Permanent-Magnet SLM Drive System Using AMRRSPNNB Control System with DGWO

Because permanent-magnet synchronous linear motors (SLM) still exhibit nonlinear friction, ending effects and time-varying dynamic uncertainties, better control performances cannot be achieved by using common linear controllers. We propose a backstepping approach with three adaptive laws and a beating function to control the motion of permanent-magnet SLM drive systems that enhance the robustness of the system. In order to reduce greater vibration in situations with uncertainty actions in the aforementioned control systems, we propose an adaptive modified recurrent Rogers–Szego polynomials neural network backstepping (AMRRSPNNB) control system with three adaptive laws and reimbursed controller with decorated gray wolf optimization (DGWO), in order to handle external bunched force uncertainty, including nonlinear friction, ending effects and time-varying dynamic uncertainties, as well as to reimburse the minimal rebuild error of the reckoned law. In accordance with the Lyapunov stability, online parameter training method of the modified recurrent Rogers–Szego polynomials neural network (MRRSPNN) can be derived by utilizing an adaptive law. Furthermore, to help reduce error and better obtain learning fulfillment, the DGWO algorithm was used to change the two learning rates in the weights of the MRRSPNN. Finally, the usefulness of the proposed control system is validated by tested results.

Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System

Due to a good ability of learning for nonlinear uncertainties, a mixed modified recurring Rogers-Szego polynomials neural network (MMRRSPNN) control with mended grey wolf optimization (MGWO) by using two linear adjusted factors is proposed to the six-phase induction motor (SIM) expelling continuously variable transmission (CVT) organized system for acquiring better control performance. The control system can execute MRRSPNN control with a fitted learning rule, and repay control with an evaluated rule. In the light of the Lyapunov stability theorem, the fitted learning rule in the MRRSPNN control can be derived, and the evaluated rule of the repay control can be originated. Besides, the MGWO by using two linear adjusted factors yields two changeable learning rates for two parameters to find two ideal values and to speed-up convergence of weights. Experimental results in comparisons with some control systems are demonstrated to confirm that the proposed control system can achieve better control performance.

Two expansion formulas involving the Rogers–Szegő polynomials with applications

Using two expansion formulas for the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials, we give new proofs of a variety of important classical formulas including Bailey's 6ψ6 summation, the Askey–Wilson integral and its extension. Furthermore, we give nontrivial extensions of the Andrews multiple version of the Rogers–Selberg identity, as well as the Sylvester identity.