Projection Methods for Balanced Model Reduction

这个项目提出了通过对某些系统语法的低阶逼近来发展形式为dx/dt=Ax+Bu，y=Cx的大规模线性时不变(LTI)系统的简化模型。这种形式的复杂系统既存在于电路仿真中，也存在于某些依赖时间的偏微分方程控制系统的空间离散化中。我们将研究一种新的模型降阶方法，以解决现有降维技术中的一些基本困难。这些问题对于健壮和广泛应用的软件的潜在开发至关重要。我们打算开发一种计算方法，它将：(1)对简化系统的响应误差提供严格的界限；(2)自然地保持系统的基本属性，如稳定性；(3)一旦指定了所需的误差容限，就完全自动。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重新启动方法，迭代地产生对可控性、可观性和交叉文法的低阶近似，这些低阶近似是降阶到完全文法的近乎最佳的近似。所提出的方法将提供大型状态空间系统的平衡部分实现。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，对系统的语法进行非常低阶的近似是可能的，并且将产生准确的低阶简化模型。我们期望在电路模拟和偏微分方程组的控制中有重要的应用，特别是受边界控制的抛物型方程。这类问题在大规模降维方面有很大的潜力。这将使复杂系统的实时控制器的设计成为可能。

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.