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

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.