American Institute of Mathematical Sciences

Advanced
June  2009, 4(2): 267-285. doi: 10.3934/nhm.2009.4.267

Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems

Didier Georges 1,

1. 

Control systems department, Gipsa-lab, Grenoble, Grenoble INP-Ense3, BP 46, 38402 Saint-Martin d’Hères, France

Received  November 2008 Revised  January 2009 Published  June 2009

A nonlinear predictive control design based on Saint Venant equations is presented in this paper in order to regulate both water depth and water flow rate in a single pool of an open-channel hydraulic system. Thanks to variational calculus, some necessary optimality conditions are given. The adjoint partial differential equations of Saint Venant partial differential equations are also derived. The resulting two-point boundary value problem is solved numerically by using both time and space discretization and operator approximations based on nonlinear time-implicit finite differences. The practical effectiveness of the control design is demonstrated by a simulation example. A extension of the predictive control scheme to a multi-pool system is proposed by using a decomposition-coordination approach based on two-level algorithm and the use of an augmented Lagrangian, which can take advantage of communication networks used for distributed control. This approach may be easily applied to other problems governed by hyperbolic PDEs, such as road traffic systems.
Keywords: open-channel hydraulic systems., Nonlinear model predictive control, nonlinear hyperbolic partial differential equations.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks & Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267
[1]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[2]

Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211
[3]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[4]

João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303
[5]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[6]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[7]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[8]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[9]

Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739
[10]

Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127
[11]

Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165
[12]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[13]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[14]

Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765
[15]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295
[16]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[17]

Tatiana Filippova. Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. Conference Publications, 2011, 2011 (Special) : 410-419. doi: 10.3934/proc.2011.2011.410
[18]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[19]

Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
[20]

Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks & Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy