn−1,2 = fn−1 (Z f , Z˙ f ). Finally, for the n-th couple of equations on the Payne-Whitham model it holds ∂ zn,2 ∂t

∂ zn,1 zn,1 z3,2 −zn−1,1 zn−1,2 ∆x ∂t = − z −z V (z )−z c20 zn,1 −zn−1,1 = −zn,2 n,2 ∆xn−1,2 − zn,1 + n,1τ n,2 ∆x

(14)

By solving the first row of the above equation with respect to zn,2 one finally gets zn,2 =

(∆x˙zn,1 −zn−1,1 zn−1,2 ) −zn,1

(15)

and using that zn,1 and zn−1,1 are by definition components of the flat output vector as well as that zn−1,2 is a differential function of the flat output, one gets that zn,2 is a function of the flat output and its derivatives, or equivalently zn,2 = fn (Z f , Z˙ f ). According to the above, all state variables of the Payne-Whitham model can be written as differential functions of the previously defined flat output Z f . Consequently, the Payne-Whitham model is a differentially flat one.

Derivative-free nonlinear Kalman Filter for the Payne-Whitham PDE At point x1 , the Payne-Whitham PDE model is rewritten as: ∂ z1,1 ∂t

=−

∂ z1,2 ∂t

z1,1 z1,2 ∆x

+ d1

(16)

= v1,2

where the disturbance term d1 and the control input v1 are defined as ρ0 v0 ∆x c20 z1,1 −ρ0 z1,1 ∆x

d1 = v1 =

z −v −z1,2 1,2∆x 0

−

+

V (z1,1 )−z1,2 τ

(17)

At point x2 : ∂ z2,1 ∂t

=−

z2,1 z2,2 −z1,1 z1,2 ∆x

∂ z1,2 ∂t

(18)

= v2

where the control input v2 are defined as v2 = −z1,2

z2,2 −z1,2 ∆x

c2 z2,1 −z1,1 ∆x

0 − z2,1

+

V (z2,1 )−z2,2 τ

(19)

At point x3 : ∂ z3,1 ∂t

=−

z3,1 z3,2 −z2,1 z2,2 ∆x

∂ z3,2 ∂t

(20)

= v3 ··· ···

where the control input v3 is defined as v3 = −z3,2

c2 z

z3,2 −z3,2 ∆x

−z

2,1 0 3,1 − z3,1 + ∆x ··· ··· ···

V (z3,1 )−z3,2 τ

(21)

At point xn−1 : ∂ zn−1,1 ∂t

=−

zn−1,1 zn−1,2 −zn−2,1 zn−2,2 ∆x

∂ zn−1,2 ∂t

(22)

= vn−1

where the control input vn−1 is defined as vn−1 = −zn−1,2

zn−1,2 −zn−2,2 ∆x

c2

0 − zn−1,1

zn−1,1 −zn−2,1 ∆x

+

V (zn−1,1 )−zn−1,2 τ

(23)

At point xn : ∂ zn,1 ∂t

=−

zn,1 zn,2 −zn−1,1 zn−1,2 ∆x

∂ zn,2 ∂t

(24)

= vn

Thus, one arrives at a linearized description for the Payne-Whitman state-space model Z˙ = AZ + BV + D

(25)

where the state vector is defined as Z = [z1,1 , z1,2 , z2,1 , z2,2 , · · · , zn,1 , zn,2 ]T , the control input vn is defined as vn = −zn,2

zn,2 −zn−1,2 ∆x

c2 zn,1 −zn−1,1 ∆x n,1

−z0

+

V (zn,1 )−zn,2 τ

(26)

the disturbances vector D describes the effects of initial conditions to this model and is given by D = [ρ0 v0 /∆x, 0, 0, 0, 0, 0, · · · , 0, 0, 0, 0]T , while matrices A and B are defined as



        A=        

0 0 0 0 0 0 ··· 0 0 0 0

z

1,1 − ∆x 0

z1,1 ∆x

0 0 0 ··· 0 0 0 0

0 0 0 0 0 0 ··· 0 0 0 0

0 0

0 0 0 0 0 0 ··· 0 0 0 0 

z2,1 − ∆x

0

z2,1 ∆x

0 ··· 0 0 0 0

        B=       

0 0 0 0

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

z

3,1 − ∆x 0 ··· 0 0 0 0

0 1 0 0 0 0 ··· 0 0 0 0

0 0 0 1 0 0 ··· 0 0 0 0

0 0 0 0 0 1 ··· 0 0 0 0

0 0 0 0 0 0 ··· 0 0 0 0

0 0 0 0 0 0 ···

0 0 0 0 0 0 ··· 0 0 0 0

zn−2,1 ∆x

0 0 0

0 0 0 0 0 0

··· ··· ··· ··· ··· ···

0 0 0 0 0 0

0 0 0 0 0 0 ··· −

zn−1,1 ∆x

0

zn−1,1 ∆x

0

0 0 0 0 0 0 ··· 0 0 zn,1 − ∆x 0

                 

