MODIFICATION OF PSB QUASI-NEWTON UPDATE AND ITS GLOBAL CONVERGENCE FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

Nonlinear problems mostly emanate from the work of engineers, physicists, mathematicians and many other scientists. A variety of iterative methods have been developed for solving large scale nonlinear systems of equations. A prominent method for solving such equations is the classical Newton’s method, but it has many shortcomings that include computing Jacobian inverse that sometimes fails. To overcome such drawbacks, an approximation with derivative free line is used on an existing method. The method uses PSB (Powell-Symmetric Broyden) update. The efficiency of the proposed method has been improved in terms of number of iteration and CPU time, hence the aim of this research. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

Symmetric and Positive Definite Broyden Update for Unconstrained Optimization

Broyden update is one of the one-rank updates which solves the unconstrained optimization problem but this update does not guarantee the positive definite and the symmetric property of Hessian matrix. In this paper the guarantee of positive definite and symmetric property for the Hessian matrix will be established by updating the vector which represents the difference between the next gradient and the current gradient of the objective function assumed to be twice continuous and differentiable .Numerical results are reported to compare the proposed method with the Broyden method under standard problems.

Programming-by-Demonstration and Adaptation of Robot Skills by Fuzzy Time Modeling

Robot skills are motion or grasping primitives from which a complicated robot task consists. Skills can be directly learned and recognized by a technique named programming-by-demonstration. A human operator demonstrates a set of reference skills where his motions are recorded by a data-capturing system and modeled via fuzzy clustering and a Takagi–Sugeno modeling technique. The skill models use time instants as input and operator actions as outputs. In the recognition phase, the robot identifies the skill shown by the operator in a novel test demonstration. Finally, using the corresponding reference skill model the robot executes the recognized skill. Skill models can be updated online where drastic differences between learned and real world conditions are eliminated using the Broyden update formula. This method was extended for fuzzy models especially for time cluster models.