Iterative Rational Krylov Algorithm Research Articles

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

STRUCTURE-PRESERVING MODEL ORDER REDUCTION OF RANDOM PARAMETRIC LINEAR SYSTEMS VIA REGRESSION

We investigate model order reduction (MOR) of random parametric linear systems via the regression method. By sampling the random parameters contained in the coefficient matrices of the systems, the iterative rational Krylov algorithm (IRKA) is used to generate sample reduced models corresponding to the sample data.We assemble the resulting reduced models by interpolating the coefficient matrices of reduced sample models with the regression technique, where the generalized polynomial chaos (gPC) is adopted to characterize the random dependence coming from the original systems. Noting the invariance of the transfer function with respect to restricted equivalence transformations, the regression method is conducted based on the controllable canonical form of reduced sample models in such a way to improve the accuracy of reduced models greatly.We also provide a posteriori error bound for the projection reduction method in the stochastic setting. We showcase the efficiency of the proposed approach by two large-scale systems along with random parameters: a synthetic model and a mass-spring-damper system.

Relative error-based time-limited [formula omitted] model order reduction via oblique projection

In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval is not that important. The error incurred when a reduced-order model is used as a surrogate for the original model can be quantified in absolute or relative terms to access the performance of the model reduction algorithm. The relative error is generally more meaningful than an absolute error because if the original and reduced systems’ responses are of small magnitude, the absolute error is small in magnitude as well. However, this does not necessarily mean that the reduced model is accurate. The relative error in such scenarios is useful and meaningful as it quantifies percentage error irrespective of the magnitude of the system’s response. In this paper, the necessary conditions for a local optimum of the time-limited H2 norm of the relative error system are derived. Inspired by these conditions, an oblique projection algorithm is proposed that ensures small H2-norm relative error within the desired time interval. Unlike the existing relative error-based model reduction algorithms, the proposed algorithm does not require solutions of large-scale Lyapunov and Riccati equations. The proposed algorithm is compared with time-limited balanced truncation, time-limited balanced stochastic truncation, and time-limited iterative Rational Krylov algorithm. Numerical results confirm the superiority of the proposed algorithm over these existing algorithms.

Optimizing Oblique Projections for Nonlinear Systems using Trajectories

Reduced-order modeling techniques, including balanced truncation and $\mathcal{H}_2$-optimal model reduction, exploit the structure of linear dynamical systems to produce models that accurately capture the dynamics. For nonlinear systems operating far away from equilibria, on the other hand, current approaches seek low-dimensional representations of the state that often neglect low-energy features that have high dynamical significance. For instance, low-energy features are known to play an important role in fluid dynamics, where they can be a driving mechanism for shear-layer instabilities. Neglecting these features leads to models with poor predictive accuracy despite being able to accurately encode and decode states. In order to improve predictive accuracy, we propose to optimize the reduced-order model to fit a collection of coarsely sampled trajectories from the original system. In particular, we optimize over the product of two Grassmann manifolds defining Petrov--Galerkin projections of the full-order governing equations. We compare our approach with existing methods including proper orthogonal decomposition, balanced truncation-based Petrov--Galerkin projection, quadratic-bilinear balanced truncation, and the quadratic-bilinear iterative rational Krylov algorithm. Our approach demonstrates significantly improved accuracy both on a nonlinear toy model and on an incompressible (nonlinear) axisymmetric jet flow with $10^5$ states.

Passivity preserving model reduction via spectral factorization

We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman–Yakubovich–Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2-optimal reduced-order models.

An Adaptive Method for Reducing Second-order Dynamical Systems

We present a method to compute reduced models of second-order dynamical systems valid in a desired frequency range without specifying the order of the reduced models prior to the reduction process. The approach is based on a second-order variant of the iterative rational Krylov algorithm (SO-IRKA) and exploits the fact, that an eigenvalue decomposition of the reduced second-order system yields twice as many eigenvalues as its order. By selecting all eigenvalues whose mirror images lie in the frequency range in which the reduced model should be valid during each iteration of SO-IRKA, the order of the reduced model is growing (or shrinking) until a reasonable reduced order is found. Additionally, we show that performing the SO-IRKA optimization steps on an intermediate-sized model yields accurate reduced-order models while the computational cost of the reduction phase is reduced. We show the effectiveness of the proposed method by applying it to numerical models of a vibro-acoustic system.

