Exponential stability of nonlinear differential repetitive processes with applications to iterative learning control

Abstract This paper studies exponential stability properties of a class of two-dimensional (2D) systems called differential repetitive processes (DRPs). Since a distinguishing feature of DRPs is that the problem domain is bounded in the “time” direction, the notion of stability to be evaluated does not require the nonlinear system defining a DRP to be stable in the typical sense. In particular, we study a notion of exponential stability along the discrete iteration dimension of the 2D dynamics, which requires the boundary data for the differential pass dynamics to converge to zero as the iterations evolve. Our main contribution is to show, under standard regularity assumptions, that exponential stability of a DRP is equivalent to that of its linearized dynamics. In turn, exponential stability of this linearization can be readily verified by a spectral radius condition. The application of this result to iterative learning control (ILC) is discussed. Theoretical findings are supported by a numerical simulation of an ILC algorithm.

[1]  Groege Weiss,et al.  Weakly /pstable linear operators are power stable , 1989 .

[2]  Ilya V. Kolmanovsky,et al.  Approximate optimal control of nonlinear systems with quadratic performance criteria , 2015, 2015 American Control Conference (ACC).

[3]  S. Foda,et al.  Control of the Metal Rolling Process: A Multidimensional System Approach , 1989, 1989 American Control Conference.

[4]  David H. Owens,et al.  Analysis and Control of Multipass Processes , 1982 .

[5]  Krzysztof Galkowski,et al.  Experimentally supported 2D systems based iterative learning control law design for error convergence and performance , 2010 .

[6]  Tryphon T. Georgiou,et al.  Input–Output Stability , 2010 .

[7]  M. Vidyasagar.  6. Input-Output Stability , 2002 .

[8]  Robert G. Landers,et al.  Repetitive Process Control of Laser Metal Deposition , 2014 .

[9]  B. Paden,et al.  Stability of learning control with disturbances and uncertain initial conditions , 1992 .

[10]  Emmanuel Moulay,et al.  Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability , 2013, IEEE Transactions on Automatic Control.

[11]  Roy M. Howard,et al.  Linear System Theory , 1992 .

[12]  J. Kurek,et al.  Iterative learning control synthesis based on 2-D system theory , 1993, IEEE Trans. Autom. Control..

[13]  Krzysztof Galkowski,et al.  Stability and stabilization of systems modeled by 2D nonlinear stochastic roesser models , 2011, The 2011 International Workshop on Multidimensional (nD) Systems.

[14]  Ilya V. Kolmanovsky,et al.  Optimal control of manifold filling during VDE mode transitions , 2013, 52nd IEEE Conference on Decision and Control.

[15]  Krzysztof Galkowski,et al.  Control Systems Theory and Applications for Linear Repetitive Processes - Recent Progress and Open Research Questions , 2007 .

[16]  Graziano Chesi,et al.  Necessary and Sufficient LMI Conditions for Stability and Performance Analysis of 2-D Mixed Continuous-Discrete-Time Systems , 2014, IEEE Transactions on Automatic Control.

[17]  R. Roesser.  A discrete state-space model for linear image processing , 1975 .

[18]  G. Marchesini,et al.  State-space realization theory of two-dimensional filters , 1976 .

[19]  K. Maciej Przyluski,et al.  The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space , 1980 .

[20]  Kira Barton,et al.  On linearized stability of differential repetitive processes and iterative learning control , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[21]  YangQuan Chen,et al.  Iterative Learning Control: A Tutorial and Big Picture View , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[22]  Jian-Xin Xu,et al.  A survey on iterative learning control for nonlinear systems , 2011, Int. J. Control.

[23]  Pedro Albertos,et al.  Iterative Identification and Control , 2002, Springer London.

[24]  A.G. Alleyne,et al.  A survey of iterative learning control , 2006, IEEE Control Systems.

[25]  P. Olver.  Nonlinear Systems , 2013 .

[26]  B. Paden,et al.  Nonlinear inversion-based output tracking , 1996, IEEE Trans. Autom. Control..

[27]  Kevin L. Moore,et al.  Iterative Learning Control: Brief Survey and Categorization , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[28]  Chen Hanfu,et al.  Iterative learning control for a class of nonlinear systems , 2008, 2008 27th Chinese Control Conference.

[29]  Alfonso Baños,et al.  Delay-dependent stability of reset systems , 2010, Autom..

[30]  Jun Liu,et al.  Lyapunov-Based Sufficient Conditions for Stability of Hybrid Systems With Memory , 2015, IEEE Transactions on Automatic Control.

[31]  Graziano Chesi,et al.  H ∞ and H 2 Norms of 2 D Mixed Continuous-Discrete-Time Systems via Rationally-Dependent Complex Lyapunov Functions , 2015 .

[32]  Krzysztof Galkowski,et al.  Stability and Stabilization of Differential Nonlinear Repetitive Processes with Applications , 2014 .

[33]  Judith B. Edwards.  Stability problems in the control of multipass processes , 1974 .

[34]  Ying Tan,et al.  On P-type iterative learning control for nonlinear systems without global Lipschitz continuity condition , 2015, 2015 American Control Conference (ACC).

[35]  Ettore Fornasini,et al.  Doubly-indexed dynamical systems: State-space models and structural properties , 1978, Mathematical systems theory.

[36]  Ye Sun,et al.  Stability of discontinuous retarded functional differential equations with applications , 2005, IEEE Transactions on Automatic Control.

[37]  Graziano Chesi,et al.  ${\cal H}_{\infty}$ and ${\cal H}_{2}$ Norms of 2-D Mixed Continuous-Discrete-Time Systems via Rationally-Dependent Complex Lyapunov Functions , 2015, IEEE Transactions on Automatic Control.

[38]  J. Warga.  Optimal control of differential and functional equations , 1972 .