In this study, we numerically explore methods of coupling sensitivity analysis to the reduced model in order to increase the accuracy of a proper orthogonal decomposition (POD) basis across a wider range of parameters. Various techniques based on polynomial interpolation and basis alteration are compared. These techniques are performed on a 1-dimensional reaction-diffusion equation and 2-dimensional incompressible Navier-Stokes equations solved using the finite element method (FEM) as the full scale model. The expanded model formed by expanding the POD basis with the orthonormalized basis sensitivity vectors achieves the best mixture of accuracy and computational efficiency among the methods compared. / Master of Science

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Model reduction is motivated by the fact that complex process models may pre- vent the application of model-based process control. While extensive research on model reduction has been done in the past few decades, model reduction of systems exhibiting two-time scale behavior as well as parametric uncertainty has received little attention to date. This work addresses these types of problems in detail. Systems with two-time scale behavior can be described by differential-algebraic equations (DAEs). A new technique based on projections and system identification is presented for reducing this type of system. This method reduces the order of the differential equations as well as the number and complexity of the algebraic equations. Additionally, the algebraic equations of the resulting system can be replaced by an explicit expression for the algebraic variables such as a feed-forward neural network or partial least squares. This last property is important insofar as the reduced model does not require a DAE solver for its solution, but system trajectories can instead be computed with regular ordinary differential equation (ODE) solvers. For systems with uncertain parameters, two types of problems are investigated, including parameter reduction and parameter dependent model reduction. The pa- rameter reduction problem is motivated by the fact that a large number of parameters exist in process models while some of them contribute little to a system's input-output behavior. This portion of the work presents three novel methodologies which include (1) parameter reduction where the contribution is measured by Hankel singular val- ues, (2) reduction of the parameter space via singular value decomposition, and (3) a combination of these two techniques. Parameter dependent model reduction investigates how to incorporate the influ- ence of parameters in the procedure of conventional model reductions. An approach augmenting the input vector to include the parameters are developed to solve this problem. Finally, a nonlinear model predictive control scheme is developed in which the reduced models are used for the controller. Examples are investigated to illustrate these techniques. The results show that excellent performance can be obtained for the reduced models.

NR 20140805

A model reduction technique that is optimal in the H_∞-norm has long been pursued due to its theoretical and practical importance. We consider the optimal H_∞ model reduction problem broadly from an interpolation-based approach, and give a method for finding the approximation to a state-space symmetric dynamical system which is optimal over a family of interpolants to the full order system. This family of interpolants has a simple parameterization that simplifies a direct search for the optimal interpolant. Several numerical examples show that the interpolation points satisfying the Meier-Luenberger conditions for H₂-optimal approximations are a good starting point for minimizing the H_∞-norm of the approximation error. Interpolation points satisfying the Meier-Luenberger conditions can be computed iteratively using the IRKA algorithm [12]. We consider the special case of state-space symmetric systems and show that simple sufficient conditions can be derived for minimizing the approximation error when starting from the interpolation points found by the IRKA algorithm. We then explore the relationship between potential theory in the complex plane and the optimal H_∞-norm interpolation points through several numerical experiments. The results of these experiments suggest that the optimal H_∞ approximation of order r yields an error system for which significant pole-zero cancellation occurs, effectively reducing an order n+r error system to an order 2r+1 system. These observations lead to a heuristic method for choosing interpolation points that involves solving a rational Zolatarev problem over a discrete set of points in the complex plane. / Master of Science

Few model reduction techniques exist for dynamical systems whose parameters vary with time. We have particular interest here in linear time-periodic dynamical systems; we seek a structure-preserving algorithm for model reduction of linear time-periodic (LTP) dynamical systems of large scale that generalizes from the linear time-invariant (LTI) model reduction problem. We extend the familiar LTI system theory to analogous concepts in the LTP setting. First, we represent the LTP system as a convolution operator of a bivariate periodic kernel function. The kernel suggests a representation of the system as a frequency operator, called the Harmonic Transfer Function. Second, we exploit the Hilbert space structure of the family of LTP systems to develop necessary conditions for optimal approximations. Additionally, we show an a posteriori error bound written in terms of the $\\mathcal H_2$ norm of related LTI multiple input/multiple output system. This bound inspires an algorithm to construct approximations of reduced order. To verify the efficacy of this algorithm we apply it to three models: (1) fluid flow around a cylinder by a finite element discretization of the Navier-Stokes equations, (2) thermal diffusion through a plate modeled by the heat equation, and (3) structural model of component 1r of the Russian service module of the International Space Station. / Master of Science

