This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.
Alizadeh Moghadam, A., Aksikas, I., Dubljevic, S. and Forbes, J. (2011). LQR control of an in finite dimensional timevarying CSTR-PFR system, 18th IFAC World Congress, Milan, Italy.
Athans, M., Fekri, S. and Pascoal, A. (2005). Issues on robust adaptive feedback control, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic.
Banerjee, A., Arkun, Y., Pearson, R. and Ogunnaike, B. (1995). H∞ control of nonlinear processes using multiple linear models, Proceedings of the European Control Conference, Rome, Italy, pp. 2671-2676.
Bhagwat, A., Srinivasan, R. and Krishnaswamy, P.R. (2003). Multi-linear model-based fault detection during process transitions, Chemical Engineering Science 58: 1649-1670.
Blesa, J., Puig, V. and Bolea, Y. (2010). Fault detection using interval LPV models in an open-flow canal, Control Engineering and Practice 18(5): 460-470.
Cordier, S., Buet, C. and Dos Santos, V. (2004). A conservative and entropy scheme for a simplified model of granular media, Transport Theory and Statistical Physics 33(2): HYKE 2003-021.
Coron, J. M., d’Andr´ea Novel, B. and Bastin, G. (2007). A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Transactions on Automatic Control 52(1): 2-11.
Curtain, R. and Zwart, H. (1995). An Introduction to Infinite Dimensional Linear Systems, Springer Verlag, New York, NY.
Dos Santos Martins, V. and Rodrigues, M. (2011). A proportional integral feedback for open channels control trough LMI design, 18th IFAC World Congress, Milan, Italy.
Dos Santos, V. (2004). Contrˆole Fronti`ere par Mod`ele Interne de Syst`emes Hyperboliques: Application `a la R´egulation de Canaux d’Irrigation, Ph.D. thesis, University of Orl´eans, Orl´eans.
Dos Santos, V., Bastin, G., Coron, J.-M. and d’Andr´ea Novel, B. (2008). Boundary control with integral action for hyperbolic systems of conservation laws: Lyapunov stability analysis and experimental validation, Automatica 44(5): 1310-1318.
Dos Santos, V. and Prieur, C. (2008). Boundary control of open channels with numerical and experimental validations, IEEE Transactions on Control Systems Technology 16(99): 1252-1264.
Dos Santos, V. and Toure, Y. (2005). Irrigation multireaches regulation problem by internal model boundary control, 44th CDC-ECC05, IEEE Control Systems Society Conference, Seville, Spain, pp. 1905-1910.
Dos Santos, V., Toure, Y., Mendes, E. and Courtial, E. (2005). Multivariable boundary control approach by internal model, applied to irrigations canals regulation, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic.
Dulhoste, J.-F., Besanc¸on, G. and Georges, D. (2001). Nonlinear control of water flow dynamics by input-output linearisation based on a collocation model, European Control Conference, Porto, Portugal.
Gatzke, E. and Doyle, F. (2002). Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis, Journal of Process Control 12(2): 339-352.
Georges (2002). Automatique pour la Gestion des Ressources en Eau, Edts IC2, Syst`emes automatis´es, Herm`es, Paris.
Greenberg, J.-M. and Li, T. (1984). The effect of boundary damping for the quasilinear wave equations, Journal of Differential Equations 52(1): 66-75.
Hamdi, H., Rodrigues, M., Mechmeche, C., Theilliol, D. and Braiek, N. B. (2011). Fault detection and isolation in linear parameter varying descriptor systems via proportional integral observer, International Journal of Adaptive Control and Signal Processing 26(3): 224-240, DOI: 10.1002/acs.1260.
Hante, F. and Sigalotti, M. (2010). Existence of common Lyapunov functions for infinite-dimensional switched linear systems, 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, pp. 5668-5673.
Leith, D.J. and Leithead, W.E. (2000). Survey of gainscheduling analysis and design, International Journal of Control 73(11): 1001-1025.
Li, T. (1994). Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, Masson and Wiley, Paris/Milan/Barcelona.
Litrico, X., Fromion, V., Baume, J.-P., Arranja, C. and Rijo, M. (2005). Experimental validation of a methodology to control irrigation canals based on Saint-Venant equations, Control Engineering Practice 13(11): 1425-1437.
Litrico, X. and Georges, D. (1999). Robust continuous-time and discrete-time flow control of a dam-river system, I: Modelling, Journal of Applied Mathematical Modelling 23(11): 809-827.
Lopez-Toribio, C., Patton, R. and Daley, S. (1999). A mutiplemodel approach to fault-tolerant control using Takagi-Sugeno fuzzy modelling: Real application to an induction motor drive system, European Control Conference, ECC 99, Karlsruhe, Germany.
Malaterre, P.-O., Rogers, D. and Schuurmans, J. (1998). Classification of canal control algorithms, Journal of Irrigation and Drainage Engineering 124(1): 3-10.
Mareels, I., Weyer, E., Ooi, S., Cantoni, M., Li, Y. and Nair, G. (2005). Systems engineering for irrigation systems: Successes and challenges, Annual Reviews in Control 29(2): 191-204.
Murray-Smith, R. and Johansen, T. (1997). Multiple Model Approaches to Modelling and Control, Taylor and Francis, London.
Narendra, K., Balakrishnan, J. and Kermal, M. (1995). Adaptation and learning using multiple models, switching and tuning, IEEE Control Systems 15(3): 37-51.
Ouarit, H., Lefevre, L. and Georges, D. (2003). Robust optimal control of one-reach open channels, European Control Conference 03, Cambridge, UK.
Papageorgiou, M. and Messmer, A. (1989). Flow control of a long river stretch, Automatica 25(2): 177-183.
Porfirio, C. R., Neito, E. A. and Odloak, D. (2003). Multi-model predictive control of an industrial c3/c4 splitter, Control Engineering Practice 11(7): 765-779.
Rodrigues, M., Theilliol, D., Aberkane, S. and Sauter, D. (2007). Fault tolerant control design for polytopic LPV systems, International Journal of Applied Mathematics and Computer Science 17(1): 27-37, DOI: 10.2478/v10006-007-0004-5.
Rodrigues, M., Theilliol, D., Adam-Medina, M. and Sauter, D. (2008). A fault detection and isolation scheme for industrial systems based on multiple operating models, Control Engineering Practice 16(2): 225-239.
Sakawa, Y. and Matsushita, T. (1975). Feedback stabilization of a class of distributed systems and construction of a state estimator, IEEE Transactions on Automatic Control AC-20 (6): 748-753.
Tour´e, Y. and Rudolph, J. (2002). Controller design for distributed parameter systems, Encyclopedia of LIFE Support on Control Systems, Robotics and Automation, Vol. I, Eolss Publishers, Oxford, pp. 933-979.
Triggiani, R. (1975). On the stability problem in Banach space, Journal of Mathematical Analysis and Applications 52(3): 383-403.
Wang, J.-W., Wu, H.-N. and Li, H.-X. (2011). Distributed fuzzy control design of nonlinear hyperbolic PDE systems with application to nonisothermal plug-flow reactor, IEEE Transactions on Fuzzy Systems 19(3): 514-526.
Weyer, E. (2002). Decentralised PI controller of an open water channel, 15th IFAC World Congress, Barcelona, Spain.
Zaccarian, L., Li, Y., Weyer, E., Cantoni, M. and Teel, A.R. (2007). Anti-windup for marginally stable plants and its application to open water channel control systems, Control Engineering Practice 15(2): 261-272.