(27)



           0 0   1 0   0 0  0 1

··· ··· ··· ···

0 0 0 0 0 0 ··· 0 0 0 0

(28)

For the linearized equivalent model of the Payne-Whitham PDE one can perform state estimation using the Kalman Filter recursion. The discrete-time equivalents of matrices A,B and C are denoted as Ad , Bd and Cd , and the estimation error covariance matrix is denoted as P. Then the recursion of the Kalman Filter is applied [17]-[19]. measurement update : K(k) = P− (k)CdT [Cd ·P− (k)CdT + R]−1 x(k) ˆ = xˆ− (k) + K(k)[z(k) − Cd xˆ− (k)] P(k) = P− (k) − K(k)Cd P− (k)

time update : P− (k + 1) = Ad (k)P(k)ATd (k) + Q(k) xˆ− (k + 1) = Ad (k)x(k) ˆ + Bd (k)u(k)

A Brunovsky canonical form for the PDE model Elaborating on the previous results it also possible to show that the Payne-Whitham PDE model can be also written in the linear canonical Brunovsky form. The first couple of equations of the discretized Payne-Whitham PDE is: z1,1 z1,2 −ρ0 v0 ∆x c20 z1,1 −ρ0 V (z1,1 =z1,2 ) z −v −z1,2 1,2∆x 0 − z1,1 ∆x + τ

z˙1,1 = −

z˙1,2 =

(29)

By deriving the first row of Eq. (29) with respect to time one gets z˙

z

z

z˙

z¨1,1 = − 1,1∆x1,2 − 1,1∆x1,2 By substituting the first and the second row of Eq. (29) into Eq. (30) one gets

(30)

z z −ρ v z1,2 z1,1 z˙ z z z˙ (31) z¨1,1 = [− 1,1 1,2∆x 0 0 ] ∆x − ∆x [− 1,1∆x1,2 + 1,1∆x1,2 ] where as proven above z1,2 = f1 (Z f , Z˙ f ). Next, one can define the right part of Eq. (31) as the new control input v˜1 (Z f , Z˙ f ), which gives z¨1,1 = v˜1 . In a similar manner, for the i-th couple of equations of the discretized PayneWhitham PDE description one has zi,1 zi,2 −zi−1,1 zi−1,2 ∆x z −z c2 z −ρ V (z =z ) −zi,2 i,2 ∆xi−1,1 − zi,10 i,1∆x 0 + i,1τ i,2

z˙i,1 = −

z˙i,2 =

(32)

By deriving the first row of Eq. (32) with respect to time one obtains z˙ z

z z˙

z˙

z

z

z˙

z¨i,1 = − i,1∆xi,2 − i,1∆xi,2 + i−1,1∆xi−1,2 + i−1,1∆xi−1,2 By substituting the first and the second row of Eq. (32) into Eq. (33) one gets z1,1 z1,2 −zi−1,1 zi−1,2 zi,2 zi,1 z −z c2 z −z V (z )−z ] ∆x − ∆x [−zi,2 i,2 ∆xi−1,2 − zi,10 i,1 ∆xi−1,1 + i,1τ i,2 ]− ∆x z z −z z z zi−1,1 zi−1,2 −zi−2,2 c20 zi−1,1 −zi−2,1 V (z )−z −[− i−1,1 i−1,2∆x i−2,1 i−2,2 ] i−1,1 − zi−1,1 + i−1,1τ i−1,2 ] ∆x + ∆x [−zi−1,2 ∆x ∆x

z¨i,1 = −[−

(33)

(34)

where zi,1 , zi−1,1 are elements of the flat output vector and zi−1,2 = fi−1 (Z f , Z˙ f ), while also zi−2,1 = fi (Z f , Z˙ f ). Next, one can define the right part of Eq. (34) as the new control input v˜i (Z f , Z˙ f ), which gives z¨i,1 = v˜i , for i = 1, 2, · · · , n. In this manner, the states-space description of the Payne-Whitham PDE is transformed into the equivalent description z¨1,1 = v˜1 , z¨2,1 = v˜3 , z¨3,1 = v˜3 , · · · , z¨n−1,1 = v˜n−1 , z¨n,1 = v˜n−1 , which stands for the linear canonical Brunovsky form of the system.

