We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured perturbations), and also a closely related algorithm for computing the structured distance of a given passive system to non-passivity. Both problems are addressed by solving eigenvalue optimization problems for Hamiltonian matrices that are constructed from perturbed system matrices. The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow. They optimize over the perturbation size in the outer iteration, which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis. For large systems, we propose a variant of the algorithm that takes advantage of the inherent low-rank structure of the problem. Numerical experiments illustrate the behavior of the proposed algorithms.
翻译:我们提出并研究一种算法,用于计算一个最近的被动系统,以计算一个最接近的非被动线性线性时变系统(在选择“最远”的衡量定义方面有很大的自由,这种定义可能限于结构的扰动),以及一个密切相关的算法,用以计算一个特定被动系统的结构距离至非被动性。解决这两个问题的方法是解决从扰动系统基质构造的汉密尔顿基矩阵的机能价值优化问题。提议的算法是两个层次的方法,它优化了汉密尔顿的最小正值实际部分相对于内部迭代中固定大小的扰动,使用一个有限的梯度流。它们优化了外部迭代的扰动尺寸,这在典型情况下显示,在接近想象中轴的简单精度值的缺陷的煤凝固方面,这表现为四面形汇合。对于大型系统,我们提出了一种替代算法,利用问题内在的低级结构。数量实验说明了拟议算法的行为。