Browsing by Author "Sinani, Klajdi"
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- Finite Horizon Optimality and Operator Splitting in Model Reduction of Large-Scale Dynamical SystemSinani, Klajdi (Virginia Tech, 2020-07-15)Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when employed in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. There exist a plethora of methods for reduced order modeling of linear systems, including the Iterative Rational Krylov Algorithm (IRKA), Balanced Truncation (BT), and Hankel Norm Approximation. However, these methods generally target stable systems and the approximation is performed over an infinite time horizon. If we are interested in a finite horizon reduced model, we utilize techniques such as Time-limited Balanced Truncation (TLBT) and Proper Orthogonal Decomposition (POD). In this dissertation we establish interpolation-based optimality conditions over a finite horizon and develop an algorithm, Finite Horizon IRKA (FHIRKA), that produces a locally optimal reduced model on a specified time-interval. Nonetheless, the quantities being interpolated and the interpolant are not the same as in the infinite horizon case. Numerical experiments comparing FHIRKA to other algorithms further support our theoretical results. Next, we discuss model reduction for nonlinear dynamical systems. For models with unstructured nonlinearities, POD is the method of choice. However, POD is input dependent and not optimal with respect to the output. Thus, we use operator splitting to integrate the best features of system theoretic approaches with trajectory based methods such as POD in order to mitigate the effect of the control inputs for the approximation of nonlinear dynamical systems. We reduce the linear terms with system theoretic methods and the nonlinear terms terms via POD. Evolving the linear and nonlinear terms separately yields the reduced operator splitting solution. We present an error analysis for this method, as well as numerical results that illustrate the effectiveness of our approach. While in this dissertation we only pursue the splitting of linear and nonlinear terms, this approach can be implemented with Quadratic Bilinear IRKA or Balanced Truncation for Quadratic Bilinear systems to further diminish the input dependence of the reduced order modeling.
- Iterative Rational Krylov Algorithm for Unstable Dynamical Systems and Genaralized Coprime FactorizationsSinani, Klajdi (Virginia Tech, 2016-01-08)Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable system by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the ℋ₂-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA's behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency.