Multivariable Steffensen’s Accelerator in Adaptive Sliding Mode Control

Abstract

Due to their internal mathematical structure Sliding Mode Controllers (SMC) are robust against the imprecisions of the dynamic model they use for the calculation of the necessary control forces as well as against a priori not known external disturbances. However, this robustness has natural limits. To widen the range of stable operation various particular constructions have been developed and tested in the recent decades. Key element of stability is the more or less precise realization of the invented kinematic design. To guarantee that natural possibility seems to be the integration of SMC technique with adaptive ones that also aim at the precise realization of the kinematic designs in the possession of some available, normally imprecise dynamic model. Due to its simple mathematical structure the Fixed Point Iteration (FPI)-based adaptive control seems to be a good candidate for this purpose. Like the SMC control, FPI-based adaptive controllers also suffer from certain limitations due to which they cannot be considered as some “panacea” against modeling errors. They work on the basis of iterative sequences generated by a contractive map in a Banach space, and their speed of convergence is limited by the fact that during one digital control step only one step in this iteration can be executed. Beside decreasing the cycle time of the digital control this convergence can be speeded up by a witty trick suggested by Steffensen in 1936 for single variable functions. In this paper it is shown how Steffensen’s convergence accelerator can be generalized for multiple variable Banach sequences. Following that simulation results are presented that testify that the simplest SMC and the simplest variant of the FPI-based adaptive controller using Steffensen’s generalized accelerator can be successfully integrated in the precise control of a strongly nonlinear benchmark system consisting of two nonlinearly coupled generalized van der Pol oscillators. Robustness of the integrated controller is also tested against the modification of the control parameters. Furthermore, it is shown that the sensitivity of the control to the measurement noise efficiently can be reduced by reducing the cycle time of the digital control.