On the Power of Standard Information for Tractability for L2- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

Identification of Robot Joint Torsional Stiffness Based on the Amplitude of the Frequency Response of Asynchronous Data

The difficulty of adding external excitation and the asynchronous data collection from the industrial robot operation limited the online parameter identification of industrial robots. In this regard, this study proposes an identification method that only uses the amplitude of the frequency response function (FRF) of the system to identify robot joint torsional stiffness and dynamic parameters. The error criterion function shows that this method is feasible and comparable to applying the complete frequency response for identification. The Levenberg–Marquardt (L-M) algorithm is used to find the global optimal value of the error criterion function. In addition, an operational excitation method is proposed to excite the system. The speed profile is set as a triangle wave to excite the system using rectangular wave electromagnetic torques. The simulation results show that using the amplitude of the FRF to identify parameters applies to asynchronous data. The experiments on a single-degree-of-freedom articulated arm test bench show that the motion excitation method is effective, and both stiffness and inertia are identifiable.

Optimization of sliding mode control with PID surface for robot manipulator by Evolutionary Algorithms

In this paper, a sliding mode controller (SMC) with PID surface is designed for the trajectory tracking control of a robot manipulator using different optimization algorithms such as, Antlion Optimization Algorithm (ALO) Sine Cosine Algorithm (SCA) Grey Wolf Optimizer (GWO) and Whale Optimizer Algorithm (WOA). The aim of this work is to introduce a novel SMC-PID-ALO to control nonlinear systems, especially the position of two of the joints of a 2DOF robot manipulator. The basic idea is to determinate four optimal parameters (Kp, Ki, Kd and lamda) ensuring the best performance of a robot manipulator system, minimizing the integral time absolute error criterion (ITAE) and the integral time square error criterion (ISTE). The robot manipulator is modeled in Simulink and the control is implemented using the MATLAB environment. The obtained simulation results prove the robustness of ALO in comparison with other algorithms.

The Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion

In this paper, we conduct a convergence rate analysis of the augmented Lagrangian method with a practical relative error criterion designed in Eckstein and Silva [Mathematical Programming, 141, 319–348 (2013)] for convex nonlinear programming problems. We show that under a mild local error bound condition, this method admits locally a Q-linear rate of convergence. More importantly, we show that the modulus of the convergence rate is inversely proportional to the penalty parameter. That is, an asymptotically superlinear convergence is obtained if the penalty parameter used in the algorithm is increasing to infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if the penalty parameter is fixed but it is sufficiently large. Besides, as a byproduct, the convergence, as well as the convergence rate, of the distance from the primal sequence to the solution set of the problem is obtained.

Performance of the New Ridge Regression Parameters

A new approach is presented to find the ridge parameter k when the multiple regression model suffers from multicollinearity. This approach studied two cases, for the value k, scalar, and matrix. A comparison between this proposed ridge parameter and other well-known ridge parameters evaluated elsewhere, in terms of the mean squares error criterion, is given. Examples from several research papers are conducted to illustrate the optimality of this proposed ridge parameter k.