Dimension Reduction in Structured Dynamical Systems: Optimal-$mathcal{H}_2$ Approximation, Data-Driven Balancing, and Real-Time Monitoring}

Abstract

This dissertation considers a variety of problems pertaining to the model-order reduction, data-driven reduced-order modeling, and real-time monitoring of large-scale and structured dynamical systems. In the first part, balancing-based methods for system-theoretic model reduction of linear time-invariant systems are considered. We generalize conditions for which the balanced truncation mathcalHinfty error bound is known to hold with equality. Specifically, we show that the bound is tight for single-input, single-output systems for which the truncated part of the model is a mild generalization of state-space symmetric. After this, we develop data-driven reformulations of various kinds of balancing-based model reduction for linear first-order and second-order systems. The variants considered are balanced stochastic truncation, positive-real, bounded-real, bounded-real balanced truncation, frequency-weighted balanced truncation, and position-velocity balanced truncation. For each variant, we show how to approximately construct the balanced truncation reduced model from various kinds of input-output invariant frequency response data. In the second part of this dissertation, we consider the mathcalH2-optimal model reduction problem for linear dynamical systems with quadratic-output functions. As the significant theoretical contributions of this portion, we establish the Sylvester equation-based (Wilson) and interpolation-based (Meier-Luenberger) mathcalH2-optimality frameworks for this class of systems. These frameworks are based on two independent sets of first-order necessary conditions for optimality. We additionally show how to enforce the established necessary optimality conditions using a Petrov-Galerkin projection, and prove that the Wilson optimality conditions imply the interpolation-based optimality conditions under some mild assumptions. Based on the theoretical optimality frameworks, two iterative algorithms for the optimal-mathcalH2 approximation of linear quadratic-output systems are proposed. In the final portion, we investigate low-rank interpolatory matrix decompositions for reducing the dimensionality of large matrices of Phasor Measurement Unit (PMU) data in the real-time monitoring of electrical power networks. We propose a theoretical framework for analyzing sparse reconstructions of PMU data during online operations. Drawing upon the numerical linear algebra literature, this framework allows us to state a rigorous, computable upper bound for the interpolatory reconstruction error. This bound can be used to certify whether a collection of PMUs or time instances truly captures the low-rank character of the data, and can be leveraged toward various operational functions. Specifically, we propose a data-driven algorithm for real-time disturbance monitoring that is based upon interpolatory approximations and the discrete empirical interpolation method. Numerical results are included in each portion to validate the theoretical results of the dissertation.