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：本文研究依赖于参数解析的矩阵值函数的伪谱横坐标的最小化问题。该问题源于具有优化参数的线性控制系统的鲁棒稳定性和瞬态行为考量。我们描述了一种子空间方法，用于处理矩阵值函数规模较大的情况。所提出的子空间方法通过将矩阵值函数限制在维数逐渐增加的小子空间上，求解一系列降阶问题。该方法具有理想特性，例如降阶问题的最小值全局收敛到原始问题的最小值，以及降阶问题极小化点的误差呈超线性衰减。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，其中特征值函数出现在内部最大化问题的约束中。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特殊处理，这涉及拉格朗日函数和对偶变量的运用。与最大化距离不稳定度或最小化$\mathcal{H}_\infty$范数相比，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值可以任意选取，而优化距离不稳定度和$\mathcal{H}_\infty$范数时通常需要选择能产生渐近稳定系统的初值。