We herein investigate the boundary input-to-state stability (ISS) of a class of coupled hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) systems with respect to the presence of uncertainties and external disturbances. The boundary feedback control of the proportional type acts on the ODE part and indirectly affects the hyperbolic PDE dynamics via the boundary input. Using the strict Lyapunov function, some sufficient conditions in terms of matrix inequalities are obtained for the boundary ISS of the closed-loop hyperbolic PDE-ODE systems. The feedback control laws are designed by combining the line search algorithm and polytopic embedding techniques. The effectiveness of the designed boundary control is assessed by applying it to the system of interconnected continuous stirred tank reactor and a plug flow reactor through a numerical simulation.
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Hasan A, Aamo O M, Krstic M. Boundary observer design for hyperbolic PDE-ODE cascade systems. Automatica, 2016, 68: 75–86
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
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
Zhang L G, Prieur C. Necessary and sufficient conditions on the exponential stability of positive hyperbolic systems. IEEE Trans Automat Contr, 2017, 62: 3610–3617
Zhang L G, Prieur C, Qiao J F. Local exponential stabilization of semi-linear hyperbolic systems by means of a boundary feedback control. IEEE Control Syst Lett, 2018, 2: 55–60
Zhang L G, Prieur C. Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control. Automatica, 2017, 86: 29–37
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
Moghadam A A, Aksikas I, Dubljevic S, et al. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica, 2013, 49: 526–533
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
Krstic M. Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst Control Lett, 2009, 58: 372–377
Li J, Liu Y G. Stabilization of coupled pde-ode systems with spatially varying coefficient. J Syst Sci Complex, 2013, 26: 151–174
Karafyllis I, Jiang Z P. Stability and Stabilization of Nonlinear Systems. London: Springer-Verlag, 2011
Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201
Yang C D, Cao J D, Huang T W, et al. Guaranteed cost boundary control for cluster synchronization of complex spatio-temporal dynamical networks with community structure. Sci China Inf Sci, 2018, 61: 052203
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
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
Dashkovskiy S, Rüffer B S, Wirth F R. An ISS small gain theorem for general networks. Math Control Signals Syst, 2007, 19: 93–122
Dashkovskiy S, Mironchenko A. Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst, 2013, 25: 1–35
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
Prieur C, Mazenc F. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math Control Signals Syst, 2012, 24: 111–134
Tanwani A, Prieur C, Tarbouriech S. Input-to-state stabilization in H1-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
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
Karafyllis I, Krstic M. ISS in different norms for 1-D parabolic PDES with boundary disturbances. SIAM J Control Optim, 2017, 55: 1716–1751
Bastin G, Coron J M. Stability and boundary stabilization of 1-D hyperbolic systems. In: Progress in Nonlinear Differential Equations and Their Applications. Berlin: Springer, 2016
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
Shampine L F. Solving hyperbolic PDEs in MATLAB. Appl Num Anal Comp Math, 2005, 2: 346–358
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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Zhang, L., Hao, J. & Qiao, J. Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control. Sci. China Inf. Sci. 62, 42201 (2019). https://doi.org/10.1007/s11432-018-9437-x
- hyperbolic PDE-ODE systems
- input-to-state stability
- boundary control
- Lyapunov function
- PFR-CSTR models