Projection Methods for Balanced Model Reduction

该项目提出了大规模线性时不变（LTI）系统的简化模型的发展形式 dx/dt = Ax + Bu，y = Cx通过某些系统Grammians的低秩近似。这种形式的复杂系统出现在电路仿真中，它们也出现在某些时间依赖的PDE控制系统的空间离散化中。一种新的模型降维方法将被研究，这将解决现有降维技术的一些基本困难。这些问题是潜在的强大和广泛适用的软件开发的核心。我们打算开发一种计算方法，它将：（1）提供严格的限制减少系统的响应误差;（2）自然地保持基本的系统特性，如稳定性;（3）是完全自动的，一旦指定所需的误差容限。我们建议研究Krylov型和非Krylov型的子空间投影方法。 我们的主要目标将是开发隐式restartingmethods，将迭代产生低秩逼近的可控性，可观性，和交叉Grammians这是近最好的近似降低秩的完整Grammians.The方法提出将提供平衡的大状态空间系统的部分实现。我们对模型问题的计算经验表明，许多系统的Hankel奇异值衰减得非常快。因此，非常低的秩近似系统Grammians是可能的，准确的低阶简化模型将导致。我们希望在电路模拟和偏微分方程，特别是边界控制抛物方程控制系统中找到重要的应用。有很大的潜力，在这样的问题广泛的降维。这将使复杂系统的实时控制器的设计。

This project proposes the development of reduced models for large-scalelinear time invariant (LTI) systems of the form dx/dt = Ax + Bu, y = Cxthrough low rank approximation of certain system Grammians. Complex systems of this form arise in circuit simulation; they also arise through spatial discretization of certain time dependent PDE control systems.A new approach to model reduction will be studied, that will address some fundamental difficulties with existing dimension reduction techniques. These issues are central to the potential development of robust and widely applicable software. We intend to develop a computational methodology that will:(1) Provide rigorous bounds on the response error of the reduced system; (2) Naturally preserve fundamental system properties such as stability;(3) Be fully automatic once a desired error tolerance is specified. We propose to investigate subspace projection methods of Krylov and non-Krylov type. Our primary goal will be to develop implicit restartingmethods that will iteratively produce low rank approximations to controllability-, observability-, and cross-Grammians which are near best approximations of reduced rank to the full Grammians.The methods proposed will provide balanced partial realizations of large state space systems. Our computational experience with model problems indicates that the Hankel singular values of many systems decay extremely rapidly. Hence very low rank approximations to the system Grammians are possible and accurate low order reduced models will result. We expect to find important applications in circuit simulation and in the control of systems governed by partial differential equations,particularly parabolic equations subject to boundary control. There is great potential in such problems for extensive dimension reduction. This will enable the design of real-time controllers for complex systems.