Abstract:
Balanced truncation (BT) is one of the oldest, most popular model reduction techniques that offers a way to construct stable, balanced, reduced order matrices that preserve the original system’s dominant Hankel singular values. It also provides a priori H∞ bounds for the reduced order model. BT has been refined over the years and has been extended to mild nonlinear systems as well. Data-driven balancing or quadrature-based balanced truncation is one variant of BT that aims to approximate the Hankel singular values of the original system in a purely non-intrusive fashion using samples of the system kernels and their derivatives. In this thesis, we first define balanced truncation for a new class of systems - bilinear systems with quadratic output (BQO). The gramians (and kernels) are first defined for these systems, which are vital for the BT algorithm. Finally, we introduce truncated Gramians and use them to implement a quick, approximate balanced truncation algorithm. The second section of this thesis involves extending the data-driven balancing method (Quad BT) to two nonlinear classes of systems - linear systems with quadratic output (LQO) and quadratic-bilinear systems (QB). The recipe for Quad BT remains the same as that for the linear case. We construct a quadrature-based approximation of the gramians (or trun cated gramians in the QB case) and replace the intrusive terms in BT with these quadrature based approximations. Finally, we show that these approximate matrices can be expressed as data matrices with entries corresponding to samples of the system kernels or derivatives of the kernels. In addition to the theory, we propose schemes to improve the computational viability of these methods. The theory developed in this thesis has been put to the test with numerical experiments on benchmark datasets.
