标题： 第一和第二阶最优$\mathcal{H}_2$模型降阶对于线性连续时间系统 显示英文标题

标题： First and Second Order Optimal $\mathcal{H}_2$ Model Reduction for Linear Continuous-Time Systems

摘要： 在本文中，我们研究单输入单输出（SISO）连续时间线性时不变（LTI）系统的最优$\mathcal{H}_2$模型降阶问题。提出了一种半定松弛（SDR）方法，以确定全局最优的插值点，通过基于Krylov投影的方法提供一种有效计算降阶模型的方式。与迭代方法相比，我们使用可控性Gramian和矩匹配条件，通过引入一个上界$\gamma$来将模型降阶问题转化为凸优化，以最小化模型降阶误差系统的$\mathcal{H}_2$范数。我们还证明了对于一阶降阶模型，该松弛是精确的，并通过例子表明对于二阶降阶模型也是精确的。我们在例子中比较了我们提出的方法与其他迭代方法和选择方法的性能。重要的是，我们的方法还提供了一种验证已知局部收敛方法全局最优性的手段。 显示英文摘要

摘要： In this paper, we investigate the optimal $\mathcal{H}_2$ model reduction problem for single-input single-output (SISO) continuous-time linear time-invariant (LTI) systems. A semi-definite relaxation (SDR) approach is proposed to determine globally optimal interpolation points, providing an effective way to compute the reduced-order models via Krylov projection-based methods. In contrast to iterative approaches, we use the controllability Gramian and the moment-matching conditions to recast the model reduction problem as a convex optimization by introducing an upper bound $\gamma$ to minimize the $\mathcal{H}_2$ norm of the model reduction error system. We also prove that the relaxation is exact for first order reduced models and demonstrate, through examples, that it is exact for second order reduced models. We compare the performance of our proposed method with other iterative approaches and shift-selection methods on examples. Importantly, our approach also provides a means to verify the global optimality of known locally convergent methods.