Modeling and Iterative Learning Control of a Circular Deformable Mirror

Abstract An unconditionally stable finite difference scheme for systems whose dynamics are described by a fourth-order partial differential equation is developed using a regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization that was originally developed for second-order systems and it is used in this paper to develop a discrete in time and space model of a deformable mirror as a basis for control law design. As one example, the resulting model is used for iterative learning control law design and supporting numerical simulations are given.

[1]  Peter Steffen,et al.  Numerical iterative methods and repetitive processes , 2012, Multidimens. Syst. Signal Process..

[2]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[3]  Peter Steffen,et al.  Implicit discretization of linear partial differential equations and repetitive processes , 2009, 2009 International Workshop on Multidimensional (nD) Systems.

[4]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947 .

[5]  Krzysztof Galkowski,et al.  An unconditionally stable approximation of a circular flexible plate described by a fourth order partial differential equation , 2016, 2016 21st International Conference on Methods and Models in Automation and Robotics (MMAR).

[6]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[7]  Krzysztof Galkowski,et al.  An unconditionally stable finite difference scheme systems described by second order partial differential equations , 2015, 2015 IEEE 9th International Workshop on Multidimensional (nD) Systems (nDS).

[8]  Peter Steffen,et al.  Stability analysis for implicit second order finite difference schemes , 2011, The 2011 International Workshop on Multidimensional (nD) Systems.

[9]  Krzysztof Galkowski,et al.  Iterative learning control for spatio-temporal dynamics using Crank-Nicholson discretization , 2012, Multidimens. Syst. Signal Process..

[10]  Anton Kummert,et al.  An approach to iterative learning control for spatio-temporal dynamics using nD discrete linear systems models , 2011, Multidimens. Syst. Signal Process..