A hybrid algebraic equations of motion and neural estimator to implement the direct control method

The study of dynamic systems without resorting to or any knowledge of differential equations is known as the “direct method”. In this method, algebraic equations of motion characterize the system dynamics. The algebraic optimal control laws can be derived in an explicit form for general nonlinear time-varying and time-invariant systems by minimizing an algebraic performance measure. The essence of the approach is based on using assumed-time-modes expansions of generalized coordinates and inputs in conjunction with the variational work-energy principles that govern the physical system. However to implement these control laws an algebraic state estimator must be designed. The development of such an estimator is incorporated by utilizing neural networks within a hybrid algebraic equations of motion for general nonlinear systems. To proof of concept, computer simulations are validated on linear systems under deterministic, noisy and modeling uncertainty cases. As modeling uncertainty is concerned, both parameter uncertainty and model truncation have been considered