We consider the problem of locating a nearest descriptor system of prescribed reduced order to a descriptor system with large order with respect to the ${\mathcal L}_\infty$ norm. Widely employed approaches such as the balanced truncation and best Hankel norm approximation for this ${\mathcal L}_\infty$ model reduction problem are usually expensive and yield solutions that are not optimal, not even locally. We propose approaches based on the minimization of the ${\mathcal L}_\infty$ objective by means of smooth optimization techniques. As we illustrate, direct applications of smooth optimization techniques are not feasible, since the optimization techniques converge at best at a linear rate requiring too many evaluations of the costly ${\mathcal L}_\infty$-norm objective to be practical. We replace the original large-scale system with a system of smaller order that interpolates the original system at points on the imaginary axis, and minimize the ${\mathcal L}_\infty$ objective after this replacement. The smaller system is refined by interpolating at additional imaginary points determined based on the local minimizer of the ${\mathcal L}_\infty$ objective, and the optimization is repeated. We argue the framework converges at a quadratic rate under smoothness and nondegeneracy assumptions, and describe how asymptotic stability constraints on the reduced system sought can be incorporated into our approach. The numerical experiments on benchmark examples illustrate that the approach leads to locally optimal solutions to the ${\mathcal L}_\infty$ model reduction problem, and the convergence occurs quickly for descriptors systems of order a few ten thousands.

翻译：我们考虑在 ${\mathcal L}_\infty$ 范数意义下，寻找与大阶数描述系统最接近的指定降阶描述系统的问题。常用的方法如平衡截断和最佳 Hankel 范数逼近通常代价高昂，且所得解并非最优，甚至不是局部最优。我们提出基于通过光滑优化技术最小化 ${\mathcal L}_\infty$ 目标函数的方法。如我们所示，直接应用光滑优化技术并不可行，因为优化技术至多以线性速率收敛，需要过多计算昂贵的 ${\mathcal L}_\infty$ 范数目标函数，从而缺乏实用性。我们用一个在虚轴上插值原系统的较小阶数系统替换原大规模系统，并在替换后最小化 ${\mathcal L}_\infty$ 目标函数。该较小系统通过基于 ${\mathcal L}_\infty$ 目标函数局部最小化器确定的额外虚轴插值点进行细化，并重复优化过程。我们论证了该框架在光滑性和非退化性假设下以二次速率收敛，并描述了如何将对所求降阶系统的渐近稳定性约束融入我们的方法中。基准算例的数值实验表明，该方法能获得 ${\mathcal L}_\infty$ 模型降阶问题的局部最优解，且对于阶数达数万的描述系统，收敛速度很快。