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Exponential Boundary Observers for Pressurized Water Pipe

This paper deals with state estimation on a pressurized water pipe modeled by nonlinear coupled distributed hyperbolic equations for non-conservative laws with three known boundary measures. Our objective is to estimate the fourth boundary variable, which will be useful for leakage detection. Two approaches are studied. Firstly, the distributed hyperbolic equations are discretized through a finite-difference scheme. By using the Lipschitz property of the nonlinear term and a Lyapunov function, the exponential stability of the estimation error is proven by solving Linear Matrix Inequalities (LMIs). Secondly, the distributed hyperbolic system is preserved for state estimation. After state transformations, a Luenberger-like PDE boundary observer based on backstepping mathematical tools is proposed. An exponential Lyapunov function is used to prove the stability of the resulted estimation error. The performance of the two observers are shown on a water pipe prototype simulated example.


Introduction
Supervision of physical transport plant has been an active research topic in recent years and water distribution network (WDN) monitoring is a major concern [1], [2], [3]. Indeed a water distribution network is a set of interconnected physical components (pipes, pumps, tanks and valves) with the mission of supplying required water flow under sufficient pressure for various load conditions. The real-time knowledge of the network state is essential to improve the efficiency rate of the network which is usually around 75% mainly due to leakages [4]. For that purpose, WDN are equipped with flow and pressure sensors at certain nodes. Due to cost reasons and physical constraints, it is not possible to implement sensors in every nodes. Thus, unmeasured quantities should be estimated to provide useful information for network supervision and management. Different kinds of estimators or state observers may be used depending on the quantities to be estimated, the available sensors and the desired performance of the estimation scheme.
The pressurized pipes of a WDN are relatively long to consider a one-dimensional flow movement modeled by nonlinear coupled distributed hyperbolic equations derived from the nonconservative laws [5]. Flow and pressure measures are available at certain boundaries in a WDN. Exact boundary observability for one dimensional first order quasilinear hyperbolic systems has been studied in [6]. The observability is based on the existence of a unique semi-global solution and leads to the possibility to design an exponential boundary observer.
Two approaches may be used to design a boundary observer. 1) The indirect approach where Partial Differential Equations (PDE) operating in a functional infinite dimensional space are approximated by Ordinary Differential Equations (ODE) in a finite-dimensional space through a differentiation operator (finite differences schemes are commonly used [7], [8]). The advantage of such approach is to give access to many techniques well developed for observers design in finite dimension.
2) The direct approach that preserves all the system information for state estimation [9], [10] by designing directly PDE observers. This approach should potentially lead to better estimation results since no model approximation is made, however it is much more complex and needs mathematical background. This paper presents both observer design approaches. An exponential boundary observer for the discretized system is proposed. The Lipschitz property of the nonlinear term and Linear Matrix Inequalities (LMIs) techniques are used to prove the Lyapunov stability of the estimation error and the exponential convergence of the proposed observer. For the direct approach, a Luenberger-like PDE observer based on backstepping mathematical tools [11], [12] is proposed. An exponential Lyapunov function is used to prove the stability of the resulted dynamic estimation error. The exponential convergence of the PDE observer is guaranteed with the wise choice of a parameter µ (see section 4). Behaviors of observers based on ODE model and PDE model are shown with a length varying pipe. This paper is structured as follows. Section 2 describes the mathematical model of a pressurized water pipe. The model is discretized and an exponential boundary observer is proposed in section 3. Section 4 deals with the PDE observer. Simulations of both observers are shown and discussed in section 5. Section 6 gives concluding remarks.

Mathematical model
We consider an isothermal flow of a non-compressible fluid in a non-deformable pipeline with constant cross-sectional area. The pipe wall friction is assumed to be constant along the pipe and is supposed to be the only contributor to pressure drop along the pipe according to the Darcy-Weisbach formula [5]. Further we assume that the wave speed is large compared with the flow velocity and the length of the pipe is sufficiently large to consider a uniform flow movement in the transversal direction which is described by a set of two nonlinear hyperbolic PDE obtained by applying the mass and momentum balance conditions on a control volume of length dx [5], x  ) stands for derivative with respect to time (resp. to x axis).
We suppose that we measure three boundary conditions of (1):   Our goal is to design in healthy conditions (without leakage) an exponential boundary observer for both discretized and original PDE system to estimate the non-measured state ( ,0) Qt which will be noted ˆ( ,0) Qt in order to compare the behavior of both observers.

