基于矩匹配和optimization的线性网络系统H2模型降阶
H2 model reduction of linear network systems by moment matching and optimization
摘要：
本文研究线性网络系统的模型降阶问题。我们的目标是计算具有相同拓扑结构和最佳w.r.t.的网络的降阶稳定近似。H2模误差逼近我们的方法是基于时域矩匹配框架，在那里我们优化的参数化降阶模型匹配的一组时刻在任意插值点的家庭。低阶模型的参数化是根据自由参数和插值点来进行的。对于这个家庭的参数化模型，我们制定了一个基于优化的模型降阶问题的H2范数的误差近似为目标函数，而保存的一些结构和物理性质产生的约束。这个问题是非凸的，我们把它写在Gramians的最小实现的错误系统。我们提出了两个解决这个问题的方法。第一个解决方案假设误差系统允许块对角可观Gramian，允许一个简单的凸重新制定为半定规划，但在一些性能损失的代价。我们还得到了保证Gramian块对角化的充分条件。第二种解决方案采用了梯度投影法的光滑重新产生（局部）最优插值点和自由参数。电力网络上的方法的潜力示出。
英文摘要：
In this paper we study the problem of model reduction of linear network systems. We aim at computing a reduced order stable approximation of the network with the same topology and optimal w.r.t. H2 norm error approximation. Our approach is based on time-domain moment matching framework, where we optimize over families of parameterized reduced order models matching a set of moments at arbitrary interpolation points. The parameterization of the low order models is in terms of the free parameters and of the interpolation points. For this family of parameterized models we formulate an optimization-based model reduction problem with the H2 norm of error approximation as objective function while the preservation of some structural and physical properties yields the constraints. This problem is nonconvex and we write it in terms of the Gramians of a minimal realization of the error system. We propose two solutions for this problem. The first solution assumes that the error system admits a block diagonal observability Gramian, allowing for a simple convex reformulation as semidefinite programming, but at the cost of some performance loss. We also derive sufficient conditions to guarantee block diagonalization of the Gramian. The second solution employs a gradient projection method for a smooth reformulation yielding (locally) optimal interpolation points and free parameters. The potential of the methods is illustrated on a power network.
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