SCIENCE CHINA Information Sciences, Volume 62, Issue 4: 042201(2019) https://doi.org/10.1007/s11432-018-9437-x

Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control

More info
  • ReceivedJan 4, 2018
  • AcceptedApr 2, 2018
  • PublishedFeb 27, 2019


We herein investigate the boundary input-to-statestability (ISS) of a class of coupled hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) systemswith respect to the presence of uncertainties and externaldisturbances. The boundary feedback control of the proportional typeacts on the ODE part and indirectly affects the hyperbolic PDEdynamics via the boundary input. Using the strict Lyapunovfunction, some sufficient conditions in terms of matrix inequalitiesare obtained for the boundary ISS of the closed-loop hyperbolicPDE-ODE systems. The feedback control laws are designed by combiningthe line search algorithm and polytopic embedding techniques.The effectiveness of the designed boundary control is assessed byapplying it to the system of interconnected continuous stirred tankreactor and a plug flow reactor through a numericalsimulation.


This work was supported by National Natural Science Foundation of China (Grant Nos. 61374076, 61533002) and Beijing Municipal Natural Science Foundation (Grant No. 1182001).


[1] Hasan A, Aamo O M, Krstic M. Boundary observer design for hyperbolic PDE-ODE cascade systems. Automatica, 2016, 68: 75-86 CrossRef Google Scholar

[2] Tang Y, Prieur C, Girard A. Stability analysis of a singularly perturbed coupled ODE-PDE system. In: Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, 2015. 4591--4596. Google Scholar

[3] Zhou H C, Guo B Z. Performance output tracking for one-dimensional wave equation subject to unmatched general disturbance and non-collocated control. Eur J Contr, 2017, 39: 39-52. Google Scholar

[4] Zhang L, Prieur C. Necessary and Sufficient Conditions on the Exponential Stability of Positive Hyperbolic Systems. IEEE Trans Automat Contr, 2017, 62: 3610-3617 CrossRef Google Scholar

[5] Zhang L, Prieur C, Qiao J. Local Exponential Stabilization of Semi-Linear Hyperbolic Systems by Means of a Boundary Feedback Control. IEEE Control Syst Lett, 2018, 2: 55-60 CrossRef Google Scholar

[6] Zhang L, Prieur C. Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control. Automatica, 2017, 86: 29-37 CrossRef Google Scholar

[7] Diagne M, Bekiaris-Liberis N, Krstic M. Time- and state-dependent input delay-compensated bang-bang control of a screw extruder for 3D printing. Int J Robust Nonlin, 2017, 27: 3727--3757. Google Scholar

[8] Alizadeh Moghadam A, Aksikas I, Dubljevic S. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica, 2013, 49: 526-533 CrossRef Google Scholar

[9] Krstic M, Smyshlyaev A. Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst Control Lett, 2008, 57: 750-758 CrossRef Google Scholar

[10] Krstic M. Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst Control Lett, 2009, 58: 372-377 CrossRef Google Scholar

[11] Li J, Liu Y. Stabilization of coupled pde-ode systems with spatially varying coefficient. J Syst Sci Complex, 2013, 26: 151-174 CrossRef Google Scholar

[12] Karafyllis I, Jiang Z P. Stability and Stabilization of Nonlinear Systems. London: Springer-Verlag, 2011. Google Scholar

[13] Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201 CrossRef Google Scholar

[14] Yang C, Cao J, Huang T. Guaranteed cost boundary control for cluster synchronization of complex spatio-temporal dynamical networks with community structure. Sci China Inf Sci, 2018, 61: 052203 CrossRef Google Scholar

[15] Ito H, Dashkovskiy S, Wirth F. Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems. Automatica, 2012, 48: 1197-1204 CrossRef Google Scholar

[16] Geiselhart R, Wirth F. Numerical construction of LISS Lyapunov functions under a small gain condition. In: Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, 2012. 25--30. Google Scholar

[17] Dashkovskiy S, Rüffer B S, Wirth F R. An ISS small gain theorem for general networks. Math Control Signals Syst, 2007, 19: 93-122 CrossRef Google Scholar

[18] Dashkovskiy S, Mironchenko A. Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst, 2013, 25: 1-35 CrossRef Google Scholar

[19] Karafyllis I, Krstic M. On the relation of delay equations to first-order hyperbolic partial differential equations. Esaim Control Optim Calc Var, 2013, 20: 894--923. Google Scholar

[20] Prieur C, Mazenc F. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math Control Signals Syst, 2012, 24: 111-134 CrossRef Google Scholar

[21] Tanwani A, Prieur C, Tarbouriech S. Input-to-state stabilization in $H^1$-norm for boundary controlled linear hyperbolic PDEs with application to quantized control. In: Proceedings of the 55th IEEE Conference on Decision and Control, Vegas, 2016. 3112--3117. Google Scholar

[22] Espitia N, Girard A, Marchand N, et al. Fluid-flow modeling and stability analysis of communication networks. In: Proceedings of the 20th IFAC World Congress, Toulouse, 2017. 4534--4539. Google Scholar

[23] Karafyllis I, Krstic M. ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances. SIAM J Control Optim, 2017, 55: 1716-1751 CrossRef Google Scholar

[24] Bastin G, Coron J M. Stability and boundary stabilization of 1-D hyperbolic systems. Springer Int Publishing, 2016 DOI 10.1007/978-3-319-32062-5. Google Scholar

[25] Aksikas I, Winkin J J, Dochain D. Optimal LQ-Feedback Regulation of a Nonisothermal Plug Flow Reactor Model by Spectral Factorization. IEEE Trans Automat Contr, 2007, 52: 1179-1193 CrossRef Google Scholar

[26] Shampine L F. Solving Hyperbolic PDEs in MATLAB. Appl Num Anal Comp Math, 2005, 2: 346-358 CrossRef Google Scholar

  • Figure 1

    (Color online) Coupled hyperbolic PFR-CSTR model.

  • Table 1   PFR-CSTR model parameters
    Process parameter Notation Value
    Kinetic constant $~k$ $225.225~\times~{10^6}$ ${{\rm~s}^{~-~1}}$
    Activation energy/universal gas constant ${E/R}$ 9758.3 K
    Steady inlet flow rate ${F_{\rm~in}}$ $0.0041~$ ${{\rm~m}^3}/{\rm~s}$
    Steady cooling rate $Q_{\rm~ss}$ $-$1.36 kJ/s
    Inlet reactant concentration $C_A^{\rm~in}$ 3 ${\rm~kmol/m}^3$
    Inlet temperature $T_{\rm~in}$ 429 K
    Heat of reaction for reactions 1 $\Delta~{H}$ $-$4200 kJ/kmol
    Volume of the CSTR $V_{c}$ 0.01 ${\rm~m}^3$
    Volume of the PFR $V_p$ 0.022 ${\rm~m}^3$
    Average fluid density $\rho$ 934.2 ${\rm~kg/m}^3$
    Specific heat $c_p$ 3.01 kJ/kgK
    Heat transfer coefficient $\beta$ 0.2 ${{\rm~s}^{~-~1}}$

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1       京公网安备11010102003388号