Exponential boundary observer for the discretized system
Let us define, The corresponding exponential boundary observer is, This particular observer structure has already been studied and several design methods may be found in the literature. One of them is the non-constructive Thau's method [13]. Another one is an iterative constructive method proposed in [14]. Also a direct computation of observer gain by solving LMIs was developed in [15]. In the following described method, an exponential term is added to the aforementioned work and the Lyapunov stability of the estimation error is proven using LMIs.

Theorem 1
Consider system (6)-(7) and observer (9)- (10). If the observer gain  is chosen such that the following inequalities holds then the estimation error   tends exponentially to 0 when t tends to infinity.

Proof of theorem 1
The dynamic state estimation error is .   is Lipchitz with respect to  i.e. there exists a constant 0    such that the following inequality is satisfied represents the 2-norm.
As we mentionned in our objectives, it would be interesting to design an observer using the original PDE system. This is the aim of the next section.

PDE exponential boundary observer
System (1) is a strictly hyperbolic system with two distinct real eigenvalues 1 a   and 2 a   . The exact boundary observability of hyperbolic systems is studied in [6]. The observation is carried out from one-sided of the pipe on boundary 0 x  . We first apply changes of coordinates to relax our observer's design task.
System (1) may be rewritten w t x remain partially matched. We first search for a change of coordinate to put (27) into the following form: where  is a constant to be fixed.
Taking the spatial-derivative of (30), we get Taking its time derivative we obtain

Observer setup and stability analysis
A Luenberger-like observer is designed. This observer is built as a copy of the system with a correction function of the output estimation error is a function to be designed to stabilize the state estimation error From the system and observer equations, we obtain the error equation

Simulations and results
Results are validated in simulation on a pipe with parameters presented in table 1. We have considered outlet pressure and flow rate respectively at 0.656x10 5 Pa and 0.0082544 31 . ms  as input of the system.
The observer gain for the discretized system is computed according to LMIs (11)-(12), we get   The estimated inlet pressure and flow rate with the two observers are represented in figures 2a and 2.b. Both observers converge exponentially; however, the convergence of the PDE observer is faster than the nonlinear observer based on the discretized system.
The length of the pipe is changed to   We note a relative long convergence time for the discretized observer. Figure 4b shows with a PDE observer, a stabilization of the inlet pressure estimation error around 3 100 10 bar Pa    and an accuracy error of 0.48% (shown in figure 5b). Compared with the sensitivity of pressure sensors on distributed water network, we can conclude that both observers converge exponentially to the desired value. Furthermore, the implementation of PDE boundary observer seems to be suitable for relatively long pipes in order to alleviate the computation of observer gain and to guarantee a real time monitoring of distributed water network.

Conclusion
State estimation on a pressurized water pipe with three known measures at boundary has been addressed. Two approaches have been explored. The indirect approach uses the discretized system; an exponential boundary nonlinear observer is designed by using the Lipschitz property of the nonlinear term and by solving LMIs to ensure the stability of estimation error. The direct approach, where some state transformations are used in order to design a Luenberger-like PDE boundary observer based on backstepping mathematical tools. An exponential Lyapunov function is used to prove the stability of the error equation.
Simulations have been carried out for a water pipe prototype. Results have shown the effectiveness of both approaches. The convergence time of the nonlinear observer increases with the length of pipe, The PDE observer seems more efficient. These two observers may be used to detect a leakage in the pipe using a reduced number of sensors. We will study in future work the efficiency of both observers for leakage detection. Early studies and simulations show that the leakage detection is more efficient using a PDE observer. Moreover, the PDE observer may also be easily extended to localize the leakage in the pipe if the four boundary conditions are measured.