Distributed-parameter second-order sliding-mode control techniques for the uncertain heat and wave equations

Abstract In the present paper, preliminary results towards the generalization to the infinite-dimensional setting of some finite-dimensional second-order sliding mode control algorithms are illustrated. The main focus of the present paper is on the rejection of non-vanishing external disturbances. The “Twisting” and “SuperTwisting” second-order sliding-mode control algorithms (see (Levant (1993))) are properly generalized to address the state tracking problem for the wave and heat equations. Constructive proofs of stability are developed via Lyapunov functional technique, which leads to simple tuning rules for the controller parameters. Simulation results are discussed to verify the effectiveness of the proposed schemes.

[1]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[2]  Emilia Fridman,et al.  An LMI approach toH∞ boundary control of semilinear parabolic and hyperbolic systems , 2009 .

[3]  Yuri V. Orlov,et al.  Discontinuous unit feedback control of uncertain infinite-dimensional systems , 2000, IEEE Trans. Autom. Control..

[4]  Vadim I. Utkin,et al.  Sliding mode control in indefinite-dimensional systems , 1987, Autom..

[5]  Yury Orlov,et al.  Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control , 2004 .

[6]  Tingxiu Wang Stability in Abstract Functional Differential Equations. Part II. Applications , 1994 .

[7]  Miroslav Krstic,et al.  Adaptive control of PDES , 2007 .

[8]  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..

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

[10]  A. Levant Sliding order and sliding accuracy in sliding mode control , 1993 .

[11]  Jaime A. Moreno,et al.  A Lyapunov approach to second-order sliding mode controllers and observers , 2008, 2008 47th IEEE Conference on Decision and Control.

[12]  Emilia Fridman,et al.  An LMI approach to Hinfinity boundary control of semilinear parabolic and hyperbolic systems , 2009, Autom..

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

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

[15]  L. Levaggi Sliding modes in Banach spaces , 2002 .