Comparison of Model Order Reduction Methods in Thermo-Acoustic Stability Analysis

Abstract Model order reduction (MOR) can play a pivotal role in reducing the cost of repeated computations of large thermo-acoustic models required for comprehensive stability analysis and optimization. In this proof-of-concept study, acoustic wave propagation is modeled with a one-dimensional (1D) network approach, while acoustic–flame interactions are modeled by a flame transfer function (FTF). Three reduction techniques are applied to the acoustic subsystem: firstly modal truncation (MT) based on preserving the acoustic eigenmodes, and then two approaches that strive to preserve the input–output transfer behavior of the acoustic subsystem, i.e., truncated balanced realization (TBR) and iterative rational Krylov algorithm (IRKA). After reduction, the reduced-order models (ROMs) are coupled with the FTF. Results show that the coupled reduced system from MT accurately captures thermo-acoustic cavity modes with weak influence of the flame, but fails for cavity modes strongly influenced by the flame as well as for intrinsic thermo-acoustic (ITA) modes. On the contrary, the coupled ROMs generated with the other two methods accurately predict all types of modes. It is concluded that reduction techniques based on preserving transfer behavior are more suitable for thermo-acoustic stability analysis.

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications

<p style='text-indent:20px;'>This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deflecting the algebraic equations the second-order index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper.</p>

Interpolatory projection technique for Riccati-based feedback stabilization of index-1 descriptor systems

The work aims to stabilize the unstable index-1 descriptor systems by Riccati-based feedback stabilization via a modified form of Iterative Rational Krylov Algorithm (IRKA), which is a bi-tangential interpolation-based technique. In the basic IRKA, for the stable systems the Reduced Order Models (ROMs) can be found conveniently, but it is unsuitable for the unstable ones. In the proposed technique, the initial feedback is implemented within the construction of the projectors of the IRKA approach. The solution of the Riccati equation is estimated from the ROM achieved by IRKA and hence the low-rank feedback matrix is attained. Using the reverse projecting process, for the full model the optimal feedback matrix is retrieved from the low-rank feedback matrix. Finally, to validate the aptness and competency of the proposed technique it is applied to unstable index-1 descriptor systems. The comparison of the present work with two previous works is narrated. The simulation is done by numerical computation using MATLAB, and both the tabular method and graphical method are used as the supporting tools of comparative analysis.

SVD-Krylov based techniques for structure-preserving reduced order modelling of second-order systems

<abstract><p>We introduce an efficient structure-preserving model-order reduction technique for the large-scale second-order linear dynamical systems by imposing two-sided projection matrices. The projectors are formed based on the features of the singular value decomposition (SVD) and Krylov-based model-order reduction methods. The left projector is constructed by utilizing the concept of the observability Gramian of the systems and the right one is made by following the notion of the interpolation-based technique iterative rational Krylov algorithm (IRKA). It is well-known that the proficient model-order reduction technique IRKA cannot ensure system stability, and the Gramian based methods are computationally expensive. Another issue is preserving the second-order structure in the reduced-order model. The structure-preserving model-order reduction provides a more exact approximation to the original model with maintaining some significant physical properties. In terms of these perspectives, the proposed method can perform better by preserving the second-order structure and stability of the system with minimized $ \mathcal{H}_2 $-norm. Several model examples are presented that illustrated the capability and accuracy of the introducing technique.</p></abstract>

Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to Mechatronics

This paper studies the model order reduction of second-order index-1 descriptor systems using a tangential interpolation projection method based on the Iterative Rational Krylov Algorithm (IRKA). Our primary focus is to reduce the system into a second-order form so that the structure of the original system can be preserved. For this purpose, the IRKA based tangential interpolatory method is modified to deal with the second-order structure of the underlying descriptor system efficiently in an implicit way. The paper also shows that by exploiting the symmetric properties of the system the implementing computational costs can be reduced significantly. Theoretical results are verified for the model reduction of the piezo actuator based adaptive spindle support which is second-order index-1 differential-algebraic form. The efficiency and accuracy of the method are demonstrated by analyzing the numerical results.

