This work concerns the minimization of the pseudospectral abscissa of a matrix-valued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control system that has optimization parameters. We describe a subspace procedure to cope with the setting when the matrix-valued function is of large size. The proposed subspace procedure solves a sequence of reduced problems obtained by restricting the matrix-valued function to small subspaces, whose dimensions increase gradually. It possesses desirable features such as a superlinear convergence exhibited by the decay in the errors of the minimizers of the reduced problems. In mathematical terms, the problem we consider is a large-scale nonconvex minimax eigenvalue optimization problem such that the eigenvalue function appears in the constraint of the inner maximization problem. Devising and analyzing a subspace framework for the minimax eigenvalue optimization problem at hand with the eigenvalue function in the constraint require special treatment that makes use of a Lagrangian and dual variables. There are notable advantages in minimizing the pseudospectral abscissa over maximizing the distance to instability or minimizing the $\mathcal{H}_\infty$ norm; the optimized pseudospectral abscissa provides quantitative information about the worst-case transient growth, and the initial guesses for the parameter values to optimize the pseudospectral abscissa can be arbitrary, unlike the case to optimize the distance to instability and $\mathcal{H}_\infty$ norm that would normally require initial guesses yielding asymptotically stable systems.

翻译：本文研究依赖于解析参数的矩阵值函数的伪谱横坐标最小化问题。该问题源于具有优化参数的线性控制系统的鲁棒稳定性与暂态行为考量。针对矩阵值函数规模较大的情形，我们提出一种子空间处理框架。该子空间框架通过将矩阵值函数限制在维度逐渐增大的小子空间上，求解一系列降维子问题。该方法具有超线性收敛等优良特性，表现为降维问题极小化点误差的衰减。从数学角度，我们所考虑的问题是一个大规模非凸极小极大特征值优化问题，其中特征值函数作为内层极大化问题的约束出现。针对当前这种特征值函数位于约束内的极小极大特征值优化问题，设计与分析子空间框架需要特殊处理，需借助拉格朗日函数及对偶变量。相较于最大化不稳定距离或最小化$\mathcal{H}_\infty$范数，最小化伪谱横坐标具有显著优势：优化后的伪谱横坐标为最坏情况暂态增长提供量化信息；且优化伪谱横坐标时参数初始猜测可任意选取，而优化不稳定距离与$\mathcal{H}_\infty$范数通常需要能产生渐近稳定系统的初始猜测。