Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems

Xu, Cheng-Zhong; Sallet, Gauthier

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 421-442.

In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.

Classification : 93D09, 93D25, 80A20, 35L50
Mots clés : heat exchangers, symmetric hyperbolic equations, exponential stability, regular systems, transfer functions

[1] C.D. Benchimol, A note on weak stabilizability of contraction semigroups. SIAM J. Control Optim. 16 (1978) 373-379. | MR | Zbl

[2] H. Bounit, H. Hammouri and J. Sau, Regulation of an irrigation canal system through the semigroup approach, in Proc. of the International Workshop Regulation of Irrigation Canals: State of the Art of Research and Applications. Marocco (1997) 261-267.

[3] S.X. Chen, Introduction to partial differential equations. People Education Press (in Chinese) (1981).

[4] V.T. Chow, Open channel hydraulics1985).

[5] J.M. Coron, B. D'Andréa-Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations, in European Control Conference ECC'99. Karlsruhe (1999).

[6] R.F. Curtain, Equivalence of input-output stability and exponential stability for infinite-dimensional systems. Math. Systems Theory 21 (1988) 19-48. | MR | Zbl

[7] C. Foias, H. Özbay and A. Tannenbaum, Robust Control of Infinite Dimensional Systems. Frequency Domain Methods. Springer, Hong Kong, Lecture Notes in Control and Inform. Sci. 209 (1996). | MR | Zbl

[8] B.A. Francis and G. Zames, On $H^{\infty }$-optimal sensitivity theory for SISO feedback systems. IEEE Trans. Automat. Control 29 (1984) 9-16. | MR | Zbl

[9] J.C. Friedly, Dynamic Behavior of Processes. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1972).

[10] J.P. Gauthier and C.Z. Xu, $H^{\infty }$-control of a distributed parameter system with non-minimum phase. Int. J. Control 53 (1991) 45-79. | MR | Zbl

[11] K.M. Hangos, A.A. Alonso, J.D. Perkins and B.E. Ydstie, Thermodynamic approach to the structural stability of process plants. AIChE J. 45 (1999) 802-816.

[12] H. Hoffman, Banach Spaces of Analytic Functions. Prentice-Hall Inc., Englewood Cliffs (1962). | MR | Zbl

[13] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differential Equations 1 (1985) 43-56. | MR | Zbl

[14] H.O. Kreiss, O.E. Ortiz and O.A. Reula, Stability of quasi-linear hyperbolic dissipative systems. J. Differential Equations 142 (1998) 78-96. | MR | Zbl

[15] P.D. Lax and R.S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13 (1960) 427-455. | MR | Zbl

[16] T.S. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, edited by P.G. Ciarlet and J.-L. Lions. John Willey & Sons, New York (1994). | Zbl

[17] H. Logemann, E.P. Ryan and S. Townley, Integral control of infinite-dimensional linear systems subject to input saturation. SIAM J. Control Optim. 36 (1998) 1940-1961. | MR | Zbl

[18] H. Logemann and S. Townley, Low gain control of uncertain regular linear systems. SIAM J. Control Optim. 35 (1997) 78-116. | MR | Zbl

[19] K.A. Morris, Justification of input/output methods for systems with unbounded control and observation. IEEE Trans. Automat. Control 44 (1999) 81-85. | MR | Zbl

[20] O.E. Ortiz, Stability of nonconservative hyperbolic systems and relativistic dissipative fluids. J. Math. Phys. 42 (2001) 1426-1442. | MR | Zbl

[21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl

[22] S.A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems. IEEE Trans. Automat. Control 27 (1985) 17-30. | MR | Zbl

[23] J. Prüss, On the spectrum of $C_0$-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847-857. | MR | Zbl

[24] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Amer. Math. Soc. 291 (1985) 167-187. | MR | Zbl

[25] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J. 24 (1974) 79-86. | MR | Zbl

[26] R. Rebarber, Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Trans. Automat. Control 38 (1993) 994-998. | MR | Zbl

[27] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR | Zbl

[28] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | MR | Zbl

[29] O.J. Staffans, Feedback representations of critical controls for well-posed linear systems. Int. J. Robust Nonlinear Control 8 (1998) 1189-1217. | MR | Zbl

[30] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. | MR | Zbl

[31] G. Weiss, Regular linear systems with feedback. Math. Control, Signals & Systems 7 (1994) 23-57. | MR | Zbl

[32] G. Weiss, Transfer functions of regular linear systems. Part I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR | Zbl

[33] G. Weiss and R.F. Curtain, Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Control 42 (1997) 4-21. | MR | Zbl

[34] C.Z. Xu and D.X. Feng, Linearization method to stability analysis for nonlinear hyperbolic systems. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 809-814. | MR | Zbl

[35] C.Z. Xu and J.P. Gauthier, Analyse et commande d'un échangeur thermique à contre-courant. RAIRO APII 25 (1991) 377-396. | Zbl

[36] C.Z. Xu, J.P. Gauthier and I. Kupka, Exponential stability of the heat exchanger equation, in Proc. of the European Control Conference. Groningen, The Netherlands (1993) 303-307.

[37] C.Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems. Int. J. Control 61 (1995) 33-45. | MR | Zbl

[38] C.Z. Xu, Exponential stability of a class of infinite dimensional time-varying linear systems, in Proc. of the International Conference on Control and Information. Hong Kong (1995).

[39] C.Z. Xu, Exact observability and exponential stability of infinite dimensional bilinear systems. Math. Control, Signals & Systems 9 (1996) 73-93. | MR | Zbl

[40] C.Z. Xu and G. Sallet, Proportional and Integral regulation of irrigation canal systems governed by the Saint-Venant equation, in 14th IFAC World Congress. Beijing, China (1999).

[41] C.Z. Xu and D.X. Feng, Symmetric hyperbolic systems and applications to exponential stability of heat exchangers and irrigation canals, in Proc. of the MTNS'2000. Perpignan (2000).

[42] B.E. Ydstie and A.A. Alonso, Process systems and passivity via the Clausius-Planck inequality. Systems Control Lett. 30 (1997) 253-264. | Zbl