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

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  • 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).


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  • 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 ${{\rm~s}^{~-~1}}$

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