作为结构化eigenvalue赋值的Marx生成器的设计
Design of Marx generators as a structured eigenvalue assignment
摘要：
我们考虑的设计问题，马克思发电机电气网络，脉冲功率发生器。该设计的工程规范是，电路满足适当的谐振条件，使得最初存储在多个存储电容器中的能量在有限时间内转移到单个负载电容器，该单个负载电容器然后可以存储总能量并传递脉冲。我们表明，组件的设计可以方便地铸造作为一个结构化的实特征值分配显着低于马克思电路的状态大小的尺寸。然后，我们评论的非平凡性，这种结构化的实特征值分配问题，并提出了两种可能的方法来确定其解决方案。第一种符号方法包括使用Gröbner基表示，它允许我们计算所有的（许多）解决方案。第二种方法是基于多项式约束的非凸最优化问题的凸化。我们表明，符号方法很容易提供解决方案的网络多达六个阶段，而数值方法可以达到七个和八个阶段。我们还评论的猜想，对于任何数量的阶段的问题有许多解决方案，这是一个必要的假设所提出的方法收敛。我们认为这个猜想的证明是一个有趣的挑战，在真正的代数几何领域的普遍兴趣。
英文摘要：
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.
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