Projection Methods for Balanced Model Reduction

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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.