# American Institute of Mathematical Sciences

December  2014, 4(4): 501-520. doi: 10.3934/mcrf.2014.4.501

## Multivariable boundary PI control and regulation of a fluid flow system

 1 Université de Lyon, LAGEP, Bât. CPE, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France 2 Université de Lorraine and Inria Nancy-Grand Est, Bâtiment A, Ile du Saulcy, F-57045 Metz, France

Received  July 2013 Revised  March 2014 Published  September 2014

The paper is concerned with the control of a fluid flow system governed by nonlinear hyperbolic partial differential equations. The control and the output observation are located on the boundary. We study local stability of spatially heterogeneous equilibrium states by using Lyapunov approach. We prove that the linearized system is exponentially stable around each subcritical equilibrium state. A systematic design of proportional and integral controllers is proposed for the system based on the linearized model. Robust stabilization of the closed-loop system is proved by using a spectrum method.
Citation: Cheng-Zhong Xu, Gauthier Sallet. Multivariable boundary PI control and regulation of a fluid flow system. Mathematical Control & Related Fields, 2014, 4 (4) : 501-520. doi: 10.3934/mcrf.2014.4.501
##### References:
 [1] G. Bastin and J. M. Coron, On boundary feedback stabilization of non-uniform linear $2 \times 2$ hyperbolic systems over a bounded interval,, Systems & Control Letters, 60 (2011), 900.  doi: 10.1016/j.sysconle.2011.07.008.  Google Scholar [2] H. Bounit, H. Hammouri and J. Sau, Regulation of an irrigation channel through the semigroup approach,, Proceedings of the workshop on Regulation and Irrigation Canals, (1997), 261.   Google Scholar [3] J. M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2.  doi: 10.1109/TAC.2006.887903.  Google Scholar [4] E. J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems,, IEEE Transactions Automatic Control, 21 (1976), 25.   Google Scholar [5] V. Dos Santos and C. Prieur, Boundary control of open channels with numerical and experimental validations,, IEEE Transactions on Control Systems Technolgy, 16 (2008), 1252.   Google Scholar [6] D. Georegs, Infinite-dimensional nonlinear predictive control design for open hydraulic systems,, Networks and Heterogeneous Media - American Institute of Mathematical Sciences, 4 (2009), 267.  doi: 10.3934/nhm.2009.4.267.  Google Scholar [7] J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasi-linear wave equation,, Journal of Differential Equations, 52 (1984), 66.  doi: 10.1016/0022-0396(84)90135-9.  Google Scholar [8] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. of Differential Equations, 1 (1985), 43.   Google Scholar [9] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1976).   Google Scholar [10] I. Lasiecka and R. Triggiani, Finite rank, relative bounded perturbation of semigroups - Part I : Well-posedness and boundary feedback hyperbolic dynamics,, Annali Scuola Normale Superior-Pisa, 12 (1985), 641.   Google Scholar [11] X. Litrico and V. Fromion, Boundary control of hyperbolic conservation laws using a frequency domain approach,, Automatica, 45 (2009), 647.  doi: 10.1016/j.automatica.2008.09.022.  Google Scholar [12] H. Logemann and S. Townley, Low gain control of uncertain regular linear systems,, SIAM J. Control Optim., 35 (1997), 78.  doi: 10.1137/S0363012994275920.  Google Scholar [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [14] S. A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems,, IEEE Transaction Automatic Control, 27 (1982), 17.  doi: 10.1109/TAC.1982.1102887.  Google Scholar [15] S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups,, Journal of Mathematical Analysis and Applications, 111 (1985), 622.  doi: 10.1016/0022-247X(85)90239-2.  Google Scholar [16] J. Prüss, On the spectrum of $C_0$- semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [17] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain,, Indiana University Mathematics Journal, 24 (1974), 79.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar [18] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations : Recent progress and open questions,, Siam Review, 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar [19] D. L. Russell and G. Weiss, A general necessary condition for exact observability,, SIAM J. Control and Optimization, 32 (1994), 1.  doi: 10.1137/S036301299119795X.  Google Scholar [20] D. Salamon, Realization theory in Hilbert space,, Math. Systems Theory, 21 (1989), 147.  doi: 10.1007/BF02088011.  Google Scholar [21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar [22] G. Weiss, Regular linear systems with feedback,, Math. Control, 7 (1994), 23.  doi: 10.1007/BF01211484.  Google Scholar [23] C. Z. Xu and D. X. Feng, Symmetric hyperbolic systems and applications to exponential stability of heat exchangers and irrigation canals,, Proceedings of the Mathematical Theory of Networks and Systems, (2000).   Google Scholar [24] C. Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems,, Int. J. Control, 61 (1995), 33.  doi: 10.1080/00207179508921891.  Google Scholar [25] C. Z. Xu and G. Sallet, Proportional and integral regulation of irrigation canal systems governed by the Saint Venant equation,, Proceedings of the 14th IFAC World Congress, (1999).   Google Scholar [26] C. Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,, ESAIM: Control, 7 (2002), 421.  doi: 10.1051/cocv:2002062.  Google Scholar

