Computing Lipschitz Constants for Hydraulic Models of Water Distribution Networks

Drinking water distribution networks (WDN) are large-scale, dynamic systems spanning large geographic areas. Water networks include various components such as junctions, reservoirs, tanks, pipes, pumps, and valves. Hydraulic models for these components depicting mass and energy balance form nonlinear algebraic differential equations (NDAE). While control theoretic studies have been thoroughly explored for other complex infrastructure such as power and transportation systems, little is understood or even investigated for feedback control and state estimation problems for the NDAE models of WDN. The objective of this letter is to showcase a complete NDAE model of WDN followed by computing Lipschitz constants of the vector-valued nonlinearity in that model. The computation of Lipschitz constants of hydraulic models is crucial as it paves the way to apply a plethora of control-theoretic studies for water system applications. In particular, the computation of Lipschitz constant is explored through closed-form, analytical expressions as well as via numerical methods. Case studies reveal how such computations fare against each other for various water networks.

[1]  David A. Chin,et al.  Water-Resources Engineering , 1999 .

[2]  Yi Shen,et al.  Observer design for discrete-time descriptor systems: An LMI approach , 2012, Syst. Control. Lett..

[3]  Damien Koenig,et al.  Observer design for unknown input nonlinear descriptor systems via convex optimization , 2006, IEEE Transactions on Automatic Control.

[4]  Yan Wang,et al.  Multi-Objective Nonlinear Observer Design using BMIs , 2018, 2018 Annual American Control Conference (ACC).

[5]  R. K. Linsley,et al.  Water - Resources Engineering , 1964 .

[6]  Saïd Mammar,et al.  Observer for Lipschitz nonlinear systems: Mean Value Theorem and sector nonlinearity transformation , 2012, 2012 IEEE International Symposium on Intelligent Control.

[7]  Thomas Berger,et al.  On Observers For Nonlinear Differential-Algebraic Systems , 2019, IEEE Transactions on Automatic Control.

[8]  Larry W. Mays,et al.  Battle of the network models: Epilogue , 1987 .

[9]  M. Rehan,et al.  Observer design for one-sided Lipschitz descriptor systems , 2016 .

[10]  Vicenç Puig,et al.  Real-time Monitoring and Operational Control of Drinking-Water Systems , 2017 .

[11]  T. O’Neil.  Geometric Measure Theory , 2002 .

[12]  Sergei S. Kucherenko,et al.  Application of Deterministic Low-Discrepancy Sequences in Global Optimization , 2005, Comput. Optim. Appl..

[13]  R. Rajamani,et al.  Observer design for Lipschitz nonlinear systems using Riccati equations , 2010, Proceedings of the 2010 American Control Conference.

[14]  Junjian Qi,et al.  Characterizing the Nonlinearity of Power System Generator Models , 2019, 2019 American Control Conference (ACC).

[15]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[16]  Michel Zasadzinski,et al.  Observer design for one-sided Lipschitz discrete-time systems , 2012, Syst. Control. Lett..

[17]  A. Alessandri.  Design of observers for lipschitz nonlinear systems using LMI , 2004 .

[18]  R Rajamani,et al.  The bounded Jacobian approach to nonlinear observer design , 2010, Proceedings of the 2010 American Control Conference.

[19]  Ahmad F. Taha,et al.  Nonlinear Dynamic Systems Parameterization Using Interval-Based Global Optimization: Computing Lipschitz Constants and Beyond , 2020, ArXiv.

[20]  Javad Lavaei,et al.  Multiplier-based Observer Design for Large-Scale Lipschitz Systems , 2018 .

[21]  Housheng Su,et al.  Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations , 2012 .

[22]  Ahmad Afshar,et al.  Observer-based tracking controller design for a class of Lipschitz nonlinear systems , 2018 .

[23]  Nikolaos Gatsis,et al.  A New Derivative‐Free Linear Approximation for Solving the Network Water Flow Problem With Convergence Guarantees , 2020, Water Resources Research.

[24]  Ali Zemouche,et al.  On LMI conditions to design observers for Lipschitz nonlinear systems , 2013, Autom..