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 the global convergence of the minimal values of the reduced problems to the minimal value of the original problem, and 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：本文研究解析依赖于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瞬态行为考量。我们提出一种子空间方法以应对矩阵值函数规模庞大的情形。所提出的子空间方法通过将矩阵值函数限制在逐渐增大的小维子空间上，求解一系列简化问题。该方法具有理想特性，包括简化问题的最小值全局收敛至原问题的最小值，以及简化问题极小点误差呈超线性衰减的收敛表现。从数学角度而言，我们考虑的问题是一个大规模非凸极小极大特征值优化问题，其特征值函数出现在内层极大化问题的约束中。针对此类约束中包含特征值函数的极小极大特征值优化问题，设计并分析子空间框架需要借助拉格朗日函数与对偶变量进行特殊处理。相较于最大化与不稳定性的距离或最小化$\mathcal{H}_\infty$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能提供最坏情况瞬态增长的定量信息，且伪谱横坐标优化的参数初始猜测可任意选取——这与优化距离不稳定性和$\mathcal{H}_\infty$范数时通常需要能生成渐近稳定系统的初始猜测形成鲜明对比。