State and parameter estimation in 1-D hyperbolic PDEs based on an adjoint method

Abstract

International audienceAn optimal estimation method for state and distributed parameters in1-D hyperbolic system based on adjoint method is proposed in thispaper. A general form of the partial differential equations governingthe dynamics of system is first introduced. In this equation, theinitial condition or state variable as well as some empiricalparameters are supposed to be unknown and need to be estimated. TheLagrangian multiplier method is used to connect the dynamics of thesystem and the cost function defined as the least square error betweenthe simulation values and the measurements. The adjoint state method isapplied to the objective functional in order to get the adjoint systemand the gradients with respect to parameters and initial state. Theobjective functional is minimized by Broyden–Fletcher–Goldfarb–Shanno(BFGS) method. Due to the non-linearity of both direct and adjointsystem, the nonlinear explicit Lax–Wendroff scheme is used to solvethem numerically. The presented optimal estimation approach isvalidated by two illustrative examples, the first one about state andparameter estimation in a traffic flow, and the second one in anoverland flow system

Calculus of variations approach for state and parameter estimation in switched 1D hyperbolic PDEs

“…For the reason of simplicity, the notations , f ( u ( x , t ), t ), λ ( x , t ), γ ( t ), ϕ a ( t , p , u ), g 1 ( u , x , t , p , y ), g 2 ( u , x , t , p , y ), h 1 ( y , p ), h 2 ( y , p ), y ( t ), and ξ ( t , p , u ) are shortly denoted by , u , f , λ , γ , ϕ a , g 1 , g 2 , h 1 , h 2 , y , and ξ , respectively, except in some special cases. The cumbersome and formal calculations of the first variation, which are already presented in other publications,() are moved to Appendix A. The variations of Lagrangian functional with respect to the variations of all possible directions δ u ( x , t ), , δ y ( t ), δ α i ( x ), and δ p are written under the form of the Gâteaux derivative of .…”

“…By using real field data, its applicability was verified on the Tondi Kiboro catchment . The state and distributed parameter identification for a class of a hyperbolic system with 2 numerical examples was addressed in our previous work . The present paper can be seen as an extension of the aforementioned work, where the source term was considered to be perfectly known and no switching was considered.…”

“…In the present work, the switching characteristic is added to the source term of system dynamics, and the switching condition is not only a function of time but also depends on the parameters or state of the systems. This work is thus an extension of the work of Nguyen et al, with additional difficulties related to the presence of the switch on the one hand and the fact that the switch condition depends on parameters, which are not known (and to be estimated), on the other hand. The main contribution in that respect is to show how to adapt the similar techniques of the calculus of variations to those particular difficulties and illustrate this with a couple of application examples.…”

“…In the example of free overland flow, due to the discontinuity of Green‐Ampt infiltration rate under the constant rainfall process, some works such as our previous work assume that the ponding time, ie, the time from the beginning to the ponding moment, is small and can be neglected. As a result, the overall system is not switched anymore.…”

“…Notice that Lattanzio et al also modified the original LWR model by adding an ODE, which represents the moving bottleneck to create a PDE‐ODE coupled model of the traffic flow. The original LWR model was used in our other works() but without the right term. The location of the relief route is normally fixed and well known.…”