A power grid network is an interconnected network of coupled devices that generate, transmit and distribute power to consumers. These complex and usually large-scale systems have high dimensional models that are computationally expensive to simulate especially in real time applications, stability analysis, and control design. Model order reduction (MOR) tackles this issue by approximating these high dimensional models with reduced high-fidelity representations. When the internal description of the models is not available, the reduced representations are constructed by data. In this dissertation, we investigate four problems regarding the MOR and data-driven modeling of the power networks model, particularly the swing equations. We first develop a parametric MOR approach for linearized parametric swing equations that preserves the physically-meaningful second-order structure of the swing equations dynamics. Parameters in the model correspond to variations in operating conditions. We employ a global basis approach to develop the parametric reduced model. We obtain these local bases by $mathcal{H}_2$-based interpolatory model reduction and then concatenate them to form a global basis. We develop a framework to enrich this global basis based on a residue analysis to ensure bounded $mathcal{H}_2$ and $mathcal{H}_infty$ errors over the entire parameter domain. Then, we focus on nonlinear power grid networks and develop a structure-preserving system-theoretic model reduction framework. First, to perform an intermediate model reduction step, we convert the original nonlinear system to an equivalent quadratic nonlinear model via a lifting transformation. Then, we employ the $mathcal{H}_2$-based model reduction approach, Quadratic Iterative Rational Krylov Algorithm (Q-IRKA). Using a special subspace structure of the model reduction bases resulting from Q-IRKA and the structure of the underlying power network model, we form our final reduction basis that yields a reduced model of the same second-order structure as the original model. Next, we focus on a data-driven modeling framework for power network dynamics by applying the Lift and Learn approach. Once again, with the help of the lifting transformation, we lift the snapshot data resulting from the simulation of the original nonlinear swing equations such that the resulting lifted-data corresponds to a quadratic nonlinearity. We then, project the lifted data onto a lower dimensional basis via a singular value decomposition. By employing a least-squares measure, we fit the reduced quadratic matrices to this reduced lifted data. Moreover, we investigate various regularization approaches. Finally, inspired by the second-order sparse identification of nonlinear dynamics (SINDY) method, we propose a structure-preserving data-driven system identification method for the nonlinear swing equations. Using the special structure on the right-hand-side of power systems dynamics, we choose functions in the SINDY library of terms, and enforce sparsity in the SINDY output of coefficients. Throughout the dissertation, we use various power network models to illustrate the effectiveness of our approaches. / Doctor of Philosophy / Power grid networks are interconnected networks of devices responsible for delivering electricity to consumers, e.g., houses and industries for their daily needs. There exist mathematical models representing power networks dynamics that are generally nonlinear but can also be simplified by linear dynamics. Usually, these models are complex and large-scale and therefore take a long time to simulate. Hence, obtaining models of much smaller dimension that can capture the behavior of the original systems with an acceptable accuracy is a necessity. In this dissertation, we focus on approximation of power networks model through the swing equations. First, we study the linear parametric power network model whose operating conditions depend on parameters. We develop an algorithm to replace the original model with a model of smaller dimension and the ability to perform in different operating conditions. Second, given an explicit representation of the nonlinear power network model, we approximate the original model with a model of the same structure but smaller dimension. In the cases where the mathematical models are not available but only time-domain data resulting from simulation of the model is at hand, we apply an already developed framework to infer a model of a small dimension and a specific nonlinear structure: quadratic dynamics. In addition, we develop a framework to identify the nonlinear dynamics while maintaining their original physically-meaningful structure.

The modelling of physical processes gives rise to mathematical systems of increasing complexity. Good mathematical models have to reproduce the physical process as precisely as possible while the computing time and the storage resources needed to simulate the mathematical model are limited. As a consequence, there must be a tradeoff between accuracy and computational constraints. At the present time, one is often faced with systems that have an unacceptably high level of complexity. It is then desirable to approximate such systems by systems of lower complexity. This is the Model Reduction Problem. This thesis focuses on the study of new model reduction techniques for linear systems. Our objective is twofold. First, there is a need for a better understanding of Krylov techniques. With such techniques, one can construct a reduced order transfer function that satisfies a set of interpolation conditions with respect to the original transfer function. A study of the generality of such techniques and their extension for MIMO systems via the concept of tangential interpolation constitutes the first part of this thesis. This also led us to study the generality of the projection technique for model reduction. Most large scale systems have a particular structure. They can be modelled as a set of subsystems that interconnect to each other. It then makes sense to develop model reduction techniques that preserve the structure of the original system. Both interpolation-based and gramian-based structure preserving model reduction techniques are developed in a unified way. Second order systems that appear in many branches of engineering deserve a special attention. This constitutes the second part of this thesis.

In this paper we describe an approach for generating geometrically-parameterized integrated-circuit interconnect models that are efficient enough for use in interconnect synthesis. The model generation approach presented is automatic, and is based on a multi-parameter model-reduction algorithm. The effectiveness of the technique is tested using a multi-line bus example, where both wire spacing and wire width are considered as geometric parameters. Experimental results demonstrate that the generated models accurately predict both delay and cross-talk effects over a wide range of spacing and width variation. / Singapore-MIT Alliance (SMA)

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