Revisiting IRKA: Connections with Pole Placement and Backward Stability

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order ${\mathscr{H}}_{2}$ -approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid—but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

Model Order Reduction of Multi-Terminal Direct-Current Grid Systems

This paper proposes a methodology to develop a linear low-order model of a multi-terminal direct-current (MTDC) system connected to practically-sized offshore wind farm systems. The method individually linearizes the model of wind farm systems and subsequently performs model order reduction (MOR) using iterative rational Krylov algorithm (IRKA). This is, therefore, a modular approach and expandable to any number of wind farm connections. On the other hand, the model order and non-linearity of other components are retained. Effectiveness of the proposed method is demonstrated on an MTDC system connected to two large-scale wind farm models by comparing the responses obtained from the full order model (FOM) and reduced order model (ROM). From the simulation studies, it can be seen that the ROM obtained from proposed method is able to maintain high degree accuracy of FOM responses at a much faster simulation time, and hence this can help power system operation planning. Various control functionalities and strategies, different operating conditions and faults of the MTDC system are also considered to investigate robustness of the proposed method.

Delay Compensation of Demand Response and Adaptive Disturbance Rejection Applied to Power System Frequency Control

In this paper, a modified frequency control model is proposed, where the demand response (DR) control loop is added to the traditional load frequency control (LFC) model to improve the frequency regulation of the power system. One of the main obstacles for using DR in the frequency regulation is communication delay which exists in transferring data from control center to appliances. To overcome this issue, an adaptive delay compensator (ADC) is used in order to compensate the communication delay in the control loop. In this regard, a weighted combination of several vertex compensators, whose weights are updated according to the measured delay, is employed. Generating the phase lead is the most important strategy for these compensators to eliminate the impact of communication delay. Moreover, to overcome the impact of disturbances and uncertainties in power system, an active disturbance rejection control (ADRC) is utilized as the load frequency controller. Being used instead of a PI controller, this robust controller employs an extended state observer to estimate the disturbances and uncertainties and uses a feedback controller to compensate them. The confined iterative rational Krylov algorithm (CIRKA) is employed to reduce the order of the detailed IEEE 39-bus test system model meticulously and facilitate the controller process design. Therefore, a DR control loop, ADC, and ADRC are employed in the LFC to regulate the frequency of the power system in a more efficient way. The simulation results confirm the effectiveness and robustness of the frequency control in presence of communication delay and model uncertainties.

Comparative analysis of model reduction strategies for circuit based periodic control problems

AbstractThis paper is a comparative analysis of two prominent iterative algorithms for model order reduction of linear time‐varying (LTV) periodic systems where the system's matrices are singular. Our proposed method is based on a reformulation of the LTV model to an equivalent linear time‐invariant (LTI) model using a suitable discretization procedure. The resulting LTI model is reduced in two ways, once by applying a balanced truncation method and once by applying a Krylov‐based method known as iterative rational Krylov algorithm (IRKA). During the application of balanced truncation, the low‐rank Cholesky factorized alternating directions implicit (LRCF‐ADI) method is used to estimate the solutions of the corresponding LTI form of Lyapunov equations. Since the system's matrices are singular, the concept of pseudo‐inverse is adopted to compute the shift parameters needed in the LRCF‐ADI iterations. For the Krylov‐based IRKA, our work is twofold. We solve the time‐invariant Lyapunov equation for the observability Gramian and apply a moment‐matching Krylov technique. The accuracy and effectiveness of the two proposed techniques are demonstrated with the help of frequency response graphs, bode plots, and eigenstructure of the main and reduced models.

Interpolatory Projection Techniques for H2 Optimal Structure-Preserving Model Order Reduction of Second-Order Systems

This paper focuses on exploring efficient ways to find $\mathcal{H}_2$ optimal Structure-Preserving Model Order Reduction (SPMOR) of the second-order systems via interpolatory projection-based method Iterative Rational Krylov Algorithm (IRKA). To get the reduced models of the second-order systems, the classical IRKA deals with the equivalent first-order converted forms and estimates the first-order reduced models. The drawbacks of that of the technique are failure of structure preservation and abolishing the properties of the original models, which are the key factors for some of the physical applications. To surpass those issues, we introduce IRKA based techniques that enable us to approximate the second-order systems through the reduced models implicitly without forming the first-order forms. On the other hand, there are very challenging tasks to the Model Order Reduction (MOR) of the large-scale second-order systems with the optimal $\mathcal{H}_2$ error norm and attain the rapid rate of convergence. For the convenient computations, we discuss competent techniques to determine the optimal $\mathcal{H}_2$ error norms efficiently for the second-order systems. The applicability and efficiency of the proposed techniques are validated by applying them to some large-scale systems extracted form engineering applications. The computations are done numerically using MATLAB simulation and the achieved results are discussed in both tabular and graphical approaches.

$\mathcal{H}_2$-Optimal Model Reduction Using Projected Nonlinear Least Squares

In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that are locally optimal approximations in the $\mathcal{H}_2$ norm, an approach taken by the Iterative Rational Krylov Algorithm (IRKA) among others. Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction using the projected nonlinear least squares framework previously introduced in [J. M. Hokanson, SIAM J. Sci. Comput. 39 (2017), pp. A3107--A3128]. At each iteration, we project the $\mathcal{H}_2$ optimization problem onto a finite-dimensional subspace yielding a weighted least squares rational approximation problem. Subsequent iterations append this subspace such that the least squares rational approximant asymptotically satisfies the first order necessary conditions of the original, $\mathcal{H}_2$ optimization problem. This enables us to build reduced order models with similar error in the $\mathcal{H}_2$ norm but using far fewer evaluations of the expensive, full order model compared to competing methods. Moreover, our new algorithm only requires access to the transfer function of the full order model, unlike IRKA which requires a state-space representation or TF-IRKA which requires both the transfer function and its derivative. Applying the projected nonlinear least squares framework to the $\mathcal{H}_2$-optimal model reduction problem open new avenues for related model reduction problems.

A‐priori pole selection for reduced models in vibro‐acoustics

AbstractLarge models of complex dynamic systems can be evaluated efficiently using model order reduction methods. Many techniques, for example the iterative rational Krylov algorithm (IRKA), rely on a set of expansion points chosen before the reduction procedure. The number and location of the expansion points has a major impact on the quality of the resulting reduced model and the convergence of the algorithm. Based on the system's geometry and material, the number of modes in a certain frequency range can be computed using wave equations. This mode count allows the choice of both a reasonable size for the reduced model as well as a reasonable distribution of initial expansion points, which improves the convergence of IRKA. Using the mode count in a specific frequency range, a reduced model approximating the full model only in this frequency range can be generated.

Parametric model order reduction for acoustic metamaterials based on local thickness variations

The use of lightweight materials is obligatory in the design of economic structures, but with decreasing mass, the vibro-acoustic properties of the structures become unfavorable. Acoustic metamaterials based on the concept of acoustic black holes (ABH) can improve the vibrational behavior by adding local thickness variations to the host structure. To make computational models of metamaterials also valid for high frequencies, a fine discretization has to be used. This leads to high computation times. A parametric model order reduction (PMOR) approach is presented, which is able to create an efficient reduced model of ABHs. In the first step, one structure preserving reduced model for each parameter is generated using the iterative rational Krylov algorithm (IRKA). The transfer functions of all reduced models are used in the second step to create a single parametric model using the Loewner framework. The resulting model can be used to efficiently evaluate the frequency response of a structure equipped with ABHs for different parameter sets.