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



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 ${{\rm~s}^{~-~1}}$