Corrigendum: New Form of Kane's Equations of Motion for Constrained Systems

A correction to the previously published article "New Form of Kane's Equations of Motion for Constrained Systems" is presented. Misuse of the transformation matrix between time rates of change of the generalized coordinates and generalized speeds (sometimes called motion variables) resulted in a false conclusion concerning the symmetry of the generalized inertia matrix. The generalized inertia matrix (sometimes referred to as the mass matrix) is in fact symmetric and usually positive definite when one forms nonminimal Kane's equations for holonomic or simple nonholonomic systems, systems subject to nonlinear nonholonomic constraints, and holonomic or simple nonholonomic systems subject to impulsive constraints according to Refs. 1, 2, and 3, respectively. The mass matrix is of course symmetric when one forms minimal equations for holonomic or simple nonholonomic systems using Kane s method as set forth in Ref. 4

Constraint-based, Single-point Approximate Kinetic Energy Functionals

We present a substantial extension of our constraint-based approach for development of orbital-free (OF) kinetic-energy (KE) density functionals intended for the calculation of quantum-mechanical forces in multi-scale molecular dynamics simulations. Suitability for realistic system simulations requires that the OF-KE functional yield accurate forces on the nuclei yet be relatively simple. We therefore require that the functionals be based on DFT constraints, local, dependent upon a small number of parameters fitted to a training set of limited size, and applicable beyond the scope of the training set. Our previous "modified conjoint" generalized-gradient-type functionals were constrained to producing a positive-definite Pauli potential. Though distinctly better than several published GGA-type functionals in that they gave semi-quantitative agreement with Born-Oppenheimer forces from full Kohn-Sham results, those modified conjoint functionals suffer from unphysical singularities at the nuclei. Here we show how to remove such singularities by introducing higher-order density derivatives. We give a simple illustration of such a functional used for the dissociation energy as a function of bond length for selected molecules.Comment: 16 pages, 9 figures, 2 tables, submitted to Phys. Rev.

Closed-loop optimal experiment design: Solution via moment extension

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multiple-input multiple-output (MIMO) systems in closed-loop when the true system is in the model set. The optimization is performed jointly over the controller and the spectrum of the external excitation, which can be reparametrized as a joint spectral density matrix. We have shown in [18] that the optimal solution consists of first computing a finite set of generalized moments of this spectrum as the solution of a semi-definite program. A second step then consists of constructing a spectrum that matches this finite set of optimal moments and satisfies some constraints due to the particular closed-loop nature of the optimization problem. This problem can be seen as a moment extension problem under constraints. Here we first show that the so-called central extension always satisfies these constraints, leading to a constructive procedure for the optimal controller and excitation spectrum.We then show that, using this central extension, one can construct a broader set of parametrized optimal solutions that also satisfy the constraints; the additional degrees of freedom can then be used to achieve additional objectives. Finally, our new solution method for the MIMO case allows us to considerably simplify the proofs given in [18] for the single-input single-output case

Generalized ℓ2 synthesis

A framework for optimal controller design with generalized ℓ2 objectives is presented. The allowable disturbances are constrained to be in a pre-specified set; the design objective is to ensure that the resulting output errors do not belong to another pre-specified set. The solution takes the form of an affine matrix inequality (AMI), which is both a necessary and sufficient condition for the posed problem to have a solution. In the simplest case, the resulting optimization reduces to standard ℋ∞ synthesis