SIMULATION TESTS The efficiency of the Derivative-free nonlinear Kalman filtering scheme for estimation of traffic dynamics described by the Payne-Whitman PDE model has been confirmed through simulation experiments. The sampling period was taken to be Ts = 0.01sec. The x-axis (corresponding to the traffic lane) of the vehicles’ spatiotemporal variables ρ (x,t) and v(x,t) was divided into n = 50 equidistant points. Estimates of the vehicles’ density ρˆ (x,t) and of the vehicles’ flow speed v(x,t) ˆ were obtained with the use of the Derivative free nonlinear Kalman Filter, through the processing of measurements of ρ (xi ,t) at specific points of the grid. The obtained results are first depicted in Fig 1, Fig. 2, where the real and estimated values of ρ (x,t) and v(x,t) are given. Moreover, in Fig. 3, Fig. 4 and Fig. 5 one can see the convergence of the estimated state variables of the vehicles PDE dynamics to the associated real state variables. Results are given for the first 12 grid points of the spatiotemporal representation of the system. From the simulation experiments it can be observed that the Derivative-free nonlinear Kalman Filter was capable of computing fast and with high accuracy the estimates of the non-measurable state variables of the PDE traffic model. This is important for the assessment of the condition of highways in real-time, because it helps to develop a lowcost monitoring infrastructure [20]. Actually, to extract information about the highway’s condition there is no need to deploy a sensor network that comprises a large number of sensors (cameras, induction loops, GPS devices, etc.). It suffices to use a smaller number of suitably placed sensors (to assure that the system’s observability and detectability is preserved), which will provide input to the Derivative-free nonlinear Kalman Filter.

(a)

(b)

FIGURE 1. (a) Spatiotemporal variation of the vehicles’ density ρ (x,t) (state variable zi,1 , i = 1, · · · , n), (b) Estimation of the spatiotemporal variation of the vehicles’ density ρˆ (x,t) provided by the Derivative-free nonlinear Kalman Filter

CONCLUSIONS The paper has introduced a new nonlinear filtering method, under the name Derivative-free nonlinear Kalman Filter, that is capable of providing estimates of highway traffic dynamics through the processing of measurements from a

(a)

(b)

FIGURE 2. (a) Spatiotemporal variation of the vehicles’ flow speed v(x,t) (state variable zi,2 , i = 1, · · · , n), (b) Estimation of the spatiotemporal variation of the vehicles’ flow speed v(x,t) ˆ provided by the Derivative-free nonlinear Kalman Filter

5

10 t (sec)

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0

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10 t (sec)

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state variable z

50

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10 t (sec)

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20

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i,2

state variable z

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state variable z

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i,2

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state variable z

state variable z

i,1

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(a)

20

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(b)

FIGURE 3. State variables (blue line) and their estimation provided by the Derivative free nonlinear Kalman Filter (green line) (a) z1,1 ,z1,2 , at grid point 1 and z2,1 ,z2,2 at grid point 2 (b) z3,1 ,z3,2 , at grid point 3 and z4,1 ,z4,2 at grid point 4

limited number of spatially distributed sensors. First it was proven that the Payne-Whitham PDE which is used to describe the spatiotemporal dynamics of highway traffic is a differentially flat system. Moreover, it was shown that the Payne-Whitham PDE can be expressed in a linear canonical state-space form. For the linearized equivalent description of the traffic’s PDE it was shown that state estimation with Kalman Filtering recursion does not require the computation of Jacobian matrices and partial derivatives and does not introduce cumulative numerical errors. This is the Derivativefree nonlinear Kalman Filter which succeeds improved estimation comparing to the Extended Kalman Filter, both in terms of numerical accuracy and in terms of computational robustness. The efficiency of the proposed filtering method

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10 t (sec)

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state variable zi+1,1

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(a)

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state variable z

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i,2

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state variable z

state variable zi,1

200

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(b)

FIGURE 4. State variables (blue line) and their estimation provided by the Derivative free nonlinear Kalman Filter (green line) (a) z5,1 ,z5,2 , at grid point 5 and z6,1 ,z6,2 at grid point 6 (b) z7,1 ,z7,2 , at grid point 7 and z8,1 ,z8,2 at grid point 8

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state variable z

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FIGURE 5. State variables (blue line) and their estimation provided by the Derivative free nonlinear Kalman Filter (green line) (a) z9,1 ,z9,2 , at grid point 9 and z10,1 ,z10,2 at grid point 10 (b) z11,1 ,z11,2 , at grid point 11 and z12,1 ,z12,2 at grid point 12

in the case of monitoring of highway traffic was confirmed through simulation experiments.

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