The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

A Nonlinear Optimal Control Approach for Tracked Mobile Robots

The article proposes a nonlinear optimal(H-infinity)control approach for the model of a tracked robotic vehicle.The kinematic model of such a tracked vehicle takes into account slippage effects due to the contact of the tracks with the ground.To solve the related control problem,the dynamic model of the vehicle undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm.The linearization process relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the vehicle.For the approximately linearized description of the tracked vehicle a stabilizing H-infinity feedback controller is designed.To compute the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method.The stability properties of the control scheme are proven through Lyapunov analysis.It is also demonstrated that the control method retains the advantages of linear optimal control,that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

A Nonlinear Optimal Control Method for Attitude Stabilization of Micro-Satellites

Attitude control and stabilization of micro-satellites is a nontrivial problem due to the highly nonlinear and multivariable structure of the satellites'state-space model.In this paper,a novel nonlinear optimal(H-infinity)control approach is developed for this control problem.The dynamic model of the satellite's attitude dynamics undergoesfirst approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm.The linearization process relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the satellite's attitude dynamics.For the approximately linearized description of the satellite's attitude a stabilizing H-infinity feedback controller is designed.To compute the controller's feedback gains,an algebraic Riccati equation is solved at each time-step of the control method.The stability properties of the control scheme are proven through Lyapunov analysis.It is also demonstrated that the control method retains the advantages of linear optimal control that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.