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##### References:
 [1] G. Bastin and J. M. Coron, On boundary feedback stabilization of non-uniform linear $2 \times 2$ hyperbolic systems over a bounded interval,, Systems & Control Letters, 60 (2011), 900.  doi: 10.1016/j.sysconle.2011.07.008.  Google Scholar [2] H. Bounit, H. Hammouri and J. Sau, Regulation of an irrigation channel through the semigroup approach,, Proceedings of the workshop on Regulation and Irrigation Canals, (1997), 261.   Google Scholar [3] J. M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2.  doi: 10.1109/TAC.2006.887903.  Google Scholar [4] E. J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems,, IEEE Transactions Automatic Control, 21 (1976), 25.   Google Scholar [5] V. Dos Santos and C. Prieur, Boundary control of open channels with numerical and experimental validations,, IEEE Transactions on Control Systems Technolgy, 16 (2008), 1252.   Google Scholar [6] D. Georegs, Infinite-dimensional nonlinear predictive control design for open hydraulic systems,, Networks and Heterogeneous Media - American Institute of Mathematical Sciences, 4 (2009), 267.  doi: 10.3934/nhm.2009.4.267.  Google Scholar [7] J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasi-linear wave equation,, Journal of Differential Equations, 52 (1984), 66.  doi: 10.1016/0022-0396(84)90135-9.  Google Scholar [8] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. of Differential Equations, 1 (1985), 43.   Google Scholar [9] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1976).   Google Scholar [10] I. Lasiecka and R. Triggiani, Finite rank, relative bounded perturbation of semigroups - Part I : Well-posedness and boundary feedback hyperbolic dynamics,, Annali Scuola Normale Superior-Pisa, 12 (1985), 641.   Google Scholar [11] X. Litrico and V. Fromion, Boundary control of hyperbolic conservation laws using a frequency domain approach,, Automatica, 45 (2009), 647.  doi: 10.1016/j.automatica.2008.09.022.  Google Scholar [12] H. Logemann and S. Townley, Low gain control of uncertain regular linear systems,, SIAM J. Control Optim., 35 (1997), 78.  doi: 10.1137/S0363012994275920.  Google Scholar [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [14] S. A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems,, IEEE Transaction Automatic Control, 27 (1982), 17.  doi: 10.1109/TAC.1982.1102887.  Google Scholar [15] S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups,, Journal of Mathematical Analysis and Applications, 111 (1985), 622.  doi: 10.1016/0022-247X(85)90239-2.  Google Scholar [16] J. Prüss, On the spectrum of $C_0$- semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [17] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain,, Indiana University Mathematics Journal, 24 (1974), 79.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar [18] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations : Recent progress and open questions,, Siam Review, 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar [19] D. L. Russell and G. Weiss, A general necessary condition for exact observability,, SIAM J. Control and Optimization, 32 (1994), 1.  doi: 10.1137/S036301299119795X.  Google Scholar [20] D. Salamon, Realization theory in Hilbert space,, Math. Systems Theory, 21 (1989), 147.  doi: 10.1007/BF02088011.  Google Scholar [21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar [22] G. Weiss, Regular linear systems with feedback,, Math. Control, 7 (1994), 23.  doi: 10.1007/BF01211484.  Google Scholar [23] C. Z. Xu and D. X. Feng, Symmetric hyperbolic systems and applications to exponential stability of heat exchangers and irrigation canals,, Proceedings of the Mathematical Theory of Networks and Systems, (2000).   Google Scholar [24] C. Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems,, Int. J. Control, 61 (1995), 33.  doi: 10.1080/00207179508921891.  Google Scholar [25] C. Z. Xu and G. Sallet, Proportional and integral regulation of irrigation canal systems governed by the Saint Venant equation,, Proceedings of the 14th IFAC World Congress, (1999).   Google Scholar [26] C. Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,, ESAIM: Control, 7 (2002), 421.  doi: 10.1051/cocv:2002062.  Google Scholar
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