Multivariable boundary PI control and regulation of a fluid flow system

Previous Article
Observability and controllability analysis of blood flow network
MCRF Home
This Issue
Next Article
Asymptotic stability of Webster-Lokshin equation

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

Multivariable boundary PI control and regulation of a fluid flow system

Cheng-Zhong Xu ^1, and Gauthier Sallet ^2,

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

Abstract
Related Papers

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.

Keywords: robust stabilization, spectrum assignment., spatially heterogeneous equilibrium states, exponential stability, Hyperbolic partial differential equations.

Mathematics Subject Classification: 93C20, 93D15, 93D21, 93D3.

Citation: Cheng-Zhong Xu, Gauthier Sallet. Multivariable boundary PI control and regulation of a fluid flow system. Mathematical Control and 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), 900-906. doi: 10.1016/j.sysconle.2011.07.008.
[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), 261-267, Marrakesh, Maroc.
[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), 2-11. doi: 10.1109/TAC.2006.887903.
[4]	E. J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Transactions Automatic Control, 21 (1976), 25-34.
[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), 1252-1264.
[6]	D. Georegs, Infinite-dimensional nonlinear predictive control design for open hydraulic systems, Networks and Heterogeneous Media - American Institute of Mathematical Sciences, 4 (2009), 267-285. doi: 10.3934/nhm.2009.4.267.
[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), 66-75. doi: 10.1016/0022-0396(84)90135-9.
[8]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Differential Equations, 1 (1985), 43-56.
[9]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976.
[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, Classe di Scienze, Serie IV, XII (4), 12 (1985), 641-668.
[11]	X. Litrico and V. Fromion, Boundary control of hyperbolic conservation laws using a frequency domain approach, Automatica, 45 (2009), 647-656. doi: 10.1016/j.automatica.2008.09.022.
[12]	H. Logemann and S. Townley, Low gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116. doi: 10.1137/S0363012994275920.
[13]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[14]	S. A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems, IEEE Transaction Automatic Control, 27 (1982), 17-30. doi: 10.1109/TAC.1982.1102887.
[15]	S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups, Journal of Mathematical Analysis and Applications, 111 (1985), 622-636. doi: 10.1016/0022-247X(85)90239-2.
[16]	J. Prüss, On the spectrum of $C_0$- semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[17]	J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain, Indiana University Mathematics Journal, 24 (1974), 79-86. doi: 10.1512/iumj.1975.24.24004.
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations : Recent progress and open questions, Siam Review, 20 (1978), 639-739. doi: 10.1137/1020095.
[19]	D. L. Russell and G. Weiss, A general necessary condition for exact observability, SIAM J. Control and Optimization, 32 (1994), 1-23. doi: 10.1137/S036301299119795X.
[20]	D. Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011.
[21]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Boston, 2009. doi: 10.1007/978-3-7643-8994-9.
[22]	G. Weiss, Regular linear systems with feedback, Math. Control, Signals & Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.
[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, Perpignan, France.
[24]	C. Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems, Int. J. Control, 61 (1995), 33-45. doi: 10.1080/00207179508921891.
[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, Beijing, China.
[26]	C. Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM: Control, Optimization and Calculus of Variations, 7 (2002), 421-442. doi: 10.1051/cocv:2002062.

show all references

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), 900-906. doi: 10.1016/j.sysconle.2011.07.008.
[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), 261-267, Marrakesh, Maroc.
[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), 2-11. doi: 10.1109/TAC.2006.887903.
[4]	E. J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Transactions Automatic Control, 21 (1976), 25-34.
[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), 1252-1264.
[6]	D. Georegs, Infinite-dimensional nonlinear predictive control design for open hydraulic systems, Networks and Heterogeneous Media - American Institute of Mathematical Sciences, 4 (2009), 267-285. doi: 10.3934/nhm.2009.4.267.
[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), 66-75. doi: 10.1016/0022-0396(84)90135-9.
[8]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Differential Equations, 1 (1985), 43-56.
[9]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976.
[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, Classe di Scienze, Serie IV, XII (4), 12 (1985), 641-668.
[11]	X. Litrico and V. Fromion, Boundary control of hyperbolic conservation laws using a frequency domain approach, Automatica, 45 (2009), 647-656. doi: 10.1016/j.automatica.2008.09.022.
[12]	H. Logemann and S. Townley, Low gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116. doi: 10.1137/S0363012994275920.
[13]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[14]	S. A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems, IEEE Transaction Automatic Control, 27 (1982), 17-30. doi: 10.1109/TAC.1982.1102887.
[15]	S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups, Journal of Mathematical Analysis and Applications, 111 (1985), 622-636. doi: 10.1016/0022-247X(85)90239-2.
[16]	J. Prüss, On the spectrum of $C_0$- semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[17]	J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain, Indiana University Mathematics Journal, 24 (1974), 79-86. doi: 10.1512/iumj.1975.24.24004.
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations : Recent progress and open questions, Siam Review, 20 (1978), 639-739. doi: 10.1137/1020095.
[19]	D. L. Russell and G. Weiss, A general necessary condition for exact observability, SIAM J. Control and Optimization, 32 (1994), 1-23. doi: 10.1137/S036301299119795X.
[20]	D. Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011.
[21]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Boston, 2009. doi: 10.1007/978-3-7643-8994-9.
[22]	G. Weiss, Regular linear systems with feedback, Math. Control, Signals & Systems, 7 (1994), 23-57. doi: 10.1007/BF01211484.
[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, Perpignan, France.
[24]	C. Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems, Int. J. Control, 61 (1995), 33-45. doi: 10.1080/00207179508921891.
[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, Beijing, China.
[26]	C. Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM: Control, Optimization and Calculus of Variations, 7 (2002), 421-442. doi: 10.1051/cocv:2002062.

[1]	Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[2]	Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[3]	Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
[4]	Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27
[5]	Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[6]	Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[7]	Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555
[8]	Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
[9]	Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[10]	Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045
[11]	Arzu Ahmadova, Nazim I. Mahmudov, Juan J. Nieto. Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022008
[12]	Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[13]	Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[14]	Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310
[15]	Louis Tebou. Stabilization of some coupled hyperbolic/parabolic equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1601-1620. doi: 10.3934/dcdsb.2010.14.1601
[16]	Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control and Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
[17]	Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics and Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004
[18]	Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[19]	Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
[20]	Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2022, 11 (4) : 1191-1200. doi: 10.3934/eect.2021040

2021 Impact Factor: 1.141

Tools

Metrics

Other articles
by authors

on AIMS
Cheng-Zhong Xu Gauthier Sallet
on Google Scholar
Cheng-Zhong Xu Gauthier Sallet

[Back to Top]