Design of Marx generators as a structured eigenvalue assignment

Abstract

We consider the design problem for a Marx generator electrical network, a pulsed power generator. The engineering specification of the design is that a suitable resonance condition is satisfied by the circuit so that the energy initially stored in a number of storage capacitors is transferred in finite time to a single load capacitor which can then store the total energy and deliver the pulse. We show that the components design can be conveniently cast as a structured real eigenvalue assignment with significantly lower dimension than the state size of the Marx circuit. Then we comment on the nontrivial nature of this structured real eigenvalue assignment problem and present two possible approaches to determine its solutions. A first symbolic approach consists in the use of Gröbner basis representations, which allows us to compute all the (finitely many) solutions. A second approach is based on convexification of a nonconvex optimization problem with polynomial constraints. We show that the symbolic method easily provides solutions for networks up to six stages while the numerical method can reach up to seven and eight stages. We also comment on the conjecture that for any number of stages the problem has finitely many solutions, which is a necessary assumption for the proposed methods to converge. We regard the proof of this conjecture as an interesting challenge of general interest in the real algebraic geometry field.

Optimization on linear matrix inequalities for polynomial systems control

“…The outer approximation result of Proposition 3.7 follows from the convergence proof of Lasserre's relaxation [28] and elementary duality arguments. Finally, the structured eigenvalue assignment problem of Sections 1.1 and 3.11 is comprehensively described in [14].…”

Dispersionless charge transfer on lumped-element transmission lines synthesized via resonant frequency assignment

“…If mirror symmetry is imposed on the line such that right and left sides of the line have identical components, the natural resonant modes of the line will then be either spatially even or spatially odd. The even harmonic frequencies may be assigned to the spatially even modes, and the odd harmonic frequencies may be assigned to the spatially odd modes by methods similar to those used for Marx networks [1], [2], [3], [4]. The resulting network can then be shown to have several important properties: (a) All voltages and currents will be periodic in time with the fundamental period.…”