Differential equations generally define the relationship between two items by relating functions to their derivatives, where the function represents the physical quantities while the derivative represents the rate of change. Thus, the equations are widely used to understand various problems across all disciplines such as dynamical processes. Unlike linear differential equations, nonlinear equations are far much complex and time-consuming to solve. However, they are the most appropriate for describing various systems hence they have attracted significant attention of researchers who are looking for alternative easier and efficient solution methods.
Among the available methods for analysis and solving nonlinear differential equations, integral inequalities are widely preferred owing to their simple and efficient nature. Consequently, they enhance the stability conditions of uncertain dynamical systems like the engineering controls. Unfortunately, the unknown nonlinearities in the uncertain nonlinear control dynamic systems have not been fully explored. This has led to less understanding of such systems that have further resulted in their failure. To this note, researchers have been looking for alternative methods to meet the requirements of the control theory as well as take into consideration the unknown nonlinearities and have identified a new integral inequality combining an adaptive approach as a promising solution.
In a recent research paper published in International Journal of Robust and Nonlinear Contro. Professor Hansheng Wu at Prefectural University of Hiroshima developed a new integral inequality technique combined with an adaptive approach. He resolved the adaptive robust control problems especially for complex nonlinear systems and to deal with the nonlinearities in the nonlinear dynamical systems. Furthermore, he hoped that the new method would be suitable for engineering control theory.
This method allows the user to approximate the unknown nonlinearities to efficiently synthesis stabilizing control problems. Consequently, the resulting feedback control systems can be treated as linear systems due to their simple structures in relations to the time-varying control gains. The applicability of the developed method to control theory was validated by considering the stabilization of uncertain feedback nonlinear dynamical systems with unknown and time delay dead-zone nonlinearities as inputs.
Professor Hansheng Wu inequality method can be efficiently integrated with the adaptive approach to develop new design methods for synthesis of complex uncertain nonlinear systems and to deal with unknown nonlinearities in the systems. However, the author elaborates that the new inequality is not only applicable to the engineering control problems but also in the synthesis of dynamical systems in other fields. Therefore, it forms the fundamental basics and guidelines in the crackdown of complex uncertain nonlinear problems. As such, it will advance the future work which will incorporate the new inequality in conjunction with other control methods to obtain more efficient methods for handling a big class of uncertain nonlinear dynamical problems.
Wu, H. (2018). A new integral inequality and its applications to robust control problems of uncertain nonlinear systems. International Journal of Robust and Nonlinear Control.Go To International Journal of Robust and Nonlinear Control