Sliding mode control for N‐coupled reaction‐diffusion PDEs with boundary input disturbances

Funding information National Natural Science Foundation of China, Grant/Award Number: 61673061 Summary This paper develops the sliding mode control (SMC) design for N-coupled reaction-diffusion parabolic PDEs with boundary input disturbances. In order to reject the disturbances, the backstepping-based boundary SMC law is constructed to steer the system trajectory to a suitable sliding surface and then maintain sliding motion on the surface thereafter, resulting in the exponential convergence to the zero equilibrium state. The well-posedness of the closed-loop system is established based on a detailed spectral analysis and Riesz basis generation. Finally, a simulation example is provided to illustrate the effectiveness of the SMC design.

[1]  Yury Orlov,et al.  Continuous state-feedback tracking of an uncertain heat diffusion process , 2010, Syst. Control. Lett..

[2]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[3]  Ya-Ping Guo,et al.  Output regulation of anti-stable coupled wave equations via the backstepping technique , 2017 .

[4]  Bao-Zhu Guo,et al.  Sliding mode control and active disturbance rejection control to the stabilization of one‐dimensional Schrödinger equation subject to boundary control matched disturbance , 2014 .

[5]  C. Pao.  Coexistence and stability of a competition—diffusion system in population dynamics , 1981 .

[6]  Christiane Tretter,et al.  Boundary Eigenvalue Problems for Differential Equations Nη= λPη with λ-Polynomial Boundary Conditions , 2001 .

[7]  M. Krstić,et al.  Boundary Control of PDEs , 2008 .

[8]  Bao-Zhu Guo,et al.  The Active Disturbance Rejection Control to Stabilization for Multi-Dimensional Wave Equation With Boundary Control Matched Disturbance , 2015, IEEE Transactions on Automatic Control.

[9]  Beibei Ren,et al.  Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance , 2015, Autom..

[10]  Driss Boutat,et al.  Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions , 2016, Syst. Control. Lett..

[11]  Stevan Dubljevic,et al.  The state feedback servo-regulator for countercurrent heat-exchanger system modelled by system of hyperbolic PDEs , 2016, Eur. J. Control.

[12]  Miroslav Krstic,et al.  Boundary Control of Coupled Reaction-Advection-Diffusion Systems With Spatially-Varying Coefficients , 2016, IEEE Transactions on Automatic Control.

[13]  Shuzhi Sam Ge,et al.  Boundary Output-Feedback Stabilization of a Timoshenko Beam Using Disturbance Observer , 2013, IEEE Transactions on Industrial Electronics.

[14]  Joachim Deutscher,et al.  A backstepping approach to the output regulation of boundary controlled parabolic PDEs , 2015, Autom..

[15]  Lei Guo,et al.  Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback , 2016 .

[16]  Yury Orlov,et al.  On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters , 2014, 53rd IEEE Conference on Decision and Control.

[17]  Miroslav Krstic,et al.  Boundary observer design for coupled reaction-diffusion systems with spatially-varying reaction , 2017, 2017 American Control Conference (ACC).

[18]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[19]  Y. Orlov.  Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions , 2008 .

[20]  H. Berestycki,et al.  Quenching and Propagation in KPP Reaction-Diffusion Equations with a Heat Loss , 2005 .

[21]  Christopher Edwards,et al.  Sliding Mode Control and Observation , 2013 .

[22]  Bao-Zhu Guo,et al.  Sliding Mode and Active Disturbance Rejection Control to Stabilization of One-Dimensional Anti-Stable Wave Equations Subject to Disturbance in Boundary Input , 2012, IEEE Transactions on Automatic Control.

[23]  V. Utkin,et al.  Unit sliding mode control in infinite dimensional systems , 1998 .

[24]  Feng-Fei Jin,et al.  The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance , 2013, Autom..

[25]  Jun-Jun Liu,et al.  Boundary stabilization of a cascade of ODE‐wave systems subject to boundary control matched disturbance , 2017 .

[26]  Yury Orlov,et al.  Boundary control of coupled reaction-diffusion processes with constant parameters , 2015, Autom..

[27]  Emilia Fridman,et al.  Sliding mode control of Schrödinger equation-ODE in the presence of unmatched disturbances , 2016, Syst. Control. Lett..

[28]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[29]  Denis Dochain,et al.  Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor , 2002, IEEE Trans. Autom. Control..

[30]  C. Cosner,et al.  Stable Coexistence States in the Volterra–Lotka Competition Model with Diffusion , 1984 .

[31]  Bao-Zhu Guo,et al.  BOUNDARY FEEDBACK STABILIZATION OF A THREE-LAYER SANDWICH BEAM: RIESZ BASIS APPROACH ∗ , 2006 .

[32]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

[33]  Yury Orlov,et al.  Output Feedback Stabilization of Coupled Reaction-Diffusion Processes with Constant Parameters , 2017, SIAM J. Control. Optim..

[34]  Wu-Chung Su,et al.  Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties , 2011, Autom..

[35]  Jun-Min Wang,et al.  Sliding Mode Control of the Orr-Sommerfeld Equation Cascaded by Both the Squire Equation and ODE in the Presence of Boundary Disturbances , 2018, SIAM J. Control. Optim..

[36]  Wei Guo,et al.  Performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance , 2016, Autom..

[37]  Yury Orlov,et al.  On General Properties of Eigenvalues and Eigenfunctions of a Sturm–Liouville Operator: Comments on “ISS With Respect to Boundary Disturbances for 1-D Parabolic PDEs” , 2017, IEEE Transactions on Automatic Control.

[38]  Yury Orlov,et al.  Tracking Control of the Uncertain Heat and Wave Equation via Power-Fractional and Sliding-Mode Techniques , 2011, SIAM J. Control. Optim..